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arXiv:1304.7599

Expectation Values of Local Fields in an Integrable Theory after a Quantum Quench

Giuseppe Mussardo

Published Apr 29, 2013• 2 citations
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Abstract

The expectation values of local fields of any interacting quantum theory after a quench process are key quantities for matching theoretical and experimental results. For quantum integrable field theories, we argue that they can be obtained by a generalization of the Leclair-Mussardo formula and a Bethe Ansatz result of Caux and Konik. Specializing to the Sinh-Gordon model and taking the non-relativistic limit, one can recover the results of Kormos et al. for the Lieb-Liniger model.

DOI
10.48550/arxiv.1304.7599

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The expectation values of local fields of any interacting quantum theory after a quench process are key quantities for matching theoretical and experimental results. For quantum integrable field theories, we argue that they can be obtained by a generalization of the Leclair-Mussardo formula and a Bethe Ansatz result of Caux and Konik. Specializing to the Sinh-Gordon model and taking the non-relativistic limit, one can recover the results of Kormos et al. for the Lieb-Liniger model.
Abstract. We consider quantum quenches to the sinh-Gordon integrable quantum field theory from a particular class of initial states. Our analysis includes the case of mass and interaction quenches starting from a non-interacting theory. By means of the recently developed quench action method, we fully characterize the stationary state reached at long times after the quench in terms of the corresponding rapidity distribution. We also provide exact results for the expectation values of arbitrary vertex operators in the post-quench stationary state by proposing a formula based on the analogy with the standard thermodynamic Bethe ansatz. Finally, we comment on the behavior of the post-quench stationary state under the mapping between the sinh-Gordon field theory and the one-dimensional Lieb–Liniger model.
We consider expectation values of local operators in (continuum) integrable models in a situation when the mean value is calculated in a single Bethe state with a large number of particles. We develop a form factor expansion for the thermodynamic limit of the mean value, which applies whenever the distribution of Bethe roots is given by smooth density functions. We present three applications of our general result: (i) in the framework of integrable quantum field theory (IQFT) we present a derivation of the LeClair–Mussardo formula for finite temperature one-point functions. We also extend the results to boundary operators in boundary field theories. (ii) We establish the LeClair–Mussardo formula for the non-relativistic 1D Bose gas in the framework of the algebraic Bethe ansatz (ABA). This way we obtain an alternative derivation of the results of Kormos et al for the (temperature dependent) local correlations using only the concepts of the ABA. (iii) In IQFT we consider the long-time limit of one-point functions after a certain type of global quench. It is shown that our general results imply the integral series found by Fioretti and Mussardo. We also discuss the generalized eigenstate thermalization hypothesis in the context of quantum quenches in integrable models. It is shown that a single mean value always takes the form of a thermodynamic average in a generalized Gibbs ensemble, although the relation to the conserved charges is rather indirect.
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 develop a variational framework for addressing two-dimensional non-integrable quantum field theories through the exact structure of their integrable counterparts. Concentrating on the φ4\varphi^4φ4 Landau-Ginzburg model, we use the analytical Vacuum Expectation Values and Form Factors of local operators in the sinh-Gordon theory as the foundation of a variational ansatz. In this way, we obtain controlled estimates of central physical quantities of the φ4\varphi^4φ4 theory - such as the finite-volume ground-state energy and the physical mass as a function of the coupling constant. The strengths of the variational methods are leveraged in combination with the Hamiltonian truncation techniques and the LeClair-Mussardo formula, which also allow to probe the accuracy of the variational approximation varying the system size. Within the weak-coupling regime, a detailed numerical analysis reveals the behaviour of the finite-volume spectrum, the ground-state energy, and the elastic part of the scattering matrix, showing how the rigorous machinery of integrable models can serve as a guiding light into the complex landscape of non-integrable quantum field dynamics.