Chaos is an important characterization of classical dynamical systems. How is chaos linked to the long-time dynamics of collective modes across phases and phase transitions? We address this by studying chaos across Ising and Kosterlitz-Thouless transitions in classical XXZ model. We show that spatio-temporal chaotic properties have crossovers across the transitions and distinct temperature dependence in the high and low-temperature phases which show normal and anomalous diffusions, respectively. Our results also provide new insights into the dynamics of interacting quantum systems in the semiclassical limit.
We study chaotic dynamics and anomalous transport in a Bose-Hubbard chain in the semiclassical regime (the limit when the number of particles goes to infinity). We find that the system has mixed phase space with both regular and chaotic dynamics, even for long chains with up to 100 wells. The consequence of the mixed phase space is strongly anomalous diffusion in the space of occupation numbers, with a discrete set of transport exponents. After very long times the system crosses over to the hydrodynamic regime with normal diffusion. Anomalous transport is quite universal and almost completely independent of the parameters of the model (Coulomb interaction and chemical potential): It is mainly determined by the initial distribution of particles along the chain. We corroborate our findings by analytical arguments: scaling analysis for the anomalous regime and the Langevin equation for the normal diffusion regime.
In quantum systems with a classical limit, advanced semiclassical methods provide the crucial link between phase-space structures, reflecting the distinction between chaotic, mixed or integrable classical dynamics, and the corresponding quantum properties. Well established techniques dealing with ergodic wave interference in the usual semiclassical limit
ℏ→0, where the classical limit is given by Hamiltonian mechanics of particles, constitute a now standard part of the toolkit of theoretical physics. During the last years, these ideas have been extended into the field theoretical domain of systems composed of
N indistinguishable particles, aka quantum fields, displaying a different type of semiclassical limit
ℏeff=1/N→0 and accounting for genuine many-body quantum interference. The foundational concept behind this idea of many-body interference, the many-body version of the van Vleck-Gutzwillers semiclassical propagator, is explained in detail. Based on this the corresponding semiclassical many-body theory is reviewed. It provides a unified framework for understanding a variety of quantum chaotic phenomena addressed, including random-matrix spectral correlations in many-body systems, the universal morphology of many-body eigenstates, interference effects kin to mesoscopic weak localization, and the key to the scrambling of many-body correlations characterized by out-of-time-order correlators.
We study the chaotic dynamics in a classical many-body system of interacting spins on the kagome lattice. We characterize many-body chaos via the butterfly effect as captured by an appropriate out-of-time-ordered commutator. Due to the emergence of a spin-liquid phase, the chaotic dynamics extends all the way to zero temperature. We thus determine the full temperature dependence of two complementary aspects of the butterfly effect: the Lyapunov exponent, $\ensuremath{\mu}$, and the butterfly speed,
vb, and study their interrelations with usual measures of spin dynamics such as the spin-diffusion constant,
D, and spin-autocorrelation time, $\ensuremath{\tau}$. We find that they all exhibit power-law behavior at low temperature, consistent with scaling of the form $D\ensuremath{\sim}{v}_{b}^{2}/\ensuremath{\mu}$ and ${\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremath{\sim}T$. The vanishing of $\ensuremath{\mu}\ensuremath{\sim}{T}^{0.48}$ is parametrically slower than that of the corresponding quantum bound, $\ensuremath{\mu}\ensuremath{\sim}T$, raising interesting questions regarding the semiclassical limit of such spin systems.
We analyze the propagation of experimentally relevant two-particle correlations for one-dimensional interacting bosons, and give evidence that many-body chaos induces the emergence of an effective diffusive regime for the fully coherent correlation dynamics, characterized by an interaction dependent diffusion coefficient, which we estimate. This result supports very recent experimental observations, and paves the way toward an efficient description of the dynamical behavior of nonintegrable complex many-body systems. Furthermore, we show that the dynamical features within experimentally accessible time scales of a conveniently defined two-particle correlation transport distance provide a direct and unambiguous characterization of many-body quantum chaos in perfect agreement with its spectral identification.
We study many-body chaos in a (2+1)D relativistic scalar field theory at high temperatures in the classical statistical approximation, which captures the quantum critical regime and the thermal phase transition from an ordered to a disordered phase. We evaluate out-of-time ordered correlation functions (OTOCs) and find that the associated Lyapunov exponent increases linearly with temperature in the quantum critical regime, and approaches the non-interacting limit algebraically in terms of a fluctuation parameter. OTOCs spread ballistically in all regimes, also at the thermal phase transition, where the butterfly velocity is maximal. Our work contributes to the understanding of the relation between quantum and classical many-body chaos and our method can be applied to other field theories dominated by classical modes at long wavelengths.