Abstract
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 $K(t)$ 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 $K(t)$ explicitly 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.
Highlights
Random matrix theory (RMT) was introduced into physics in the 1950s by Wigner [1] to provide a statistical description of nuclear resonance or excitation spectra
Since the random phase model (RPM) prediction for long times becomes equivalent to RMT and is expected to match KðtÞ well, provided the model is nonintegrable and ergodic, the order parameter ψ becomes independent of tmax as long as tà ≪ tmax ≪ tH
Our results have a direct relevance for understanding the vast body of numerical experiments, simulations, and in the near future possibly experimental spectra of highly excited simple many-body systems, which correspond to ever longer accessible observation times of perfectly coherent out-of-equilibrium quantum systems
Summary
Random matrix theory (RMT) was introduced into physics in the 1950s by Wigner [1] to provide a statistical description of nuclear resonance or excitation spectra. In the early 1980s a much more surprising fact was revealed, namely, that RMT works extremely well for capturing spectral fluctuations of simple single-particle systems whose corresponding classical dynamics are completely chaotic, such as dispersive (Sinai) billiards or hydrogen or Rydberg atoms in external magnetic or microwave fields These observations [4,5,6], termed the quantum chaos conjecture, which has been concisely stated in. Recent studies of out-of-time-ordered correlations in many-body systems, some of which establish exponential growth [in particular in (0 þ 1)-dimensional systems such as the Sachdev-Ye-Kitaev model], have no clear connection to Lyapunov instability, as it is understood in classical dynamical systems theory, and is the only mathematically meaningful definition of chaos This has to do with the lack of the concept of classical orbits and the corresponding unstable (nonlinear) equations of motion, which result in sensitive dependence on initial conditions (e.g., the butterfly effect). We identify the nonsemiclassical analog of the Sieber-Richter pairing mechanism [11] and exactly reproduce the subleading RMT term −2t2=tH as well
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