Many-Body Quantum Chaos: 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). Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in the 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. A 400, 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 the 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 to 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 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 nonintegrable, 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 $K(t)\simeq 2t$ is a consequence of translation and reflection symmetry of the Ising partition function.

Popular Summary

In recent years, the dynamics of many interacting quantum particles has become increasingly relevant. Experiments involving ensembles of ultracold atoms, for example, now allow physicists to prepare and study such systems over long periods of time. These experiments can help verify some fundamental aspects of statistical mechanics. For example, systems of many interacting particles exhibit a behavior known as the ergodic, or thermal, phase wherein only a few parameters (such as temperature or pressure) are needed to characterize the state of matter. In the ergodic phase, spectral fluctuations are well described by random matrix theory, a mathematical framework involving matrices with random elements, though it is not clear why. Here, we provide the first theoretical explanation for this observation.

Considering the energy spectrum as a fictitious gas, we analyze its pair correlation function and compute its Fourier transform, the so-called spectral form factor. We show that the form factor (specifically, the first two leading orders of expansion in the time parameter) is in agreement with results from random matrix theory for many-body systems that are nonintegrable and invariant to time reversal. Most importantly, and unlike in previous approaches, our theory applies to systems without any meaningful classical counterparts; that is, the quantum nature of these systems cannot be ignored.

With further development, our method offers a deeper understanding of the quantum ergodic phase. For example, we envision an explicit computation of dynamical response functions of local observables. This could lead to a dynamical mechanism behind the celebrated eigenstate thermalization hypothesis, which says that almost any eigenstate is a good representative of a thermal state.

Figure 1

Time-integrated spectral form factor for the kicked Ising model for up to ten Heisenberg times and at four different values of the transverse field $h$. The black dashed lines show predictions of the random phase model, while dotted (solid) lines give the Poissonian (RMT OE) result. The colored dots show numerical data averaged over 100 realizations of $J$ sampled uniformly in the interval [5.5, 55], for the kicked Ising model introduced in Eq. (26). In the inset we show that averaging is only necessary for very short times. Thin blue lines are particular realizations of the model for $J=10,11,\dots 19$ and the shaded area shows the second to ninth decile assuming exponential distribution for realizations of $K(t)$ resulting in a hypoexponential distribution for the integrated spectral form factor. Other parameters are fixed to $\ell =14$, $a=1$, $b=5$, $\alpha =1.5$.

Figure 2

Spectral form factor for the kicked Ising model at short times. (a) Comparison of a moving average over ten consecutive time steps and fixed model realization ($J=10$) with the averaging over ten realizations $J=10,11,\dots ,19$. All other parameters are fixed to $\ell =14$, $a=1$, $b=5$, $\alpha =1.5$. Shaded area shows the second to ninth decile assuming exponential distribution for realizations of $K(t)$ resulting in the gamma distribution for the averages. (b) $J$-averaged spectral form factor (now averaged over a sample of 100 values of $J\in [5.5,55]$) at short times shows deviations from the RMT, and the deviations are well captured by the random phase model.

Figure 3

(a) Spectral form factor for the kicked Ising model at short times for different values of locality exponent $\alpha$. When the model becomes increasingly short ranged, it develops deviations at short times different from the prediction of the random phase model. (b) The order parameter as defined in Eq. (29) for different values of $\alpha$, which shows that the model starts to develop deviations from the random phase model when $\alpha \lesssim 0.5$ or $\alpha \gtrsim 2.5$. The shaded area guides the eye to the section with small, statistically insignificant deviations. Averaging over 500 realizations of $J\in [5.5,255]$ is performed, other parameters are fixed to $\ell =14$, $a=1$, $b=5$, and $t_{\max }=100$.

Figure 4

Partition function $Z_{\pi }$ of the twisted one-dimensional Ising model for different permutation families is plotted against the field parameter $h$. Since in the expression the contribution occurs at a large power $\ell$, only the largest contribution matters in the first order (red), which corresponds to (anti)cyclic permutations. The first subleading corrections are given by the $X$ diagrams for $\tau =1$, 2. The colored lines are numerical data for $t=13$, 14 shown in (a) and (b), respectively. Dashed lines are the exact expressions for $Z_X(\tau )$ given by Eq. (a1). Note that all terms have multiplicities which are multiples of $2t$.