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arXiv:1707.07176 · Physical review. B./Physical review. B

Sensitivity of the entanglement spectrum to boundary conditions as a characterization of the phase transition from delocalization to localization

Mohammad Pouranvari, Afshin Montakhab

Published Jul 17, 2017• 9 citations
arXivHTMLPDF

Abstract

Sensitivity of entanglement Hamiltonian spectrum to boundary conditions is considered as a phase detection parameter for delocalized-localized phase transition. By employing one-dimensional models that undergo delocalized-localized phase transition, we study the shift in the entanglement energies and the shift in the entanglement entropy when we change boundary conditions from periodic to antiperiodic. Specifically, we show that both these quantities show a change of several orders of magnitude at the transition point in the models considered. Therefore, this shift can be used to indicate the phase-transition points in the models. We also show that both these quantities can be used to determine mobility edges separating localized and delocalized states.

DOI
10.1103/physrevb.96.045123

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Sensitivity of entanglement Hamiltonian spectrum to boundary conditions is considered as a phase detection parameter for delocalized-localized phase transition. By employing one-dimensional models that undergo delocalized-localized phase transition, we study the shift in the entanglement energies and the shift in the entanglement entropy when we change boundary conditions from periodic to antiperiodic. Specifically, we show that both these quantities show a change of several orders of magnitude at the transition point in the models considered. Therefore, this shift can be used to indicate the phase-transition points in the models. We also show that both these quantities can be used to determine mobility edges separating localized and delocalized states.
We introduce characterizations for many-body phase transitions between delocalized and localized phases based on the system's sensitivity to boundary conditions. In particular, we change boundary conditions from periodic to antiperiodic and calculate a shift in the system's energy and shifts in the single-particle density-matrix eigenvalues in the corresponding energy window. We employ the typical model for studying MBL, a one-dimensional disordered system of fermions with nearest-neighbor repulsive interaction where disorder is introduced as randomness on on-site energies. By calculating numerically the shifts in the system's energy and eigenvalues of the single-particle density matrix, we observe that in the localized regime, both shifts are vanishing; while in the extended regime, both shifts are on the order of the corresponding level spacing. We also applied these characterizations of the phase transition to the case of having next-nearest-neighbor interactions in addition to the nearest-neighbor interactions and studied its effect on the transition.
In general, extended states are more sensitive to boundary conditions (BCs) than localized ones. We expand an eigenstate of a Hamiltonian with periodic BC as the superposition of all eigenstates of the same Hamiltonian but with antiperiodic BC, i.e. , , where and N is the Hilbert space dimension. Thus, we propose a novel quantum Shannon entropy , which simultaneously relates to periodic and antiperiodic BCs. It measures the information of contained in all the N eigenstates . Based on it, we define a reduced quantum Shannon entropy , where . We apply it to several typical models. Results show that are relatively smaller for delocalized states and are less than and close to one for localized states as N increases. There are rapid changes in around Anderson transition points. Therefore, can be used as a sensitivity measure to characterize Anderson transitions.
By using the measure of von Neumann entropy, we numerically investigate quantum entanglement of an electron moving in the one-dimensional Harper model and in the one-dimensional slowly varying potential model. The delocalized and localized eigenstates can be distinguished by von Neumann entropy of the individual eigenstates. There are drastic decreases in von Neumann entropy of the individual eigenstates at mobility edges. In the curve of the spectrum averaged von Neumann entropy as a function of potential parameter λ, a sharp transition exists at the metal–insulator transition point λ c = 2. It is found that the von Neumann entropy is a good quantity to reflect localization and metal–insulator transition.
We study the entanglement Hamiltonian (EH) associated with the reduced density matrix of free fermion models in a delocalized-localized Anderson phase transition. We show numerically that the structure of the EH matrix differentiates the delocalized from the localized phase. In the delocalized phase, EH becomes a long-range Hamiltonian but is short-range in the localized phase, no matter what the configuration of the system's Hamiltonian is (whether it is long- or short-range). With this view, we introduce the entanglement conductance (EC), which quantifies how much EH is long-range, and we propose it as an alternative quantity to measure entanglement in the Anderson phase transition, by which we locate the phase transition point of some one-dimensional free fermion models. In addition, by applying the finite-size method to the EC, we find three-dimensional Anderson phase transition critical disorder strength.
In one dimension, noninteracting particles can undergo a localization-delocalization transition in a quasiperiodic potential. Recent studies have suggested that this transition transforms into a many-body localization (MBL) transition upon the introduction of interactions. It has also been shown that mobility edges can appear in the single particle spectrum for certain types of quasiperiodic potentials. Here, we investigate the effect of interactions in two models with such mobility edges. Employing the technique of exact diagonalization for finite-sized systems, we calculate the level spacing distribution, time evolution of entanglement entropy, optical conductivity, and return probability to detect MBL. We find that MBL does indeed occur in one of the two models we study, but the entanglement appears to grow faster than logarithmically with time unlike in other MBL systems.
Single-particle entanglement entropy (SPEE) is calculated for entanglement Hamiltonian eigenmode in a one-dimensional free fermion model that undergoes a delocalized–localized phase transition. In this numerical study, we show that SPEE of entanglement Hamiltonian eigenmode has the same behavior as SPEE of Hamiltonian eigenmode at the Fermi level: as we go from delocalized phase toward localized phase, SPEE of both modes decrease in the same manner. Furthermore, fluctuations of SPEE of entanglement Hamiltonian eigenmode — which can be obtained through the calculation of moments of SPEE — signature very sharply the phase transition point. These two modes are also compared by calculation of single-particle Rényi entropy (SPRE). We show that SPEE and SPRE of entanglement Hamiltonian eigenmode can be used as phase detection parameters.