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.

Using a dynamic functional renormalization group treatment of driven elastic interfaces in a disordered medium, we investigate several aspects of the creep-type motion induced by external forces below the depinning threshold fc: i) We show that in the experimentally important regime of forces slightly below fc the velocity obeys an Arrhenius-type law v ∼ exp[−U (f )/T ] with an effective energy barrier U (f ) ∝ (fc − f ) vanishing linearly when f approaches the threshold fc. ii) Thermal fluctuations soften the pinning landscape at high temperatures. Determining the corresponding velocity-force characteristics at low driving forces for internal dimensions d = 1, 2 (strings and interfaces) we find a particular non-Arrhenius type creep v ∼ exp[−(fc(T )/f ) µ ] involving the reduced threshold force fc(T ) alone. For d = 3 we obtain a similar v-f characteristic which is, however, non-universal and depends explicitly on the microscopic cutoff.

We study the elastic (1 + 1)-dimensional string subject to a random gaussian potential on scales smaller than the correlation radius of the disorder potential (Larkin problem). We present an exact calculation of the probability function P [F (u, L)] for the free energy F of a string starting at (0, 0) and ending at (u, L). The function P(F ) is strongly asymmetric, with the left tail decaying exponentially (ln P(F → −∞) ∝ F ) and the right tail vanishing as ln P(F → +∞) ∝ −F 3 . Our analysis defines a strategy for future attacks on this class of problems.

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