Chiral Luttinger Liquids at the Fractional Quantum Hall Edge

A.M. Chang Department of Physics Purdue University, West Lafayette, IN 47907-1396

This article contains a comprehensive review of the chiral Tomonaga- Luttinger liquid. Although the main emphasis is on the key experimental findings on electron-tunneling into the chiral Tomonaga-Luttinger liquid, in which the tunneling is utilized to probe the power- law tunneling density of states in this interacting 1-dimensional system at the edge of the fractional quantum Hall fluid, this review also contains a basic description of the theoretical aspects. The inclusion of the theory section provides a suitable framework and language for discussing the unique and novel features of the chiral Tomonaga-Luttinger liquid. The key experimental results will be contrasted against the predictions of the ”standard theory” based on the effective, Chern-Simon field theories of the fractional Hall fluid edge. Possible ramifications of the differences between experiment and theory are highlighted. In addition, for completeness a brief survey of other 1-dimensional systems exhibiting the interaction physics associated with the (non-chiral) Tomonaga-Luttinger liquid is also provided.

1 Contents

I INTRODUCTION 3

II THEORETICAL BACKGROUND 5 A LandauFermiliquid...... 6 1 Dynamics: TheBoltzmannkineticequation ...... 10 B Many-bodyanalysis...... 11 1 Green’sfunctions ...... 11 2 Lehman representation: spectral density, tunneling densityofstates ...... 14 3 IntuitiveIdeaofaQuasi-Particle...... 15 4 TwoParticleGreen’sFunction ...... 16 5 Dynamical equation of motion for the single particle Green function ...... 17 6 Quasi-particleinteraction ...... 20 C Breakdown of the Fermi liquid Picture in 1d and the Tomonaga-Luttinger liquid ...... 24 1 Bosonization...... 27 2 Power law behavior in the single particle Green’s function ...... 29 3 g-ology ...... 32 D ChiralLuttingerliquid ...... 36 1 Wen’shydrodynamicformulation ...... 37 2 1-d effective field theory of the chiral Luttinger liquid ...... 43 3 RoleofDisorder ...... 46 4 Compressiblefluidedges ...... 47 5 Scaling functions for electron tunneling ...... 49 6 Resonanttunneling ...... 56 7 Shot noise and fractional charge: quasi-particle tunneling...... 59

III EXPERIMENTS ON CHIRAL LUTTINGER LIQUIDS– TUNNELING INTO THE FRACTIONAL QUANTUM HALL EDGE 59 A HistoricalOverview ...... 61 B Cleaved-edgeovergrowth ...... 63 C Measurementtechniques ...... 65 D Samplepreparation ...... 66 E Datapresentation ...... 67 1 Establishment of Chiral Luttinger Behavior in Electron Tunneling into Fractional Quantum Hall Edges ...... 67 2 An apparent g=1/2 chiral Luttinger liquid at the edge of the ν =1/2compressiblecompositeFermionliquid ...... 73 3 A continuum of chiral Luttinger liquid behavior ...... 77 4 Plateau behavior in the chiral Luttinger liquid exponent ...... 79 5 Discussion: Is the chiral Luttinger liquid exponent universal?...... 83 6 Resonant tunneling into a biased fractional quantum Hall edge ...... 87 7 Other Manifestations of Chiral Luttinger Liquid Characteristics in the Edge Properties ofFractionalQuantumHallFluids...... 91

IV OTHER LUTTINGER LIQUID SYSTEMS 95 A Ballisticsingle-channelwires ...... 95 B Carbonnanotubes...... 96 C 1-dchaingoldatomchains ...... 99 D Quasi-1-dimensionalconductors ...... 100

APPENDIXES 104

A Edge Density Proﬁle 104

2 I. INTRODUCTION

This article deals with electron transport in a new state of matter, specifically electron tunneling into the chiral Luttinger liquid (CLL). The chiral Luttinger liquid, also known as the chiral Tomonaga-Luttinger liquid (Wen, 1990a, 1990b, 1991a, 1991b, 1992, 1995; Kane et al., 1994; Kane and Fisher, 1995; Fendley et al., 1995a, 1995b; Moon, 1993; Chang et al., 1996, and 2001; Grayson et al. 1998). is a particularly clean realization of a new class of strongly interacting 1-dimensional (1-d) metallic systems, distinguished from conventional 3-dimensional (3-d) metals by the absence of a single-particle pole in the spectral density. Instead, the usual pole is replaced by power law dependences in momentum and frequency. Experimentally, what this means is that in a transport measurement where electrons are injected into or removed from the 1-d correlated metal across a tunneling barrier, a power law behavior is observed in either the tunnel current or in the differential conductance as a function of energy, where this energy may be set by a bias voltage or by temperature. From the perspective of measurements, the CLL is an extremely clean system. Residing at the 1-dimensional edge of the 2-dimensional fractional quantum Hall fluid, the CLL is realized in devices grown by the molecular beam epitaxy (MBE) growth technique, which offers extremely precise, atomic-scale control. This control extends beyond the growth of the metallic electron system itself to include the tunnel barrier as well. Furthermore, the nature of the CLL can be tuned by changing the magnetic field and hence the filling factor of the fractional quantum Hall fluid. This tunability leads to a rich variety of similar yet different Luttinger liquids. In Fig. 1, we show the first set of current-voltage (I-V) data with sufficient dynamic range to establish a power law dependence, obtained for electron tunneling from a 3-d, highly-doped bulk GaAs metal into the edge of a ν =1/3 fractional quantum Hall fluid (Chang et al., 1996). Note that in the log-log plot, the low bias voltage range below 15µ eV is dominated by the thermal energy and is therefore linear. Above this regime a power law behavior with a range exceeding 3 decades in current and 1 1/2 decades in voltage is observable. This power law stands in direct contrast to conventional metals which exhibit an energy independent tunneling conductance and a linear I-V in the clean limit reflecting the energy independent tunneling density of states. To date, both on-resonance tunneling and off-resonance tunneling have revealed signatures of this unusual power-law of behavior (Chang et al., 1996, 1998, 2001; Grayson et al., 1998, 2001). The present strong interest in Luttinger liquids comes from two major directions. First of all interacting 1-d systems are extremely interesting in their own right, particularly in light of recent observations of anomalous behaviors in the properties of 1-d and quasi-1-d systems, such as transport properties including DC and frequency dependent conductivities, photoemission spectra, NMR relaxation, etc. These systems are quite diverse and include the CLL, 1-d ballistic semiconductor wires (Tarucha et al., 1995; Yacoby et al., 1996; Auslaender et al., 2000), 1-d organic conductors (Basista et al., 1990; Dardel et al., 1993; Schwartz et al., 1998; Zwick et al., 1997) or blue bronze type conductors (Dardel et al., 1991; Sing et al., 1999; Denlinger et al., 1999), and carbon nanotubes (Bockrath et al., 1999; Yao et al., 1999; Bachtold et al., 2001). Secondly, phenomenologically there appears to be considerable and unmistakable similarities between the high temperature superconductors (Bednorz and Muller, 1986; Wu et al., 1987) in their normal state (Ando et al., 1995; Harris et al., 1992; Anderson, 1987, 1990, 1992; Varma et al., 1989) and the 1-d Luttinger liquids, most notably the absence of single particle pole as determined from Angle-Resolved Photoemission measurements (Shen and Schrieffer, 1997; Ding et al., 1997; Anderson, 1987, 1990, 1992). This unambiguous and unusual feature has led to a tremendous amount of speculation and investigation on two dimensional Luttinger-like correlated systems where the coupling to a Gauge field (Baskaran and Anderson, 1988; Zou and Anderson, 1988; Lee et al., 1998; Kopietz, 1996) leads to a power law correlations (Ranter and Wen, 2001; Ren and Anderson, 1993), or to models which assume outright the existence of coupled-1-d Luttinger stripes (Emery et al., 1999; Carlson et al., 2000) where the stripes form as a result of phase separation. Beyond these two reasons there are other possibilities which down the road can become significant. One possibility pertains to the relevance of 1-d Luttinger liquids to 1+1 dimensional conformal field theories (CFT). See for example Voit (1995). Conformal field theories are central to the theory of super strings which are modeled as one dimensional objects. Furthermore, it is not entirely unthinkable that the physics associated with the chiral Luttinger liquid which exists at the edge of the 2-dimensional quantum Hall fluid will bear similarity to the embedding of our 3+1 dimension space-time at the boundary of higher dimensional spaces.

3 FIG. 1. Current-voltage (I V ) characteristics for tunneling from the bulk-doped n+ GaAs into the edge of a ν =1/3 fractional quantum− Hall effect for Sample 1.1 in a log-log plot at B = 13.4T (crosses), and for Sample 2 at B = 10.8T (solid circles). The solid curves represent fits to the theoretical universal form of Eq. (289) for α =2.7 and 2.65, respectively. [From Chang, Pfeiffer, and West, 1996.]

The CLL belongs to a larger class of new, correlated systems which are termed collectively as non-Fermi- liquid systems. By a non-Fermi-liquid, we mean in general terms interacting Fermion systems in which the elementary excitations cannot be described by the venerable Fermi-liquid picture of long-lived quasi-particles which are related to bare electrons with a one-to-one correspondence via the adiabatic principle. Non-Fermi liquids come in many varieties and are encountered in diverse systems. Outstanding examples include: 1) 1-d Luttinger liquids and Hubbard models, 2) 2-d systems coupled to Gauge fields, 3) systems with strong Fermi-surface nesting, 3) systems in the vicinity of critical fluctuations or soft modes, and 4) Kondo systems or Anderson Hamiltonian systems where a local degree of freedom couples to a continuum. In all these 2 2 systems, the requirement that the quasi-particle life time diverges as (∆E)− = (E E )− when the − F energy approaches the Fermi energy, EF , no longer holds and is often replaced by a substantially weaker dependence. It is the purpose of this article to provide a self-contained, and here I emphasize self-contained, description of the current status in the field of the chiral Luttinger liquid. This review is intended to be readily accessible to both experimentalists and theorists alike. To accomplish this it is necessary to include in the contents in addition to the major experimental results a theory section in order to provide the framework for introducing the key concepts and novel features associated with the Luttinger liquid. Written from an experimentalist’s perspective, it is hoped that this theory section could also serve as a starting point for those readers interested in more in depth theoretical studies. The experimental sections will discuss the experimental techniques, the uniqueness and advantages of the CLL system both from the perspective of the phenomenon itself and from the perspective of device realization, highlighted the major results of the clear cut observation of the unusual power-law tunneling density of states in both off-resonance and on-resonance tunneling. We will also discuss recent tunneling current noise measurements which shed light on the quasi-particle fractional charge, and briefly survey other 1-d systems such as 1-d quantum wires, carbon nanotubes, and quasi-1-d

4 organic and blue bronze conductors. This review is organized as follows: Section II contains the theoretical section devoted to introducing the basic theoretical framework for understanding the chiral Luttinger liquid. Section III contains the the main experimental Section summarizing the major results in electron tunneling into the fractional quantum Hall edge. This section will include a discussion of the conditions under which the existence of a chiral Luttinger liquid is established, particularly of relevance to the edge of the ν = 1/3 fractional Hall fluid. We will present evidence which demonstrate that Luttinger liquid-like tunneling behavior is not restricted to the edge of the special, incompressible fractional quantum Hall fluid given by the Laughlin (1983), hierarchical (Haldane, 1983; Halperin, 1984) or Jain series (Jain, 1989a, 1989b, 1990), but can be observed at general filling fractions including those corresponding to compressible fluids. We will discuss the clear evidence of non-Fermi-liquid behavior in resonant tunneling (as opposed to off-resonance tunneling) where a resonant, impurity level mediates the tunneling process. Furthermore, we will describe other aspects associated with the chiral Luttinger liquid picture, specifically, quantum shot noise measurements which probe the fractional charge of the quasi particles (Saminadayar et al., 1997; De-Picciotto et al., 1997; Reznikov et al., 1999). In Section IV, the review concludes with a survey of the latest developments in other 1-d conducting systems such as quantum wires, single-walled nanotubes, as well as quasi 1-d organic and blue bronze conductors.

II. THEORETICAL BACKGROUND

This entire theoretical section is devoted to an introductory review of the key theoretical concepts of a chiral Luttinger liquid, presented from the point of view of an experimentalist. The hope is to provide a starting point, particularly for experimentalist and in general for those readers who are not familiar with the theory of 1-d interacting systems, to grasping the novel and unusual aspects of this new type of correlated electron fluid. The chiral Luttinger liquid in 1-d is characterized by bosonic elementary excitations (plasmons) etc. rather than the familiar Fermi-liquid (FL) quasi particle/hole excitations, and the absence of single-particle pole which is replaced by power-law behavior in the correlation functions. However, the difference can be subtle. In particular, in the sector of Hilbert space spanning neutral excitations at low energy (ω 0) and small wave-vector (q 0) a Fermi-liquid description is still workable (Castellani et al., 1994; Carmelo→ and Castro Neto, 1996; Castellani→ and Di Castro, 1999; Ng, 1997). However, in the charged sector in which the addition or removal of charged carriers (electrons or holes) is involved, the Fermi-liquid description breaks down as evidenced in the destruction of the quasi-particle pole and the vanishing of the quasi-particle renormalization factor, zk. To set the stage we begin with a review of the key features of Landau’s phenomenological theory of a Fermi liquid (Landau, 1956, 1957, 1958; Nozieres, 1964), proceed to sketching its justification in 3-dimensions based on key concepts in many-body Green’s function techniques (Landau, 1958; Nozieres, 1964), and describe why this Fermi liquid picture breaks down in 1-d. These preliminary sections enable us to bring forth the unusual phase space structure in 1-d which leads to the Tomonaga-Luttinger liquid, or Luttinger liquid (LL) for short. We will describe Haldane’s formulation (Haldane, 1979, 1981) based on the bosonization technique (Tomonaga, 1950; Matthis and Lieb, 1965; Luther and Peschel, 1974; Haldane, 1981) which solves the original Luttinger model (Luttinger, 1963; Tomonaga, 1950) exactly, and the alternative g-ology model renormalization group analysis of the Luttinger fixed point (Solyom, 1979; Metzner et al., 1998; Voit, 1995). At this point, we are finally and most importantly in a position to capture the essential aspects of the theory of the chiral Luttinger liquid. We start with the seminal formulation due to Wen (1990a, 1990b, 1991a, 1991b, 1992, 1995) for incompressible fluid fractional Hall fluid edges, discuss the role of disorder (Kane et al., 1994; Kane and Fisher, 1995) before moving to a discussion of the compressible fractional fluids (Shytov et al., 1998; Levitov et al., 2001). We summarize the universal scaling functions for the tunneling current and conductance and in particular the Chamon-Fradkin scaling function (Chamon and Fradkin, 1997) for off-resonance tunneling. This scaling function, which covers the entire range from weak to strong tunneling for the case of incoherent, multiple point contacts, has proven extremely useful in providing systematic fits to the experimental data to extract reliable power-law exponents. Lastly we will touch on the case of on resonance tunneling through an impurity level (Chamon and Wen, 1993; Kane, 1998; Geller et al., 1996; Kane and Fisher, 1994; Fendley et al., 1995b).

5 Although the field of the chiral Luttinger liquid at the fractional quantum Hall edge is relative young, approximately 12 years, the field of the Luttinger liquid has had a long history dating back to the 1950’s and 60’s, at least as far as theory is concerned. Many authors have contributed to this exciting field and more complete reviews of different aspects are available (Solyom, 1979; Emery, 1979; Schulz, 1991; Voit, 1995; Metzner, 1998). Excellent reviews of the theory of the chiral Luttinger liquid can be found in the articles of Wen (1992, and 1995). Once again, our goal here is to provide a self-contained, fairly complete overview of the transport properties with an emphasis on describing the key ideas in relatively simple terms. For more thorough discourses of the theoretical aspects we refer the readers to these excellent reviews.

A. Landau Fermi liquid

The key feature of the Landau Fermi liquid is the existence of long lived quasi-particle/hole excitations as the energy of excitation approaches zero or equivalently as the energy approaches the Fermi energy, where the quasi-particles (quasi-holes) can be traced back to bare electrons (holes) with a one to one correspondence, originating from the non-interacting situation and subsequently adiabatically turning on the electron-electron interaction. Phenomenologically supposing the existence of such low energy excitations, Landau was able to account for a rich variety of physical phenomena exhibited by conventional metals in the presence of non- negligible interaction (Landau 1956, 1957, 1958). The Fermi Liquid picture ﬁnds more rigorous justiﬁcation in many-body perturbation theory to be discussed is subsequent sections (Nozieres, 1964). Our discourse on the Fermi Liquid begins with the familiar case of a non-interacting Fermi gas. To illustrate the key ideas, we will restrict our discussions to the case of a spin independent, isotropic, and disorder free system throughout. In the simple case of a uniform, non-interacting Fermi-gas of N spin 1/2 particles in a volume V with 2 2 single particle dispersion ǫo(k)=¯h k /2m, at zero temperature the system can be characterized by a Fermi wave vector, kF, with density, n:

3 N 4π kF n = = (2) , k = kF , (1) V 3 (2π)3 F | |

and total energy, Eo:

Eo = ǫo(k)no(k), (2) Xk where no(k) is the occupation of state k at zero temperature given by:

2 2 1 ǫo(k) EF ¯h kF no(k)= ≤ , EF = . (3) 0 ǫo(k) > EF 2m

no(k) exhibits a jump of unit height at the Fermi wave vector, kF, and the sharp Fermi surface is determined by the condition ǫo(k)= EF . Excitations above the ground state involve single particle excitations yielding an energy change in the system, δE, of:

δE = E E = ǫ (k)[n(k) n (k)] = ǫ (k)δn(k), (4) − o o − o o Xk Xk where resulting from the Pauli exclusion principle,

1 or 0 ǫ (k) E δn(k)= − o ≤ F (5) +1 or 0 ǫo(k) > EF subjected to the additional constraint, k δn(k) = 0 to account for a ﬁxed particle number, N. In fact, the single particle energy, ǫo(k), can be thought of as the ﬁrst functional derivative of E with respect to the change in occupation, δn(k). From this simpleP description, all thermodynamic and transport quantities can

6 be computed through the use of the Fermi distribution function accounting for the temperature dependence in occupation and the Boltzmann equation for transport. Now imagine introducing interaction between the particles by turning on the interaction adiabatically. Landau hypothesized that although it is no longer possible in general to have well-deﬁned, long-lived particle or hole states, if we conﬁne our attention to low energy excitations above the ground state, it will still be possible to identify long-lived particle-like (hole-like) quasi-particle (quasi-hole) states. Each long-lived quasi- particle state will have a one-to-one correspondence to the bare electron state which existed before the interaction was turned on and can be labeled by the same momentum vector, k. In the dilute limit of few quasi-particles or holes, the additional energy of the excited state is given by:

δE = E E = ǫ(k)δn(k). (6) − o Xk Here, in analogy to the non-interacting case, ǫ(k) is the quasi-particle energy at wavevector, k, and δn(k) the change in the quasi-particle occupation. Therefore ǫ(k) is the first functional derivative of the energy, E, with respect to n(k) : ǫ(k)= δE/δn(k). Because these quasi-particle (quasi-holes) behave as Fermions and are long-lived, it is still possible to define a sharp Fermi energy, EF, and Fermi wave vector, kF, yielding a sharp Fermi surface in k space. It is important to note that since the quasi-particles are well-defined only near the Fermi surface, it is in general not possible to express the total energy of the ground state simply as Eo = k ǫ(k)n(k)! The quantities ǫ(k) and n(k) are only meaningful for k kF. When the quasi-particle number is increased, interaction between particles≈ necessitates the addition of higherP order terms to the total energy. δE is then given by: 1 δE = ǫ(k)δn(k)+ f(k, k’)δn(k)δn(k’) (7) 2 Xk Xk Xk’ 1 = ǫ(k)+ f(k, k’)δn(k’) δn(k),  2  Xk Xk’   where f(k, k’) represents the second functional derivative of E with respect to δn(k) and δn(k’), and f(k, k’)= f(k’, k) by construction. We can identify the quasi particle energy,ǫ ˜(k):

ǫ˜(k)= ǫ(k)+ f(k, k’)δn(k’), (8) Xk’

with the second term representing the eﬀect of quasi-particle quasi-particle interaction. Near kF the quasi- particle occupation is well-deﬁned and at zero temperature, we have as in the non-interacting case:

1ǫ ˜(k) E or k k n(k)= ≤ F ≤ F (9) 0ǫ ˜(k) > EF or k > kF We again have a sharp Fermi surface in k-space with a quasi-particle distribution function, n(k), which exhibits a unit step discontinuity at kF at zero temperature. At a finite but low temperature and under equilibrium conditions, we can use the standard method to find the occupation probability of a given quasi-particle state k. For a system of fixed volume V and particle number N, the probability of finding a state (total system configuration at total energy E is (β = 1/kBT) (Ashcroft and Mermin, 1976):

βE′ E′ e− δ(E E′) pN (E)= βE′− (10) ′ e P E − ′ βE βFN P where the denominator E′ e− = ZN = e− , ZN being the partition function and FN the Helmholtz free energy of the system. To compute the probability of ﬁnding the quasi-particle state k with energyǫ ˜(k) occupied, we may write:P

7 N nN (k)= pN (E withǫ ˜(k) occupied) (11)

N =1X p (E′ withǫ ˜(k) not occupied) − N N+1 =1 X p (E′ withǫ ˜(k) occupied ˜ǫ(k)) − N − β[˜ǫ(k) µ] =1 Xe − n (k) − N+1 Here EN and EN+1 refer to the energy with a system with N and N+1 particles, respectively. The third line follows from the second due to Pauli exclusion and the fact that at low temperatures only excited states with few quasi-particle excitations are present. As a result, for the relevant states which contribute to the N N N N sum the energy is given by: E = Eo + δE and δE is found in Eq. (7). Therefore with theǫ ˜(k) state occupied, we have:

N+1 N+1 N+1 N with ǫ˜(k) not occupied E = Eo + δE = E (12) 1 + ǫ(k)+ f(k, k’)δn(k’)+ f(k’, k)δn(k’) = EN +˜ǫ(k),  2    Xk’ Xk’    where we have made use of the fact that f(k, k’) = f(k’, k) and Eq. (8). In this way, we can generate all EN states with ǫ(k) not occupied through states EN+1 with it occupied. The last line in Eq. (11) follows from:

N+1 N+1 βǫ˜(k) N+1 Z [pN+1(E′ withǫ ˜(k) occupied)e ] p (E′ withǫ ˜(k) occupied ǫ˜(k)) = (13) N − ZN β[˜ǫ(k) µ] N+1 = e − pN+1(E′ ),

N+1 N with µ = FN+1 FN = kBT[lnZ lnZ ] as the chemical potential. When N is large, nN(k) = nN+1(k) to order 1/N. We− thus have− for the quasi-particle− occupation: 1 nN (k)= . (14) β[˜ǫ(k) µ] e − +1 This distribution reproduces the unit step function occupation at T=0 and is reminiscent of the Fermi-Dirac distribution for free electrons in the non-interacting system. It bears re-emphasis that this is a distribution function of quasi-particles, not electrons! At this point, with the supposition of long-lived quasi-particle states labeled by the momentum vector, k, for k near kF, in conjunction with the group velocity relation which deﬁnes the renormalized quasi-particle mass, m∗: 1 ¯hk vk = k˜ǫ(k)= (15) ¯h∇ m∗ we are in position to compute thermodynamic quantities such as the speciﬁc heat, cV, electron current density, J, compressibility, κ, and hence speed of ordinary sound, vS, the spin susceptibility, etc. In computation it is necessary to bear in mind that when the distribution function changes through a non zero δn(k), the quasi-particle energy,ǫ ˜(k), also can change due to the interaction term in Eq. (8). To illustrate the success of the Fermi liquid picture, we describe the computation of thermodynamic and transport quantities, cV, and J. Together, these examples bring forth the importance of the second order interaction term in Eq. (8) and demonstrate how the phenomenological Landau parameters characterizing f(k, k’) can be deduced from experimentally measurable quantities. In three spatial dimensions (3-d) for which the Fermi liquid theory is valid, the density of states, ρ(E) is readily obtained from Eqs. (6) and (15):

2 2 d(N/V ) kF dkF kF 1 m∗kF ρ(E)= = 2 = 2 = 2 . (16) dE π dǫ˜ π ¯hvk π2¯h At low temperatures, the specific heat at constant volume can be deduced as follows. A change in temperature will modify the distribution function, n(k) for k near kF: n(k) = no(k)+ δn(k). Since the temperature in low to first order we neglect the modification in the quasi-particle energy,ǫ ˜(k) due to δn(k). We find:

8 1 ∂E δn(k) cV = V = lim ˜ǫ(k) + higher order terms... (17) V ∂T δT 0  δT  → Xk’   subject to the constraint of ﬁxed total particle number: δn(k)=0. (18) X Using the expression Eq. (14) for the distribution function and carrying through the standard computation leads to:

2 2 2 π kB T kBT cV = ρ(µ)= kF m∗. (19) 3 3¯h2

To relate m∗ to the interaction function, f(k, k’), we compute the particle current, J, in the presence of a change in the distribution, δn(k), at zero temperature. In the ground state, the net current should be zero due to reﬂection symmetry in our isotropic, disorder-free system. Net current arises from the change in the distribution from the grounds state distribution, no(k). δn(k) can arise from the application of an external electric ﬁeld displacing the Fermi sphere, or introduction of localized charges in the system. To illustrate the ideas, we follow Landau’s original derivation (1956). Starting with the intuitive relation for J:

J = vkn(k), (20) Xk a modification in the distribution, δn(k), will have two effects on the current density–(i) vkδn(k), and (ii) a contribution due to the modification of vk, denoted by δvk:

J = Jo + vkδn(k)+ δvkno(k) , (21) k X

where Jo = 0. Recalling thatǫ ˜(k) and n(k) and therefore v(k) as well are not well-deﬁned except near the Fermi surface, we can restrict our computation to situations where δn(k) is non-zero only for k in the vicinity of kF while converting the second term to a surface term using the relationship: 1 1 δv = δ ˜ǫ(k)= f(k, k’)δn(k’) (22) k ¯h∇k ¯h∇k Xk’ and integration by parts, yielding:

1 J = v δn(k) f(k, k’) n (k)δn(k’) (23)  k − ¯h ∇k o  Xk Xk’   = v δn(k)+ f(k, k’)v δ(˜ǫ(k) µ)δn(k) ,  k k’ −  Xk Xk’   where we have used the symmetry f(k, k’) = f(k’, k) and interchanged the labels k and k’ in the second term. The ﬁrst term is obvious and indicates that each quasi-particle contributes a current of evk, while the second represents the well-known back-ﬂow contribution associated with the electron cloud which dresses the quasi-particle. To make contact with the renormalized mass, m∗, we specialize to the case where one particular k state is occupied above the ground state. δJ become: k ¯hk k δJ = v [1 + F f(θ, φ)cos θdΩ] = [1 + F f(θ, φ)cos θdΩ]. (24) k 8π3¯h2 m 8π3¯h2 Z ∗ Z

9 In this expression the components perpendicular to vk or to k average out to zero. On the other hand, the total current can be shown to be equal to the noninteracting value ¯hk/m where m is the bare electron mass since turning on particle pair interaction does not aﬀect the single particle current density operator. This is a consequence of the fact that the interaction Hamiltonian commutes with the current operator. The term in the bracket containing an integral over f(kF, k’F) therefore gives the ratio of m∗/m.

1. Dynamics: The Boltzmann kinetic equation

The success of the Fermi liquid theory goes beyond thermodynamic and quasi-static properties. Spatially dependent collective modes and responses to time dependent and spatially varying external perturbations can be analyzed with the aid of the Boltzmann equation. To do so, the energy functional Eq. (7) must be generalized to a spatially dependent case (Landau, 1956 and 1957; Nozieres 1964):

3 1 3 3 δE = d rǫ(k)δn(k, r)+ d rd r′f(k, k’; r r’)δn(k, r)δn(k’, r’), (25) 2 − Xk Z kX,k’ ZZ

where n(k,r) is the local distribution function, and δn(k, r) = n(k, r) no, while f depends on r r’ only and not on r, r’ separately. This f dependence is a reﬂection of the similar− r r’ in the interaction− potential due to an absence of disorder in the system. To make the equation tractable,− it is necessary to restrict the range of interaction to short range. Coulomb interaction can be accounted by separating out the long range Hartree contribution leaving a short ranged screen potential. In the simplest scenario, we approximate the short range potential by a δ function leading to a simpliﬁcation of δE: 1 δE = d3rǫδn(k, r)+ d3rf(k, k’)δn(k, r)δn(k’, r). (26) 2 Xk Z kX,k’ Z To make a connection with the Boltzmann equation, we rewrite this expression as:

δE = d3rǫ˜(k, r)δn(k, r) (27) kX,k’ Z with

ǫ˜(k, r)= ǫ(k)+ f(k, k’)δn(k, r), (28) Xk being the local quasi-particle energy, and make an identiﬁcation of this expression as the classical Hamiltonian of the quasi-particle. A force can then be derived as F=~ ǫ˜(k, r). Substituting this force in the Boltzmann equation which equates the loss rate of particles from a−∇ local phase-space volume to the collision integral, I(n):

∂n(k, r,t) 1 + rn.¯hv + n.F~ = I(n) (29) ∂t ¯h ∇ k ∇k n o leads to: ∂n(k, r,t) 1 + rn. ǫ˜ n. rǫ˜ = I(n). (30) ∂t ¯h ∇ ∇k − ∇k ∇ External driving forces can then be incorporated as additional contributions to the force term. As before, to avoid the interpretive difficulty associate with n(k, r) for k away from kF, it is necessary to consider deviations from the equilibrium value, no(k). Linearization to first order finally gives:

10 ∂δn(k, r,t) + v . rδn + v .δ(˜ǫ µ) f(k, k’) rδn(k’, r,t)= I(n) (31) ∂t k ∇ k k − ∇ Xk’ with a corresponding Fourier space expression,

1 3 i(q.r ωt) (q.v ω)δn(k; q,ω)+ q.v δ(˜ǫ µ) f(k, k’)δn(k’; q,ω)= d r dte − I(n), (32) k − k k − 2π Xk’ Z Z where

1 3 i(q.r ωt) δn(k, q,ω)= d r dte − δn(k, r,t). (33) 2π Z Z This equation is particular useful at low temperatures where collisions are rare and disturbance are conﬁned near the Fermi surface. In fact it turns out for a Fermi liquid the quasi-particle decay rate behaves as Γ (k k )2 leading to a collision rate T 2 tending to zero at low temperatures. Therefore it is possible to ∼ − F ∼ approximate the equation above by setting the collision integral, I(n), to zero. For q kF and ν ω µ, where ν is the collision frequency, it is possible to deduce the natural oscillations as well≪ as the response≪ ≪ of the quasi-particles to a periodic external electric ﬁeld, yielding the well-known zero-sound associated with the oscillations of the Fermi surface, and the current response in transport, etc. The zero sound behavior may be contrasted with the ordinary sound for which ω ν so that the system is always in quasi-equilibrium and deformations occur in an adiabatic manner. ≪

B. Many-body analysis

The starting point for a many-body analysis of the properties of an interacting electron system is based on two important quantities, the single-particle Green’s function (SPGF) and the two-particle Green’s function (TPGF). The SPGF measures the probability amplitude of adding (creating) a bare particle with momentum k > kF (or a bare hole with momentum k < kF) at a position r and time t, and finding it at another position r’ and time, t’. In the context of our discussion, the particular significance of the SPGF lies in the fact that it measures the properties of the many-body interacting system associated with the addition/removal of bare electrons (holes). In the cases where there exist well-defined, long-lived single quasi-particle (quasi- hole) states, it also provides information on the energy and lifetime of such states, as well as the overlap between these states and the corresponding unperturbed states of the non-interacting system. Furthermore, the imaginary part of the SPGF in the k, ω representation provides information on the spectral distribution of the single particle excitations in terms of the unperturbed k-states, as well as the tunneling density of states (TDOS) when adding and removing a bare electron. The TDOS and spectral distribution represent physical quantities which are directly measured in electron tunneling or photoemission experiments. The two-particle Green’s function (TPGF) measures the interaction between two elementary excitations of the interacting system. These can be particle-particle (hole-hole) interactions relevant for cooper-pairing and superconductivity, or electron-hole interaction in a normal system of direct relevance to our discussion at hand. In 3-d the SPGF makes connection to the qp/qh creation operators, while the TPGF makes connection to Landau’s f(k,k’) through the so-called ”interaction operator.”

1. Green’s functions

Given a Hamiltonian:

H = Ho + HI , (34)

11 where Ho is the free-part with eigenstates labeled by ~k, and HI represents the perturbation containing the electron-electron interaction and may be dependent on time (e.g. for the adiabatic turning on of the interaction). 1 The Green’s functions are deﬁned in the Heisenberg representation.1 The SPGF is given by:

G(r,t; r’,t′)= i < 0 T ψ(r,t)ψ†(r’,t′) 0 > , (45) H | { }| H and the TPGF given by:

K(x1, x2, x3, x4)= (46) < 0 T ψ(x )ψ(x )ψ†(x )ψ†(x ) 0 > , H | { 1 2 3 4 }| H

1The three commonly utilized Schroedinger, Heisenberg, and interaction representations are related as follows. In the familiar Schroedinger representation, the evolutions of the wave function and an operator O are given by: d i¯h |Ψ >S= H|Ψ >S (35) dt and d ∂ OS = OS . (36) dt ∂t In the Heisenberg representation, these read: d i¯h |Ψ >H = 0 (37) dt and d i ∂ OH = [H, OH ]+ OH . (38) dt ¯h ∂t

In the case the HI is also time independent, the Heisenberg representation is directly obtainable from the Schroedinger representation via the transformations:

i Ht |Ψ H = e h¯ |Ψ >S (39) and

i Ht − i Ht OH = e h¯ OS e h¯ . (40)

In the interaction representation convenient for the implementation of perturbation theory: d i¯h |Ψ >I = HI |Ψ >I (41) dt and d i ∂ OI = [HI , OI ]+ OI . (42) dt ¯h ∂t

Since Ho is independent of time, we immediately have the transforms:

i H t |Ψ >I = e h¯ O |Ψ >S (43) and

i H t − i H t OI = e h¯ o OS e h¯ o . (44)

12 where T denotes the time-ordering operator signifying an ordering of the ﬁeld operators in the curly brackets with the one at the earliest time to the right. At the same time when two adjacent operators need to be exchanged in position, a negative sign appears to account for Pauli exclusion and Fermi statistic. Here the abbreviated notations xi′s denote ri, ti (and spin, σi ). Depending on the sequencing of the times, t1, t2, t3, and t4, the TPGF can describe the propagation of an electron or hole pair relevant for the investigation of superconductivity, or the propagation of an electron-hole pair of relevance to our discussion of the Fermi liquid. The ψ’s satisfy the equal time anti-commutation relation,

ψ(r,t), ψ†(r’,t) = iδ(r r’). (47) { } − Since we will only consider pure systems which are isotropic, i.e. which possess translational and rotational invariance, it is natural to work in the Fourier transformed, k,ω representation. First transforming into k † Fourier -space and using second quantized notation where ak acting on the unperturbed vacuum state creates a unit occupancy in that k-state, the ψ and ψ† operators are expressed as: 1 ψ(r,t)= eik.ra (t) (48) √2π k Xk 1 ik.r ψ†(r,t)= e a† (t) √2π k Xk The inverse transforms are given by:

1 3 ik.r a (t)= d re− ψ(r,t) (49) k √2π Z 1 3 ik.r a† (t)= d re− ψ†(r,t). k √2π Z These ak(t), ak† (t) operators in the Heisenberg representation are related to their Schroedinger counterparts iHt iHt iHt iHt ak, ak† via ak(t) = e ake− and ak† (t) = e ak† e− (settingh ¯ to 1), where the Schroedinger ak’s and ak† ’s obey the anti-commutation relations:

a ,a† = δ . (50) { k k} k,k’ The k-space SPGF becomes:

G(kt; k’t′)= i< 0 T a (t)a† (t′) 0 > (51) | { k k’ }| = G(kt; kt′)δk,k’ = G(k,t t′)δ − k,k’ [G(k,t”) = i< 0 T a (t”)a† (0) 0 >]. | { k k’ }|

In the non-interacting case G(k, t) takes the familiar form, denoted by Go(k, t), iǫ t ie− k t> 0 k > kF 0 t< 0 k > kF Go(k,t)=  (52) 0 t> 0 k < kF  iǫ t  ie− k t< 0 k < k − F  Completing the time transform into ω space yields:

G(k,ω)= G(k,t)eiωtdt (53) Z 1 iωt G(k,t)= G(k,ω)e− dω, 2π Z

13 and the expression for the non-interacting SPGF, Go(k,ω):

+ 1 0 k > kF Go(k,ω)= ; η = (54) ǫ ω iη 0− k < kF k − − η ensures convergence in the inverse ω transform. We can account for η by adding an infinitesimal imaginary 2 2 part to ǫk. In other words, ǫk ǫk iη. For free electrons, ǫk = ¯h k /2m. This familiar expression readily provides us with an important→ intuition.− η represents an imaginary part of the ”quasi-particle” energy and can be viewed as the decay rate proportional to the inverse lifetime of the particle at momentum, k, and energy ǫk, which is infinitesimal for a non-interacting system. In the presence of interaction, we can expect the imaginary part to acquire a finite value and the lifetime to be finite except exactly at the Fermi energy. In the general case with interaction, the SPGF, G(k, t), has the following properties near t=0 arising from the discontinuity of G(r, t):

+ G(k, 0 )= i(1 mk); G(k, 0−)= imk; ∆G0+ 0− = i, (55) − →

where m =< φ a† a φ > measures the occupancy of the plane wave k state in the ground state, φ > k o| k k| o | o for bare particles.

2. Lehman representation: spectral density, tunneling density of states

To gain further insight into the general case of the interacting system, we go into the Lehman representation for G(k, t) by inserting a complete set of eigenstates for the full Hamiltonian between the operators ak(t) and ak† (0), where

iHt iHt G(k,t) G(k,t; k, 0) = i< 0 T e a e− a† 0 > . (56) ≡ | { k k}| The only surviving matrix elements involve eigenstates of N 1 particles: ± N N+1 iEo t iEn t i n < 0 ak n>e e− t> 0 G(k,t)= | | N N−1 | | (57) iEo t iE ′ t ( i ′ < 0 a† n′ >e− e n t< 0 −P n | k| | k| 2 i(ǫ +µ)t i P e− n t> 0 = n | | k| | 2 +i(ǫn′ µ)t i ′ e − t< 0 ( −P n | | k| | N+1 N+1 P N 1 N 1 in which ǫ =E E (ǫ ′ =E ′− E − ) represents the excitation energy relative to the N+1 (N-1) n n − o n n − o particle ground state. Note that ǫn, ǫn′ > 0. Fourier transforming into a full k,ω representation, we ﬁnd:

2 1 G(k,ω)= (58) | | k| | ǫ + µ iη ω n n X − − 2 1 + k ′ ′ | | | | ǫn + µ + iη ω Xn − − ∞ A+(k,ω′) A (k,ω′) = dω′[ + − ]. ω ω + µ iη ω ω + µ + iη Z0 ′ − − − ′ − Here

2 A (k,ω′)= δ(ω′ ǫ ) (59) + | | k| | − n n X 2 A (k,ω′)= δ(ω′ ǫn′ ) − k ′ | | | | − Xn

14 are the spectral densities of the bare k-states in the exact, N+1 or N-1 particle eigenstates of the full interacting Hamiltonian. These spectral densities represent directly measurable quantities in experiments such as photoemission measurements. Of course in the non-interacting situation, A (k,ω) = δ(ω ǫ ) as ± − k expected, where A+ corresponds to k > kF, and A to k < kF. Furthermore, using the relation: − 1 1 = P + iπδ(ω′ ω + µ), (60) ω ω + µ iη ω ω + µ − ′ − − ′ −

the tunneling density of states, ρTDOS, relevant for the addition or removal of bare electrons can be related to the imaginary part of the SPGF as well as the spectral densities integrated over k-space: 1 ρ (ω)= dkImG(k,ω) (61) TDOS −π Z = dkA (k,ω). (62) ± Z It is precisely this tunneling which is measured in the type of electron tunneling experiments described in this review article.

3. Intuitive Idea of a Quasi-Particle

As pointed out above for non-interacting particles the pole of the single particle Green’s function, Go(k,ω), gives the energy, ǫk of the single particle/hole state at momentum-k. In this case since these states are exact, non-decaying eigenstates, given by a† 0 > for k > k and a 0 > for k < k , they are inﬁnitely long-lived k| F k| F and the poles in the ω-complex plane can be represented by a complex energy, ǫ iη, where: k − 0+ k > k η = F (63) 0− k < kF The lifetime is given by ¯h/ η . Associated with G (k,ω) are the spectral densities, A (k,ω)= δ(ω ǫ ) | | → ∞ o + − k for k > kF and A (k,ω)= δ(ω ǫk) for k < kF. Following these− ideas one can− surmise that in the presence of interaction and under appropriate conditions, the SPGF may possess a structure similar to the non-interacting case. More speciﬁcally, if the spectral density can be decomposed into a sharply peaked ”quasi-particle” contribution centered about some energy,ǫ ˜k, with a lifetime τ ¯h/Γ plus an incoherent, broadly distributed background, i.e.: k ∼ k

A (k,ω)= A qp/qh(k,ω)+ A inc(k,ω), (64) ± ± ± where the quasi-particle contribution takes the form:

zk Γk A qp/qh(k,ω)= , (65) ± π (ω ǫ˜ )2 +Γ2 − k k then the SPGF will correspondingly contain an incoherent piece and a quasi-particle(hole) piece,

G(k,ω)= Gqp/qh(k,ω)+ Ginc(k,ω) (66)

Notice that in Eq. (65), other then the quasi-particle/hole renormalization factor, z ( 1), the Lorentzian k ≤ (Γ )/π[(ω ǫ˜ )2 +Γ2 ] approaches a delta function when Γ 0. In other words, if the state at momentum k − k k k → (k) and energy (˜ǫk) is long lived, then we have a well-deﬁned quasi-particle excited state which is labeled by k just as in the non-interacting case. However, this quasi-particle (hole) has a reduced spectral weight, zk 1. In short, corresponding to a quasi-particle (hole) there is pole in the SPGF at the position ω =ǫ ˜ iΓ≤ . k ∓ k See Fig. 2(a). Note that for k = kF, the quasi-particle renormalization, zkF , also gives the discontinuity in

15 the distribution function, n(k), for bare particles, at zero temperature as shown in Fig. 2(b). Of course, in the non-interacting case, this jump is unity.

((a) b) n(k) 1.0 ~ ε k - iΓk z h h ω k k ~ε - iΓ k µ=EF kp kp F

FIG. 2. (a) Quasi particle (hole) pole in the ω-complex plane, denoted by a solid circle (open circle); (b) zk, the quasi-particle renormalization factor at kF, which is equal to the jump in the bare particle distribution function, n(k), at zero temperature.

The success of the Landau Fermi liquid theory hinges on the fact that in 2 or 3 dimension, the quasi 2 2 particle decay rate goes rapidly to zero. In fact Γk (k kF ) as k approach kF or (ω µ) as ω approaches the chemical potential µ. In the vicinity of the Fermi∼ surface,− the quasi-particles and− quasi-hole states are

long-lived. In fact it turns out the analytic properties of the SPGF requires that at kF, ΓkF = 0. However, in diﬀerent spatial dimensions, the phase space constraint produces diﬀerent functional dependences in the lifetime. In particular, in 1-d, Γ scales as k kF . This leads to logarithmic divergences in a perturbative calculation of the SPGF. | − |

4. Two Particle Green’s Function

The structure of the TPGF can be broken down into a nearly “free” quasi-particle contribution involving products of the SPGF’s only, and an interacting part involving multiple scattering of the particles through their interaction. Going directly into the (k,ω) representation, K(k ,t ; k ,t ; k ,t ; k ,t ) K(1, 2, 3, 4) (67) 1 1 2 2 3 3 4 4 ≡ = < 0 T a (t )a (t )a† (t )a† (t ) 0 >, k1 1 k2 2 3 4 | { k3 k1 }| we have: K(1, 2, 3, 4) = K + K = G(1, 3)(G(2, 4) G(1, 4)G(2, 3) (68) F int − 4 + dt′ G(k ,t t′ )G(k ,t t′ )γ(k ,t′ ; k ,t′ ; k ,t′ ; k ,t′ ) i i 1 − 1 2 2 − 2 1 1 2 2 3 3 4 4 i=1 Z Y G(k ,t t′ )G(k ,t t′ ), × 3 3 − 3 4 4 − 4 where γ accounts for the interaction. Further Fourier transform into ω space yields:

K(k1,ω1; k2,ω2; k3,ω3; k4,ω4)= (69)

G(k1,ω1)δ δ(ω1 ω3)G(k2,ω2)δ δ(ω2 ω4) k1,k3 − k2,k4 − − h G(k1,ω1)δ δ(ω1 ω4)G(k2,ω2)δ δ(ω2 ω3) k1,k4 − k2,k3 − +G(k1,ω1)G(k2,ω2)G(k3,ω3)G(k4,ω4) i

Γ(k1,ω1; k2,ω2; k3,ω3; k4,ω4)δ(ω1 + ω2 ω3 ω4)δ . − − k1+k2,k3+k4 h i 16 Here Γ is known as the interaction operator deﬁned by:

γ(k1,ω1; k2,ω2; k3,ω3; k4,ω4)= (70)

Γ(k1,ω1; k2,ω2; k3,ω3; k4,ω4)δ(ω1 + ω2 ω3 ω4)δ . − − k1+k2,k3+k4 For electron-hole interactions, Γ is a regular function in 3 dimensions. Detailed investigation of Γ will enable the justiﬁcation of the interaction term in the Fermi liquid picture.

5. Dynamical equation of motion for the single particle Green function

To actually compute the single particle Green’s function and gain further insight, we need the dynamical equation of motion which can be obtained in the Heisenberg representation in the standard way:

∂G(r,t) i = [G(r,t), H]. (71) ∂t Accounting for the discontinuity of G at t=0, we arrive at:

∂ i 2 ∇ G(r,t; r’,t′) (72) ∂t − 2m 3 + d r”V (r r”)K(r”,t; r,t; r”,t ; r’,t′) − − Z = iδ(t t′)δ(r r’), − − which relates the SPGF to the TPGF. By introducing the quantity, Σ, this expression can be recast into a more intuitive form:

Σ(r,t; r”’,t”′)=ΣF +Σint (73)

3 + = i d r”V (r r”) G(r”,t; r”,t )δ(r r”’)δ(t t”′) − − − − Z + G(r,t; r”,t )δ(r” r”’)δ(t t”′) − − − 3

i d3r” d3r dt V (r r”) − i i − i=1 Z ZZZ Y G(r”,t; r1,t1)G(r,t; r2,t2)G(r3,t3; r”,t)γ(r1,t1; r2,t2; r3,t3; r”’,t”′).

This allows us to write Eq. (72) in the form:

∂ i 2 ∇ G(r,t; r’,t′) (74) ∂t − 2m 3 i d r”′dt”′Σ(r,t; r”’,t”′)G(r”’,t”′; r’,t′) − Z = iδ(t t′)δ(r r’). − − By transforming into (k,ω) space, we arrive at the very intuitively suggestive form of this Dyson equation:

k2 G(k,ω) ω Σ(k,ω) =1. (75) 2m − − or,

17 1 G(k,ω)= 2 (76) k ω Σ(k,ω) iη 2m − − − +0 k > k η = F 0 k < kF − Just as in the case of Go we expect the pole of G(k,ω) to yield the single particle excitations,ǫ ˜k, with a quasi-particle weight, zk, given by the residue of the pole: 1 z = . (77) k ∂Σ ω=˜ǫk 1+ ∂ω | The quantity Σ simply takes into account the modification of the single particle excitation energy by the interaction, and hence the nomenclature, the self-energy. Furthermore, the imaginary part of Σ, ImΣ, is related to the decay to the single particle excited state (Eqs. (63), (82) - (84)). In fact, through the use of perturbation theory, one finds that Im(Σ) must be continuous when k crosses the Fermi surface but changes sign (Fig. 2(a)). Therefore at k=kF, the decay rate is exactly zero. Furthermore, one finds a quasi-particle 2 decay rate, Γk, proportional to k kF for d 2. This slow rate of decay for excitations near kF is essential for the success of the Fermi liquid| − picture.| However,≥ in 1-d the reduced phase space leads to Γ k k k ∝ | − F| instead, and logarithmic divergence in Re(Σ) at k = kF, signaling the breakdown of the Fermi liquid picture (Luttinger, 1960; Voit, 1995; Metzner et al., 1998). For the purpose of actual computation, it is useful to identify the irreducible self-energy parts, representing terms in the perturbation series in which the corresponding Feynman diagrams cannot be cut into disjoint pieces by simply cutting a single propagator line, as depicted in Fig. 3(a). The full propagator is built up by summing up an infinite series of terms each containing such irreducible parts linked together by a single propagator line. Figure 3(b) presents the usual diagrammatic representation of this well-know series, where the unperturbed SPGF is denoted by a thin line, the full propagator a heavy line, and the total contributions of the irreducible parts by a shaded circle. The resultant expression for the full propagator:

G(k,ω)= Go(k,ω)+ Go(k,ω)S(k,ω)G(k,ω), (78)

is the well-know Dyson equation. Solving for G, we arrive at:

1 G(k,ω) = [ǫ ω S(k,ω)]− , (79) k − − which is identical in form to Eq. (76) above. We can immediately identify the S, the sum of the irreducible parts, as the self- energy, Σ(k,ω). Beyond the computation of the self-energy, Σ(k,ω), the concept of such ”irreducible self-energy insertions” depicted in Fig. 3(a) will be needed for analyzing the structure of quasiparticle/hole interaction in Section II B 6.

18 cut

(a)

(b)

FIG. 3. (a) A reducible diagram for the single-particle Green’s function (SPGF) which can be cut into two irreducible parts; (b) the SPGF Dyson Equation. A heavy line represents the full propagator, while the light line represents the unrenormalized propagator.

In d = 3, the existence of well-defined quasi-particles(holes) signifies that the spectral densities, A (k,ω), contain a coherent part associated with the long-lived quasi-particles (holes) on top of an incoherent± part as discussed in Section II B 3. Since there is a well-defined, one-to-one correspondence of these quasi-particle (hole) states with the unperturbed bare electron (hole) states labeled by the momentum k, we can imagine constructing quasi-particle (hole) creation operators from the bare electron (hole) creation operators by filtering out the incoherent portion of the spectral weight. Such a filter, however, must not disturb the decay of the excitation on the time scale of 1/Γk. In effect, this can be accomplished by an adiabatic turn on of the bare particle operator on a time scale, tadiab < 1/Γk, but tadiab > tincoh, the latter representing a microscopic time scale beyond which the incoherent contribution becomes small. In fact such a quasi-particle (hole) creation operator, αk† for k > kF:

0 1 iǫ˜ t ′ k t /tadiab αk† dt′e− e− ak† , (80) ≈ tadiab√zk Z−∞ and a corresponding operator for k < kF with this type of ﬁltering can be rigorously constructed in perturbation theory which represents a major triumph of the Landau Fermi liquid picture. When operated on the ground state, α† creates an approximate quasi-particle eigenstate, φ >: k | k

φ >= α† 0 >, (81) | k k| and likewise for αk and a quasi-hole eigenstate. The coherent part of the SPGF corresponding to the quasi-particle (hole) is: z G = k , (82) coh,qp/qh ǫ˜ ω iη k − − with k2 ǫ˜ = ǫ Σ(k,ω)= Re[Σ(k,ω)] iΓ , (83) k k − 2m − ∓ k

19 and

Γ = Im[Σk,ω]; (84) k ± > 0,k>k Im[Σ(k,ω)] = F . < 0,k

6. Quasi-particle interaction

The SPGF has given us information about the single particle excitations and in particular, under appropriate conditions (e.g. in 3-d in the absence of superconducting instability), the Fermi liquid long-lived quasi-particles (holes) for k with k near kF. To complete the picture and obtain full justiﬁcation of the Landau Fermi liquid, it is necessary to study the interaction between the quasi-particles (holes) embodied in the TPGF and the interaction operators, γ and Γ. Here again, many body perturbation techniques prove extremely powerful. We sketch some of the key points which lead to a justiﬁcation of the Landau interaction term, f(k, k’) (cf. Eq. (8)), and the Boltzmann equation (Eq. (31)). The analysis is quite involved and include several key ingredients: 1) Multiple scattering in the TPGF, K, analyzed in terms of the interaction operator, Γ (Eqs. (68) - (70)), and its irreproducible parts, I, via the Bethe-Salpeter equation:

s s Γ(p,p′; ω)= I(p,p′; ω) (85) s s + I(p,p”; ω)G(p” ω/2)G(p”+ ω/2) Γ(p”,p′; ω), e e− Xp” e e e e where I corresponds to diagrams which cannot be cut across 2 legs (Fig. 4). The quantities p, p’, p”, and ω represent the four momenta: p = (ω, k), p′ = (ω′, k’), p” = (ω”, k”), and ω = (ǫ, q). For our discussion, we are again interested only in the electron-hole scattering terms. The inclusion of spin is s r accountede for by decoupling the singlet-triplet channels. We will focus on Γ in the singlete channel, Γ , where r = q/ǫ, parameterize the exchange of energy, ǫ, and momentum , q in the interaction. 2) The behavior of the SPGF product pair in the limit of small energy- momentum transfer, ω 0, where ω = (ǫ, q): → e 2 2 q.vk e G(p ω/2)G(p + ω/2) = G (p)+2iπz δ(µ ω)δ(µ ǫ˜ ), (86) − k ǫ q.v + iα − − k − k e e z [G(k,ω)= G (k,ω)+ k ]. inc ǫ˜ ω iη k − −

3) Vertex functions, Λµ, in the response, Rµ, to external driving scalar and vector potentials, Aµ, expressible in terms of the TPGF, K. The external potentials couple to the current, Jµ = (ρ, J), contributing a perturbing Hamiltonian:

µ Hext = J AµdV = ρΦdV + J.AdV, (87) Z Z Z

where the Fourier transform of Jµ is expressible as:

20 Jµ, q = vµa† a , (88) − k q k − Xk

with vµ = (1, ki/m). The response is given by:

Rµ(k,t,t′; ω) (89)

∞ ǫt′ = dt′e− < 0 T a† a Jµ, q(t”) 0 >, e | { k+q k − }| Z−∞ leading to:

s Rµ(k,ω; ω)= vµ′ K(p,p′; ω) (90) ′ Xp e Λµ(p, ω)G(p + ω/e 2)G(p ω/2). ≡ −

The quantities, Λµ, thus defined via Eq. (90) aree the vertexe functions.e 4) Ward Identities reflecting underlying conservation laws, e.g. continuity equations reflection the conservation of charge, which finally enables relating the vertex functions, Λµ, and the interaction operators, s 0 s o Γ , Γ∞ to the Landau parameters: Λ z , J , v , Γ f(k, k’). µ ⇔ k k k ⇔ We begin by discussing the (reduced) interaction operator, Γ, associated with the TPGF (Eq. (68) - (70)). Particularly important for our discussion is the operator embodying the electron-hole scattering relevant for the Fermi-liquid properties. In perturbation theory it is possible to derive an equation analogous to the Dyson equation for SPGF discussed in Section IIB5, Eq. (78), in terms of an irreducible part, I, which cannot be isolated into separate parts by cutting a pair of propagator lines:

i i i Γ(p,p′; ω)= I(p,p′; ω)+ I(p,p′′; ω) (91) ′′ Xp i G(p′′ ω/e 2)G(p′′ + ω/e2) Γ(p′′,p′; ω), e − with a corresponding diagrammatic representation shown in Fig. 4. The left superscript, i, is an index e e e associated with the spin. This equation, known as the Bethe-Salpeter equation, is an integral equation and its solution in general is more difficult compared to the Dyson equation for the SPGF. The reduced interaction operator, iΓ, can be decoupled into separate equations for the spin singlet (i=s) and spin triplet (i=t) parts. i A detailed study of the properties of Γ enables the identification of the Fermi interaction function, f(k, k′), with the singlet reduced-interaction-operator in the appropriate low energy/momentum limit, sΓo (often denoted as oΓo), as well as a full microscopic justification of the Boltzmann transport equation Eq. (31).

I Γ Γ I I I I

FIG. 4. Bethe-Salpeter equation for Γ, the reduced interaction operator.

21 In analyzing sΓ(k,ω), the ﬁrst step lies in understanding the behavior of the extremely singular product term, G(p ω/2)G(p + ω/2). Due to the fact that the coherent part of G(p) can have a pole in either − the upper or lower half of the complex-ω plane determined by whether or not k exceeds kF, in other words (Eq. (82)): e e z G(ω, k) k (92) coh ≈ ǫ˜ ω iη k − −

+0 k > k η = F , (93) 0 k < kF − the product G(p ω)G(p + ω) can exhibit very singular behavior depending on which regime of phase space in which the momentum− vectors, k q/2, lie. In specifics, it depends on whether these vectors lie inside or ± 2 outside of the Fermie sphere.e Consequently, G(p ω/2)G(p + ω/2) is not in general given simply by G (p) when ω 0, but instead: − → G(p ω/2)G(p + ω/2)e e (94) e − 2 2 q.vk = G (p)+2πiz δ(µ ω)δ(µ ǫ˜k). e k ǫe q.v + iα − − − k Since we are interested in the ω 0 limit where ω = (ǫ, q), by rewriting the above expression in terms of the vector r = q/ǫ, we can recast→ it as: e G(p ω/2)Ge(p + ω/2) (95) − r.v = G2(p)+2πiz2 k . e k 1e r.v + iα − k It becomes apparent that the contribution of the singular second term depends on how the ω 0 limit is approached through r. Note that since the infinitesimal, α, is taken to zero before ǫ, for convenience→ we have replaced α/ǫ by α. For 3-d it turns out in the expression for the reduced interaction operator, the irreducible s e contribution, I(p, p′; ω), is not singular as ω 0 for both short ranged interaction or long range interaction → once effects of screening is properly taken into account (by subtracting out the term Vq/2πi(Volume)). The singular terms arise solelye from the G G producte and its repetition in an infinite series. Utilizing the last expression for the G G product, the Bethe-Salpeter equation for Γ takes the useful form:

s r s r s Γ (p,p′) lim Γ (p,p′; ω)= I(p,p′) (96) ≡ ω 0 → r.v s e 2 2 k” s r + I(p,p′) G (p”)+2πizk” δ(µ ω′′)δ(µ ǫ˜k”) Γ (p′′,p′) e 1 r.vk” + iα′′ − − Xp” −

This form proves useful in help establishing the connection with Landau Fermi liquid quantities, f(k, k’), Jk, vk, etc., through the Ward identities. To make these connections, we need to relate the response functions to external electromagnetic perturbations to the TPGF and the vertex functions associated with the response. We consider an external perturbation in the form of:

3 Hext = d r[Φ(r,t)ρ(r,t)+ A(r,t).J(r,t)], (97) Z where Φ and A represent external scalar and vector potentials and ρ(r, t), J(r, t) the density and current operators. Its Fourier representation can be written as:

Hext = Φq(t)ρ q + Aq(t).J q, (98) q − q − X X

22 with ρq = a† a and Jq = (k/m)a† a . The linear response will be given by the density- k k+q k k k+q k density andP current- current correlations,P Soo and Sij where i and j denote the three spatial coordinates:

Soo(q,t)=2π< 0 T ρq(t)ρ q(t) 0 > (99) | { − }| and

Sij (q,t)=2π< 0 T Ji,q(t)Jj, q(t) 0 > . (100) | { − }|

In their Fourier representation, these correlation functions, Soo(ω) and Sij(ω) are related to the TPGF by:

s Soo(ω)= K(p,p′; ω) e e (101) ′ Xp,p and e e

ki kj s Sij (ω)= K(p,p′; ω). (102) ′ m m Xp,p It is therefore useful to study the quantities:e e

kµ s Rµ = K(p,p′; ω) (103) ′ m Xp Λµ(p, ω)G(p + ω/e 2)G(p ω/2) ≡ − s with k (m, k , k , k ) and µ =0, 1, 2, 3. The fact that K(p, p′; ω) can be written as (Eq. (69)): µ ≡ 1 2 3 e e e K(p ,p ,p ,p ) (104) 1 2 3 4 e = G(p )G(p )[δ δ δ δ ]+ G(p )G(p )G(p )G(p )γ(p ,p ,p ,p ), 1 2 p1,p3 p2,p4 − p1,p4 p2,p3 1 2 3 4 1 2 3 4 where γ(p , p , p , p ) = Γ(p , p , p , p )δ(ω + ω ω ω )δ (Eq. (70)), yields: 1 2 3 4 1 2 3 4 1 2 3 4 k1+k2 k3 k4 − − − −

r kµ kµ′ s r Λµ(p, ω)= + Γ (p,p′; ω)G(p′ + ω/2)G(p′ ω/2) (105) m ′ m − Xp s r e = λµ + I(p,p′; ω)G(ep′ + ω/2)Ge (p′ ω/2)Λe µ(p′, ω). ′ − Xp e e e e with λµ (1, k1/m, k2/m, k3/m) These expressions remain valid for long-range Coulomb interaction for ≡ 2 2 r s r which Vq = 4πe /q in 3d, if one takes account of the screening effects by replacing Λµ and Γ by their ”screened” counterparts. r The final step to making the connection lies in the Ward identities which relate the vertex functions, Λµ, to the self-energy, Σ(k,ω). The Ward identities are a consequence of conservation laws such as the continuity equation for charge or spin (Metzner et al. 1998) and can also be deduced from an analysis of so-called skeleton diagrams in perturbation theory in which the self-energy insertions in the propagator (SPGF) are implicit and suppressed within the diagrams. Such analysis is based on a study of the variation of the SPGF as a function of infinitesimal changes in the momentum, k, or energy, ω or µ. By studying these changes in perturbation theory based on the structure of the Feynman diagrams wherein any propagator line is understood to be the already renormalized propagator, and therefore no self-energy insertion is allowed, it is possible to show that certain combinations of quantities related to the self-energy, Σ(k,ω), obey the same r integral equations as the vertex functions, Λµ. The end result is the all important connection between the vertex functions, Γ, and the Landau Fermi liquid quantities on the Fermi surface where ω = µ (Nozieres, 1964; Landau, 1958):

sΓo o Γo =2πiz z f(k, k’), (106) ≡ k k’

23 o 1 ∂Σ(k,ω) Λ4 = =1+ , (107) zk ∂ω

vk dkF Λ4∞ = , (108) zk dµ

J o kα Λα = , (109) zk

v kα Λα∞ = . (110) zk These expressions complete the justification of the Landau picture. As a final step in order to justify the Boltzmann equation, it is sufficient to utilize the quasi-particle creation operator (Eq. (80)) to construct the local quasi-particle density operator, n(k, r, t), based on an analogy with the electron density operator:

< φ ρ (r) φ>= δn(k, r,t), (111) | qp | Xk where

iqr δn(k, r,t)= < φ α† α φ>e (112) k q/2 k+q/2 q | − | X and φ> is the state of the system under the external perturbation. By considering the dynamical evolution of this| density operator where the full Hamiltonian contains the electron-electron interaction as well as perturbation due to coupling to external electromagnetic ﬁelds, and utilizing generalized response functions for quasi-particles analogous to those for the electrons, Eq. (103), the Boltzmann equation, Eq. (31), can readily be derived as an alternate form of the Bethe-Salpeter equation (Nozieres, 1964; Landau, 1957).

C. Breakdown of the Fermi liquid Picture in 1d and the Tomonaga-Luttinger liquid

The Fermi liquid picture breaks down in 1 dimension. This is a direct consequence of the unique phase space structure in 1-d, notably the fact that the Fermi surface consists of two discrete points ( kF) rather than a line (lines) or surface (surfaces), and that for each branch of the momentum-energy dispersion–left-± or right-moving branch–the 1-d wave-vector uniquely determines the energy. Some of the most significant, inter-related consequences include: 1) Im[Σ(k,ω)] ω rather than ω2 in 2- and 3-d. This is one key signature indicating the absence of well-defined quasi-particles∝ in 1-d. Im[Σ] is identifiable as the quasi-particle scattering rate and its rapidly vanishing behavior as the Fermi energy is approached is responsible for the existence of well-defined quasi- particles and holes in 3-d. 2) Logarithmic divergences in the two-particle interaction operator commencing in second order perturbation theory, and consequently also in the real part of the self-energy, Re[Σ(k,ω)] as well. Note that the real and imaginary parts of Σ are related by the Kramers-Kronig relations. 3) The consequence of a logarithmic divergence at EF directly implies that the quasi-particle renormaliza- 1 tion, or quasi-particle weight, vanishes: zk kF [1+∂[Re[Σ]/∂ω]− EF 1/log(ω) 0! Hence the one-to-one correspondence of the unperturbed k electron| ∼ state to the elementary| ∼ excitations→ of the interacting system is lost. 4) Low energy electron-hole excitations arising from interaction can occur about the two discrete Fermi points at k with either small momentum transfer, q << k , or across the Fermi sea with momentum ± F F

24 transfer q 2kF only. This indicates that as the excitation energy approaches zero, there is a forbidden region for electron-hole≈ excitations between 0 q 2k . Generalization to multiple electron-hole processes ≤ ≤ F yield a series of forbidden regions, 2(n 1)kF q 2nkF, where n denotes the number of electron-hole pairs. − ≤ ≤ and 5) Spin-charge separation, i.e. the spin and charge bosonic elementary excitations propagate at different velocities. The presence of low energy, ungapped bosonic charge and spin modes are a direct consequence of the finiteness of the spin and charge density responses at low q and ω. All of these well-known behaviors point to the inescapable fact that the Fermi liquid picture is not appropriate in 1-d once electron-electron interaction is turned on, and the Fermi liquid cannot serve as an adequate starting point for understanding certain aspects of this strongly-correlated system, particularly those processes involving the so-called charged sector where injection or removal of bare electron takes place. In contrast, thermodynamic properties which depend on the neutral excitations (e.g. e-h pairs, collective modes, plasmons, etc., at low ω, q(ω ǫF , q kF ) can still adequately be described by the Fermi liquid, albeit with the caveat that the quasi-particle≪ ≪ to bare electron one-to-one correspondence is lost ( Solyom, 1979; Schulz, 1991; Voit, 1995; Metzner et al., 1998; Carmelo and Castro Neto, 1996; Castellani and Di Castro, 1999; Ng, 1997) . Historically there have been two major parallel and complementary approaches to the theoretical investigation of interacting metallic 1-d systems, with their focus centered about the idealized Tomonaga-Luttinger models ( Tomonaga, 1950; Luttinger, 1963). One approach is based on the bosonization technique ( Tomon- aga, 1950; Matthis and Lieb, 1965; Luther and Peschel, 1974; Luther and Emery, 1974; Haldane, 1981), and the other on the so-called g-ology model in connection with renormalization group treatment (Solyom 1979; Menyhard and Solyom, 1973; Metzner et al., 1998). Note that related models in the field theoretic context have also been explored ( Heidenreich et al., 1975). A general discussion of the topic of interaction in 1-d encompasses rather diverse range of systems and phenomena. In addition to metallic systems other related systems such as the 1-d Hubbard model or models with strong back-scattering are of great interest in their own right. Here we confine our attention to metallic systems with weak back-scattering to avoid complications introduced by insulating tendencies or by gapped behavior in the spectrum of low energy excitation. The bosonization approach, first introduced by Tomonaga (1950), Matthis and Lieb (1965), and later on expanded upon by Luther and Peschel (1974) and Haldane (Haldane, 1979; Haldane, 1981) indicates that a bosonic description is appropriate in 1-d for interacting systems dominated by forward scattering. The renormalization group approach is basically a method for going beyond perturbation theory and in many instances has the effect of summing up the most relevant (logarithmic divergent) diagrams in a systematic and controlled way. Because its implementation is invariably founded on many-body perturbation theory, the usual many-body techniques play a natural and useful role. These including the Ward identities arising out of conservation laws connecting the vertex functions with physically measurable quantities, equation of motion methods, etc. As a consequence, this type of analysis has proven helpful in elucidating the relation between a Fermi liquid and a Tomonaga-Luttinger liquid, while at the same time allowing clear-cut differ- entiation of the two systems. The aforementioned signatures of the difficulties in conventional treatments of the 1-d problem are simply indicative of the fact that the Fermion representation is not the appropriate basis for describing the essential low energy physics and dynamics. The Tomonaga and Luttinger models are idealized interacting 1-d models which are soluble. The key characteristics of the models enabling solubility are (i) linearized dispersion about the two Fermi points at kF; and (ii) forward scattering interaction only with no back-scattering terms. These original models do not± include electron spin. Generalization to include spin is straight forward and will be discussed in the g-ology model. Specifically the Tomonaga Hamiltonian is given in second-quantized form by (Tomonaga, 1950; Dzyaloshinskii and Larkin, 1973) :

k2 1 HTomo = a† ak + λqa† ak+qa† ′ ak′ q, (113) k k k − 2m 2 ′ Xk k,kX,q where q is restricted to q << 2k , or equivalently λ is appreciable only for small q values and approaches | | F q zero when q is of order 2kF. The Luttinger model introduces massless Dirac Fermions with linear dispersion which renders the problem more mathematically tractable . The introduction of Dirac Fermions and a sea

25 of negative energy states does not qualitatively change the low energy physics since excitations to the Fermi level require large energies. The Luttinger Hamiltonian deﬁned on a line of length L is given by ( Luttinger, 1963; Matthis and Lieb, 1965; Luther and Peschel, 1974):

HLutt = Ho + HI (114)

L Ho = vF Ψ†(x)σzpΨ(x)dx (115) Z0 L = v [ψ†(x)pψ (x) ψ†(x)pψ (x)]dx F r r − l l Z0 = v k(a† a a† a ), F r,k r,k − l,k l,k Xk

λvF HI =2 uq(a† ar,k+qa† ′ al,k′ q). (116) r,k l,k − L ′ k,kX,q

Here vF is the magnitude of the Fermi velocity, Ψ† = (ψl, ψr), σz is the z-component of the Pauli spin matrix, and the subscripts l and r refer to the left and right moving branches of the energy-momentum dispersion, respectively. Note that there are no backscattering terms of the form ar†,kal,k+qal†,k′ ar,k′ q which scatter particles between left- and right-moving branches. Second order perturbation calculation for− the self energy, Σ, corresponding to the diagram of the type shown in Fig. 5 yields the well know results that Im[Σ] ω and Re[Σ] ωlogω leading to the vanishing of the quasi-particle normalization factor, z 0 ( Luttinger,∼ 1960; Metzner∼ et al., 1998). This is a clear signature that conventional perturbative treatment→ so successful in demonstrating the validity of the Fermi liquid in 3-d is no longer valid in 1-d. The two models exhibit essentially the same low energy physics. Tomonaga demonstrated that this type of 1-d interacting model supports bosonic sound waves excitation which are plasmon oscillations with a linear dispersion ω = vqq where the interaction dependent velocity is given by:

2 2vF vq = vF + λq. (117) r π

FIG. 5. Cooper (left) and Zero Sound (right) diagrams for 1-d. Solid lines denote the unrenormalized propagator for right-moving electrons while the dashed denote the left-moving electrons.

26 A similar conclusion was reached by Matthis and Lieb (1965) for the Luttinger model. The Tomonaga and Luttinger models can be solved two ways, by bosonization ( Tomonaga, 1950; Matthis and Lieb, 1965; Luther and Peschel, 1974; Haldane, 1979; Haldane, 1981) and by non-perturbative many-body techniques such as renormalization group analysis and Ward identities (Dzyaloshinskii and Larkin, 1973; Everts and Schulz, 1974; Solyom, 1979; Metzner and Di Castro, 1993; Metzner et al., 1998; Voit, 1995). It is convenient to work with the Luttinger model since the introduction of the massless Dirac Fermions enables a fully rigorous mathematical solution of the problem.

1. Bosonization

In a sense, bosonization is a natural way to address the 1-d interacting system since for models dominated by forward scattering, the elementary excitations are linearly dispersing boson modes. The key idea of bosonization lies in the central role played by the density operator:

ρi,q = ai,k† +q ai,k (118) Xk

ρi, q = ai,k† qai,k (119) − − Xk q 0 and i=l,r. This definition is not fully rigorous and leads to ambiguities for q = 0 as pointed out by ≥ Haldane (1979, 1981). For q = 0 it is sufficient to subtract out the expectation value of ρi,0 in the ground state of the filled Fermi sea:

ρ = [a† a < 0 a† a 0 >]. (120) i,0 i,k+q i,k− | i,k+q i,k| Xk This well-defined operator also properly ensures that quantities arising from the cancellation of two divergent quantities when the continuum limit is taken to reflect an infinite degrees of freedom are uniquely and correctly accounted for. When used in a computation of commutation relations between operators, this subtraction is equivalent to a normal ordering with respect to the ground state of a filled Fermi sea in which also annihilation operators which annihilation the ground state are placed to the right before the cancellation of divergent quantities is carried out (Matthis and Lieb, 1965; Haldane, 1979; Haldane, 1981). In so doing one obtains the all important commutation relation for the ρ’s:

sgn(i)Lq [ρi,q,ρj, q′ ]= δi,j δq,q′ , (121) − − 2π where sgn(i)=+1 for i=r, and sgn(i)=-1 for i=l. This commutation relationship immediately leads to the deﬁnition of harmonic oscillator type raising- lowering operators for q > 0:

2π a† = ρ (122) q L q r,q s | |

2π a† q = ρl, q, (123) − L q − s | | and

2π aq = ρr, q (124) L q − s | |

27 2π a q = ρl,q, (125) − L q s | | respectively. The lowering operators annihilate the ground state. Note that there is no q=0 bosonic mode. The q=0 mode is represented by the number operator in each of the two left- and right-moving branches for which

ρ = N = [a† a < 0 a† a 0 >], (126) i,q=0 i i,k i,k− | i,k i,k| Xk

as mentioned above regarding normal ordering. In terms of these boson operators a† q and a q, after ± ± normal ordering to properly subtract out divergent contributions the kinetic term of the Hamiltonian, Ho = v k[a† a a† a ], can be written in terms of the boson raising and lowering operators as: F k r,k r,k − l,k l,k

P vF 2 2 Ho = vF q (aq†aq + a† qa q)+ [(Nr) + (Nl) ]. (127) | | − − L q>0 X

A generalized interaction term of the Tomonaga-Luttinger type describing forward scattering only, Hfor, can be written as: π Hfor = [Vq(ρr,qρr, q + ρl,qρl, q)+ Uq(ρr,qρl, q + ρr, qρl,q)]. (128) L − − − − q X In terms of the boson operators, the total Hamiltonian takes the quadratic form (Haldane, 1979):

H = Ho + Hfor (129) 1 π k L = v q + [v ( F + N + N )2 + v (N N )2] −2 F | | 2L N π r l J r − l q X 1 + q [(vF + Vq)(aq†aq + aqaq†)+ Uq(aq†a† q + aqa q)], 2 | | − − q X where the velocities v = v + V + U and v = v + V U characterize the propagation of the N F 0 0 J F 0 − 0 charge and current modes associated with the symmetric and anti-symmetric combinations of Nr and Nl, N = (kFL)/π + Nr + Nl and J = Nr Nl, at low energies, respectively. This quadratic form is readily diagonalized by a Bogoliubov transformation:−

bq† = cosh(φq )aq† sinh(φq)a q , (130) − − 1 1 with φ = tanh− [ U /(v + V )], yielding a Hamiltonian of non-interacting boson modes: q 2 − q F q 1 π H = ω b b + (ω v q )+ [v N 2 + v J 2], (131) q †q q 2 q − F | | 2L N J q X X where ω = (v + V )2 U2 q , while the boson (plasmon) velocity, v v = (v + V )2 U2 as q F q − q| | q → o F 0 − 0 2 q 0. Noteq that the relationship vo = vNvJ holds. This relation is important andp will remain valid in the→ case of the Luttinger liquid when interaction between boson modes are present. For this Hamiltonian both the quantum numbers N, reflecting total charge, and J, reflecting the total current are conserved. This form of the Hamiltonian clearly illustrates that the low energy excitations of this 1-d interacting Fermion system are bosonic modes. Furthermore, it can be shown that the boson representation generates a complete Hilbert space equivalent to the original Fermionic Hilbert space ( Overhauser, 1965; Haldane, 1981). What remains is the need to introduce a well-defined fermion operator, ψi(x), in terms of the boson operators. Luther and Peschel (1974) first discovered such a representation, although the original expression suffers

28 some ambiguities, it is nevertheless useful for illustrating the main features. The Luther-Peschel expression for the Fermion ﬁeld, ψLP,i(x), is given by: 1 ψLP,i(x)= exp[ikF x + iφi(x)], (132) (2πα) where i=r,l and p

i 1 iqx iqx φi(x)= − [ρi, qe ρi,qe− ]. (133) L q − − q>0 X

Note that the density operator in real space, ρi(x), can be written as:

2π iqx iqx 1 ρi(x)= [ρi, qe + ρi,qe− ]= ∂xφi. (134) L − 2π q>0 X The above expression for the Fermion ﬁeld can readily be shown to obey the appropriate anti-commutation B A relation by utilizing the standard identity, eAeB = e[A,B]e e for [A,B] equaling a c-number. ¿From such an expression it is possible to deduce all the power-law correlations in the SPGF and multi-particle GF’s ( Luther and Peschel, 1974; Theumann, 1967; Dover, 1968; Efetov and Larkin, 1975; Finkelstein, 1977). The quantity φ, utilized in the Bogoliubov transformation, governs the power law at large distances. For the SPGF, the power-law exponent is given by:

vF + Vo α = cosh(2φo)= . (135) (v + V )2 U 2 F o − o We will defer the derivation of this result utilizingp the alternative equation of motion method below. The Luther-Peschel expression suﬀers some ambiguities as it is necessary to take the limit of α 0. A full, well-deﬁned expression which properly takes into account the q=0 mode is quite involved and→ will not be discussed here. Instead, we refer the interested reader to the original papers by Haldane (1979, 1981).

2. Power law behavior in the single particle Green’s function

Alternatively, Ward identities derivable from either the equation of motion ( Everts and Schulz, 1974) or skeleton diagrams in a perturbation diagrammatic analysis (Dzyaloshinskii and Larkin, 1973) enables the solution of the correlation functions, e.g. Green’s functions, in closed form. These Ward identities are direct consequences of conservation laws in the Tomonaga-Luttinger model. In addition to the usual conservation of total charge, the separate conservation of charge in the left and right-moving channels, as a result of an absence of backscattering processes that couples the channels, leads to additional Ward identities which together make possible an exact solution. To illustrate these points, consider the charge density in the left or right channel, ρi(x), i=r,l:

1 iqx ρ (x)= ρ e− . (136) i L i,q q X The evolution of this operator is governed by the equation of motion:

∂ρ (x) i i = [ρ (x), H]. (137) ∂t i

Using the commutation relation between ρi,q and the Fermion operators, ai,q:

[ρi,q,ai′,q′ ]= δi,i′ aq′ q, (138) − −

29 and the commutator for the ρi,q’s (Eq. (121)), yields the continuity equations:

∂ρ (x) i r = q[(v + V )ρ + U ρ ] (139) dt − F q r,q q l,q

∂ρ (x) i l = q[(v + V )ρ + U ρ ], (140) dt F q l,q q r,q

where the currents in the right and left channels are proportional to the respective ρi,q’s. These equations reﬂecting separate right and left channel charge conservation give rise to the important Ward identities for the density correlation functions, Ri(k1, t1;q, t; k2, t2):

Ri(k1,t1; q,t; k2,t2)=< 0 T ar,k (t1)ρi,q(t)a† (t2) 0 >, (k1, k2) > 0, (141) | { 1 r,k2 }| and

Ri(k1,t1; q,t; k2,t2)=< 0 T al,k (t1)ρi,q(t)a† (t2) 0 >, (k1, k2) < 0. (142) | { 1 l,k2 }|

Note that we are only interested in k1, k2 in the neighborhood of kF. This restriction also avoids complications from the positron branch which is irrelevant to the physics of interest. To arrive at the Ward identities we again use the usual relationship, i∂O(t)ˆ /∂t = [Oˆ , H] on the Ri’s, and Eqs. 139 and 140 to ﬁnd:

[i∂t + sgn(i)q(vF + Vq)]Ri(k1,t1; q,t; k2,t2)+ qUqRj (k1,t1; q,t; k2,t2) (143)

= δi,rδk2,k1 q[δ(t t1)G(k2,t1 t2) δ(t t2)G(k1,t1 t2)], (k1, k2) > 0, − − − − − − and

[i∂ sgn(i)q(v + V )]R (k ,t ; q,t; k ,t )+ qU R (k ,t ; q,t; k ,t ) (144) t − F q i 1 1 2 2 q j 1 1 2 2 = δi,lδk2,k1 q[δ(t t1)G(k2,t1 t2) δ(t t2)G(k1,t1 t2)], (k1, k2) < 0, − − − − − − where (i,j)= r,l but i = j, sgn(i=r)=+1, sgn(i=l)=-1, and G(k, t) is the SPGF. The right hand side in the equations comes{ } from the6 discontinuity of the time-ordered product at equal time. Here the SPGF is given by:

G(k,t)= i< 0 T a (t)a† (0) 0 >,k > 0, (145) | { r,k r,k }| and

G(k,t)= i< 0 T a (t)a† (0) 0 >,k < 0. (146) | { l,k l,k }| These Ward identities relating the ”density-single particle” correlation, in conjunction with the usual equation of motion for the SPGF (Eq. (72)), form a closed set of equations:

[i∂ v k + µ]G(k,t)= δ(t) (147) t − F − π + +i [V (R (k q,t ; q,t; k, 0)+ R (k + q,t−; q,t; k, 0) L q r − − r q X + +U R (k q,t ; q,t; k, 0)+ R (k + q,t−; q,t; k, 0)], (k q, k) > 0, q l − − l ± and

[i∂ v k + µ]G(k,t)= δ(t) (148) t − | F | − π + +i [V (R (k q,t ; q,t; k, 0)+ R (k + q,t−; q,t; k, 0) L q l − − l q X + +U R (k q,t ; q,t; k, 0)+ R (k + q,t−; q,t; k, 0)], (k q, k) < 0. q r − − r ±

30 Note that for these expression, we are interested in the situation where k kF, and q kF. Upon transforming into ω-Fourier representation and eliminating the R’s, one ﬁnds| | ∼ an integral| equation| ≪ for the SPGF ( Dzyaloshinskii and Larkin, 1973; Everts and Schulz, 1974; Metzner et al., 1998; Voit, 1995):

[ω + vF k µ]G(k,ω) (149) | |− 2 1 Vq[ω′ + sgn(k)q(vF + Vq)] sgn(k)qUq =1+ i dω′G(k q,ω ω′) − . L − − ω2 + q2(v + V )2 q2U 2 q>0 F q q X Z − −

For the simplest case where the q dependence of Vq and Uq are neglected but an ultraviolet cutoﬀ Λ is introduced, this integral equation is solved by transforming to real space-time with

dω dk ikx iωt G˜(x, t)= G(k,ω)e − , (150) 2π 2π Z Z and

2 dω dk Vq[ω′ + sgn(k)q(vF + Vq) sgn(k)qUq ikx iωt K(x, t)= − )e − , (151) 2π 2π ω2 + q2(v + V )2 q2U 2 Z Z − F q − q yielding e

[∂ v ∂ ]G˜(x, t)= δ(t)δ(r)+ K(x, t)G˜(x, t). (152) t ± F r Equation (152) is solved by the ansatz: e G˜(x, t) = exp[L(x, t) L(0, 0)]G˜ (x, t) (153) − o with

L(x, t) = log(r v t + is(t)/Λ) (154) − F [(α + 1)/2] log(x v t + is(t)/Λ) − − o [(α 1)/2] log(x + v t + is(t)/Λ). − − o Here L(0, 0) = ηlog(Λ), and 1 1 G˜ (x, t)= , (155) o 2π x v t + i0+s(t) − F where s(t)=+1 for t>0, and -1 for t<0. The ﬁnal form for = G(x˜ , t) exhibits the well-known, power-law behavior: 1 1 G(x, t)= α 1 α+1 α−1 (156) 2πΛ − [x ut + is(t)/Λ] 2 [x + ut + is(t)/Λ] 2 − 2 2 with the renormalized velocity vo = (vF + Vo) Uo , where Vo = Vq, q 0 and Uo = Uq, q 0, and the exponent, α, is given by: − → → p 1 α =1+ [vF + Vo vo]. (157) 2vo − The resulting density of state, D(ω), also exhibits the hallmark power-law behavior:

α 1 D(ω) ω − (158) ∝ Straightforward generalization to include spin yields:

31 1 G(x, t)= (159) αc+αs 2πΛq 1 αc+1 αc−1 [x v t + is(t)/Λ] 4 [x + v t + is(t)/Λ ] 4 − c c q 1 αs+1 αs−1 [x v t + is(t)/Λ] 4 [x + v t + is(t)/Λ ] 4 − s s q where the indices c and s refer to the charge and spin modes, respectively. Extensions beyond the idealized Tomonaga-Luttinger model to take into account nonlinear energy- momentum dispersion and weak back-scattering lead to generalized models which can be analyzed either by bosonization (Haldane, 1981; Metzner et al., 1998; Voit, 1995; Luther and Emery, 1974) or many-body techniques of renormalization group and Ward identities ( Solyom, 1979; Metzner and Di Castro, 1993; Metzner et al., 1998). The nomenclature of the ”Luttinger liquid” was coined to reflect the fact that the non-idealities lead to an interaction between the boson modes away from q=0, much in the spirit that interaction between quasi-particles arise in a Fermi liquid ( Haldane, 1981; Nozieres, 1964). Here we highlight the outstanding features and ideas. The interested reader is referred to the literature for details. Bosonization has been utilized in two directions–(i) to analyze the effect of non-linear dispersion (Haldane, 1981), and (ii) to solve a specific type of back-scattering problem ( Luther and Emery, 1974; Haldane, 1979). By introducing a quadratic energy-momentum dispersion plus a third order term to ensure stability of the Luttinger liquid state, Haldane showed that that Hamiltonian can be written in terms of the boson operators, bq , and number and current operators, N and J, to include cubic and quartic terms for q = 0 not present in the Luttinger model signify an interaction between modes. Nevertheless, several key features6 of the Luttinger model are preserved for situations where J is still a good quantum number. Haldane further argued that even if J is not a good quantum number, the low energy structure is preserved: 1) vo = √vN vJ , 2) vN = vo exp( 2φo), 3) vJ = vo exp(2φo), and 4) φo still controls the power law fall-off of Fermi Green’s functions at large− distance. However, the Fermi velocity is renormalized and becomes q-dependent. The Luther-Emery backscattering problem will be briefly mentioned at next Section’s end.

3. g-ology

The alternative route to a similar conclusion on general, 1-d interacting systems including weak backscattering is based on renormalization group analysis. As we have seen, forward scattering in the Tomonaga- Luttinger model leads to a logarithmic contribution in the second-order perturbative calculation of the self-energy of the self-energy, and a logarithmic divergence in the interaction operator at low energies, Γ ln(ω/Λ). The effect of a renormalization of the theory is to re-sum the perturbation series to all- orders,∝ rendering the divergent physical quantities finite, by rescaling the divergences away with some cut-off dependent divergent factors. The success of the renormalization in this context of 1-d interacting systems relies heavily on Ward identities reflecting conservation laws, and skeleton diagram analysis in many-body theory. A successful renormalization group analysis was first put forth as an ansatz by Solyom (1979) and coworkers (Menyhard and Solyom, 1973; Solyom, 1973; Solyom and Zawadowski, 1974), based on a band-width renormalization scheme within a model of linearized energy-momentum dispersion about the two Fermi points. The model used is the so-called g-ology model with a spin-dependent Hamiltonian which may include a backscattering term (H1), in addition to the usual forward scattering terms H2 containing density operators

for both the right and left-moving branches, ρr,q,σ = k ar†,k+q,σar,k,σ and ρl,q,σ = k al†,k+q,σal,k,σ, respectively, where σ denotes the spin, and H4 term involving density operators in one branch only. Umklapp P P processes (H3) relevant for lattice Hubbard type models at half ﬁlling will not be included here. The g-ology Hamiltonian is given by (Solyom, 1979; Metzner and Di Castro, 1993; and Metzner et al., 1998):

H = Ho + H1 + H2 + H4, (160)

where

32 H = v ka† a v ka† a (161) o F r,k,σ r,k,σ − F l,k,σ l,k,σ σ k k

1 ′ ′ ′ H1 = L− g1 δσ, σ ar,k† +q,σal,k 2kF ,σ al,k† ′ q,σ′ ar,k +2kF ,σ (162) ⊥ − − − ′ q ′ Xσ,σ X Xk Xk

1 H2 = (L)− [g2 δσ,σ′ + g2 δσ, σ′ ] ar,k† +q,σar,k,σ al,k† ′ q,σ′ al,k′ ,σ′ (163) k ⊥ − − ′ q ′ Xσ,σ X Xk Xk 1 = L− [g2 δσ,σ′ + g2 δσ, σ′ ]ρr,q,σρl, q,σ′ , k ⊥ − − ′ q Xσ,σ X

1 H4 = (2L)− [g4 δσ,σ′ + g4 δσ, σ′ ] (164) k ⊥ − ′ q Xσ,σ X

a† a a† ′ a ′ + a† ′ a ′ a† ′ a ′ ′ r,k1+q,σ r,k1,σ r,k2 q,σ r,k2,σ l,k +q,σ l,k1,σ l,k q,σ′ l,k2,σ { − 1 2− } k k ′ ′ X1 X2 Xk1 Xk2 1 = (2L)− [g4 δσ,σ′ + g4 δσ, σ′ ][ρr,q,σρr, q,σ′ + ρl,q,σρl, q,σ′ ] k ⊥ − − − ′ q Xσ,σ X where for the right branch the summation is for k k < k < k +k , while for the left branch, k k < F − o F o − F − o k′ < k +k , with a band-width cutoﬀ of E =v k . See Fig. 6. The notations and denote scattering − F o o F o k ⊥ processes between parallel and anti-parallel spins, respectively. The band width cut-oﬀ, Eo, turns out to generate a multiplicative renormalization and was shown to hold order by order using perturbation theory. The renormalization involves the multiplicative parameters, z, associated with the renormalization of the SPGF, G, and the zj′s, associated with the renormalization of the various dimensionless interaction vertices, Γj, j=1,2,4:

G zG, (165) −→ and

1 Γ z− Γ , (166) j −→ j j

) (a) E (b E

Eo Eo k k -k k -Eo F -Eo F -kF kF -kF -ko -kF +ko kF -ko kF +ko

+k -k

FIG. 6. Linearized dispersion for Tomonaga-Luttinger model, about the Fermi points at k . For con- ± F venience the Fermi energy is set to zero. The band width 2Eo is equal to 2vFko. The allowed momenta is restricted to the region within k k k . | ± F|≤ o

33 when the bandwidth cut-oﬀ is reduced from Eo Eo′ . The coupling constants, gj’s, then transform as 2 −→ g z− z g . The renormalizability amounts to demonstrating that the quantities z, z depend only on j −→ j j j the ratio, Eo′ /Eo, and are independent of the energy and momentum variables, (ω, k), while G and the Γi′s preserve their analytic form before and after the renormalization transformation. Even with the scaling analysis, the computed scaling trajectories are valid only in the weak-coupling limit of weak-backscattering, g1 g2 , while strong coupling ﬂows out of range of validity of scaling hypothesis. (See text and Fig. 7 below.)| ⊥|≤ k A full, modern day RG analysis is discussed by Metzner and Di Castro (1993) and Metzner et al. (1998). Only two renormalization parameters, z, and zρ, in addition to the coupling constants, gi’s, are necessary to renormalize the most singular correlation functions of the Fermion, and boson (density-density) and mixed types:

2n,l G ( k ,t , k′,t′ ; q ,t” )= (167) { i i i i} { j j } n+l ( i) < 0 T ak1 (t1)...akn (tn)ak† ′ (tn′ )...a† ′ (t1′ )ρq1 (t”1)...ρql (t”l) 0 >, − | { n k1 }| where it is convenient to work in the Fourier transformed representation using the associated vertex functions, Γ2n,l(K, Ω), which equals the G2n,l(K, Ω) divided by the external legs. Here (K, Ω) denotes all the incoming and outgoing external energy-momenta. It turns out that only the four most singular vertices, Γ2,Γ4,Γ(0,2), and Γ(2,1) need to be renormalized since all other higher order vertices do not exhibit logarithmic divergences in a perturbation series. The renormalization is deﬁned by:

1 1 (2) (2) G− zG− ;Γ zΓ (168) −→ −→

Γ(4) z2Γ(4) (169) −→

Γ(0,2) z2Γ(0,2) (170) −→ ρ

Γ(2,1) z2z Γ(2,1) (171) −→ ρ

g g˜ (172) −→ where the renormalization in occurs as the energy-momentum cut-oﬀ is changed from Λ to Λ′, and the renormalization factors, z, zρ, and couplings, gi’s, only depend on gi’s and the ratio Λ′/Λ, and not on the external energy momenta, (K,Ω), e.g.

z = z(Λ′/Λ, g˜). (173)

The renormalization group is deﬁned by:

˜2n,l n l 2n,l Γ (K, Ω;˜g, λ) lim z (˜g,λ/Λ)zρ(˜g,λ/Λ)Γ (K, Ω; g, Λ), (174) ≡ Λ →∞ with

˜2n,l n l ˜2n,l Γ (K, Ω;˜g′, λ′)= z (˜g, λ′/λ)zρ(˜g, λ′/λ)Γ (K, Ω;˜g, λ), (175) and

z(g, λ′/λ) = lim z(˜g, λ′/Λ)/z(˜g,λ/Λ). (176) Λ →∞ The differential form for the renormalization group flow follows from Eq. (175) to (176) by considering infinitesimal changes, λ′ = λ + ∂λ:

34 [λ∂ + β(˜g)∂ nγ(˜g)+ lγ (˜g)]Γ˜(2n,l)(K, Ω;˜g, λ)=0 (177) λ g˜ − ρ with β(˜g) = ∂˜g′/∂(λ′/λ) λ′=λ, γ(˜g) = ∂lnz/∂(λ′/λ) λ′=λ, and γρ(˜g) = ∂lnzρ/∂(λ′/λ) λ′=λ. Note that a fixed point of the renormalization| group signifies that| ˜gdoes not change with λ, or β| = 0. Under such circumstance the renormalization group flow immediately yields:

(2n,l) nγ(˜g∗) lγ (˜g∗) Γ˜ (K, Ω;˜g, λ)= λ − ρ (178) where ˜g∗ is the fixed point. The fact that the removal of all divergences with these finite number of renormalization parameters indicates the renormalizability of the theory. In this renormalization group analysis, full usage is made of the Ward identities reflecting the separate conservation of charge and spin in the right and left branches which severely constrain the theory and number of free parameters. Renormalization is usually computed perturbatively. For the simple case of only g2 coupling with all other couplings zero (a special case of the Tomonaga-Luttinger model), computation to second order (two loops) leads to unrenormalized g2 and zρ, and a renormalization of z ( Solyom, 1979; Metzner and Di Castro, 1993; Metzner et al., 1998):

2 g2 λ 3 z(λ/Λ,g)=1+ 2 2 ln + O(g ) (179) 8π vF Λ

It turns out that for g1 = 0(g1 = 0), the set of arbitrary g2 and g4, namely the Tomonaga-Luttinger model, forms a ﬁxed-line of the g-ology⊥ model. For the general case, however, all couplings scale (Solyom, 1979):

dg2 1 1 1 k 2 2 = [ g1 + 2 2 g1 g1 + ...] (180) dx x πvσ ⊥ 2π vσ k ⊥

dg1 1 1 1 2 3 ⊥ = [ g2 g1 + 2 2 (g2 g1 + g1 )+ ...] (181) dx x πvσ k ⊥ 4π vσ k ⊥ ⊥

d(g2 2g2) k − =0 (182) dx

d(g4 g4 ) 1 1 k − ⊥ 2 = [ 2 2 g2 g1 + ...] (183) dx x 2π vF k ⊥ and

d(g4 + g4 ) k ⊥ =0 (184) dx where the last two expressions imply:

dg4 1 1 2 ⊥ = [ 2 2 g1 g1 + ...] (185) dx −x 4π vF k ⊥

These scaling trajectories plotted for a cut in the g2, g1, plane is shown in Fig. 7. For weak-coupling k − ⊥ (weak back-scattering) where g1 g2 the trajectories scale to the Tomonaga-Luttinger fixed line of no | ⊥| ≤ k backscattering, i.e. g1∗ = 0. On the other hand for strong-coupling the trajectories scale away to a regime where the perturbative⊥ treatment is no longer valid. Intuitively, under sufficiently strong back-scattering, insulating behavior can be expected. For a specific strong-coupling (strong backscattering) problem, in particular under the specific condition of:

g2 3 − k = , (186) 2πvF + g4 g4 5 k − ⊥

35 Luther and Emery (1974) showed that there is a decoupling of the charge and spin sectors, resulting in a gapped behavior in the spin excitation spectrum. It has however been argued that the Luther-Emergy problem does not represent a true backscattering problem (Haldane, 1979). In addition to the solution of the weak-back-scattering problem, renormalization group analysis has been successful in showing that nonlinear energy-momentum dispersion can be scaled away as it generates irrelevant terms in the renormalization group sense (Metzner and Di Castro, 1993).

2 2

FIG. 7. Renormalization group (RG) flow diagram for the g2 , g1, g4 g-ology model, in a cut within k the g2 -g plane from (a) second order perturbation calculations, and (b) third order calculations. For k ⊥ g1 < g2 corresponding to weak back-scattering, the trajectories scale to the Tomonaga-Luttinger fixed | ⊥| k line for which g1 = 0 (no back-scattering). For large g1 , the trajectories flow away from the T-L fixed line into regimes where⊥ the perturbation calculations are no⊥ longer valid. [From Solyom (1974).]

D. Chiral Luttinger liquid

The chiral Luttinger liquid (CLL) theory of the fractional quantum Hall (FQH) edge pioneered by Wen (1990a, 1990b, 1991a, 1991b, 1992, 1995) provides an effective way to describe the low energy dynamics at the boundary of the strongly-correlated 2-d electron gas system. In the case of the incompressible fluids such as the Laughlin or the hierarchical states, the bulk excitation contains an energy gap and the edge is truly one-dimensional in nature. The existence of a Tomonaga-Luttinger-liquid type behavior, characterized by power-law correlations and gapless bosonic excitations, arises as a natural consequence of the strong correlation and one-dimensionality. Here in contrast to the ordinary case, the magnetic field introduces a sense of rotation in the propagation of edge modes, dictated by the motion of semiclassical skipping orbits at the boundary in the direction given by ExB, where E is the effective electric field and includes the contribution of the edge confinement potential. The resulting Tomonaga-Luttinger liquid is therefore chiral. A unique feature of the idealized chiral Luttinger liquid investigated in effective theories is the universality of the electron tunneling exponent, α. For incompressible fractional Hall fluids belonging to the Laughlin fractions, ν = 1/m, where ν is the filling factor and m is an odd integer, α, defined by the tunneling current-voltage (I-V) relationship I Vα, is believed to be given by the dimensionless Hall resistance, i.e. 2 ∝ α = ρxy/(h/e ). For the 1/m fractional fluids it is exactly equal to m. In other words for the ν = 1/3 Laughlin fluid, α would equal 3 exactly! This line of thought is based on the topological properties and characterizations of the fractional quantum Hall states (Wen and Zee, 1992; Wen, 1995; Frohlich and Zee, 1991; Blok and Wen, 1990a; Blok and Wen, 1990b) and as such is insensitive to the specific form of the electron-electron interaction potential. In complex incompressible fractional Hall fluids involving the hierarchical sequence ( Halperin, 1984; Haldane, 1983; Jain, 1989a, 1989b, 1990), the tunneling characteristics has been proposed to be an effective way to classify and distinguish different states based on the topological order of the states. This topological order embodied in the K-matrix formulation ( Wen and Zee, 1992; Wen, 1995; Frohlich and Zee, 1991)

36 in principle enables different incompressible fluids exhibiting the same Hall resistance (conductance) to be distinguished one from another. The tunneling exponents for many hierarchical states with complex, multi- branch edge modes are a priori non-universal. The value of the exponent depends on the details of the inter-mode interaction which mixes the modes. However, Kane, Fisher, and Polchinski ( Kane et al., 1994; Kane and Fisher, 1995) have shown that residual disorder can restore the exponent to universal values when sufficiently strong. In particular, for the Jain sequence, ν = n/(np + 1), where n=1,2,3,etc. and p an even integer, mode-mixing gives rise to 1 charged mode and (n-1) neutral modes. In this universal limit, the charged mode propagates at one velocity while the (n-1) neutral modes obeying an SU(N), N=n, symmetry all propagate at a different but identical velocity. The charged and neutral modes thus behave analogously to the charge and spin modes in the ordinary 1-d Tomonaga-Luttinger liquid case. When all modes co- propagate (i.e. travel in the same direction), e.g. for p=2, and 1 n corresponding to 1/3 ν 1/2, the exponent remains constant and equal to p+1 equaling 3. On the≤ other≤ ∞ hand, for p=-2, before≤ mixing≤ the inner hole-like edge modes propagate in a direction opposite to the outermost mode. For example at ν =2/3 the outermost edge mode is the usual electron mode familiar in the integer quantum Hall fluid, while a 1/3 charged, inner hole-like branch propagates in the opposing direction. These counter-propagating inner and outer modes when mixed lead to a shake-up process and to a renormalization of the exponent. At a given filling fraction, this renormalization tends to a universal value when sufficient disorder is present. The experimental observation of a Luttinger liquid-like behavior for compressible states (Chang et al., 1998) came as somewhat of a surprise. At the current juncture, the most consistent theoretical description for the edge dynamics of compressible fractional quantum Hall fluids and for their edge tunneling characteristics was put forth by Shytov, Levitov and Halperin (Shytov et al., 1998; Levitov et al., 2001) based on a composite Fermion, Chern-Simon effective field theory formulation. The result is a Luttinger liquid-like edge dynamic where the finite, non-zero diagonal resistivity, ρxx, of the compressible fluid slightly changes the exponent value from the interpolated, universal value valid for ρxx = 0. The idea is as follows. Since for high quality samples ρxx is typically small, of order (1/10)ρxy or less regardless of whether the fractional Hall fluid 2 2 is compressible or incompressible, the corresponding diagonal conductance, σxx = ρxx/[ρxx + ρxy], is also 2 2 small, σxx 0.1σxy. Here σxy = ρxy/[ρxx + ρxy] 1/ρxy is the Hall conductance. As a result, when an external electron≤ is injected at the edge, its spreading≈ is much faster along the boundary compared to the spreading perpendicular to the boundary into the bulk. Therefore, the tunneling characteristics will still appear Luttinger liquid-like. For compressible fluids residual disorder or interaction also tends to drive the exponent to universal values. The inclusion of disorder results in an exponent for electron tunneling, α, 2 which when plotted as a function of reduced Hall resistance, ρxy/(h/e ), exhibits step-like plateaus on which α is roughly constant. In between plateaus it varies in a linear fashion. The plateaus are not perfectly flat due to the non-zero ρxx and occur when all edge modes co-propagate, while the linear regions occur when counter-propagating modes are present. The functional dependence is an extension of the Kane, Fisher result to continuous values of the Hall resistance. What is summarized above constitutes the “standard” descriptions of the theory of the fractional quantum Hall edge. These descriptions are expected to hold rigorously when the wavefunction is of the Laughlin form or its hierarchical derivatives even in the presence of the boundary. This means that nominally the interaction between electrons is of the extreme short ranged form of δ′′(r), a second derivative of a delta function. The leap to realistic, long-ranged Coulomb or partially screened (e.g. dipolar) interactions can only be treated in an approximate way (Zülicke and MacDonald, 1997; Imura and Nagaosa, 1997; Zheng and Yu, 1997; Levitov et al., 2001). Based on the topological nature of the quantum Hall fluids, it is usually argued that the range of the interaction is irrelevant. This remains an open and extremely important issue, however. Recent cumulative experimental as well as suggestive numerical evidence are raising the possibility of deviations from universality, which will be fully discussed below in Section III E 5.

1. Wen’s hydrodynamic formulation

As was seen in Section II C, for the Tomonaga-Luttinger liquids in 1 dimension the commutator relationship between density operators, [ρq,ρq′ ]=cqδq, q′ where c is a constant (the so-called Kac-Moody algebra), and − the existence of a well-deﬁned of Fermion operator, Ψ(x) eiφ(x) where φ(x) is a boson operator related to ∼

37 ρ(x) by ∂xφ =2πρ(x), lead to bosonic sound-like low energy neutral excitations and power law correlations in the charged sector when electrons are added or removed, provided forward scattering dominates and backscattering is irrelevant. The one-dimensional edge of the fractional quantum Hall fluid is a strongly interacting system in which backscattering is nearly completely suppressed, particularly in the quantized regime. Therefore the existence of Luttinger-liquid-like edge dynamics is quite natural. Wen introduced several equivalent formulations to motivate and derive the effective law energy physics of the fractional quantum Hall edge, based on hydrodynamics or Gauge invariance of the effective field theory. The most intuitive is the hydrodynamic approach which we will describe in this Section. We start with the simplest case of the ν = 1/m Laughlin Hall fluid, where m is an odd integer. Based on the observation that the fractional Hall fluid is an irrotational, incompressible fluid, Wen first wrote down a classical hydrodynamical theory of a surface wave which travels in one direction only, followed by canonical quantization of the momentum and coordinate variables to obtain a quantum theory. Using the 1-d edge density, ρ(x), to describe the disturbance depicted in Fig. 8, the propagation of this disturbance is describe by the wave equation: ∂ ρ(x) v∂ ρ(x)=0. (187) t − x

Edge Plasmon + at k=2π/L

2DEG Undisturbed Boundary

−

FIG. 8. Edge density wave-plasmon at the boundary of a fractional quantum Hall ﬂuid. In the absence of a density disturbance the boundary is depicted as a straight line. In the presence of a plasmon wave at wave vector, k = 2π/L, where L is the length of the boundary, some regions accumulate excess charge while others suﬀer a depletion of charge.

In an incompressible fluid, the vanishing of dissipation (longitudinal conductivity σ 0) and the presence xx → of a non-zero Hall conductance, σHall = σxy, generates a persistent current along the edge due to the boundary electric field: e2 j = σ zˆ E = σ zˆ [ V ] E = V σ = ν , (188) xy × xy × −∇ −∇ xy h where V is the edge potential. At the boundary the electrons execute classical skipping orbits and drift with the velocity v=(E/B)c. Introducing the vertical displacement, h(x), at position x along the edge, related to the edge density by: eB h(x)= ρ(x)n,n = ν , (189) hc where n is the fractional Hall fluid average bulk density, the Hamiltonian associated with the chiral edge wave is given by:

38 1 1 e H = dx eρ(x)h(x)E = dx[ρ(x)]2 (190) 2 2 n Z Z v = π¯h dx[ρ(x)]2 ν Z Transforming to momentum representation where

1 iqx ρq = dxe ρ(x) (191) √L Z

1 iqx ρ(x)= ρ e− , (192) √ q L q X we have, v H =2π¯h ρqρ q. (193) ν − q>0 X This yields a corresponding wave equation in k-space of

ρ˙ = ivqρ . (194) q − q Hamilton’s equations for coordinates and momenta, ∂H ∂H q˙q = , p˙q = , (195) ∂pq −∂qq allows us to identify: ih qq = ρq k> 0 pq = ρ q k< 0. (196) νq − The ﬁnal step is to quantize the Hamiltonian by requiring the canonical commutation relation:

[qq, pq′ ]= i¯hδq,q′ , (197)

giving: ν [ρq,ρq′ ]= qδq, q′ , (198) 2π −

[H,ρ ]= qvρ . (199) q − q Aside from the extra factor ν in Eq. (198), these expressions bear considerable similarity to the corresponding expressions for the ordinary Luttinger-liquid. We can draw parallel with the ordinary Luttinger liquid. Expressing the density operator in terms of fermion creation-annihilation operators:

ρq = ck†+qck (200) Xk

1 iqx ρ(x)= ρ e− , (201) √ q L q X and writing the Fermion ﬁeld operator, ψ(x), in terms of the annihilation operator,

39 1 ikx ψ(x)= cke− , (202) √L Xk we arrive at the important commutation relationship

[ρ(x), ψ(x′)] = δ(x x′)ψ†(x), (203) − which is satisﬁed by a representation of ψ(x) in the form, ψ˜(x):

i 1 φ ψ˜(x) e ν , ∂ φ =2π¯hρ, (204) ∝ x where φ(x) obeys the commutation relationship:

[φ(x), φ(x′)] = iπνsgn(x x′). (205) − − Note the similarity of Eq. (204) with Eqs. (132) and (134) for the ordinary Luttinger liquid. For this form to be a valid Fermionic operator, ψ˜(x) must satisfy the usual anticommutation relationship, ψ˜(x), ψ˜(x′) = 0. Making use of the operator identity { }

eAeB = e[A,B]eBeA (206) between two operators A and B when their commutator, [A,B], is a c-number, we ﬁnd

1/ν ψ˜(x)ψ˜(x′) = ( 1) ψ˜(x′)ψ˜(x), (207) − which requires 1/ν to be an odd integer, consistent with the condition for a simple fractional Hall fluid of the Laughlin type (Laughlin, 1983; Tsui et al., 1982). Just as the case of the ordinary Luttinger liquid, the Green’s functions can be evaluated in terms of the φ boson fields ( Luther and Peschel, 1974; Wen, 1992, 1995). For the 1/m Laughlin fluids, we have:

iHt iHt < φ(x, t)φ(0, 0) >=< 0 e φ(x)e− φ(0) 0 > (208) | | i[ v ∂ ]t =< 0 e− i x φ(x)φ(0) 0 > | | i[vp ]t =< 0 e− x φ(x)φ(0) 0 >= νln(x vt) + Constant. | | − − The SPGF is then given by:

< 0 T ψ†(x, t)ψ(o, o) 0 > (209) | { }| 1 1 =< 0 exp[ < φ(x, t)φ(0, 0) >] , | ν2 ∝ (x vt)1/ν − established using the operator relationships:

eAeB = eA+B+[A,B]/2 (210)

and

2 = e/2 (211)

where A and B are linear combinations of harmonic oscillator-type creation and annihilation operators, and the average is in a harmonic-oscillator type ensemble. Notice that for the ν = 1 integer ﬂuid edge, the propagator is the conventional fermion propagator, and the edge dynamics is Fermi-liquid like albeit still chiral. On the other hand, for ν < 1 or m > 1, the propagator exhibits the unusual, power-law correlation which is the Hallmark of Luttinger liquid behavior. In the (k,ω) representation, the SPGF takes the form:

(vk + ω)m 1 G(k,ω) − , (212) ∝ vk ω − 40 leading to the power-law density of states with the universal exponent value of m-1:

m 1 D(ω) ω − . (213) ∝ | | In particular, for the strongest, most robust fractional Hall fluid ν =1/3, m=3. This theory predicts an exact value for the exponent in the electron tunneling density of states of 2. This in turn leads to an exponent, α, of exactly 3 in the current voltage (I-V) characteristics for tunneling from a normal metal into the ν =1/3 fractional fluid edge! This is a central result and a dramatic prediction of the chiral Luttinger liquid model of edge dynamics. It is possible to generalize this type of dynamics to the edges of more complex fractional Hall fluids. The simplest examples are the spin-polarized ν = 2/5 and ν = 2/3 fluids, which each contains two fluid components resulting in two edge modes. In the case of the 2/5 fluid consisting of a ν1 = 1/3 outer component and a ν2 =1/15 inner component of quasi-particle condensate, where 2/5=1/3+1/15, the edge modes consist of an e/3-charged outer mode closest to the boundary, and an e/5-charged inner mode with both modes propagating in the same direction. In the case of the 2/3 fluid, the outer component is a ν1 =1 fluid, while the inner component consists of a ν = 1/3 e /3 quasi-hole condensate corresponding to a e/3 h | | quasi- particle condensate at ν2 = 1/3. Therefore the edge modes consist of an outer, e-charged electron, while the inner mode is an e/3 quasi-− particle mode which propagates in the opposite direction as the outer mode (MacDonald, 1990; Kane et al., 1994; Kane and Fisher, 1995; Wen, 1992, 1995)! Note this important distinction of co- versus counter-propagating modes in the two cases. For instance, in the 2/5 fluid it does not matter whether the modes propagate both in the positive or negative directions, e.g. under magnetic field reversal, or for a hole rather than an electron gas. What matters is that they propagate in the same direction. Generalizing the edge Hamiltonian, we have: [Note for convenience, from here on we suppress the constant, ¯h.]

vI H =2π ρI,qρI, q, (214) ν − I q>0 XI X

where I=1,2 and refers to the Ith edge mode and I νI = ν. For the Hamiltonian to be bounded from below to yield a well-deﬁne ground state, it is necessary to have the condition νIvI > 0. The commutation P relationship between ρI,q’s are given by:

νI [ρI,q,ρJ,q′ ]= qδI,J δq, q′ . (215) 2π − The electron ﬁeld operators can be written as:

1 i ν φI (x) ΨI (x)= e I (216)

where ρI(x) = (1/2π)∂φI(x). A straightforward computation of the electron propagators yields: 1 < 0 T Ψ†(x, t)Ψ (o, o) 0 > = < 0 exp[ < φ (x, t)φ (0, 0) >] 0 > (217) | { I I }| | ν2 I I | 1 = eikI x (x v t)1/νI − I 2 where kI = rI/lo and lo = ¯hc/eB the magnetic length. Being charged the two edges can interact either by Coulomb interaction, or through the mediation of disorder giving rise to an interaction terms between ρ p 1 and ρ2. The resulting total Hamiltonian can be written as:

H =2π[ UII ρI,qρI, q + VIJ ρI,qρJ, q], (218) − − I q>0 I=J q>0 X X X6 X where U contains the diagonal elements only and V contains the oﬀ-diagonal elements, i.e. VII = 0. At this point it is convenient to introduce the scaled density operators:

41 1 ρ˜I ρI . (219) ≡ ν | I | The Hamiltonian becomes: p

H =2π[ UII νI ρ˜I,qρ˜I, q + VIJ νI νJ ρ˜I,qρ˜J, q] (220) | | − | | − I q>0 I=J q>0 X X X6 p X =2π[ uII ρ˜I,qρ˜I, q + vIJ ρ˜I,qρ˜J, q], − − I q>0 I=J q>0 X X X6 X

where the matrix elements uII = UII νI and vIJ = VIJ νIνJ , have the unit of a velocity. The commutation relationships become: | | | | p

sgn(νI ) [˜ρI,q, ρ˜J,q′ ]= qδI,J δq, q′ . (221) 2π −

H can be rediagonalized with the following transformation of theρ ˜′s while preserving the above form of the commutation relationship: 1 R˜1,q = cos(√ηθ)˜ρ1,q + sin(√ηθ)˜ρ2,q (222) √η

R˜ = √η sin(√ηθ)˜ρ + cos(√ηθ)˜ρ , (223) 2,q − 1,q 2,q where

η v12 tan(2√ηθ)=2 | | , (224) u ηu 11p− 22 with η = sgn(ν1ν2) and the convention that √ 1 = i. The resulting Hamiltonian and commutation relations are given by: −

H =2π sgn(νI )VI R˜I,qR˜I, q (225) − q>0 XI X

sgn(νI ) [R˜I,q, R˜J,q′ ]= qδI,J δq, q′ , (226) 2π − where the velocities of the two modes of the rediagonalized edge excitations are:

1 2 2 sgn(νI )VI = [cos (√ηθ)uII η sin (√ηθ)uJJ ] (227) cos(2√ηθ) −

for J = I. These new velocities preserve the direction of propagation for the corresponding modes before rediagonalization.6 The electron propagator for the I-th mode takes the form:

< 0 T Ψ (x, t)Ψ† (0, 0) 0 > (228) | { I I }| 1 1 = eikI x 1 cos2(√ηθ) 1 η sin2(√ηθ) (x V t) |νI | (x V t) |νI | − I − J J = I. A most general electron propagation process may be accompanied by a transfer of any integer number, n,6 of quasi-particles between the inner and outer edges across the outer fractional ﬂuid. In the 2/5 case, the outer ﬂuid is a ν1 =1/3 condensate and the quasi-particles have charge e/3, while for the 2/3 case, the

42 other ﬂuid is a ν1 = 1 condensate with e-charged quasi-particles. This quasi-particle transfer operator, Tν1 , may be written as:

ν1 Tν1 = [Ψ1Ψ2†] . (229) The most general electron ﬁeld operator, Ψ, then can be formed as a linear combination of terms containing n a product of the ΨI’s and Tν1 :

n Ψ=Ψ1 cnTν1 , (230) n X 1 noting that n may take on negative values and from Eq. (216) we have ΨI† = [ΨI]− . or residual disorder (Kane et al., 1994; Kane and Fisher, 1995) which mix the counter-propagating modes drive the exponent to a universal value of α = 2 as discuss the next section. One ﬁnal remark is in order, for ψ˜(x) to satisfy the anticommutation relationship, ψ˜(x), ψ˜(x′) = 0, the topological quantum numbers, λn, must be odd integers, yielding: { }

1/λ ψ˜(x)ψ˜(x′) = ( 1) n ψ˜(x′)ψ˜(x)= ψ˜(x′)ψ˜(x). (231) − −

2. 1-d eﬀective ﬁeld theory of the chiral Luttinger liquid

For other fillings corresponding to incompressible fluids, in particular the Jain series, one can generalize this type of discussion. To proceed in the most expedient and rigorous way, it is necessary to introduce the generalized chiral boson description in terms of the boson fields, φi’s, arising as natural extension of the φ fields above. The chiral boson action is given by (Wen, 1992, 1993) 1 S = dtdx[K ∂ φ ∂ φ (232) edge 4π IJ t I x J Z U ∂ φ ∂ φ V ∂ φ ∂ φ ], − II x I x I − IJ x I x J with the Hamiltonian: 1 H = dx[U ∂ φ ∂ φ + V ∂ φ ∂ φ ] (233) 4π II x I x I IJ x I x J Z 1 = dxW ∂ φ ∂ φ , 4π IJ x I x J Z where the matrix, WIJ: W U δ + V , (234) IJ ≡ II IJ IJ is a positive-definite symmetric matrix with positive-definite eigenvalues to ensure that H is bounded from below. This edge action is derived from the the K-matrix formulation of the bulk 2-d effective action of fractional Hall fluids ( Wen and Zee, 1992; Wen, 1995; Frohlich and Zee, 1991; Blok and Wen, 1990a; Blok and Wen, 1990b) in terms of the gauge fields aIµ: 1 1 S = dtdxdy[ K a ∂ a ǫννλ + et A ∂ a ǫµνλ], (235) bulk −2π 2 IJ Iµ ν Jλ I µ ν Iλ Z after inputing the information about the boundary potential via the UII, VIJ terms. By identifying the edge 1 densities, ρI′ s, with the derivative of the boson fields, ρI = 2π ∂xφI, the Hamiltonian form of Eq. (237) above is recovered. Here K is an nxn matrix with integer elements containing information about the topological properties of the n condensates and n different varieties of quasi-particle contained in the fractional Hall fluid. t is an n-component, charge vector from which the fluid filling factor, ν can be computed:

43 T 1 ν = tI [K− ]IJ tJ . (236)

Aν is an external potential coupled to the electron 3 current: 1 jµ = t ∂νa ǫµνλ. (237) 2π I Iλ In the “symmetric” basis for Abelian fractional Hall ﬂuids, i.e. ﬂuids with quasi-particle excitations obeying Abelian statistics, K is a symmetric matrix with odd integer diagonal elements while tI = 1, i.e. tT = (1, 1, ..., 1), yielding:

1 ν = [K− ]IJ . (238) XIJ For the ν = n/(np+1) Jain series, where n is a positive integer and p an even integer, K is of dimension n with KIJ = δIJ + p. For instance, the 2/5 ﬂuid is represented by n=2, p=2 with K given by:

(K11,K12; K21,K22)=(3, 2;2, 3), (239) while the 2/3 “hole fluid” with ν = 2/3 by n=2, p=-2 with: − (K ,K ; K ,K ) = ( 1, 2; 2, 1). (240) 11 12 21 22 − − − − (Note that the 2 x 2 matrix of the ν = 1 1/m fractional Hall electron fluid can also be represented with − K11 =1, K22 = m and zero off diagonal elements.) In the edge dynamics, n also gives the number of edge modes while the− relative signs of the eigenvalues of the K matrix gives the relative direction of propagation for the modes. In the 2/5 case, the eigenvalues, λ1 =1, λ2 = 5, are both positive, and the two edge modes co-propagating, while in the the 2/3 fluid the eigenvalues of λ1 = 1, λ2 = 3, indicate that the modes counter- propagate. − The edge action Eq. (236) with the Hamiltonian Eq. (237) describe the low energy dynamics of the n edge modes associated with the n condensates where the edge density of the I-th condensate is given by: 1 ρ = ∂ φ . (241) I 2π x I

The ρ′s satisfy the Kac-Moody commutation relationship:

1 k [ρIk,ρJ k′ ] = [K− ]IJ δk,k′ . (242) − 2π The electron charge density is:

ρe = e ρI . (243) XI

The quasi-particle operator, ΨI(x), associated with the I-th condensate satisﬁes the relation:

1 [ρ (x), Ψ (x′)] = l [K− ] δ(x x′)Ψ (x), (244) I I J JI − I where the elements of the n-component l vector, lJ , are arbitrary integers, and the quasiparticle carries a charge

1 QI = e lJ [K− ]JI tI . (245) XIJ

In terms of the chiral boson ﬁelds, φI′ s, ΨI has the representation: Ψ exp(iφ l ). (246) I ∝ I I

44 In particular the electron operator is given by:

ΨeL exp i φI lI , (247) ∝ ! XI with the requirement:

lI = KIJ LJ , LI =1 (248) XJ XI to yield a unit charge. There are many possible choices possible for the n-component vector L with integer valued elements. The true electron operator, Ψe, can be written as a linear combination of such operators, ΨeL. The requirement of anticommutation for the electron operator yields λ = IJ LIKIJLJ = odd integer. In the symmetric basis because both the K matrix and the W matrix (WIJ = UIIδIJ + VIJ) are symmetric while W is positive deﬁnite, it is possible to simultaneously diagonalized them usingP transforms:

W SWST ,K SKST , (249) → → where the S transformation is a product of orthogonal transforms and diagonal transforms containing positive-valued diagonal elements which rescale the ρ ﬁelds. The resultant Ith eigenvalue of the trans- th formed K matrix carries a sign, σI , which indicates the direction of propagation of the I mode, while the corresponding eigenvalue of the transformed W matrix yields the respective velocity (speed of sound), σ V = V . In terms of the transformed R˜ ’s: 1 1 | I | I

e e RI = SIJ ρI , (250) XIJ e the Hamiltonian and RI commutation are given by:

e H =2π V RI,kRI, k (251) | | − XI,k e e e and k [RI,k, RJ, k′ ]= σI δI,J δk,k′ . (252) − 2π The propagator for the Ith quasi-particlee takese the form:

il k x γ < 0 T Ψ (x, t)Ψ† (0, 0) 0 > e I I (x V t + iσ δ)− I , (253) | { I I }| ∝ − I I YI where

1 √γI = lJ [S− ]IJ , (254) XJ and γI satisfy the sum rule:

1 σ γ λ = l [K− ] l . (255) I I ≡ I I IJ J XI XIJ For electron tunneling in the Jain sequence, for which the nxn K matrix takes the form: KIJ = δIJ + p, with 1 [K− ]IJ = δIJ p/(np + 1), the tunneling process is dominated by the smallest exponent. In the case p > 0, e.g. 1/3, 2/5,− 3/7, 4/9 ... series, all modes co-propagate, i.e. in the move same direction. The minimum exponent constrained by the sum rule corresponds to lI = p;I = n, ln = p + 1, with the exponent value, α = p+1. For the above Jain series p=2, and all ﬂuids within this6 series is predicted to exhibit the identical,

45 universal exponent value of α = 3. On the other hand, for the electron-hole symmetric, p=-2 series, 2/3, 3/5, 4/7, ..., the presence of counter-propagating modes means that the sum rule no longer constrain the exponent to a universal value, even for a given fluid, e.g. 2/3. In fact, it is necessary to include the effect of residual disorder to drive the exponents to universal values as discussed by Kane, Fisher, Polchinski (Kane et al., 1994; Kane and Fisher, 1995). Long-range Coulomb interaction may also have a similar effect (Wen, 1992). What is required is a mechanism to equilibrate the edge modes, and in particular those that propagate in opposite directions, leading to a universal exponent value for each fluid in the p < 0 Jain sequence given by (Kane and Fisher, 1995): 2 α =1+ p . (256) | |− n Note that for the above series, α approaches the value, 3, as ν 1/2 (n ). → → ∞

3. Role of Disorder

The detailed mathematical analysis which demonstrated that residual disorder drives the exponent to universal values given by Eq. (256) for the p < 0 Jain series as a consequence of the equilibration of the edge modes is beyond the spirit and scope of this article. Here we summarize the basic ideas. The ﬁrst has to do with the existence of an SU(N) symmetry among the neutral modes which are decoupled from the single charged mode. Starting from the action for the edge dynamics, Eq. (236), Kane and Fisher showed that if the interaction term characterized by the W matrix, is separated into a diagonal contribution, D: DIJ =vδIJ, and a traceless contribution, W: W = W vδ , the action containing the K-matrix and D-matrix: IJ IJ − IJ 1 S =f f dtdx [K ∂ φ ∂ φ + vδ ∂ φ ∂ φ ] (257) o 4π IJ t I x J IJ x I x J Z XIJ describe edge modes which can be transformed via an orthogonal transformation, O, into a single charged mode, φ = √nΦ = φ , decoupled from n-1 neutral modes with boson ﬁelds, Φ =;I=1, ..., n 1, where ρ n I I I − ΦI =OIJφJ. The resultant action is: P

So = Scharge + Sneu, (258)

1 1 S = dtdx[ ∂ φ ∂ φ + v∂ φ ∂ φ ] (259) charge 4π ν t ρ x ρ x ρ x ρ Z and

n 1 1 − S = dtdx [∂ Φ ∂ Φ + v∂ Φ ∂ Φ ]. (260) neu 4π t I x I x I x I i=1 Z X Note that since p < 0 therefore ν < 0, and the charge and neutral modes propagate in opposite directions, while all neutral modes propagate at the same velocity, v. Introduction of an auxiliary ﬁeld completes the SU(N) symmetry of the neutral sector. The remarkable result follows where it can be shown that adding randomness to the action, So, does not destroy the SU(N) symmetry (Kane and Fisher, 1995). Furthermore, the remaining mode-mode coupling through the traceless, W = W vI matrix can be either be absorbed into a velocity renormalization of the charged mode, or is else is an irrelevant− perturbation. In the absence of disorder, however, these terms are relevant and change thef exponent of the propagators. Therefore, disorder has the eﬀect of restoring the SU(N) symmetry of the neutral modes, and maintaining the decoupling between the charged and neutral

46 sectors. This disorder driven approach to universality occurs as a Kosterlitz-Thouless phase transition in the disorder parameter as is shown in Fig. 9 for the edge of the 2/3 ﬂuid.

FIG. 9. Kosterlitz-Thouless type phase transition for the ν =2/3 composite edge. When the disorder, W, is weak (√W small), the scaling trajectories ﬂow to non-universal values of the interaction parameter, ∆. On the other hand, when √W is suﬃciently large, the system scales to the universal value of ∆ = 1. Here ∆ is related to the tunneling exponent. [From Kane, Fisher, and Polchinski (1994).]

4. Compressible ﬂuid edges

Thus far, we have exclusively focused on edge tunneling into the edge of incompressible fraction quantum Hall fluids, specifically into the Laughlin (1983) and Jain series (Jain, 1989a, 1989b, 1990). (See also the Haldane/Halperin series (Haldane, 1983; Halperin, 1983).) One of the early experimental surprises was the observation of power-law tunneling characteristics for electron tunneling into the compressible, ν = 1/2 composite-Fermion fluid edge (Chang, 1998). At first sight it was not entirely clear that electron tunneling into the composite Fermion edge necessarily would entail to an orthogonality catastrophe. To account for the experimental findings and to investigate the tunneling behavior for general fillings, compressible or incompressible, Shytov, Levitov and Halperin (Shytov et al., 1998; and Levitov et al., 2001) proposed a theory based on an effective edge action derived from the bulk 2-d composite-Fermion effective action and computed the equal space SPGF relevant for electron tunneling. They found an approximate power-law behavior for all fillings. The behavior of the exponent exhibits plateau (steps) when plotted against the dimensionless 2 Hall resistance, ρxy/(h/e ), as shown in Fig. 10, and is driven close to universal values in between steps varying approximately in a linear manner by residual disorder. Small deviations from the universal values arise from interaction and a non-zero longitudinal resistivity, ρxx 0. Note that for compressible fluids 2 ≥ ρxy/(h/e ) 1/ν, where ν is the Landau level filling. Their result thus basically fills in the continuous sections between≈ the discrete points of incompressible fluids previously investigated by Kane and Fisher (1995), and Wen (1992). This somewhat surprising result of a power-law behavior at all fillings, ν, has its origin in the fact that for high quality samples such as those used in experiment, the longitudinal resistance is invariably small, ρxx 0.1ρxy, in the fractional quantum Hall regime. Therefore a charge introduced into the edge necessarily propagates≤ a large distance along the boundary before it is able to penetrate into the 2-d bulk.

47 FIG. 10. The electron tunneling exponent, α, as a function of the dimensionless Hall resistance, ρxy. A 1 constant Hall angle, tan− [ρxy/ρxx] is assumed. For ρxx = 0.05ρxy, α is plotted for three values of the short-range interaction U = 1/κ 1/κ . Note that at ρ = 0 the exponent is universal (no U dependence), − o xx but at ﬁnite ρxx it can be either larger or smaller than the universal result. [From Shytov, Levitov, and Halperin, (1998).]

The actual computation is extremely technical (Shytov et al., 1998, Levitov iet al., 2001). We sketch the ideas. For a model of short ranged electron-electron interaction:

U(r)= Uδ(r), (261)

with an eﬀective composite-Fermion interaction: 1 UCF (r)= U + δ(r) , (262) κo 2 where κo = 2πm∗/¯h is the free composite-Fermion compressibility, integrating out the degree of freedom perpendicular to the edge yields an action for the edge:

1 2 ω Sedge = [σxx ω k + | | + iσxyωk]φ ω, kφω,k + J( ω, k)φω,k, (263) 1 − − 2 | | U + κ ! − − Xω,k o

where φω,k is the (ω,k)-Fourier transform of the boundary boson ﬁeld, φ(x, t), and J is Fourier transform of the source term J(x, t) = eδ(x x )[δ(t t ) δ(t t )] at the boundary. This action reduces to the − o − 1 − − 2 standard expression, Eq. (236), in the limit of incompressibility where σxx = 0. The equal space electron SPGF relevant for electron tunneling processes is computed using the composite Fermion Green’s function coupled to gauge ﬁeld, aµ:

∞ G(t ,t ,a )= G (t t )exp i d2rdtaµ(r,t)jfree(r,t) (264) 1 2 µ CF 2 − 1 µ Z−∞ where GCF(t) 1/t is the composite fermion Green’s function in the absence of slow gauge ﬁeld ﬂuctuations, free ≈ and jµ (r, t) is a current describing the spreading of free composite fermion density. The electron SPGF is approximated by

G(t)= G (t)exp i[S (t) S (t)] (265) CF edge − free

48 where the subscript free refers to the action of non-interacting composite fermions. The end result yields an expression for the electron tunneling exponent:

o 2 1 ρxy 1 ρxy o α =1+ tan− ρ˜ tan− ρ˜ (266) π ρ xy − ρo xy xx xx ρ˜xx σxx + ln[(1 + κoU)] o , π σxx with

o 2 2 o ρxy = ρxy + ph/e = h/νe , ρxx = ρxx, (267) p being the number of ﬂux quanta attached to CF, and ρ ρ˜ = xx (268) xx e2/h

ρ ρ˜ = xy (269) xy e2/h are the dimensionless diagonal and Hall resistivities. Please see Fig. 10. Accounting for the long-range Coulomb interaction only slightly modifies the exponent α. In particular, a logarithmic correction is found and the effective exponent increases as the energy associated with the tunneling decreases. For edge tunneling from a normal metal into a ν = 1/3 fractional fluid edge, this predicted that the exponent will exceed 3 as the energy decreases towards 0.

5. Scaling functions for electron tunneling

To probe the chiral Luttinger liquid, transport measurements of electron tunneling into the edge of a fractional quantum Hall fluid can be made by placing contacts on opposite sides of the tunnel barrier as shown in Fig. 18 in the experimental Section III D. Several measurement schemes are possible: (i) a direct measurement of the tunneling current through the barrier, Itun, for a voltage bias, V, applied between the two sides to deduce the current-voltage (I-V) characteristics. This should yield a power law dependence, α α 1 Itun V reflecting the power-law tunneling density of states, D(ω) ω − ; (ii) a differential conductance ∝ α 1 ∝ measurement, dI/dV V − ; and (iii) a zero-bias measurement of the linear, tunneling conductance, G(T), ∝ α 1 which should exhibit a non-linear, G T − , dependence on the temperature, T, again reflecting the power law tunneling density of states. The value∝ α extracted from all these measurements must be consistent. The experimental finding that in fact the tunneling behavior as a function of V and temperature, T, does indeed obey a universal scaling function predicted by theory represents an extremely significant result. Therefore, to make contact with experiments, it is necessary to compute the tunneling current under the conditions where the energy-scale is set either by V (E=eV), or T (E=kT). The tunnel current response is usually computed in perturbation theory by considering the operator which annihilation an electron on one side of the tunnel barrier at an initial time, t, and creates it on the opposite side at a different time,

t’, ψ1(x, y, t)ψ2†(x′, y′, t′). To be specific and make contact with the experimental results presented in the experimental section on electron tunneling from a heavily-doped, 3-d n+GaAs metal into the edge of a fractional quantum Hall fluid, we will use the notation where the subscript 1 refers to the normal metal and subscript 2 to the edge of the fractional Hall fluid. The direction x (x’) parameterize the direction along the boundary while y (y’) the direction perpendicular to the boundary. A tunneling event must conserve momentum parallel to the interface (boundary). For instance in the Landau gauge, .A = 0, the wavefunction, φ(x, y), takes the form of a running wave in the x-direction, eikx, where the wavevector,∇ k, is 2 proportional to the y position of the guidance center, yo,k=yo/lo, where lo = ¯hc/eB is the magnetic length with the value of 8.1nm at B=10T. Tunneling across the barrier involves a change in the y-position, and therefore a change in x-momentum. If the system is completely clean and withoutp defect or impurities,

49 translational invariance in this parallel direction alone would suppress tunneling. Therefore it is essential to mediate tunneling via momentum non-conserving processes. The presence of impurities and point-defects can mediate tunneling at point-like (size lo) positions. Point tunneling eﬃciently mediates tunneling since it necessarily involves a broad range of momenta≤ as dictated by the uncertainty principle. Therefore, the tunneling probability which in the general case can be k dependent must be averaged over k. In other words, the spectral density function, A (ω, k), will be averaged, yielding the tunneling density of states: ±

D (ω)= A dk, (270) ± ± Z where denote the tunneling of electrons and holes, respectively. Note that the power-law dependence is expected± to be the same for the tunneling of electrons and holes (Eq. (212) and (213)). The task at hand is therefore to compute the tunneling current for point tunneling. The simplest approximation to point tunneling is made by setting the coordinates equal to the respective values at the tunneling point, (x, y) = (0, y1), (x′, y′)=(0, y2), where for convenience we have chosen x=x’=0, and y2 y1 = b equals the barrier width. As a further approximation, the eﬀect of the ﬁnal barrier width, b, is| incorporated− | in the ”bare” tunneling amplitude, Γ, only. Suppressing the y-coordinates and choose the initial time, t=0, the tunnel coupling is given by:

Htun =Γψ2†(0)ψ1(0) + h.c.. (271) The current response to an applied bias, V, at zero-temperature is given in the weak-coupling (weak tunneling) limit by ﬁrst-order perturbation theory (Nozieres, 1964):

2 ∞ Itun(t)= eΓ dt′θ(t) (272) Z−∞ t′ t′ i eV (t”)dt” i eV (t”)dt” e t < [A(t), A†(0)] > e− t < [A(t)A†(t), A(0)] > ×{ − } R R where A(t) = c2(0, t)c1† (0, t) are the electron operators on the edges 1 and 2. In general, the electron operator for the ith edge can be written as:

1 αi αi Gi(x =0,t)= ai− ωi− t− , (273) where ai is a cutoff length, and ωi a cutoff frequency. Note that this is an equal space propagator. For the Laughlin series with filling factor, ν=1/m, m odd integer, and α =1/ν,

i α kF x (x v t)− i e∓ νi i (274) 1,2 1,2 ∝ ± i where ai is a cutoﬀ length, and ωi a cutoﬀ frequency. This form leads to: 1 1 1 < A(t)A†(0) >= (275) a a [ ω (t iδ)]α1 [ ω (t iδ)]α2 1 2 − 1 − − 2 −

1 1 1 < A†(0)A(t) >= (276) a a [ ω (t + iδ)]α1 [ ω (t + iδ)]α2 1 2 − 1 − 2

< A(x, t)A†(0, 0) > (277)

i i α1 1 α2 1 1 1 kF x kF x = a − a − e ν1 1 e ν2 2 1 2 [x v (t iδ)]α1 [x v (t + iδ)]α2 − 1 − − 2

< A†(0, 0)A(x, t) > (278)

i i α1 1 α2 1 1 1 kF x kF x = a − a − e− ν1 1 e− ν2 2 . 1 2 [x v (t + iδ)]α1 [x + v (t + iδ)]α2 − 1 2

50 The zero-temperature tunneling current is then given by: π I (V )= 2eΓ2Im i dω (279) tun − − (α + α 1)! 1 2 − Z α1 1 α2 1 a1 − a2 − α1+α2 1 f(ω,t) ω − , × vα1 vα2 1 2 where f(ω, t) is deﬁned by:

t′ i eV (t”)dt” iω(t t′) e t = dωf(ω,t)e − . (280) R Z For DC bias this expression simpliﬁes to:

α1 1 α2 1 2 π a1 − a2 − α1+α2 1 I (V )= 2eΓ Im i [ V − ] . (281) tun − − (α + α 1)! vα1 vα2 1 2 − 1 2 At ﬁnite temperature, the expressions become:

α1 1 α2 1 a1 − a2 − α1+α2 < A(0,t)A†(0, 0) >= α1 α2 (πT ) (282) v1 v2 1 1 α1+α2 eiπ 2 sgn(t) ×[ sinh(πTt)]α1 sinh[πTt]α2 −

α1 1 α2 1 a1 − a2 − α1+α2 < A†(0, 0)A(0,t) >= α1 α2 (πT ) (283) v1 v2 + 1 1 iπ α1 α2 sgn(t) e− 2 ×[ sinh(πTt)]α1 sinh[πTt]α2 − yielding a tunnel current:

α1 1 α2 1 2 a1 − a2 − α1+α2 1 Itun =2eΓ α1 α2 (2πT ) − (284) v1 v2 ω ω ω sin π(α12 + i 2πT ) B(α12 i , α12 i ) , × − 2πT − 2πT cos πα12

where α12 = (α1 + α2)/2. Explicit expressions for a DC current can be obtained in the two limits of eV 2πT, where the current is linear in V and power-law in T: ≪

α1 1 α2 1 a1 − a2 − π α 2 eV α 1 Itun α1 α2 T [Γ(α12)] T − eV, (285) ∝ v1 v2 Γ(2α12) 2πT ∝ and for eV 2πT, where the current exhibits a power-law in V: ≫ α1 1 α2 1 α a − a − π eV I 1 2 T α (eV )α. (286) tun ∝ vα1 vα2 Γ(2α ) 2πT ∝ 1 2 12 Here α =2α 1= α + α 1, Γ(s) denotes the gamma function with argument, s, and note that we have 12 − 1 2 − set the Boltzmann constant, kB, to 1 for convenience. Correspondingly, the diﬀerential conductance, Gtun, in the two limits are given by:

α1 1 α2 1 dItun a1 − a2 − π α 2 e α 1 Gtun α1 α2 T [Γ(α12)] T − (287) ≡ dV ∝ v1 v2 Γ(2α12) 2πT ∝

51 and:

α1 1 α2 1 α 1 a1 − a2 − π α αe eV − α 1 T (eV ) − . (288) vα1 vα2 Γ(2α ) 2πT 2πT ∝ 1 2 12 The dimensionless variable, x = eV/2πT, appears as a natural variable for which the condition x=1 denotes an approximate cross-over condition between a situation where the energy scale for tunneling is determined by the thermal energy for x < 1 and the tunneling current approaches linearity in the bias voltage, V, and a α 1 power law in temperature, I T − , and the opposite situation where the energy scale is set by eV for x > 1 and I is nonlinear in V approaching∝ the power law I Vα functional form. For the diﬀerential conductance, ∝ α 1 G approaches a form independent of V while proportional to T − for x < 1, and a power-law versus bias α 1 voltage, G V − , for x > 1. By measuring the voltage bias power law for x >> 1 and temperature power law for x <<∝ 1 and deducing the exponent, α, in these two independent measurements, one can check the consistency of the two values deduce for the exponent, α. Besides DC characteristics, based on expression, Eq. (283), under an additional AC excitation at frequency, ω, akin to the Josephson frequency, ω = e∗V/¯h, interesting resonance structures in the tunneling current and in the tunneling current quantum shot-noise are predicted to be observable. At typical accessible base temperatures of a few 10’s of milli-Kelvin, to exceed the thermal energy scale, excitation voltages in the few µV range corresponding to Josephson frequencies in the GHz regime are needed. Other than the scaling expressions due to Wen, a similar but alternative formulation of the has been put forth by Kane and Fisher based on a renormalization group analysis of the back-scattering between left- and right-moving Luttinger liquid channels by a single impurity. Furthermore, they uncovered a duality relation between the strong and weak-tunneling limits. In the weak-tunneling limit relevant to the experimental conditions discussed in this review, Kane, Fisher (1992b, 1992c) give an alternative expression of:

I [Γ(α )]2x + xα. (289) ∝ 12 This expression exhibits the same limiting behaviors for x 1 and x 1 as the Eqs. 285 and 286. The cross voltage where the linear and power-law contributions≪ are equal≫ to each other is given by eV = 2/(α 1) [Γ(α12)] − (2πT). For tunneling between the ν1 = 1 and ν2 = 1/3, α12 = 2 implying Γ(α12)=1!=1 and the cross over occurs at eV = 2πT. It turns out that for tunneling through a single impurity or contact point, a complete solution with a full universal curve spanning the entire range of weak and strong-coupling (tunneling) can also be obtain based on the Bethe-Ansatz. Fendley, Ludwig, and Saleur (1995a, 1995b) were able to demonstrate the integrability of the problem for tunneling between chiral Luttinger edges at filling fractions 1/4 <ν< 1, assuming a single edge mode for each edge while at the same time proved the exact duality relation between weak and strong tunneling. In the context of the fractional Hall edge, the duality corresponding to the physical situation of electron-tunneling in the weak limit, and quasi-particle tunneling in the strong limit. It turns out that even in the presence of the back-scattering between the left and right-moving ν channels coupling occurring at a point via and impurity or point contact, the problem is integrable and can be mapped onto known field theory models of the boundary Sine-Gordon type (Ghoshal and Zamolodchikov, 1994), which in term is directly related to the Kondo problem. This entire class of problems is soluble via the Bethe ansatz. As a consequence of this integrability and the associated presence of an underlying quantum critical point, the differential conductance can be expressed in terms of two dimensionless variables, eV/T and TK/T, formed out of the three energy scales, the bias voltage, V, the temperature, T, and the Kondo energy scale, TK which characterizes the strength of the point contact coupling: dI eV T G = = G( , K ). (290) dV T T

In the limit for weak coupling for which eV, T << TK, the expression for the current has the functional form of Eq. (289) with the scale of the current set by TK:

e2 T 2πT α I = ν K [Γ((α + 1)/2)]2x + xα . (291) tun h 2[Γ((α + 1)/2)]2 T K 52 In the special case of tunneling between two ν = 1/2 edges, the entire scaling curve can be expressed in closed form ( Fendley et al., 1995b; Kane and Fisher, 1992c):

2 ∞ dω ω Itun = [f(ω ωo) f(ω)] (292) 2π ω2 + ( TK )2 − − Z−∞ 2 and

2 2 1 e ∞ ω Gtun = dω [ f ′(ω ωo)]. (293) 2 h ω2 + ( TK )2 − − Z−∞ 2

where ωo = eV/2 and f is the Fermi-Dirac distribution. These expressions can be recast in terms of the digamma function, ψ, and its derivative (Fendley et al., 1995b). For general 1/4 <ν< 1, these solutions yield for G in the T=0 limit (Chamon and Fradkin, 1997):

2n( 1 1) V ν − V δ 2 ∞ cn(1/ν) e  − 1 2TK 2TK where  P

n 1 Γ(nν + 1) Γ(1/2) c (ν) = ( 1) − , (295) n − Γ(n + 1) Γ(n(ν 1)+1/2) − and δ = [νlnν + (1 ν)ln(1 ν)]/2(ν 1). To further make− contact with− experiments− in which tunneling takes place between a 3-d n+GaAs normal metal and a chiral Luttinger edge, Chamon and Fradkin (Chamon and Fradkin, 1997; Fradkin, 2000) made extensive use of the exact solutions provided by Fendley, Ludwig, and Saleur, applied to a case of multiple, weak- coupling point contacts which are incoherent with each other. They ﬁrst mapped the 3-d metal to a ν1 = 1 chiral Fermion mode resulting in an action for tunneling at a point contact into a ν2 = 1/m chiral Luttinger: [Please see Chklovskii and Halperin (1998) for a somewhat diﬀerent view of such a mapping.]

1 1 1 S = dtdx[ (∂tφ1 v1∂xφ1)∂xφ1 + (∂tφ2 v2∂xφ2)∂xφ2 (296) 4π ν1 − ν2 − Z 1 1 iωt i[( φ1(0,t) φ2(0,t)] +Γδ(x)e e− ν1 − ν2 + h.c.]

where the two edge Luttinger modes interact at the point x=0 and the two modes live in diﬀerent spaces of x 0 and x 0 respectively. As a result the x coordinate for each ﬁeld can be rescaled to yield the ≥ ≤ same velocity for the two modes, v = v1 =v2. The respective electron operator for the modes are given by 1 i ν φI ψI(x, t)=e I . An orthogonal transformation:

φ˜1 = cos θφ1 + sin θφ2 (297)

φ˜ = sin θφ + cos θφ , (298) 2 − 1 2 where

1 1/ν + 1/ν cos θ = 1 2 (299) √2 p 1/ν1 +1p/ν2 p 1 1/ν 1/ν sin θ = 1 − 2 , (300) √2 p 1/ν1 +1p/ν2 p 53 yields a tunneling action between identical transformed ﬁllings,ν ˜: 1 ν˜ = 1 (301) [(ν− + 1)/2] of 1 1 S = dtdx [(∂ φ˜ v∂ φ˜ )∂ φ˜ + (∂ φ˜ v∂ φ˜ )∂ φ˜ (302) 4π ν˜ t 1 − x 1 x 1 t 2 − x 2 x 2 Z 1 iωt i [φ˜1(0,t) φ˜2(0,t)] +Γδ(x)e e− ν˜ − + h.c.].

For ν = 1 to ν = 1/3 tunneling, this results in tunneling between twoν ˜ = 1/2 edges!! This case therefore corresponds to the exactly soluble case discussed by Fendley, Ludwig, and Saleur for which the entire scaling function is known.

FIG. 11. Multi (N)-impurity scattering, assembled from the one impurity building block. It is crucial that the voltages on the fractional quantum Hall ﬂuid side are maintained in between scattering events, whereas the electrons from the reservoir side always come into the scattering process at VR. [From Chamon and Fradkin (1997).]

The ﬁnal step is to assume that in the cleaved-edge tunneling experiment, electron tunneling takes place under the condition of weak-tunneling at multiple contact points, where the tunneling at successive points are incoherent in nature as depicted in Fig. 11. At each point, the receiving chiral Luttinger edge mode is characterized by a voltage, Vi, while the injecting ν = 1 chiral Fermion edge mode is always residing at the voltage of the 3-d, normal metal reservoir. In other words, energy relaxation is fast within the 3-d, n+ GaAs normal metal compared to the time between successive tunneling events at the incoherent tunneling points. Whatever voltage the eﬀective chiral Fermion channel ends up with after passing a given point contact tunneling point is replenished by fast energy exchange with the rest of the bulk 3d metal. A full scaling curve extending from the weak to strong tunneling regime relevant to the experimental situation can now be obtained from two relationships: 1) with the chiral Luttinger edge between two successive tunneling points, we have:

Ii Vi Vi 1 = , (303) − − νe2/h where Ii is the tunneling current at the i-th point. The total tunneling current is then the sum of the currents, Ii, Itun = i Ii, and 2) at each tunneling point, the tunneling is weak and we have from Eq. (291): P 54 e2 T 2πT α I = ν Ki [Γ((α + 1)/2)]2x + xα . (304) i h 2[Γ((α + 1)/2)]2 T Ki where that (α + 1)/2= α12. The ﬁrst equation, Eq. (303), is strictly correct only when the dissipative conductance, Gxx, is zero as in the case of a well-developed quantized Hall state. It is still a good approximation as long as Gxx << GHall as is the case for the high quality samples used in the experiments at general ﬁllings. Combining these two equations yields:

α 1 2πT 1 α 2πT ∆xi = TKi xi 1 + α+1 xi 1 , (305) − 2 T − [Γ( )]2 − Ki 2 By ﬁne-graining the successive tunneling points, Chamon and Fradkin obtained a diﬀerential equation:

α 1 dx 1 2πT − 1 α = xi 1 + α+1 xi 1 . (306) − di 2 T − [Γ( )]2 − Ki 2 Imposing the boundary conditions valid for 1 to ν tunneling:

e(V V ) eV x = R − o = (307) o 2πT 2πT and e2 2πT I = I = ν (x x ), (308) tun i h e o − N i X where N indexes the last tunneling point, one obtains with β = α 1: −

1 2 β 2 ( πT ) e e− 2 Ts Itun = ν V 1 1  , (309) β h  − 1 β 2πT eV β β   2 ( Ts )   1+ α+1 2 (1 e− ) 2πT  [Γ( 2 )] −  h i  yielding:   dI G = tun (310) tun dV

1 2 β 2 ( πT ) e e− 2 Ts = ν 1 α .  β  h − 1 β 2πT eV β β  2 ( Ts )   1+ α+1 2 (1 e− ) 2πT  [Γ( 2 )] −  h i  Here, Ts denotes a cross over temperature to the strong tunneling regime where: 

1 β N 1 β = . (311) T T s i=1 Ki X

Although strictly speaking this expression is valid only for ν1 = 1/m1, ν2 = 1/m2, mi is an odd integer, so that tunneling occurs between a single edge mode for each ﬂuid, in practice one is able to interpolate to continuous values of νi. This is the expression used extensively in the data analysis to extract the tunneling exponent, α.

55 6. Resonant tunneling

In the discussion above, we focused on tunneling events in which the individual event is in the weak- tunneling regime, and the total tunneling current is the incoherent sum of weakly tunneling events. In fact, it is often possible to observe tunneling resonances as the magnetic ﬁeld is swept (Milliken et al., 1996; Maasilta and Goldman, 1997; Grayson et al., 2001). In resonant tunneling, the process is presumably mediated by a resonant bound-level or impurity state situated spatially close to the two chiral edges. In fact the tunneling current in resonant tunneling can be computed is an manner similar to the discussions 2 above in both the incoherent, sequential resonant-tunneling case for which Itun << ν(e /h)V (Chamon and 2 Wen, 1993; Furusaki et al., 1993), and the fully coherent case for which Itun approaches ν(e /h)V (Kane and Fisher, 1992c; Moon et al., 1993; Fendley et al., 1995b). To gain physical insight and to make contact with the experimental results presented below, we address the key features of sequential-tunneling treated in ﬁrst-order perturbation theory. The coupling between a given edge, i=R,L, and the impurity, I, can be written as (Chamon and Wen, 1993; Furusaki et al., 1993):

H =Γ Ψ†Ψ + h.c., (312) I,i i I i|x=0 where the first terms contributes to filling the impurity level while the second (h.c.) to the emptying of the level. The field operators can be both electrons and quasi-particles of e∗ In this sequential tunneling limit and neglecting higher-order virtual processes, the tunneling current will contain contributions from both the filling of the impurity level when empty (off) or emptying when occupied (on). In analogy to the off-resonance case between two chiral edges (Eq. (276)), using the standard first-order perturbation theory the current from the i-th chiral edge onto the impurity, is given by (Chamon and Wen, 1993):

2 Γi ∞ Ii = e∗ | | dt′Θ(t t′) exp[ i(e∗Vi EI )(t t′)] (313) aI − − − − Z−∞ < [Ψ† (x =0,t)Ψ (x =0,t), Ψ†(x =0,t′)Ψ (x =0,t′)] >, × I i i I where the energy level of the impurity level is denoted by EI, and aI is a length. As before the expectation of the chiral edge operator is given by (Eq. (286) and (287)):

αi 1 αi ai − πT π < Ψ†(x =0,t)Ψ (x =0,t) >= exp[ i α sgn(t t′)], (314) i i vαi sinh[πT (t t )] ± 2 i − i ′ − and

< ΨI†(t)ΨI (t′) >= nI (315)

< Ψ (t)Ψ† (t′) >=1 n , (316) I I − I where nI is the average occupation of the impurity level. In terms of the ﬁll current when the impurity level is entirely empty, Ii,ﬁll, and the removal current when the level is fully occupied, Ii,rem, the i-th current can be written as (see inset to Fig. 12):

I = [I (1 n )+ I n ], (317) i i,fill − I i,rem I where

α 1 i− 2 ai αi 1 Ii,fill = e∗ri,fill = e∗ Γi αi (2πT ) − (318) − | | aI vi αi iω αi iω ω B[ , + ] exp[ ] ω= (e∗V E ), × 2 − 2πT 2 2πT 2T | − i− I

56 and α 1 i− 2 ai αi 1 Ii,rem = e∗ri,reme∗ Γi αi (2πT ) − (319) | | aI vi αi iω αi iω ω B[ , + ] exp[ ] ω= (e∗V E ), × 2 − 2πT 2 2πT −2T | − i− I where ri,fill and ri,rem denote the rate of filling and removal, respectively. The total combined rate of filling, Ifill, and removal, Irem, from both the R and L channels are:

I = [I + I ]= e∗[r + r ], (320) fill R,fill L,fill − R,fill L,fill and

Irem = [IR,rem + IL,rem]= e∗[rR,rem + rL,rem]. (321)

The respective times to fill when empty, τoff , and to empty when filled, τon, are:

1 e∗ τoff = = − (322) rR,fill + rL,fill Ifill and

1 e∗ τon = = . (323) rR,rem + rL,rem Irem

The average occupancy, nI, of the impurity level is: τ I n = on = − fill (324) I τ + τ I I on off rem − fill and τ I 1 n = off = rem . (325) − I τ + τ I I on off rem − fill The ﬁnal form of the tunnel current is given by: 1 I =< I >= [I I I I ]. (326) tun R I I R,fill L,rem − R,rem L,fill rem − fill

Specializing to the νR = 1 to νL = 1/m resonant electron-tunneling case for which αR = 1 and αL = 3, we have: 2 e ΓR 1 iω 1 iω ω IR,fill = − | | B , + exp ω= (e∗V E ) , (327) a v 2 − 2πT 2 2πT 2T − R− I I R h i

2 e ΓR 1 iω 1 iω ω IR,rem = | | B , + exp ω= (e∗V E ) , (328) a v 2 − 2πT 2 2πT −2T − R− I I R h i

2 2 e ΓL aL 2 3 iω 3 iω ω IL,fill = − | | (2πT ) B , + exp ω= (e∗V E ) , (329) a v3 2 − 2πT 2 2πT 2T − L− I I L h i and 2 2 e ΓL ai 2 3 iω 3 iω ω IL,rem = | | (2πT ) B , + exp ω= (e∗V E ) . (330) a v3 2 − 2πT 2 2πT −2T − L− I I L h i

57 This expression gives rise to an asymmetry as one sweeps through the resonance, and a power-law integrated area versus T in the differential conductance, dI/dV, as well as a non-preservation of integrated area, as shown in Fig. 12. As a result of this first order perturbation treatment based on sequential tunneling, the expressions Eq. (326) - (330) are valid when the the on and off times in the impurity level, τon and τoff , 1 1 1 1 satisfy the respective condition, τ >> min( e∗V E − , T− ) and τ >> min( e∗V E − , T− ), in on | R − I| off | L − I| the case where e∗VR > e∗VL. Beyond the sequential regime the resonant-tunneling current, including the case of perfect resonant transmission, can be calculated by renormalization group and quantum Monte-Carlo methods (Moon et al., 1993) and exact Bethe-Ansatz calculations (Fendley, Luwig, Saleur, 1995b). In the opposite limits of off-resonance tunneling occurring in the tail region of the tunneling resonance, second order virtual processes contribute. This leads to off-resonance tunneling with an effective coupling between the R and L chiral Luttinger modes of:

HR,R′ =Γ′ΨR† ΨL + h.c., (331)

where the coupling, Γ = Γ Γ† /∆E, where ∆E = e∗(V + V )/2 E when far oﬀ-resonance and is L R | R L − I | essentially independent of the bias voltage, VR VL for small voltages VR VL << max( ∆E/e∗ , T). This corresponds to the case treated in the previous− Section IID5 on scaling| functions.− | | |

−3 −2 −1 0 1 2 −3 −2 −1 0 1 2 3 4 4 eV 3 µ R 3 +30 V EI 2 eV L 2 1 1 0 0

3 FL 0 ν=1 to 1/3 3 tunneling 2 2 1 1 dI/dV 0 0 −3 −2 −1 0 1 2−3 3 −2 −1 0 1 2 3 3 −30 3 2 2 1 1 0 0 −3 −2 −1 0 1 2 −3 −2 −1 0 1 2 3

E I/kT

FIG. 12. The diﬀerential conductance, dI/dV, under bias, for the resonant tunneling of electrons (holes). Left panels–tunneling into a Fermi liquid (FL) edge, right panels–tunneling into a ν =1/3 chiral Luttinger liquid edge, plotted as a function of the impurity level position referenced to the thermal energy, EI/kT. Dashed curves in the top-right panel show the separate contributions for the right and left leads. Here the conventional is that the right lead refers to the n+ GaAs metallic lead, while the left lead refers to either the Fermi liquid or Luttinger liquid edge lead. The linear portions of the left dashed peak reﬂects the energy derivative of the chiral Luttinger liquid tunneling density of states, D (ω) ω2. Inset shows the energy tun ∼ positions of the n+ GaAs normal metal, eVR, FL or CLL edge, eVL, and the impurity level position, EI.

58 7. Shot noise and fractional charge: quasi-particle tunneling

In addition to the Hallmark signatures of the CLL in the tunneling current, remarkable signatures are present in the current and voltage fluctuations resulting in quantum current shot noise and voltage noise. These noise fluctuations can be measured or computed under equilibrium as well as non-equilibrium conditions. In fact, generalization of the fluctuation-dissipation theorem to its non-equilibrium analogue within the chiral Luttinger liquid model imply that in the case of weak backscattering and strong coupling for which quasi-particle tunneling dominates, the low frequency current shot noise provides a measure of the fractional charge, e∗, of the tunneling quasi-particle, where the shot noise fluctuations reflect the graininess of the charge carriers in units of e∗ (Kane and Fisher, 1994; Fendley et al., 1995c; Chamon et al., 1995). Furthermore, high frequency noise is predicted to exhibit singularities in its power spectrum at frequencies related to the Josephson frequencies of the quasi-particles (Chamon et al., 1995, and represent fluctuations in the AC current. In mesoscopic conductors, recent theoretical (Lesovik, 1989; Buttiker, 1990) and experimental work ( Li et al., 1990; Dekker et al., 1991; Liefrink et al., 1994) have shown that the zero-frequency quantum shot noise in a 1-dimensional conductor scales as: e2 S(ω 0) = t(1 t)(eV ), (332) → h − where t is the transmission probability of the electron. In the limit t << 1, the classical, uncorrelated shot e noise result is recovered with I h t(eV). In the opposite limit of (1 t) << 1, one obtains the shot noise of ”holes”. Similar results can be≈ deduced for the tunneling of electrons− and quasi-particles between identical fractional Hall edges. In particular Kane and Fisher (Kane and Fisher, 1994) showed that for tunneling between two Laughlin, ν =1/m edges, under non-equilibrium conditions with current flow and voltage bias across the tunnel junction an analogue of the fluctuation-dissipation theorem takes the form:

ω νe2 2 ω c (ω) coth R (ω)= c (ω) coth R (ω) , (333) I − 2T I h V − 2T V h i where cI(cV) and RI (RV) are the respective correlation and response functions for I and V. Both sides of this expression becomes identical to 0 only under equilibrium, yielding the ﬂuctuation dissipation theorem. Remarkably as ω 0 this expression implies: → νe2 2 c (ω 0) = c (ω), (334) I → h V relating the power of the current and voltage shot noises with the extra factor ν2. ¿From such expressions one obtains the quantum shot noise for quasi-particle tunneling under weak backscattering of: (νe)2 c (ω 0) = eV. (335) I → h The above results were obtained using the lowest order terms, inclusion of higher order terms lead to ”interaction eﬀects” and singularities in the noise power spectrum at Josephson frequencies, ω =e∗V (Chamon et al., 1994). Due to the integrability of the single point contact backscattering model, it is possible to derive the exact expression for any amount of backscattering at T=0 using the Bethe ansatz (Fendley et al., 1995c).

III. EXPERIMENTS ON CHIRAL LUTTINGER LIQUIDS– TUNNELING INTO THE FRACTIONAL QUANTUM HALL EDGE

The most outstanding physical characteristic which distinguishes a Luttinger liquid, chiral or non-chiral, from a conventional Fermi liquid metal is its low energy property when an external ”bare” electron is

59 added to or removed out of the correlated Luttinger liquid at energies near the ”Fermi” energy. The associated orthogonality catastrophe which occurs between the state consisting of a bare particle added to (or removed from) a highly correlated N electron ground state and the ground state of the N + 1 (N- 1) electron system gives rise to a power law suppression of the tunneling current as the energy from the Fermi surface, E EF, approaches zero. To study this unique low energy property, it is natural to perform a tunneling experiment− in transport. Although lacking the ability to resolve momentum due to the fact that in reality most tunneling takes place at point-like contact points, the distinct advantage of a transport experiment in the precise control of the low energy scale, set either by an external voltage bias across the tunnel junction down to the 1µeV level, or by temperature down to 25mK for which kT 2.15µeV, enables truly low energy behavior to be studied in detail. These energies are a factor of 102 ≈ 103 smaller than − the relevant characteristic energies of either the Fermi energy, EF 4meV, or the quasi-particle gap, ∆, of 0.1 1meV in the most robust incompressible fractional Hall fluid∼ s such as ν = 1/3. In contrast, the powerful∼ − and complementary techniques such as angle-resolved photoemission spectroscopy (ARPES), while offering the ability to resolve k-dependences, nevertheless requires the usage of energetic photons of energy 20eV to eject surface electrons while attempting to determine the low energy properties down to the meV ∼level. For example, it is often technically challenging to precisely locate the Fermi level to meV accuracy, as well as the position and width of the quasi-particle peak (or the absence of such a peak) in the spectral function. Damage to the specimen can also result from the radiation of energetic photons. In the electron-tunneling transport measurements several key conditions must be met to achieve a successful demonstration of chiral Luttinger liquid behavior. These include: i) the requirement that the observed nonlinearity in the current-voltage (I-V) characteristics arise from the tunneling density of states (TDOS), and not from residual energy dependences in the tunneling matrix element across the tunnel barrier. This amounts to the requirement that the tunnel barrier be insensitive to the voltage bias applied across the barrier, as well as an insensitivity towards thermal smearing. This requirement can be satisfied if the height of the tunnel barrier is tall (large) compared to these external energy scales; ii) the existence of a power-law regime in the I-V relationship with a substantial dynamic range in both the current and the excitation bias voltage. A small dynamic range cannot reliably differentiate between a power law functional form from other competing forms such as exponential, Ahrenius, or variable-range hopping, unless the measurements are performed with extremely high precision as well as accuracy so as to be free of either random or systematic sources of error;

iii) consistency in the power law tunneling density of states, ρTDOS, deduced independently from measurements of the I-V relation, temperature dependence of the low-bias linear conductance, G(T), and the differential conductance under bias, dI/dV; and finally iv) from a practical point of view, it is desirable to have chiral Luttinger liquid characteristics where the exponent, α, where I Vα is in the range α 1.5 4. To establish a good non-linear behavior with good dynamic range∝ in both the current and∼ the− excitation energy scale (either bias voltage or temperature), given the current noise floor of the order of femto-amperes, exponents in this range is desirable. Our approach is to first convincingly establish the presence of a power-law functional dependence in the I-V characteristics The clear-cut observation of power-law behavior with unsurpassed quality–the Hallmark signature of Luttinger liquid behavior–unequivocally establishes the fractional quantum Hall edge as a chiral Luttinger liquid system. Subsequently, we provide detailed investigations of the nature of the chiral Luttinger liquid, in particular its dependence on magnetic field for samples with fixed electron densities. This amounts to a study of the different chiral Luttinger liquids at the edge of fractional Hall fluids at different filling factors. This section is structured as follows. After a brief synopsis of the history in the experimentation on the chiral Luttinger liquid, we proceed with a discussion of the special, Cleaved-Edge Overgrowth (CEO) devices which has enabled our fruitful experimentation. Next, the measurement technique for sensitive 15 current-voltage measurements, down to a few femto-amperes (10− A) in current and 1 µV is described.

60 The ensuing section contains the main results of this review. We present evidence for the I-V power law dependence in the appropriate regime of voltage bias exceeding the thermal scale but below the saturation regime, while rejecting other competing functional forms. This establishes the chiral Luttinger liquid nature of the fractional quantum Hall edge. Next, we show that the observed I-V dependence, in addition to exhibiting excellent power-laws, often can be fitted to theoretical universal scaling expressions discussed in Section IID5. In the best case, an I-V trace spanning 9 orders of magnitude in current and 5 orders of magnitude in excitation voltage can be fitted to the Chamon-Fradkin expression using two free parameters. Moreover, the fit contains a maximum deviation from data of at most 10% in value over the entire range aside from the lowest current region where random measurement noise becomes significant! We follow by presenting results at continuous Landau level filling factors, ν, by varying the magnetic field. We are able to establish power-law behavior at general fillings, for both incompressible and compressible fluids. These results, in particular the functional dependences of the I-V exponent, α, are contrasted with the standard theoretical predictions of Wen (1992, 1995), Kane and Fisher (Kane et al., 1994; Kane and Fisher, 1995), and Shytov, Levitov, and Halperin (Shytov et al., 1998; Levitov et al., 2001). The overall significance of these outstanding experimental results and their implications are discussed in detail. In addition we report evidence of unusual line shapes in resonant tunneling under bias into the 1/3 fractional Hall edge which provide further compelling evidence for chiral Luttinger liquid behavior. Following these presentations, we will describe noise signatures in the fractional quantum Hall regime in experiments performed by other workers.

A. Historical Overview

The first attempts to investigate Luttinger liquid behavior in semiconductor based systems were initiated in 1-dimensional quantum wire systems at zero magnetic field. Two approaches were attempted, lateral (side) gating to form relatively long channel quantum point contacts (Tarucha et al., 1995), and the formation of a novel cleaved-edge overgrown quantum wire (Yacoby et al., 1996). The results were inconclusive. Although an intriguing suppression of the conductance below the quantized value of e2/h were observed, clear signatures of power law dependences in the deviation were not observable. In the case of a 1-d conductor at B=0, the power law exponent is determined by the reduced conductance g G/(e2/h), where g < 1 for repulsive interaction potentials and is dependent on the exact nature of the interaction≡ and is therefore not universal. Experiments to observe LL behavior must deal with the complications of localization effects which tend to obscure the power law characteristics, resulting from the back-scattering of electrons by residual disorder or non-ideality in the one-dimensionality. Consequently, only indirect hints were initially observed (Tarucha et al., 1995; Yacoby et al., 1996). (Recent notable advances will be described in Section IV.) In contrast, as discussed in detail in the theory section, in the fractional quantum Hall effect the edge is expected to behave as a chiral Luttinger liquid, where the chirality arises from the presence of the magnetic field and the formation of skipping orbit states along the two-dimensional electron gas (2DEG) boundary. In this system, the forward and backward propagating edge modes are spatially separated, minimizing backscattering and localization effect. Impurities and imperfections only cause the one-dimensional boundary to meander, and have negligible effect on the nature of the chiral Luttinger liquid. Furthermore, here g is well defined and is expected to be simply related to the reduced quantized Hall conductance, at least within the context of the standard theories. Because of the importance of finding a readily accessible and well-characterized non-Fermi liquid system, early experiments (Milliken et al., 1996) on tunneling into fractional quantum Hall edges greatly catalyzed interest in this field. These experiments (Milliken et al., 1996; Kouwenhoven and McEuen, 1995; Alphenaar et al., 1995) studied the tunneling between two ν =1/3 edges through a potential barrier created by electrostatic gating. In the leading work by Milliken et al. (1996), a power law was often found in the conductance as well as an anomalous temperature dependence of the resonant tunneling line shape. Nevertheless, these intriguing results invariably suffered from a limited temperature range ( a factor of 2 to 3) (Milliken et al., 1996; Turley et al., 1998), rendering it difficult to distinguish between a∼ power law versus other competing functional forms (Kouwenhoven and McEuen, 1995; Alphenaar et al., 1995; Maasilta and Goldman, 1997; Turley et al., 1998), and in the case of resonant tunneling from difficulties in resolving the subtle difference

61 between a Fermi-liquid and Luttinger-liquid resonance line shape. Reports of a T 2/3 dependence in the width of tunneling resonance have proven problematic to reproduce. Furthermore, meaningful current-voltage (I-V) relationships were not obtained. These difficulties are a consequence of the geometry. Due to the large, vertical spatial separation (> 100nm) between the metal gate (situated on the sample top surface) and the electron gas, the boundary of the 2DEG is necessarily smooth. This smooth boundary potential has two important but undesirable effects. First, it gives rise to a phase separation of the 2DEG into strips of alternating compressible and incompressible quantum Hall fluids as one approaches the boundary from within the bulk (Beenakker, 1990; Chang, 1990; Chklovskii et al., 1992). Consequently, an electron gas which exhibits a ν =1/3 effect in the bulk may be bordered by ν = 1/4 and 1/5 fluids, etc. Electrons tunneling into the 1/3 edge must cross the other phases, giving rise to complex tunneling characteristics. Secondly, the barrier potential defined by gating is low and broad. In other words, the slope is quite shallow. When a voltage bias is applied, or when thermal broadening of the Fermi surface occurs, the tunneling position can shift substantially, causing the tunneling matrix element to change. From this effect alone, one would expect a limited range for any power law behavior to occur, where the maximum range in voltage or temperature is set by the small window in energy with roughly constant tunneling transmission coefficient. Consequently, the outcome is suggestive but controversial; within the limited temperature range competing functional forms can equally well be ( To )1/2 accommodated, e.g. power-law, e− T variable-range hopping, etc. In contrast, the author and co-workers (Chang et al., 1996; Levi, 1996) reported the first clear evidence of power law characteristics in the I-V relation and in the temperature dependence of the tunneling conductance by making use of a novel geometry implemented by the CEO growth technique ( Pfeiffer et al., 1990; Grayson et al., 1996). In particular, they achieved a geometry where tunneling takes place from a 3-d, bulk n+ doped GaAs metal overgrown on the (011) plane into the edge of a fractional quantum Hall fluid within a quantum well in the (100) plane as discussed below. (See Figs. 13- 15 below.) Here the boundary of the 2DEG is nominally atomically sharp and the tunneling barrier tall and thin, in direct contrast to the low and broad barrier previously achieved by electrostatic gating. This ability to create well-controlled, sharp boundaries, and a tall and thin barrier has opened up new possibilities for studying the chiral Luttinger liquid.

FIG. 13. Several unusual electron gas geometries made available by the Cleaved Edge Overgrowth (CEO) technique. Clockwise from upper left: 2DEG on (011) plane, 1D quantum wire, T-wire, and L-shaped electron gas.

62 B. Cleaved-edge overgrowth

Pfeiffer et al. (1990) pioneered the Cleaved-Edge Overgrowth (CEO) technique for the molecular beam epitaxy (MBE) growth of GaAs/AlxGa1 xAs on the unconventional (011) cleavage plane. The overgrowth takes place after an initial growth in the− conventional (100) direction and subsequent in-situ cleaving along the (011) direction. Since the (100) and (011) planes are perpendicular to each other, the overgrowth achieves structures which contains an element of 3-dimensionality, going beyond the 2-dimensional layered growth in the (100) direction alone. In Fig. 13 we show some of the unusual structures which can be obtained using this CEO technique. By the ingenious combination of growths in the two directions and suitable modulation doping, it has been possible to fabricate 1-d wires where the walls of confinement are nominally uniform to one mono-layer of atoms ( Wegscheider et al., 1994a; Zaslavsky et al., 1991; Wegscheider et al., 1994b; Someya et al., 1995; Yacoby et al., 1996; Kurdak et al., 1994; De Picciotto et al., 2001). Such uniformity is nearly impossible by any state-of-the-art lithographic techniques. To date the most successful application of CEO to technology is the invention of the T-wire laser based on electronic transitions between 1-d sub-bands in the 1-d quantum wires ( Wegscheider et al., 1994a, 1994b; Someya et al., 1995). Here, the 1-d quantum wire is formed at the T-junction of quantum wells grown separately in the (100) and (011) directions. Going one step further by successive cleaved-edge-overgrowth in two orthogonal directions, a cleaved-edge quantum dot has now been invented as well (Wegscheider et al., 1997). In addition, a novel surface resonant tunneling diode structure has been demonstrated on the (011) direction where tunneling occurs through a 1-d quantum wire (Zaslavsky et al., 1991; Kurdak et al., 1994). By the same virtue which makes it possible to grow quantum wires with minimal width fluctuations, one can interchange the GaAs and AlxGa1 xAs to form energy barriers of unparalleled uniformity as shown in Fig. 14 left. Not only can one design structures− where tunneling takes place between different electron gases within the (011) plane, it is also possible to achieve tunneling between electron gases residing separately in the (100) and (011) planes, through a barrier grown in the (011) plane (Fig. 14 right). A variation of these types of structures have been proven to be extremely useful for edge tunneling in the fractional quantum Hall regime.

FIG. 14. Left–CEO structure for 2D to 2D tunneling in the (011) plane, and right–from the (011) plane to (100) plane.

The tunneling experiments are performed on samples grown by the Cleaved-Edge-Overgrowth technique ( Pfeiﬀer et al., 1990; Grayson et al., 1998), producing devices in which a tall, thin Al0.1Ga0.9As barrier separates the structurally atomically sharp edge of the 2DEG, conﬁned within a quantum well, from a heavily doped n+ GaAs bulk metal layer. The sharp edge is created by in-situ cleaving along the (011) direction followed by a regrowth of the thin barrier, a 15nm region of undoped GaAs, and the heavily doped n+GaAs metal on this (011) plane perpendicular to the conventional (100) growth plane. The barrier, which results from a band-gap discontinuity between the Al0.1Ga0.9As and GaAs, imposes a nearly atomically sharp potential of 100meV in height on the electrons relative to band bottom. The high mobility 2DEG in the (100) plane is terminated in the (011) direction by this abrupt barrier, giving rise to a structurally sharp 2DEG edge structure in direct contrast to the situation in a gated, smooth boundary described above. The

63 barrier thickness is of order 5-25nm while its height rises 70meV above the 2DEG chemical potential far exceeding the 2DEG Fermi energy of 4meV. This tall, thin barrier has proven essential by enabling access to a significant dynamic range in the∼ tunneling bias voltage with only minimal distortions of the barrier shape. A typical structure consists of a delta-doped quantum well on a GaAs (100) substrate situated 600nm below the surface, followed by the cleaved-edge-overgrowth of the barrier on the (011) plane, the 15nm of undoped GaAs, and 485nm of highly doped n+ GaAs layer. Please see Fig. 15. The n+ doping density is 18 3 in the 0.5 2.2x10 cm− range. It is essential to use a quantum well to confine the 2DEG rather than a single heterojunction− to prevent leakage through a second channel. In the undesirable case where a single heterojunction is used in place of the quantum well in the initial (100) growth, this second channel will form at a second heterojunction on the (011) plane between the GaAs of the first growth and the cleaved-edgeovergrowth Al0.1Ga0.9As barrier. Although in principle this second heterojunction is undoped and devoid of carriers, under even a small voltage bias carriers can readily tunnel across the Al0.1Ga0.9As barrier from the 3d n+ GaAs layer into the (011) 2-d layer and subsequently trickle down into the 2DEG in the (100) heterojunction, thereby shorting out the highly-suppressed tunneling path into the chiral Luttinger liquid at the 2DEG edge.

FIG. 15. (a) Device geometry showing the cleaved-edge Al0.1Ga0.9As tunnel barrier and the heavily doped 3d n+ GaAs metal; (b) geometry for the tunneling current measurements.

Because the bare tunneling matrix element across the barrier is strongly magnetic field dependent (Sec- tion III E 1), to obtain data spanning a sizable range of Landau level filling factor, ν, corresponding to different fractional quantum Hall fluids, it is necessary to perform measurements on a set of samples with varying 2DEG densities and barrier thicknesses. In Table I we summarize the device characteristics, while the substrate growth parameters are summarized in Table II. For example the sample set of Samples 1.1, 1.2, 1.3, and 1.4 were all grown from the same high quality quantum well substrate, but with different tunnel bar- 11 2 6 2 rier thicknesses. For these samples, the 2DEG is of density 1.08x10 cm− and mobility 3x10 cm /Vs. 11 2 ∼ 6 2 ∼ Sample 2 has a 2DEG density of 0.87x10 cm− and a mobility of 1.8x10 cm /Vs, etc. The Al0.1Ga0.9As barrier thickness for Samples 1.1, 1.2, 1.3, and 1.4, are 9nm, 22.5nm, 12.5nm, and 24.5nm, respectively. The 18 3 n+ GaAs is doped to 0.5 2x10 cm− carrier density yielding a chemical potential of 29-85meV from the GaAs band bottom (34-90meV− from the impurity band bottom), while the chemical potential of the 2DEG is approximately 27meV (Eo +EF) above band bottom for Samples 1.1, etc. Charge redistribution can take place across the barrier due to the difference chemical potential. The actual density profile will also depend on whether residual silicon dopants penetrate into the 15nm undoped GaAs buffer layer during the regrowth process. (Please see the Appendix for a full discussion.)

64 C. Measurement techniques

The tunneling experiment requires the measurement of ultra-low currents down to the few femto-amperes 15 (10− A) level, at low voltage bias excitations as low as 1µV. To achieve the conditions to enable such measurements several key features and safeguards must be incorporated into the measurement circuitry and low temperature dilution refrigerator cryostat. Several similar but complementary ways to perform a high sensitivity I-V tunneling measurement are utilized. The most straight forward is a DC measurement. A floating voltage source is used for supplying the DC-bias excitation voltage across the tunnel junction, while a DC current meter with a high-gain pre-amplifier is used to measure the tunnel current. The entire circuitry is grounded at a single point, typically at the input low of the current amplifier. This arrangement avoids ground loops and the associated noise currents, as well as undesirable shunting of currents through unwanted paths. Typical noise floor of the order of 30-50fA is achievable with an integration time constant of 10-30sec per point. To improve beyond the DC noise floor and achieve a few fA sensitivity, it is necessary to utilize an AC lock-in technique. There are two basic methods: (i) a small AC sinusoidal excitation is superimposed on top of a DC bias to yield a measure of the differential conductance, dI/dV, and (ii) a symmetric, square- wave excitation about zero bias voltage is applied to generate a square-wave output current. This is a viable method when the I-V relationship is odd-symmetrical (anti-symmetrical) under reversal of the bias voltage (V V). For our experiments on tunneling into the fractional quantum Hall edge, this anti-symmetry requirement→ − turns out to be satisfied at low excitations, typical for V 5meV. | bias|≤

FIG. 16. AC measurement circuit for both square-wave excitation at zero DC bias, and for dI/dV measurements with a ﬁnite DC bias plus a small AC sinusoidal excitation superimposed on top. The isolation of this measurement circuit from external electrical circuitry is achieved by the use of an isolation transformer, battery DC supply, and a single ground point applied at the input to the lock-in ampliﬁer in order to avoid any ground loop problem.

Our AC lock-in measurements are performed at 2.3Hz frequency. A typical circuitry excitation is shown in Fig. 16. As in the DC case, it is desirable to float the voltage source and ground at a single point. The floating is achieved with an isolation transformer. The current is fed into the negative input of the operational amplifier which performs the current to voltage conversion (inverting amplifier). Typical feedback resistances 15 is in the MegaΩ to GigaΩ range for current values in the µA to fA (10− A) range. This type of circuitry is relatively insensitive to parasitic capacitance to ground, since one end of the tunnel junction is driven by the source which ideally would have low source impedance, and the other end is at the negative input of the op-amp and is therefore at virtual ground, being driven by the feed back loop. On the other hand, parasitic capacitance between the leads connected to these two sides of the tunneling junction will end up shunting the current, by-passing the tunnel junction. This leads must be brought out of the cryogenic system to room temperature and are therefore 2m in length and will have capacitances in the 100pF range. As a result, at 2.3 Hz, the in phase and out-out-phase∼ components of the current response typically become equal around a tunnel resistance of 0.3GΩ. By properly setting the phase of the lock-in it is possible to

65 extend the range and measure up of 1GΩ tunnel resistance with reliability. A more complete solution would be to use coaxial cables for each lead and twisting the coaxes together. The outer ground shield of the coax will provide shielding to eliminate the mutual capacitance between leads, while by twisting the coaxes, inductive pick up will be minimized. The inner and outer conductors must be separately thermally anchored at some low temperature point, at which the shielding will be broken. A further consideration is mechanical vibrations. Microphonics must be reduced by proper vibrational isolation. One final and important feature is line filtering. This is absolutely necessary in order to both achieve the lowest electron temperature and to prevent extraneous noise voltage signals from reaching the device thereby overwhelming the low voltage bias down at the µV level. Such noise can either arise from pick-up or from room-temperature Johnson noise radiated down the lines. Aside from the basic low current, low excitation techniques described above, one additional feature of the measurement warrants discussion. The range of voltage bias typically spans up to 5 orders of magnitude, from 1µV to 100mV. The very first measurements which were carried out ( Chang et al., 1996, 1998) were performed using a set of discrete points. This turned out to be inconvenient. Subsequently, a continuous sweep was employed (Grayson et al., 1998; Chang et al., 2001). To span 5 orders of magnitude, an exponential ramp is necessary. Furthermore a square-wave at 2.3Hz must pass through the isolation transformer without distortion. A non-ideal transformer, coupled with a wave form generator of finite (non-zero) source impedance, typically has reduced response at low frequencies. To compensate for the 30% drop off in voltage at the tail end of each square-wave step contained in a half cycle, it was necessary to compensate by adding a linear ramp of the exponentiated output. Such a circuit is depicted in Fig. 17. Very recently, a new generation of commercially ultra low-noise current preamplifiers with noise figures of 30aA/ (Hz) at 0.1Hz have become available. With such advanced instrumentation, current measurements with sub fA resolution will likely be achievable, further expanding the dynamic range of the tunneling current-voltagep measurements.

FIG. 17. A continuous sweep, exponential ampliﬁer ramp via a linear Vramp input. The exponential output is used to bias the tunnel junction in a sample. The output spans several decades. To compensate for the poor frequency response of the isolation transformer circuitry (Fig. 16) at the 2.3 Hz lock-in frequency, an integrated signal of the exponentiated output is added on to achieve ﬂatness during each 1/2 cycle of the square-wave to better than 5%.

D. Sample preparation

A typical device used in measurement is of physical dimensions 1mm x 3mm x 0.15mm (width x length x thickness) where the length of the Al0.1Ga0.9As tunnel barrier spans the sample width of 1mm. Because of this long barrier, in oﬀ-resonance tunneling processes electrons are most likely injected and removed at many point-like contacts along the 1mm length. Moreover, the tunneling events at diﬀerent points are

66 believed to be largely incoherent. This is to be contrasted to the case of resonant-tunneling discussed in Sections II D 6 and III E 6. To make ohmic contact to the 2DEG in the (100) quantum well and to the 3-d bulk n+ GaAs on the cleaved (011) edge, indium metal is used in two separate steps. These two separate steps are necessary to avoid unwanted shorting of the tunnel barrier due to uncontrolled indium diffusion. The contact arrangements are shown in Fig. 18. In the first step, indium is diffused into the quantum well at various contact positions located away (> 0.5mm) from the tunnel junction. This enables ohmic contacts to be made to the 2DEG while avoiding degradation of the tunnel barrier due to the penetration of residual amounts of indium. These 2DEG contacts are utilized for in-situ measurements of the transport coefficients, the longitudinal (Rxx) and Hall resistances (Rxy), enabling the characterization of the electron density and mobility. The second step involves contacting the 3d n+ GaAs layer on the cleaved edge. To accomplish this indium is applied to the n+ GaAs using a soldering iron at 200oC. Note the indium melting point o ∼ 18 3 is 156 C. Typically, the n+ GaAs is sufficiently highly doped (0.5 2.2x10 cm− ) that an ohmic contact is routinely made. This ohmic behavior of the n+ GaAs contact persists− down to mK temperatures. It has turned out absolutely necessary that the indium on the n+ GaAs not be annealed at temperatures exceeding 350oC to avoid accidental diffusion into the tunnel barrier. Even trace amounts of indium will short out the∼ power law tunneling behavior! With careful same preparation, high quality electron tunneling data are readily obtainable.

FIG. 18. Sample with indium contacts. The dark contacts are first annealed to enable diffusion into the GaAs quantum well where the 2DEG resides. Subsequently, the light colored contacts were cold-soldered onto the n+ GaAs metal at 180oC to prevent any diffusion and shorting of the tunnel barrier.

E. Data presentation

1. Establishment of Chiral Luttinger Behavior in Electron Tunneling into Fractional Quantum Hall Edges

We begin the data presentation with the first clear evidence of power law characteristics for the electron tunnel into the edge of a fractional quantum Hall effect. Making use of the cleaved-edge overgrowth devices we studied the tunneling conductance (G(T)), current-voltage (I-V) relationship, as well as the differential conductance (dI/dV), for electron tunneling between the bulk-doped n+ GaAs metal and the edge of various 2.7 .06 incompressible fractional quantum Hall fluids. For tunneling into the ν =1/3 edge we found that I V ± ∝

67 1.75 .08 and G T ± , where the two dependences yielded nearly identical values for the exponent, α of 2.7. In contrast,∝ tunneling into a ν = 1 edge was essentially linear in the I V while G was temperature independent.≈ These results strongly indicated that the 1/3 fractional edge behaves− like a chiral Luttinger liquid, while the ν = 1 edge behaves as a one-dimensional Fermi liquid. In Fig. 19(a), we show the longitudinal resistance (Rxx) and Hall resistance (Rxy), and in (b) the tunneling conductance (Gtun), for Sample 1.1 versus magnetic field at a temperature of 50mK. The ν =1/3 fractional quantum Hall effect occurred at 13.4T. Gtun exhibited an abrupt drop above a magnetic field of 9.5T. This reduction arises from two contributions. The first comes from the chiral Luttinger liquid nature of the edge states as will be shown in subsequent figures. The second comes from the tunneling matrix element across the Al0.1Ga0.9As barrier. In the Landau gauge where the vector potential A=-Byˆx, for a perfect, infinitely long barrier the matrix element is separable into a product of three components associated with the vertical (B) direction, z, and the directions along the barrier, x, and normal to the barrier, y. The z matrix element couples the 3d n+ GaAs electrons into the 2d quantum well (x-y) plane and is insensitive to B. At low and intermediate B, the y matrix element is dominated by the 9nm thick barrier and is also

FIG. 19. Magnetic ﬁeld traces of (a) Longitudinal resistance (Rxx) and Hall resistance (Rxy), (b) Tunneling conductance (Gtun), at low bias for Sample 1.1. The temperature is 50mK. [From Chang et al., 1996.]

insensitive. The x component, however, can exhibit substantial B dependence. The x-eigenfunction, eikx, is indexed by the momentum k which is proportional to the y center of coordinate, y ; i.e. k y l2. o ∼ o o Tunneling through the barrier involves a displacement in y, of ∆yo 9nm, accompanied by a change in k 2 ∼ of ∆k ∆yolo B. Since the barrier potential is translationally invariant in x it cannot couple states of different∼ k; the∝ x matrix vanishes and tunneling is forbidden. However, the presence of imperfections and disorder in a real sample as well as a finite extent breaks the x translational invariance and tunneling becomes possible. Nevertheless, ∆k is proportional to B. At higher magnetic fields, a larger momentum change is required. The x matrix element which measures the x-Fourier transform of the total potential, V (∆k,y,z), is expected to fall off with B ∆k. Clearly, for the case of ionized impurities or interface roughness mediating the tunneling process,∝ this fall off does occur. Moreover, at high B the tunneling matrix may become limited by the tunneling barrier. The tunneling probability, Ptunn, drops exponentially with B due 1/2 (x /l )2 B/B to its dependence on the magnetic length, l = (¯h/eB) : P e− o B e− o . For example, for B tunn ∼ ∼

68 a b=9nm Al0.1Ga0.9As barrier, we measure a crossover to this exponential behavior at B=7.5-9.0 T or lB = 8.6-9.4nm for three different samples, roughly corresponding to lB b. This relation appeared to hold to within 10% for barriers up to 12.5nm thick. ∼ To demonstrate the tunneling-density-of-states contribution to the reduction of the tunnel current, in Fig. 20 we show the I V characteristics in a log-log plot for Samples 1.1 and 2 at the filling factor ν =1/3. The respective− magnetic fields were 13.4T and 10.8T, and the temperature was 25mK. At voltages below 12µV , the tunneling was thermally dominated (kT/e = 2.15µV ) and I V exhibited a linear relationship.∼ The respective tunneling resistances were 100MΩ and 300MΩ. Above− a cross-over voltage of 6kT/e 12µV , the I V followed a nonlinear power law given by I V α where α = 2.7 .06 and ∼2.65 .06,∼ respectively. The− power law persisted over 1 decade in V and 2.7∝ decades in I beyond± which I was observed± to fall below the power law. This power law behavior was predicted by the chiral Luttinger theory due to Wen (1992, 1995), Kane and Fisher (Kane and Fisher, 1992b; Kane and Fisher, 1995) Moon et al. (1993), and Fendley et al. (1995a, 1995b) and arises from the power law tunneling density of states. However, the observed exponent was smaller than the theoretical prediction of exactly 3. The discrepancy

FIG. 20. Current-voltage (I V ) characteristics for tunneling from the bulk-doped n+ GaAs into the edge of a ν =1/3 fractional quantum− Hall eﬀect for Sample 1.1 in a log-log plot at B = 13.4T (crosses), and for Sample 2 at B = 10.8T (solid circles). The solid curves represent ﬁts to the theoretical universal form of Eq. (289) for α =2.7 and 2.65, respectively. Note same as Fig. 1. [From Chang et al., 1996.]

69 FIG. 21. The data in Fig. 20 for Sample 1.1 plotted in different functional forms–(a) log I versus V, and (b) log I versus 1/V. In (b), the solid dots represent the same data plotted with the x-axis expanded by a factor of 4. The lack of any straight portion demonstrates the poorness of these functional forms as fits to ′ V/Vo V /V the data. The corresponding functional forms are I e− and I e− o . ∝ ∝ will be addressed in detail in the ensuing sections. Beyond the prediction of a simple power law, the data could 2 be fitted to the Kane and Fisher universal scaling form which holds in the limit Gtun << GHall = e /3h. For tunneling into the 1/3 effect from a normal metal, it is approximately given by (Kane and Fisher, 1992b, 1992c):

I T α[x + xα] (336) ∝ where eV x ; α =3. (337) ≡ 2πkT Note that this expression is an approximation to Eq. (289) since for α between 2 and 3, the extra factor Γ[(α +1/2)]2 1, and actually falls between the values 0.78 and 1. Because our exponent was different from 3, we substituted≈ α by 2.7 and 2.65, respectively and plotted the results as the solid curves. The two fitting parameter were the exponent, α, and the proportionality constant between the I and V . The fits appear to be excellent. In other words, the predicted 1/2π scale factor between the bias voltage and the temperature was born out by experiment. The power law region exceeded 3 decades in current and 1.4 decades in bias. The large dynamic range enabled us to rule out other competing functional forms, such as exp[ V/V ], or − o exp[ vo/V], which gave substantially poorer fits as shown in Fig. 21. Next we demonstrate the cross-over voltage− from linear to power law behavior scales with temperature by plotting the I-V characteristics between 26mK and 840mK in Fig. 22(a) and the collapsed curves normalized to the 26mK curve in Fig. 22(b). In Fig. 22(b), data points beyond the break-off voltage of 1mV were removed for clarity; all other data points fell on a universal curve as required. ∼

70 FIG. 22. (a) Log-log plot of I-V characteristics for Sample 1.1 at ν = 1/3 at six diﬀerent temperatures. (b) Collapsed curves for the data in (a) where Icoll(V ′) = I(V )[G(To)/G(T )]V >0 , To = 26mK and − V ′ = VT/To. [From Chang et al., 1996.]

In Fig. 23(a) we plotted the diﬀerential conductance, dI/dV , measured independently. The agreement with theory is also quite good. Here, the solid curves represent the dI/dV of the corresponding theoretical curves in Fig. 20. In Fig. 23(b), we plot Gtun for Samples 1.1 and 2 at low voltage bias versus temperature in a log-log plot. Power law behavior was again observed with an exponent of 1.75 .08 and 1.5 0.08, respectively, although the data exhibited a slight meander about ideal behavior. The± dynamic range± was roughly 1 decade in temperature and 1.7 (1.5) in Gtun. These exponents yielded values for α close to those obtained from I V and dI/dV as required by theory. As a check, we attempted an Arrhenius plot of log − Gtun versus 1/T as well as various variable-range-hopping functional forms instead which yielded a large curvature (Fig. 24), clearly indicating that neither a simple activated process over an energy barrier nor standard variable range hopping is appropriate.

71 FIG. 23. (a) Log-log plot of the diﬀerential tunneling conductance, dI/dV , for Samples 1.1(crosses) and 2 (solid circles) at ν = 1/3, at a temperature of 25mK. The solid curves represent the theoretical dI/dV obtained from Fig. 20. (b) Log-log plot of the temperature dependence at low voltage bias for Samples 1.1 (upper curve) and 2 (lower curve) at ν = 1/3. The respective voltage biases are 4.97µV and 2.64µV . The solid straight lines represent power laws with the respective exponents, α 1, of 1.75 and 1.5. [From Chang et al., 1996.] −

FIG. 24. The data in Fig. 23(b) replotted in an Ahrenius plot, (a), and in variable hopping forms of 1/2 ′ 1/3 (To/T) 1/2 (TO/T) 1/3 G e− (logG/Go (To/T) ) and G e− (logG/Go′ (To′ /T) ), (b). The clear absence∝ of a straight portion∝ − in any of the three∝ plots demonstrates that∝ these − functional forms do not adequately describe the data. The dashed line in (b) is included for the purpose of comparison. As the variable range hopping exponent becomes smaller, from 1 to 1/2 and 1/3, the ﬁt improves. This is to be expected since a power law which corresponds to the functional dependence, logG/Go logT/To, is obtained in the limit this exponent approaches zero. ∝

In Fig. 25, we plotted the I V characteristic for the ν = 1 effect for Samples 1.2 and 3.1, and for the ν = 2/3 effect for Samples 3.2− and 4. In direct contrast to tunneling into the ν = 1/3 effect, the I V characteristics were nearly linear for the ν = 1 case, and were slightly non-linear for ν = 2/3, for voltage− bias beyond the temperature dominated regime. The corresponding values of α were 1.2 and 1.14 for ν = 1, and 1.2 and 1.42 for ν = 2/3. Note that for tunneling into the ν = 1 edge, the experiment indicated that the edge behaves as a chiral Fermi liquid. This is a rare example of a strongly interacting 1-d system with Fermi liquid rather than Luttinger liquid behavior! In the case of tunneling into the 2/3 edge, the power law exponent is non-universal, ranging from 1.2 α 1.42 where I V α. The ν =2/3 result will be discussed in the context of edge tunneling at general filling≤ ≤ factors. ∝

72 FIG. 25. (a) Log-log plot of the I V characteristics for tunneling into the ν = 1 quantum Hall edge for Samples 1.2 at B =4.8T (crosses) and− 3.1 at B =8.5T (solid circles). The temperature is 24mK. (b) Log-log plot of I V for tunneling into the ν = 2/3 edge for Samples 3.2 at 12.0T (crosses) and 4 at 11.1T (solid circles) at− a temperature of 25mK. A slightly non-linear behavior is observed above 12µV voltage bias. The respective exponents, α, are 1.2 and 1.42. [From Chang et al., 1996.] ∼

2. An apparent g=1/2 chiral Luttinger liquid at the edge of the ν = 1/2 compressible composite Fermion liquid

The evidence in the previous section and its overall reasonable agreement with theory substantially elu- cidated the physics of the edge of incompressible fractional quantum Hall fluids. On the other hand, the physical properties at the edge of a compressible fluid were yet to be explored, for instance the edge of the ν =1/2 composite-Fermion quantum Hall liquid and its tunneling properties. A composite Fermion, initially proposed by Jain, is composed of a real electron (Fermion) and an integral number of flux tubes (Jain, 1989a; Halperin et al., 1993; Willett et al., 1990, 1993; Kang et al., 1993; Du et al., 1993, 1994). Due to these attached fluxes, the tunneling of electrons into the bulk could be substantially different from into the edge. For bulk tunneling, the difficulty in bringing in extra fluxes to attach to the added electrons gives rise to a pseudo gap in the tunneling density of states and an exponential suppression of tunneling current with bias (He et al., 1993; Eisenstein et al., 1992). Tunneling into the edge, however, was often surmised to be linear in its I-V characteristics since extra flux lines can readily enter from the boundary. In the 2-d bulk region of the incompressible ν = 1/3 fractional quantum Hall fluid, the excitation spectrum contains a gap above the ground state and there are no zero energy excitations. The edge of the fluid and its low energy excitations decoupled from the bulk at low temperatures and the edge can thus be rigorously treated as a one-dimensional system. On the other hand, the edge of a compressible fractional quantum Hall fluid such as the composite Fermion fluid at filling factor, ν = 1/2, is more complicated. The absence of a bulk excitation gap in a compressible fluid enables the edge dynamics to couple to the bulk excitations and a Luttinger liquid description may not be appropriate. It was far from clear that the tunneling of electrons into a composite Fermi system would necessarily entail a suppression of the tunneling density of states at low energies. Here we show that a for electron tunneling into the edge of a ν =1/2 FQH fluid, non-linear I-V characteristics as well as a temperature dependent low bias tunneling conductance, G, can be obtained. In fact for three different samples, the I-V exhibited a power law behavior reminiscent of a chiral Luttinger liquid, with an exponent, α of 1.80 0.05, 2.10 0.10, and 1.83 0.05, respectively. The exponent deduced from G(T) from the first sample yielded± α =1.±77 .07 which is± consistent with the value of 1.80 deduced from the I-V ±

73 curve. In essence, α was roughly given by α =1/g=1/ν = 2. These results indicated that the edge of the compressible ν =1/2 fluid behaves almost like a one dimensional chiral Luttinger liquid, and provided the first compelling evidence that Luttinger liquid behavior can exist at the edge of a compressible fractional Hall fluid! Our observation of a clear non-linearity in the case of the gapless ν =1/2 composite Fermion fluid fueled intense theoretical efforts to understand the edge properties of compressible fluids at general fillings and led to further systematic experimental studies described below.

FIG. 26. (a) The Cleaved-Edge Overgrowth sample geometry. (b) Magnetic ﬁeld traces of the 2 terminal 2DEG resistance (R2pt) at 50mK for Sample 1.3. [From Chang et al., 1998.]

FIG. 27. Current-voltage (I V ) characteristics for tunneling from the bulk-doped n+ GaAs into the edge of a ν = 1/2 composite Fermion− liquid for Sample 1.3 in a log-log plot at B = 9.28T . The solid curve represent a ﬁt to Eq. (289) for α =1.80. [From Chang et al., 1998.]

74 As before, in Fig. 26 we first present the two terminal resistance, R2pt, for Sample 1.3 versus magnetic field at 50mK temperature. The ν =1/2 composite Fermion state occurred at 9.3T. The high quality of the 2DEG is reflected both in the high electron mobility of 3.2x106cm2/Vs (Table I) and the presence of the well developed 1/3, 2/3, 2/5 quantized Hall plateaus as well as additional plateau structures at 4/3 and 5/3. Fig. 27 shows our main result of a non-linear tunneling characteristic for Sample 1.3 at 26mK. Below 10µV bias, the tunneling was thermally dominated and consequently linear in the I-V, yielding a conductance∼ of 0.023µS corresponding to a tunneling resistance of 43MΩ. Above 15µV, a power law behavior with and exponent 1.80 was observed up to an excitation of 5.6mV. This large∼ excitation exceeded the voltage scale set by the∼ Fermi energy of 3.9mV. We again fitted the data to the Kane/Fisher functional form appropriate for a chiral Luttinger liquid with a dimensionless conductance, g, of 1/α(e2/h) (Eq. (289)) (Kane and Fisher, 1992c; Chamon and Fradkin, 1997): α +1 I = γT α([Γ( )]2x + xα) (338) 2 where x = eV/2πkT, Γ is the Gamma function, and γ is a proportionality constant. A best fit was achieved with α =1.80 and is shown as the solid curve in Fig. 27. In Fig. 28(a) we show the temperature evolution of the I-V curve. Rescaling the voltage by V(To/T) and the current by I(V)[G(To)/G(T)]V 0, data at all temperatures can be collapsed onto a universal curve as depicted in Fig. 28(b). →

FIG. 28. (a) Log-log plot of I-V characteristics for Sample 1.3 at ν = 1/2 at six diﬀerent temperatures. (b) Collapsed curves for the data in (a) where Icoll(V ′) = I(V )[G(To)/G(T )]V >0 , To = 26mK and − V ′ = VT/To. [From Chang et al., 1998.]

In Fig. 29 we plotted the temperature dependence of the small bias linear conductance, G, in a log-log plot. 0.77 0.07 A power law with G T ± was observable between 26mK and 900mK. The exponent of α 1=0.77 is consistent with the∝ value of α =1.80 0.05 deduced from the I-V curve in Fig. 27, and satisﬁes− the requirement of Eq. (342), provided Eq. (342)± is relevant and appropriate to tunneling into the ν =1/2 edge. Two additional samples of diﬀering electron mobility or tunneling barrier thickness exhibited similar behavior in I-V. We plotted the results in Fig. 30. Sample 1.4 possessed the same high quality 2DEG as Sample 1.3, but contained a thicker tunnel barrier, 24.5nm versus 12.5nm for Sample 1.3, and was grown several months earlier. For this sample the exponent, α, was 2.10 0.10. Sample 5.1 contained a 2DEG of substantially lower mobility, 0.5x106cm2/Vs versus 3.2x106cm2/Vs± and a tunnel barrier of 16nm. Here α =1.83 0.05, close to the value for the other samples. The average for all three is 1.91. The insensitivity to electron± mobility rendered our main conclusion of a Luttinger-liquid-like behavior extremely robust.

75 FIG. 29. Log-log plot of the temperature dependence at low voltage bias for Samples 1.3 at ν =1/2. The solid straight line represents a power law with an exponent, α 1, of 0.77. [From Chang et al., 1998.] −

FIG. 30. Log-log plot of the I V characteristics for tunneling into the ν =1/2 quantum Hall edge for Samples 1.4 at B = 10.5T (crosses)− and 5.1 at B = 10.2T (solid circles). The temperature is 28mK. The solid lines represent power laws with the exponents, α, of 2.10 (Sample 1.4) and 1.83 (Sample 5.1), respectively. [From Chang et al., 1998.]

76 3. A continuum of chiral Luttinger liquid behavior

The observation of power-law electron tunneling characteristics at ν =1/2 led us naturally to inquire about the behavior at general filling factors, ν. In fact it turned out that power-law tunneling I-V is a generic feature for the fractional quantum Hall fluid edge. To demonstrate this, we performed a systematic characterization of the power law behavior over a continuum of fractional filling factors, spanning both compressible and incompressible liquids. For the best traces, power law behavior with dynamic range exceeding 4 1/2 decades in current and 1 1/2 decades in voltage was observed. Two major results emerged: (1) there is a continuum of power law I-V behavior, and (2) the I-V exponent is approximately given by 1/ν, with the edge appearing to behave as a single mode Luttinger liquid with reduced conductance parameter g ν. Our results came as a major surprise, first because the observation of Luttinger liquid behavior at all∼ fillings was not fully anticipated (Chang et al., 1998) as incompressibility of the bulk fluid (gapped behavior) was considered crucial to the existence of a Luttinger liquid; and second because the power law exponent exhibited the lack of a clear-cut plateau features, in direct contrast to theoretical analyses based on the inter-mixing of co- versus counter-propagating edge modes (Wen, 1992; Kane and Fisher, 1995; Shytov et al., 1998). The tunneling exponent, α, was extracted from the I-V data utilizing the theory of Chamon and Fradkin (1993) for a single mode CLL with g = ν which models the wide tunnel junction as a sequence of incoherent, point-like tunnel junctions, while treating the 3-d metal as a chiral Fermi liquid. Although the justification for a single mode CLL at arbitrary ν is still lacking, this model was successful in fitting our data. Ideally, an I-V curve consists of 3 regimes: a) a low voltage bias regime with a linear I-V where the thermal energy, kT, dominates over the voltage bias energy, eV, (eV 2πkT ), b) an intermediate voltage bias regime (2πkT eV T ) exhibiting the important power law≤ I-V behavior, and c) a high bias saturation regime ≤ ≤ S (eV >kTS) in which I-V approaches linearity again and where the tunneling conductance saturates to the 2 terminal conductance of the 2DEG as the tunnel barrier becomes transparent. Here TS represents a crossover temperature with kTS the cross-over energy above which saturation takes place. Since T is determined by experimental conditions, and ν from the Hall measurement, the only adjustable parameters were α and 2πT TS (β = α 1, r = ): − TS

2 1 rβ e  e− 2  I = ν 1 α dV (339) β β h  − V  Z rT β β  S 2 r   2 α+1 1 e− +1   " Γ ( 2 ) − #      Eq. (339) (Eq. (314)) is expected to be appropriate for a single mode Luttinger liquid with reduced conductance g = 1/α. At B = 11.0 T (ν =1/3), it fitted the data with remarkable precision (Fig. 31 top, dotted line). For comparison we also plotted the series resistance model used to guide our intuition (Fig. 31 top, dashed line) and noted that the knee of the crossover region at high bias was far too soft. Next we examined the series of log-log I-V curves over the whole range of B field for Sample 2 in Fig. 31 bottom. At the higher B fields (13.0 T) we observed a power law up to 6 decades in current, whereas at the lowest field (7.0 T) the curve was approximately linear over the entire range. At lower B < 10.0 T (corresponding to high ν > 2/5, the fit of the Chamon-Fradkin theory to each trace was still good, and we are able to extract α and TS. Nonetheless the fit was not as exact as indicated by the larger error bars in Fig. 32. Performing similar I-V measurements on the three Samples 1.1, 5.2, and 1.2 (Table I), we summarize the full result of the exponent, α, versus 1/ν in Fig. 32. Samples 1.1 and 5.2 yielded sufficient decades of power law to fit to the Chamon-Fradkin theory, whereas Sample 1.2 exhibited a strong power law only at the highest magnetic fields, settling to the weak power law of about 1.1 over the 1/ν range of 1 to 1.4. Error bars for representative data points are provided at various fillings. For Samples 2 and 1.1 above 1/ν > 2.8, and Sample 5.2 above 1/ν > 2.4 the error is negligible.

77 FIG. 31. a) (Top) Log-log I-V for Sample 2 at 11.0 T, ν = 1/3. Theory of Chamon and Fradkin (1997), Eq. (339), (dotted line) and simple series resistance model (dashed line) are overlaid for comparison. b) (Bottom) Log-log I-V for Sample 2 at diﬀerent values of B from 7.0 T to 15.0 T in 0.5 T steps. [From Grayson et al., 1998.]

Based on our results we make the following observations. First, the plot shows a remarkable continuum of power law exponent values spanning the entire range 1 <α< 4. This was the first experimental evidence that the characteristic CLL coupling constant, g, may in fact assume a whole continuum of values. Second, the trend in α versus 1/ν is linear for 1/ν > 1.4 with α 1.16/ν 0.58. This linear behavior appeared to roughly characterize all four samples studied regardless≃ of electron− mobility, carrier density and tunneling barrier thickness. It is in striking contrast to theoretical expectations that α would reflect the bulk transport and therefore exhibit plateaus whenever the Hall conductance is quantized. Finally, for 1/ν < 1.4, the exponent saturated at a lower limit, α=1.1, indicating an approach to Fermi liquid behavior. For certain samples, e.g. 1.1 and 5.2, hints of a possible plateau feature in the exponent near 1/ν =1/3 can be seen in Fig. 32. However, the limited range in 1/ν (or B) precludes a definitive conclusion. This important issue is

78 addressed in the next section.

FIG. 32. Power law exponent α vs. 1/ν, the reciprocal of the ﬁlling factor, for four samples. The data from Chang et al. (1996) are included for reference. (Inset) TS vs. 1/ν for three samples whose traces spanned high excitations. [From Grayson et al., 1998.]

4. Plateau behavior in the chiral Luttinger liquid exponent

The basic observation that the power law behavior is not restricted to incompressible quantum Hall fluids and is in fact present for general filling factors with the exponent, α, varying in a continuous manner roughly as 1/ν for 1/ν > 1.3 (Chang et al., 1996; Grayson et al., 1998) presented a puzzle. While the hierarchical picture of incompressible fluids is able to produce power law behavior at a set of rational filling fractions (Kane and Fisher, 1995), and the Shytov, Levitov, Halperin theory (1998) based on the composite Fermion effective field theory predicts power laws at continuous values of inverse filling, 1/ν, (more precisely Hall resistivity, ρxy,) the predicted step-like plateau features in α stand in contrast to the featureless linear behavior of the experiment. These theories are based directly on our understanding of the relation between the edge mode structure and the topological characterization of fractional Hall states in effective field theories, as well as the inter-mixing of co- and counter-propagating edge modes into charged and neutral varieties. Two major issues arose. Firstly, the absence of plateaus for 1/ν > 1.3 was difficult to reconcile with the theoretical expectation even without accounting for the finite widths of quantized Hall plateaus. Secondly, the lack of a plateau near bulk filling, ν =1/3 (1/ν = 3) despite the appearance of a plateau in ρxy indicated that edge tunneling characteristic is not solely dictated by the bulk Hall resistivity, again in contradiction to expectation. This absence of structure was even invoked by some workers as evidence that the existence of Luttinger liquid behavior was not conclusive (Bockrath et al., 1999; Egger, 1999; Altland et al., 1999). Because the ν =1/3 fractional quantum Hall fluid possesses the largest gap and is robust, evidence for plateauing in the exponent was of critical importance. In this section we report clear observation of a plateau feature for the α versus

79 1/ν dependence with an α value close to 3. Our results were obtained based on careful analysis of I-V tunneling data with precise ﬁtting to the Chamon and Fradkin (1997) expression, followed by a statistical F-test for the χ2 of the α versus 1/ν ﬁts. However, the 1/ν position where this plateau occurs can shift depending on sample.

10−6

Sample 1.1 10−8

B=12T 10−10 I (amp) 10−12 B=19T T=195mK 10−14

10−6 10−4 10−2 100 102 104 V (Volt)

FIG. 33. Log-log plot of the I-V characteristics (solid lines) for electron tunneling from the fractional quantum Hall edge into the bulk doped n+ GaAs in Sample 1.1 at various magnetic fields from 12T to 19T in steps of 0.5T, 18T and 18.5T excluded. Corresponding filling factors vary from 2.69 to 4.26. Dashed lines represent best fits to the Chamon-Fradkin expression, Eq. (339). Successive curves are displaced by 0.3 units (a factor of 2) in the horizontal-direction for clarity. [From Chang et al., 2001.]

In Figs. 33- 35 we present log-log plots of the tunneling I-V characteristics (solid curves) for Samples 1.1, 1.3, and 6, respectively, in a wide range of magnetic fields/filling factors in order to deduce the behavior of the power law exponent, α, as a function of 1/ν. Successive curves are shifted in the positive direction on the horizontal axis by 0.3 units (a factor of 2) for clarity and curves for which sufficient dynamic range is available to yield a meaningful exponent are included. The dashed curves represent best fits to data as described below. Essentially all traces for Samples 1.1 and 6 exhibited the expected behavior with a low-bias linear region, an intermediate power-law region, and a large-bias saturation region. Note that the thermally dominated low-voltage-bias, linear regime was visible only when it was above the noise floor ( 4fA). Aside from the traces for which the saturation region is not included, the only significant exceptions∼ occurred for Sample 6 in a few traces at the largest 1/ν values where there is evidence of an additional parallel channel 8 8 for tunneling when the conductance falls below 0.5x10− S (R exceeds 2x10 Ω). In contrast Sample 1.3 did not show an ideal saturation regime. This nonideality∼ could arise from the opaqueness of the thicker tunnel barrier of 12.5nm. To establish the presence of a plateau feature in the exponent, α, we again extracted α in a systematic way by fitting the entire I-V range containing the three bias regimes to the Chamon-Fradkin expression for the tunnel current, in conjunction with the added constraint that Va = V + IRs where Va is the voltage applied on the device across contacts and Rs a 2DEG series resistance. Here, the addition of the extra parameter, Rs, significantly improved the fits away from ν =1/3. Since the temperature was known, 3 parameters were

80 needed: α, TS, and a 2DEG series resistance Rs. The additional parameter Rs was needed to properly fit the saturation regime since at small filling factors (large 1/ν) the background from the longitudinal resistance of the 2DEG (ρxx) can be substantial (or order 100kΩ). This parameter is justifiable since often the edge and bulk densities can be different as discussed in the Appendix on Edge Density Profile.

10−6 Sample 1.3 10−8 B=7.24T

10−10

I (amp) B=12.9T 10−12 T=25mK

10−14

10−6 10−4 10−2 100 102 104 V (Volt)

FIG. 34. Log-log plot of the I-V characteristics (solid lines) for edge tunneling in Sample 1.3 at B=7.24, 7.5, 7.76, 8.02, 8.28, 8.53, 8.79, 9.31, 9.83, 10.34, 10.86, 11.38, 11.9, 12.4, and 12.9T, with 1.611 < 1/ν < 3. Dashed lines represent best ﬁts. Note a displacement of successive traces in the horizontal direction for clarity. [From Chang et al., 2001.]

For Samples 1.1 and 6, a full fit over the entire range of I-V could be accomplished. The fitted I-V curves were plotted as dashed lines in Figs. 33 and 35. As evidenced by the near complete overlap between the fit and the data in a majority of the traces, excellent fits were achieved. For Sample 1.3, the absence of a true saturation regime precluded a full fit and the fitting was only performed within the low and intermediate bias regimes containing the linear and power law dependences. The affect on α was found to be minimal (< 0.1) in traces which exhibit a sizable power law dynamic range occurring for 1/ν > 2. For 1/ν 2 where α< 2, nonlinearity arising from bias-induced reshaping of the tunnel barrier distorted the fits, leading≤ to systematic errors in α estimated to be less than 0.15. However, these 1/ν 2 data points turned out not to affect our conclusions. In Sample 6 data from the largest 1/ν where 1/ν >≤ 4.9 needs explanation. In these traces the intermediate bias regime contained two sub-regions of differing exponents with α1 characterizing the lower portion crossing over to α2 which characterizes the higher portion. The relevant exponent was α2 for the following reasons. First of all, for traces with lower 1/ν, within noise α was always either monotonically increasing or nearly flat with increasing 1/ν and this trend is consistent with theoretical expectations. 8 Secondly the lower, α1, sub-region occurred at low conductances (below 0.5x10− S) where residual parallel tunneling channels can have an effect. In fact at 1/ν =4.446 corresponding to B=11.2T excess conductance was already visible at bias voltages 10µV V 100µV (note the displacement a factor of log(27) for clarity), suggesting the presence of a parallel≤ channel.≤ In the traces in question, the low conductance regime was reached at higher values of voltage bias due to the large value of the exponent, α2, and one may expect the effect of parallel channels to be more pronounced. Despite residual parallel conduction, the vast majority of traces were fully fitted by Eq. (339) with the additional series resistance fitting parameter, Rs.

81 10−6 Sample 6 10−8 B=8.55T

10−10 B=12.8T

I (amp) T=25mK 10−12

10−14

10−6 10−4 10−2 100 102 104 V (Volt)

FIG. 35. Log-log plot of the I-V characteristics (solid lines) for edge tunneling in Sample 6 at B=8.55, 8.75, 9.15, 9.58, 10., 10.4, 10.8, 11.2, 11.6, 12.0, 12.2, 12.4, 12.8T, with 3.605 < 1/ν < 5.082. Dashed lines represent best ﬁts. Note a displacement of successive traces in the horizontal direction for clarity. [From Chang et al., 2001.]

Fig. 36 summarizes the fitting parameters α, TS, and Rs deduced for the two sets of samples versus 1/ν. Results for Samples 1.1 and 1.3 are presented together since they contain the identical 2DEG. We focus our attention on α in panels (a) and (c). To establish unequivocally the presence of a plateau we first fitted our data to curves containing a) three line segments where the middle exhibits a reduced slope, b) two line segments, and c) a single straight lines, indexed by 3, 2, and 1, respectively. We next applied the statistical 2 2 2 F-test on the random variable Fj,i(n, m) = [(χi χj )/χj ]/[(dfi dfj)/dfj] (see for example Daniel and Wood, 2 − − 1980) to compare the χ ’s, where (i,j)=1,2,3. Here dfi = N pi represents the degree of freedom of the i-th fit with N being the number of data points and p the number− of fit parameters, while n = df df and i i − j m = dfj. When F exceeds the criterion for an integrated probability of 99% in the F(n,m) distribution with n and m degrees of freedom, the j-th fit is established while the i-th fit was rejected with confidence exceeding 99%. Following this procedure, we were able to conclusively demonstrate that for the first sample set (S1.1 and 1.3) the functional form with 3 line segments joined continuously (6 parameters) was appropriate while rejecting a simple linear fit (2 parameters) as shown in Fig. 36(a) as well as rejecting a two line (4 parameter) fit. For the second data set (S6), both the 3 segment and 2 segment fits were viable while the single line fit was rejected (Fig. 36(c)). In both sets the plateau region was found to occur at α 2.7 with corresponding reduced slopes of 0.15 0.15 and 0.14 0.18. In fact the respective F values for∼ the two sets were 31.1 ± − ± 2 2 compared to the 99% confidence level criterion of F(4,45)=3.79 (N=51, χ3 = 0.47, χ1 = 1.77) and 17.9 2 2 2 compared to the average [F(4,15)+F(2,17)]/2=5.5 (N=21, χ3 = 0.45, χ2 = .66, χ1 = 2.26). The result for the first set was robust even when the lowest seven points where 1/ν 2 were removed! In terms of 1/ν the plateau region occurred at 2.76 < 1/ν < 3.33 and 4.12 < 1/ν <≤4.76, respectively. These positions were shifted to higher values compared to theoretical prediction of 2 < 1/ν < 3.3 (Kane and Fisher, 1995; Shytov et al., 1998) taking into account the finite Hall plateau width. In the first sample set, the position corresponded well with the bulk ν =1/3 quantum Hall plateau. On the other hand, in the second set it was shifted substantially beyond the position of the ν =1/3 Hall plateau. At present it is not fully understood how this shift occurs, although one possibility arises from edge-reconstruction due to density gradients (Lee and Wen, 1998). Nevertheless, given the quality of the vast majority of the I-V curve fits to the Chamon-Fradkin

82 expression (Eq. (339)) and the robust results of the F-test on the α versus 1/ν ﬁts, we conclusive established the presence of a plateau feature.

0 1 2 3 4 5 0 1 2 3 4 5 6

) 100 100 Ω ( 80 (b) 11.1+ 1.3 (d) 6 80 S R 60 S 60 TS 40 RS 40 S

T (K), R T 20 S 20 0 0 4 (a) 11.1+ 1.3 (c) 6 4 3 3

α Theory Average 2 2 + 3− 2 3−seg fit seg fit 1 line fit 1 line fit 0 0 0 1 2 3 4 5 0 1 2 3 4 5 6 1/ν

FIG. 36. The chiral Luttinger liquid exponent, α, for Sample 1.1 and 1.3, versus 1/ν, in (a), for Sample 6 in (c). Representative error bars are as shown and solid curves are as labeled. The parameters TS and Rs versus 1/ν for Samples 1.1 and 1.3, panel (b), for Sample 6, panel (d). [From Chang et al., 2001.]

5. Discussion: Is the chiral Luttinger liquid exponent universal?

The cumulative evidence from electron tunneling measurements presented in the previous sections indicate that the chiral Luttinger liquid power-law exponent, α, for tunneling into the fractional quantum Hall edge deviates substantially from the universal behavior predicted by theory. Let us summarize, again, the outstanding experimental findings to date. The results which we have established are: (1) a power law behavior for electron tunneling into the fractional quantum Hall edge observed in the I-V characteristics at all filling factors between ν =1 to 5, indicative of chiral Luttinger behavior for the fractional Hall edge, (2) the power law exponent, α, defined∼ by I Vα, when plotted versus 1/ν exhibits a plateau for exponent values near α 2.7 for the highest quality samples,∝ and otherwise behaves roughly as 1/ν, and (3) in some samples the 1/ν≈ position where the plateauing in α occurs can be shifted to higher 1/ν, where ν refers the filling deduced from the bulk 2DEG carrier density, when compared to the expected position center about 1/ν = 3 corresponding to the ν =1/3 fractional quantum Hall effect. These results stand in stark contrast to the predictions of the standard theoretical picture of the chiral Luttinger liquid based on effective theories of the edge for a short-ranged interaction between electrons described in the Section II D. The key differences are as follows: (1) the exponent for tunneling into the ν =1/3 edge is always observed to be less than 3. This is a first indication of non-universal behavior. The systematic error for the quoted plateau value in α of 2.7 is unlikely to exceed 0.15, while the uncertainty based on random noise is smaller, of order 0.05. The≈ measured exponent therefore falls below the value

83 of exactly 3 as required by the topological nature of the exponent. Note that the long range Coulomb interaction can lead to a logarithmic correction to the power law dependence yielding an effective, energy dependent exponent which exceeds 3 (Zülicke and MacDonald, 1997; Imura and Nagaosa, 1997; Zheng and 2 Yu, 1997; Levitov et al., 2001), (2) accounting for the quantized Hall plateau width at ρxy/(h/e ) = 3, and 2 the difference between edge and bulk densities (and hence filling factors), the α versus ρxy/(h/e ) plot (as 2 opposed to versus 1/ν) behaves roughly as 1/[ρxy/(h/e )], or as 1/ν in the absence of a plateau feature in α, in contrast to the step-like dependence shown in Figs. 10. In any event, even generously allowing for experimental error in the determination of both the edge filling factor, νedge, and in α, α shows no evidence of reaching the predicted value of 3 (Shytov et al., 1998; Levitov et al., 2001) at 1/ν = 2 (ν =1/2), and (3) the edge tunneling exponent, α, in not directly tied to the bulk filling factor as evidenced by the large shifts in 1/ν position in some samples (Chang et al., 2001; Hilke et al., 2001). First we need to address the issue of the difference in the edge versus bulk 2DEG density as discussed at length in the Appendix. Due to the chemical potential imbalance between the 2DEG and the 3d n+ doped GaAs, charge transfer can occur across the tunnel barrier, leading to inhomogeneous density profile near the tunneling edge. It is therefore not surprising to have a different edge density than the bulk. Whereas the tunneling experiment probes a spatial region within a few magnetic lengths, lo, of the boundary and is therefore sensitive to the edge density which may differ from the bulk 2DEG density, the Hall resistance (conductance), on the other hand, directly reflects the bulk density and should be insensitive to the edge density so long as the sample boundary is sufficiently long to allow for a full equilibrium of the edge modes. These edge modes may include, in addition to the boundary modes at the outermost edge of the sample, those which exist in the transition region between the edge and bulk density regions. In the absence of a direct method to independently determine the edge density, we propose the following method to estimate the edge density which we argue should be accurate to 5-10%. Since by all reasonable analysis and sensible argument the tunneling exponent must remain nearly constant when the Hall resistance 2 ρxy is quantized at 3(h/e ) (note ρxx << ρxy always), and at the same time α is expected to be able to 2 exceed 3 only when ρxy/(h/e ) exceeds 3, we can determine the 1/νedge = 3 (νedge = 1/3) position by the 1/νbulk value where α first exceeds 3, or more accurately exceeds the experimental plateau value of 2.7. This position value must be reduced by roughly 5% equaling one-half of a typical 1/3 Hall-plateau-width. In samples which do not exhibit a plateau in α, the corresponding 1/νedge = 3 position is simply given by the 1/νbulk value where α = 2.7. For example in Samples 1.1 and 2, this yields an edge density roughly equal to 1.05 the bulk density. Based on this type of estimate, it would appear that the plateau feature in α is more likely ascribable to the finite width of the Hall plateau rather than to a step of the type predicted by the standard theories. In any event, the exponent at νedge = 1/2 is highly unlikely to reach the value, 3, as noted above. Furthermore, even if there might be legitimate concern that the edge density profile can be highly inhomogeneous, i.e. not constant on the length scale of a few magnetic lengths, the prediction that the exponent must remain unchanged within a rather large range from 1/νedge = 2 to 3 (more precisely 2 2 ρxy/(h/e ) 3) means that even if some type of weighted averaging over density need be employed, the exponent≤ will most≤ likely still take on the topological value of 3, albeit over a reduced range! At this point it is necessary to examine all possibilities or non-ideality which could lead to the discrepancy between experiment and theory. Two major issues come to mind–(i) long range nature of the Coulomb interaction, and (ii) a non-constant density profile near the tunneling edge leading to edge reconstruction. Based on the effective field theories, long range Coulomb interaction will lead to a log(V) correction in the power law relation, with an increase in the effective exponent at low energies (Zülicke and MacDonald, 1996, 1997; Wen, 1992; Shytov et al., 1998; Levitov et al., 2001). However, interestingly enough no evidence of this type is observable in the tunneling data despite the large dynamic range in the I-V, as evidenced by a nearly perfect straight line in the power law region of the data. Furthermore, the observation of α 1/ν, in one interpretation, is an indication that only the charged mode is observable in tunneling, while the∼ neutral modes (see Section II D 3) become effectively decoupled from the tunneling process. On the other hand, based on the large dynamic range spanned by the data, a log(V) increase in the propagation velocity of the charged mode relative to the neutral modes appears to insufficient to separate out the respective energy scales and lead to the apparent absence of a contribution from the neutral modes (Lee and Wen, 1998). Edge reconstruction, however, can lead to the formation of extra edge modes which can renormalize the exponent when counter-propagating varieties are present. A number of works have attempted to explain the discrepancy between the experiment and the ”standard”

84 theory. The follwoing approaches have been explored: (i) finding sensible mechanisms to ensure that only the charged mode (see Section II D 3) contributes to the tunneling exponent, yielding α 1/ν, while the neutral modes are undetectable. These mechanisms include a separation of energy scales≈ for the different types of modes (Lee and Wen, 1998, (Zülicke and MacDonald, 1996, 1997, 1999; Zülicke et al., 1998), and topological constructions (Lopez and Fradkin, 1999); (ii) the introduction of extra edge modes which results from edge reconstruction (Lee Wen, 1998; Chamon and Wen, 1994) or smooth disorder which can produce local pockets of differing fillings (Pruisken et al., 1999; Skoric and Pruisken, 1999); and (iii) coupling to additional impurity-levels located nearby the chiral edge (Alekseev et al., 2000); (iv) continuum elasticity theory (Conti and Vignale, 1996, 1997, 1998; Han 1997, Han and Thouless, 1997a, 1997b), and (v) various other scenarios and possibilities ( Khveshchenko, 1999, 2000; Imura, 1999; Yu et al., 1997; Yu, 2000; Yang et al., 2000). Thus far, no clear consensus has emerged as to what the correct picture should be.

ρ 2.6 ρ L 2.5 0 0.1 0.2 0.3 1/N

FIG. 37. The ratio of the angular momentum occupation number, ρ(m), for N interacting electrons on the disk. This ratio measures the power-law decay exponent of the equal-time correlation function, which is believed to be identical to the equal-space exponent measured in experiment. Shown are the Laughlin state ρL for a short-range interaction, and the exact ground state for the Coulomb interaction ρC. [From Goldman and Tsiper (2001).]

In view of the lack of a coherent picture it is necessary to seriously consider scenarios which go beyond mere extensions of the standard analysis. The key pieces of physics which may be missing from the idealized model of the FQH fluid edge include extra edge modes from edge reconstruction, and the renormalization of exponent due to long range potential V(r), or due even to any type of interaction potential which differ from the idealized δ”(r) potential. Several recent and independent numerical, finite-systems studies are beginning to shed evidence indicating that, indeed, a renormalization of the edge tunneling exponent is possible. These include exact numerical diagonalization in a disk geometry (Goldman and Tsiper, 2001) as well as calculations based on the mixing of composite-Fermion Landau-level wavefunctions at the sample edge (Mandal and Jain, 2001). It was found that for a 3-d Coulomb interaction the exponent ν =1/3 is no longer universal and instead takes on a value in the 2.5-2.75 range. In the exact diagonalization calculation by Goldman and Tsiper (2001), the exponent value for up to 12 particles is shown in Fig. 37. Extrapolating to an infinite system yielded a value of α which falls between 2.58 and 2.75. In a separate and related work, Mandal and Jain (2001) studied a system of up to 40 particles using composite Fermion wavefunctions and found a similar renormalization of the exponent at ν =1/3. In a separate work investigating the effect of the edge confinement potential, evidence

85 for reconstruction of the edge leading to extra counter-propagating modes and a possible renormalization of the exponent was also uncovered (Wan et al., 2002). In all such numerical calculations, it was found that extra oscillations in the density occur near the edge which are absent in a Laughlin-type edge wavefunction. Although the largest systems studied is in the few tens of electrons and cannot be taken as rigorous proof for the thermodynamic limit, it is nevertheless clear that in all situations where the interacting potential or edge confinement favors the Laughlin-type edge function, the universal exponent value of 3 is recovered for the ν =1/3 fractional Hall edge. Deviation from this universal value arises only when the Laughlin-type edge wavefunction is no longer exact and is modified. These new developments suggest that the edge dynamics in the fractional quantum Hall regime may be more complex than previously thought and potentially will lead to further discoveries of more novel and interesting physics. Taken as a whole, our experimental results in conjunction with these recent numerical work suggest that the existing standard analyses based on effective 2-d Chern-Simon field theories (Girvin and MacDonald, 1987; Read, 1989; Zhang et al., 1989; Wen and Niu, 1990; Wen, 1995; Frohlich and Zee, 1991) deserve careful reexamination when applied to the dynamics at the Hall fluid edge. These results are raising questions regarding our fundamental understanding of the connection between the edge dynamics and the topological characterization of the bulk fluids, even though the basic Hallmark feature of the Luttinger liquid, i.e. power- law tunneling, is unequivocally established. Despite the fact that finite-size calculations cannot be taken as definitive proofs, the combined experimental and computational evidence should stimulate a reexamination of the detailed properties of the rich and novel physics at the edge of the fractional quantum Hall fluids. Very recently, Mandal and Jain (2002) extended their work on the mixing of composite Fermion Landau levels to the ν = 2/5 and 3/7 fractional Hall fluid edge. They found that in the absence of composite Fermion Landau level interaction, the exponent is quantized at exactly 3 in agreement with the standard theories (Sections II D). However, as soon as interaction is introduced, the exponent is renormalized downward below 3. In Fig. 38 we show their results plotted alongside experiment data taken from Section III E. The good quantitative agreement is striking. They went on to point out that one important possibility for the appearance of non-universality– it may represent a signature that in an effective 1-d theory of the edge, the electron operator (see Section IID1) may become a non-local in character. If proven correct, this is a major and significant new development in our understanding of the chiral Luttinger liquid and in 1-d Tomonaga-Luttinger liquids in general!

4.5 4.0 3.5 3.0 α 2.5

2.0 interacting CFs 1.5 non−interacting CFs , ,, experiment 1.0 2.0 2.5 3.0 3.5 4.0 1/ν

86 FIG. 38. Tomonaga-Luttinger exponent, α, for the fractional quantum Hall edge fluid as a function of the inverse filling factor, 1/ν. The filled circles (filled triangles) show theoretical values for interacting (non-interacting) composite Fermions at ν = 1/3, 2/5, and3/7. The experimental results (empty symbols) are taken from the following sources: square from Chang et al. (1996) (Sample 2); circles and triangles from Grayson et al. (1998) (Samples 1.1 and 2 in this review); inverted triangles from Chang et al. (2001) (Samples 1.1 and 1.3 in this review). [From Mandal and Jain (2002).]

6. Resonant tunneling into a biased fractional quantum Hall edge

Up to this point we have reviewed the off-resonance electron-tunneling characteristics in great detail. It was mentioned in Section II D 6 that CLL manifestation should be observable in resonance tunneling as well. In this section we discuss the first study of resonant tunneling into a voltage biased fractional quantum Hall effect (FQHE) edge, possible only with the atomically sharp tunneling barriers unique to cleaved- edge overgrown devices. In the resonances observed, we identify different tunnel coupling strengths to the metallic lead and to the FQHE edge. The introduction of the energy scale associated with the application of an external voltage bias, other than the energy scale dictated by temperature alone can reveal dramatic, non-Fermi liquid signatures. In particular, weak coupling to the FQHE edge produces clear non-Fermi liquid behavior with a six-fold increase in resonance area under bias resulting from the power-law density of states at the sharp FQHE edge. Such a striking non-conservation of resonance area has never before been observed. A simple device model combined with the resonant tunneling formalism for chiral Luttinger liquid theory successfully describes all the features in the data. The discovery and characterization of the resonance in this Section represents a new direction for understanding the physics of the fractional quantum Hall edge. This new type of tunneling resonance occurs between a bulk doped metal and a fractional quantum Hall edge (Kane, 1998) as we sweep the magnetic field, and is likely due to an accidentally introduced impurity state in the barrier. As in the case of off-resonance tunneling the sharp and energetically tall (100 meV) barrier permits us to apply a substantial DC bias across the tunnel junction and characterize the response of the resonance under bias.

FIG. 39. Conductance resonance for Samples 2, 1.1A, and 1.1B (S2, S1.1A, and S1.1B, respectively). Inset, right: 1.1A plotted with expanded scale against a derivative Fermi function. Device inset, left. [From Grayson et al., 2001.]

87 The samples studied are from the same set of cleaved-edge overgrown samples described above for off- resonance edge tunneling measurements (Grayson et al., 2001; Chang et al., 1996; Grayson et al., 1998; Chang et al., 2001). In particular Samples 1.1 and 2 were studied. Please see Table I for sample parameters. The zero-bias differential conductance, dI/dV at V=0, was obtained by applying an 8 µV AC square wave to the sample and measuring the resulting AC current with a lock-in amplifier while sweeping the magnetic field, B (Fig. 39). The AC voltage amplitude was chosen to be of the same order as the thermally limited resolution at base temperature, 25 mK (VAC 2πkB T ) (Kane, 1998). For the samples examined, the inferred edge density differs from the bulk values only∼ slightly, of the order of 5-10% (please see Sections IIIE5), as before all filling factors will refer to the bulk values, νbulk. Sample 1.1 was studied in two cooldowns labeled respectively as 1.1A and 1.1B, and different resonances are identified by appending an additional suffix (1.1Ba, 1.1Bb, etc.).

FIG. 40. Temperature dependence of the resonance 1.1A. [From Grayson et al., 2001.]

Sample 2 in Fig. 39 exhibited a resonance at νbulk =0.294, and Sample 1.1A showed the strongest resonance at νbulk=0.338. For the second cooldown 1.1B, four resonances were obtained at νbulk = 0.346, 0.333, 0.307, and 0.303 corresponding to 1.1Ba, 1.1Bb, 1.1Bc, and 1.1Bd, respectively. Other samples measured showed no detectable resonances. The temperature evolution of the resonance 1.1A was also measured as shown in (Fig. 40). With increasing T the resonance linewidth broadens and the peak height decreases relative to a rising background conductance. These trends are plotted in the inset of Fig. 40. To measure bias dependence, a fixed DC bias was added to the AC square wave and differential conductance was measured as a function of B sweep (Fig. 41). We employ the sign convention where the 2DEG electrode resided at ground while the signed voltage was applied to the n+ electrode. With bias the background conductance increases due to the power law density of states. Starting with 1.1A (Fig. 41 right), the peak splits under bias into two peaks of different height with a separation in B proportional to the applied voltage. The total area subtended by the two peaks above the background conductance increases only slightly. The bias dependence of resonance peak 1.1Ba (Fig. 41, left), instead of splitting, this peak broadens into a lopsided single peak that leans to the right for positive bias and to the left for negative bias. Most notably, the area subtended by the resonance relative to the background conductance increases by a factor of 6 under the application of a 30 µV DC bias, an altogether different qualitative behavior than 1.1A. ∼ The understanding of this unique set of data was achieved by first relating the magnetic field to an energy scale. Interestingly enough it can be shown (Fig. 39, inset), that the lineshape for 1.1A can be scaled to 1 2 fit the derivative of a Fermi function, f ′(Er) = sech (E/2kBT ), the exact lineshape expected for a − 4kB T

88 resonance fed by a Fermi liquid lead. Dividing the calculated thermal linewidth, ∆E =4kBT arccosh(√2) by the linewidth of the observed resonance ∆B, we deduce from thermal measurements a scaling× coefficient, βT : dE ∆E β = = =0.42 meV/T. (340) | T | |dB | ∆B Alternatively, since the B-field separation between peaks in Fig. 41 is linear with the applied bias, we can empirically assign from voltage bias measurements a coefficient βV of:

FIG. 41. dI/dV vs. B at fixed DC bias for resonances 1.1Ba and 1.1A: data (thin lines), and fit (heavy lines). [From Grayson et al., 2001.] dE eV β = = bias = 0.43meV/T (341) | V | |dB | B B peak2 − peak1 The clear agreement of these coefficients indicates that the energy of the resonant state, Er, is being swept dEr through the chemical potential in proportion to the magnetic field by a single coefficient β = dB , with the resonant state producing a conductance peak whenever it passes the chemical potential in either lead. We next turn to the temperature dependent measurements of 1.1A in Fig. 40. The resonance broadens roughly linearly in T (inset Fig. 40) and the peak height drops relative to the background as 1/T consistent with thermal broadening of a Fermi liquid-resonance. The background conductance, however, behaves as a power law in temperature (Fig. 40, inset) with the same Luttinger liquid-like exponent reported previously ( Chang et al., 1996; Grayson et al., 1998; Chang et al., 1998; Chang et al., 2001), implying we have a FL-like resonance on top of a LL-like background. To understand this puzzling behavior requires comparing the relative coupling strength of the resonance + to the two leads with the notation of ΓFL as the tunnel coupling strength from the resonant state to the n lead, and ΓQH to the quantum Hall edge. Since the strongest peak is a factor of 100 smaller than the perfect resonant conductance, e2/3h, overall the resonance is in the weakly coupled limit . The tunnel conductance

89 will be determined entirely by the weakly-coupled lead. We begin by discussing the limiting case where the strongly coupled lead combines with the resonance to act as a delta function at the resonance energy Er. Under bias V, a current Ir results from the weakly coupling Γ to the density of states in the remaining lead E/kT 1 [with f(E) = (e + 1)− ]: e dn I (E )=Γ (E ) [1 f(E )]f(E eV ) r r h dE r × − r r − n f(E )[1 f(E eV )] (342) − r − r − o The real space position of the resonance in the tunnel junction causes the resonance energy, Er, to depend on bias. Assuming the resonance is bound to the local band structure inside the barrier with energy E0, and deﬁning the lever arm parameter λ as the fraction of the applied bias that falls to the weakly coupled side of the resonance yields:

Er(V,B)= λeV + βB + E0 (343)

Eq. (342) and (343) and diﬀerentiation with respect to V yields the diﬀerential conductance:

2 2 dIr e d n dn (E )=Γ λ (E ) λ (E )f ′(E ) dV r h dE2 r · · · − dE r r h dn (1 λ) (E )f ′(E eV ) (344) − − dE r r − i where the expression in braces is the same as that in Eq. (342). · · · In the Fermi-liquid coupled (FLC) limit, ΓFL << ΓQH, the Fermi-liquid density of states enters. Neglecting d2n dn the first term in Eq. (344) relative to the others since for a FL dE2 << dE f ′(E) at low temperatures and dn approximating the density of states by dE = n0, the second term in Eq. (344) produces a peak weighted by λ when the resonance is aligned with the n+ chemical potential, while the third term produces a peak weighted by (1 λ) when the resonance is aligned with the chemical potential of the FQHE edge. These behaviors are qualitatively− similar to the data in Fig. 41. In the contrasting quantum Hall coupled (QHC) limit ΓQH << ΓFL, the power law density of states enters. Here the resonance peak from the second term in dn α 1 Eq. (344) is completely suppressed because the density of states dE (Er) Er=0 = Er − = 0, vanishes when the resonance is aligned with the FQHE chemical potential. We expect| only a single peak from the third + term when Er is aligned with the n chemical potential. The first term in Eq. (344), now contributes a d2n conductance proportional to the derivative of the density of states dE2 between the two resonance centers. dI α 2 For α = 3 at a 1/3 FQHE edge, this additional conductance goes as dV Er − Er, a contribution that rises linearly with B as it approaches the single resonance peak, seen clearly∼ in F∼ig. 41, left. (See also Fig. 12.) Beyond differences in lineshape, the most striking qualitative difference between the QHC and FLC limits is the (non)conservation of resonance area under bias. In the FLC limit, the total integral of the resonance dI e2 area is a constant, dV (Er)dEr = Γ h n0, independent of bias. In the contrary QHC limit the power law dn α 1 density of states dE E − leads to a simple power law dependence of the resonance area on voltage bias R∼ dI α 1 (at zero temperature): dV (Er)dEr V − . At finite temperature, Fig. 41, left, shows that this strong bias dependence remains, with a factor∼ of 6 increase in resonance area at 30 µV. Together, the asymmetric lineshape and non-conservationR of resonance∼ area under bias convincingly demonstrate the presence of a never before observed non-Fermi liquid resonance. More quantitatively a reasonable fit of the data was achieved by adopting the resonance formalism of Chamon and Wen (1993) for sequential tunneling between biased chiral Luttinger liquids (Section II D 6), in conjunction with the lever arm model. Specializing to the situation at hand where tunneling takes place from a highly doped n+GaAs metal into the edge of the ν =1/3 fractional Hall fluid, the Luttinger parameter of one channel needed to be changed to g=1 to represent a Fermi liquid while retaining the other at g=1/3. The resulting fitting parameters are shown in Table III. Here, the fit was not sufficiently precise to distinguish between a tunneling exponent of α = 3 or 2.7, however. The device specific lever arm parameter, λ′, denotes

90 the fraction of voltage bias on the n+ side of the resonant state. Up to a scaling factor for the Γ’s, our model has a unique solution for a given sign of β since the three independent quantities β, λ′, and ΓQH/ΓFL are parametrically determined by fitting three structural features: the left and right peak positions, and the ratio of peak heights. We include a background conductance that has no fit parameters since it obeys the well known power law behavior with voltage bias and exponential decay with B ( Chang et al., 1996, 1998, 2001; Grayson et al., 1998). The simulated resonance curves are plotted with heavy lines against the data in Fig. 41, and the four fit parameters are listed in Table III along with the increase in resonance area with bias for both data and fit. Using the coupling constants and area conservation as an indicator of the coupling limit, we see that resonances 1.1A and 1.1Bb are near the FLC limit with ΓFL < ΓQH, and a modest 50% and 70% increase in area with bias, respectively. On the other hand, 1.1Ba approaches the QHC limit with ΓQH ΓFL and the previously noted factor of 6 increase in area with bias. For all resonances, the model calculation∼ fits the data with a unique solution. The sign of β is uniquely determined for 1.1Ba because of the lopsided single peak structure, and for 1.1A and 1.1Bb, only the negative β solution can account for the observed increase in area. It was also possible to gain an understanding of the values of β as well as the observed sign change between 2D neighboring resonances 1.1Ba and 1.1Bb by considering how the 2D ground energy E0 (B) affects the lever 1 eB arm model. In the compressible state increasing B increases the 2D ground energy, E0(B)= 2 ¯h m∗ , thereby lowering the 2D conduction band and proportionately lowering the resonance energy by the lever arm factor, dEr 1 he¯ λ′: βcomp = dB = λ′ 2 m∗ . The resulting βcomp = 0.26meV/T exactly matches the fit parameter for 1.1Ba suggesting that− this resonance couples to a compressi− ble edge. On the other hand in the incompressible state, the ground energy drops by the mobility gap ∆ over the width of the plateau B¯, changing the sign of β and setting a lower bound β +∆/B¯. Using measured values of ∆ = 450µeV and B¯=2T, we get incomp ≥ βincomp 0.22 meV/T, consistent with the values for 1.1A and 1.1Bb, and suggesting that these resonances couple to≥ incompressible edges. With the negative β (1.1Ba) and and positive β (1.1Bb) resonances having the same ΓFL, it is even possible that these neighboring resonances arise from the same resonant state which is first lowered and then raised through the chemical potential by leveraged 2D ground energy oscillations. The differences in λ′ and ΓQH for these resonances are to be expected from differences in the screening length and tunneling character of compressible and incompressible edges. If the resonance pair 1.1Ba/1.1Bb results from the νedge = 1/3 transition and 1.1Bc/1.1Bd from νedge = 2/5, this is consistent with an edge density enhancement of ν 1.05 ν . edge ∼ bulk

7. Other Manifestations of Chiral Luttinger Liquid Characteristics in the Edge Properties of Fractional Quantum Hall Fluids

Aside from a determination of the power-law tunneling density of states discussed in detail in the above Sections, there are other consequences of a chiral Luttinger liquid description of the edge dynamics. Particu- larly intriguing is the possibility of a direct measurement of the quasi-particle fractional charge predicted for the Laughlin incompressible states (Laughlin, 1983). There are two approaches–(i) transport through anti- dots in which the edge states encircling an antidot can mediate back-scattering between counter-propagating edge modes on opposing boundaries, at opposite sides of a device, and, (ii) a direct measurement of the quantum shot noise in the regime of quasi-particle tunneling between counter-propagating edge modes at opposing boundaries when the boundaries are brought into close contact a. Transport through anti-Dots– the Aharonov-Bohm effect and fractional charges Simmons et al. (1989) first attempted to determine the fractional charge of the Laughlin quasi-particles in the ν = 1/3 fractional quantum Hall fluid by studying the Aharonov-Bohm, quantum interference effect in narrow wires. They observed a suggestive tripling of the magnetic field period in the Aharonov-Bohm (AB) oscillations when going from the ν = 1 integer Hall effect to the ν = 1/3 fractional Hall effect. Such AB oscillation are a result of the presence of edge states encircling an accidental impurity site mediating the tunneling between edges. Although suggestive, it was argued however that the period tripling is more likely associated with a difference in energy scales, rather than a direct manifestation of the quasi-particle charge

(Lee, 1990; Thouless and Gefen, 1991). Subsequently, Goldman and Su (1995) reported an extremely interesting determination by introducing a submicron size antidot (potential hill) between counterpropagating fractional Hall edges to mediate tunneling, implemented in a conventional planar geometry. By combining the Aharonov- Bohm period in the tunneling conductance and the backgate voltage period which modulates the carrier density, and relating the area of the antidot determined in the two measurements, the quasi-particle charge for the ν = 1/3, 2/5, and higher order states could be determined. An alternative interpretation of this type of experiment has been advanced by Franklin et al. (1996), however. b. Shot-noise characteristics and fractional charges

L C R o /Hz) 2 A

-28 preamp 6 QPC (10 i

3 T=57 mK Current Noise, S S (G=0)=1.1 10 -28A2/Hz i 0

0 100 200 Conductance (µmho)

FIG. 42. The total current noise inferred to the input of the preamplifier as a function of the input conductance at equilibrium (circles). The measured noise is a sum of thermal noise and the (conductance-independent) noise of the amplifier (intercept of the vertical axis at zero conductance). Inset, the quantum point contact (QPC) embedded in the 2DEG is shown to be connected to an LCR tank circuit at the input of a cryogenic preamplifier. [From de-Picciotto et al. (1997).]

More directly relevant to the chiral Luttinger liquid description of the fractional Hall edges, two groups have independently succeeded in accomplishing the technically impressive feat of tunneling-current shot- noise measurements in the fractional Hall regime discussed in theory Section II D 7 of the theory section. By alternatively using a tank circuitry at 4MHz coupled to a low noise GaAs preamp operated at low temperatures (Fig. 42) or by using noise correlation techniques, De Picciotto et al. (1997) and Saminadayar et 14 al. (Samninadayar et al., 1997) independently measured the ”low” frequency noise down to the 10− A/√Hz level and below. The resultant tunneling current noise curves for tunneling between two counter-propagating ν = 1/3 edges are shown in Figs. 43 - 45. In the fractional Hall regime where -e/3 charged quasi-particles are expected to dominate the tunneling between counter-propagating the ν = 1/3 edge modes, the noise characteristics as a function of backscattered current could no longer be accounted for by the conventional shot-noise expression involving charge e carriers. Instead, the data were successfully ﬁtted by extending the conventional expression via a replacement of the electron charge e by the fractional charge, e∗ (De Picciotto et al., 1997; Reznikov et al., 1999; Samninadayar et al., 1997; Fendley et al., 1995c):

e∗V S =2g t(1 t) (e∗coth 2kT +4kTg t, (345) I o − 2kT − o where g t(1 t)V = I t I at low temperatures. Note that to be more precise, the second and third terms o − B ≈ B

92 in the above expression are sometimes expressed in terms of dIB/dV which accounts for the non-linearity of the tunneling characteristics of chiral Luttinger liquids (Samninadayar et al., 1997). This expression closely approximate the exact expression computed numerically from the Bethe-ansatz solution (Fendley et al., 1995c). The data shown in the Figs. 43 - 45 indicate that remarkably, the quantum shot noise follow the expressions with e∗ =e/3 rather than e! This is a striking and signiﬁcant result. More recently, additional measurements on other fractional ﬂuids yielding the e/5 fractional charge has also been reported (Reznikov et al., 1999; Comforti et al., 2002).

FIG. 43. Tunneling noise at ν = 1/3 as a function of the backscattering current, IB. The slopes for e/3 quasiparticles (dashed line) and electrons (dotted line) are shown. The temperature is 25mK. Inset: data showing electron tunneling but in the integer quantum Hall regime at ν = 4. The data follow the expected slope for charge e. [From Saminadayar et al. (1997).]

These experiments were carried out in the conventional planar geometry with top gates to define the edges and tunneling point-contact. Therefore associated with these experiments the smooth edge issue mentioned in Section III A must be carefully accounted for. Fortunately in these noise measurements, the voltage excitation were often low, < 0.5mV. As a result the distortion of the broad and smooth edge potential defined by electrostatic top-gating is likely small. Nevertheless, several complications can arise. For instance, the tunneling I-V characteristics often do not fully display the expected power law exponent (Glattli, 2002). Another is the often observed and rather substantial shift in the filling factor at which charge e shot-noise signatures can be recovered, often taking place not at ν = 1 but rather significantly higher fillings factors, ν. as well as the observability of anomalous charge values (non-standard fractions) in various regimes.

FIG. 44. Crossover from Johnson-Nyquist to shot noise. The arrow indicates the data for which e∗Vds = 2kBT. A comparison with Eq. (345) (solid curve) and a similar expression for electrons (dotted curve) is shown. [Saminadayar et al. (1997).]

7 e e/3 /Hz)

2 6 A

-29 t=0.82 5 (10 i

Current Noise, S Noise, Current t=0.73

0 200 400 Backscattered Current, I (pA) B

94 FIG. 45. Quantum shot noise as function of the backscatter current, IB, in the fractional quantum Hall regime at ν =1/3 for two diﬀerent transmission coeﬃcients through the quantum point contact (circles and squares). The solid lines correspond to Eq. (345) with a charge Q=e/3 and the appropriate transmission probability, t. For comparison the expected behavior of the noise for Q=e is shown by the broken line. [From de-Picciotto et al. (1997).]

IV. OTHER LUTTINGER LIQUID SYSTEMS

Aside from the edge of the fractional quantum Hall fluid, evidence for 1-dimensional physics has been observed in several other interesting systems. These systems are emerging as fertile grounds for the investigation of the unique properties associated with an interacting 1-d electron system. In this final section we present a brief summary of the most promising examples. These include ballistic 1-d wires, carbon nanotubes, 1-d atom chains, and the venerable quasi-1-d system of Bechgaard and Fabre organic salts and blue bronze conductors. We have seen that for the fractional quantum Hall edge, the chiral Luttinger liquid tunneling exponent in an idealized model is given by universal values determined by the topological characterization of the bulk, 2-d fluid, as discussed in Section II D . Moreover in the case of a single edge branch, e.g. the ν = 1/3, and 1/5 edges, the electron tunneling exponent, is related to the Hall conductance: α =1/g, where g is the dimensionless Hall conductance, g = ν. In the case of the Luttinger liquid in quantum wires, the exponent is again related to the conductance, g. However, in general it is not universal since g is interaction dependent (Eq. (117), Section II C). Moreover, depending on the tunneling geometry, whether tunneling into the end of a Luttinger liquid or into the side from a normal metal electrode, the exponent value takes the respective forms (Egger and Gogolin, 1997; Kane et al., 1997; Kane and Fisher, 1992a):

1 1 α =1+[g− 1]/n , α 1 = [g− 1]/n (346) − ch − − ch

1 1 α =1+[g + g− 2]/2n , α 1 = [g + g− 2]/2n , (347) − ch − − ch where nch is the number of conducting channels (modes), including spin. As will be seem below in the case of carbon nanotubes, both types of tunneling are realizable.

A. Ballistic single-channel wires

Tarucha et al. pioneered the study of single channel ballistic nanowires in top-gated devices implemented in the planar geometry (Tarucha et al., 1995). By using very high quality GaAs/AlxGa1 xAs crystals, they were able to observe a suppression in the quantized conductance below e2/h for a single-channel− ballistic point contact with channel length as long as 10µm. To date the best evidence for Luttinger liquid behavior in semiconductor ballistic nanowires is provided by cleaved-edge-overgrowth nanowires (Auslaender et al., 2000; Yacoby et al., 1996). When the carrier density of the single-channel cleaved-edge wire is reduced, accidental imperfections along the wire lead to the formation of double tunnel barriers. Yacoby and colleagues were able to study the resultant resonances in the tunnel conductance. Although a non-Fermi liquid line shape was not observable, the temperature dependence of the resonance line-width, Γi, exhibited a non-linear dependence on temperature expected for a Luttinger liquid as shown in Fig. 46 (Auslaender et al., 2000; Furusaki, 1998). This is an interesting and promising result. Down the road with improved dynamic range and consistency 1 between independently measured power law exponent values from the resonance line-width, αLW = [g− 1], 1 − and from 1-d wire tunneling conductance, where α 1 = 2[g− 1] for end to end tunneling between two Luttinger liquid segments, as well as the dimensionless− conductance,− g, this system has the potential for fruitful investigations of momentum-resolved Luttinger liquid properties (Altland et al., 1999; Auslaender et al., 2002).

95 FIG. 46. The intrinsic linewidth of the resonance, Γi, versus temperature (in units of gate voltage). A power law behavior is observed indicating Luttinger-liquid behavior. Dashed lines are ﬁts to the the data. [Auslaender et al. (2000).]

B. Carbon nanotubes

An extremely promising system for the investigation of the conventional, non-chiral Luttinger liquid in a different realization of ballistic nanowires is the single wall carbon nanotube (SWCNT)/multi-walled nanotube (MWCNT) system. In the SWCNT, due to the fact that the electronic wave function is spread over several atoms, the characteristic energy scale for Peierls distortion and the associated formation of an energy gap is exponentially suppressed and below accessible temperatures. Therefore the system is a nearly ideal 1-d conductor albeit with a 4-fold degeneracy due to the presence of two metallic bands and the spin degree of freedom (Kane et al., 1997; Egger and Gogolin, 1997). since typical nanotubes are a few microns in length, 2 this finite length sets a lower cut-off energy, given by the Coulomb charging energy, Ec = e /2C 2meV, below which the 0-dimensional Coulomb blockade phenomenon dominates the transport. Note that∼ here C is the nanotube capacitance to the environment. There are two relevant tunneling geometries readily achievable by clever fabrication methods and manipulation of the nanotubes on the surface of a substrate:

96 (i) tunneling into the side wall, and (ii) tunneling into the end of a nanotube. In Figs. 47 to 48 we show the data obtained for SWCNT bundles by Bockrath et al. (1999) for tunneling from a metallic, gold contact. Above the Coulomb charging energy scale, a behavior suggestive of power-law I-V, and power-law tunneling conductance was observed. (See Fig. 47.) In a log-log plot, after accounting for contact resistance, a power- law type behavior was observed in the two contact geometries of side wall and end tunneling. In addition, Yao et al. (Yao et al., 1999) have demonstrated that it is possible to achieve nanotube to nanotube tunneling in these geometries of side wall to side wall and end to end tunneling. The tunneling data exhibiting power-law signatures are shown in Figs. 49 and 50. In either case of metal to nanotube or nanotube to nanotube tunneling it is possible to deduce a consistent value of the conductance based on the expressions above for the tunneling exponent, Eqs. 346 and 347. The deduced values of g 0.22 is also consistent with theoretical estimates (Egger and Gogolin, 1997; Kane et al., 1997). These initial∼ findings bode well for future studies. Particular noteworthy is the current trend towards the growth of extremely long SWCNT–up to mm in length. These long tubes, once proven to be ballistic, should greatly expand the available low energy range for Luttinger liquid behavior by reducing the limitation set by the lower energy cut-off resulting from the finite size.

285 K 20 20 129 K 15 S)

15 µ 10 G ( 5 S) 40 K µ

10 0 G ( 16 K 0 100 200 300 T (K) 5 8 K

1.3 K

0 0.0 0.9 1.8 2.7 3.6

Vg (V)

FIG. 47. The two-terminal linear-response conductance, G, versus gate voltage, Vg, for a sidewall-contacted metallic nanotube rope at diﬀerent temperatures. The data show signiﬁcant temperature dependence for energy scales above the Coulomb charging energy, EC. Inset: average conductance as a function of temperature. [From Bockrath et al. (1999).]

0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 ααα

10 10 S) µ

1 1 10 100 10 100 T(K) T(K)

97 FIG. 48. Log-log plots of the conductance, G, versus temperature, T, for individual nanotube ropes. (a) Data for ropes that are deposited over pre-deﬁned leads (sidewall-contacted); (b) data for ropes that are contacted by evaporating the leads on top of the ropes (end-contacted). Insets show the respective geometries. The plots show both the raw data (solid line) and the data corrected for the temperature dependence of the Coulomb blockade contribution (dashed line). After correction, the dependences follow a power-law form with a diﬀerent exponent for the two geometries. [From Bockrath et al. (1999).]

101

100 S) m ( G G

10-1 Segment I Segment II Across the kink

10-2 50 100 200 300 T (K)

FIG. 49. Linear-response two-probe conductances, G, for tunneling into an individual single-walled carbon nanotube plotted against temperature, T, on a log-log scale. The data are ﬁtted (solid lines) by the power law, G(T) Tβ. The exponent, β, for the two straight segments (I and II) are roughly 0.34, corresponding to sidewall∝ tunneling. Across the kink where tunneling occurs between ends of two single-walled nanotube tube segments and an exponent, β, of 2.2 is obtained. [From Yao et al. (1999).]

98 250 a 298 K 200 150 K 200 0 50 K (nA) I I

150 -200 100 K (nA)

I I -200 -100 0 100 200 100 V (mV) 50 K

0 0 50 100 150 200 V (mV)

10-1 a

)

V 50 K d

/ -2 I 10 100 K

(d 150 K

10-3 10-1 100 101

eV / k T B

FIG. 50. Large voltage-bias transport characteristics. (a) Nonlinear I-V characteristics; (b) Scaled diﬀerent conductance plotted against the dimensionless voltage bias, eV/kBT, for tunneling across the kink (end to end tunneling). The data for diﬀerent temperatures collapse onto a single universal curve. The dashed line represents theoretical expectation corresponding to the exponent, β=2.2. [From Yao et al. (1999).]

C. 1-d chain gold atom chains

Another unique and promising 1-d system has recently been created, that of single atom wide, 1-d Au atom chains on the vicinal silicon [111] surface (9.45o mis-cut in the [1¯12]¯ direction) (Segovia et al., 1999; Bertel and Lehmann, 1988; Hill and McLean, 1997). In an intriguing work, Sergovia et al. reported angle- resolved photoemission measurements at low temperatures ( 10K) which probe the spectral density of the 1-d chains as a function of momentum. The found evidence for∼ the absence of a quasi-particle peak as shown in Fig. 51 where in all traces, the absence of a sharp quasi-particle peak is apparent in the spectral density. They attempted to further probe the spin-charge separation of spinon and holon excitations. Although two distinct, separately dispersing peaks were observed, the interpretation that these represent evidence for spin- charge separation into spinon and holon modes need further substantiation and diﬀerentiation from band

99 hybridization eﬀects with the underlying silicon substrate. This hybridization may well account for the two distinct peaks. Nevertheless, this work opens up a new avenue. Future work on suitable substrates inert to band hybridization may well lead to new and signiﬁcant advances.

θ = -12¡ e φ e 3¡

2¡

1¡

0¡ Intensity (arb. units)

-1¡

-2¡

-0.6 -0.4 -0.2 0.0 EF 0.2 Energy (eV)

FIG. 51. Angle-resolved photoemission spectra with varying surface wave vector k perpendicular to the o k chains. The polar angle θe in the emission plane was ﬁxed at -12 and the angle φe in the perpendicular plane is varied. The lack of dispersion in this direction appears to conﬁrm the one-dimensional nature of the system. [From Segovia et al. (1999).]

D. Quasi-1-dimensional conductors

The final system or set of systems we will mention arguably represents the first experimental system for which 1-d interaction physics was evident. This includes the organic salts of the TTF-TCNQ, (TMTSF)2X, where X = PF6, AsF6, etc., (Bechgaard Salts) as well as the (TMTTF)2X salts. (tetra thia fulvalene- tetra cyano quinodimethane; tetra methyl tetra selena fulvalene ) series (Basista et al., 1990; Jerome and Schulz, 1990; Voit, 1995; Solyom, 1979; Emery, 1979), the blue bronze metals K0.3MoO3 and (TaSe4)2I, NbSe3 (Dardel et al., 1991; Sing et al., 1999), and the non-charge-density-wave 1-d metal Li0.9Mo6O17 (Denlinger et al., 1999). The organic salts have chain structures which naturally lead to large anisotropy in the transport and other properties ( Emery, 1979; Jerome and Schulz, 1990; Voit, 1995). The bulk of these highly anisotropic, quasi-1-d conductors, both the organic salt and blue bronze variety, by and large undergo CDW (charge-density-wave) or SDW (spin-density-wave) transitions at sufficiently low temperatures. Some salts also undergo a superconducting transition, e.g. (TMTSF)2ClO4. An exception is the non-CDW blue bronze metal, Li0.9Mo6O17. Evidence that 1-d interaction physics is relevant comes form a variety of measurement, including transport, frequency dependent conductivity measurements from the microwave to optical and uv range, NMR, and photoemission. Evidence from all of these measurements point to strong deviations

100 from conventional 3-d behavior. See Voit (1995) for a recent comprehensive review of this exciting ﬁeld. Particularly noteworthy are evidence for optical absorption where substantial deviation from simple Drude behavior is observable (Basista et al., 1990; Dardel et al., 1991; Schwartz et al., 1998), anomalous NMR relaxation (Behnia, 1995; Bourbonnais and Jerome, 1998), and photoemission data where the quasi-particle peak is absent (Dardel et al., 1993; Zwick et al., 1997; Denlinger et al., 1999) in a manner reminiscent of the normal state in high Tc superconductor materials (Shen and Schrieﬀer, 1997; Ding et al., 1997). Some of these examples are presented below.

FIG. 52. Real part of the optical conductivity of TTF-TCNF at a temperature of 85K contrasted against the expected conventional Drude curve (dashed line). These data represent one of the ﬁrst indications that 1-d interaction physics may be relevant in this type of systems. [From Basista et al. (1991).]

This field has a long and venerable history. Perhaps the very first system in which 1-d physics was thought to be relevant is the TTF-TCNQ system. Figure 52, reproduced from a seminal work by Basista et al. (1990), shows the optical conductivity (Solid curves) which is suppressed below the Drude conductivity (dashed line). This suppression likely arises from the formation of a pseudogap due to umklapp processes in a 1-d electron gas.(Lee et al., 1973). This important finding was followed closely by the discovery of unusual photoemission spectra at temperatures just above the Peierls CDW transition in inorganic quasi-1-d conductors K0.3MoO3 and (TaSe4)2I where an absence of the Fermi liquid quasi-particle peak was observed (Dardel et al., 1991), as shown in Fig. 53. More recently, in non-CDW organic conductor, (TMTSF)2ClO4, which undergoes a SDW transition at 12K, a non-dispersive feature was found along the 1-d direction at 150K, (Zwick et al., 1997). In addition electrodynamics response measurements from microwave to uv frequencies in the (TMTSF)2X Bechgaard salts were found to be consistent with a Luttinger liquid behavior, with a power law conductivity as a function of photon energy, ω, as shown in Fig. 54 (Schwartz et al., 1998). The discovery of a strongly temperature dependent Hall coefficient down to 12K in (TMTSF)2PF6, again with a power law behavior (Fig. 55) lent further support to the relevance of 1-d physics (Bechgaard2, PRL2000).

101 FIG. 53. Angle-integrated photoemission spectra of the K0.3MoO3 and (TaSe4)2I quasi-1-d conductors measured just above the Peierls charge-density-wave transition temperature. For The absence of a sharp Fermi-edge is apparent. For comparison, the spectra os 2-d (TaSe2) and 3-d (Rh) metals are also shown. [From Dardel et al. (1991).]

102 FIG. 54. The normalized frequency-dependent conductivities for 1-d organic conductors (TMTSF)2X, X = PF6, AsF6, andClO4, in a log-log plot to demonstrate the power-law frequency dependence. The solid line shows a ﬁt of the form σ(ω) ωδ. [From Schwartz et al. (1998).] ∼ The last example is the non-CDW inorganic, quasi-1-d blue bronze conductor (Sing et al., 1999). In this system, extensive ARPES studies clearly demonstrated the absence of a quasi-particle pole in the spectral function consistent with Luttinger liquid behavior, as shown in Fig. 56 (Denlinger et al., 1999; Gweon et al., 2001, 2002).

FIG. 55. Hall constant of UCLA-Gruner group (full circles) and Riso-Bechgaard group (open circles) samples versus temperature. The dotted line indicates the Hall constant derived in a band model, and the dashed line is a Tβ power law ﬁt with β = 0.73. Inset: Magnetic ﬁeld data for the UCLA sample. [From Moser et al. (2000).]

Although many of these systems are complex, exhibiting a variety of phase as the temperature or pressure is varied, as well as diﬀerent signatures arising from Coulomb interaction, it is undoubtedly the case that conventional 3-d scenarios is not adequate in describing the essential features of the diverse observations, and 1-d physics, and particularly Tomonaga-Luttinger liquid physics is relevant in the appropriate regime. All these examples show that the unique and novel physics associated with interaction in the 1-d world is not limited to the realm of pure theoretical and mathematical discourse, and are relevant in a diverse variety of physically realization system.

103 Li 0.9 Mo 6O17 TL model T=250 K holon peak T=250 K vc/vs=2 spinon edge α=0.9

k=k F a b Intensity (arb. units) Intensity

-0.4 -0.2 0 -0.4 -0.2 0 E – (eV)

FIG. 56. High resolution angle-resolved photoemission spectroscopy (ARPES) data for the Li purple bronze taken at T=250K, with photon energy 30eV, energy resolution 49meV and angle resolution 0.36o–(a), and Tomonaga-Luttinger model simulation–(b). [From Gweon et al. (2002).]

ACKNOWLEDGMENTS

I am truly grateful to Virginia Pang-Ying Yeh Chang for her many contributions. I gratefully acknowledge the contributions of my collaborators C.C. Chi, M. Grayson, L.N. Pfeiﬀer, D.C. Tsui, K.W. West, and M.K. Wu. I would also like to thank the many colleagues with whom I have had stimulating conversations: C. Chamon, M.P.A. Fisher, E. Fradkin, S.M. Girvin, V.J. Goldman, B.I. Halperin, Song He, J.K. Jain, C.L. Kane, Yong-Baek Kim, L. Levitov, A.H. MacDonald, and X.-G. Wen. Work at Purdue supported in part by NSF Grant DMR-9801760 and DMR-0135931.

APPENDIX A: EDGE DENSITY PROFILE

This Appendix section discusses a first attempt to estimate the edge density profile based on Thomas- Fermi screening and classical electrostatics considerations. However, as a cautionary note, we remark that due to uncertainties in the sample growth process, specifically silicon dopant diffusion, it is not yet possible to obtain a reliable quantitative estimate. Nevertheless, this Section is included to illustrate some of the considerations necessary for such an estimation. As discussed in detail in Section II D, within the standard picture of idealized short-ranged interaction the edge tunneling properties are determined by the topological quantities characterizing the bulk FQH fluid. This topological property was shown to be rigorously valid for the case of the incompressible fluids in the Laughlin series and Jain series (Wen, 1992, 1995; Kane and Fisher, 1995). It was then argued that even for long ranged Coulomb type interaction, the topological properties

104 are still valid, aside from logarithmic corrections. Because of the topological nature it was further argued that the edge tunneling characteristics should be relatively insensitive to the details and imperfections at the edge. However, experiment clearly indicates otherwise. In fact, even before attempting to address the issue of non-universality in the tunneling exponent manifest in the unmistakable absence of wide plateau regions and the consistently low value of the exponent for tunneling into the ν = 1/3 edge for which α 2.7 < 3, ≈ there is the issue of the apparent edge electron density, nedge distinct from the bulk density, nbulk, obtained from standard measurements on the 2D electron gas. In fact data from several samples examined show that the deviation of the edge density can result in either an enhancement above (Chang et al., 2001) or reduction below (Hilke et al., 2001) the bulk value by as much as 60-70%. Intuitively, this kind of difference can arise from charge redistribution due to a chemical potential imbalance across the tunnel barrier and to the long- range Coulomb potential. The existence of a difference in the edge and bulk densities is not an unusual occurrence. One can even imagine an extreme case where an electrostatic gate is artificially patterned onto the edge region with a width many times the magnetic length, e.g. a 200nm wide edge strip of 20lo. By gating this ”edge” region to an entirely different density from the bulk, the edge tunneling properties∼ will reflect the resultant νedge rather than νbulk, while at the same time a standard transport measurement on a macroscopic device will still exhibit the quantum Hall effect reflecting the bulk filling, since the macroscopic distances along the boundary will be sufficiently long to enable the edge modes between different regions to establish full equilibrium. In our devices the actual edge density profile will depend on various factors, including the chemical potential imbalance between the 2DEG and the highly-doped 3-d bulk n+ GaAs, the distance of the 2DEG from the top surface (s) in the (100) crystal direction of the initial growth, the spacer distance of the silicon dopants from the 2D layer (d), and to a much lesser extent the barrier thickness (b), as well as the n+ GaAs doping profile. To obtain a quantitative estimate of the edge profile, as a first approximation a Thomas-Fermi type calculation can be employed (Levitov et al., 2001; Chang, 2001). This approach neglects entirely the contribution of correlation and exchange (which changes with magnetic field, B) and quantum effects. The first step is to pin down the chemical potentials on the two sides of the tunnel barrier before any charge redistribution has taken place. For the 2DEG within the quantum well of the initial (100) growth, this amounts to estimating the lowest subband energy level position, Eo, for the z confinement. In addition to the quantum well width (25nm for all devices studied), the remote silicon doping profile, e.g., delta doping versus continuous doping or asymmetric (one side) versus symmetric (both sides of quantum well) doping, as well as the amount of residual acceptors (e.g. C, O, N) will determine this position (Stern and Das Sarma, 1984). With the exception of Sample 6, our quantum wells are asymmetrically doped on the top side only as 13 14 3 indicated in Table II, while at the same time the residual acceptor density is in the 10 10 cm− range. − The resultant energy level, Eo, is listed in Table I. On top of Eo, we need to add the Fermi energy of the 10 2 2DEG at 2.9x10 cm− /meV. To obtain an estimate of the chemical potential within the bulk GaAs at ∼ the Al0.1Ga0.9As tunnel barrier position, µb, it is necessary to account for the exact n+ doping profile in the GaAs layer. To prevent unintentional back-diffusion of the silicon dopant into the tunnel barrier, the 485nm thick uniformly doped bulk GaAs was grown on top of an 15nm undoped region to separate it from direct contact with the tunnel barrier. The presence of this 15nm undoped GaAs region leads to a redistribution of charge within the GaAs bulk layer, and to an effective local chemical potential distinct from the bulk n+ GaAs chemical potential. We start by first experimentally determining the n+ doping density of the 485nm uniformly doped GaAs region, accomplished via a low temperature Hall measurement either directly on the cleaved-edge of the devices when the tunnel barrier is opaque, or on (011) monitor wafers grown at the same time as the cleaved-edge samples. This yields the chemical potential far away from the Al0.1Ga0.9As tunnel barrier. Next, we need to estimate the local GaAs chemical potential adjacent to the tunnel barrier, µb. To do so, we employ a combination of classical electrostatics and Fermi sea filling which relates the local electron density to the local chemical potential. Since the affected region is of order 15nm which exceeds the 2 2 Bohr radius of the dopants and the screening radius of the electron gas, ao (ǫ¯h /m∗e ) 10nm, while the doping is degenerate with a corresponding Fermi wavelength in the 10-20nm∼ range, for simplicitly∼ we model the doping n+ doping profile as a step function: n+(y)= n Θ(y y ) (A1) d − o where nd is the uniform doping level in the 485nm thick n+ GaAs layer and y=0 at the tunnel barrier while yo=15nm. The undoped GaAs therefore resides between y=0 and yo = 15nm. Note that the dielectric

105 constant, ǫ 13, and the eﬀective mass m∗ 0.067me. In the spirit of the Thomas-Fermi approach, we deﬁne a local chemical∼ potential for the electrons,∼µ(y), with the constraint that the sum of the local electrostatic potential energy, eΦ(y), and µ(y) remains constant across the entire bulk GaAs bulk region:

eΦ(y)+ µ(y)= µo (A2) where µ(y) is related to the local electron density, n(y), via a ﬁlling of the Fermi sea:

¯h2 µ(y)= (3π2)2/3[n(y)]2/3 (A3) 2m∗ and

2 ¯h 2 2/3 2/3 µo = (3π ) [nd] (A4) 2m∗ Applying Poisson’s equation we have:

d2Φ(y) 4πe 2Φ(y)= = [n+(y) n(y)] (A5) ∇ dy2 ǫ − Note e is the electron charge. This equation is supplemented by the condition for charge neutrality:

L L n(y)dy = n+(y)dy = n (L y ) (A6) d − o Z0 Z0

where is the thickness of the entire bulk GaAs layer. Deﬁning a characteristic length scale, do, where:

2 2 1 ǫ ¯h 2 2/3 1 do 2 (3π ) 1/3 (A7) ≡ 2 4πe 2m∗ nd

18 3 so that for n = 10 cm− , d 4.34nm, and defining a dimensionless density, s(y) n(y)/n , we obtain: d o ≈ ≡ d 2 2 2 1/3 d s 1 4/3 ds 1 1 n + (y s− s− + s = 0 (A8) 3 dy2 − 3 dy 2 d2 n − " # o d Although this equation does not readily yield a closed form solution, it is straight forward to find the value of s at y=0, in other words the electron density at the barrier and hence the corresponding chemical 2/3 potential, µb. Equivalently, defining a dimensionless chemical potential, u(y) [s(y)] and using the fact that n+(y) = n Θ(y y ), for y y corresponding to the doped n+ GaAs region,≡ we have: d − o ≥ o d2u 1 (u3/2 1) 2 = 2− (A9) dy 2 do with the boundary condition at y=L (L 500nm) far from the barrier of u(L) = 1 and [du/dy] = 0, giving: ≈ L

du 2 5/2 do = 1 u (1 u ) (A10) dy r − − 5 − For the undoped GaAs region where 0 y < y , we have: ≤ o d2u 1 u3/2 2 = 2 (A11) dy 2 do

Continuity of u and du/dy at y = yo lead to:

106 du 3 2 5/2 do = uo + u (A12) dy r5 − 5 where u u(y = y ). The value of u at the barrier where y=0, u(0) u , is deduced from the condition: o ≡ o ≡ b [du/dy]y=0 = 0, which corresponds to requiring the electric ﬁeld to vanish, yielding:

u = [u 3/5]2/5 (A13) b o −

uo still needs to be determined. This can be done numerically by integrating the equations. For a given value of uo > 3/5, we can calculate the value yo/do for which du/dy vanishes at y=0 using expression Eq. (A12). By setting yo equal 15nm, the width of the undoped region, we obtain uo. Therefore ub as a function of yo/do can be deduced. In Fig. 57 top, we plot ub versus (yo/do), while in Fig. 57 bottom, we plot the chemical potential at the barrier, µb, as a function of the n+ doping density, nd, for several values of yo =9, 12, and15nm. Although nominally the growth structure was designed at yo = 15nm, back diﬀusion of the Silicon dopants can reduce the eﬀective yo. The value of µb thus obtained may be used to estimate the chemical potential imbalance across the tunnel barrier, ∆µ = µ µ . However since the value of b − 2DEG yo is not fully certain, there is a substantial uncertainty associated with the value of ∆µ.

1.0 b

u 0.5

0.0 0.0 2.0 4.0 6.0 8.0 yo /d o 30 9nm 20 12nm (meV) b

µ 10 b=15nm

0 0.0 0.5 1.0 1.5 2.0 2.5 18 −3 nd (10 cm )

FIG. 57. Top–ub, the normalized carrier chemical potential at the Al0.1Ga0.9As tunnel barrier position on the bulk GaAs side versus the dimensionless GaAs undoped region thickness, yo/do. Bottom–The chemical potential at the barrier, µb, as a function of n+ doping density, nd, for various GaAs undoped region thickness, yo=9, 12, and 15nm.

To estimate the density enhancement and its spatial proﬁle, we assume electrostatics plays the dominant role and solve a classical problem neglecting quantum corrections and correlation eﬀects. For Sample 1.1 our device is modeled as follows. There are three length scales: 1) the tunnel barrier thickness of b=9nm, 2) the silicon dopant spacer layer distance from the GaAs/AlGaAs heterojunction, d =60nm, and 3) the distance to the top surface, s = 600nm. Please see Fig. 58. Next all conducting layers are assumed to each form an equipotential. These include: 1) the 2DEG, 2) the n+ GaAs bulk metal, 3) the top surface which has a high density of defect states that pins the Fermi level at midgap, and 4) the dopant layer. The latter two layers are only conducting at room temperature and are poor conductors even there. Furthermore, the

107 dopant (donor) layer is an equipotential only if electrons still reside on the donors. In spatial regions where all donors are ionized, the requirement of an equipotential is replaced by the enforcement of Gauss’ Law which relates the electric ﬂux through a Gaussian surface to the ionized donor concentration. The values of the equipotentials are assigned as follows: 1) the 2DEG is held a ground potential, 2) the n+ GaAs is held at ∆µ = µb µ2DEG to simulate the imbalance in chemical potential between the 2DEG and n+ GaAs metal at the barrier,− 3) the top surface is held at –770mV to simulate the pinning at midgap, where the minus sign signiﬁes that negative charges must be transferred to this surface, and 4) for most samples the donors turn out to be fully ionized. As a result, near the tunnel barrier the donor layer is no longer an equipotential.

FIG. 58. Sample 1.1 growth structure and parameters.

An approximate analytic solution can be obtained if we for the moment assume that a residual amount of donors always remain un-ionized so that the donor layer can be viewed as an equipotential. Using the method of superposition, Levitov, Shytov, and Halperin (2001) obtained the following solution. In the end, the accuracy of such a solution can subsequently be checked by a numerical calculation based on the relaxation method (Chang, 2001). Assuming an inﬁnitely long edge region, the charge density proﬁle in the perpendicular direction to the edge, the y-direction, is given by a sum of three contributions:

ntotal(y)= ndonor(y)+ nsurf (y)+ n∆µ(y) (A14) where ndonor represents the contribution from the silicon donors, nsurf the contribution from the eﬀective top surface potential due to pinning of the Fermi level at midgap, and n∆µ the contribution from the chemical potential imbalance between the n+ GaAs and the 2DEG. ndonor is estimated by removing the top surface to inﬁnity, a reasonable approximation since s >> d, resulting in:

2n2DEG 1 y n (y)= tan− (A15) donor π d where n2DEG is the carrier density in the quantum well far from the tunnel barrier. The contribution of the surface potential, Vsurf 770meV, can be obtained by setting the 2DEG and ∆µ/e potentials to zero in the absence of a donor layer.≈− In the limit of b << d appropriate for most devices, this is the standard problem of a half-open slit with one side at potential Vsurf and the other side and end at ground, yielding:

ǫVsurf πy nsurf (y)= tanh (A16) 4πd1 s Lastly, the eﬀect of the chemical potential imbalance, ∆µ, will be accounted for by ignoring the top surface and donors. The resultant contribution is: ǫ∆µ/e 1 n∆µ(y)= (A17) 2π2 (y + b)2 b2 − p 108 This solution can provide a valuable guide.

FIG. 59. Top–Equipotential lines for the Thomas-Fermi/classical electrostatics solution of for the charge redistribution across the tunnel barrier in Sample 1.1. The chemical potential imbalance is ∆µ = 10meV (higher on the n+ GaAs side). Bottom–Density proﬁle on the 2DEG quantum well side as a function of distance from the Al0.1Ga0.9As/bulk-GaAs interface for various values of chemical potential imbalance.

As it turns out, in the crucial region adjacent to the tunnel barrier the donors are in most cases fully ionized, a conclusion reached based on the numerical solution. The resultant equipotential and density profile obtained from the numerical calculation for Sample 1.1 and various values of chemical potential imbalance are plotted in Fig. 59. In terms of the electrostatic potential profile, intuitively, far from the tunnel barrier we expect to have parallel plate capacitors with the 2DEG at ground as one plate and the top surface s=600nm away at -770mV as the other. The equipotential surfaces in between run parallel to these plates but with a layer of ionized donors at d=60nm above the 2DEG. The sum of the negative sheet charge density in the top surface and in the 2DEG must equal the ionized donor concentration. On the other hand, in the vicinity of the corner where the top surface and the n+ GaAs metal meet (see Fig. 59 top panel), the equipotential surfaces form a fan where V = 770 + (770 + 25)θ/(π/2)mV . The spatial extent of this fan region is roughly s, the surface distance from the− 2DEG. By joining the two regions in a continuous manner using a numerical relaxation method, we find that within a lateral distance of s/2 from the barrier, the electron sheet density can be enhanced or reduced from its value in the 2D bulk∼ as shown in the bottom panel in Fig. 59, dependent on the chemical potential imbalance. In most instances, the density profile varies

109 rather continuously as we approach the bulk from the edge without a well-deﬁned region of ﬁxed value.

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117 TABLE I. Sample Parameters

Sample n µ EF Eo QW Barrier n + Doping µn+ ν :∆ Figures − − (1011cm 2) (cm2/V s) (meV ) (meV ) (nm) (nm) (1018cm 3) (meV ) (µeV ) 1.1 1.08 2.9x106 3.9 23 25 9.0 2.1x1018 90 1/3: 450 1,19-24,32 33,36,39-41 1.2 1.16 3.3x106 4.1 23 25 22.5 2.1x1018 90 1: 1450 25,32

1.3 1.13 3.2x106 4.0 23 25 12.5 2.1x1018 90 26-29,34,36

1.4 1.27 2.9x106 4.5 24 25 24.5 2.1x1018 90 30

2 0.87 1.4x106 3.2 23 25 9.0 2.1x1018 90 1/3: 320 1,19,23 31,32,39 3.1 2.06 1.6x106 7.3 31 25 12.5 2.1x1018 90 1: 2600 25

3.2 1.94 1.5x106 6.9 31 25 9.0 2.1x1018 90 2/3: 220 25

4 1.80 2.0x106 6.4 30 25 9.0 2.1x1018 90 2/3: 140 25

5.1 1.24 0.5x106 4.4 24 25 16.0 2.1x1018 90 30

5.2 1.09 0.5x106 3.9 24 25 9.0 2.1x1018 90 1/3: 380 32

6 0.61 1.0x106 2.2 8 25 5.0 2.1x1018 90 35-36 aEstimates based on Stern and Das Sarma (1994).

TABLE II. Substrates − Sample n(1011cm 2) d, spacer distance (nm) s, surface distance (nm) 1 1.08-1.27 60 600 2 0.87 39 600 3 1.94-2.06 40 450 4 1.80 40 580 5 1.09-1.24 40 590 6a 0.61 aSymmetrically-doped on both sides of the quantum well (QW).

TABLE III. Parameters for lever arm model. 1.1A 1.1Ba 1.1Bb (FLC) (QHC) (FLC) ′ λ 0.20 0.30 0.20 Lever arm meV meV meV r β +0.43 T −0.26 T +0.59 T dE /dB + ΓFL 1.3 1.0 1.0 Coupling to n ΓQH 19.7 3.5 14.0 Coupling to QH edge

×1.5 ± .1 ×6 ± 1 ×1.7 ± .1 Area Increase - Data (× 1.54 ) (× 5.9) (× 1.58) (Area Increase - Fit) 39µeV 30µeV 30µeV at bias voltage

118