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arXiv:2307.11733

Conserved Quantities in Models of Classical Chaos

Henry Ando, David A. Huse

Published Jul 17, 2023• 0 citations
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Abstract

Quantum chaos is a major subject of interest in condensed matter theory, and has recently motivated new questions in the study of classical chaos. In particular, recent studies have uncovered interesting physics in the relationship between chaos and conserved quantities in models of quantum chaos. In this paper, we investigate this relationship in two simple models of classical chaos: the infinite-temperature Heisenberg spin chain, and the directed polymer in a random medium. We relate these models by drawing analogies between the energy landscape over which the directed polymer moves and the magnetization of the spin chain. We find that the coupling of the chaos to these conserved quantities results in, among other things, a marked transition from the rough perturbation profiles predicted by analogy to the KPZ equation to smooth, triangular profiles with reduced wandering exponents. These results suggest that diffusive conserved quantities can, in some cases, be the dominant forces shaping the development of chaos in classical systems.

DOI
10.48550/arxiv.2307.11733

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Quantum chaos is a major subject of interest in condensed matter theory, and has recently motivated new questions in the study of classical chaos. In particular, recent studies have uncovered interesting physics in the relationship between chaos and conserved quantities in models of quantum chaos. In this paper, we investigate this relationship in two simple models of classical chaos: the infinite-temperature Heisenberg spin chain, and the directed polymer in a random medium. We relate these models by drawing analogies between the energy landscape over which the directed polymer moves and the magnetization of the spin chain. We find that the coupling of the chaos to these conserved quantities results in, among other things, a marked transition from the rough perturbation profiles predicted by analogy to the KPZ equation to smooth, triangular profiles with reduced wandering exponents. These results suggest that diffusive conserved quantities can, in some cases, be the dominant forces shaping the development of chaos in classical systems.
We study a nonreciprocal generalization [EPL 60, 418 (2002)] of the classical Heisenberg spin chain, in which the exchange coupling is nonsymmetric, and show that it displays a ballistic spreading of chaos as measured by the decorrelator. We show that the interactions are reciprocal in terms of transformed variables, with conserved quantities that can be identified as magnetization and energy, with a Poisson-bracket algebra and Hamiltonian dynamics. For strictly antisymmetric couplings in the original model the conserved quantities diffuse, the decorrelator spreads symmetrically, and a simple hydrodynamic theory emerges. The general case in which the interaction has symmetric and antisymmetric parts presents complexities in the limit of large scales. Ballistic propagation of chaos survives the inclusion of interactions beyond nearest neighbours, but the conservation laws in general do not.
We study the quantum-classical correspondence for systems with interacting spin particles that are strongly chaotic in the classical limit. This is done in the presence of constants of motion associated with the fixed angular momenta of individual spins. Our analysis of the Lyapunov spectra reveals that the largest Lyapunov exponent agrees with the Lyapunov exponent that determines the local instability of each individual spin moving under the influence of all other spins. Within this picture, we introduce a rigorous and simple test of ergodicity for the spin motion, and use it to identify when classical chaos is both strong and global in phase space. In the quantum domain, our analysis of the Hamiltonian matrix in a proper representation allows us to obtain the conditions for the onset of quantum chaos as a function of the model parameters. From the comparison between the quantum and classical domains, we demonstrate that quantum quantities, such as the local density of states (LDoS) and the shape of the chaotic eigenfunctions written in the noninteracting many-body basis, have well-defined classical counterparts. We also find a relationship between the Kolmogorov-Sinai entropy and the width of the LDoS, which is useful for studies of many-body dynamics.
Many-body quantum chaos at ℏ∼1\hbar\sim 1ℏ∼1: Analytic connection to random matrix theory
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT). For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry (1985) within the so-called diagonal approximation of semiclassical periodic-orbit sums. Derivation of the full RMT spectral form factor K(t)K(t)K(t) from semiclassics has been completed only much later in a tour de force by Mueller et al (2004). In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains. Such systems display two universal types of behaviour which are termed as `many-body localized phase' and `ergodic phase'. In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder. Here we provide the first theoretical explanation for these observations. We compute K(t)K(t)K(t) explicitly in the leading two orders in ttt and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.
We investigate disordered one- and two-dimensional Heisenberg spin lattices across the transition from integrability to quantum chaos from both statistical many-body and quantum-information perspectives. Special emphasis is devoted to quantitatively exploring the interplay between eigenvector statistics, delocalization, and entanglement in the presence of nontrivial symmetries. The implication of the basis dependence of state delocalization indicators (such as the number of principal components) is addressed, and a measure of relative delocalization is proposed in order to robustly characterize the onset of chaos in the presence of disorder. Both standard multipartite and generalized entanglement are investigated in a wide parameter regime by using a family of spin- and fermion-purity measures, their dependence on delocalization and on energy spectrum statistics being examined. A distinctive correlation between entanglement, delocalization, and integrability is uncovered, which may be generic to systems described by the two-body random ensemble and may point to a new diagnostic tool for quantum chaos. Analytical estimates for typical entanglement of random pure states restricted to a proper subspace of the full Hilbert space are also established and compared with random matrix theory predictions.
Previous work [Gong and Brumer, Phys. Rev. Lett. 97, 240602 (2006)] motivates this study as to how asymmetry-driven quantum ratchet effects can persist despite a corresponding fully chaotic classical phase space. A simple perspective of ratchet dynamics, based on the Heisenberg picture, is introduced. We show that ratchet effects are in principle of common origin in classical and quantum mechanics, although full chaos suppresses these effects in the former but not necessarily the latter. The relationship between ratchet effects and coherent dynamical control is noted.
The conventional wisdom suggests that transports of conserved quantities in nonintegrable quantum many-body systems at high temperatures are diffusive. Recent study suggests a potential counterexample of this paradigm. We independently complement this counterexample by directly uncovering anomalous infinite-temperature hydrodynamics in a spin-1/2 XXZ chain with power-law couplings. This model, classified as nonintegrable due to its Wigner-Dyson level-spacing statistics in the random matrix theory, exhibits a surprising superdiffusive-ballistic-superdiffusive transport transition by varying the power-law exponent of couplings for a fixed anisotropy. Our findings are verified by multiple observables, including the spin-spin autocorrelator, mean-square displacement, and spin conductivity. Interestingly, we further quantify the degree of quantum chaos using the Kullback-Leibler divergence between the entanglement entropy distributions of the model's eigenstates and a random state. Remarkably, an observed local maximum in the divergence near the transition boundary suggests a link between anomalous hydrodynamics and a suppression of quantum chaos. This work offers another deep understanding of emergent anomalous transport phenomena in a wider range of nonintegrable quantum many-body systems.