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Detailed Calculation of the Average Work Done in a Ground State Quench of the Quantum Ising Model

Francis N.C. Paraan

Published Dec 1, 2016• 0 citations
OpenAlex

Abstract

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.

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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.
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We consider the non-equilibrium dynamics after a sudden quench of the magnetic field in the transverse field Ising chain starting from excited states of the pre-quench Hamiltonian. We prove that stationary values of local correlation functions can be described by the generalized Gibbs ensemble. Then we study the full time evolution of the transverse magnetization by means of stationary phase methods. The equal-time two-point longitudinal correlation function is analytically derived for a particular class of excited states for quenches within the ferromagnetic phase, and studied numerically in general. The full time dependence of the entanglement entropy of a block of spins is also obtained analytically for the same class of states and for arbitrary quenches.
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 non-equilibrium dynamics after a sudden quench of the magnetic field in the transverse field Ising chain starting from excited states of the pre-quench Hamiltonian. We prove that stationary values of local correlation functions can be described by the generalised Gibbs ensemble (GGE). Then we study the full time evo-lution of the transverse magnetisation by means of stationary phase methods. The equal time two-point longitudinal correlation function is analytically derived for a par-ticular class of excited states for quenches within the ferromagnetic phase, and studied numerically in general. The full time dependence of the entanglement entropy of a block of spins is also obtained analytically for the same class of states and for arbitrary quenches. ar
We analyze the fidelity per lattice site for two different ground states of the one-dimensional quantum Ising model in a transverse field near the critical point. It is found that, in the thermodynamic limit, the fidelity per lattice site is singular, and the derivative of its logarithmic function with respect to the transverse field strength is logarithmically divergent at the critical point. The scaling behavior is confirmed numerically by performing a finite-size scaling analysis for systems of different sizes, consistent with the conformal invariance at the critical point. This allows us to extract the correlation length critical exponent, which turns out to be universal in the sense that the correlation length critical exponent does not depend on either the anisotropic parameter or the transverse field strength.