Decay of metastable states: Sharp transition from quantum to classical behavior

Abstract

The decay rate of metastable states is determined at high temperatures by thermal activation, whereas at temperatures close to zero quantum tunneling is relevant. At some temperature Tc the transition from classical to quantum-dominated decay occurs. The transition can be first-order like, with a discontinuous first derivative of the Euclidean action, or smooth with only a second derivative developing a jump. In the former case the crossover temperature Tc cannot be calculated perturbatively and must be found as the intersection point of the Euclidean actions calculated at low and high temperatures. In this paper we present a sufficient criterion for a first-order transition in tunneling problems and apply it to the problem of the tunneling of strings. It is shown that the problem of the depinning of a massive string from a linear defect in the presence of an arbitrarily strong dissipation exhibits a first-order transition.

Negative modes and the decay-rate transition

“…As will be shown, the criterion derived from this viewpoint exactly concides with that of Ref. [12], in which the criterion is derived by computing the variation of the period near sphaleron. In Sec.II we examine the conjecture of Ref.…”

First-order phase transitions in quantum-mechanical tunneling models

“…We envisage that considerations like those presented above and in ref. [2] can be useful in wider contexts, and also in field theory models, such as those with Skyrme terms which imply an effective field dependent mass. Fig.…”

“…at the bottom of the well of the inverse potential, as advocated in ref. [2]. If the frequency of oscillation about the sphaleron point is ω s and oscillations different from ω s about it are possible, a first order transition requires ω 2 > ω 2 s or τ − τ s < 0.…”

“…In particular, a sharp first order transition is there shown to appear in the plot of action S versus temperature T which is very analogous to that of the free enthalpy versus pressure of a van der Waals gas, whose equation of state plotted as pressure versus volume corresponds to the plot of the period P (E) (of the periodic bounce or instanton) versus energy in the consideration of quantum mechanical systems. It is well known [2] that in the case of a periodic problem in Euclidean time the derivative of the action with respect to the energy E is the negative of the oscillation time τ (E) or period P (E) at that energy which has to be identified with −h/T . If τ (E) is a monotonically decreasing function with increasing E, one has a second order transition; if τ (E) has a minimum and then rises again within the domain 0 < E < barrier height, one has a first order transition, i.e.…”

“…A significant step forward in this direction on the basis of higher dimensional arguments, was achieved in ref. [2], where an expansion about the time independent saddle point configuration is considered. However, many considerations of basic interest are best illustrated by quantum mechanical models, which serve as useful prototypes for more complicated theories, and it is natural to enquire about criteria for the occurrence of first order transitions in these, and hence to verify the conditions obtained heuristically as in the models referred to above [3,4,5].…”