Recently it was suggested that certain perturbations of integrable
spin chains lead to a weak breaking of integrability in the sense that
integrability is preserved at the first order in the coupling. Here we
examine this claim using level spacing distribution. We find that the
volume dependent crossover between integrable and chaotic level spacing
statistics {which marks the onset of quantum chaotic behaviour, is
markedly different for weak vs. strong breaking of integrability. In
particular}, for the gapless case we find that the crossover coupling as
a function of the volume LL
scales with a 1/L^{2}1/L2
law for weak breaking as opposed to the 1/L^{3}1/L3
law previously found for the strong case.
We develop a truncated Hamiltonian method to investigate the dynamics of the (1 + 1)d φ 4 theory following quantum quenches. The results are compared to two different semi-classical approaches, the self-consistent Gaussian approximation and the truncated Wigner approximation, and used to determine the range of validity of these widely used approaches. We show that the self-consistent approximation is strongly limited in comparison to the truncated Hamiltonian method which for larger cutoffs is practically exact for the parameter range studied. We find that the self-consistent approximation is only valid when the effective mass is in the vicinity of the renormalised mass. Similarly to the self-consistent approximation, the truncated Wigner approximation is not able to capture the correct mass renormalisation, and breaks down for strong enough interactions where the bare mass becomes negative. We attribute the failure of TWA to the presence of a classical symmetry broken fixed point. Besides establishing the truncated Hamiltonian approach as a powerful tool for studying the dynamics of the φ 4 model, our results on the limitation of semi-classical approximations are expected to be relevant for modelling the dynamics of other quantum field theories.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.