Bifurcation of a periodic instanton in a decay-rate transition

Effective Landau theory for crossover from thermal hopping to quantum tunnelling

“…second-order transition is the boundary between the first-order transition and the second-order one. With this theory [17,[28][29][30][31][32][33], not only is a new kind of global broken symmetry pointed out, but also the first-order transition is easily studied, especially the phase-coexistence of crossover is first proposed. After a detailed discussion, we find that there is no discontinuity of the transition rate's first derivative during the first-order transition.…”

“…Some years ago Grabert and Weiss discussed the transition rate in the presence of dissipative effects of the environment in some detail. In addition to the zero mode near the transition point they found an unstable mode and calculated it carefully [17,[28][29][30][31][32][33]. Near the phase transition point the fluctuation modes about the saddle points include two dangerous modes which cannot be calculated by the Gaussian semiclassical approximation and it is necessary to consider higher-order couplings between the two dangerous modes [17,[28][29][30][31][32][33].…”

“…Above T c , the decay process is dominated by the saddle point, called a sphelaron x = x b . Considering the fluctuation modes around it, we have the periodic paths near the saddle point [17,[28][29][30][31][32][33]…”

“…There is another one φ p = Mω (b)2 1 /4B 4 which represents the periodic instanton solution x = √ 2φ p sin(ω R τ ). It is well known that there is a universal law in the crossover region of a second-order transition [17,[28][29][30][31][32][33]. The crossover region is defined as…”

Hidden functional relation in large-N quark–monopole system at finite temperature

“…This fact is shown explicitly by computing the spectrum of the fluctuation operator numerically [9]. It is also possible to prove the multiple zero modes at the bifurcation point analytically [10].…”