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Bulk-boundary correspondence in non-equilibrium dynamics of one-dimensional topological insulators

Philipp Jaeger

Published May 24, 2017• 1 citation
Publisher

Abstract

Dynamical phase transitions (DPT) are receiving a rising interest. They are known to behave analogously to equilibrium phase transitions (EPT) to a large extend. However, it is easy to see that DPT can occur in finite systems, while EPT are only possible in the thermodynamic limit. So far it is not clear how far the analogy of DPT and EPT goes. It was suggested, that there is a relation between topological phase transitions (TPT) and DPT, but many open questions remain. Typically, to study DPT, the Loschmidt echo (LE) after a quench is investigated, where DPT are visible as singularities. For one-dimensional systems, each singularity is connected to a certain critical time scale, which is given by the dispersion in the chain. In topological free-fermion models with winding numbers 0 or 1, only the LE in periodic boundary conditions (PBC) has been investigated. In open boundary conditions (OBC), these models are characterized by symmetry protected edge modes in the topologically non-trivial phase. It is completely unclear how these modes affect DPT. We investigate systems with PBC governed by multiple time scales with a Z topological invariant. In OBC, we provide numerical evidence for the presence of bulk-boundary correspondence in DPT in quenches across a TPT.

DOI
10.5281/zenodo.1183396

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Dynamical phase transitions (DPT) are receiving a rising interest. They are known to behave analogously to equilibrium phase transitions (EPT) to a large extend. However, it is easy to see that DPT can occur in finite systems, while EPT are only possible in the thermodynamic limit. So far it is not clear how far the analogy of DPT and EPT goes. It was suggested, that there is a relation between topological phase transitions (TPT) and DPT, but many open questions remain. Typically, to study DPT, the Loschmidt echo (LE) after a quench is investigated, where DPT are visible as singularities. For one-dimensional systems, each singularity is connected to a certain critical time scale, which is given by the dispersion in the chain. In topological free-fermion models with winding numbers 0 or 1, only the LE in periodic boundary conditions (PBC) has been investigated. In open boundary conditions (OBC), these models are characterized by symmetry protected edge modes in the topologically non-trivial phase. It is completely unclear how these modes affect DPT. We investigate systems with PBC governed by multiple time scales with a Z topological invariant. In OBC, we provide numerical evidence for the presence of bulk-boundary correspondence in DPT in quenches across a TPT.
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.
Dynamical quantum phase transitions occur in dynamically evolving quantum systems when nonanalyticities occur at critical times in the return rate, a dynamical analog of the free energy. This extension of the concept of phase transitions can be brought into contact with another, namely, that of topological phase transitions in which the phase transition is marked by a change in a topological invariant. Following a quantum quench dynamical quantum phase transitions can happen in topological matter, a fact which has already been explored in one-dimensional topological insulators and in two-dimensional Chern insulators. Additionally, in one-dimensional systems a dynamical bulk boundary correspondence has been seen, related to the periodic appearance of zero modes of the Loschmidt echo itself. Here we extend both of these concepts to two-dimensional higher-order topological matter in which the topologically protected boundary modes are corner modes. We consider a minimal model which encompasses all possible forms of higher-order topology in two-dimensional topological band structures. We find that dynamical quantum phase transitions can still occur, and can occur for quenches which cross both bulk and boundary gap closings. Furthermore, a dynamical bulk boundary correspondence is also found, which takes a different form to that in one dimension.
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.
We study the Loschmidt echo for quenches in open one-dimensional lattice models with symmetry protected topological phases. For quenches where dynamical quantum phase transitions do occur we find that cusps in the bulk return rate at critical times tc{t}_{c}tc​ are associated with sudden changes in the boundary contribution. For our main example, the Su-Schrieffer-Heeger model, we show that these sudden changes are related to the periodical appearance of two eigenvalues close to zero in the dynamical Loschmidt matrix. We demonstrate, furthermore, that the structure of the Loschmidt spectrum is linked to the periodic creation of long-range entanglement between the edges of the system.
Dynamical phase transitions (DPTs) occur after quenching some global parameters in quantum systems, and are signalled by the nonanalytical time evolution of the dynamical free energy, which is calculated from the Loschmidt overlap between the initial and time evolved states. In a recent Letter [M. Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)], it was suggested that DPTs are closely related to equilibrium phase transitions (EPTs) for the transverse field Ising model. By studying a minimal model, the XY chain in a transverse magnetic field, we show analytically that this connection does not hold generally. We present examples where DPT occurs without crossing any equilibrium critical lines by the quench, and a nontrivial example with no DPT but crossing a critical line by the quench. Although the nonanalyticities of the dynamical free energy on the real time axis do not indicate the presence or absence of an EPT, the structure of Fisher lines for complex times reveals a qualitative difference.
Abstract We provide strong numerical evidence for a dynamical bulk-boundary correspondence in two-dimensional topological matter which manifests itself as boundary contributions to the dynamical free energy and is governed by a two-dimensional non-Hermitian dynamical Loschmidt matrix—a setting largely unexplored beyond one dimension. Following a quantum quench, in-gap bands emerge in the spectrum of the Loschmidt matrix between successive dynamical quantum phase transitions when the time-evolving Hamiltonian is topological, while they are absent for quenches into the trivial phase in all cases we have studied. By fitting these in-gap bands, we show that they account for the observed boundary contributions to the dynamical free energy thus supporting a direct connection between the spectrum of a non-Hermitian dynamical matrix and topological boundary contributions. Taken together with earlier studies of the one-dimensional case, our results provide a framework to understand and classify dynamical topological phenomena based on the spectral properties of certain non-Hermitian matrices.