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Acta Physica Sinica

Topological invariant in quench dynamics

Chao Yang, Shu Chen

Published Jan 1, 2019• 6 citations
Publisher

Abstract

In this review, we give a brief review on the recent progress in the theoretical research of quench dynamics in topological band systems. Beginning with two band models, we introduce conception of dynamical Chern number and give the connection between the dynamical Chern number and topological invariant in the corresponding equilibrium systems. Then by studying the 1 + 1 dimensional parent Hamiltonian, we show the complete dynamical classification of Altland-Zirnbauer classes, and show the crossing of entanglement spectrum as a feature of dynamical bulk edge correspondence. Furthermore, we consider the impact of the disorder and band dispersion. At last, we show the experimental simulation of dynamical Chern number by a superconducting qubit system.

DOI
10.7498/aps.68.20191410

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In this review, we give a brief review on the recent progress in the theoretical research of quench dynamics in topological band systems. Beginning with two band models, we introduce conception of dynamical Chern number and give the connection between the dynamical Chern number and topological invariant in the corresponding equilibrium systems. Then by studying the 1 + 1 dimensional parent Hamiltonian, we show the complete dynamical classification of Altland-Zirnbauer classes, and show the crossing of entanglement spectrum as a feature of dynamical bulk edge correspondence. Furthermore, we consider the impact of the disorder and band dispersion. At last, we show the experimental simulation of dynamical Chern number by a superconducting qubit system.
We unveil the stable (d+1)-dimensional topological structures underlying the quench dynamics for all of the Altland-Zirnbauer classes in d=1 dimension, and we propose to detect such dynamical topology from the time evolution of entanglement spectra. Focusing on systems in classes BDI and D, we find crossings in single-particle entanglement spectra for quantum quenches between different symmetry-protected topological phases. The entanglement-spectrum crossings are shown to be stable against symmetry-preserving disorder and faithfully reflect both Z (class BDI) and Z_{2} (class D) topological characterizations. As a by-product, we unravel the topological origin of the global degeneracies temporarily emerging in the many-body entanglement spectrum in the quench dynamics of the transverse-field Ising model. These findings can experimentally be tested in ultracold atoms and trapped ions with the help of cutting-edge tomography for quantum many-body states. Our work paves the way towards a systematic understanding of the role of topology in quench dynamics.
We study the effect of strong disorder on topology and entanglement in quench dynamics. Although disorder-induced topological phases have been well studied in equilibrium, the disorder-induced topology in quench dynamics has not been explored. In this paper, we predict a disorder-induced topology of postquench states characterized by the quantized dynamical Chern number and the crossings in the entanglement spectrum in (1+1)(1+1)(1+1) dimensions. The dynamical Chern number undergoes transitions from zero to unity and back to zero when increasing the disorder strength. The boundaries between different dynamical Chern numbers are determined by delocalized critical points in the postquench Hamiltonian with the strong disorder. An experimental realization in quantum walks is discussed.
In a recent work by Gong and Ueda [Phys. Rev. Lett. 121, 250601 (2018)], the classification of (1+1)-dimensional quench dynamics for the ten Altland-Zirnbauer classes is achieved, and entanglement-spectrum crossings of the time-dependent states for the topological classes (AIII, DIII, CII, BDI, and D) are discovered as a consequence of the bulk-edge correspondence. We note that their classification scheme for the (1+1)-dimensional bulk topology as well as the adoption of entanglement-spectrum crossings as a diagnosis of topological edge states focus on the limit that the spectrum of the postquench Hamiltonian is flat. When band dispersion of the postquench Hamiltonian is considered, even though the bulk topology is still stable, the entanglement-spectrum crossing at the edge might be unstable due to the fact that the band dispersion can be regarded as a ``symmetry-breaking disorder'' in the frequency domain. Such a symmetry-breaking disorder lowers the symmetry for the edge side, and the conventional bulk-edge correspondence could not be applied directly for the quench dynamics. We show that because of the reduction of symmetry by finite energy dispersion the gapless entanglement-spectrum crossing in the flatband limit in classes AIII, DIII, and CII is unstable and could be gapped without closing the bulk gap. The entanglement-spectrum crossing in classes BDI and D is still stable against energy dispersion. We show that the quench process for classes BDI and D can be understood as a Z2{\mathbb{Z}}_{2}Z2​ fermion parity pump, and the entanglement-spectrum crossing for this case is protected by the conservation of fermion parity.
A dynamical equivalent of the bulk-boundary correspondence has been suggested to occur in one and two dimensional topological models following sudden quenches. Depending on the topological invariant of the time evolving and initial phases involved large boundary contributions to a dynamical free energy occur. Moreover they occur periodically between the critical times at which this dynamical free energy becomes non-analytic, \emph{i.e.}~at dynamical quantum phase transitions. At these critical times the eigenvalue spectrum of the Loschmidt matrix which underlies the dynamical free energy closes its gap. The boundary contributions are understood to be due to zero-modes or in-gap bands of this matrix, forming a close analogy with equilibrium topological models and their edge modes. The exact cause of this phenomena and its generality remain unknown. In this article we test the dynamical bulk-boundary correspondence for a complicated two dimensional topological superconductor with a rich phase diagram, allowing quenches between many different Chern numbers. We show that there is no straightforward correspondence between the equilibrium phases quenched between and the dynamical bulk boundary correspondence. Furthermore the correspondence can depend on the orientation of the edges, suggesting a possible weak topological variant.
We classify the topology of the quench dynamics by homotopy groups. A relation between the topological invariant of a postquench order parameter and the topological invariant of a static Hamiltonian is shown in d+1d+1d+1 dimensions (d=1,2,3d=1,2,3d=1,2,3). The midgap states in the entanglement spectrum of the postquench states reveal their topological nature. When a trivial quantum state is under a sudden quench to a Chern insulator, the midgap states in the entanglement spectrum form rings. These rings are analogous to the boundary Fermi rings in the Hopf insulators. Finally, we show a postquench order parameter in 3+1 dimensions can be characterized by the second Chern number. The number of Dirac cones in the entanglement spectrum is equal to the second Chern number.