Universal power law in crossover from integrability to quantum chaos

Abstract

We study models of interacting fermions in one dimension to investigate the crossover from integrability to non-integrability, i.e., quantum chaos, as a function of system size. Using exact diagonalization of finite-sized systems, we study this crossover by obtaining the energy level statistics and Drude weight associated with transport. Our results reinforce the idea that for system size L → ∞ non-integrability sets in for an arbitrarily small integrability-breaking perturbation. The crossover value of the perturbation scales as a power law ∼ L −3 when the integrable system is gapless and the scaling appears to be robust to microscopic details and the precise form of the perturbation. We conjecture that the exponent in the power law is characteristic of the random matrix ensemble describing the non-integrable system. For systems with a gap, the crossover scaling appears to be faster than a power law. How isolated quantum systems thermalize, hitherto investigated theoretically in a few special cases [1][2][3], is now the subject of active experimental study thanks to the advent of cold-atom systems [4,5]. Recall that in isolated classical systems that thermalize, a phase space trajectory samples all possible microstates at a given energy spending equal amounts of time in each, yielding the microcanonical prescription. On the other hand, for a system which does not thermalize, the trajectory typically follows regular, not chaotic, orbits constrained by conservation laws and samples only a low-dimensional subspace. This notion of thermalization underpins the Fermi-PastaUlam problem of a classical system of masses connected by springs [6]. For harmonic springs the system does not thermalize and, even upon the introduction of anharmonicity, thermalization occurs only above an energy threshold which, however, scales to zero with increasing system size, as a power-law characterizing the nature of anharmonicity [7]. Signatures of lack of thermalization in classical systems can also be seen in transport [8] (but note that singular size-dependence of thermal conductivity is distinct from a failure to thermalize [9]).In this paper we investigate analogous issues for quantum systems. Later in the paper we compare our study to the related work of Rabson et al. [10]. As quantum mechanics lacks a notion of phase space, we identify thermalization with non-integrability, i.e., quantum chaos, a now-standard prescription, and use the corresponding diagnostic tools. We would like to emphasize that while the integrable systems we study a) are exactly solvable, b) have an infinity of conservation laws in the thermodynamic limit and c) display Poissonian level-spacing statistics it is perhaps only the last one that is important to prevent thermalization: localized phases of disordered systems lacking properties a) and b) have been argued to not thermalize [11].Our main result is that the characteristic value of control parameter at which significant non-integrability is seen scales to zero with increasing system size as in the classi...

Many-body localization due to correlated disorder in Fock space

“…Only in the case of a linear variation of the covariance with Hamming distance the level spacing statistics show Poissonian statistics implying nonthermal behavior in the system. However that can imply either emergence of MBL phase which is believed to have emergent conservation laws 6,[19][20][21] or conventional integrability of integrable system with dynamical symmetries that inhibit ergodicity [22][23][24][25] . After the introduction of additional interactions which can break the integrability of the system, the system shows a thermal to MBL transition as a function of the slope of the linear variation of covariance ( Fig.…”

From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics

“…This suggests that, at least for this class of models, an infinitesimal integrability breaking perturbation is sufficient to generate quantum chaos in the thermodynamic limit. Recent numerical studies have attempted to quantify how the strength of the integrability breaking terms should scale with the system size for the GOE predictions to hold in one dimension [105,106]. These works suggest that the strength needs to be ∝ L −3 for this to happen, but the origin of such a scaling is not understood.…”

Integrals of motion for one-dimensional Anderson localized systems

“…It can be seen that the m h decrease rapidly with increasing order of power series m indicating convergence. We have also checked the convergence of the power series for i  drawn from the eigenvalues of non-integrable t t V -¢model, which also follow a GOE distribution [33,52].…”

“…Additionally the usual space-time symmetries result in degeneracies in the energy level spectrum, and hence a lack of level repulsion [32]. The addition of perturbations destroys such conservation laws and restores level repulsion, although the strength of the perturbations has a non-trivial finitesize dependence [33][34][35].…”