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Proceedings of the Samahang Pisika ng Pilipinas

Exact calculation of the average work done in a quenched quantum Ising model

Salvador T. Laurente, Francis N. C. Paraan

Published Jun 3, 2015• 0 citations
OpenAlex

Abstract

The main aim of this paper is to calculate the average work done in a ground state quench of the XY Heisenberg model in the transverse field quantum Ising limit. We obtain an exact closed form expression for the average work done in terms of complete elliptic integrals. This result reveals an inflection point at a pre-quench field that satisfies |h0| = 1, which signals quantum criticality.

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The main aim of this paper is to calculate the average work done in a ground state quench of the XY Heisenberg model in the transverse field quantum Ising limit. We obtain an exact closed form expression for the average work done in terms of complete elliptic integrals. This result reveals an inflection point at a pre-quench field that satisfies |h0| = 1, which signals quantum criticality.
A detailed derivation of the exact expression for the average work done in a transverse field quench of the quantum Ising model ground state is presented. In the thermodynamic limit, it is proved that the average work done generally has inflection points at the critical field for the pre-quench quantum phase transition. The first divergent field derivative is calculated and is shown to diverge logarithmically. We also demonstrate that the average work done is equal to the product of the transverse magnetization of the pre-quench ground state and the change in the magnetic field.
We derive exact analytic expressions for the average work done and work fluctuations in instantaneous quenches of the ground and thermal states of a one-dimensional anisotropic XY model. The average work and a quantum fluctuation relation is used to determine the amount of irreversible entropy produced during the quench, eventually revealing how the closing of the excitation gap leads to increased dissipated work. The work fluctuation is calculated and shown to exhibit nonanalytic behavior as the prequench anisotropy parameter and transverse field are tuned across quantum critical points. Exact compact formulas for the average work and work fluctuation in ground state quenches of the transverse field Ising model allow us to calculate the first singular field derivative at the critical field values.
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.
We consider the time evolution of observables in the transverse-field Ising chain 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 understood in terms of a GGE. We present a conjecture on how these results generalize to particular quenches in other integrable models.
A complete understanding of the statistics of the work done by quenching a parameter of a quantum many-body system is still lacking in the presence of an initial quantum coherence in the energy basis. In this case, the work can be represented by a class of quasiprobability distributions. Here, we try to clarify the genuinely quantum features of the process by studying the work quasiprobability for an Ising model in a transverse field. We consider both a global and local quench by focusing mainly on the thermodynamic limit. We find that, while for a global quench there is a symmetric noncontextual representation with a Gaussian probability distribution of work, for a local quench we can get quantum contextuality as signaled by a negative fourth moment of the work. Furthermore, we examine the critical features related to a quantum phase transition and the role of the initial quantum coherence as a useful resource.
We investigate the behavior of quantum coherence of the ground states of 2D Heisenberg XY model and 2D Ising model with transverse field on square lattices, by using the method of Quantum Renormalization Group (QRG). We show that the non-analytic behavior of quantum coherence near the critical point, can detect quantum phase transition (QPT) of these models. We also use the scaling behavior of maximum derivative of quantum coherence, with system size, to find the critical exponent of coherence for both models and also the length exponent of the Ising model. The results are in close agreement with the ones obtained from entanglement analysis, that is while quantum coherence needs less computational calculations in comparison to entanglement approaches.