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Journal de physique

Symmetry breaking in the double-well model: ohmic dissipation with unrelaxed (Feynman-Vernon) initial conditions

Vladislav Čápek

Published Jan 1, 1989• 4 citations
Publisher

Abstract

The problem of asymptotic-time symmetry breaking for a particle in a symmetric double-well, interacting with phonons, is formulated for a standard minimal two-site model and for initially asymmetric particle distribution and thermal but unrelaxed phonons. The timeconvolutionless Generalized Master Equations can be handled explicitly in the Born approximation. Formally extending the solution to the infinite time, the criteria for the asymptotic-time symmetry breaking comply with those known for the relaxed initial condition.

DOI
10.1051/jphys:01989005007077500

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The problem of asymptotic-time symmetry breaking for a particle in a symmetric double-well, interacting with phonons, is formulated for a standard minimal two-site model and for initially asymmetric particle distribution and thermal but unrelaxed phonons. The timeconvolutionless Generalized Master Equations can be handled explicitly in the Born approximation. Formally extending the solution to the infinite time, the criteria for the asymptotic-time symmetry breaking comply with those known for the relaxed initial condition.
The real-time dynamics of a particle in a slightly asymmetric double-well potential in the presence of ohmic dissipation is analysed by using standard quantum-statistical methods. At zero temperature, the particle eventually goes towards the lower potential well; the relaxation has an overall exponential character with possibly small oscillations; at finite temperature, the particle thermalizes over the two wells; two successive exponential regimes are observed in this case. The results about the final average position of the particle agree with the detailed balance principle and are in fact independent of the ohmic assumption. For an ohmic dissipation law, the localization phenomenon which occurs in the perfectly symmetric double-well potential at zero temperature disappears when an asymmetry is present, although a precursor behaviour is displayed for very low values of the asymmetry.
The author applied the generalised master equation (GME) to the analysis of the object function, s(t), which describes the motion of the tunnelling particle in a double-well system coupled to a reservoir consisting of independent bosonic-type elementary excitation. First, using the 'unrelaxed' initial condition and a suitable projection operator we give the exact formal solution of the GME. Then, the memory operator in the GME is explicitly calculated in the weak-tunnelling regime and the function s(t) is obtained. An independent derivation is given which enables us to obtain the exact expansion of the function s(t) by means of a purely algebraic method. This expansion forms a basis on which this GME method and the well known functional-integral approach are compared: the GME-weak-tunnelling approximation is shown to be identical to the commonly used non-interacting-blip approximation ensuing from the functional-integral method. A new approximation is analysed which is weaker than the non-interacting-blip one.
One necessary criterion for the thermalization of a nonequilibrium quantum many-particle system is ergodicity. It is, however, not sufficient in cases where the asymptotic long-time state lies in a symmetry-broken phase but the initial state of nonequilibrium time evolution is fully symmetric with respect to this symmetry. In equilibrium, one particular symmetry-broken state is chosen as a result of an infinitesimal symmetry-breaking perturbation. From a dynamical point of view the question is: Can such an infinitesimal perturbation be sufficient for the system to establish a nonvanishing order during quantum real-time evolution? We study this question analytically for a minimal model system that can be associated with symmetry breaking, the ferromagnetic Kondo model. We show that after a quantum quench from a completely symmetric state the system is able to break its symmetry dynamically and discuss how these features can be observed experimentally.
We propose a generalization of the eigenstate thermalization hypothesis accountin g for the emergence of symmetry-breaking phases. It consists of two conditions that any system with a degenerate spectrum must fulfill in order to thermalize. The failure of each condition generates a different nonthermalizing scenario. One is due to the absence of chaos and may indicate that extra constants of motion are required to describe equilibrium states. The other one implies the existence of initial conditions evolving toward symmetry-breaking equilibrium states. If it spreads across an entire spectral region, then this region gives rise to a symmetry-breaking phase. We explore the applicability of this formalism by means of numerical experiments on a three-site Bose-Hubbard model with two noncommuting discrete symmetries.
The ground-state phase transition and the phonon dispersion relation of the quantum double-well model are studied by means of the time-dependent variational approach combined with a Hartree-type many-body trial wavefunction. The single-particle state is taken to be a frozen Jackiw–Kerman wavefunction. Under the condition of minimum uncertainty relation, we obtain an effective classical Hamiltonian for the system and equations of motion for the particle's expectation values. It is shown that the effective substrate potential transits from a symmetric double-well potential to a symmetric single-well potential, and the ground state exhibits a transition from a broken symmetry phase to a restored symmetry phase as increasing the strength of quantum fluctuations. We also obtain the phonon dispersion relations and the phonon gaps at the two phases.
A field theory approach for the nonequilibrium relaxation dynamics in open systems at late times is developed. In the absence of conservation laws, all excitations are subject to dissipation. Nevertheless, ordered stationary states satisfy Goldstone's theorem. It implies a vanishing damping rate at small momenta, which in turn allows for competition between environment-induced dissipation and thermalization due to collisions. We derive the dynamic theory in the symmetry-broken phase of an O(N)O(N)O(N)-symmetric field theory based on an expansion of the two-particle irreducible (2PI) effective action to next-to-leading order in 1/N1/N1/N and highlight the analogies and differences to the corresponding theory for closed systems. A central result of this approach is the systematic derivation of an open-system Boltzmann equation, which takes a very different form from its closed-system counterpart due to the absence of well-defined quasiparticles. As a consequence of the general structure of its derivation, it applies to open, gapless field theories that satisfy certain testable conditions, which we identify here. Specifically for the O(N)O(N)O(N) model, we use scaling analysis and numerical simulations to show that interactions are screened efficiently at small momenta and, therefore, the late-time evolution is effectively collisionless. This implies that fluctuations induced by a quench dissipate into the environment before they thermalize. Goldstone's theorem also constrains the dynamics far from equilibrium, which is used to show that the order parameter equilibrates more quickly for quenches preserving the O(N)O(N)O(N) symmetry than those breaking it explicitly.