---
abstract: 'We analyse the finite temperature charge stiffness $D(T>0)$, by a generalization of Kohn’s method, for the problem of a particle interacting with a fermionic bath in one dimension. We present analytical evidence, using the Bethe ansatz method, that $D(T>0)$ is finite in the integrable case where the mass of the particle equals the mass of the fermions and numerical evidence that it vanishes in the nonintegrable one of unequal masses. We conjecture that a finite $D(T>0)$ is a generic property of integrable systems.'
address:
- |
     (1) Institut Romand de Recherche Numérique en Physique des Matériaux (IRRMA),\
    PHB-Ecublens, CH-1015 Lausanne, Switzerland\
    (2) Département de Physique de la Matière Condensée,\
    24,quai E. Ansermet, CH-1211 Genève, Switzerland
- |
    J. Stefan Institute, University of Ljubljana,\
    61111 Ljubljana, Slovenia
author:
- 'H. Castella$^{(1,2)}$, X. Zotos$^{(1)}$'
- 'P. Prelovšek'
date: Received           
title:  Integrability and ideal conductance at finite temperatures
---

This work relates the finite temperature charge transport to the response of energy levels to an infinitesimal flux and proposes a connection to the integrability of quantum systems.

Starting with the formulation by Kohn [@Kohn], the charge stiffness (or Drude weight) representing the weight of the delta function contribution $D\delta(\omega)$ part in the dynamical conductivity $\sigma(\omega)$, has been investigated as a criterion for a metallic, superconducting or insulating state [@Suth; @Scal]. The original approach [@Kohn] relates $D$ to the response of the ground state energy to a magnetic flux and thus requires only the calculation of an equilibrium property, bypassing a complete evaluation of the Kubo formula[@Kubo].

A similar concept appears also in the study of the conductance of disordered metallic systems. Starting with the work of Thouless[@Thoul] a close relation has also being established between the conductance, the sensitivity on boundary conditions (being equivalent to the introduction of a flux) and the statistical properties of [*single-particle*]{} energy spectra [@w] of metallic systems.

Recently it has also been observed that the level statistics in [*many-particle*]{} correlated systems is closely related to the integrability of the system[@Bell; @DiSZ]. Still the implications of this fact on transport quantities in correlated fermion systems at finite temperatures $T>0$, e.g. a nonzero d.c. resistivity $\rho(T)>0$ or on the contrary a possible finite charge stiffness $D(T)>0$, have not been known so far.

In this work we analyse the relation between integrability and transport in the generic problem of a single tagged particle moving in a bath of fermions by a generalization of Kohn’s method at finite temperatures.

[*Kohn’s method at*]{} $T > 0$: We consider a general tight-binding Hamiltonian of the form $$\hat H=-t\sum_i( e^{i\phi} d^{\dagger}_{i+1} d_{i} + H.c.)+\hat H_{int}=
\hat T + \hat H_{int}, \label{eq1}$$ representing a 1D system of length $L$ with periodic boundary conditions pierced by a flux $L \phi$, using the Peierls construction. $\hat T$ is the kinetic and $\hat H_{int}$ the interaction part of the Hamiltonian. From the Kubo formula[@mald; @Suth] we can relate the imaginary part of the dynamical conductivity $\sigma''(\omega)$ to the charge stiffness $D$, $$D= \frac{1}{2}[\omega \sigma''(\omega)]_{\omega \to 0}
=\frac{1}{L} \{ \frac{1}{2} \langle  -\hat T \rangle -
\sum_{m\neq n} p_n \frac{ \mid \langle n \mid \hat j \mid m \rangle \mid ^2}
{\epsilon_m - \epsilon_n} \}, \label{eq2}$$

where $\hat j$ is the current operator, $\langle \hat T\rangle$ the thermal expectation value of the kinetic energy and $p_n=e^{-\epsilon_n/T}/Z$ the Boltzmann weight for an eigenstate $\mid n\rangle$ of Hamiltonian (\[eq1\]) with energy $\epsilon_n$.

On the other hand[@Kohn] we can evaluate, by second order perturbation theory, for $\phi \to 0$ a shift of the level $\mid n\rangle$: $$\epsilon_n(\phi)= \langle n\mid \hat H(\phi=0)\mid n\rangle
-\phi \langle n \mid \hat j \mid n \rangle
-\phi^2 \sum_{m\neq n} \frac{ \mid \langle n \mid \hat j \mid m \rangle
\mid ^2}{\epsilon_m - \epsilon_n} -
\frac{1}{2} \phi^2 \langle n \mid \hat T \mid n \rangle .  \label{eq3}$$

Extracting second order terms in $\phi$ (the curvature of levels) we see that: $$D =
\frac{1}{L} \sum_n p_n D_n=
\frac{1}{L} \sum_n p_n
\frac{1}{2} \frac {\partial ^2 \epsilon_n (\phi) }{\partial \phi^2} .
\label{eq5}$$

In this demonstration we assumed that the levels $\mid n\rangle$ are nondegenerate but the same procedure can be used for degenerate levels after applying first order degenerate perturbation theory to lift the level degeneracy. This expression for $D(T)$ generalizes Kohn’s method to $T>0$ and reduces its evaluation to the calculation of curvature of levels under an external flux. Easier than the complete evaluation of Kubo formula, it can be performed either numericaly on finite size systems followed by finite size scaling or in some cases (e.g. integrable systems) analytically[@peter]. It also provides a basis for analysing the conditions for the occurence of ideal conductance.

The ground state of the metal should be characterized by $D_0=D(T=0)>0$ [@Kohn; @Suth; @Scal], where $D_0$ is a measure of of the charge carriers [*coherent*]{} motion. On physical grounds it is commonly assumed [@Scal] that $D(T>0)=0$, for generic macroscopic interacting systems (involving Umklapp scattering), except in the superconducting phase. This statement, never really proved to our knowledge for interacting fermions, implies according to our Eqs.(\[eq2\],\[eq5\]) that the positive curvature of the ground state $D_0$ is cancelled by (mostly) negative curvatures of low lying states, a situation reminiscent of the mechanism for phase transitions and the dissapearance of long range order.

[*Model system:*]{} The Hamiltonian describing a particle in a bath of spinless fermions is:

$$\hat H=-t_h\sum_i (e^{i\phi} d^{\dagger}_{i+1} d_i +H.c.)
-t\sum_i (c^{\dagger}_{i+1} c_i + H.c. ) +
U\sum_i d^{\dagger}_i d_i c^{\dagger}_i c_i ,\label{eq6}$$

where $c_i ( c^{\dagger}_i )$ are annihilation (creation) operators for $N$ spinless fermions and $d_i ( d^{\dagger}_i )$ for the (tagged) particle on an $L$ site chain with periodic boundary conditions. The interaction comes only through the on-site repulsion $U>0$. The current $\hat j$ in (\[eq2\]) refers to the particle only ( $\hat j=-it_h\sum_i d^{\dagger}_{i+1} d_i + H.c $) and $[\hat j,\hat H]\neq 0$. The volume normalization $1/L$ in (\[eq5\]) is absent as the current refers only to one particle.

The model (\[eq6\]) is well adapted to our study: it is integrable by the Bethe ansatz method in the case of equal masses, i.e. $t_h = t$ [@mcg] (equivalent to the problem of a Hubbard chain in a nearly polarized state $S^z=S_{max}^z-1$) and nonintegrable for unequal masses ($t_h\neq t$). We have also previously studied $D_0$[@xz] of the particle as well as its quasiparticle properties[@cz].

We first present the results for $D(T)$ from exact numerical diagonalization of the Hamiltonian (\[eq6\]) on finite size systems. We will consider only the $N=L/2$ case, which in this system should be analogous to all other fillings. By numerical calculation of the energy spectrum with and without a small flux $\phi$ (typically $\phi \simeq 10^{-4}$) we deduce the energy level curvature from: $$D_n=\frac{1}{2}
\frac{\epsilon_n(\phi)+\epsilon_n(-\phi)-2\epsilon(0)}{\phi^2} .\label{eq7}$$

In Fig.1 we present $D(T)$ as a function of $t_h/t$, for different temperatures $T$, size systems (from 6 to 14 sites) but fixed $U/t=2$. We observe: [*a)*]{} a nonmonotonic behavior of $D(T)$ as we sweep through the integrable $t_h=t$ point for $T>0$, [*b)*]{} a very weak dependence of $D(T)$ with system size for the integrable case $t_h=t$ while a rather strong one for the nonintegrable ones.

These observations are the first hints that something particular happens at the integrable point. To further study this behavior, we present in Fig.2 $D(T)/D_0$ as a function of $1/L$ for $t_h=t$. We indeed observe a scaling in $1/L$ which we also know that it is the relevant one to recover the ground state $D_0$[@xz]. The circle at $1/L=0$ indicates the result of the Bethe ansatz calculation for $t_h=t$ in very good agreement with the extapolated numerical results.

The same plot in Fig.2 of $\ln D(T)/D_0$ as a function of $L^2$ for $t_h=0.5, 1.2, 2.0t$ seems to indicate a strong $L$ dependence following a law $D(T)\simeq e^{-\gamma L^2}$ for these nonintegrable cases. This is consistent with a rapid decrease of $D(T)$ for systems larger than the mean free path.

Next to obtain an impression of the overall behavior of $D(T)$ as function of temperature we present in Fig.3 $D(T)/D_0$ for $t_h=t, 0.5t$ and different size systems. For $t_h=0.5t$ $D(T)/D_0$ seems to scale to zero for any $T>0$ as we found above while for $t_h=t$ there is a smooth decrease with T. At high temperatures $D(T) \propto 1/T$, vanishing for $T\to \infty$ a general feature of systems on a lattice. Finally the discrepancy from the free particle $D(T)$ indicates that the temperature effects cannot be accounted for by a simple scaling of $D_0$.

[*Bethe ansatz analysis:*]{} We present here an analytical approach for the calculation of $D(T)$ for the integrable case using the Bethe ansatz method along the line of reference [@xz]. The starting point is the Lieb-Wu[@lw] solution of the Hubbard model adapted for one spin up fermion (the tagged particle) and $N$ spin down fermions. The Bethe ansatz wavefunctions, in the presence of flux $\phi$, are then characterized by $N+1$ quantum numbers $k_j$ given by the following equations (we take $t_h=t=1$): $$Lk_j=2\pi I_j+\theta(\sin k_j-\Lambda),~~~j=1,\ldots,N+1 \label{eq8}$$ $$\theta(p)=-2\tan^{-1}(4p/U) \label{eq9}$$ $$L\sum_{j=1}^{N+1}k_j=2\pi \sum_{j=1}^{N+1}I_j+2\pi J + L\phi .
\label{eq10}$$

Every state is characterized by a set of half-odd integers $\{I_j\}$ and the (half-odd) integer $J$ for (even) odd number of fermions. However this set of states(regular) does not constitute a complete set; to obtain a complete set we must include states representing bound states of energy of order U. The simplest way to include these states is by considering an electron-hole transformation ($\tilde c_i=c^{\dagger}_i$, $\tilde d_i=(-1)^i d_i$) and then solving Eq.(\[eq8\]) but for a $L-N$ number of fermions (equal to the number of holes).

The total energy of a regular state is given by : $$E=\sum_{j=1}^{N+1} \epsilon(k_j)=\sum_j (-2\cos k_j) ,\label{eq11}$$ and for a bound state: $$E=U+\sum_{j=1}^{L-N+1} 2\cos k_j .\label{eq12}$$

In the following we present explicitely only the analysis for the contribution of regular states as it is clear how to include the bound ones by the prescription described above[@fut] (although we include both contributions in the results presented in Fig. 2 and 3).

To order $1/L$ and for $UL\gg 1$ the values of $k_j$ are given by: $$k_j=k_j^0+\frac{1}{L}\theta(\sin k_j^0-\Lambda),
{}~~~k_j^0=\frac{2\pi I_j}{L} .\label{eq13}$$

To order one then the energy and momentum of a state can be written as: $$E=\sum_j \epsilon(k_j^0)+\frac{2}{L}\sin k_j^0~
\theta(\sin k_j^0-\Lambda) \label{eq14}$$ $$\frac{1}{L}\sum_j \theta(\sin k_j^0-\Lambda)=
\frac{2\pi J}{L} +\phi .\label{eq15}$$

In this scheme the interaction between the particle and the fermions is represented through second term in Eq.(\[eq14\]), a correlation energy; the coupling to the flux $\phi$ is through the collective coordinate $\Lambda$. In the thermodynamic limit, defining a density $\rho(k)$ $(-\pi \leq k \leq +\pi)$ we obtain: $$E(\rho(k),\Lambda)=\frac{L}{2\pi}\int dk \rho(k)
( -2\cos k+\frac{2}{L}\sin k~ \theta(\sin k-\Lambda) )\label{eq16}$$ $$\frac{1}{2\pi} \int dk \rho(k) \theta(\sin k-\Lambda)=
P+\phi,~~~ P=\frac{2\pi J}{L} .\label{eq17}$$

To calculate the thermal average of the curvature of energy levels we will work in the grand canonical ensemble for the system plus the one particle. Assuming that the equilibrium distribution of $k's$ is not affected by the presence of the one particle we use the Fermi-Dirac distribution for free fermions $f(k)=1/(1+\exp( (\mu-\epsilon(k))/T)$ ($\mu$ is the chemical potential). We then assume that the distribution of $\Lambda's$ is determined by the average correlation energy $\epsilon_c(\Lambda)$,

$$\epsilon_c(\Lambda)=\frac{1}{2\pi} \int dk f(k)
2\sin k~ \theta(\sin k-\Lambda) ,\label{eq18}$$

through a Boltzmann weight: $w(\Lambda)=\exp(-\epsilon_c(\Lambda)/T)$.

Finally $D(T)$ is given by: $$D=
\frac{1}{2\pi Z_{\Lambda}}
\int_{-\infty}^{+\infty} d\Lambda g(\Lambda) w(\Lambda)
\frac{1}{2\pi} \int dk f(k) D(\Lambda,k) ,\label{eq19}$$

with $g(\Lambda)=\partial P/\partial\Lambda$ determined from Eq.(\[eq17\]):

$$\frac{\partial P}{\partial \Lambda}=\frac{1}{2\pi}\int dk f(k)
\frac{\partial \theta(\sin k-\Lambda)}{\partial \Lambda} \label{eq20}$$

$$Z_{\Lambda}=\frac{1}{2\pi}\int d\Lambda g(\Lambda) w(\Lambda) ,
\label{eq21}$$

(in the total partition function $Z_{\Lambda}$ we add the contribution from both the regular and bound states).

$$D(\Lambda,k)=\frac{1}{2}2\sin k
( \frac{\partial^2 \theta}{\partial \Lambda^2}
(\frac{\partial\Lambda}{\partial\phi})^2+
\frac{\partial \theta}
{\partial\Lambda} \frac{\partial^2\Lambda}{\partial\phi^2} ) .
\label{eq22}$$

By successive differentiation of Eq.(\[eq17\]) we can determine $\partial \Lambda/\partial\phi=1/g(\Lambda)$ and $\partial^2 \Lambda/\partial \phi^2$.

Evaluating the expressions (\[eq19\]-\[eq22\]) we obtain the results presented in Fig.2, 3 in very good agreement with the numerical results, providing support for our approach[@fut]. We also verified that the agreement remains (within a couple of percent) for other values of the interaction (e.g. $U/t=8$).

[*Comments:*]{} From our formulation of $D(T)$ as a thermal average over level curvatures, we can try to understand the difference in behavior between integrable and nonintegrable systems. In nonintegrable cases the level repulsion prevents level crossings, i.e. crossings are statistically negligible for macroscopic systems. Then each level $\epsilon_n(\phi)$ fluctuates only on the scale $\Delta \epsilon
\propto 1/{\cal N}(E)$, where ${\cal N}(E)$ is the many-body density of states. Therefore the curvature $D_n$ averaged over $\phi$ or over different $k$ vectors in the thermodynamic limit should vanish. On the other hand, in integrable systems levels in general cross, so fluctuations of $\epsilon_n(\phi)$ do not necessarily vanish for $L\to \infty$. Hence there is no restriction on the average $D(T)$, except that $D>0$. The difference between both cases is intimately related to level statistics.

This connection between integrability and finite temperature charge transport born out of this model calculation, we can conjecture to hold true for other quantum (as well as classical) integrable systems. We can trace it to the existence of a macroscopic number of conservation laws. It is plausible that with respect to transport one should distinguish two types of models within the class of one-dimensional (1D) correlated systems where some solvable models[@Emer] are available:

a\) In few (mostly solvable) 1D models the current is a conserved quantity. This is generically the case for models without Umklapp scattering, e.g. the Luttinger model, 1D Bose gas etc, but also for the $U=\infty$ Hubbard model [@Brin]. In these cases one expects at $T>0$ ideal conductance of the system characterized by $\rho(T)=0$ or $D(T)>0$.

b\) Nontrivial answers are expected for models with Umklapp scattering, e.g. for the 1D Hubbard model, the $t-V$ model etc. If our conjecture is correct we expect integrable models as the Hubbard, $t-V$ or supersymmetric $t-J (J=2t)$ models to behave as ideal conductors at finite temperatures. We can then argue that even 1D nonintegrable models as the $U-V$ model (with longer range interactions), as they are characterized by the integrable Luttinger liquid Hamiltonian at low energies, should behave as nearly ideal conductors at low T; the $\delta$-peak then would broaden to a narrow Drude peak of weight $D(T)$[@Giam].

We should stress that our study is based on the Kubo linear response theory[@Kubo] whose applicability in the context of Luttinger liquids has recently been debated[@pwa]. Further work is necessary on other systems to lend further support for these ideas.

We would like to thank H. Schulz, H. Kunz and U.Eckern for useful discussions. This work was supported by the Swiss National Fond Grants No. 20-39528.93, the University of Geneva, University of Fribourg and by the Ministry of Science and Technology of Slovenia.

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