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arXiv:hep-ph/0005224 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Nonequilibrium quantum dynamics of second order phase transitions

Sang Pyo Kim, Chul H. Lee

Published Nov 28, 2000• 51 citations
arXivHTMLPDF

Abstract

We use the so-called Liouville--von Neumann (LvN) approach to study the nonequilibrium quantum dynamics of time-dependent second order phase transitions. The LvN approach is a canonical method that unifies the functional Schr\"odinger equation for the quantum evolution of pure states and the LvN equation for the quantum description of mixed states of either equilibrium or nonequilibrium. As nonequilibrium quantum mechanical systems we study a time-dependent harmonic and an anharmonic oscillator and find the exact Fock space and density operator for the harmonic oscillator and the nonperturbative Gaussian Fock space and density operator for the anharmonic oscillator. The density matrix and the coherent, thermal, and coherent-thermal states are found in terms of their classical solutions, for which the effective Hamiltonians and equations of motion are derived. The LvN approach is further extended to quantum fields undergoing time-dependent second order phase transitions. We study an exactly solvable model with a finite smooth quench and find the two-point correlation functions. Because of the spinodal instability of long wavelength modes, the two-point correlation functions lead to the ${t}^{1/4}$-scaling relation during the quench and the Cahn-Allen scaling relation ${t}^{1/2}$ after completion of the quench. Further, after the finite quench the domain formation shows a time-lag behavior at the cubic power of the quench period. Finally we study the time-dependent phase transition of a self-interacting scalar field.

DOI
10.1103/physrevd.62.125020

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We use the so-called Liouville--von Neumann (LvN) approach to study the nonequilibrium quantum dynamics of time-dependent second order phase transitions. The LvN approach is a canonical method that unifies the functional Schr"odinger equation for the quantum evolution of pure states and the LvN equation for the quantum description of mixed states of either equilibrium or nonequilibrium. As nonequilibrium quantum mechanical systems we study a time-dependent harmonic and an anharmonic oscillator and find the exact Fock space and density operator for the harmonic oscillator and the nonperturbative Gaussian Fock space and density operator for the anharmonic oscillator. The density matrix and the coherent, thermal, and coherent-thermal states are found in terms of their classical solutions, for which the effective Hamiltonians and equations of motion are derived. The LvN approach is further extended to quantum fields undergoing time-dependent second order phase transitions. We study an exactly solvable model with a finite smooth quench and find the two-point correlation functions. Because of the spinodal instability of long wavelength modes, the two-point correlation functions lead to the t1/4{t}^{1/4}t1/4-scaling relation during the quench and the Cahn-Allen scaling relation t1/2{t}^{1/2}t1/2 after completion of the quench. Further, after the finite quench the domain formation shows a time-lag behavior at the cubic power of the quench period. Finally we study the time-dependent phase transition of a self-interacting scalar field.
A quenched second order phase transition is modeled by an effective ${\ensuremath{\Phi}}^{4}$ theory with a time-dependent Hamiltonian $\ifmmode \hat{H}\else \^{H}\fi{}(t),$ whose symmetry is broken spontaneously in time. The quantum field evolves out of equilibrium (nonequilibrium) during the phase transition as the density operator significantly deviates from $\stackrel{^}{\ensuremath{\rho}}{(t)=e}^{\ensuremath{-}\ensuremath{\beta}\ifmmode \hat{H}\else \^{H}\fi{}(t)}{/Z}_{H}.$ The recently developed Liouville--von Neumann method provides various quantum states for the phase transition in terms of a complex solution to the mean-field equation, which is equivalent to the Gaussian effective potential in the static case and to the time-dependent Hartree-Fock equation in the nonequilibrium case. Using multiple-scale perturbation theory we solve the mean-field equation analytically to the first order of the coupling constant and find the quantum states during the quenched second order phase transition. We propose a renormalization scheme during the process of phase transition to regularize the divergences, which originate from the mode coupling between hard and hard modes or between soft and hard modes. The effect of mode coupling is discussed.
We study nonequilibrium evolution in a self-interacting quantum field theory invariant under space translation only by using a canonical approach based on the recently developed Liouville--von Neumann formalism. The method is first used to obtain the correlation functions both in and beyond the Hartree approximation, for the quantum mechanical analogue of the ${\ensuremath{\varphi}}^{4}$ model. The technique involves representing the Hamiltonian in a Fock basis of annihilation and creation operators. By separating it into a solvable Gaussian part involving quadratic terms and a perturbation of quartic terms, it is possible to find the improved vacuum state to any desired order. The correlation functions for the field theory are then investigated in the Hartree approximation, and those beyond the Hartree approximation are obtained by finding the improved vacuum state corrected up to $O({\ensuremath{\lambda}}^{2}).$ These correlation functions take into account next-to-leading and next-to-next-to-leading order effects in the coupling constant. We also use the Heisenberg formalism to obtain the time evolution equations for the equal-time, connected correlation functions beyond the leading order. These equations are derived by including the connected four-point functions in the hierarchy. The resulting coupled set of equations form part of an infinite hierarchy of coupled equations relating the various connected n-point functions. The connection with other approaches based on the path integral formalism is established, and the physical implications of the set of equations are discussed with particular emphasis on thermalization.
We use the Liouville--von Neumann approach to study the dynamics and the adiabaticity of a time-dependent driven anharmonic oscillator as an example of nonequilibrium quantum dynamics. We show that the adiabaticity is minimally broken in the sense that a Gaussian wave packet at the past infinity evolves to coherent states; however slowly the potential changes, its coherence factor is of the order of the coupling. We also show that the dynamics are governed by an equation of motion similar to the Kepler motion, which satisfies angular momentum conservation.
The dynamics of a quantum system coupled to a classical environment and subject to constraints that drive it out of equilibrium are described. The evolution of the system is governed by the quantum-classical Liouville equation. Rather than evaluating the evolution of the mixed quantum-classical density operator, we derive exact equations of motion for the nonequilibrium average values of a set of operators or variables, along with correlation function expressions for the dissipative coefficients that enter these equations. These equations are obtained by requiring that the exact nonequilibrium averages are equal to local nonequilibrium averages that depend on auxiliary fields whose values satisfy evolution equations obtained using projection operator methods. The results are illustrated by deriving reaction-diffusion equations coupled to fluid hydrodynamic equations for a solution of quantum particles that can exist in two metastable states. Nonequilibrium steady states are discussed along with the reaction rate and diffusion correlation functions that characterize such states.