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Chiral Luttinger Liquids at the Fractional Quantum Hall Edge

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Chiral Luttinger Liquids at the Fractional Quantum Hall Edge 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
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