Tunneling Conductance of Long-Range Coulomb Interacting Luttinger Liquid

Tunneling conductance of long-range Coulomb interacting Luttinger liquid

DinhDuy Vu,1 An´ıbal Iucci,1, 2, 3 and S. Das Sarma1 1Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742, USA 2Instituto de F´ısica La Plata - CONICET, Diag 113 y 64 (1900) La Plata, Argentina 3Departamento de F´ısica, Universidad Nacional de La Plata, cc 67, 1900 La Plata, Argentina. The theoretical model of the short-range interacting Luttinger liquid predicts a power-law scaling of the density of states and the momentum distribution function around the Fermi surface, which can be readily tested through tunneling experiments. However, some physical systems have long- range interaction, most notably the Coulomb interaction, leading to significantly different behaviors from the short-range interacting system. In this paper, we revisit the tunneling theory for the one- dimensional electrons interacting via the long-range Coulomb force. We show that even though in a small dynamic range of temperature and bias voltage, the tunneling conductance may appear to have a power-law decay similar to short-range interacting systems, the effective exponent is scale- dependent and slowly increases with decreasing energy. This factor may lead to the sample-to-sample variation in the measured tunneling exponents. We also discuss the crossover to a free Fermi gas at high energy and the effect of the finite size. Our work demonstrates that experimental tunneling measurements in one-dimensional electron systems should be interpreted with great caution when the system is a Coulomb Luttinger liquid.

I. INTRODUCTION This power-law tunneling behavior is considered a signature of the Luttinger liquid since in Fermi liquids, G is simply a constant for small values of T and V (as long Luttinger liquids emerge from interacting one- 0 as k T, eV E , where E is the Fermi energy). In- dimensional many-electron systems where the Fermi sur- B 0 F F deed, tunneling experiments have confirmed this behav- face is two discrete points rather than a connected sur- ior in many physical systems. Earliest attempts include face as in higher-dimensional cases. Two and three- chiral Luttinger liquids found in the edge mode of frac- dimensional interacting systems are conventionally de- tional quantum Hall fluids [7–10] and the power law in scribed by the Fermi liquid model where the excitations the optical response from quasi-one-dimensional conduc- are individual quasi-particles with renormalized proper- tors [11, 12], suggesting the Luttinger liquid nature of ties (e.g. effective mass) from the bare particle. Fermi these systems. Recently, studies have focused on car- liquids manifest qualitative resemblance to the free Fermi bon nanotubes where extreme isolating conditions can gas, for example, the discontinuity of the momentum dis- be obtained to create a strongly correlated non-chiral 1D tribution through the Fermi momentum and singulari- electron system. Tunneling experiments on carbon nan- ties in the spectral function representing quasi-particles. otubes also show evidences for power laws characterizing However, in one-dimensional systems, quasi-particle ex- Luttinger liquids [13–15]. citations are replaced by collective excitations even for very weak interaction, leading to the complete disap- It is instructive to review the theoretical description of pearance of the one-to-one correspondence with the non- the Luttinger liquid based on the bosonization method, interacting Fermi gas. Most remarkably, the momentum which in principle can give exact solutions for any types distribution function is continuous through the Fermi of interactions provided that the back-scattering is ig- point and the density of states displays a pseudo gap at nored and the dispersion can be linearized. If we as- the Fermi energy. These features indicate the breakdown sume that the system has only contact charge-charge in- R 2 of the quasi-particle picture. More non-Fermi liquid phe- teraction with a strength U, i.e. Hint = U/2 ρ(x) dx; nomenon include charge-spin separation, and power-law the plasmon velocity is renormalized by g = vF /vρ = −1/2 scaling of charge and spin correlations [1–4]. Assuming a (1 + NU/(πvF )) where N is system degeneracy (e.g. zero-range (or short-range) interaction (e.g. a Dirac delta N = 2 for spinful systems or N = 4 for carbon nan- α otubes including spin and valley without spin-orbit cou- arXiv:1912.10379v2 [cond-mat.str-el] 29 May 2020 function), the density of states decays as (E−EF ) where α is a finite and nonuniversal constant since it depends on pling). The exponent of the density of states is then −1 the actual interaction strength. In addition, this constant αbulk = (g + g − 2)/2N at the bulk of the system and −1 α also shows up in the tunneling conductance, giving it αbound = (g −1)/N at the open boundary [16, 17]. For a distinct power-law decaying behavior. Specifically, if systems with repulsive interaction, U > 0, so g < 1 and the tunneling is between a Luttinger liquid characterized αbound > αbulk > 0. by the exponent α and an ordinary Fermi-liquid metal, The simplicity of the short-range interacting Luttinger the tunneling conductance at temperature T and bias liquid model might be attractive, but the model of zero- α voltage V0 is G = dI/dV0 ∝ V0 for eV0 kBT and range interaction is only an approximation. Most real α G ∝ T for eV0 kBT . For tunneling between two systems have long-range interactions, most notably the Luttinger liquids, the exponent is simply doubled [5–7]. electronic Coulomb interaction. If the Fourier transform 2 of the interaction approaches a finite constant at the within even a given type of 1D physical systems. Since zero transfer momentum limit, the low-energy physics the Coulomb Luttinger liquid by definition does not have can be described by a short-range interaction model a constant exponent (i.e. the exponent varies slowly over without great loss of accuracy. However, the Coulomb the energy scale of measurements), it is important to an- interaction is special because of its logarithmic diver- alyze the tunneling experiment in depth using the long- gence at small momentum. One way to obtain a short- range interaction model to figure out how this scale de- range interaction is to consider an appropriately screened pendence might manifest in the tunneling spectroscopy. Coulomb interaction [18, 19], imagining the carbon nan- This is the main goal of the current work. A secondary otube is placed inside a larger metallic tube with a ra- goal is to investigate the role of the finite length of the dius Rs much larger than the radius R of the nanotube . 1D system (e.g. carbon nanotube or semiconductor quan- Then, the interaction potential is totally screened out and tum wire) in the tunneling experiments to check whether can be considered as the classical energy of the E-field an implicit or explicit length dependence affects the tun- trapped between the nanotube and the metallic cylinder neling exponent, particularly in the context of the scale- 2 R 2 Hint = e ln(Rs/R) ρ(x) dx. This is exactly the form dependent exponent in the Coulomb Luttinger liquid. of the zero-range interaction, so even if the nanotube The rest of the paper is organized as follows. In Sec. II has long-range interaction, the Luttinger liquid consider- we provide the theoretical description for a Coulomb Lut- ations from the short-range case still apply with an ap- tinger liquid with the open boundary condition and com- propriate g provided, of course, the nanotube is indeed pare its properties with the short-range interacting Lut- enclosed in a metallic cage. However, the ambiguity here tinger liquid counterpart. We also discuss the crossover is the value Rs since in reality, no specific metal tube to the free Fermi gas behavior at high energy, which may encloses the nanotube. Note that the logarithmic diver- not be obvious in the short-range model but appears nat- gence of the effective interaction in Hint as Rs goes to urally in the long-range Coulomb Luttinger liquid. In infinity is the well-known logarithmic divergence of the Sec. III, we study the effect of the finite system, finding it 1D Coulomb interaction arising from its long-range na- to be irrelevant as long as the 1D system is not too short. ture, which is simply cut off by taking Rs to be finite. We conclude in Sec. IV summarizing our main findings A more rigorous approach is not to make an ad hoc and discussing possible experimental implications of our short-range interaction approximation, and instead use results. the long-range 1D Coulomb interaction itself in the Lut- tinger liquid theory, i.e. performing bosonization with the logarithmically divergent interaction originating from II. THEORETICAL MODEL the 1D 1/x Coulomb interaction. Within this description, the power law behavior is no longer valid. Indeed, A. Bosonization in open-boundary systems for the density-density√ correlation, the 4kF oscillation decays as x− ln x, slower than any power law [20, 21]. The We first present the bosonization study of a 1D N−fold 1D Coulomb Luttinger liquid is thus an effective Wigner degenerate system of size L and open boundaries at x = 0 crystal at finite length with the 4k density oscillation F and x = L which are appropriate for tunneling measure- decaying very slowly over distance (since the correlation ments. Due to the open boundary condition, the left and dies out eventually there cannot be any real long-range right-moving electrons are no longer independent opera- order). Moreover, due to the log-divergence, the “effec- tors. Accordingly, the fermion field can be decomposed tive exponent” of a Coulomb Luttinger liquid is scale- as dependent, i.e. α is also a function of energy [22]. This scale-dependence of the effective Luttinger exponent in ψs(x) = ψL,s(x) + ψR,s(x) the Coulomb Luttinger liquid, in contrast to the constant i X −ikx i X ikx (1) exponent for the short-range interaction model, compli- = √ e ck,s − √ e ck,s, cates the description of tunneling measurement as the 2L k 2L k effective exponents also depend on the temperature or voltage scale being studied. It is unclear apriori how one where s ∈ {s1, s2, . . . , sN } is the electron species (e.g can investigate the Coulomb Luttinger liquid simply by spin, valley, etc.) index. The construction implies that mapping it into a corresponding short-range interaction ψR,s(x) = −ψL,s(−x). The linearized Hamiltonian with model although this is often done in the interpretation of only charge-charge interaction in the unit system of ~ = 1 the experimental 1D results. is In this paper, we investigate the effect of the long- Z Z X i∂ψs U(x − y)ρ(x)ρ(y) range Coulomb interaction on the tunneling conductance H = v ψ† dx + dxdy, F s ∂x 2 using the long-range interaction description of a scale- s dependent exponent. One specific consequence of the (2) P scale-dependent exponent of the Coulomb Luttinger liq- where ρ(x) = s ρs(x) is the charge density. It is noted uid may be the manifestation of the sample-to-sample that for an open boundary system, the discretized mo- variations in the measured tunneling exponent often seen mentum is k = nπ/L instead of 2π/L as in the periodic 3 one. We define bosonic creation and annihilation opera- After the Bogoliubov transformation, the Luttinger in- tors with q = mπ/L > 0 as teraction parameter is defined as

r 1/2 † π X † 2θ vF aq,s = ck+q,sck,s g(q) = e = , (8) qL vF + NU(q)/π k (3) r π X a = c† c , and the collective plasmon mode velocity is q,s qL k−q,s k,s k s v NU(q) v (q) = F = v 1 + . (9) † ρ g(q) F πv and [aq,s, aq0,s0 ] = δq,q0 δs,s0 . The right-moving chiral F fermion ﬁeld is For a short-range interaction, U(q) is a constant and −i F iπxNs s φs(x) hence the constancy of g as a Luttinger exponent for ψR,s(x) = lim √ √ e L e ; where →0+ 2 2π a given model, i.e., a given U. For a long-range inter- (4) action, U(q) is obviously scale-dependent as it depends X r π φ (x) = e−q/2 eiqxa − e−iqxa† , explicitly on the momentum q, resulting in the remark- s qL q,s q,s q able divergence g(q) ∼ ln(1/q) in the case of 1D Coulomb interaction. The fermion correlation function is Ns is the number operator of the s−electrons. Fs is an † † operator such that Fs commutes with all bosonic opera- hψs(x, t)ψs(y, 0)i = 2 hψR,s(x, t)ψR,s(y, 0)i † tors, Fs |Nsi = |Ns − 1i and Fs |Nsi = |Ns + 1i. One − hψ (x, t)ψ† (−y, 0)i − hψ (−x, t)ψ† (y, 0)i . can perform a unitary transformation to separate the R,s R,s R,s R,s (10) charge channel from other channels Assuming hN i=N /N with N being the total number N s e e 0 1 X 2iπmn of electrons, the chiral correlation is a = √ e N aq,s , (5) q,n N m m=1 † ikF (x−y)−iEF t hψR,s(x, t)ψR,s(y, 0)i = e C(x, y; t); (11) The Hamiltonian is then the sum of separated channels PN−1 where H = n=0 Hn. The interacting charge/plasmon chan- iπ (x−y−v t) nel (n = 0) is described by the Hamiltonian e 2L c C(x, y; t) = lim →0+ 4π 2 πv0|N0| X NU(q) 0† 0 N−1 H0 = + q vF + a a , F (x, y; t) + iK(x, y; t) πt N 2L 2π q,0 q,0 × exp . q>0 (6) N β sinh(πt/β) NU(q) (12) + q a0† a0† + a0 a0 , 4π q,0 q,0 q,0 q,0 Here, we deﬁne the Fermi momentum kF = (Ne/N + L R iqx 1/2)π/L, the Fermi energy EF = v0Neπ/(NL) and the where v0 = vF +NU(0)/2π, and U(q) = −L U(x)e dx. For the other non-interacting channels (n = 1,...,N −1), charge gap velocity vc = [v0 + (N − 1)vF ]/N (i.e. the average over all channels). For brevity, we drop the q 1/2 2 argument in vρ(q) and g(q), and use cosh θ = (g + πvF |Nn| X H = + v a0† a0 . (7) g−1/2)/2 and sinh θ = (g1/2 − g−1/2)/2. We have n 2L F q,n q,n q

X π K(x, y; t) = e−q cosh2 θ sin q(x − y − v t) − sinh2 θ sin q(x − y + v t) qL ρ ρ q>0

− sinh 2θ sin q(x + y − vρt)/2 + sinh 2θ sin q(x + y + vρt)/2] , (13) X π F (x, y; t) = − e−q [1 + 2f (v q)] × [cosh2 θ(1 − cos q(x − y − v t)) + sinh2 θ(1 − cos q(x − y + v t)) qL B ρ ρ ρ q>0

− sinh 2θ(cos 2qx + cos 2qy − cos q(x + y − vρt) − cos q(x + y + vρt))/2],

βz −1 where fB(z) = e − 1 is the Bose-Einstein distribution coming from the bosonic plasmon with z being 4 the excitation energy and β = 1/kBT being the inverse is unknown and arbitrary in the experimental systems, electron temperature. Equations (12) and (13) are the and often used simply as a ﬁtting parameter uncritically. exact expressions and do not assume any speciﬁc form of We begin by studying the tunneling density of states in the interaction. the bulk and the boundary of a Coulomb Luttinger liquid. We can perform a quick check on the power law for T = The following argument is based on Wang et al. [22], P R 0, L → ∞ (in that case q π/L → dk) and zero-range and extended to include the open boundary condition of interaction g(q) = g. The spinless (N = 1) dynamic relevance to tunneling spectroscopy. The bulk density of chiral correlation function is states in a carbon nanotube is given by

g+g−1 g−1−g Z ∞ 2 2 4 1 1 − cos ωt 1 4x −Φbulk C(x, x; t) ∝ . (14) ρbulk(ω) = Im e dt; 2 2 2 π 0 vF t t |4x − vρt | Z ∞ dq g−1 + g ivρqt ivF qt This term is the primary contribution to the correlation Φbulk(t) = 1 − e − 1 − e . 0 4q 2 function because the other term C(x, −x; t) has fast 2kF (18) oscillation of e2ikF x and is further suppressed by x−g. −g−1 For vρt x (near the boundary), C(x, x; t) ∝ t In the phase factor Φbulk(t), when q < A/t with A is some −1 corresponding to the density of states ρ(ω) ∝ ωg −1 number, the factor 1 − eiqvt ≈ 0 and when q > A/t, this (ω = E − EF ) for ω vρ/x. For vρt x (far from factor is fast oscillating and negligible, thus the leading −1 the boundary), C(x, x; t) ∝ t−(g +g−2)/2. As a result, order term in the phase factor is (g−1+g−2)/2 ρ(ω) ∝ ω for ω vρ/x. These are of course 1 Z ∞ g(q) + g(q)−1 − 2 well-known results provided here for the sake of complete- Φbulk(t) ≈ dq 4 A/t 2q ness and to set a context for our work. √ U q t 1 q t 1 q t ≈ 0 ln3/2 s + √ ln1/2 s − ln s , 12 A 4 U0 A 4 A B. Coulomb Luttinger eﬀective exponent (19)

In this section, we assume the semi-infinite 1D limit where we have used the low-q asymptotic form g(q) = √ 1/2 2 L → ∞ (the effect of finite L is discussed later). We U0 ln (qs/q) with U0 = 8e /(κπvF ) and qs ≈ study two models of interactions in the four-fold degener- 1.12e1/U0 /d in the integration instead of the full form ate carbon nanotube system: (i) short-range interaction Eq. (16). The power law of the density of states is then with constant g up to a cut-off Λ, i.e. ( ω γbulk(ω) g for q ≤ Λ ρbulk(ω) ≈ ; where g(q) = . (15) ωs 0 for q > Λ √ U0 1/2 ωs 1 −1/2 ωs 1 γbulk(ω) = ln + √ ln − , As long as ω < E0 = ΛvF , the density of states ρ(ω) ∝ 12 ω 4 U0 ω 4 −1 −1 ω(g −1)/4 at the boundary and ρ(ω) ∝ ω(g +g−2)/8 in (20) the bulk. It is noted that the cut-off Λ is purely artificial, √ in fact, many theoretical works only consider the large and the scale ωs = 20vF qs U0. Extending the argument to the limit x → 0, the boundary phase factor is distance asymptote and set Λ ≈ 1/ √→ ∞. (ii) The 1D Coulomb interaction U(x) = e2/(κ x2 + d2) where 1 Z ∞ g(q)−1 − 1 κ is the dielectric constant and d is proportional to the Φbound(t) ≈ dq transverse size of the nanotube, which regularizes the 1D 4 A/t q √ (21) Coulomb coupling. The corresponding g is U q t 1 q t ≈ 0 ln3/2 s − ln s . −1/2 6 A 4 A 8e2 g(q) = 1 + K0(qd) , (16) κπvF Accordingly, the local density of states at the open

5 boundary is with vF = 8 × 10 m/s and K0 is the Bessel function.

Note that in Ref. [18], the Coulomb Luttinger liquid γbound(ω) ω is approximated by an effective short-range interaction ρbound(ω) ≈ , where ωs model with a constant g as √ (22) U0 1/2 ωs 1 8e2 R −1/2 γbound(ω) = ln − . g = 1 + ln s , (17) 6 ω 4 πv R F As γ depends on the energy ω, strictly speaking this is where R is the radius of the tube and Rs is some screening not a power law. However, we can define an effective ex- length; for Rs ∼ 100R, g ∼ 0.2. We emphasize that Rs ponent α = d ln(ρ(ω))/d ln ω for the bulk and boundary 5 tunneling density of states as as L-M, with C2(x, x; t) ∼ 1/t (assuming the temperature √ is much less than the metal Fermi temperature) U0 1/2 ωs 1 −1/2 ωs 1 αbulk = ln + √ ln − , 1− 1 8 ω 8 U0 ω 4 Z ∞ N √ cos(eV0t) πt GL−M ∝ 1 − 2 U0 1/2 ωs 1 t β sinh (πt/β) αbound = ln − . 0 (27) 4 ω 4 F (x, x; t) K(x, x; t) (23) × exp sin dt. N N We expect these exponents to appear in the tunneling conductance of the Coulomb Luttinger liquid. We note From Eqs. (26) and (27), we can expect that the exponent that the energy dependence of α as reflected in the ex- of the L-L tunneling is two times as large as that of the plicit appearance of ω in the right hand side of Eq. (23) tunneling through L-M contact. We recall that F (x, x; t) leads to an ill-defined scale-dependent exponent in the and K(x, x; t) are given in Eq. (13). Coulomb Luttinger liquid in contrast to the constant In Fig. 1, we show the calculated L-M tunneling con- (but nonuniversal) exponent in the short-range interac- ductance of a short-range interacting Luttinger liquid tion model. with g = 0.2 in the inset and the effective exponent in the main plot as a function of voltage bias and temperature. There are three energy scales in the plot: the satura- C. Tunneling conductance tion regime (eV0 kBT or kBT eV0) where G is independent of V0 (or T ), thus the effective exponent ap- Suppose electrons tunnel between systems 1 and 2 at proaches zero, the boundary regime (eV0, kBT vρ/x) voltage bias V0, the tunneling Hamiltonian is given by where the power law is given by the boundary relation −1 [5, 7, 23] α = (g − 1)/4 = 1.0 and the bulk regime (eV0, kBT vρ/x) where the power law is given by the bulk relation X † −1 Htunnel Γs,s0 (ψ1,s(x)ψ2,s0 (x) + h.c.), (24) α = (g + g − 2) = 0.4. Note that, although there s,s0 are noisy fluctuations in the exponent in the crossover regimes, the bulk and boundary exponents clearly mani- 0 where ψ1,s and ψ2,s0 correspond to the s and s −electron fest themselves as constants in Fig. 1. wavefunction in the system 1 and 2 respectively. Our tun- Because the scale-dependent Luttinger parameter in neling Hamiltonian corresponds to a perturbative point- the Coulomb Luttinger liquid is an explicit function of contact, which is a common experimental setup. One can the momentum, we can naturally define an energy scale refer to other works for more rigorous tunneling condi- E0 = vF /d. In Fig. 2, we show the directly calculated L- tions, for instance, non-perturbative contact [24], back- M effective exponent of a Coulomb Luttinger liquid with scattering defects [25] or a Y −junction of three nanowires U0 = 5. Similar to the short-range case, there are also [26–31]. Within the scope of this paper, the electron three distinct regimes. However, the striking difference species degeneracy reduces the problem to the tunneling is the continuously increasing effective exponent with de- between two single-mode Luttinger liquids because the creasing energy. For the conductance measurement with electron correlation function is identical for all electron respect to the voltage bias eV0 (see Fig. 2(a)), the numer- species (see Eq. 12 and (13)). For our purpose, there- ical result is consistent with Eq. (23). In addition, when fore, we do not need to worry about the symmetry or eV0 kBT (see Fig. 2(b)), the effective exponent of G degeneracy aspects of the Luttinger liquid. The tunnel- with respect to T has the same form as Eq. (23) with ω ing current is then given by √ substituted by kBT and ωs replaced by Ts = 7vF qs U0. Z ieV0t I ∝ e Im [C1(x, x; t)C2(x, x; t)] dt. (25) D. Universal scaling function Equation (25) is basically the Fermi golden rule expressed in the Fourier transform. We specify the differential tun- Another contrasting property between the Coulomb neling conductance G = dI/dV in two distinct experi- 0 and the short-range Luttinger liquid is the universal scal- mental setups: the two systems 1 and 2 are two identical ing with eV /k T , i.e. the tunneling conductances of Luttinger liquids, denoted as L-L, 0 B a short-range Luttinger liquid at different temperatures 2− 2 can collapse into a single function of eV0/kBT . Using the Z ∞ cos(eV t) πt N G ∝ 1 − 2 0 asymptotic correlation function (no cut-off), the scaling L−L t β sinh (πt/β) 0 (26) function with respect to µ = eV0/kBT is [5, 6, 14] 2F (x, x; t) 2K(x, x; t) × exp sin dt; α 2 N N GL−L ∝ T sinh(µ/2) |Γ(1 + α/2 + iµ/2π)| and, system 1 is the Luttinger liquid sample and system 2 × {coth(µ/2)/2 − Im [Ψ(1 + α/2 + iµ/2π)] /π} , (28) α 2 is a conventional 3D Fermi liquid metal contact, denoted GL−M ∝ T cosh(µ/2) |Γ((1 + α)/2 + iµ/2π)| ; 6

G G 100 3 (a) 100 (a) Λx = 500 Λx = 2000 G (a) x/d = 0 x/d = 5000 G -2 Λx = 1000 Λx = 5000 10-2 x/d = 500 x/d → ∞ 10 ) eV / E ) 0 0 0 -4 0 10-4 eV0 / E0 V 10 V 2 1 -6 -4 -2 0 ln( 10 10 10 10 ln( 10-6 10-4 10-2 100 G/d G/d 1 ln ln d d 0 0 10-6 10-5 10-4 10-3 10-2 10-1 100 10-5 10-4 10-3 10-2 10-1 100

eV0/E0 eV /E G 0 0 G 0 3 0 (b) Λx = 500 Λx = 2000 10 G (b) x/d = 0 x/d = 5000 10 G 10-2 Λx = 1000 Λx = 5000 10-2 x/d = 500 x/d → ∞ 10-4 ) kBT / E0 ) kBT / E0 -4 -6

T 2 1 10 T 10 -6 -4 -2 0 -6 -4 -2 0 ln(

10 10 10 10 ln( 10 10 10 10 G/d G/d 1 ln ln d d

0 0 -6 -5 -4 -3 -2 -1 0 10 10 10 10 10 10 10 10-6 10-5 10-4 10-3 10-2 10-1 100

kB T/E0 kB T/E0

Figure 1. The effective exponent of the L-M tunneling con- Figure 2. The effective exponent of the L-M tunneling conduc- ductance for a short-range interacting Luttinger liquid with tance for a Coulomb Luttinger liquid with U0 = 5 (a) with re- −5 g = 0.2 (a) with respect to eV0/E0 at fixed temperature spect to eV0/E0 at fixed temperature kB T = 10 E0 and (b) −5 −5 kB T = 10 E0 and (b) with respect to kB T/E0 at fixed volt- with respect to kB T/E0 at fixed voltage bias eV0 = 10 E0. −5 age bias eV0 = 10 E0. The energy scale is E0 = vF Λ. The The energy scale is E0 = vF /d. The dashed lines are the the- insets show the tunneling conductance in arbitrary unit for oretical effective exponent and the insets show the tunneling Λx = 2000. conductance for x/d = 2000 in arbitrary unit.

10 where Γ and Ψ are the gamma and digamma functions. A Scaling function g = 0.2 single scaling function is a direct result of a scale invari- k T = 10-4E -2 ant interaction constant g where a conformal transfor- B 0 kBT = 10 E0 -3 -1 mation can transform between the voltage and the tem- G kBT = 10 E0 kBT = 10 E0 perature (corresponding to the real time and imaginary

time boundary respectively). Clearly, the existence of a Scaled scale-dependent exponent for the Coulomb Luttinger liquid rules out such a universal scaling function uniquely determined by eV /k T . In Fig. 3, we plot the scaled L- 1 0 B 0.1 1 10 M tunneling conductance of a Coulomb Luttinger liquid eV /k T at different temperatures, which clearly does not con- 0 B verge to a single function as in the short-range case. The Figure 3. The scaled L-M tunneling conductances of a predictions of Fig. 3 should also be directly experimen- Coulomb Luttinger liquid with U0 = 5 at different temper- tally verifiable using tunneling data provided that the atures. varying range of the temperature and the bias voltage is of several orders of magnitudes. We conclude this section by comparing the tunneling ponent. However, within a narrow range of one order of conductance of a short-range Luttinger liquid with two magnitude, the two are almost indistinguishable. More- Coulomb Luttinger liquids characterized by different pa- over, Coulomb Luttinger liquids of different parameter rameters at T = 0. As can be seen from Fig. 4, over sets are also indistinguishable within one order of magni- a range of three orders of magnitude, the difference be- tude of the tuning parameter. This suggests that we can tween short-range and long-range cases is obvious: in the only conclusively study the Coulomb Luttinger liquid if log-log scale the tunneling conductance of a short-range the dynamic range of the independent variable (i.e. T,V0) interacting system is a line corresponding to an ideal is more than 2 orders of magnitude. We believe that the power law while that for a Coulomb system is a down- existing experimental literature on tunneling measure- ward bending curve characterizing a scale-dependent ex- ments is limited to a very narrow voltage and temper- 7

(a) (b) x/d = 0 0 kBT = 0.05 E0 kBT = 0.3 E0 eV = 0.05 E0 eV = 0.3 E0 10 0.4 0.4 x/d = 500 kBT = 0.1 E0 G ∝V eV = 0.1 E0 G ∝T bulk - metal 100 100 10-1

10-2 boundary - metal G

10-1 10-1 10-3 Scaled G 10-2 10-1 100 101 10-2 10-1 100 101 Long range, U0 = 5, E0 = 0.5 V -4 (c) (d) 10 kBT = 0.05 E0 kBT = 0.3 E0 eV = 0.05 E0 eV = 0.3 E0

Long range, U0 = 3, E0 = 8.0 V Scaled γ ∝ γ kBT = 0.1 E0 G ∝ ( V0 / ωs) eV = 0.1 E0 G ( T / Ts) Short range, g = 0.22 0 100 10-5 10 10-3 10-2 10-1 100 Voltage bias (V)

Figure 4. The scaled L-M tunneling conductances of a short-

-1 range and two Coulomb Luttinger liquids (with diﬀerent inter- 10-1 10 action parameters) at the bulk and at the boundary. Within -2 -1 0 1 -2 -1 0 1 one order of magnitude variation of the independent variable, 10 10 10 10 10 10 10 10 the distinction among the three is hardly noticeable. eV0/E0 kB T/E0

Figure 5. The scaled L-M bulk tunneling conductances of a ature range and is thus insufficient to fully characterize short-range Luttinger liquid with g = 0.2 as a function of eV0 the long-range interacting nature of Coulombic systems. (a) and kB T (b). The scaled L-M bulk tunneling conductances of a Coulomb Luttinger liquid with U = 5 as a func- A true verification of the theory necessitates the obser- 0 tion of eV0 (c) and kB T (d). The dashed lines are theoretical vation of the scale dependence shown in Figs. 2, 3, and predictions without regarding the high energy crossover. 4, which would require a large variation in the dynamical range of temperature and bias voltage. We suggest tunneling experiments be carried out with a substantial but still less than the crossover scale E . As a result, if increase in the dynamical range of temperature and volt- 0 eV is close to E , one cannot extract the power law with age so that the predicted scale-dependent deviation from 0 0 respect to the temperature and vice versa. Moreover, a constant Luttinger exponent can manifest itself in the in the case of Coulomb Luttinger liquid, the theoretical measurement. scale-dependent exponent already intrinsically contains a decay at high energy even though it is derived using low energy assumption. This is because the definition of the E. High-energy crossover to free Fermi gas long-range interaction already contains the ultra-violet regularization through the transverse size d. Therefore, If we assume that the interaction vanishes at very high the consistency between numerical results and theoreti- momentum, which is true for all interactions with ultra- cal predictions extends to much higher energy than the violet regularization, then the Luttinger liquid can cross short-range interacting Luttinger liquid. The fact that over to the free Fermi gas defined by g → 1. In this the Coulomb Luttinger liquid has a built-in physical ul- section, we investigate the system near this high-energy traviolet regularization, in contrast to a completely arbi- crossover. For the short-range model defined previously, trary cut-off-dependent regularization in the short-range it is obvious that the crossover energy must be associated model, makes the long-range interaction model more the- with the cut-off momentum Λ while for the Coulomb in- oretically meaningful. teraction, the interaction potential K0(qd) decays expo- nentially for q 1/d so the crossover energy must be related to 1/d. Therefore, for the Coulomb Luttinger liquid, the crossover energy scale is physically defined III. FINITE-SIZE EFFECT whereas for the short-range model, it depends on an ad hoc cut off. Hence, the energy scale we define earlier, i.e. In this section, we study the effect of finite size (i.e. the E0 = vF Λ for short-range interaction and E0 = vF /d for finite length of the 1D system in the experimental tun- Coulomb interaction, is also the high-energy crossover neling measurements) on the experimentally extracted scale. In Fig. 5, we show the L-M tunneling conduc- properties of the Luttinger liquid, e.g. the momentum tance as a function of the voltage bias and temperature. distribution and the density of states. We will show that The usable region where one can extract a meaningful the finite system (as long as the system length is much power law is for eV0(kBT ) much higher than kBT (eV0) longer than the ultraviolet cutoff length d) does not al- ΛL = 400, σ/Λ = 10-3 ΛL = 800, σ/Λ = 5x10-4

8 ter the Luttinger liquid properties except for inducing -3 0.6 (a) ΛL = 400, Δk/Λ = 10 discrete peaks pattern to the measured spectrum. This ΛL = 800, Δk/Λ = 5x10-4 effect, however, can only be observed if experiments have 0.4 resolution much better than the level splitting, which is 0.2 typically not the case. Since this discreteness has not 2 / been reported in experiments, the finite-size effect may 1 0 − not be relevant to experiments. Beside the finite resolu- ) 0 0.05 0.1 0.15 k tion, finite temperature most likely smoothens the level ( L/d = 400, dΔk = 10-3 1 (b) spacing induced peak structures in the actual experi- exp -4 n L/d = 800, dΔk = 5x10 ments. For very short wires, the system would behave as a quantum dot dominated by Coulomb blockade and the 0.5 Luttinger liquid behavior becomes irrelevant. For simplicity, we consider a 4-fold degenerate 1D system with 0 size L and periodic boundary condition. The use of the 0 0.05 0.1 0.15 periodic boundary condition in this section (in contrast k/k0 to the rest of this work) is to separate the finite-size effect from the non-trivial perturbation of the open boundary. Figure 6. The normalized momentum distribution of (a) short-range interacting Luttinger liquid with g = 0.2 and (b) Coulomb Luttinger liquid with U0 = 5. The momentum scale k0 = Λ for the short-range and k0 = 1/d for the Coulomb A. Momentum distribution function Luttinger liquid. The dashed lines are the corresponding momentum distribution in the limit L → ∞. We first consider the chiral static correlation function

iπx/L e − P∞ H(n) In practice, measurements have finite resolution (and C(x, 0) = lim e n=1 , (29) →0+ L 1 − e(ix−)q0 finite temperature) that smooths out the physical quan- tity and removes fine details. The effect of the finite where resolution can be presented by a convolution −1 1 − cos(qx) g(q) + q (q) − 2 Z H(n) = , (30) 0 0 0 n 8 nexp(k) = n(k − k )S(k )dk (33) q0 = 2π/L and q = nq0. The momentum distribution at Z = e−ikxC(x, 0)S˜(x)dx momentum k with respect to kF is Z n(k) = e−ikxC(x, 0)dx where S(k) is a distribution function and S˜(x) is the Z (31) Fourier transform of S(k). If we assume S(k) is a Gaus- 1 1 sin kx − P H(n) = − e n dx. sian function with standard deviation ∆k, then S˜(x) = 2 2L sin(q x/2) 2 2 0 e−∆k x /2. If ∆kL 1, there are several periods of To retrieve the infinite-size limit, one can take the limit C inside the integral interval, leading to the peak pat- 2 q0 → 0, in which q0/(2 sin(q0x/2)) ≈ 1/x + O(qx) and tern in nexp(k) at k = (n + 1/2)q0 with n being an in- the sum in the exponential is replaced by the correspond- teger. We show in Fig. 6 the momentum distribution of ing integral. The difference between the sum and the finite-length short-range and Coulomb Luttinger liquids integral counterpart is given by the Euler-Maclaurin for- compared with their infinite-length counterparts. At the mula resonant momentum kL = (2n + 1)π, the finite-size correction ∼ 1/((2n + 1)π)2 is insignificant. Therefore, the ∞ X Z ∞ peaks at resonant momentum match the momentum dis- H(n) − H(n)dn 0 tribution of the corresponding infinite system as shown n=1 in Fig. 6. Z 1 H(1) H0(1) = H(n)dn + − + O(H00(1)) (32) Next, we consider the second case when ∆kL 1 and 0 2 12 the measured spectrum is smooth. Then, we can cal- q2x2α(q ) culate the effective exponent by taking the first deriva- = − 0 0 + O(q x)3 24 0 tive d ln(nexp(k) − 1/2)/d ln(k) and display the results in Fig. 7. An interesting feature is how the finite resolu- −1 with α(q0) = (g(q0) + g (q0) − 2)/8. We empha- tion affects even the distribution in infinite-length cases. size that even for the logarithmically divergent α(q0) ∼ It is known that n(k) − 1/2 ∼ kα has a singularity in 1/2 ln (qs/q0) in the Coulomb Luttinger liquid, this diver- the first-order derivative. The convolution, regardless of 2 gence is much weaker than the quadratic decay q0. As a the exact form of the distribution function, always sup- result, the finite-size correction to the momentum density presses this singularity, thus making nexp(k) − 1/2 ∼ k function has the order of O(1/kL)2. and the effective exponent approaches 1 for k < ∆k. For 9

1.5 (a) L → ∞, Δk/Λ = 0 L → ∞, Δk/Λ = 10-3 For the short-range interacting Luttinger liquid, we can Δ Λ -2 easily see that the C(0, t) has two periods: t1 = L/vF for ) 1.0 L → ∞, k/ = 10 k -2 the three un-renormalized channels and the high-energy ΛL = 1000, Δk/Λ = 10 ln( 0.5 plasmon channel when ω vF Λ, and t2 = L/vρ for the

/d renormalized plasmon channel. As a result, the Fourier

2) 0.0 / -3 -2 -1 0 transform for L∆E/v 1 shows two groups of peaks 1 10 10 10 10

− in Fig. 8a. The group of major peaks has the gap of

) 1.5 -3 k (b) L → ∞, dΔk = 0 L → ∞, dΔk = 10 q0vρ, corresponding to the plasmon excitation; the sec- ( L → ∞, dΔk = 10-2 ond one of minor peaks has the gap of q0vF reﬂecting exp 1.0 n ΛL = 1000, dΔk = 10-2 excitations in the other channels. Due to this interfer-

ln( 0.5 ence, it is not possible to compare the finite-size ρexp(ω) d with its infinite-size counterpart. In fact, if we revert to 0.0 -3 -2 -1 0 the spinless model, i.e. removing other excitation chan- 10 10 10 10 nels, and push Λ → ∞ one can fit the peak pattern of k/k0 the finite-L into ρ(ω) computed in the infinite-size model. For the Coulomb Luttinger liquid as in Fig. 8b, the plas- Figure 7. The effective exponent of the momentum distribution for (a) short-range interacting Luttinger liquid with mon velocity is already scale-dependent, as a result, the density of state shows intricate interference pattern. g = 0.2 and (b) Coulomb Luttinger liquid with U0 = 5. This exponent always approaches 1 for k < ∆k. The scale k0 is We now study the smooth regime for L∆E/v 1. similarly defined as in Fig. 6. The density of states is given by

Z −∆E2t2/2 vF e cos ωt ρexp(ω)vF = 1 − the finite-size case, the non-trivial behavior only mani- L sin(q0vF t/2) (38) fests for k > ∆k but the finite-size correction is of the P × Im e− J(n)−i(Eρ−Eu)t dt. order 1/(kL)2 < 1/(∆kL)2 1, rendering its effective exponent almost identical to that of the corresponding Applying the same technique for the estimating the mo- infinite-size case. mentum distribution,

∞ X Z ∞ B. Finite-size density of states J(n) − J(n)dn n=1 0 Z 1 J(1) J 0(1) We now study the finite-size effect on the density of = J(n)dn + − + O(J 00(1)) states, which has direct relation to tunneling experi- 0 2 12 −1 ments. Analogous to the momentum distribution, we −iq0t vρ(q0)(g(q0) + g (q0)) vF mimic the finite resolution effect by introducing a de- = − 2 2 2 8 4 caying function e−∆E t /2 into the dynamic correlation 2 " 2 −1 2 # (q t) v (q0)(g(q0) + g (q0)) v function − 0 ρ − F + O(q v t)3. 24 8 4 0 F 2 2 −i(3Eu+Eρ)t−∆E t /2 e P∞ −J(n) (39) C(0, t) = lim e n=1 , (34) →0+ L 1 − e(−ivF t−)q0 The first-order term of Eq. (39) exactly cancels the charge where gap term i(Eρ − Eu)t in Eq. (38), thus the finite-size effect only introduce corrections of the order of (v/ωL)2 1 − e−ivρqt g(q) + g−1(q) 1 − e−ivF qt J(n) = − , (35) or higher into the density of states. As shown in Fig. 9, n 8 4n even for infinite-size systems, the effective exponent approaches zero for ω < ∆E. This is because the convolu- and the charge gaps tion fills out the singular pseudo gap at ω = 0, making −2 this gap effectively a non-zero constant. For finite-size 1 q0vF 1 q0vF 1 + g(q0) Eu = ,Eρ = . (36) systems, non-trivial behavior only appears for ω > ∆E, 4 2 4 2 2 with a negligible correction proportional to (v/∆EL)2. The first term corresponds to three unrenormalized We note that for a finite-system with open boundaries, spin/valley channels, while the second one represent the the same conclusion can be made; however, the transi- renormalized charge excitation. The density of states as tion from the bulk to boundary exponent happens with πv(sin(πx/L)L)−1 replacing v/x as in the limit L → ∞. a function of ω = E − EF is defined through the Fourier transform In fact, the suppressed influence of the finite size on the observed density of states has been mentioned in the liter- 1 Z ature before [16–18], but our results show quantitatively ρ (ω) = eiωt [C(0, t) + C(0, −t)] dt. (37) exp 2π that the finite size effects are not a serious problem for 10

0.8 -3 (a) ΛL = 400, ΔE / E0 = 2x10 ductance is more visible over the range of two or more 0.6 orders of magnitude in the independent variable. We believe that the clear theoretical diﬀerence between short- 0.4 and long-range Luttinger liquids established in this pa- 0.2 per should be experimentally observable provided that F

v 0 the experimental tuning variables (temperature and bias )

ω 0 0.1 0.2 0.3 0.4

( voltage) are varied over a large dynamical range. -3

exp (b) L/d = 400, ΔE / E0 = 2x10 We also study the eﬀect of the high energy crossover

ρ 1.0 to the free Fermi gas and the finite-size effect. This high-energy crossover is due to short-distance behavior 0.5 of the interaction and depends on the microscopic ultraviolet regularization. Near the high-energy crossover, the 0.0 observed tunneling conductance deviates from the low- 0 0.1 0.2 0.3 0.4 energy theoretical prediction and approaches a constant ω/E0 value. In addition, we show that the finite-size of the Figure 8. The density of states of (a) short-range interacting system only introduces corrections of the second order or Luttinger liquid with g = 0.2 and (b) Coulomb Luttinger higher, which are negligible in the measured spectrum. liquid with U0 = 5. The interference pattern is caused by On the other hand, the finite resolution of the measure- different existing excitation propagating velocities. ment may replace the true Luttinger power law by trivial laws, i.e. the exponents of the momentum distribution 1.0 and the density of states may approach one and zero re- Δ Δ -3 (a) L → ∞, E / E0 = 0 L → ∞, E / E0 = 8x10 -3 -3 spectively. These features should be taken into account L → ∞, ΔE / E0 = 10 ΛL = 2000, ΔE / E0= 8x10 to ensure that the measured power law is physical and 0.5

) not artifacts of measurement protocols. ω Having considered all factors that may aﬀect the tun- ln( 0.0 neling conductance, we ﬁnally conclude by considering /d -3 -2 -1 0 1

)) 10 10 10 10 10 (Fig. 10) a comparison between theory and experiments.

ω (b) ( We focus on carbon nanotube experiments in Refs. [13– L → ∞, ΔE/E = 0 -3 (b) 0 L → ∞, ΔE / E0 = 8x10 exp 1.0 -3 -3 15]. The Coulomb Luttinger liquid has two parame- ρ L → ∞, ΔE/E0 = 10 L/d = 3000, ΔE / E = 8x10 0 ters: the interaction strength U and the energy scale

ln( 0

d 0.5 or crossover energy E0 = vF /d. Unfortunately, all the reported data have dynamic range of less than 2 or- 0.0 ders of magnitude, hence ﬁtting to ﬁnd both U0 and -3 -2 -1 0 1 10 10 10 10 10 E0 is not possible as shown earlier. (In addition, such

ω/E0 a small dynamic range makes any distinction between long-range and short-range Luttinger liquid models es- Figure 9. The effective exponent of the density of states for sentially impossible.) However, we assume that the in- (a) short-range interacting Luttinger liquid with g = 0.2 and teraction strength is universal for all carbon nanotubes (b) Coulomb Luttinger liquid with U0 = 5. This exponent while the crossover energy scale can vary depending on vanishes for ω < ∆E. the sample. Therefore, we fix the value of U0 = 1.7 and treat E0 as the fitting parameter. It is noted that the choice U0 = 1.7 is arbitrary, as the data range is insuffi- 1D Lutinger liquid studies, either for the short-range or cient for an accurate fitting; we can also choose another the long-range model. U0 and the values of E0 will change accordingly. We also add a rescaling factor so that V → ηV where η(< 1) ac- counts for the real voltage across the tunneling contact IV. CONCLUSION after subtracting out the voltage drop along conductors. In Fig. 10, we show the fitted parameters along with the In this paper, we theoretically compare the tunneling value g obtained by fitting the data to a line. This result conductance in 1D systems between the short-range and suggests that different values of the measured interaction the long-range interacting Luttinger liquid models. The parameter g may rise entirely from the sample-to-sample logarithmic divergence of the long-range Coulomb inter- variations in the crossover scale E0. This should be taken action gives raise to a scale-dependent effective exponent into account in future experiments on Luttinger liquids. which increases at lower energy. However, this exponent Our theoretical analysis for carbon nanotube assumes varies slowly, giving an impression of an actual power law SU(2)×SU(2) symmetry, which is valid under most ex- when the dynamic range (of temperature or bias voltage) perimental conditions. For example, in Ref. [13], a pair of is around one order of magnitude or less. The difference pentagon and heptagon is inserted to the hexagonal car- between short-range and long-range Luttinger liquid con- bon lattice to create a kink that acts as a semiconductor 11 junction between two straight nanotube segments. How- can be easily generalized to the multiple non-degenerate ever, the bulk of each segment is still pristine and the val- electron species situation. As a result, the scale depen- ley symmetry is not broken in the bulk. The degeneracy dence of the Luttinger exponent with decreasing energy, of the system indeed simplifies the problem significantly. which is the signature of long-range interaction, should In principle, tunneling between multiple (more than 2) still manifest qualitatively although some of the quanti- Luttinger liquids is complicated due to a large number of tative details of our work will change depending on the possible tunneling channels and possibly different behav- precise details of which symmetry is broken and how it iors in each channel [26–37]. However, in a degenerate is broken. system as in our work, the symmetry of electron species enforces the same V0,T −dependence on all the channels, thus effectively reducing the problem to the tunneling between only two Luttinger liquid modes. Although beyond ACKNOWLEDGMENTS the scope of this paper, the symmetry breaking situation (e.g. valley polarization) can be straightforwardly incor- This work is supported by the Laboratory for Physical porated in the Luttinger liquid formalism. Hence, our Sciences. A. I. acknowledges financial support from the theoretical treatment of long-range Coulomb interaction Fulbright Foundation.

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3 2.0 40 10 -3 (a) E0 = 32 V, η = 0.79, U0 = 1.7 (c) E0 = 0.14 V (e) E0 = 1.2x10 V 1.5 η = 0.29, U0 = 1.7 η = 0.087, U = 1.7 T = 150 K 30 0 2 10 1.0 T = 15 K T = 50 K 20 T = 15 K 1 10 g = 0.24 g = 0.31 g = 0.76

) 0.5 0.01 0.10 0.01 0.1 0.01 0.1 µS ( 80 3.0 60 G -3 (b) E0 = 3.0 V, η = 0.18, U0 = 1.7 (d) E0 = 0.023 V (f) E0 = 3.7x10 V 50 2.0 η = 0.36, U0 = 1.7 η = 0.16, U0 = 1.7 60 T = 150 K 40 1.0 T = 20 K T = 100 K T = 15 K 30 g = 0.27 g = 0.35 g = 0.54 40 0.1 0.01 0.1 0.01 0.1 Voltage bias (V)

Figure 10. Coulomb Luttinger liquid tunneling conductance ﬁtted to experimental data: (a) L-L boundary tunneling from Fig. 4b Ref.[13], (b) L-L bulk tunneling from Fig. 3a Ref.[14], (c) L-L bulk tunneling from Fig. 3b Ref.[15], (d) L-L bulk tunneling from Fig. 3c Ref.[15], (e) L-M boundary tunneling from Fig. SM4a supplement Ref.[15], (f) L-M tunneling from Fig. SM4b supplement Ref.[15]