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arXiv:1603.01526 · Physical review. B./Physical review. B

Dynamical conductivity and its fluctuations along the crossover to many-body localization

O. S. Barišić, J. Kokalj, Ivan Balog, Peter Prelovšek

Published Jul 20, 2016• 72 citations
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Abstract

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 $W$. 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.

DOI
10.1103/physrevb.94.045126

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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 \ensuremathσ(\ensuremathω)\ensuremath{\sigma}(\ensuremath{\omega})\ensuremathσ(\ensuremathω) 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≤28L\ensuremath{\le}28L\ensuremath≤28 and the frequency resolution. We identify differences between the general interacting case \ensuremathΔ>0\mathrm{\ensuremath{\Delta}}>0\ensuremathΔ>0 and the anisotropy \ensuremathΔ=0\mathrm{\ensuremath{\Delta}}=0\ensuremathΔ=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-\ensuremathω\ensuremath{\omega}\ensuremathω dependence of the conductivity. Results for the dc value \ensuremathσ0{\ensuremath{\sigma}}_{0}\ensuremathσ0​ 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 \ensuremathα\ensuremath{\alpha}\ensuremathα exhibits a small departure from \ensuremathα\ensuremath∼1\ensuremath{\alpha}\ensuremath{\sim}1\ensuremathα\ensuremath∼1 only.
We theoretically study transport properties in one-dimensional interacting quasiperiodic systems at infinite temperature. We compare and contrast the dynamical transport properties across the many-body localization (MBL) transition in quasiperiodic and random models. Using exact diagonalization we compute the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ and the return probability $R(\ensuremath{\tau})$ and study their average low-frequency and long-time power-law behavior, respectively. We show that the low-energy transport dynamics is markedly distinct in both the thermal and MBL phases in quasiperiodic and random models and find that the diffusive and MBL regimes of the quasiperiodic model are more robust than those in the random system. Using the distribution of the dc conductivity, we quantify the contribution of sample-to-sample and state-to-state fluctuations of $\ensuremath{\sigma}(\ensuremath{\omega})$ across the MBL transition. We find that the activated dynamical scaling ansatz works poorly in the quasiperiodic model but holds in the random model with an estimated activation exponent $\ensuremath{\psi}\ensuremath{\approx}0.9$. We argue that near the MBL transition in quasiperiodic systems, critical eigenstates give rise to a subdiffusive crossover regime on finite-size systems.
We study the frequency dependence of the optical conductivity $\text{Re}\phantom{\rule{0.16em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega})$ of the Heisenberg spin-12\frac{1}{2}21​ chain in the thermal and near the transition to the many-body localized phase induced by the strength of a random zzz-directed magnetic field. Using the method of dynamical quantum typicality, we calculate the real-time dynamics of the spin-current autocorrelation function and obtain the Fourier transform $\text{Re}\phantom{\rule{0.16em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega})$ for system sizes much larger than accessible to standard exact-diagonalization approaches. We find that the low-frequency behavior of $\text{Re}\phantom{\rule{0.16em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega})$ is well described by $\text{Re}\phantom{\rule{0.16em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\approx}{\ensuremath{\sigma}}_{\text{dc}}+a{|\ensuremath{\omega}|}^{\ensuremath{\alpha}}$, with $\ensuremath{\alpha}\ensuremath{\approx}1$ in a wide range within the thermal phase and close to the transition. We particularly detail the decrease of ${\ensuremath{\sigma}}_{\text{dc}}$ in the thermal phase as a function of increasing disorder for strong exchange anisotropies. We further find that the temperature dependence of ${\ensuremath{\sigma}}_{\text{dc}}$ is consistent with the existence of a mobility edge.
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.
An important experimental signature of localization is the low-frequency ac conductivity, which typically vanishes as ${\ensuremath{\omega}}^{\ensuremath{\phi}}$. The exponent $\ensuremath{\phi}=2$ for Anderson insulators, whereas for many body-localized insulators $\ensuremath{\phi}$ is a continuously varying exponent, $1\ensuremath{\le}\ensuremath{\phi}\ensuremath{\le}2$. In this work, we study the low-frequency ac conductivity of localized superconductors, in which disorder is strong enough to localize all quasiparticles, while remaining weak enough to leave superconductivity intact. We find that, while the ac conductivity still follows the general form $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{\phi}}$, the exponent $\ensuremath{\phi}$ can be markedly different from the characteristic value for localized insulators. This difference occurs due to singularities in the low-energy density of states, permitted by the effective particle-hole symmetry around the Fermi level. In particular, in certain symmetry classes at zero temperature, we obtain $\ensuremath{\phi}>2$. We further identify an interesting temperature-dependent crossover in the scaling form of the ac conductivity, which could be useful for the experimental characterization of localized superconductors.
We explore the high-temperature dynamics of the disordered, one-dimensional XXZ model near the many-body localization (MBL) transition, focusing on the delocalized (i.e., "metallic") phase. In the vicinity of the transition, we find that this phase has the following properties: (i) local magnetization fluctuations relax subdiffusively; (ii) the ac conductivity vanishes near zero frequency as a power law; and (iii) the distribution of resistivities becomes increasingly broad at low frequencies, approaching a power law in the zero-frequency limit. We argue that these effects can be understood in a unified way if the metallic phase near the MBL transition is a quantum Griffiths phase. We establish scaling relations between the associated exponents, assuming a scaling form of the spin-diffusion propagator. A phenomenological classical resistor-capacitor model captures all the essential features.
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.