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arXiv:hep-th/0312270

Dynamics of phase transition in (3+1) dimensional scalar $ϕ^4$ theory

Hyeong-Chan Kim, Jae Hyung Yee

Published Dec 23, 2003• 0 citations
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Abstract

We use the variational approximation with double Gaussian type trial wave-functional approximation, in which we use the square root of the dispersion of the zero-mode wave-function as an order parameter, to study the out of equilibrium quantum dynamics of time-dependent second order phase transitions in (3+1) dimensions. We study the time evolution of symmetric states of scalar $λϕ^4$ theory in several situations by properly treating the effect of the $λϕ^4$ interaction. We also calculate the effective action and the effective potential of the theory with the precarious renormalization. We show that the presence of a quenching of the mass-squared leads to second order phase transition nontrivially since the vacuum structure changes by absorbing the energy required for quenching, even though there is no symmetry breaking in the effective potential of the theory without quenching process. We also calculate the equal time correlation function, and then evaluate the correlation length as a function of the mass-squared. The time dependence of the correlation length varies depending on how the mass-squared changes in time. For constant mass-squared it gives the classical Cahn-Allen relation, and it leads to different relations for other time-dependence of the mass-squared. We also show that there exists a propagating spatial correlation after termination of the phase transition process in addition to the correlation corresponding to the formation and growth of domains.

DOI
10.48550/arxiv.hep-th/0312270

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Dynamics of phase transition in (3+1) dimensional scalar ϕ4ϕ^4ϕ4 theory
We use the variational approximation with double Gaussian type trial wave-functional approximation, in which we use the square root of the dispersion of the zero-mode wave-function as an order parameter, to study the out of equilibrium quantum dynamics of time-dependent second order phase transitions in (3+1) dimensions. We study the time evolution of symmetric states of scalar λϕ4λϕ^4λϕ4 theory in several situations by properly treating the effect of the λϕ4λϕ^4λϕ4 interaction. We also calculate the effective action and the effective potential of the theory with the precarious renormalization. We show that the presence of a quenching of the mass-squared leads to second order phase transition nontrivially since the vacuum structure changes by absorbing the energy required for quenching, even though there is no symmetry breaking in the effective potential of the theory without quenching process. We also calculate the equal time correlation function, and then evaluate the correlation length as a function of the mass-squared. The time dependence of the correlation length varies depending on how the mass-squared changes in time. For constant mass-squared it gives the classical Cahn-Allen relation, and it leads to different relations for other time-dependence of the mass-squared. We also show that there exists a propagating spatial correlation after termination of the phase transition process in addition to the correlation corresponding to the formation and growth of domains.
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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 present a variational method which uses a double-Gaussian type trial wave-function to describe systems in second order phase transitions. We introduce a new physical variable yyy which is appropriate to describe the shapes of wave-packets for systems in phase transition, and calculate the effective action as a function of both the dispersion sqrt<hatq2>\\sqrt{< \\hat{q}^2>}sqrt<hatq2> and yyy. The effective potential successfully describes the transition of the system from the false vacuum to the true vacuum, and the equation of motion shows that highly non-classical states periodically appear after the quench transition. The numerical computation shows that these results properly describe time evolution of the system.
We consider the out-of-equilibrium evolution of a classical condensate field and its quantum fluctuations for a ${\ensuremath{\Phi}}^{4}$ model in 1+11+11+1 dimensions with a symmetric and a double well potential. We use the two-particle point-irreducible (2PPI) formalism and go beyond the Hartree approximation by including the sunset term. In addition to the mean field $\ensuremath{\varphi}(t)=〈\ensuremath{\Phi}〉$ the 2PPI formalism uses as a variational parameter a time dependent mass M2(t){\mathcal{M}}^{2}(t)M2(t) which contains all local insertions into the Green's function. We compare our results to those obtained in the Hartree approximation. In the symmetric ${\ensuremath{\Phi}}^{4}$ theory we observe that the mean field shows a stronger dissipation than the one found in the Hartree approximation. The dissipation is roughly exponential in an intermediate time region. In the theory with spontaneous symmetry breaking, i.e., with a double well potential, the field amplitude tends to zero, i.e., to the symmetric configuration. This is expected on general grounds: in (1+1)(1+1)(1+1)-dimensional quantum field theory there is no spontaneous symmetry breaking for T>0,T>0,T>0, and so there should be none at finite energy density (microcanonical ensemble), either. Within the time range of our simulations the momentum spectra do not thermalize and display parametric resonance bands.
Abstract We study dynamics of quantum entanglement in smooth global quenches with a finite rate, by computing the time evolution of entanglement entropy in 1 + 1 dimensional free scalar theory with time-dependent masses which start from a nonzero value at early time and either crosses or approaches zero. The time-dependence is chosen so that the quantum dynamics is exactly solvable. If the quenches asymptotically approach a critical point at late time, the early-time and late-time entropies are proportional to the time and subsystem size respectively. Their proportionality coefficients are determined by scales: in a fast limit, an initial correlation length; in a slow limit, an effective scale defined when adiabaticity breaks down. If the quenches cross a critical point, the time evolution of entropy is characterized by the scales: the initial correlation length in the fast limit and the effective correlation length in the slow limit. The entropy oscillates, and the entanglement oscillation comes from a coherence between right-moving and left-moving waves if we measure the entropy after time characterized by the quench rate. The periodicity of the late-time oscillation is consistent with the periodicity of the oscillation of zero modes which are zero-momentum value of two point functions of a fundamental field and its conjugate momentum.
In this paper, we study the dynamics of the Bose-Hubbard model by using time-dependent Gutzwiller methods. In particular, we vary the parameters in the Hamiltonian as a function of time, and investigate the temporal behavior of the system from the Mott insulator to the superfluid (SF) crossing a second-order phase transition. We first solve a time-dependent Schr\"odinger equation for the experimental setup recently done by Braun et al. [Proc. Natl. Acad. Sci. USA 112, 3641 (2015)] and show that the numerical and experimental results are in fairly good agreement. However, these results disagree with the Kibble-Zurek scaling. From our numerical study, we reveal a possible source of the discrepancy. Next, we calculate the critical exponents of the correlation length and vortex density in addition to the SF order parameter for a Kibble-Zurek protocol. We show that beside the ``freeze'' time $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{t}$, there exists another important time, teq{t}_{\mathrm{eq}}teq​, at which an oscillating behavior of the SF amplitude starts. From calculations of the exponents of the correlation length and vortex density with respect to a quench time ${\ensuremath{\tau}}_{\mathrm{Q}}$, we obtain a physical picture of a coarsening process. Finally, we study how the system evolves after the quench. We give a global picture of dynamics of the Bose-Hubbard model.