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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Free energy in thermo field dynamics

H. Matsumoto, Yoshimasa Nakano, Hiroomi Umezawa

Published Mar 15, 1985• 29 citations
Publisher

Abstract

A formula to obtain the free energy within the framework of thermo field dynamics (a real-time quantum field theory at finite temperature) is presented. The relation to the Matsubara method is discussed. Examples of a perturbation calculation are presented for the ${\ensuremath{\varphi}}^{4}$ model up to the two-loop order and are compared with the corresponding result obtained in the Matsubara formalism.

DOI
10.1103/physrevd.31.1495

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A formula to obtain the free energy within the framework of thermo field dynamics (a real-time quantum field theory at finite temperature) is presented. The relation to the Matsubara method is discussed. Examples of a perturbation calculation are presented for the \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 model up to the two-loop order and are compared with the corresponding result obtained in the Matsubara formalism.
In this project we develop a quantum algorithm to realize finite temperature simulation on a quantum computer. As quantum computers use real-time evolution we did not use the imaginary time methods popular on classical algorithms. Instead, we implemented a real-time therom field dynamics formalism, which has the added benefit of being able to compute quantities that are both time-and temperature-dependent. To implement thermo field dynamics we apply a unitary transformation [1] to discrete quantum mechanical operators to make new Hamiltoni-ans with encoded temperature dependence. The method works normally for fermions, which have a finite representation, but needs some modification to work with bosons. These Hamilto-nians are then processed into a Pauli matrix representation in order to be used as input for IBM's Qiskit package [2]. We then use IBM's quantum simulator to calculate an approximation to the Hamiltonaian's ground state energy via the variational quantum eigensolver (VQE) algorithm [3]. This approximation is then compared to a classically calculated value for the exact energy. The thermo field dynamics quantum algorithm has general applications to material science, high-energy physics and nuclear physics, particularly in those situations involving realtime evolution at high temperature.
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.
The Gaussian wavefunctional approach is developed in thermofield dynamics. We construct the thermal vacuum wavefunctional, its creation as well as annihilation operators, and accordingly the thermo-particle excited states. For a ( D + 1)-dimensional scalar field system with an arbitrary potential whose Fourier representation exists in the sense of tempered distributions, we calculate the finite-temperature Gaussian effective potential (FTGEP), one- and two-thermo-particle-state energies. The zero-temperature limit of each of them is just the corresponding result in quantum field theory, and the FTGEP can lead to the same one for some concrete models as calculated by the imaginary-time Green function.