We study the slow quenching dynamics (characterized by an inverse rate, τ −1 ) of a onedimensional transverse Ising chain with nearest neighbor ferromagentic interactions across the quantum critical point (QCP) and analyze the Loschmidt overlap measured using the subsequent temporal evolution of the final wave function (reached at the end of the quenching) with the final time-independent Hamiltonian. Studying the Fisher zeros of the corresponding generalized "partition function", we probe non-analyticities manifested in the rate function of the return probability known as dynamical phase transitions (DPTs). In contrast to the sudden quenching case, we show that DPTs survive in the subsequent temporal evolution following the quenching across two critical points of the model for a sufficiently slow rate; furthermore, an interesting "lobe" structure of Fisher zeros emerge. We have also made a connection to topological aspects studying the dynamical topological order parameter (νD(t)), as a function of time (t) measured from the instant when the quenching is complete. Remarkably, the time evolution of νD(t) exhibits drastically different behavior following quenches across a single QCP and two QCPs. In the former case, νD(t) increases step-wise by unity at every DPT (i.e., ∆νD = 1). In the latter case, on the other hand, νD(t) essentially oscillates between 0 and 1 (i.e., successive DPTs occur with ∆νD = 1 and ∆νD = −1, respectively), except for instants where it shows a sudden jump by a factor of unity when two successive DPTs carry a topological charge of same sign.

We study quenching dynamics of a one-dimensional transverse Ising chain with nearest neighbor antiferromagentic interactions in the presence of a longitudinal field which renders the model nonintegrable. The dynamics of the spin chain is studied following a slow (characterized by a rate) or sudden quenches of the longitudinal field; the residual energy, as obtained numerically using a t-DMRG scheme, is found to satisfy analytically predicted scaling relations in both the cases. However, analyzing the temporal evolution of the Loschmidt overlap, we find different possibilities of the presence (or absence) of dynamical phase transitions (DPTs) manifested in the non-analyticities of the rate function. Even though the model is non-integrable, there are periodic occurrences of DPTs when the system is slowly ramped across the quantum critical point (QCP) as opposed to the ferromagnetic (FM) version of the model; this numerical finding is qualitatively explained by mapping the original model to an effective integrable spin model which is appropriate for describing such slow quenches. Furthermore, concerning the sudden quenches, our numerical results show that in some cases, DPTs can be present even when the spin chain is quenched within the same phase or even to the QCP while in some other situations they completely disappear even after quenching across the QCP. These observations lead us to the conclusion that it is the change in the nature of the ground state that determines the presence of DPTs following a sudden quench. Following the remarkable advancement of the experimental studies of ultracold atoms trapped in optical lattices [1,2], there is a recent upsurge in the studies of nonequilibrium dynamics of closed quantum systems, in particular from the viewpoint of quantum quenches across a quantum critical point (QCP) [3,4]. The relaxation time of the quantum system diverges at the QCP resulting in a non-adiabatic dynamics and proliferation of topological defects in the final state reached after the quench.According to the Kibble-Zurek (KZ) scaling relation [5,6], generalized to quantum critical systems [7,8], when a d-dimensional quantum system, initially prepared in its ground state, is driven across an isolated QCP, by changing a parameter of the Hamiltonian in a linear fashion as t/τ , the density of defect satisfies the KZ scaling τ −dν/(zν+1) ; here, ν and z are the correlation length and the dynamical exponent associated with the QCP respectively [9][10][11]. Subsequently several modifications of the scaling have been proposed [12][13][14]. Similarly when the system is quenched to the gapless QCP, the residual energy (the excess energy over the ground state of the final Hamiltonian) scales as τ −(d+z)ν/(zν+1) ; on the contrary, when quenched to the gapped phase, the residual energy follows a scaling relation identical to that of the defect density. Similar scaling relations for the residual energy and the defect density have also been derived using an adiabatic perturbation theory for a sudden quench of sma...

Abstract. We study the work statistics of a periodically-driven integrable closed quantum system, addressing in particular the role played by the presence of a quantum critical point. Taking the example of a one-dimensional transverse Ising model in the presence of a spatially homogeneous but periodically timevarying transverse field of frequency ω 0 , we arrive at the characteristic cumulant generating function G(u), which is then used to calculate the work distribution function P (W ). By applying the Floquet theory we show that, in the infinite time limit, P (W ) converges, at zero temperature, towards an asymptotic steady state value whose small-W behaviour depends only on the properties of the small-wavevector modes and on a few important ingredients: the time-averaged value of the transverse field, h 0 , the initial transverse field, h i , and the equilibrium quantum critical point hc, which we find to generate a sequence of non-equilibrium critical points h * l = hc + lω 0 /2, with l integer. When h i = hc, we find a "universal" edge singularity in P (W ) at a threshold value of W th = 2|h i − hc| which is entirely determined by h i . The form of that singularity -Dirac delta derivative or square root -depends on h 0 being or not at a non-equilibrium critical point h * l . On the contrary, when h i = hc, G(u) decays as a power-law for large u, leading to different types of edge singularity at W th = 0. Generalizing our calculations to the finite temperature case, the irreversible entropy generated by the periodic driving is also shown to reach a steady state value in the infinite time limit.

An expression for four-tangle is obtained by examining the negativity fonts present in a four-way partial transpose under local unitary operations. An alternate derivation of three tangle is also given.Comment: 5 pages, No figure

We study the Loschmidt echo (LE) in a central spin model in which a central spin is globally coupled to an environment (E) which is subjected to a small and sudden quench at t = 0 so that its state at t = 0 + , remains the same as the ground state of the initial environmental Hamiltonian before the quench; this leads to a non-equilibrium situation. This state now evolves with two Hamiltonians, the final Hamiltonian following the quench and its modified version which incorporates an additional term arising due to the coupling of the central spin to the environment. Using a generic short-time scaling of the decay rate, we establish that in the early time limit, the rate of decay of the LE (or the overlap between two states generated from the initial state evolving through two channels ) close to the quantum critical point (QCP) of E is independent of the quenching. We do also study the temporal evolution of the LE and establish the presence of a crossover to a situation where the quenching becomes irrelevant. In the limit of large quench amplitude the non-equilibrium initial condition is found to result in a drastic increase in decoherence at large times, even far away from a QCP. These generic results are verified analytically as well as numerically, choosing E to be a transverse Ising chain where the transverse field is suddenly quenched. The emergence of the classical world from the quantum world, namely decoherence, or the quantum-classical transition through reduction of a pure state to a mixed state has been a subject of perpetual interest to the physics community 1-4 . The concept of the LE has been proposed in connection to this quantum-classical transition in quantum chaos to describe the hypersensitivity of the time evolution of a system to the perturbation experienced by its surrounding 5-12 . The LE is defined as follows: if a quantum state |ψ evolves with two Hamiltonians H and H ′ , respectively, the LE is the measure of the overlap given byIn recent years, the temporal evolution of the LE has been studied in the vicinity of a QCP. In this context, the central spin model (CSM) where a central spin (CS) is coupled globally to all the spins of an environment (E) which is chosen to be a transverse Ising chain has been introduced 13 . The CS is assumed to be in a pure state initially while the spin chain is in the ground state. The interaction between the central spin and the environment effectively leads to two Hamiltonians which provide two channels of time evolution of the environmental ground state and lead to a decay in the LE. It has been reported that in the limit of weak coupling between the central spin and the environment, the LE shows a sharp decay close to the QCP of the spin chain and right at the QCP, it shows a collapse and revival as a function of time t. This collapse and revival of the LE can be taken to be an indicator of the proximity to a QCP. It can also be shown that the CS makes a transition to a mixed state when the LE vanishes.The CSM has been generalized to a more generalized e...