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arXiv:1511.05255 · Physical review. B./Physical review. B

Universal nonanalytic behavior of the Hall conductance in a Chern insulator at the topologically driven nonequilibrium phase transition

Pei Wang, Markus Schmitt, Stefan K. Kehrein

Published Feb 24, 2016• 33 citations
arXivHTMLPDF

Abstract

We study the Hall conductance of a Chern insulator after a global quench of the Hamiltonian. The Hall conductance in the long time limit is obtained by applying the linear response theory to the diagonal ensemble. It is expressed as the integral of the Berry curvature weighted by the occupation number over the Brillouin zone. We identify a topologically driven nonequilibrium phase transition, which is indicated by the nonanalyticity of the Hall conductance as a function of the energy gap ${m}_{f}$ in the post-quench Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{f}$. The topological invariant for the quenched state is the winding number of the Green's function $W$, which equals the Chern number for the ground state of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{f}$. In the limit ${m}_{f}\ensuremath{\rightarrow}0$, the derivative of the Hall conductance with respect to ${m}_{f}$ is proportional to $ln|{m}_{f}|$, with the constant of proportionality being the ratio of the change of $W$ at ${m}_{f}=0$ to the energy gap in the initial state. This nonanalytic behavior is universal in two-band Chern insulators such as the Dirac model, the Haldane model, or the Kitaev honeycomb model in the fermionic basis.

DOI
10.1103/physrevb.93.085134

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We study the Hall conductance of a Chern insulator after a global quench of the Hamiltonian. The Hall conductance in the long time limit is obtained by applying the linear response theory to the diagonal ensemble. It is expressed as the integral of the Berry curvature weighted by the occupation number over the Brillouin zone. We identify a topologically driven nonequilibrium phase transition, which is indicated by the nonanalyticity of the Hall conductance as a function of the energy gap mf{m}_{f}mf​ in the post-quench Hamiltonian H\ifmmode^\elseˆ\fif{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{f}H\ifmmode^\elseˆ\fi​f​. The topological invariant for the quenched state is the winding number of the Green's function WWW, which equals the Chern number for the ground state of H\ifmmode^\elseˆ\fif{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{f}H\ifmmode^\elseˆ\fi​f​. In the limit mf\ensuremath→0{m}_{f}\ensuremath{\rightarrow}0mf​\ensuremath→0, the derivative of the Hall conductance with respect to mf{m}_{f}mf​ is proportional to ln∣mf∣ln|{m}_{f}|ln∣mf​∣, with the constant of proportionality being the ratio of the change of WWW at mf=0{m}_{f}=0mf​=0 to the energy gap in the initial state. This nonanalytic behavior is universal in two-band Chern insulators such as the Dirac model, the Haldane model, or the Kitaev honeycomb model in the fermionic basis.
We identify a new class of phase transitions when calculating the Hall conductance of two-band Chern insulators in the long-time limit after a global quench of the Hamiltonian. The Hall conductance is expressed as the integral of the Berry curvature in the diagonal ensemble. Even if the Chern number of the unitarily-evolving wave function is conserved, the Hall conductance as a function of the energy gap in the post-quench Hamiltonian displays a continuous but nonanalytic behavior, that is it has a logarithmically divergent derivative as the gap closes. The coefficient of this logarithmic function is the ratio of the change of the Chern number for the ground state of the post-quench Hamiltonian to the energy gap in the initial state. This nonanalytic behavior is universal in two-band Chern insulators.
We develop a method to characterize topological phase transitions for strongly correlated Hamiltonians defined on two-dimensional lattices based on the many-body Berry curvature. Our goal is to identify a class of quantum critical points between topologically nontrivial phases with fractionally quantized Hall (FQH) conductivity and topologically trivial gapped phases through the discontinuities of the many-body Berry curvature in the so-called flux Brillouin zone (fBZ), the latter being defined by imposing all possible twisted boundary conditions. For this purpose, we study the finite-size signatures of several quantum phase transitions between fractional Chern insulators and charge-ordered phases for two-dimensional lattices by evaluating the many-body Berry curvature numerically using exact diagonalization. We observe degeneracy points (nodes) of many-body energy levels at high-symmetry points in the fBZ, accompanied by diverging Berry curvature. We find a correspondence between the number and order of these nodal points, and the change of the topological invariants of the many-body ground states across the transition, in close analogy with Weyl nodes in noninteracting band structures. This motivates us to apply a scaling procedure, originally developed for noninteracting systems, for the Berry curvature at the nodal points. This procedure offers a useful tool for the classification of topological phase transitions in interacting systems harboring FQH like topological order.
We investigate the transport properties of Chern insulators following a quantum quench between topological and nontopological phases. Recent works have shown that this yields an excited state for which the Chern number is preserved under unitary evolution. However, this does not imply the preservation of other physical observables, as we stressed in our previous work. Here we provide an analysis of the Hall response following a quantum quench in an isolated system, with explicit results for the Haldane model. We show that the Hall conductance is no longer related to the Chern number in the postquench state, in agreement with previous work. We also examine the dynamics of the edge currents in finite-size systems with open boundary conditions along one direction. We show that the late-time behavior is captured by a Generalized Gibbs Ensemble, after multiple traversals of the sample. We discuss the effects of generic open boundary conditions and confinement potentials.
The observed robustly quantized Hall conductance in quantum Hall systems and Chern insulators (CI) is normally understood in terms of the bulk topology of isolated systems, not coupled to leads. It is assumed that the leads act as inert reservoirs. Within a model of a CI coupled to leads with a cylindrical geometry, we show that this is not always true. We identify the Hall conductance with a boundary invariant, the winding number of the phase of the reflection coefficient. We find anomalous topological phases where the boundary invariant is not the same as the bulk Chern number, even in the limit of weak lead coupling.
We study the nonequilibrium time evolution of a variety of one-dimensional (1D) and two-dimensional (2D) systems (including SSH model, Kitaev-chain, Haldane model, p+ipp+ipp+ip superconductor, etc.) following a sudden quench. We prove analytically that topology-changing quenches are always followed by nonanalytical temporal behavior of return rates (logarithm of the Loschmidt echo), referred to as dynamical phase transitions (DPTs) in the literature. Similarly to edge states in topological insulators, DPTs can be classified as being topologically protected or not. In 1D systems the number of topologically protected nonequilibrium time scales are determined by the difference between the initial and final winding numbers, while in 2D systems no such relation exists for the Chern numbers. The singularities of dynamical free energy in the 2D case are qualitatively different from those of the 1D case; the cusps appear only in the first time derivative.
Chern insulators exhibit fascinating properties, which originate from the topologically nontrivial state characterized by the Chern number. How these properties change if the system is quenched between topologically distinct phases is, however, not fully understood. In this paper, we investigate the quench dynamics of the prototypical massive Dirac model for topological insulators in two dimensions. We consider both dissipationless dynamics and the effect of electron-phonon interactions, and ask how the transient dynamics and nonequilibrium steady states affect simple observables. Specifically, we discuss a time-dependent generalization of the Hall effect and the dichroism of the photoexcitation probability between left and right circularly polarized light. We present optimized schemes based on these observables, which can reveal the evolution of the topological state of the quenched system.