We study the multidimensional erratic structure of correlation functions produced by local operator quenches in finite-volume free massive scalar field theory in dimensions 2 and 3. The basic observable is the subtracted equal-time two-point function in the locally excited state and its spatiotemporal patterns of extrema. We analyze these extrema by the multidimensional diagnostics recently introduced for chaotic scattering amplitudes and related problems: all-pair distance distributions, nearest-neighbor spacings, greedy-path spacing ratios, and the extrema form factor. For the
1+1-dimensional local quench we find that, in the regime of small Euclidean smearing, the fitted extremum statistics move close to the
β=1 random-matrix benchmark, while increasing the smearing scale softens the effective repulsion and moves the distributions away from the GOE-like value. For the
2+1-dimensional local quench we find that the nearest-neighbor statistics of the refined extrema are close to, or above, the
β=1 benchmark, and the greedy-path ratio statistics are described by even larger effective
β values. Finally we studied the all-pair extrema spatial form factor and found that, in the one-, two-, and three-dimensional cases, its main structure is controlled by the corresponding uniform interval, rectangle, or cuboid geometry of the extrema cloud and found the dip-ram-plateau structure in the last two cases. Thus the form factor provides a complementary global diagnostic of how the extrema fill their effective metric support, while the genuinely nontrivial local and mesoscopic organization is carried by the nearest-neighbor and greedy-path statistics.
We numerically study the spectral statistics of open quantum many-body systems (OQMBS) as signatures of quantum chaos (or the lack thereof), using the dissipative spectral form factor (DSFF), a generalization of the spectral form factor to complex spectra. We show that the DSFF of chaotic OQMBS generically displays the
quadratic ramp-plateau behaviour of the Ginibre ensemble from random matrix theory, in contrast to the linear ramp-plateau behaviour of the Gaussian ensemble in closed quantum systems. Furthermore, in the presence of many-body interactions, such RMT behaviour emerges only after a time scale
τdev, which generally increases with system size for sufficiently large system size, and can be identified as the non-Hermitian analogue of the
many-body Thouless time. The universality of the random matrix theory behavior is demonstrated by surveying twelve models of OQMBS, including random Kraus circuits (quantum channels) and random Lindbladians (Liouvillians) in several symmetry classes, as well as Lindbladians of paradigmatic models such as the Sachdev-Ye-Kitaev (SYK), XXZ, and the transverse field Ising models. We devise an unfolding and filtering procedure to remove variations of the averaged density of states which would otherwise hide the universal RMT-like signatures in the DSFF for chaotic OQMBS. Beyond chaotic OQMBS, we study the spectral statistics of non-chaotic OQMBS, specifically the integrable XX model and a system in the many-body localized (MBL) regime in the presence of dissipation, which exhibit DSFF behaviours distinct from the ramp-plateau behaviour of random matrix theory. Lastly, we study the DSFF of Lindbladians with the Hamiltonian term set to zero, i.e. only the jump operators are present, and demonstrate that the results of RMT universality and scaling of many-body Thouless time survive even without coherent evolution.
We study the behaviour of two-time correlation functions at late times for finite system sizes considering observables whose (one-point) average value does not depend on energy. In the long time limit, we show that such correlation functions display a ramp and a plateau determined by the correlations of energy levels, similar to what is already known for the spectral form factor. The plateau value is determined, in absence of degenerate energy levels, by the fluctuations of diagonal matrix elements, which highlights differences between different symmetry classes. We show this behaviour analytically by employing results from Random Matrix Theory and the Eigenstate Thermalisation Hypothesis, and numerically by exact diagonalization in the toy example of a Hamiltonian drawn from a Random Matrix ensemble and in a more realistic example of disordered spin glasses at high temperature. Importantly, correlation functions in the ramp regime do not show self-averaging behaviour, and, at difference with the spectral form factor the time average does not coincide with the ensemble average.
The Yukawa-Sachdev-Ye-Kitaev (YSYK) model, a solvable model of non-Fermi liquids with a proposed cavity-QED realization, couples
N complex fermions to
M bosons of frequency
ω0 through random Yukawa couplings of strength
g, and its dynamics depends only on
R≡ω0/g2/3. For
R≪1 the bosons are slow and act as a static random background in which the fermions are free, while for
R≫1 they are only virtually excited and generate a random four-fermion interaction of rank
M, which is generically chaotic for
M≥2. Using exact diagonalization at infinite temperature, we follow the emergence of random-matrix statistics between these limits with unfolded spectral statistics and the Krylov complexity of the thermofield-double state. For complex couplings the level-spacing distribution evolves from Poisson to Gaussian-unitary-ensemble (GUE) statistics, the spectral form factor develops a correlation hole and a ramp, and the Krylov complexity evaluated on the unfolded spectrum develops a peak above its late-time plateau whose height reaches the GUE value. All diagnostics locate the onset around
R≈0.3, and adding boson modes moves it to smaller
R. This delineates the parameter regime in which a cavity-QED realization would display random-matrix spectral correlations. On the other hand, real couplings give the same crossover towards Gaussian-orthogonal-ensemble (GOE) statistics, and along an interpolation between the two the peak height follows the mean gap ratio, so the peak identifies the symmetry class as well as the onset of chaos. Without unfolding the peak stays about 20
% below its random-matrix value in both classes, which we trace to the non-semicircular density of states, and at larger
R the boson-number bands slow down the spreading of the state in Krylov space.
Abstract The spectral form factor is a powerful probe of quantum chaos that diagnoses the statistics of energy levels, but is blind to other features of a theory such as matrix elements of operators or OPE coefficients in conformal field theories. In this paper, we introduce generalized spectral form factors: new probes of quantum chaos sensitive to the dynamical data of a theory. These quantities can be studied using random matrix theory and an effective theory of quantum chaos. We focus our attention on a particular combination of heavy-heavy-heavy OPE coefficients that generalizes the genus-2 partition function of two-dimensional CFTs, for which we define a form factor. Assuming that random matrix theory applies to chaotic CFTs, we probe heavy-heavy-heavy OPE coefficients and find statistical correlations that agree with the OPE Randomness Hypothesis: these coefficients have a random tensor component. The EFT of quantum chaos predicts that the genus-2 form factor displays a ramp and a plateau. Our results suggest that this is a common property of generalized spectral form factors.
We introduce randomness into a class of integrable models and study the spectral form factor as a diagnostic to distinguish between randomness and chaos. Spectral form factors exhibit a characteristic dip-ramp-plateau behavior in the N>2 SYK2 model at high temperatures that is absent in the N=2 SYK2 model. Our results suggest that this dip-ramp-plateau behavior implies the existence of random eigenvectors in a quantum many-body system. To further support this observation, we examine the Gaussian random transverse Ising model and obtain consistent results without suffering from small N issues. Finally, we demonstrate numerically that expectation values of observables computed in a random quantum state at late times are equivalent to the expectation values computed in the thermal ensemble in a Gaussian random one-qubit model.