Dynamical quantum phase transitions and broken-symmetry edges in the many-body eigenvalue spectrum

Abstract

Many body models undergoing a quantum phase transition to a broken-symmetry phase that survives up to a critical temperature must possess, in the ordered phase, symmetric as well as non-symmetric eigenstates. We predict, and explicitly show in the fully-connected Ising model in a transverse field, that these two classes of eigenstates do not overlap in energy, and therefore that an energy edge exists separating low-energy symmetry-breaking eigenstates from high-energy symmetry-invariant ones. This energy is actually responsible, as we show, for the dynamical phase transition displayed by this model under a sudden large increase of the transverse field. A second situation we consider is the opposite, where the symmetry-breaking eigenstates are those in the high-energy sector of the spectrum, whereas the low-energy eigenstates are symmetric. In that case too a special energy must exist marking the boundary and leading to unexpected out-ofequilibrium dynamical behavior. An example is the fermonic repulsive Hubbard model Hamiltonian H. Exploiting the trivial fact that the high energy spectrum of H is also the low energy one of −H, we conclude that the high energy eigenstates of the Hubbard model are superfluid. Simulating in a time-dependent Gutzwiller approximation the time evolution of a high energy BCS-like trial wave function, we show that a small superconducting order parameter will actually grow in spite of the repulsive nature of interaction.

Transient Dynamics ofd-Wave Superconductors after a Sudden Excitation

“…It is easy to see that in equilibrium, i.e. no quench of the pairing field ∆ i = ∆ f , we recover the standard result G < k (t − t ) = i n(E ki ) cos 2 θ ki e −iE ki (t−t ) + (1 − n(E ki )) sin 2 θ ki e iE ki (t−t ) (35) which in Fourier space gives…”

“…Theoretical investigations along these directions have appeared in the literature in recent years [19][20][21][22] addressing questions like the thermalization of pump-excited Mott or Kondo insulators [23,24] and the role of lattice vibrations [25], orbital degrees of freedom [26] and competing orders [27,28] in the relaxation dynamics. The dynamics of conventional superconductors has also been studied with reference to pump-probe experiments [29][30][31] or in the presence of electron-phonon interactions [32][33][34] and Coulomb repulsion [35]. In the context of ultracold atomic systems, the research has focused on the s-wave weak coupling BCS regime [36][37][38][39], with recent attempts to extend the analysis to the crossover into the BEC regime [40,41] and to exotic order-parameter symmetries [42,43].…”

Thermalization in a periodically driven fully connected quantum Ising ferromagnet

“…Hence, for N = ∞ their dynamics is that of a one-dimensional classical nonlinear Hamiltonian system H(Q, P, t) [39]. At variance with the perfectly regular classical dynamics observed after a quantum quench [39,40], we will show how rich is the periodic driving case: we can see classically regular motion, chaos and even full ergodicity by choosing appropriately the parameters of the driving. We will solve numerically the Schrödinger equation at finite "large" N .…”

“…The non-equilibrium quantum dynamics of these models has so-far been discussed in the cases of quantum annealing [10][11][12] across the QCP [42,[47][48][49], and of a sudden quench of Γ(t) [39,40], in the context of dynamical phase transitions. Here we will consider its non-equilibrium coherent dynamics under a periodic transverse field, more specifically Γ(t) = Γ 0 + A sin (ω 0 t).…”

Selective Transient Cooling by Impulse Perturbations in a Simple Toy Model

“…How- ever, this is not sufficient; one also needs the conserved energy E 1 (τ ) to be lower than the top of the barrier separating the two minima E * (τ ) = V (0) = −N 1 (τ ) h 1 , the broken-symmetry edge of Ref. [14]. If this indeed happens, then the system will end up after the pulse into one of the two equivalent wells and keep oscillating around the minimum, which would imply a finite time-average value of the order parameter.…”