We study the effects of symmetry-breaking perturbations in the out-of-equilibrium quantum dynamics of many-body systems slowly driven by a time-dependent symmetry-preserving parameter, across the quantum critical regime associated with a continuous quantum transition (CQT). For this purpose, we analyze the out-of-equilibrium dynamics arising from generalized Kibble-Zurek (KZ) protocols, within a dynamic renormalization-group framework allowing for finite-size systems. We show that the time dependence of generic observables develops an out-of-equilibrium finite-size scaling (FSS) behavior, arising from the interplay between the timescale
ts of the parameter variations in the KZ protocol, the size
L of the system, and the strength
h of the symmetry-breaking perturbation, in the limit of large
ts and
L. Moreover, scaling arguments based on the first-order adiabatic approximation of slow variations in quantum systems allow us to characterize the approach to the adiabatic regimes for some limits of the model parameters (for example, when we take ${t}_{s}\ensuremath{\rightarrow}\ensuremath{\infty}$ before $L\ensuremath{\rightarrow}\ensuremath{\infty}$), predicting asymptotic power-law suppressions of the nonadiabatic behaviors in the adiabatic limits. This out-of-equilibrium FSS is supported by numerical analyses for the paradigmatic quantum Ising chain along generalized KZ protocols, with a time-dependent transverse field crossing its CQT, in the presence of a static longitudinal field breaking the
Z2 symmetry.
Spontaneous symmetry breaking (SSB) occurs when a many-body system governed by a symmetric Hamiltonian, and prepared in a symmetry-broken state by the application of a field coupling to its order parameter O, retains a finite O value even after the field is switched off. SSB is generally thought to occur only in the thermodynamic limit N→∞ (for N degrees of freedom). In this limit, the time to restore the symmetry once the field is turned off, either via thermal or quantum fluctuations, is expected to diverge. Here, we show that SSB can also be observed in finite-size quantum spin systems, provided that the spin is half-integer and three conditions are met: (1) the ground state of the system has long-range correlations, (2) the Hamiltonian conserves the (spin) parity of the order parameter, and (3) N is odd. Using a combination of analytical arguments and numerical results (based on time-dependent variational Monte Carlo and rotor + spin-wave theory), we show that SSB on finite-size systems can be achieved via a quasiadiabatic preparation of the ground state-which, in U(1)-symmetric systems, is shown to require a symmetry-breaking field vanishing over timescales τ∼O(N). In these systems, the symmetry-broken state exhibits spin squeezing with Heisenberg scaling.
We consider a quantum many-body system on a lattice which exhibits a spontaneous symmetry breaking in its infinite volume ground states, but in which the corresponding order operator does not commute with the Hamiltonian. Typical examples are the Heisenberg antiferromagnet with a Néel order, and the Hubbard model with a (superconducting) off-diagonal long range order. In the corresponding finite system, the symmetry breaking is usually “obscured ” by “quantum fluctuation ” and one gets a symmetric ground state with a long range order. In such a situation, Horsch and von der Linden proved that the finite system has a low-lying eigenstate whose excitation energy is not more than of order N −1, where N denotes the number of the sites in the lattice. Here we study the situation where the broken symmetry is a continuous one. For a particular set of states (which are orthogonal to the ground state and with each other), we prove bounds for their energy expectation values. The bounds establish that there exist ever increasing numbers of low-lying eigenstates whose excitation energies are bounded by a constant times N −1. A crucial feature of the particular low-lying