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Energy transport in the Anderson insulator

We study the heat conductivity in Anderson insulators in the presence of 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 2D and 3D spin systems with $1/r^3$ dipole-dipole interaction (originating from the conventional $1/r$ Coulomb interaction between electrons), the found thermal conductivity $κ$ scales with temperature as $κ\propto T^3 $ and $κ\propto T^{4/3}$, respectively. Our results may be of relevance also to other realizations of random spin Hamiltonians with long-range interactions.