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arXiv:1608.05325 · Physical review. B./Physical review. B

Criticality revealed through quench dynamics in the Lipkin-Meshkov-Glick model

Steve Campbell

Published Nov 4, 2016• 65 citations
arXivHTMLPDF

Abstract

We examine the dynamics after a sudden quench in the magnetic field of the Lipkin-Meshkov-Glick model. Starting from the ground state and by employing the time-dependent fidelity, we see manifestly different dynamics are present if the system is quenched through the critical point. Furthermore, we show that the average work shows no sensitivity to the quantum phase transition; however, the free energy and irreversible work show markedly different rates of change in each phase. Finally, we assess the spectral function showing the fundamental excitations that dictate the dynamics of the postquenched system, further highlighting the qualitative differences between the dynamics in the two phases.

DOI
10.1103/physrevb.94.184403

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We examine the dynamics after a sudden quench in the magnetic field of the Lipkin-Meshkov-Glick model. Starting from the ground state and by employing the time-dependent fidelity, we see manifestly different dynamics are present if the system is quenched through the critical point. Furthermore, we show that the average work shows no sensitivity to the quantum phase transition; however, the free energy and irreversible work show markedly different rates of change in each phase. Finally, we assess the spectral function showing the fundamental excitations that dictate the dynamics of the postquenched system, further highlighting the qualitative differences between the dynamics in the two phases.
We examine the dynamics after a sudden quench in the magnetic field of the Lipkin-Meshkov-Glick model. Starting from the groundstate and by employing the Loschmidt echo, we see manifestly different dynamics are present if the system is quenched through the critical point. Furthermore, we show that the average work shows no sensitivity to the quantum phase transition, however the free energy and irreversible work show markedly different rates of change in each phase. Finally, we assess the spectral function showing the fundamental excitations that dictate the dynamics of the post-quenched system, further highlighting the qualitative differences between the dynamics in the two phases.
We examine how the presence of an excited-state quantum phase transition manifests in the dynamics of a many-body system subject to a sudden quench. Focusing on the Lipkin-Meshkov-Glick model initialized in the ground state of the ferromagnetic phase, we demonstrate that the work probability distribution displays non-Gaussian behavior for quenches in the vicinity of the excited-state critical point. Furthermore, we show that the entropy of the diagonal ensemble is highly susceptible to critical regions, making it a robust and practical indicator of the associated spectral characteristics. We assess the role that symmetry breaking has on the ensuing dynamics, highlighting that its effect is only present for quenches beyond the critical point. Finally, we show that similar features persist when the system is initialized in an excited state and briefly explore the behavior for initial states in the paramagnetic phase.
Abstract We use the spread complexity (SC) of a time-evolved state after a sudden quantum quench in the Lipkin–Meshkov–Glick (LMG) model prepared in the ground state as a probe of the quantum phase transition when the system is quenched toward the critical point. By studying the growth of the effective number of elements of the Krylov basis that contributes to the SC more than a preassigned cutoff, we show how the two phases of the LMG model can be distinguished. We also explore the time evolution of spread entropy after both non-critical and critical quenches. We show that the sum contributing to the spread entropy converges slowly in the symmetric phase of the LMG model compared to that in the broken phase, and for a critical quench, the spread entropy diverges logarithmically at late times.
Abstract The dynamical properties of the Lipkin–Meshkov–Glick model with longitudinal and transverse fields are studied by considering quantum quenches. The model describes a many-body quantum system with long-range interactions. A semiclassical analysis is performed to obtain the critical points of quantum phase transitions. These critical points consist in the static separatrix, which is found by analyzing the stability structure of the semiclassical potential and is useful in the identification of ground state and excited states quantum phase transitions, and in the dynamical separatrix, which depends on initial conditions and is found by studying the classical equations of motion for the Hamiltonian. The static and dynamical separatrices divide the parameter space in regions of different dynamical behavior, and it is found that both separatrices play an important role in the determination of the occurrences of the dynamical quantum phase transitions. Graphical abstract
Long-range interacting spin systems are ubiquitous in physics and exhibit a variety of ground-state disorder-to-order phase transitions. We consider a prototype of infinite-range interacting models known as the Lipkin-Meshkov-Glick model describing the collective interaction of N spins and investigate the dynamical properties of fluctuations and correlations after a sudden quench of the Hamiltonian. Specifically, we focus on critical quenches, where the initial state and/or the postquench Hamiltonian are critical. Depending on the type of quench, we identify three distinct behaviors where both the short-time dynamics and the stationary state at long times are effectively thermal, quantum, and genuinely nonequilibrium, characterized by distinct universality classes and static and dynamical critical exponents. These behaviors can be identified by an infrared effective temperature that is finite, zero, and infinite (the latter scaling with the system size as N^{1/3}), respectively. The quench dynamics is studied through a combination of exact numerics and analytical calculations utilizing the nonequilibrium Keldysh field theory. Our results are amenable to realization in experiments with trapped-ion experiments where long-range interactions naturally arise.
Investigating the time evolution of complexity in quantum systems entails evaluating the spreading of the system's state across a defined basis in its corresponding Hilbert space. Recently, the Krylov basis has been identified as the one that minimizes this spreading. In this study, we develop a numerical exploration of the Krylov complexity in quantum states following a quench in the Lipkin-Meshkov-Glick model. Our results reveal that the long-term averaged Krylov complexity acts as an order parameter for this model. It effectively discriminates between the two dynamic phases induced by the quench, sharing a critical point with the conventional order parameter. Additionally, we examine the inverse participation ratio and the Shannon entropy in both the Krylov basis and the energy basis. A matching dynamic behavior is observed in both bases when the initial state possesses a specific symmetry. This behavior is analytically explained by establishing the equivalence between the Krylov basis and the prequench energy eigenbasis.