Energy transport in the Anderson insulator

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 $\kappa$ scales with temperature as $\kappa \propto T^3$ and $\kappa \propto T^{4/3}$, respectively. Our results may be of relevance also to other realizations of random spin Hamiltonians with long-range interactions.

Figure 1

Formation of a resonant network of pseudospins (Sec. 2b). Each pseudospin $\tau$ consists of two (resonating) spins $\vec{S}$ at distance $R_1$. Two pseudospins at distance $R_2$ are at resonance provided that the condition (6) is fulfilled. To construct the ${\mathrm{pseudo}}^2$-spin network, this construction is iterated once more (Sec. 2c).

Figure 2

Number of pseudospins of size $R_1$ in the shell $R_2<r<2R_2$ that are in resonance with a given pseudospin of size $R_1$.

Figure 3

Phase diagram of the system. We concentrate on the parameter range $\alpha \ge \beta$, which is the part of the phase diagram limited by the dotted line. The full line $\beta =2d$ designates the phase transition to the many-body localized phase. Thick dashed lines divide the delocalized phase into the regions I, II, and III that are distinguished by the basic delocalized low-energy degrees of freedom. Specifically, the lowest-level resonant network is that of spins in region I, of pseudospins in region II, and of ${\mathrm{pseudo}}^2$-spins in region III. Thin dashed lines subdivide region II into regions with different structure of the optimal pseudospin and/or ${\mathrm{pseudo}}^2$-spin networks. In region ${\mathrm{II}}_{\mathrm{a}}$, the pseudospins of any size become delocalized by themselves, in full analogy with delocalization of spins in region I. In parts ${\mathrm{II}}_{\mathrm{b}}$ and ${\mathrm{II}}_{\mathrm{c}}$ the delocalization mechanism is much more intricate, and only pseudospins of sufficiently large size $R_1$ become connected. In region ${\mathrm{II}}_{\mathrm{c}}$ delocalized pseudospin and ${\mathrm{pseudo}}^2$-spin networks coexist. In this paper, we focus on regions ${\mathrm{II}}_{\mathrm{b}}, {\mathrm{II}}_{\mathrm{c}}$, and III.

Figure 4

Schematic representation of the random banded matrix approximation to the Hamiltonian of the network of pseudospins of a given size $R_1$. Each block has a size $R_2^{*}\left(R_1\right)$. Pseudospins within each block and between adjacent blocks are strongly coupled by interaction. Couplings at longer distances are neglected.

Figure 5

Characteristic energy scales ($W$: disorder; $T$: temperature; $E_{*}$: excitation energy in the optimal network) and corresponding pseudospin sizes. The optimal network is constructed and studied in Sec. 3. Transport by “high-energy” excitations (which get delocalized only via the interaction with the optimal network) is explored in Sec. 4. The ultra-low-energy excitations are discussed in Sec. 5.

Figure 6

A process contributing to dephasing of high-energy pseudospins by those with lower energies. Here a spin with a high energy $E$ is dephased by a decay into two spins of roughly equal energy $\sim E/2$. This process gives a subleading contribution in comparison with that in Fig. 7.

Figure 7

Dominant process responsible for delocalization of networks of high-energy pseudospins. This is a “flip-flop” process of two high-energy (small $R_1$) spins assisted by an excitation of the optimal network $\left[R_1\left(T\right)\right]$.