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arXiv:1803.01158 · SciPost Physics

Quench dynamics of the Ising field theory in a magnetic field

Kristóf Hódsági, Márton Kormos, G. Takács

Published Sep 25, 2018• 54 citations
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Abstract

We numerically simulate the time evolution of the Ising field theory after quenches starting from the E 8 integrable model using the Truncated Conformal Space Approach.The results are compared with two different analytic predictions based on form factor expansions in the pre-quench and post-quench basis, respectively.Our results clarify the domain of validity of these expansions and suggest directions for further improvement.We show for quenches in the E 8 model that the initial state is not of the integrable pair state form.We also construct quench overlap functions and show that their high-energy asymptotics are markedly different from those constructed before in the sinh/sine-Gordon theory, and argue that this is related to properties of the ultraviolet fixed point.

DOI
10.21468/scipostphys.5.3.027

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We numerically simulate the time evolution of the Ising field theory after quenches starting from the E 8 integrable model using the Truncated Conformal Space Approach.The results are compared with two different analytic predictions based on form factor expansions in the pre-quench and post-quench basis, respectively.Our results clarify the domain of validity of these expansions and suggest directions for further improvement.We show for quenches in the E 8 model that the initial state is not of the integrable pair state form.We also construct quench overlap functions and show that their high-energy asymptotics are markedly different from those constructed before in the sinh/sine-Gordon theory, and argue that this is related to properties of the ultraviolet fixed point.
In analytic descriptions of quantum quenches, the overlaps between the initial pre-quench state and the eigenstates of the time evolving Hamiltonian are crucial ingredients. We construct perturbative expansions of these overlaps in quantum field theories where either the pre-quench or the post-quench Hamiltonian is integrable. Using the E8 Ising field theory for concrete computations, we give explicit expressions for the overlaps up to second order in the quench size, and verify our results against numerical results obtained using the Truncated Conformal Space Approach. We demonstrate that the expansion using the post-quench basis is very effective, but find some serious limitations for the alternative approach using the pre-quench basis.
Abstract In analytic descriptions of quantum quenches, the overlaps between the initial pre-quench state and the eigenstates of the time evolving Hamiltonian are crucial ingredients. We construct perturbative expansions of these overlaps in quantum field theories where either the pre-quench or the post-quench Hamiltonian is integrable. Using the E 8 Ising field theory for concrete computations, we give explicit expressions for the overlaps up to second order in the quench size, and verify our results against numerical results obtained using the Truncated Conformal Space Approach. We demonstrate that the expansion using the post-quench basis is very effective, but find some serious limitations for the alternative approach using the pre-quench basis.
Abstract We apply the linked cluster expansion as well as the quench action approach to study the time evolution of one-point functions after a quantum quench in integrable field theories. We argue that the relaxation towards the stationary value fundamentally differs depending on the locality properties of the considered observable: while for local operators both exponential and power-law decaying terms are present in the dynamics, for semi-local operators the latter are absent. We explicitly confirm this for the Ising field theory, the sinh-Gordon model and the repulsive sine-Gordon model.
We present a numerical computation of overlaps in mass quenches in sine-Gordon quantum field theory using truncated conformal space approach (TCSA). To improve the cut-off dependence of the method, we use a novel running coupling definition which has a general applicability in free boson TCSA. The numerical results for the first breather overlaps are compared with the analytic continuation of a previously proposed analytical Ansatz for the initial state in a related sinh-Gordon quench, and good agreement is found between the numerical data and the analytical prediction in a large energy range.
In contrast to lattice systems where powerful numerical techniques such as matrix product state based methods are available to study the non-equilibrium dynamics, the non-equilibrium behaviour of continuum systems is much harder to simulate. We demonstrate here that Hamiltonian truncation methods can be efficiently applied to this problem, by studying the quantum quench dynamics of the 1 + 1 dimensional Ising field theory using a truncated free fermionic space approach. After benchmarking the method with integrable quenches corresponding to changing the mass in a free Majorana fermion field theory, we study the effect of an integrability breaking perturbation by the longitudinal magnetic field. In both the ferromagnetic and paramagnetic phases of the model we find persistent oscillations with frequencies set by the low-lying particle excitations not only for small, but even for moderate size quenches. In the ferromagnetic phase these particles are the various non-perturbative confined bound states of the domain wall excitations, while in the paramagnetic phase the single magnon excitation governs the dynamics, allowing us to capture the time evolution of the magnetisation using a combination of known results from perturbation theory and form factor based methods. We point out that the dominance of low lying excitations allows for the numerical or experimental determination of the mass spectra through the study of the quench dynamics.
We consider the problem of determining the initial state of integrable quantum field theory quenches in terms of the post-quench eigenstates. The corresponding overlaps are a fundamental input to most exact methods to treat integrable quantum quenches. We construct and examine an infinite integral equation hierarchy based on the form factor bootstrap, proposed earlier as a set of conditions determining the overlaps. Using quenches of the mass and interaction in Sinh-Gordon theory as a concrete example, we present theoretical arguments that the state has the squeezed coherent form expected for integrable quenches, and supporting an Ansatz for the solution of the hierarchy. Moreover we also develop an iterative method to solve numerically the lowest equation of the hierarchy. The iterative solution along with extensive numerical checks performed using the next equation of the hierarchy provides a strong numerical evidence that the proposed Ansatz gives a very good approximation for the solution.