Preparing an integrable system in a mixed state described by a thermal density matrix , we subject it to a sudden quench and explore the subsequent unitary dynamics. Defining a version of the generalised Loschmidt overlap amplitude (GLOA) through the purifications of the time evolved density matrix, we claim that non-analyiticies in the corresponding "dynamical free energy density" persist and is referred to as mixed state dynamical quantum phase transitions (MSDQPTs). Furthermore, these MSDQPTs are uniquely characterised by a topological index constructed by the application of the Pancharatnam geometry on the purifications of the thermal density matrix; the quantization of this index however persists up to a critical temperature. These claims are corroborated analysing the non-equilibrium dynamics of a transverse Ising chain initially prepared in a thermal state and subjected to a sudden quench of the transverse field.

We study the effects of δ-function periodic kicks on the Floquet energy-momentum dispersion in graphene. We find that a rich variety of dispersions can appear depending on the parameters of the kicking: at certain points in the Brillouin zone, the dispersion can become linear but anisotropic, linear in one direction and quadratic in the perpendicular direction, gapped with a quadratic dispersion, or completely flat (called dynamical localization). We show all these results analytically and demonstrate them numerically through the dynamics of wave packets propagating in graphene. We propose experimental methods for producing these effects.

We introduce the notion of a dynamical topological order parameter (DTOP) that characterises dynamical quantum phase transitions (DQPTs) occurring in the subsequent temporal evolution of two dimensional closed quantum systems, following a quench (or ramping) of a parameter of the Hamiltonian, which generalizes the notion of DTOP introduced in Budich and Heyl, Phys. Rev. B 93, 085416 (2016) for one-dimensional situations. This DTOP is obtained from the "gaugeinvariant" Pancharatnam phase extracted from the Loschmidt overlap, i.e., the modulus of the overlap between the initially prepared state and its time evolved counterpart reached following a temporal evolution generated by the time-independent final Hamiltonian. This generic proposal is illustrated considering DQPTs occurring in the subsequent temporal evolution following a sudden quench of the staggered mass of the topological Haldane model on a hexagonal lattice where it stays fixed to zero or unity and makes a discontinuous jump between these two values at critical times at which DQPTs occur.Recent experimental advances in realisation of closed condensed matter systems via cold atoms [1][2][3] [23,24]. (For review, we refer to [25][26][27][28][29].)The proposal of a DQPT was put forward by Heyl et al. [30], in close connection to a thermal phase transition in an equilibrium classical system; the latter can be detected, as proposed by Fisher [31] (See also, [32,33]), by analyzing the zeros of the canonical partition function in a complex temperature plane (or in a complex magnetic field [32]). In DQPTs, on the other hand, nonanalytic behavior occurs at critical times in the subsequent real time evolution (following the quench) generated by the time independent final Hamiltonian; these non-equilibrium transitions can be analyzed by locating the zeros of the "dynamical" partition function generalized to the complex time plane. It is noteworthy that Fisher (or Yang-Lee) zeros have been experimentally observed within a central field where a qubit is coupled to the bath in such a way that it experiences an effective complex magnetic field [34].Let us first elaborate on the basic notion of a DQPT focussing on the sudden quenching of a one dimensional model and treating the overlap amplitude or the Loschmidt overlap (LO) as the analogue of the "partition" function. Denoting the ground state of the initial Hamiltonian as |ψ 0 and the final Hamiltonian reached through the quenching process as H f , the Loschmidt overlap is defined as G(t) = ψ 0 |e −iH f t |ψ 0 . Generalizing G(t) to G(z) defined in the complex time (z) plane, one can introduce the notion of a dynamical free energy density, f (z) = − lim L→∞ ln G(z)/L, where L is the linear dimension of the system. One then looks for the zeros of the G(z) (or non-analyticities in f (z)) to define a dynamical phase transition. For a transverse Ising chain, it has been observed [30] that when the system is suddenly quenched across the quantum critical point (QCP), the lines of Fisher zeros cross the imaginary time a...

We perform a sudden quench on the Haldane model with long range interactions, more specifically generalising to the next to next nearest neighbour hopping, referred to as the N 3 model in our work. Such a model possesses both isotropic and multiple anisotropic (satellite) Dirac points which lead to a rich topological phase diagram consisting of phases with higher Chern number (C). Quenches between the topological and the non-topological phases of such an infinite system probe the effect of the presence of the anisotropic Dirac points on the non-equilibrium response of the topological system. Interestingly, the Chern number remains the same before and after the quench for both the quenching protocols, even when the quench of the system is carried out between two different topological phases. However, for a finite system, we establish that the initial edge current asymptotically decays to zero when the system is quenched to the non-topological phase although the Chern number for the corresponding periodically wrapped system remains unaltered; what is remarkable is that when the Hamiltonian is quenched from |C| = 2 phase to the non-topological phase the edge current associated with the inner channel decays at a faster rate than the outer channel resembling a situation in which the system passes through the phase with |C| = 1 before ending up in the phase C = 0.

The presence of algebraically decaying long-range interactions may alter the critical as well as topological behaviour of a quantum many-body systems. However, when the interaction decays at a faster rate, the short-range behaviour is expected to be retrieved. Similarly, the long-range nature of interactions has a prominent signature on the out of equilibrium dynamics of these systems, e.g. in the growth of the entanglement entropy following a quench, the propagation of mutual information and non-equilibrium phase transitions. In this review, we summarize the results of long-range interacting classical and quantum Ising chains mentioning some recent results. Thereafter, we focus on the recent developments on the integrable long-range Kitaev chain emphasising the role of long-range superconducting pairing term on determining its topological phase diagram and out of equilibrium dynamics and also incorporate relevant discussions on the corresponding Ising version.