Dynamical detection of quantum phases and phase transitions (QPT) in quenched systems with experimentally convenient initial states is a topic of interest from both theoretical and experimental perspectives. Quenched from polarized states, longitudinal magnetization decays exponentially to zero in time for the short-range transverse-field Ising model (TFIM) and hence, has a featureless steady state regime, which prevents it from exhibiting dynamical phase transitions of type-I. In this paper, we ask whether the transient regimes of such non-equilibrium processes probed by single-site observables, that is magnetization per site, could encode information about the underlying QPT. The decay rates of time-dependent and single-site observables exhibit a dynamical crossover that separates two dynamical regions, ordered and disordered, both of which have distinct nonequilibrium responses. We construct a dynamical order parameterlike quantity that exhibits a scaling law at the vicinity of the crossover. Our results reveal that scaling law exponent in short times at the close vicinity of the dynamical crossover is significantly different than the one predicted by analytical theory for long times. When integrability is strongly broken, the crossover boundary turns into a region that separates two other dynamical regions that act like dynamically-ordered and -disordered regimes.
A crossover between different power-law relaxation behaviors of many-body periodically driven integrable systems has come to light in recent years. We demonstrate using integrable quantum systems, that similar kinds of dynamical transitions may also occur in the relaxation of such systems following a sudden quench. Particularly, we observe two distinct power-law relaxation behaviors following a sudden quench in the integrable XY model, depending upon whether the quenched Hamiltonian lies in the commensurate or the incommensurate phase. The relaxation behavior for quenches at and near the bound…
Quantum many body systems with long range interactions are known to display many fascinating phenomena experimentally observable in trapped ions, Rydberg atoms and polar molecules. Among these are dynamical phase transitions which occur after an abrupt quench in spin chains with interactions decaying as $r^{-α}$ and whose critical dynamics depend crucially on the power $α$: for systems with $α<1$ the transition is sharp while for $α>1$ it fans out in a chaotic crossover region. In this paper we explore the fate of critical dynamics in Ising chains with long-range interactions when the transver…
We study analytically and numerically quench dynamics and defects formation in the quantum Ising model in the presence of a time-dependent transverse magnetic field. We generalize the Landau-Ziner formula to the case of non-adiabatic evolution of the quantum system. For a quasi-static magnetic field, with a slow dependence on time, our outcomes are similar to the results predicted by the Landau-Zener formula. However, a quench dynamics under a shock-wave load is more complicated. The final state of the system depends on the amplitude and pulse velocity, resulting in the mixture of ground and e…
We present a quantitative semi-classical theory for the non-equilibrium dynamics of transverse Ising chains after quantum quenches, in particular sudden changes of the transverse field strength. We obtain accurate predictions for the quench dependent relaxation times and correlation lengths, and also about the recurrence times and quasiperiodicity of time dependent correlations in finite systems with open or periodic boundary conditions. We compare the quantiative predictions of our semi-classical theory (local magnetization, equal time bulk-bulk and surface-to-bulk correlations, and bulk auto…
Dynamic correlation and response functions of classical and quantum systems in thermal equilibrium are connected by fluctuation-dissipation theorems, which allow an alternative definition of their (unique) temperature. Motivated by this fundamental property, we revisit the issue of thermalization of closed many-body quantum systems long after a sudden quench, focussing on the non-equilibrium dynamics of the Ising chain in a critical transverse field. We show the emergence of distinct observable-dependent effective temperatures, which rule out Gibbs thermalization in a strict sense but might st…
Based on the obtained exact results we systematically study the quench dynamics of a one-dimensional spin-1/2 transverse field Ising model with zero- and finite-temperature initial states. We focus on the magnetization of the system after a sudden change of the external field and a coherent time-evolution process. With a zero-temperature initial state, the quench magnetic susceptibility as a function of the initial field strength exhibits strongly similar scaling behaviors to those of the static magnetic susceptibility, and the quench magnetic susceptibility as a function of the final field st…
We study the time evolution of the local magnetization in the critical Ising chain in a transverse field after a sudden change of the parameters at a defect. The relaxation of the defect magnetization is algebraic and the corresponding exponent, which is a continuous function of the defect parameters, is calculated exactly. In finite chains the relaxation is oscillating in time and its form is conjectured on the basis of precise numerical calculations.
The kicked Ising model with both a pulsed transverse and a continuous longitudinal field is studied numerically. Starting from a large transverse field and a state that is nearly an eigenstate, the pulsed transverse field is quenched with a simultaneous enhancement of the longitudinal field. The generation of multipartite entanglement is observed along with a phenomenon akin to quantum resonance when the entanglement does not evolve for certain values of the pulse duration. Away from the resonance, the longitudinal field can drive the entanglement to near maximum values that is shown to agree …
Dynamics of quenching temperature is studied in pure and random Ising chains. Using the Kibble-Zurek argument, we obtain for the pure Ising model that the density of kinks after quenching decays as 1/\sqrtτ with the quench rate of temperature 1/τfor large τ. For the random Ising model, we show that decay rates of the density of kinks and the residual energy are 1/\lnτand 1/(\lnτ)^2 for large τrespectively. Analytic results for the random Ising model are confirmed by the Monte-Carlo simulation. Our results reveal a clear difference between classical and quantum quenches in the random Ising chai…
We study the quantum relaxation of the two-dimensional transverse-field Ising model after global quenches with a real-time variational Monte Carlo method and address the question whether this non-integrable, two-dimensional system thermalizes or not. We consider both interaction quenches in the paramagnetic phase and field quenches in the ferromagnetic phase and compare the time-averaged probability distributions of non-conserved quantities like magnetization and correlation functions to the thermal distributions according to the canonical Gibbs ensemble obtained with quantum Monte Carlo simul…
We consider the time evolution of the gaps of the entanglement spectrum for a block of consecutive sites in finite transverse field Ising chains after sudden quenches of the magnetic field. We provide numerical evidence that, whenever we quench at or across the quantum critical point, the time evolution of the ratios of these gaps allows to obtain universal information. They encode the low-lying gaps of the conformal spectrum of the Ising boundary conformal field theory describing the spatial bipartition within the imaginary time path integral approach to global quenches at the quantum critica…
We consider the stationary state properties of the reduced density matrix as well as spin-spin correlation functions after a sudden quantum quench of the magnetic field in the transverse field Ising chain. We demonstrate that stationary state properties are described by a generalized Gibbs ensemble. We discuss the approach to the stationary state at late times.
The Kibble-Zurek (KZ) hypothesis identifies the relevant time scales in out-of-equilibrium dynamics of critical systems employing concepts valid at equilibrium: It predicts the scaling of the defect formation immediately after quenches across classical and quantum phase transitions as a function of the quench speed. Here we study the crossover between the scaling dictated by a slow quench, which is ruled by the critical properties of the quantum phase transition, and the excitations due to a faster quench, where the dynamics is often well described by the classical model. We estimate the value…
We investigate the operator growth dynamics of the transverse field Ising spin chain in one dimension as varying the strength of the longitudinal field. An operator in the Heisenberg picture spreads in the extended Hilbert space. Recently, it has been proposed that the spreading dynamics has a universal feature signaling chaoticity of underlying quantum dynamics. We demonstrate numerically that the operator growth dynamics in the presence of the longitudinal field follows the universal scaling law for one-dimensional chaotic systems. We also find that the operator growth dynamics satisfies a c…
In this paper we have studied analytically and numerically dynamics of spin crossover induced by time-dependent pressure. We show that quasi static pressure, with a slow dependence on time, yields a spin crossover leading to transition from the state of quantum system with high spin (HS) to the low spin (LS). However, a quench dynamics under shock-wave load is more complicated. The final state of the system depends on the amplitude and pulse velocity, resulting in the mixture of the HS and LS states.
We study the dynamics of the transverse field Ising chain after a local quench in which two independently thermalised chains are joined together and are left to evolve unitarily. In the emerging non-equilibrium steady state the Rényi mutual information with different indices are calculated between two adjacent segments of the chain, and are found to scale logarithmically in the subsystem size. Surprisingly, for Rényi indices $α>2$ we find cases where the prefactor of the logarithmic dependence is negative. The fact that the naively defined Rényi mutual information might be negative has been po…
Work Statistics Under Quantum-Jump and Quench Dynamics in Monitored Ising Chains
International School for Advanced Studies (SISSA)（国际高等研究院）；Institute for Cross-Disciplinary Physics and Complex Systems (IFISC) UIB-CSIC（跨学科物理与复杂系统研究所）
Driven quantum systems coupled to an environment typically exhibit effectively thermal behavior with relaxational dynamics near criticality. However, a different qualitative behavior might be expected in the weakly dissipative limit due to the competition between coherent dynamics and weak dissipation. In this work, we investigate a driven-dissipative infinite-range Ising model in the presence of individual atomic dissipation, a model that emerges from the paradigmatic open Dicke model in the large-detuning limit. We show that the system undergoes a dynamical crossover from relaxational dynami…
We study the Ising spin chain with a Dzyaloshinskii-Moriya interaction focusing on the static and dynamic properties of the entanglement entropy, following a quantum quench. We show that the effects of the additional anisotropic interaction on the phase diagram and on the dynamics of the system are captured by the properties of the entanglement entropy. In particular, the model provides a way to study the quench dynamics in a system with an energy current. We consider quenches starting from an initial excited state of the Ising spin chain, and we analyze the effects of different initial condit…
We study the non-equilibrium dynamics of a homogeneous quantum Ising ring after a quench, in which the transverse field $g$ suddenly changes from zero to a nonzero value. The long-timescale reduced dynamics of a single spin and of two nearest-neighbor spins, which involves the evaluation of expectation values of odd operators that break the fermion parity, is exactly obtained for finite-size but large rings through the use of a recently developed Pfaffian method [N. Wu, Phys. Rev. E 101, 042108 (2020)]. Time dependence of the transverse and longitudinal magnetizations, single-spin purity, expe…
A general framework is proposed to tackle analytically local quantum quenches in integrable impurity systems, combining a mapping onto a boundary problem with the form factor approach to boundary-condition-changing operators introduced in Phys. Rev. Lett. 80, 4370 (1998). We discuss how to compute exactly two central quantities of interest: the Loschmidt echo and the distribution of the work done during the quantum quench. Our results display an interesting crossover physics characterized by the energy scale T_b of the impurity corresponding to the Kondo temperature. We discuss in detail the n…
We consider the time evolution of observables in the transverse field Ising chain (TFIC) after a sudden quench of the magnetic field. We provide exact analytical results for the asymptotic time and distance dependence of one- and two-point correlation functions of the order parameter. We employ two complementary approaches based on asymptotic evaluations of determinants and form-factor sums. We prove that the stationary value of the two-point correlation function is not thermal, but can be described by a generalized Gibbs ensemble (GGE). The approach to the stationary state can also be underst…
We consider the problems of calculating the dynamical order parameter two-point function at finite temperatures and the one-point function after a quantum quench in the transverse field Ising chain. Both of these can be expressed in terms of form factor sums in the basis of physical excitations of the model. We develop a general framework for carrying out these sums based on a decomposition of form factors into partial fractions, which leads to a factorization of the multiple sums and permits them to be evaluated asymptotically. This naturally leads to systematic low density expansions. At lat…
We study the generation and propagation of local perturbations in a quantum many-body spin system. In particular, we study the Ising model in transverse field in the presence of a local field defect at one edge. This system possesses a rich phase diagram with different regions characterized by the presence of one or two Majorana zero modes. We show that their localized character {\it i}) enables a characterization of the Ising phase transition through a local-only measurement performed on the edge spin, and {\it ii}) strongly affects the propagation of quasiparticles emitted after the sudden r…
We consider the $d$-dimensional transverse-field Ising model with power-law interactions $J/r^{d+σ}$ in the presence of a noisy longitudinal field with zero average. We study the longitudinal-magnetization dynamics of an initial paramagnetic state after a sudden switch-on of both the interactions and the noisy field. While the system eventually relaxes to an infinite-temperature state with vanishing magnetization correlations, we find that two-time correlation functions show aging at intermediate times. Moreover, for times shorter than the inverse noise strength $κ$ and distances longer than $…
The transverse field in the quantum Ising chain is linearly ramped from the para- to the ferromagnetic phase across the quantum critical point at a rate characterized by a quench time $τ_Q$. We calculate a connected kink-kink correlator in the final state at zero transverse field. The correlator is a sum of two terms: a negative (anti-bunching) Gaussian that depends on the Kibble-Zurek (KZ) correlation length only and a positive term that depends on a second longer scale of length. The second length is made longer by dephasing of the state excited near the critical point during the following r…
Simulating the real-time evolution of quantum spin systems far out of equilibrium poses a major theoretical challenge, especially in more than one dimension. We experimentally explore the dynamics of a two-dimensional Ising spin system with transverse and longitudinal fields as we quench it across a quantum phase transition from a paramagnet to an antiferromagnet. We realize the system with a near unit-occupancy atomic array of over 200 atoms obtained by loading a spin-polarized band insulator of fermionic lithium into an optical lattice and induce short-range interactions by direct excitation…
We study the quench dynamics of the topological quantum phase transition in the two-dimensional transverse Wen-plaquette model, which has a phase transition from a Z2 topologically ordered to a spin-polarized state. By mapping the Wen-plaquette model onto a one-dimensional quantum Ising model, we calculate the expectation value of the plaquette operator Fi during a slowly quenching process from a topologically ordered state. A logarithmic scaling law of quench dynamics near the quantum phase transition is found, which is analogous to the well-known static critical behavior of the specific heat…
We consider a linear quench from the paramagnetic to ferromagnetic phase in the quantum Ising chain interacting with a static spin environment. Both decoherence from the environment and non-adiabaticity of the evolution near a critical point excite the system from the final ferromagnetic ground state. For weak decoherence and relatively fast quenches the excitation energy, proportional to the number of kinks in the final state, decays like an inverse square root of a quench time, but slow transitions or strong decoherence make it decay in a much slower logarithmic way. We also find that fideli…
The near-critical unitary dynamics of quantum Ising spin chains in transversal and longitudinal magnetic fields is studied using an artificial neural network representation of the wave function. A focus is set on strong spatial correlations which build up in the system following a quench into the vicinity of the quantum critical point. We compare correlations observed following reinforcement learning of the network states with analytical solutions in integrable cases and tDMRG simulations, as well as with predictions from a semi-classical discrete Truncated Wigner analysis. While the semi-clas…
We measure the position- and momentum- space breathing dynamics of trapped one-dimensional Bose gases. The profile in real space reveals sinusoidal width oscillations whose frequency varies continuously through the quasicondensate to ideal Bose gas crossover. A comparison with theoretical models taking into account the effect of finite temperature is provided. In momentum space, we report the first observation of a frequency doubling in the quasicondensate regime, corresponding to a self-reflection mechanism. The disappearance of this mechanism through the quasicondensation crossover is mapped…
Linear quench of the transverse field drives the quantum Ising chain across a quantum critical point from the paramagnetic to the ferromagnetic phase. We focus on normal and anomalous quadratic correlators between fermionic kink creation and annihilation operators. They depend not only on the Kibble-Zurek (KZ) correlation length but also on a dephasing length scale, which differs from the KZ length by a logarithmic correction. Additional slowing down of the ramp in the ferromagnetic phase further increases the dephasing length and suppresses the anomalous correlator. The quadratic correlators …
We study the real-time dynamics of a quantum Ising chain driven periodically by instantaneous quenches of the transverse field (the transverse field varying as rectangular wave symmetric about zero). Two interesting phenomena are reported and analyzed: (1) We observe dynamical many-body freezing or DMF (Phys. Rev. B, vol. 82, 172402, 2010), i.e. strongly non-monotonic freezing of the response (transverse magnetization) with respect to the driving parameters (pulse width and height) resulting from equivocal freezing behavior of all the many-body modes. The freezing occurs due to coherent suppre…
