Tunneling conductance of long-range Coulomb interacting Luttinger liquid

Due to the low dimensionality, one-dimensional interacting electron liquids, called the Luttinger liquid, manifest radically different behaviors from the conventional Fermi liquid. A simple model of the short-range interacting Luttinger liquid suggests a power-law decaying density of states as well as a smooth momentum distribution function with no discontinuity around the Fermi surface, which can readily be tested through tunneling experiments. However, most physical systems have long-range interaction, most notably the Coulomb interaction which can significantly modify predictions from the short-range interaction model. In this paper, we revisit the tunneling theory for the one dimensional interacting electrons for the long-range Coulomb interaction. 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 defined by a Luttinger exponent similar to short-range interacting systems, the effective exponent is scale-dependent and slowly increases with decreasing energy. This implies that the measured exponent may vary with the experimental conditions. In particular, any measured exponent varies with the temperature and bias voltage ranges used in obtaining the experimental data. 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
Luttinger 1D liquids emerge from interacting onedimensional many-electron systems where the Fermi surface is two discrete points rather than a connected surface as in higher dimensional cases. Two and threedimensional interacting systems are conventionally described by the Fermi liquid model where the excitations are individual quasi-particles with modified or renormalized properties (e.g. effective mass) from the bare particle. Fermi liquids manifest qualitative resemblance to the free Fermi gas, for example, the discontinuity of the momentum distribution through the Fermi momentum and singularities in the spectral function representing quasiparticles. However, in one-dimensional systems, quasiparticle excitations are replaced by collective excitations even for very weak interaction, leading to the complete disappearance of the one-to-one correspondence with the non-interacting Fermi gas characterizing Fermi liquids. Most remarkably, the momentum distribution function is continuous through the Fermi point and the density of states displays a pseudo gap at the Fermi energy, indicating the breakdown of the quasi-particle picture. More non-Fermi liquid phenomenon include charge-spin separation, and power-law scaling of charge and spin correlations [1][2][3][4]. Assuming a zero-range (or short-range) interaction (a Dirac delta function), the density of states decays as (E − E F ) α where α is a nonuniversal exponent characterized by the interaction strength. The exponent is a true exponent (i.e. a constant for a given interaction model), but is nonuniversal since it depends on the actual interaction strength. In addition, this constant α also shows up in the tunneling conductance, giving it a distinct power-law decaying behavior. Specifically, if the tunneling is between a Luttinger liquid characterized by the exponent α and an ordinary Fermi-liquid metal, the tunneling conductance at temperature T and bias voltage V 0 is G = dI/dV 0 ∝ V α 0 for eV 0 k B T and G ∝ T α for eV 0 k B T . For tunneling between two Luttinger liquids, the exponent is simply doubled [5][6][7]. 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 0 (as long as k B T, eV 0 E F , where E F is the Fermi energy). Indeed, tunneling experiments have confirmed this behavior in many physical systems. Earliest attempts include chiral Luttinger liquids found in the edge mode of fractional quantum Hall fluids [7][8][9][10] and the power law in the optical response from quasi-one-dimensional conductors [11,12], suggesting the Luttinger liquid nature of these systems. Recently, studies have focused on carbon nanotubes where extreme isolating conditions can be obtained to create a strongly correlated non-chiral 1D electron system. Tunneling experiments on carbon nanotubes also show evidences for power laws characterizing Luttinger liquids [13][14][15].
It is instructive to review the theoretical description of the Luttinger liquid based on the bosonization method, which in principle can give exact solutions for any types of interaction provided that the back-scattering is ignored and the dispersion can be linearized. If we assume that the system has only contact charge-charge interaction with a strength U : H int = U/2 ρ(x) 2 dx, the plasmon velocity is renormalized by g = v F /v ρ = (1 + N U/(πv F )) −1/2 where N is system degeneracy, e.g. N = 2 for spinful systems or N = 4 for carbon nanotubes (including spin and valley without spin-orbit coupling). The exponent of the density of states is then α bulk = (g + g −1 − 2)/2N at the bulk of the system and α bound = (g −1 − 1)/N at the open boundary [16,17]. For systems with repulsive interaction, U > 0, so g < 1 and α bound > α bulk > 0.
The simplicity of the short-range interacting Luttinger liquid model might be attractive, but the model of zerorange interaction is only an approximation. Most real systems have long-range interactions, most notably the electronic Coulomb interaction. If the Fourier transform of the interaction approaches a finite constant at the zero transfer momentum limit, the low-energy physics can be described by a short-range interaction model without great loss of accuracy. However, the Coulomb interaction is special because of its logarithmic divergence at small momentum. One way to obtain a short-ranged interaction is to consider an appropriately screened Coulomb interaction [18,19], imagining the carbon nanotube is placed inside a larger metallic tube with a radius R s much larger than the radius R of the nanotube . Then, the interaction potential is totally screened out and can be considered as the classical energy of the E-field trapped between the nanotube and the metallic cylinder H int = e 2 /(4π ) ln(R s /R) ρ(x) 2 dx. This is exactly the form of the zero-range interaction, so even if the nanotube has long-range interaction, the Luttinger liquid considerations from the short-range case still apply with an appropriate g provided, of course, the nanotube is indeed enclosed in a metallic cage. However, the ambiguity here is the value R s since in reality no specific metal tube encloses the nanotube. Note that the logarithmic divergence of the effective interaction in H i nt as R s goes to infinity is the well-known log divergence of the 1D Coulomb interaction arising from its long-range nature, which is simply cut off by taking R s to be finite.
A more rigorous approach is not to make an ad hoc short-range interaction approximation, and instead use the long-range 1D Coulomb interaction itself in the Luttinger liquid theory, i.e. performing bosonization with the logarithmically divergent interaction originating from the 1/x 1D Coulomb interaction. Within this description, the power law behavior is no longer valid. Indeed, for the density-density correlation, the 4k F oscillation decays as x − √ ln x , slower than any power law [20,21]. The 1D Coulomb Luttinger liquid is thus an effective Wigner crystal at finite length with the 4k F density oscillation decaying very slowly over distance (since the correlation dies out eventually there cannot be any real long-range order). Moreover, due to the log-divergence, the "effective exponent" of a Coulomb Luttinger liquid is scaledependent, i.e. α is also a function of energy [22]. This scale-dependence of the effective Luttinger exponent in the Coulomb Luttinger liquid, in contrast to the constant exponent (depending only on the effective interaction strength g) for the short-range interaction model, sets apart the long-range Coulomb interaction case, making it also rather a difficult technical problem since the exponent depends on energy or momentum or length scales being studied in the problem. It is unclear apriori how one can investigate the Coulomb Luttinger liquid simply by mapping it into a corresponding short-range interaction model although this is often done in the interpretation of the experimental 1D results.
In this paper, we investigate the effect of the longrange Coulomb interaction on the tunneling conductance using the long-range interaction description of a scaledependent exponent. One specific consequence of the scale-dependent exponent of the Coulomb Luttinger liquid may be the manifestation of the sample-to-sample variations in the measured tunneling exponent often seen within even a given type of 1D physical systems. Since the Coulomb Luttinger liquid by definition does not have a constant exponent (i.e. the exponent varies slowly over the energy scale of measurements), it is important to analyze the tunneling experiment in depth using the longrange interaction model to figure out how this scale dependence might manifest in the tunneling spectroscopy. This is the main goal of the current work. A secondary goal is to investigate the role of the finite length of the 1D system (e.g. carbon nanotube or semiconductor quantum wire) in the tunneling experiments to check whether an implicit or explicit length dependence affects the tunneling exponent, particularly in the context of the scaledependent exponent in the Coulomb Luttinger liquid.
The rest of the paper is organized as follows. In Sec. II we provide the theoretical description for a Coulomb Luttinger liquid with the open boundary condition and compare its properties with the short-range interacting Luttinger liquid counterpart. We also discuss the crossover to the free Fermi gas behavior at high energy, which may not be obvious in the short-range model but appears naturally in the long-range Coulomb Luttinger liquid. In Sec. III, we study the effect of the finite system, finding it to be irrelevant as long as the 1D system is not too short. We conclude in Sec. IV summarizing our main findings and discussing possible experimental implications of our results.

A. Bosonization in open-boundary systems
We first present the bosonization study of a 1D spinless system of size L and open boundaries at x = 0 and x = L which are appropriate for tunneling measurements. Due to the open boundary condition, we only have one type of fermionic operator instead of two independent left and right-moving electron species as in systems with periodic boundary condition. Accordingly, the fermion field can be decomposed as The construction implies that ψ R (x) = −ψ L (−x). The linearized Hamiltonian with = 1 is It is noted that for an open boundary system, the discretized momentum is k = nπ/L instead of 2π/L as in the periodic one. We define bosonic creation and annihilation operators with q = mπ/L > 0 as and [a q , a † q ] = δ q,q . The right-moving chiral fermion field is N e is the total number of electrons, and F is the Majorana fermion operator such that F commutes with all bosonic operators, [N e , F ] = −F and F 2 = 1.
Within the bosonization framework, the interacting Hamiltonian is rewritten as where v N = v F +U (0)/(2π), and U (q) = L −L U (x)e iqx dx.
After the Bogoliubov transformation, the Luttinger interaction parameter is defined as and the collective plasmon mode velocity is For a short-range interaction, U (q) is a constant and hence the constancy of g as a Luttinger exponent for a given model, i.e., a given U ; while for a longrange interaction U (q) is obviously scale-dependent as it depends explicitly on the momentum q, resulting in the remarkable divergence g(q) ∼ ln(1/q) in the case of Coulomb interaction. The correlation For brevity, we drop the q argument in v ρ (q) and g(q), and use cosh θ = (g 1/2 + g −1/2 )/2 and sinh θ = (g 1/2 − g −1/2 )/2. We have where f B (z) = e βz − 1 −1 is the Bose-Einstein distribution coming from the bosonic plasmon excitation and β = 1/k B T with T is the electron temperature. Equations (8) and (9) are the exact expressions and do not assume any specific form of the interaction. The results can be readily extended to an N -fold degenerate system (e.g. spin, valley). Particularly, if the system has only charge-charge interaction, all the other channels beside the plasmon one stay unrenormalized. The chiral corre-lation function can be written as with U (q) → N U (q) in Eq. (6) and We can perform a quick check on the power law for T = 0, L → ∞ (in that case q π/L → dk) and for zero-range interaction g(q) = g. The spinless (N = 1) dynamic chiral correlation function is This term is the primary contribution to the correlation function because the other term C(x, −x; t) has fast 2k F oscillation of e 2ik F x and is further suppressed by These are of course wellknown results provided here for the sake of completeness and to set a context for our work.

B. Coulomb Luttinger effective exponent
In this section, we assume the semi-infinite 1D limit L → ∞ (the effect of finite L is discussed later). We study two models of interactions in the four-fold degenerate carbon nanotube system: (i) short-range interaction with constant g up to a cut-off Λ, i.e.
As long as ω < E 0 = Λv F , the density of states ρ(ω) ∝ ω (g −1 −1)/4 at the boundary and ρ(ω) ∝ ω (g −1 +g−2)/8 . It is noted that the cut-off Λ is purely artificial, in fact, many theoretical works only consider the large distance asymptote and set where κ is the dielectric constant and d is proportional to the transverse size of the nanotube, which is necessary to regularize the 1D Coulomb coupling. The corresponding g is with v F = 8 × 10 5 m/s and K 0 is the Bessel function. Note that in Ref. [18], the Coulomb Luttinger liquid is approximated by an effective short-range interaction model with a constant g as where R is the radius of the tube and R s is some screening length; for R s ∼ 100R, g ∼ 0.2. We emphasize that R s is unknown and arbitrary in the experimental systems, and often used simply as a fitting parameter uncritically.
We begin by studying the tunneling density of states in the bulk and the boundary of a Coulomb Luttinger liquid. The following argument is based on Wang et al. [22], and extended to include the open boundary condition of relevance to tunneling spectroscopy. The bulk density of states in a carbon nanotube is given by In the phase factor Φ bulk (t), when q < A/t with A is some number, the factor 1 − e iqvt ≈ 0 and for q > A/t, we can ignore the fast oscillating exponential, thus the leading order term in the phase factor is where we have used the low-q asymptotic form g(q) = √ U 0 ln 1/2 (q s /q) with U 0 = 8e 2 /(κπv F ) and q s ≈ 1.12e 1/U0 /d in the integration instead of the full form Eq. (13). The power law of the density of states is then where the scale ω s = 20v F q s √ U 0 . Extending the argument to the limit x → 0, the boundary phase factor is Accordingly, the local density of states at the open boundary is As γ depends on the energy ω, strictly speaking this is not a power law. However, we can define an effective exponent α = d ln(ρ(ω))/d ln ω for the bulk and boundary tunneling density of states We expect these exponents to appear in the tunneling conductance of the Coulomb Luttinger liquid. We note that the energy dependence of α as reflected in the explicit appearance of ω in the right hand side of Eq. (20) leads to an ill-defined scale-dependent exponent in the Coulomb Luttinger liquid in contrast to the constant (but nonuniversal) exponent in the short-range interaction model.

C. Tunneling conductance
Suppose electrons tunnel between systems 1 and 2 at voltage bias V 0 , the current is given by [5,7,23] where ψ 1 and ψ 2 correspond to the electron correlation function of the system 1 and 2 respectively. This is basically the Fermi golden rule expressed in the Fourier transform. The differential conductance can be computed by We specify the tunneling conductance in two distinct experimental setups: the two systems 1 and 2 are two identical Luttinger liquids, denoted as L-L, and, system 1 is the Luttinger liquid sample and system 2 is a conventional 3D Fermi liquid metal contact, denoted as L-M, with C 2 (x, x; t) ∼ 1/t (assuming the temperature is much less than the metal Fermi temperature) From Eqs. (23) and (24), we can expect that the exponent of the L-L tunneling is two times as large as that of tunneling through L-M contact. In Fig. 1, we show the calculated L-M tunneling conductance of a short-range interacting Luttinger liquid 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 saturation regime (eV 0 k B T or k B T eV 0 ) where G is independent of V 0 (or T ), thus the effective exponent approaches zero, the boundary regime (eV 0 , k B T v ρ /x) where the power law is given by the boundary relation α = (g −1 − 1)/4 = 1.0 and the bulk regime (eV 0 , k B T v ρ /x) where the power law is given by the bulk relation α = (g + g −1 − 2) = 0.4. Note that, although there are noisy fluctuations in the exponent in the crossover regimes, the bulk and boundary exponents clearly manifest themselves as constant derivatives in Fig. 1.
Because the scale-dependent Luttinger parameter in the Coulomb Luttinger liquid is an explicit function of the momentum, we can naturally define an energy scale E 0 = v F /d. In Fig. 2, we show the directly calculated L-M effective exponent of a Coulomb Luttinger liquid with U 0 = 5. Similar to the short-range case, there are also three distinct regimes. However, the striking difference is the continuously increasing effective exponent with decreasing energy. For the conductance measurement with respect to the voltage bias eV 0 (see Fig. 2(a)), the numerical result is consistent with Eq. (20). In addition, when x/d = 0 x/d = 500 x/d = 5000 x/d → ∞ (a) x/d = 0 x/d = 500 x/d = 5000 x/d → ∞ (b) eV 0 k B T (see Fig. 2(b)), the effective exponent of G with respect to T has the same form as Eq. (20) with ω substituted by k B T and ω s replaced by T s = 7v F q s √ U 0 .

D. Universal scaling function
Another contrasting property between the Coulomb and the short-range Luttinger liquid is the universal scaling with eV 0 /k B T , i.e. the tunneling conductances of a short-range Luttinger liquid at different temperatures can collapse into a single function of eV 0 /k B T . Using the asymptotic correlation function (no cut-off), the scaling function with respect to µ = eV 0 /k B T is [5,6,14] where Γ and Ψ are the gamma and digamma functions. A single scaling function is a direct result of a scale invariant interaction constant g where a conformal transformation can transform between the voltage and the temperature (corresponding to the real time and imaginary time boundary respectively). Clearly, the existence of a scale-dependent exponent rules out such a universal scaling function uniquely determined by eV 0 /k B T for the Coulomb Luttinger liquid in contrast to the short-range interaction model. In Fig. 3, we plot the scaled L-M tunneling conductance of a Coulomb Luttinger liquid at different temperatures, which clearly does not converge to a single function as in the short-range case. The predictions of Fig. 3 should also be directly experimentally verifiable using tunneling data provided that the varying range of the temperature and the bias voltage is of several orders of magnitudes. We conclude this section by comparing the tunneling conductance of a short-range Luttinger liquid with two Coulomb Luttinger liquids characterized by different parameters at T = 0. As can be seen from Fig. 4, over a range of three orders of magnitude, the difference between short-range and long-range cases is obvious: in the log-log scale the tunneling conductance of a short-range interacting system is a line corresponding to an ideal power law while that for a Coulomb system is a downward bending curve characterizing a scale-dependent exponent. However, within a narrow range of one order of magnitude, the two are almost indistinguishable. Moreover, Coulomb Luttinger liquids of different parameter sets are also indistinguishable within one order of magnitude of the tuning parameter. This suggests that we can only conclusively study the Coulomb Luttinger liquid if the dynamic range of the independent variable (i.e. T, V 0 ) is more than 2 orders of magnitude. We believe that the existing experimental literature on tunneling measurements is limited to a very narrow voltage and temperature range, explaining the apparent (and misleading) success of the short-range Luttinger liquid model with constant exponents. A true verification of the theory necessitates the observation of the scale dependence shown in Figs. 2, 3, and 4, which would require a large variation in the dynamical range of temperature and bias voltage.

E. High-energy crossover to free Fermi gas
If we assume that the interaction vanishes at very high momentum, which is true for all interactions with ultraviolet regularization, then the Luttinger liquid can cross over to the free Fermi gas defined by g → 1. In this section, we investigate the system near this high-energy crossover. For the short-range model defined previously, it is obvious that the crossover energy must be associated with the cut-off momentum Λ while for the Coulomb interaction, the interaction potential K 0 (qd) decays exponentially for q 1/d so the crossover energy must be related to 1/d. Thus, for the Coulomb Luttinger liquid, the crossover energy scale is physically defined whereas for the short-range model, it depends on an ad hoc cut off. Hence, the energy scale we define earlier, i.e. E 0 = v F Λ for short-range interaction and E 0 = v F /d for Coulomb interaction, is also the high-energy crossover scale. In Fig. 5, we show the L-M tunneling conductance as a function of the voltage bias and temperature. The usable region where one can extract a meaningful power law is for eV 0 (k B T ) much higher than k B T (eV 0 ) but still less than the crossover scale E 0 . As a result, if eV 0 is close to E 0 , one cannot extract the power law with respect to the temperature and vice versa. Moreover, in the case of Coulomb Luttinger liquid, the theoretical scaledependent exponent already intrinsically contains a decay at high energy even though it is derived using low energy assumption because the definition of the long-range interaction already contains the ultra-violet regularization through the transverse size d. Therefore, the consistency between numerical results and theoretical predictions extends to much higher energy than the short-range interacting Luttinger liquid. The fact that the Coulomb Luttinger liquid has a built-in physical ultraviolet regularization in contrast to a completely arbitrary cut-offdependent regularization in the short-range model makes the long-range interaction model much more theoretically meaningful than the artificial short-range Luttinger liquid model extensively used in the experimental literature because of its simplicity.

III. FINITE-SIZE EFFECT
In this section, we study the effect of finite size (i.e. the finite length of the 1D system in the experimental tunneling measurements) on the experimentally extracted properties of the Luttinger liquid, e.g. the momentum distribution and the density of states. We will show that the finite system (as long as the system length is much longer than the ultraviolet cutoff length d) does not alter the Luttinger liquid properties except for inducing discrete peaks pattern to the measured spectrum. This effect, however, can only be observed if experiments have resolution much better than the level splitting, which is typically not the case. Since this discreteness has not been reported in experiments, the finite-size effect may not be relevant to experiments. Beside the finite resolution, finite temperature most likely smoothens the level spacing induced peak structures in the actual experiments. For very short wires, the system would behave as a quantum dot dominated by Coulomb blockade and the Luttinger liquid behavior becomes irrelevant. For simplicity, we consider a 4-fold degenerate 1D system with size L and periodic boundary condition. The use of the periodic boundary condition in this section (in contrast to the rest of this work) is to separate the finite-size effect from the non-trivial perturbation of the open boundary.

A. Momentum distribution function
We first consider the chiral static correlation function where q 0 = 2π/L and q = nq 0 . The momentum distribution at momentum k with respect to k F is To retrieve the infinite-size limit, one can take the limit q 0 → 0, in which q 0 /(2 sin(q 0 x/2)) ≈ 1/x + O(qx) 2 and the sum in the exponential is replaced by the corresponding integral. The difference between the sum and the integral counterpart is given by the Euler-Maclaurin formula with α(q 0 ) = (g(q 0 ) + g −1 (q 0 ) − 2)/8. We emphasize that even for the logarithmically divergent α(q 0 ) ∼ ln 1/2 (q s /q 0 ) in the Coulomb Luttinger liquid, this divergence is much weaker than the quadratic decay q 2 0 . As a result, the finite-size correction to the momentum density function has the order of O(1/kL) 2 .
In practice, measurements have finite resolution (and finite temperature) that smooths out the physical quantity and removes fine details. The effect of the finite resolution can be presented by a convolution where S(k) is a distribution function andS(x) is the Fourier transform of S(k). If we assume S(k) is a Gaussian function with standard deviation ∆k, thenS(x) = e −∆k 2 x 2 /2 . If ∆kL 1, there are several periods of C inside the integral interval, leading to the peak pattern in n exp (k) at k = (n + 1/2)q 0 with an integer n. We show in Fig. 6 the momentum distribution of shortrange and Coulomb Luttinger liquids compared with its infinite-length counterpart. At the resonant momentum kL = (2n+1)π, the finite-size correction ≈ 1/((2n+1)π) 2 is insignificant. Therefore, the peaks at resonant momentum match the momentum distribution of the corresponding infinite system as shown in Fig. 6. Next, we consider the second case when ∆kL 1 and the measured spectrum is smooth. Then, we can calculate the effective exponent by taking the first derivative d ln(n exp (k) − 1/2)/d ln(k) and display the results in Fig. 7. An interesting feature is how the finite resolution affects even the distribution in infinite-length cases. It is known that n(k) − 1/2 ∼ k α has a singularity in the first-order derivative. The convolution, regardless of the exact form of the distribution function, always suppresses this singularity, thus making n exp (k) − 1/2 ∼ k and the effective exponent approaches 1 for k < ∆k. For the finite-size case, the non-trivial behavior only manifests for k > ∆k but the finite-size correction is of the order 1/(kL) 2 < 1/(∆kL) 2 1, rendering its effective exponent almost identical to that of the corresponding infinite-size case.

B. Finite-size density of states
We now study the finite-size effect on the density of states, which has direct relation to tunneling experiments. Analogous to the momentum distribution, we mimic the finite resolution effect by introducing a decaying function e −∆E 2 t 2 /2 into the dynamic correlation function where and the charge gaps The first term corresponds to three unrenormalized spin/valley channels, while the second one represent the renormalized charge excitation. The density of states as a function of ω = E − E F is defined through the Fourier transform For the short-range interacting Luttinger liquid, we can easily see that the C(0, t) has two periods: t 1 = L/v F for the three un-renormalized channels and the high-energy plasmon channel when ω v F Λ, and t 2 = L/v ρ for the renormalized plasmon channel. As a result, the Fourier transform for L∆E/v 1 shows two groups of peaks in Fig. 8a. The group of major peaks has the gap of q 0 v ρ , corresponding to the plasmon excitation; the second one of minor peaks has the gap of q 0 v F reflecting excitations in the other channels. Due to this interference, it is not possible to compare the finite-size ρ exp (ω) with its infinite-size counterpart. In fact, if we revert to the spinless model, i.e. removing other excitation channels, and push Λ → ∞ one can fit the peak pattern of the finite-L into ρ(ω) computed in the infinite-size model. For the Coulomb Luttinger liquid as in Fig. 8b, the plasmon velocity is already scale-dependent, as a result, the density of state shows intricate interference pattern.
We now study the smooth regime for L∆E/v 1. The density of states is given by Applying the same technique for the estimating the momentum distribution, The first-order term of J(n) exactly cancels the charge gap term i(E ρ − E u )t in Eq. (35), thus the finite-size effect only introduce corrections of the order of (v/ωL) 2 or higher into the density of states. As shown in Fig. 9, even for infinite-size systems, the effective exponent approaches zero for ω < ∆E. This is because the convolution fills out the singular pseudo gap at ω = 0, making this gap effectively a constant. For finite-size systems, non-trivial behavior only appears for ω > ∆E, with a correction proportional to (v/∆EL) 2 which is negligible. We note that for a finite-system with open boundaries, the same conclusion can be made; however, the transition from the bulk to boundary exponent happens with πv(sin(πx/L)L) −1 replacing v/x as in the limit L → ∞.
In fact, the suppressed influence of the finite size on the observed density of states has been mentioned in the literature before [16][17][18], but our results are the first quantitative calculations clearly showing that the finite size effects are not a serious problem for 1D Lutinger liquid studies, either for the short-range or the long-range model.

IV. CONCLUSION
In this paper, we theoretically compare the tunneling conductance in 1D systems between the short-range and the long-range interacting Luttinger liquid model. The logarithmic divergence of the long-range Coulomb interaction gives raise to a scale-dependent effective exponent which increases at lower energy. However, this exponent varies slowly, giving an impression of an actual power law when the dynamic range (of temperature or bias voltage) is around one order of magnitude or less. The difference between short-range and long-range Luttinger liquid conductance is more visible over the range of three or more  orders of magnitude in the independent variable. We believe that the clear theoretical difference between shortand long-range Luttinger liquids established in this paper should be experimentally observable provided that the experimental tuning variables (temperature and bias voltage) are varied over a large dynamical range. We also study the effect of the high energy crossover to the free Fermi gas and the finite-size effect. This high-energy crossover is due to short-distance behavior of the interaction and depends on the microscopic ul-traviolet regularization. Near the high-energy crossover, the observed tunneling conductance deviates from the low-energy theoretical prediction and approaches a constant value. In addition, we show that the finite-size of the system only introduces corrections of second order or higher, which are negligible in the measured spectrum. On the other hand, the finite resolution of the measurement may replace the true Luttinger power law by trivial laws, i.e. the exponents of the momentum distribution and the density of states may approach one and zero respectively. These features should be taken into account to ensure that the measured power law is physical and not artifacts of measurement protocols.
Having considered all factors that may affect the tunneling conductance, we finally conclude by considering ( Fig. 10) a comparison between theory and experiments. We focus on carbon nanotube experiments in Refs. [13][14][15]. The Coulomb Luttinger liquid has two parameters: the interaction strength U 0 and the energy scale or crossover energy E 0 = v F /d. Unfortunately, all the reported data have dynamic range of less than 2 orders of magnitude, hence fitting to find both U 0 and E 0 is not possible as shown earlier. (In addition, such a small dynamic range makes any distinction between longrange and short-range Luttinger liquid models essentially impossible.) However, we assume that the interaction strength is universal for all carbon nanotubes while the crossover energy scale can vary depending on the experimental details. Therefore, we fix the value of U 0 = 1.7 and treat E 0 as the fitting parameter. It is noted that the choice U 0 = 1.7 is arbitrary, as the data range is insufficient for an accurate fitting, we can also choose another U 0 and the values of E 0 will change accordingly. We also add a rescaling factor so that V → ηV where η(< 1) accounts for the applicable voltage drop in the 1D system after subtracting out the voltage drop along the contacts. In Fig. 10, we show the fitted parameters along with the value g obtained by fitting the data to a line. This result suggests that different values of the measured interaction parameter g may rise entirely from the sample to sample variations in the crossover scale E 0 . This should be taken into account in future experiments on Luttinger liquids. G(µS)