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 ∝ T 3 and κ ∝ T 4/3 , respectively. Our results may be of relevance also to other realizations of random spin Hamiltonians with long-range interactions.

The sliding Luttinger liquid approach is applied to study fractional topological insulators (FTIs). We show that FTI is the low energy fixed point of the theory for realistic spin-orbit and electronelectron interactions. We find that the topological phase pertains in the presence of interaction that breaks the spin invariance and its boundaries are even extended by those terms. Finally we show that one dimensional chiral anomaly in the Luttinger liquid leads to the emergence of topological Chern-Simons terms in the effective gauge theory of the FTI state.

In this paper we compute the entanglement, as quantified by negativity, between two blocks of length L A and L B , separated by L sites in the one-dimensional spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) model. We took the model with two different boundary conditions. We consider the case of N spin-1 particles in the bulk and one spin-1/2 particle at each boundary, which constitute a unique ground state, and the case of just spin 1, even at the end of the chain, where the degeneracy of the ground state is 4. In both scenarios we made a partition consisting of two blocks A and B, containing L A and L B sites, respectively. The separation of these two blocks is L. In both cases we explicitly obtain the reduced density matrix of the blocks A and B. We prove that the negativity in the first case vanishes identically for L 1, while in the second scenario it may approach a constant value N = 1/2 for each degenerate eigenstate depending on the way one constructs these eigenstates. However, as there is some freedom in constructing these eigenstates, vanishing entanglement is also possible in the latter case. Additionally, we also compute the entanglement between noncomplementary blocks in the case of periodic boundary conditions for the spin-1 AKLT model for which there is a unique ground state. Even in this case, we find that the negativity of separated blocks of spins is zero.