Abstract Correlations between energy levels can help distinguish whether a many-body system is of integrable or chaotic nature. The study of short-range and long-range spectral correlations generally involves quantities which are very different, unless one uses the k th nearest neighbor ( k NN) level spacing distribution. For nearest-neighbor (NN) spectral spacings, the distribution in random matrices is well captured by the Wigner surmise. This well-known approximation, derived exactly for a 2 × 2 matrix, is simple and satisfactorily describes the NN spacings of larger matrices. There have been attempts in the literature to generalize Wigner’s surmise to further away neighbors. However, as we show, the current proposal in the literature fails to accurately capture numerical data. Using the known variance of the distributions from random matrix theory, we propose a corrected surmise for the k NN spectral distributions. This surmise better characterizes spectral correlations while retaining the simplicity of Wigner’s surmise. We test the predictions against numerical results and show that the corrected surmise is systematically more accurate at capturing data from random matrices. Using the XXZ spin chain with random on-site disorder, we illustrate how these results can be used as a refined probe of many-body quantum chaos for both short- and long-range spectral correlations.
Abstract Correlations between energy levels can help distinguish whether a many-body system is of integrable or chaotic nature. The study of short-range and long-range spectral correlations generally involves quantities which are very different, unless one uses the k th nearest neighbor ( k NN) level spacing distribution. For nearest-neighbor (NN) spectral spacings, the distribution in random matrices is well captured by the Wigner surmise. This well-known approximation, derived exactly for a 2 × 2 matrix, is simple and satisfactorily describes the NN spacings of larger matrices. There have been attempts in the literature to generalize Wigner’s surmise to further away neighbors. However, as we show, the current proposal in the literature fails to accurately capture numerical data. Using the known variance of the distributions from random matrix theory, we propose a corrected surmise for the k NN spectral distributions. This surmise better characterizes spectral correlations while retaining the simplicity of Wigner’s surmise. We test the predictions against numerical results and show that the corrected surmise is systematically more accurate at capturing data from random matrices. Using the XXZ spin chain with random on-site disorder, we illustrate how these results can be used as a refined probe of many-body quantum chaos for both short- and long-range spectral correlations.
Correlations between the energies of a system's spectrum are one of the defining features of quantum chaos. They can be probed using the spectral form factor (SFF). We investigate how each spectral distance contributes in building this two-point correlation function. Specifically, starting from the spectral distribution of kth-neighbor-level spacing (knLS), we provide analytical expressions for the kth-neighbor spectral form factor (knSFF). We do so for the three Gaussian random matrix ensembles and the ``Poisson'' ensemble of uncorrelated energy levels. We study the properties of the knSFF, namely its minimum value and the time at which this minimum is reached, as well as the energy spacing with the deepest knSFF. This allows us to quantify the contribution of each individual knLS to the SFF ramp, which is a characteristic feature of quantum chaos. In particular, we show how the onset of the ramp, characterized either by the dip or the Thouless time, shifts to shorter times as contributions from longer-range spectral distance are included. Interestingly, the even and odd neighbors contribute quite distinctively, the first being the most important to built the ramp. They respectively yield a resonance or antiresonance in the ramp. All of our analytical results are tested against numerical realizations of random matrices. We complete our analysis and show how the introduced tools help characterize the spectral properties of a physical many-body system by looking at the interacting XXZ Heisenberg model with local on-site disorder that allows transitioning between the chaotic and integrable regimes.
The statistics of gap ratios between consecutive energy levels is a widely used tool-in particular, in the context of many-body physics-to distinguish between chaotic and integrable systems, described, respectively, by Gaussian ensembles of random matrices and Poisson statistics.In this work, we extend the study of the gap ratio distribution PðrÞ to the case where discrete symmetries are present.This is important since in certain situations it may be very impractical, or impossible, to split the model into symmetry sectors, let alone in cases where the symmetry is not known in the first place.Starting from the known expressions for surmises in the Gaussian ensembles, we derive analytical surmises for random matrices comprised of several independent blocks.We check our formulas against simulations from large random matrices, showing excellent agreement.We then present a large set of applications in many-body physics, ranging from quantum clock models and anyonic chains to periodically driven spin systems.In all these models, the existence of a (sometimes hidden) symmetry can be diagnosed through the study of the spectral gap ratios, and our approach furnishes an efficient way to characterize the number and size of independent symmetry subspaces.We finally discuss the relevance of our analysis for existing results in the literature, as well as its practical usefulness, and point out possible future applications and extensions.
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).Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges.For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc.R. Soc.London, A400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of spectral pair correlation function), from semiclassics has been completed by Müller et al. [Phys.Rev. Lett.93, 014103 (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 the '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 a clear theoretical explanation for these observations.We compute K(t) in the leading two orders in t 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.In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t, hence the leading order RMT behavior K(t)2t is a consequence of translation and reflection symmetry of the Ising partition function.
Quantum chaotic and integrable systems are known to exhibit a characteristic 1/f and 1/f2 noise, respectively, in the power spectrum associated to their spectral fluctuations. A recent work [R. Riser, V. A. Osipov, and E. Kanzieper, \textit{Power Spectrum of Long Eigenlevel Sequences in Quantum Chaotic Systems}, Phys. Rev. Lett. \textbf{118}, 204101 (2017)] calls into question the approximations used to derive these results from random matrix theory. In this paper we show that such approximations do remain valid under almost any circumstances. For the integrable limit, we devise a protocol to exactly recover the original results. As a corollary, we show that the theoretical predictions for other statistics are bound for failure regarding long-range correlations, due to unavoidable spurious effects emerging from the analysis. By means of a rigorous statistical test, we also show that the corrections for the chaotic case introduced in the aforementioned paper require huge statistics to become relevant ---averages over more than 1000 realizations are mandatory. As an application, we study a paradigmatic model for the crossover from the thermal to the many-body localized phase. We show that our protocol succeeds in describing the crossover. Furthermore, it also succeeds in proving that the Gaussian β-ensemble fails to account for long-range correlations between the energy levels of this paradigmatic model.
The statistics of gaps between quantum energy levels is a hallmark criterion in quantum chaos and quantum integrability studies. The relevant distributions corresponding to exactly integrable vs. fully chaotic systems are universal and described by the Poisson vs. Wigner-Dyson curves. In the transitional regime between integrability and chaos, the distributions are much less universal and have not been understood quantitatively until now. We point out that the relevant statistics that controls these distributions is that of the matrix elements of the nonintegrable perturbation Hamiltonian in the energy eigenbasis of the unperturbed integrable system. With this insight, we formulate a simple random matrix ensemble that correctly reproduces the level spacing distributions in a variety of test systems. For the distribution of matrix elements appearing in our construction, we furthermore discover surprising universal features: across a variety of physical systems with diverse degrees of freedom, these distributions are dominated by simple power laws.
In this work, the term “quantum chaos” refers to spectral correlations similar to those found in the random matrix theory. Quantum chaos can be diagnosed through the analysis of level statistics using, e.g., the spectral form factor, which detects both short- and long-range level correlations. The spectral form factor corresponds to the Fourier transform of the two-point spectral correlation function and exhibits a typical slope-dip-ramp-plateau structure (aka correlation hole) when the system is chaotic. We discuss how this structure could be detected through the quench dynamics of two physical quantities accessible to experimental many-body quantum systems: the survival probability and the spin autocorrelation function. The survival probability is equivalent to the spectral form factor with an additional filter. When the system is small, the dip of the correlation hole reaches sufficiently large values at times which are short enough to be detected with current experimental platforms. As the system is pushed away from chaos, the correlation hole disappears, signaling integrability or localization. We also provide a relatively shallow circuit with which the correlation hole could be detected with commercially available quantum computers.