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...

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. Interest in nonequilibrium dynamics has grown immensely in the past few years [1][2][3][4] thanks to experimental advances made with ultracold atomic gases. The wide controllability of these systems allows experimentalists to prepare different kinds of nonequilibrium initial states and it is also possible to study the dynamics with time resolution that is unreachable in other physical systems [5][6][7][8][9]. Some of the main questions concern when and how thermalization, or more generally, equilibration, occurs and its connection to ergodicity and integrability. These were first posed by von Neumann in 1929 [10].The nonequilibrium time evolution can be characterized in many different ways, borrowing ideas from equilibrium statistical mechanics. The ultrashort time nonequilibrium dynamics, revealing the role of high-energy excitations, is also of interest as well as the stationary state that is reached after long time evolution. The latter can be described by the diagonal ensemble, which is roughly the time averaged density matrix. The results of local measurements can be described by simpler ensembles, i.e., by the thermal Gibbs ensemble for nonintegrable (ergodic) systems [11] and by the generalized Gibbs ensemble for integrable ones [12]. The Loschmidt overlap (LO), which is the main focus of this Rapid Communication, is a nonlocal expression and is entirely determined by the diagonal ensemble, hence it characterizes the stationary state [13]. Analyzing the LO has proven to be useful in studying quantum chaos, decoherence, and quantum criticality [14][15][16][17]. It is defined as the scalar product of the initial state and the time evolved state following a sudden global quench (SQ) asand can be regarded as the characteristic function of work performed on the system during the quench. In a SQ the parameters of the Hamiltonian are changed suddenly from some initial to final values, and the system, prepared initially in the ground state |ψ of the initial Hamiltonian, is assumed to be well separated from the environmen...

We give a systematic review of the adiabatic theorem and the leading non-adiabatic corrections in periodically-driven (Floquet) systems. These corrections have a two-fold origin: (i) conventional ones originating from the gradually changing Floquet Hamiltonian and (ii) corrections originating from changing the micro-motion operator. These corrections conspire to give a Hall-type linear response for non-stroboscopic (time-averaged) observables allowing one to measure the Berry curvature and the Chern number related to the Floquet Hamiltonian, thus extending these concepts to periodically-driven many-body systems. The non-zero Floquet Chern number allows one to realize a Thouless energy pump, where one can adiabatically add energy to the system in discrete units of the driving frequency. We discuss the validity of Floquet Adiabatic Perturbation Theory (FAPT) using five different models covering linear and non-linear few and many-particle systems. We argue that in interacting systems, even in the stable high-frequency regimes, FAPT breaks down at ultra slow ramp rates due to avoided crossings of photon resonances, not captured by the inverse-frequency expansion, leading to a counter-intuitive stronger heating at slower ramp rates. Nevertheless, large windows in the ramp rate are shown to exist for which the physics of interacting driven systems is well captured by FAPT.

We developed a novel perturbative expansion based on the replica trick for the Floquet Hamiltonian governing the dynamics of periodically kicked systems where the kick strength is the small parameter. The expansion is formally equivalent to an infinite resummation of the Baker-Campbell-Hausdorff series in the undriven (nonperturbed) Hamiltonian, while considering terms up to a finite order in the kick strength. As an application of the replica expansion, we analyze an Ising spin 1/2 chain periodically kicked with a magnetic field with a strength h, which has both longitudinal and transverse components. We demonstrate that even away from the regime of high frequency driving, if there is heating, its rate is nonperturbative in the kick strength, bounded from above by a stretched exponential: e^{-const h^{-1/2}}. This guarantees the existence of a very long prethermal regime, where the dynamics is governed by the Floquet Hamiltonian obtained from the replica expansion.