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arXiv:2012.00561 · Physical review. B./Physical review. B

Dynamical phase transitions in the fully connected quantum Ising model: Time period and critical time

Arun Sehrawat, Chirag Srivastava, Ujjwal Sen

Published Aug 4, 2021• 16 citations
arXivHTMLPDF

Abstract

We study dynamical properties of the finite-size fully connected Ising model with a transverse field at zero temperature. In a quench dynamics, we study the time period and the first critical time, which play important roles in the dynamical phase transitions, based on a dynamical order parameter and the Loschmidt rate, respectively. When all the spins are initially polarized in the direction of their mutual interaction, we show that both the time period and critical time diverge logarithmically with the system size at the dynamical critical point. When all the spins are initially in the direction of transverse field, both the time period and critical time exhibit logarithmic or power-law divergences depending on the final field strength. In the case of convergence, we provide estimates for the finite-size scaling and converged value. We also investigate the equilibrium phase transition, presenting approximate ground and first excited states away from the criticality, and compare their energy gap and bipartite and multipartite entanglements with the exact eigenstates.

DOI
10.1103/physrevb.104.085105

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We study dynamical properties of the finite-size fully connected Ising model with a transverse field at zero temperature. In a quench dynamics, we study the time period and the first critical time, which play important roles in the dynamical phase transitions, based on a dynamical order parameter and the Loschmidt rate, respectively. When all the spins are initially polarized in the direction of their mutual interaction, we show that both the time period and critical time diverge logarithmically with the system size at the dynamical critical point. When all the spins are initially in the direction of transverse field, both the time period and critical time exhibit logarithmic or power-law divergences depending on the final field strength. In the case of convergence, we provide estimates for the finite-size scaling and converged value. We also investigate the equilibrium phase transition, presenting approximate ground and first excited states away from the criticality, and compare their energy gap and bipartite and multipartite entanglements with the exact eigenstates.
We construct the finite-temperature dynamical phase diagram of the fully connected transverse-field Ising model from the vantage point of two disparate concepts of dynamical criticality. An analytical derivation of the classical dynamics and exact diagonalization simulations are used to study the dynamics after a quantum quench in the system prepared in a thermal equilibrium state. The different dynamical phases characterized by the type of nonanalyticities that emerge in an appropriately defined Loschmidt-echo return rate directly correspond to the dynamical phases determined by the spontaneous breaking of Z2{\mathbb{Z}}_{2}Z2​ symmetry in the long-time steady state. The dynamical phase diagram is qualitatively different depending on whether the initial thermal state is ferromagnetic or paramagnetic. Whereas the former leads to a dynamical phase diagram that can be directly related to its equilibrium counterpart, the latter gives rise to a divergent dynamical critical temperature at vanishing final transverse-field strength.
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 strength shows a discontinuity at the quantum critical point. This discontinuity remains robust and always occurs at the quantum critical point even for the case of finite-temperature initial systems, which indicates a great advantage of employing quench dynamics to study quantum phase transitions.
The transverse field Ising model of N fully connected spins provides a mathematical accessible, yet non-trivial, system to investigate the non-thermal dynamics after a sudden quantum quench. The focus in this thesis is three-fold. First, in a 1/N expansion the dynamics of the mean magnetization within its variance is obtained analytically in the large N limit. The variance constitutes a leading order correction to the mean field limit at infinite N. While mean field theories are one of the most accessible approximations to deal with the quantum complexity, its validity in time out of equilibrium has to be checked on a case by case basis. By studying the dynamics of the variance, we see that the mean field approximation can break down at surprisingly early times scaling with the square root of N. The underlying mechanism in the fully connected Ising model is identified as a dephasing effect between nearby effective orbits. The dynamics of the magnetization variance discriminates four qualitatively different regions in the dynamical phase diagram that are indistinguishable by just looking at the mean magnetization. Second, bipartite entanglement between disjoint groups of the spins in the fully connected Ising model is studied. In the large N limit the entanglement Hamiltonian is shown to be a harmonic oscillator. A quantitative relation between the angular frequency of the oscillator and a squeezing measure of the spin state is derived. Explicit expressions for the Renyi entanglement entropies are given. In contrast to the variance, which vanishes in the thermodynamic limit, the entanglement entropy saturate to an N-independent positive value. The dynamics of the entanglement Hamiltonian and the entanglement entropies is discussed. Linear growth as a function of time is linked to the unstable fixed point and the homoclinic orbit of a pitchfork bifurcation, while logarithmic increase is explained due to a dephasing mechanism of nearby effective orbits. A generalization to other models that permit a semiclassical effective description (on a possibly higher dimensional phase space) is given. As a byproduct the connection between entangled Gaussian states and the symplectic eigenvalues of its reduced covariance is derived. An upper bound on the von Neumann entanglement entropy of any bipartite pure state is given in terms of the symplectic properties of its reduced covariance. Third, the existence of dynamical phase transitions (DPTs) in the fully connected Ising model is argued by two different approaches. DPTs are defined as non-analyticities in the time-dependence of the return probability rate function in the thermodynamic limit. Like entanglement entropy, the behavior of the return rate function is an observable-independent quantity used to characterize non-equilibrium states. A numerically based connection between the DPT and the analytic property of the infinite time averaged order parameter as a function of quench strength is discussed. The concept of Feynman history states is used to devise the notion of history entanglement. An information theoretic interpretation of the history entanglement entropy as the precision of a quantum clock follows from Holevo’s bound. The dependence of this entanglement entropy on the observation time constitutes an operator-independent characterization of the quantum dynamics in closed systems. For observation times comparable to the Heisenberg time, the history entanglement becomes sensitive to the statistics of the energy gaps, and thus distinguishes integrable and non-integrable dynamics. This is demonstrated for two examples, a random matrix ensemble, and a one-dimensional hardcore boson model with integrability breaking next nearest neighbor interaction.
We study the short-time Loschmidt dynamics after a small sudden quench near a quantum critical point. We show that the initial quadratic growth of the Loschmidt rate function is governed by the variance of the quench operator per system size. For a local quench operator, this variance density is exactly equal to the spatial sum of the equal-time connected two-point correlation function in the initial ground state. This relation connects the early-time Loschmidt response to static critical correlations. Assuming power-law correlations at criticality, we classify the finite-size scaling of the short-time coefficient by the scaling dimension of the quench operator. In one dimension, the coefficient is finite, logarithmically enhanced, or algebraically enhanced with system size. We illustrate this operator dependence in the transverse-field Ising chain, where transverse-field and longitudinal-field quenches couple to different critical operators. We also discuss the first correction beyond the quadratic regime using the fourth cumulant of the post-quench Hamiltonian.
Abstract We investigate two types of dynamical quantum phase transitions (DQPTs) in the transverse-field Ising model on ensembles of Erdős–Rényi networks of size N. These networks consist of vertices connected randomly with probability p ( 0 < p ⩽ 1 ). Using analytical derivations and numerical techniques, we compare the characteristics of the transitions for p < 1 against the fully connected network (p = 1). We analytically show that the overlap between the wave function after a quench and the wave function of the fully connected network after the same quench deviates by at most O ( N − 1 / 2 ) . For a DQPT defined by an order parameter, the critical point remains unchanged for all p. For a DQPT defined by the rate function of the Loschmidt echo, we find that the rate function deviates from the p = 1 limit near vanishing points of the overlap with the initial state, while the critical point remains independent for all p. Our analysis suggests that this divergence arises from persistent non-trivial global many-body correlations absent in the p = 1 limit.
By means of the discrete truncated Wigner approximation, we study dynamical phase transitions arising in the steady state of transverse-field Ising models after a quantum quench. Starting from a fully polarized ferromagnetic initial condition, these transitions separate a phase with nonvanishing magnetization along the ordering direction from a disordered symmetric phase upon increasing the transverse field. We consider two paradigmatic cases, a one-dimensional long-range model with power-law interactions $\ensuremath{\propto}1/{r}^{\ensuremath{\alpha}}$ decaying algebraically as a function of distance rrr and a two-dimensional system with short-range nearest-neighbor interactions. In the former case, we identify dynamical phase transitions for $\ensuremath{\alpha}\ensuremath{\lesssim}2$ and we extract the critical exponents from a data collapse of the steady-state magnetization for up to 1200 lattice sites. We find identical exponents for $\ensuremath{\alpha}\ensuremath{\lesssim}0.5$, suggesting that the dynamical transitions in this regime fall into the same universality class as the nonergodic mean-field limit. The two-dimensional Ising model is believed to be thermalizing, which we also confirm using exact diagonalization for small system sizes. Thus, the dynamical transition is expected to correspond to the thermal phase transition, which is consistent with our data upon comparing to equilibrium quantum Monte Carlo simulations. We further test the accuracy of the discrete truncated Wigner approximation by comparing against numerically exact methods such as exact diagonalization, tensor network, as well as artificial neural network states and we find good quantitative agreement on the accessible time scales. Finally, our work provides an additional contribution to the understanding of the range and the limitations of qualitative and quantitative applicability of the discrete truncated Wigner approximation.