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arXiv:1710.05322

Entanglement entropy distribution in the strongly disordered one-dimensional Anderson model

Bat-el Friedman, Richard Berkovits

Published Oct 15, 2017• 0 citations
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Abstract

The entanglement entropy distribution of strongly disordered one dimensional spin chains, which are equivalent to spinless fermions at half-filling on a bond (hopping) disordered one-dimensional Anderson model, has been shown to exhibit very distinct features such as peaks at integer multiplications of $\ln(2)$, essentially counting the number of singlets traversing the boundary. Here we show that for a canonical Anderson model with box distribution on-site disorder and repulsive nearest-neighbor interactions the entanglement entropy distribution also exhibits interesting features, albeit different than the distribution seen for the bond disordered Anderson model. The canonical Anderson model shows a broad peak at low entanglement values and one narrower peak at $\ln(2)$. Density matrix renormalization group (DMRG) calculations reveal this structure and the influence of the disorder strength and the interaction strength on its shape. A modified real space renormalization group (RSRG) method was used to get a better understanding of this behavior. As might be expected the peak centered at low values of entanglement entropy has a tendency to shift to lower values as disorder is enhanced. A second peak appears around the entanglement entropy value of $\ln(2)$,this peak is broadened and no additional peaks at higher integer multiplications of $\ln(2)$ are seen. We attribute the differences in the distribution between the canonical model and the broad hopping disorder to the influence of the on-site disorder which breaks the symmetry across the boundary.

DOI
10.48550/arxiv.1710.05322

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The entanglement entropy distribution of strongly disordered one dimensional spin chains, which are equivalent to spinless fermions at half-filling on a bond (hopping) disordered one-dimensional Anderson model, has been shown to exhibit very distinct features such as peaks at integer multiplications of ln⁡(2)\ln(2)ln(2), essentially counting the number of singlets traversing the boundary. Here we show that for a canonical Anderson model with box distribution on-site disorder and repulsive nearest-neighbor interactions the entanglement entropy distribution also exhibits interesting features, albeit different than the distribution seen for the bond disordered Anderson model. The canonical Anderson model shows a broad peak at low entanglement values and one narrower peak at ln⁡(2)\ln(2)ln(2). Density matrix renormalization group (DMRG) calculations reveal this structure and the influence of the disorder strength and the interaction strength on its shape. A modified real space renormalization group (RSRG) method was used to get a better understanding of this behavior. As might be expected the peak centered at low values of entanglement entropy has a tendency to shift to lower values as disorder is enhanced. A second peak appears around the entanglement entropy value of ln⁡(2)\ln(2)ln(2),this peak is broadened and no additional peaks at higher integer multiplications of ln⁡(2)\ln(2)ln(2) are seen. We attribute the differences in the distribution between the canonical model and the broad hopping disorder to the influence of the on-site disorder which breaks the symmetry across the boundary.
Abstract The entanglement entropy (EE) distribution of strongly disordered one dimensional spin chains, which are equivalent to spinless fermions at half-filling on a bond (hopping) disordered one-dimensional Anderson model, has been shown to exhibit very distinct features such as peaks at integer multiplications of , essentially counting the number of singlets traversing the boundary. Here we show that for a canonical Anderson model with box distribution on-site disorder and repulsive nearest-neighbor interactions the EE distribution also exhibits interesting features, albeit different than the distribution seen for the bond disordered Anderson model. The canonical Anderson model shows a broad peak at low entanglement values and one narrower peak at . Density matrix renormalization group calculations reveal this structure and the influence of the disorder strength and the interaction strength on its shape. A modified real space renormalization group method was used to get a better understanding of this behavior. As might be expected the peak centered at low values of EE has a tendency to shift to lower values as disorder is enhanced. A second peak appears around the EE value of , this peak is broadened and no additional peaks at higher integer multiplications of are seen. We attribute the differences in the distribution between the canonical model and the broad hopping disorder to the influence of the on-site disorder which breaks the symmetry across the boundary.
The properties of the entanglement entropy (EE) in one-dimensional disordered interacting systems are studied. Anderson localization leaves a clear signature on the average EE, as it saturates on the length scale exceeding the localization length. This is verified by numerically calculating the EE for an ensemble of disordered realizations using the density matrix renormalization group method. A heuristic expression describing the dependence of the EE on the localization length, which takes into account finite-size effects, is proposed. This is used to extract the localization length as a function of the interaction strength. The localization length dependence on the interaction fits nicely with the expectations.
We study the entanglement entropy(EE) of disordered one-dimensional spinless fermions with attractive interactions. With intensive numerical calculation of the EE using the density matrix renormalization group method, we find clear signatures of the transition between the localized and delocalized phase. In the delocalized phase, the fluctuations of the EE becomes minimum and independent of the system size. Meanwhile the EE's logarithmic scaling behavior is found to recover to that of a clean system. We present a general scheme of finite size scaling of the EE at the critical regime of the Anderson transition, from which we extract the critical parameters of the transition with good accuracy, including the critical exponent, critical point and a power-law divergent localization length.
We study the low-energy states of the 1D random-hopping model in a strongly disordered regime. The entanglement structure is shown to depend solely on the probability distribution for the length of the effective bonds P ( l b ), whose scaling and finite-size behavior are established using renormalization-group arguments and a simple model based on random permutations. Parity oscillations are absent in von Neumann entropy with periodic boundary conditions, but appear in the higher moments of the distribution, such as the variance. The particle-hole excited states leave the bond structure and the entanglement untouched. Nonetheless, particle addition or removal deletes bonds and leads to an effective saturation of entanglement at an effective block size given by the expected value for the longest bond.
We investigate the 1D Anderson-Hubbard model at half filling with box-disorder. The ground state phase diagram is obtained by means of real-space dynamical mean-field theory (R-DMFT) and the density matrix renormalization group (DMRG). We find Mott insulating and Anderson localized regimes as well as a strong indication of a delocalized phase for intermediate interaction and disorder strength within accessible system sizes. These phases are characterized and distinguished by qualitatively different scaling behavior of the local density of states, the energy gap in the excitation spectrum and the inverse participation number.
Abstract We consider different disordered lattice models composed of M linear chains glued together in a star-like manner, and study the scaling of the entanglement between one arm and the rest of the system using a numerical strong-disorder renormalization group method. For all studied models, the random transverse-field Ising model (RTIM), the random XX spin model, and the free-fermion model with random nearest-neighbor hopping terms, the average entanglement entropy is found to increase with the length L of the arms according to the form . For the RTIM and the XX model, the effective central charge is universal with respect to the details of junction, and only depends on the number M of arms. Interestingly, for the RTIM, decreases with M , whereas for the XX model it increases. For the free-fermion model, depends also on the details of the junction, which is related to the sublattice symmetry of the model. In this case, both increasing and decreasing tendency with M can be realized with appropriate junction geometries.