Topological classification of dynamical phase transitions

Abstract

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 + 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. [3,4] are two vividly investigated fields of physics, with no strong bonds between them. The leading role played by topology in condensed matter has only been realized recently with the discovery of topological insulators, the descendants of quantum Hall states. Some of their correlation functions are universal and are not influenced by the microscopic details of the system, but are rather determined by the underlying topology. The analysis of nonequilibrium states, on the other hand, have emerged recently in a different field: in cold atomic systems. With the unprecedented control of preparing initial states and governing the time evolution, a number of interesting phenomena has been observed such as the Kibble Zurek scaling [5], the lack of thermalization in integrable systems [6], etc. In this paper, we connect these two, seemingly unrelated fields and show that topology can be used as an organizing principle to classify out-of-equilibrium systems.The most popular setups for nonequilibrium dynamics are quench experiments in which the quantum system initially sits in the ground state of a given Hamiltonian, but its time evolution is governed by another Hamiltonian. The quench protocol can conveniently be characterized by the dynamical partition function with no reference to any particular observables, defined asFor positive real values of z this gives the partition function of a field theory with boundaries |ψ separated by z [7]. For our purposes, we use z = it with t real, which then gives the Loschmidt amplitude, that is, the overlap of the time-evolved state with the initial state G(t) = Z(it). It characterizes time evolution and the stationary state after a long waiting time [8], and also yields the characteristic function of work [9], which is accessible experimentally [10]. As the time evolution operator is highly nonlocal, it allows Z(z) to be susceptible to the topological properties of the underlying system. Similarly to the equilibrium situation, the dynamical free energy is defined as the logarithm per unit volume f (t) = −1/N d ln G(t). In the thermodynamic li...

Tuning the presence of dynamical phase transitions in a generalizedXYspin chain

“…Rather both in the isotropic and anisotropic cases, one finds restrictions on the values of λ i and λ f depending on the parameter γ determined from Eqs. (12) and (13). It is never important whether the spin chain is quenched across the QCP in the process of quenching; however, it should necessarily be swept through λ = 0, i.e., either λ i or λ f should be negative for DPTs to appear.…”

“…(13) We shall analyze the condition given in (13) for the modes k = 0 and k = π for both γ = 1 and γ = 1. In the former case (γ = 1), |ũ k=π | 2 = 1 as the off-diagonal terms of the LZ part of the Hamiltonian (7) vanish for k = π so that this mode is temporally frozen.…”

“…However, subsequently it has been shown that DPTs are not necessarily connected with the passage through the equilibrium QCP and may occur following a sudden quench even within the same phase (i.e., not crossing the QCP) for both integrable 11 as well as non-integrable models 12 . Subsequently, these studies have been generalized to twodimensional systems 13,14 and the role of topology 13 and the dynamical topological order parameter have been investigated 15 . We note in the passing that the rate function I(t) is related to the Loschmidt echo which has been studied in the context of decoherence [17][18][19][20][21][22][23][24][25][26] and the work-statistics 27,28 .…”

Overlap distributions for quantum quenches in the anisotropic Heisenberg chain

“…The behavior of the Loschmidt echo after a quantum quench has been studied in a large body of recent literature, see, e.g., [18,22,25,34,. In particular, the Loschmidt echo is central to the study of so-called dynamical phase transitions [67][68][69][70][71][72][73][74][75][76]. By expanding |Ψ 0 in terms of the eigenstates |n of the final Hamiltonian H f , we find…”

Connecting dynamical quantum phase transitions and topological steady-state transitions by tuning the energy gap

“…Similar to the free energy in equilibrium states, one can define a dynamical free energy as g(t) = − ln (|L|) /S with S being the system's area. Without considering the interaction between particles, L(t) is a product of echoes at different momentum, and then g(t) in the thermodynamic limit S → ∞ can be expressed as [23] …”