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Modern Physics Letters A

REAL-TIME FORMALISM OF THERMOFIELD DYNAMICS AND MATSUBARA APPROACH IN QUANTUM STATISTICAL PHYSICS

Alexei A. Abrikosov

Published Oct 20, 1990• 0 citations
Publisher

Abstract

When studying thermodynamic properties by means of quantum field theory methods one can deform the Matsubara integration contour in the complex time plane. The deformations are restricted by Hamiltonian singularities which are due to turning on an interaction. One should construct the real-time technique in the true vacuum taking the interaction into account.

DOI
10.1142/s0217732390002481

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When studying thermodynamic properties by means of quantum field theory methods one can deform the Matsubara integration contour in the complex time plane. The deformations are restricted by Hamiltonian singularities which are due to turning on an interaction. One should construct the real-time technique in the true vacuum taking the interaction into account.
Perturbation theory in thermofield dynamics (a real time finite temperature quantum field theory) is formulated in the interaction representation and a diagrammatic expansion for the expectation value of time ordered products is derived solely on the basis of the thermal state condition and the algebraic properties of the operators. This new formulation has a much wider domain of applicability than those presented previously. The method is applicable even in the case of purely quantum mechanical systems. When certain bosonic operators, which commute with the unperturbed Hamiltonian, contribute to the renormalization of the thermal state vacuum the vacuum diagrams do not cancel in the usual way. It will be shown how these diagrams arise and how they may be computed.
A bstract The construction of exactly-solvable models has recently been advanced by considering integrable TT‾ T\overline{T} TT T T ¯ deformations and related Hamiltonian deformations in quantum mechanics. We introduce a broader class of non-Hermitian Hamiltonian deformations in a nonrelativistic setting, to account for the description of a large class of open quantum systems, which includes, e.g., arbitrary Markovian evolutions conditioned to the absence of quantum jumps. We relate the time evolution operator and the time-evolving density matrix in the undeformed and deformed theories in terms of integral transforms with a specific kernel. Non-Hermitian Hamiltonian deformations naturally arise in the description of energy diffusion that emerges in quantum systems from time-keeping errors in a real clock used to track time evolution. We show that the latter can be related to an inverse TT‾ T\overline{T} TT T T ¯ deformation with a purely imaginary deformation parameter. In this case, the integral transforms take a particularly simple form when the initial state is a coherent Gibbs state or a thermofield double state, as we illustrate by characterizing the purity, Rényi entropies, logarithmic negativity, and the spectral form factor. As the dissipative evolution of a quantum system can be conveniently described in Liouville space, we further study the spectral properties of the Liouvillians, i.e., the dynamical generators associated with the deformed theories. As an application, we discuss the interplay between decoherence and quantum chaos in non-Hermitian deformations of random matrix Hamiltonians and the Sachdev-Ye-Kitaev model.
Methods of thermal field theory can broadly be put into two categories, according to their use of imaginary time or real time in the ensemble averages. The method of imaginary time was first proposed by Matsubara [1]. The real time method was introduced later, originally by Schwinger [2] and Keldysh [3] and followed by others [4], who applied it to study non-equilibrium processes. It was also applied to equilibrium situations by Mills [5]. The real time method was reformulated in the so-called thermofield dynamics by Umezawa and collaborators [6], who suggested a doubling of field variables in the statistical context. Niemi and Semenoff [7] clarified this apparent doubling in terms of the time contour needed in the real time formulation. A survey of early literature is available in [8]. Physical applications including later works are presented also in books [9, 10]. For systems in thermal equilibrium, one can develop perturbative expansion in both methods in parallel to the conventional method in vacuum. The difference shows up in the form of the propagators. Because of the finite interval of imaginary time, the frequencies are discrete, giving the propagator as a sum over these frequencies. On the other hand, the real time method has two infinite intervals of time, giving rise to a 2 × 2 matrix for the propagator. The majority of calculations reported in the literature are done in the imaginary time method, avoiding matrix propagators in the real time method. For systems out of thermal equilibrium, the real time method suggests itself as the appropriate framework to calculate physical quantities. However, for quantities such as the response functions (we address this problem in the last chapter) the end results relate to thermal equilibrium and so can be evaluated by either method. But, to follow the time dependence of quantities during non-equilibrium phases, such as in the early universe [11–13] – a topic too distant to discuss in this book – the real time method is the only option. In this book we adopt the real time method to calculate thermal quantities in equilibrium. This chapter is devoted to deriving the matrix form of propagators arising in this method.
Direct numerical evaluation of the real-time path integral has a well-known sign problem that makes convergence exponentially slow. One promising remedy is to use Picard-Lefschetz theory to flow the domain of the field variables into the complex plane, where the integral is better behaved. By Cauchy's theorem, the final value of the path integral is unchanged. Previous analyses have considered the case of real scalar fields in thermal equilibrium, employing a closed Schwinger-Keldysh time contour, allowing the evaluation of the full quantum correlation functions. Here we extend the analysis by not requiring a closed time path, instead allowing for an initial density matrix for out-of-equilibrium initial value problems. We are able to explicitly implement Gaussian initial conditions, and by separating the initial time and the later times into a two-step Monte-Carlo sampling, we are able to avoid the phenomenon of multiple thimbles. In fact, there exists one and only one thimble for each sample member of the initial density matrix. We demonstrate the approach through explicitly computing the real-time propagator for an interacting scalar in 0+1 dimensions, and find very good convergence allowing for comparison with perturbation theory and the classical-statistical approximation to real-time dynamics.