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arXiv:1205.5071 · Physical Review B

Signature of phase transitions in the disordered quantum spin Hall state from the entanglement spectrum

Matthew J. Gilbert, Bogdan Andrei Bernevig, Taylor L. Hughes

Published Jul 5, 2012• 13 citations
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Abstract

Of the available classes of insulators which have been shown to contain topologically nontrivial properties, one of the most important is class AII, which contains systems that possess time-reversal symmetry $T$ with ${T}^{2}=\ensuremath{-}1.$ This class has been the subject of significant attention as it encompasses nontrivial ${Z}_{2}$ topological insulators such as the quantum spin Hall (QSH) state and the three-dimensional strong topological insulator. One of the defining properties of this system is the robustness of the state under the addition of disorder that preserves $T.$ In this Rapid Communication, we explore the phase diagram of the disordered QSH state as a function of disorder strength and chemical potential by examining the entanglement spectrum for disordered class-AII symplectic systems. As for the case of the $T$-breaking Chern insulator, we show that there is a correspondence between the level-spacing statistics of the Hamiltonian and that of the level-spacing statistics of the entanglement spectrum. We observe a feature in the statistics of the entanglement spectrum that aids the identification of delocalized states and consequently critical energies across which phase transitions occur.

DOI
10.1103/physrevb.86.041401

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Of the available classes of insulators which have been shown to contain topologically nontrivial properties, one of the most important is class AII, which contains systems that possess time-reversal symmetry TTT with T2=\ensuremath−1.{T}^{2}=\ensuremath{-}1.T2=\ensuremath−1. This class has been the subject of significant attention as it encompasses nontrivial Z2{Z}_{2}Z2​ topological insulators such as the quantum spin Hall (QSH) state and the three-dimensional strong topological insulator. One of the defining properties of this system is the robustness of the state under the addition of disorder that preserves T.T.T. In this Rapid Communication, we explore the phase diagram of the disordered QSH state as a function of disorder strength and chemical potential by examining the entanglement spectrum for disordered class-AII symplectic systems. As for the case of the TTT-breaking Chern insulator, we show that there is a correspondence between the level-spacing statistics of the Hamiltonian and that of the level-spacing statistics of the entanglement spectrum. We observe a feature in the statistics of the entanglement spectrum that aids the identification of delocalized states and consequently critical energies across which phase transitions occur.
We investigate the behavior of a topological Chern insulator in the presence of disorder, with a focus on its entanglement spectrum (EtS) constructed from the ground state. For systems with symmetries, the EtS was shown to contain explicit information about the topological universality class revealed by sorting the EtS against the conserved quantum numbers. In the absence of any symmetry, we demonstrate that statistical methods such as the level statistics of the EtS can be equally insightful, allowing us to distinguish when an insulator is in a topological or trivial phase and to map the boundary between the two phases. The phase diagram of a Chern insulator is explicitly computed as function of Fermi level (EF) and disorder strength using the level statistics of the EtS and energy spectrum, together with a computation of the Chern number (C) via a new, efficient real-space formula.
We study topological insulators under dephasing noise. With examples of both a 2d2d2d Chern insulator and a 3d3d3d topological insulator protected by time-reversal symmetry, we demonstrate that there is a phase transition at finite dephasing strength between phases with nontrivial and trivial topological indices. Here, the topological index is defined through the correlation matrix. The transition can be diagnosed through the spectrum of the whole correlation matrix or of a local subsystem. Interestingly, even if the topological insulator is very close to the topological-trivial critical point in its Hamiltonian, it still takes finite strength of dephasing to change the topological index, suggesting the robustness of topological insulators under dephasing. We further consider Chern insulators in the presence of real-space disorder, which exhibit a ground-state transition between topological and Anderson insulating phases. We find that even strongly disordered Chern insulators, close to the critical disorder strength, exhibit robustness with respect to dephasing.
We discuss the effects of disorder in time-reversal-invariant topological insulators and superconductors in three spatial dimensions. For three-dimensional topological insulator in the symplectic (AII) symmetry class, the phase diagram in the presence of disorder and a mass term, which drives a transition between trivial and topological insulator phases, is computed numerically by the transfer matrix method. The numerics is supplemented by a field-theory analysis (the large-Nf{N}_{f}Nf​ expansion, where Nf{N}_{f}Nf​ is the number of valleys or Dirac cones), from which we obtain the correlation length exponent, and several anomalous dimensions at a nontrivial critical point separating a metallic phase and a Dirac semimetal. A similar-field theory approach is developed for disorder-driven transitions in symmetry classes AIII, CI, and DIII. For these three symmetry classes, where topological superconductors are characterized by integer topological invariant, a complementary description is given in terms of the nonlinear sigma model supplemented with a topological term, which is a three-dimensional analog of the Pruisken term in the integer quantum Hall effect.
We investigate topological phase transitions in Chern insulators within three-band models, focusing on the empty band and lowest band populated by spinless fermions. We consider Lieb and kagome lattices and notice phase transitions driven by the hopping integral between nearest-neighbors, which leads to the change of the lowest band Chern number from C=1C=1C=1 to C=−1C=-1C=−1. In the single-particle picture, different phases are examined by investigating corresponding entanglement spectra and the evolution of the entanglement entropy. Entanglement spectra reveal the spectral flow characteristic of topologically nontrivial systems before and after phase transitions. For the lowest band with 1/31/31/3 filling, Fractional Chern insulator (FCI) phases are identified by examining the ground state momenta, the spectral flow, and counting of the entanglement energy levels below the gap in the entanglement spectrum. A quasilinear dependence of the entanglement entropy α(nA)α(n_A)α(nA​) term is observed for FCI phase, similarly to linear behavior expected for Laughlin phase in the fractional quantum Hall effect. At the topological phase transition, for both empty and partially filled lowest bands, the energy gap closure and a discontinuity in the entanglement entropy are observed. This coincides with the divergence of a standard deviation of the Berry curvature. We notice also a phase transition driven by the nearest-neighbor interaction in the Lieb lattice, where the many-body energy gap closes. The phase transitions are shown to be stable for an arbitrary system size, thus predicted to be present in the thermodynamic limit. While our calculations are performed for an interaction energy far exceeding the gap between the two lowest energy bands, we note that the higher band does not affect the phase transitions, however destabilizes FCI phases.
By numerical exact diagonalization techniques, we obtain the quantum phase diagram of the lattice fractional quantum Hall (FQH) systems in the presence of quenched disorder. By implementing an array of local potential traps representing the disorder, we show that the system undergoes a series of quantum phase transitions as the disorder and/or the interaction is tuned. As the strength of potential traps is increased, the FQH state turns into a compressible liquid and then into a topologically trivial insulator. We use the numerically calculated energy gap, quantum degeneracy, the Chern number, the entanglement spectrum, and the fidelity metric to identify various quantum phases. The connection to continuum FQH effects is also discussed.
Interactions generically have important effects on topological quantum phases. For a quantum anomalous Hall (QAH) insulator, the presence of interactions can qualitatively change a topological phase diagram, which, however, is typically hard to measure in an experiment. Here we propose a scheme based on quench dynamics to detect the mean-field topological phase diagram of an interacting Chern insulator described by the QAH-Hubbard model, with nontrivial dynamical quantum physics being uncovered. We focus on the dynamical properties of the system at a weak to intermediate Hubbard interaction, which mainly induces a ferromagnetic order under the mean-field level. Remarkably, three characteristic times---ts,tc{t}_{s}, {t}_{c}ts​,tc​, and t∗{t}^{*}t∗---are found in the quench dynamics. The first two (ts,tc{t}_{s},{t}_{c}ts​,tc​) capture the emergences of dynamical self-consistent particle density and the dynamical topological phase transition, respectively, while the third one (t∗{t}^{*}t∗) gives a linear scaling time on the topological phase boundaries. We show generically that the characteristic times obey ts>t∗>tc{t}_{s}>{t}^{*}>{t}_{c}ts​>t∗>tc​ (t∗<ts<tc{t}^{*}<{t}_{s}<{t}_{c}t∗<ts​<tc​) in the repulsive (attractive) interacting regime, when the system is quenched from an initial nearly fully polarized state to the topologically nontrivial regimes. Moreover, the Chern number of the equilibrium phase of postquench interacting Hamiltonian can be determined by any two of the three timescales, providing a dynamical way to determine the equilibrium mean-field topological phases. Experimentally, the measurement of ts{t}_{s}ts​ is challenging, while tc{t}_{c}tc​ and t∗{t}^{*}t∗ can be directly read out by measuring the spin polarizations of four Dirac points and the time-dependent particle density, respectively, showing the feasibility of the present dynamical scheme. Our work opens a way to detect by quench dynamics the mean-field phase diagram of Chern insulators with interactions.