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arXiv:1803.05208 · Journal of Statistical Mechanics Theory and Experiment

One-half of the Kibble–Zurek quench followed by free evolution

Michał Białończyk, Bogdan Damski

Published Jul 1, 2018• 20 citations
arXivHTMLPDF

Abstract

Abstract We drive the one-dimensional quantum Ising chain in the transverse field from the paramagnetic phase to the critical point and study its free evolution there. We analyze excitation of such a system at the critical point and dynamics of its transverse magnetization and Loschmidt echo during free evolution. We discuss how the system size and quench-induced scaling relations from the Kibble–Zurek theory of non-equilibrium phase transitions are encoded in quasi-periodic time evolution of the transverse magnetization and Loschmidt echo.

DOI
10.1088/1742-5468/aad3f2

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cond-mat.stat-mechCross-listedarXiv:2605.05106•May 7, 2026•0 citations
Abstract We drive the one-dimensional quantum Ising chain in the transverse field from the paramagnetic phase to the critical point and study its free evolution there. We analyze excitation of such a system at the critical point and dynamics of its transverse magnetization and Loschmidt echo during free evolution. We discuss how the system size and quench-induced scaling relations from the Kibble–Zurek theory of non-equilibrium phase transitions are encoded in quasi-periodic time evolution of the transverse magnetization and Loschmidt echo.
Abstract We show that the symmetry-breaking gap of the quantum Ising model in the transverse field can be extracted from free evolution of the longitudinal magnetization taking place after a gradual quench of the magnetic field. We perform for this purpose numerical simulations of the Ising chains with either periodic or open boundaries. We also study the condition for adiabaticity of evolution of the longitudinal magnetization finding excellent agreement between our simulations and the prediction based on the Kibble–Zurek theory of non-equilibrium phase transitions. Our results should be relevant for ongoing cold atom and ion experiments targeting either equilibrium or dynamical aspects of quantum phase transitions. They could be also useful for benchmarking D-Wave machines.
Abstract We study the quantum Ising model in the transverse inhomogeneous magnetic field. Such a system can be approached numerically through exact diagonalization and analytically through the renormalization group techniques. Basic insight into its physics, however, can be obtained by adopting the Kibble–Zurek theory of non-equilibrium phase transitions to description of spatially inhomogeneous systems at equilibrium. We employ all these approaches and focus on derivatives of longitudinal and transverse magnetizations, which have extrema near the critical point. We discuss how these extrema can be used for locating the critical point and for verification of the Kibble–Zurek scaling predictions in the spatial quench.
We consider the finite-time quench dynamics in the quantum transverse field Ising model which exhibits a second order phase transition from a paramagnetic to a ferromagnetic phase, as the transverse magnetic field is decreased. These dynamics have been thoroughly investigated in previous studies when the critical point is crossed during the quench; here, we quench the system from deep in the paramagnetic phase to just above the critical field so that the system remains in the gapped phase throughout the quench duration. On linearly quenching the infinitely large system, we find that the behavior of mean longitudinal defect density and mean transverse magnetization at the end of the quench falls into three distinct scaling regimes as the quench time is increased. For sufficiently small quench times, these observables remain roughly constant, but for larger quench times, a crossover occurs from the Kibble-Zurek scaling law to the quadratic quench rate law when the Kibble-Zurek time is of the order of relaxation time at the final quench field. These results are shown analytically using power series and uniform asymptotic expansions of the exact solution of the model, and also compared with an adiabatic perturbation theory in the third regime. We find that the above mentioned scaling regimes hold for quenches within the ferromagnetic phase also, and provide a general scaling argument for crossover from the Kibble-Zurek regime to an adiabatic regime for slow quenches within a gapped phase.
We review quantum phase transitions of spin systems in transverse magnetic fields taking the examples of the spin-1/2 Ising and XY models in a transverse field. Beginning with an overview of quantum phase transitions, we introduce a number of model Hamiltonians. We provide exact solutions in one spatial dimension connecting them to conformal field theoretical studies. We also discuss Kitaev models and some other exactly solvable spin systems. Studies of quantum phase transitions in the presence of quenched randomness and with frustrating interactions are presented in detail. We discuss novel phenomena like Griffiths-McCoy singularities. We then turn to more recent topics like information theoretic measures of the quantum phase transitions in these models such as concurrence, entanglement entropy, quantum discord and quantum fidelity. We then focus on non-equilibrium dynamics of a variety of transverse field systems across quantum critical points and lines. After mentioning rapid quenching studies, we dwell on slow dynamics and discuss the Kibble-Zurek scaling for the defect density following a quench across critical points and its modifications for quenching across critical lines, gapless regions and multicritical points. Topics like the role of different quenching schemes, local quenching, quenching of models with random interactions and quenching of a spin chain coupled to a heat bath are touched upon. The connection between non-equilibrium dynamics and quantum information theoretic measures is presented at some length. We indicate the connection between Kibble-Zurek scaling and adiabatic evolution of a state as well as the application of adiabatic dynamics as a tool of a quantum optimization technique known as quantum annealing. The final section is dedicated to a detailed discussion on recent experimental studies of transverse Ising-like systems.
We study the quantum dynamics resulting from preparing a one-dimensional quantum system in the ground state of initially two decoupled parts which are then joined together (local quench). Specifically we focus on the transverse Ising chain and compute the time-dependence of the magnetization profile, m_l(t), and correlation functions at the critical point, in the ferromagnetically ordered phase and in the paramagnetic phase. At the critical point we find finite size scaling forms for the nonequilibrium magnetization and compare predictions of conformal field theory with our numerical results. In the ferromagnetic phase the magnetization profiles are well matched by our predictions from a quasi-classical calculation.
When a quantum system exhibiting a second order phase transition is quenched across the critical point in large but finite time, the dynamics are not adiabatic in the critical region and the Kibble-Zurek (KZ) mechanism provides a framework to determine local observables such as the mean defect density. However, to find higher-point functions, one has to go beyond the KZ paradigm asshown in recent works on one-dimensional transverse field Ising model (TFIM) following a linear quench. It has been found that (i) besides the KZ scale, the quench dynamics depend on another length scale that arises due to the finite phase difference between the low energy modes, and (ii) contrary to the expectations based on the KZ mechanism, in general, the correlation functions do not decay exponentially with distance. Motivated by these results for the linear quench, we are interested in understanding if these properties are universal, and consider the 1D TFIM when the transverse field varies algebraically in the vicinity of the critical field. We focus on the equal-time,longitudinal kink-kink correlation function at the end of the quench from the paramagnetic to the ferromagnetic phase, and find that (i) the correlator depends only on the KZ length for superlinear quenches, otherwise an additional dephasing length is required to describe it, and (ii) the dephased correlator decays as a compressed exponential with an exponent that changes continuously with the quench exponent. Our results are obtained using an adiabatic perturbation theory, analytical arguments and exact numerical integration of the relevant equations.