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arXiv:1512.06705 · Physical review. B./Physical review. B

Energy transport in the Anderson insulator

Dmitri B. Gutman, Ivan V. Protopopov, Alexander L. Burin, I. V. Gornyi, Raul A. Santos, A. D. Mirlin

Published Jun 22, 2016• 59 citations
arXivHTMLPDF

Abstract

We study the heat conductivity in Anderson insulators in the presence of a power-law interaction. Particle-hole excitations built on localized electron states are viewed as two-level systems randomly distributed in space and energy and coupled due to electron-electron interaction. A small fraction of these states form resonant pairs that in turn build a complex network allowing for energy propagation. We identify the character of energy transport through this network and evaluate the thermal conductivity. For physically relevant cases of two-dimensional and three-dimensional spin systems with $1/{r}^{3}$ dipole-dipole interaction (originating from the conventional $1/r$ Coulomb interaction between electrons), the found thermal conductivity $\ensuremath{\kappa}$ scales with temperature as $\ensuremath{\kappa}\ensuremath{\propto}{T}^{3}$ and $\ensuremath{\kappa}\ensuremath{\propto}{T}^{4/3}$, respectively. Our results may be of relevance also to other realizations of random spin Hamiltonians with long-range interactions.

DOI
10.1103/physrevb.93.245427

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We study the heat conductivity in Anderson insulators in the presence of a power-law interaction. Particle-hole excitations built on localized electron states are viewed as two-level systems randomly distributed in space and energy and coupled due to electron-electron interaction. A small fraction of these states form resonant pairs that in turn build a complex network allowing for energy propagation. We identify the character of energy transport through this network and evaluate the thermal conductivity. For physically relevant cases of two-dimensional and three-dimensional spin systems with 1/r31/{r}^{3}1/r3 dipole-dipole interaction (originating from the conventional 1/r1/r1/r Coulomb interaction between electrons), the found thermal conductivity \ensuremathκ\ensuremath{\kappa}\ensuremathκ scales with temperature as \ensuremathκ\ensuremath∝T3\ensuremath{\kappa}\ensuremath{\propto}{T}^{3}\ensuremathκ\ensuremath∝T3 and \ensuremathκ\ensuremath∝T4/3\ensuremath{\kappa}\ensuremath{\propto}{T}^{4/3}\ensuremathκ\ensuremath∝T4/3, respectively. Our results may be of relevance also to other realizations of random spin Hamiltonians with long-range interactions.
This paper describes experiments utilizing a unique property of electron glasses to gain information on the fundamental nature of the interacting Anderson-localized phase. The methodology is based on measuring the energy absorbed by the electronic system from alternating electromagnetic fields as a function of their frequency. Experiments on three-dimensional (3D) amorphous indium-oxide films suggest that, in the strongly localized regime, the energy spectrum is discrete and inelastic electron-electron events are strongly suppressed. These results imply that, at low temperatures, electron thermalization and finite conductivity depend on coupling to the phonon bath. The situation is different for samples nearing the metal-insulator transition; in insulating samples that are close to the mobility edge, energy absorption persists to much higher frequencies. Comparing these results with previously studied 2D samples [Ovadyahu, Phys. Rev. Lett. 108, 156602 (2012)] demonstrates that the mean-level spacing (on a single-particle basis) is not the only relevant scale in this problem. The possibility of delocalization by many-body effects and the relevance of a nearby mobility edge (which may be a many-body edge) are discussed.
Using variational matrix product states, we analyze the finite temperature behavior of a half-filled periodic Anderson model in one dimension, a prototypical model of a Kondo insulator. We present an extensive analysis of single-particle Green’s functions, two-particle Green’s functions, and transport functions creating a broad picture of the low-temperature properties. We confirm the existence of energetically low-lying spin excitations in this model and study their energy-momentum dispersion and temperature dependence. We demonstrate that charge-charge correlations at the Fermi energy exhibit a different temperature dependence than spin-spin correlations. While energetically low-lying spin excitations emerge approximately at the Kondo temperature, which exponentially depends on the interaction strength, charge correlations vanish already at high temperatures. Furthermore, we analyze the charge and thermal conductivity at finite temperatures by calculating the time-dependent current-current correlation functions. While both charge and thermal conductivity can be fitted for all interaction strengths by gapped systems with a renormalized band gap, the gap in the system describing the thermal conductivity is generally smaller than the system describing the charge conductivity. Thus, two-particle correlations affect the charge and heat conductivities in a different way resulting in a temperature region where the charge conductivity of this one-dimensional Kondo insulator is already decreasing while the heat conductivity is still increasing.
We study the thermal conductivity of the disordered two-dimensional electron gas. To this end, we analyze the heat density--heat density correlation function concentrating on the scattering processes induced by the Coulomb interaction in the subtemperature energy range. These scattering processes are at the origin of logarithmic corrections violating the Wiedemann-Franz law. Special care is devoted to the definition of the heat density in the presence of the long-range Coulomb interaction. To clarify the structure of the correlation function, we present details of a perturbative calculation. While the conservation of energy strongly constrains the general form of the heat density--heat density correlation function, the balance of various terms turns out to be rather different from that for the correlation functions of other conserved quantities such as the density-density or spin density--spin density correlation function.
We study electronic transport in Anderson insulators with strong Coulomb interactions in dimensions d>=2. Close to the metal insulator transition where the single particle localization length is much larger than interparticle-distance, the interactions lead to a strongly correlated quantum glass phase. Even though single particle excitations are localized and the system is insulating, there are collective electronic modes which remain delocalized down to parametrically small energies. These collective excitations serve as a continuous bath which can provide the activation energy for variable range hopping transport. This circumvents the energy conservation problem arising when only discrete particle-hole excitations are present. In contrast to the weak and material-dependent phonon-assisted hopping mechanism, the activation by an electronic bath leads to a nearly universal prefactor e^2/h of the Efros-Shklovskii conductance, as is observed in many recent experiments.
Interplay of electron correlation and randomness is studied by using the Anderson-Hubbard model within the Hartree-Fock approximation. Under the coexistence of short-range interaction and diagonal disorder, we obtain the ground-state phase diagram in three dimensions, which includes an antiferromagnetic insulator, an antiferromagnetic metal, a paramagnetic insulator (Anderson-localized insulator) and a paramagnetic metal. Although only the short-range interaction is present in this model, we find unconventional soft gaps in the insulating phases irrespective of electron filling, spatial dimensions and long-range order, where the single-particle density of states (DOS) vanishes with a power-law scaling in one dimension (1D) or even faster in two dimensions (2D) and three dimensions (3D) toward the Fermi energy. We call it soft Hubbard gap. Moreover, exact-diagonalization results in 1D support the formation of the soft Hubbard gap beyond the mean-field level. The formation of the soft Hubbard gap cannot be attributed to a conventional theory by Efros and Shklovskii (ES) owing the emergence of soft gaps to the long-range Coulomb interaction. Indeed, based on a picture of multivalley energy landscape, we propose a phenomenological scaling theory, which predicts a scaling of the DOS in perfect agreement with the numerical results. We further discuss a correction of the scaling of the DOS by the long-range part of the Coulomb interaction, which modifies the scaling of Efros and Shklovskii. Furthermore, explicit formulae for the temperature dependence of the DC resistivity via variable-range hopping under the influence of the soft gaps are derived. Finally, we compare the present theory with experimental results of SrRu_{1-x}Ti_xO_3.
We study the conductivity $\ensuremath{\sigma}(T)$ of interacting electrons in a low-dimensional disordered system at low temperature TTT. For weak interactions, the weak-localization regime crosses over with lowering TTT into a dephasing-induced ``power-law hopping.'' As TTT is further decreased, the Anderson localization in Fock space crucially affects $\ensuremath{\sigma}(T)$, inducing a transition at T=TcT={T}_{c}T=Tc​, so that $\ensuremath{\sigma}(T<{T}_{c})=0$. The critical behavior of $\ensuremath{\sigma}(T)$ above Tc{T}_{c}Tc​ is $\mathrm{ln}\ensuremath{\sigma}(T)\ensuremath{\propto}\ensuremath{-}(T\ensuremath{-}{T}_{c}{)}^{\ensuremath{-}1/2}$. The mechanism of transport in the critical regime is many-particle transitions between distant states in Fock space.