Consequences of integrability breaking in quench dynamics of pairing Hamiltonians

Abstract

We study the collisionless dynamics of two classes of nonintegrable pairing models. One is a BCS model with separable energy-dependent interactions, the other -a 2D topological superconductor with spin-orbit coupling and a band-splitting external field. The long-time quantum quench dynamics at integrable points of these models are well understood. Namely, the squared magnitude of the time-dependent order parameter ∆(t) can either vanish (Phase I), reach a nonzero constant (Phase II), or periodically oscillate as an elliptic function (Phase III). We demonstrate that nonintegrable models too exhibit some or all of these nonequilibrium phases. Remarkably, elliptic periodic oscillations persist, even though both their amplitude and functional form change drastically with integrability breaking. Striking new phenomena accompany loss of integrability. First, an extremely long time scale emerges in the relaxation to Phase III, such that short-time numerical simulations risk erroneously classifying the asymptotic state. This time scale diverges near integrable points. Second, an entirely new Phase IV of quasiperiodic oscillations of |∆| emerges in the quantum quench phase diagrams of nonintegrable pairing models. As integrability techniques do not apply for the models we study, we develop the concept of asymptotic self-consistency and a linear stability analysis of the asymptotic phases. With the help of these new tools, we determine the phase boundaries, characterize the asymptotic state, and clarify the physical meaning of the quantum quench phase diagrams of BCS superconductors. We also propose an explanation of these diagrams in terms of bifurcation theory. CONTENTS 28 b. p + ip, II-III 28 References 29 arXiv:1812.04410v1 [cond-mat.quant-gas]

Driven-dissipative dynamics of atomic ensembles in a resonant cavity: Nonequilibrium phase diagram and periodically modulated superradiance

“…Here the amplitude of the superconducting order parameter, which is the analog of |l − (t)|, either asymptotes to zero (Phase I) or to a finite constant (Phase II) or oscillates periodically (Phase III), see Ref. 67 for more on this similarity between the phase diagrams.…”

Loschmidt Echo of Far-From-Equilibrium Fermionic Superfluids

“…One is a sudden quench of the interaction strength, g i → g f with the initial state taken to be the ground state of Ĥ(g i ). Such a nonequilibrium protocol has been extensively studied previously in this and related models [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. The dynamics can classified into three distinct phases characterized by the long time behavior of the order parameter which either vanishes [Phase I], approaches a constant [Phase II] or persistently oscillates [Phase III].…”

“…We explore the significance and meaning of DQPTs for two different nonequilibrium scenarios, the quench dynamics of the ground state following a sudden change in interaction strength [39,40] and the solitonic dynamics which emerges from a range of unstable stationary states [41]. The appearance of DQPTs is then compared to the behaviour of the system at long times which has been well studied previously [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. Throughout this paper dynamical or nonequilibrium phases are understood as qualitatively distinct long time states of the system distinguished by qualitatively different behaviours of the order parameter.…”

Adiabatic transition from a BCS superconductor to a Fermi liquid and phase dynamics

“…where k and p are energy labels, J + p = J x p ± iJ y p and N norm is a normalization factor given by the number of levels. With a constant U (t) = U 0 the model becomes integrable in the thermodynamic limit, leading then to a constraint dynamics, which reflects in the peculiar properties of the model under quenching [20][21][22][23][24]30 or driving. 14 These considerations upgrade Eqs.…”

“…In the presence of a static vector potential A(q) the total current should respond according to j µ (q) = K µν (q)A ν (q) (30) where in the long-wave length limit…”