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arXiv:1009.0211 · Physical Review B

Conductivity in a disordered one-dimensional system of interacting fermions

O. S. Barišić, Peter Prelovšek

Published Oct 29, 2010• 53 citations
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Abstract

Dynamical conductivity in a disordered one-dimensional model of interacting fermions is studied numerically at high temperatures and in the weak-interaction regime in order to find a signature of many-body localization and vanishing dc transport coefficients. On the contrary, we find in the regime of moderately strong local disorder that the dc conductivity ${\ensuremath{\sigma}}_{0}$ scales linearly with the interaction strength while being exponentially dependent on the disorder. According to the behavior of the charge stiffness evaluated at the fixed number of particles, the absence of the many-body localization seems related to an increase in the effective localization length with the interaction.

DOI
10.1103/physrevb.82.161106

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Dynamical conductivity in a disordered one-dimensional model of interacting fermions is studied numerically at high temperatures and in the weak-interaction regime in order to find a signature of many-body localization and vanishing dc transport coefficients. On the contrary, we find in the regime of moderately strong local disorder that the dc conductivity \ensuremathσ0{\ensuremath{\sigma}}_{0}\ensuremathσ0​ scales linearly with the interaction strength while being exponentially dependent on the disorder. According to the behavior of the charge stiffness evaluated at the fixed number of particles, the absence of the many-body localization seems related to an increase in the effective localization length with the interaction.
Abstract We study the transport properties of a one-dimensional quantum system with disorder. We numerically compute the frequency dependence of the conductivity of a fermionic chain with nearest-neighbor interaction and a random chemical potential by using the Chebyshev matrix product state (CheMPS) method. As a benchmark, we investigate the noninteracting case first. Comparison with exact diagonalization and analytical solutions demonstrates that the results of CheMPS are reliable over a wide range of frequencies. We then calculate the dynamical conductivity spectra of the interacting system for various values of the interaction and disorder strengths. In the high-frequency regime, the conductivity decays as a power law, with an interaction-dependent exponent. This behavior is qualitatively consistent with the bosonized field theory predictions, although the numerical evaluation of the exponent shows deviations from the analytically expected values. We also compute the characteristic pinning frequency at which a peak in the conductivity appears. We confirm that it is directly related to the inverse of the localization length, even in the interacting case. We demonstrate that the localization length follows a power law of the disorder strength with an exponent dependent on the interaction, and find good quantitative agreement with the field theory predictions. In the low-frequency regime, we find a behavior consistent with the one of the noninteracting system ω2(ln⁡ω)2\omega ^{2}(\ln \omega )^{2}ω2(lnω)2 ω 2 ( ln ω ) 2 independently of the interaction. We discuss the consequences of our finding for experiments in cold atomic gases. Graphical abstract
The behavior of the dynamical structure factor S(q,ω)S(q,\omega)S(q,ω) is studied numerically in the whole range of wavevectors qqq within the prototype one-dimensional model of many-body localization (MBL). Extracted effective dynamical conductivities and current-relaxation rates confirm strong dependence on disorder but modest variation with qqq. An analytical self-consistent approximation based on the perturbation theory is shown to qualitatively account for the nontrivial features of dynamical quantities at all qqq: the emergence of the maximum in dynamical conductivities, nonanalytical low--ω\omegaω variation in the ergodic phase, and the transition to the nonergodic (MBL) phase. Finite-size scaling also reveals the possibility of the subdiffusive behavior in the ergodic regime.
Using numerical diagonalization techniques, we explore the effect of local and bond disorder on the finite-temperature spin and thermal conductivities of the one-dimensional anisotropic spin-1/2 Heisenberg model. High-temperature results for local disorder show that the dc conductivities are finite, apart from the uncorrelated XYXYXY case where dc transport vanishes. Moreover, at strong disorder, we find finite dc conductivities at all temperatures TTT except at T=0T=0T=0. The low-frequency conductivities are characterized by a nonanalytic cusp shape. Similar behavior is found for bond disorder.
We present a numerical study of the many-body localization (MBL) phenomenon in the high-temperature limit within an anisotropic Heisenberg model with random local fields. Taking the dynamical spin conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ as the test quantity, we investigate the full frequency dependence of sample-to-sample fluctuations and their scaling properties as a function of the system size $L\ensuremath{\le}28$ and the frequency resolution. We identify differences between the general interacting case $\mathrm{\ensuremath{\Delta}}>0$ and the anisotropy $\mathrm{\ensuremath{\Delta}}=0$, the latter corresponding to the standard Anderson localization. Except for the extreme MBL case when the relative sample-to-sample fluctuations became large, numerical results allow for the extraction of the low-$\ensuremath{\omega}$ dependence of the conductivity. Results for the dc value ${\ensuremath{\sigma}}_{0}$ indicate a crossover into the MBL regime, i.e., an exponential-like variation with the disorder strength WWW. For the same regime, our numerical analysis indicates that the low-frequency exponent $\ensuremath{\alpha}$ exhibits a small departure from $\ensuremath{\alpha}\ensuremath{\sim}1$ only.
We define a ``hyperconductor'' to be a material whose electrical and thermal dc conductivities are infinite at zero temperature and finite at any nonzero temperature. The low-temperature behavior of a hyperconductor is controlled by a quantum critical phase of interacting electrons that is stable to all potentially gap-generating interactions and potentially localizing disorder. In this paper, we compute the low-temperature dc and ac electrical and thermal conductivities in a one-dimensional hyperconductor, studied previously by the present authors, in the presence of both disorder and umklapp scattering. We identify the conditions under which the transport coefficients are finite, which allows us to exhibit examples of violations of the Wiedemann-Franz law. The temperature dependence of the electrical conductivity, which is characterized by the parameter ${\mathrm{\ensuremath{\Delta}}}_{X}$, is a power law, $\ensuremath{\sigma}\ensuremath{\propto}1/{T}^{1\ensuremath{-}2(2\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{X})}$ when ${\mathrm{\ensuremath{\Delta}}}_{X}\ensuremath{\ge}2$, down to zero temperature when the Fermi surface is commensurate with the lattice. There is a surface in parameter space along which ${\mathrm{\ensuremath{\Delta}}}_{X}=2$ and ${\mathrm{\ensuremath{\Delta}}}_{X}\ensuremath{\approx}2$ for small deviations from this surface. In the generic (incommensurate) case with weak disorder, such scaling is seen at high temperatures, followed by an exponential increase of the conductivity $ln\ensuremath{\sigma}\ensuremath{\sim}1/T$ at intermediate temperatures and, finally, $\ensuremath{\sigma}\ensuremath{\propto}1/{T}^{2\ensuremath{-}2(2\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{X})}$ at the lowest temperatures. In both cases, the thermal conductivity diverges at low temperatures.
We suggest that if a localized phase at nonzero temperature T>0T>0T>0 exists for strongly disordered and weakly interacting electrons, as recently argued, it will also occur when both disorder and interactions are strong and TTT is very high. We show that in this high-TTT regime, the localization transition may be studied numerically through exact diagonalization of small systems. We obtain spectra for one-dimensional lattice models of interacting spinless fermions in a random potential. As expected, the spectral statistics of finite-size samples cross over from those of orthogonal random matrices in the diffusive regime at weak random potential to Poisson statistics in the localized regime at strong randomness. However, these data show deviations from simple one-parameter finite-size scaling: the apparent mobility edge ``drifts'' as the system's size is increased. Based on spectral statistics alone, we have thus been unable to make a strong numerical case for the presence of a many-body localized phase at nonzero TTT.