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Physica A Statistical Mechanics and its Applications

4 × 4-matrix transformation for systems with spontaneous symmetry breakdown and self-consistent renormalization in nonequilibrium Thermo Field Dynamics

Takuro Oyama, Yusuke Nakamura, Y. Yamanaka

Published Dec 20, 2021• 0 citations
Publisher

Abstract

This study aims to construct a theoretical formulation of a nonequilibrium process for a system of Bose–Einstein condensate associated with a spontaneous symmetry breakdown. For this, Thermo Field Dynamics is used. We then describe the most general forms of a 4 × 4-matrix transformation and an unperturbed Hamiltonian. After calculating the 4 × 4-matrix self-energy and identifying its on-shell, we impose the renormalization condition in which the entire on-shell self-energy should vanish. This condition provides a sufficient number of independent equations to determine all of the parameters in an unperturbed Hamiltonian, among which the quantum transport equations for normal and anomalous number distributions are included.

DOI
10.1016/j.physa.2021.126732

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This study aims to construct a theoretical formulation of a nonequilibrium process for a system of Bose–Einstein condensate associated with a spontaneous symmetry breakdown. For this, Thermo Field Dynamics is used. We then describe the most general forms of a 4 × 4-matrix transformation and an unperturbed Hamiltonian. After calculating the 4 × 4-matrix self-energy and identifying its on-shell, we impose the renormalization condition in which the entire on-shell self-energy should vanish. This condition provides a sufficient number of independent equations to determine all of the parameters in an unperturbed Hamiltonian, among which the quantum transport equations for normal and anomalous number distributions are included.
This study aims to construct a theoretical formulation of a nonequilibrium process for a system of Bose--Einstein condensate associated with a spontaneous symmetry breakdown. For this, Thermo Field Dynamics is used. We then describe the most general forms of a 4x4 transformation and an unperturbed Hamiltonian. After calculating the 4×44\times 44×4 self-energy and identifying its on-shell, we impose the renormalization condition in which the entire on-shell self-energy should vanish. This condition provides a sufficient number of independent equations to determine all of the parameters in an unperturbed Hamiltonian, among which the quantum transport equations for normal and anomalous number distributions are included.
In recent experiments of Bose-Einstein condensation of a magnetically trapped dilute alkaline atomic gas, the nonequilibrium process of thermalization into the condensed state has been observed in a reasonably long time scale. This presents a new challenge to develop a theory to treat strongly space-time dependent nonequilibrium processes. In this paper, we present a nonequilibrium formulation of Bose-Einstein condensed states in the field theoretical framework of thermo-field dynamics. By use of an effective Hamiltonian method, a quasi-particle field equation is derived. This equation contains information regarding the renormalized energy, decay width and time-variation of particle distributions. It is shown that, in the present formulation, the strong space-time dependence can be treated, even including quantum corrections, as the use of the quasi-particle field equation enables us to factorize double-time functions of the self-energy.
The way to determine the renormalized energy of inhomogeneous systems of a quantum field under an external potential is established for both equilibrium and nonequilibrium scenarios based on thermo field dynamics. The key step is to find an extension of the on-shell concept valid in homogeneous case. In the nonequilibrium case, we expand the field operator by time-dependent wavefunctions that are solutions of the appropriately chosen differential equation, synchronizing with temporal change of thermal situation, and the quantum transport equation is derived from the renormalization procedure. Through numerical calculations of a triple-well model with a reservoir, we show that the number distribution and the time-dependent wavefunctions are relaxed consistently to the correct equilibrium forms at the long-term limit.
We study the thermal processes for the system of cold neutral atomic gas in the framework of nonequilibrium Thermo Field Dynamics (TFD) which is a real-time canonical formalism of quantum field theory for thermal situations [1]. The system is suitable for testing quantum many-body theories in nonequilibrium situations because temporal changes in thermal processes are so slow that various nonequilibrium phenomena can be observed experimentally. In nonequilibrium TFD, the quantum transport equation, which is the equation for a time-dependent number distribution introduced as an unknown parameter, is derived from the self-consistent renormalization condition on the self-energy [2]. The correction on the excitation energy is important both physically and in the renormalization of quantum field theory. Recently we have proposed the new self-consistent renormalization method in nonequilibrium TFD, in which the on-shell self-energy in time-dependent case is introduced as a generalization of the ordinary on-shell self-energy in stationary case and the renormalization conditions on it yields the quantum transport equation and the diagonal correction of the quasiparticle energy in a unified manner [2]. In this work, the renormalization condition is extended to the imaginary and off-diagonal parts of the self-energy. In our previous work [3], we applied nonequilibrium TFD to the one dimensional system of confined cold neutral atomic Bose gas. There the nonequilibrium situation was set up by a sudden displacement of the confining potential and a sudden switch-on of the interparticle interaction. We investigated the thermal process for the system by solving the quantum transport equation, but neglecting the correction of the quasiparticle energy. Then it was found that in some particular parameter region it gives rise to an unphysical result, namely the negative number distribution. Here we take account of the correction of the quasiparticle energy and solve the quantum transport equation again. The correction affects drastically the thermal relaxation process and resolves the difficulty in some parameter region, as will be shown.
The non-Markoffian transport equations for the systems of cold Bose atoms confined by a external potential both without and with a Bose-Einstein condensate are derived in the framework of nonequilibrium thermal filed theory (Thermo Field Dynamics). Our key elements are an explicit particle representation and a self-consistent renormalization condition which are essential in thermal field theory. The non-Markoffian transport equation for the non-condensed system, derived at the two-loop level, is reduced in the Markoffian limit to the ordinary quantum Boltzmann equation derived in the other methods. For the condensed system, we derive a new transport equation with an additional collision term which becomes important in the Landau instability.
Based on Bogoliubov's truncated Hamiltonian H B for a weakly interacting Bose system, and adding a U(1) symmetry breaking term V 1/2 (λ a 0 +λ * a 0 + ) to H B , we show by using the coherent state theory and the mean-field approximation rather than the c-number approximations, that the Bose–Einstein condensation(BEC) occurs if and only if the U(1) symmetry of the system is spontaneously broken. The real ground state energy and the justification of the Bogoliubov c-number substitution are given by solving the Schrödinger eigenvalue equation and using the self-consistent condition.