We study how two distinct signatures of quantum chaos, the level spacing ratio ⟨r⟩and a four- eigenstate correlator (FESC), behave as integrability is gradually broken in the disordered spin-1/2 Heisenberg chain. Working within the Stot z = 0 sector at system sizes N = 12, 14, and 16 via exact diagonalization, we observe that these two diagnostics do not track each other across the crossover. The level spacing ratio settles into Wigner-Dyson statistics already at rather weak disorder, but the FESC stays sparse, well below its random-matrix expectation, over a wide range of disorder strengths, and never clearly approaches a dense regime before many-body localization sets in. Taken together, these observations point to a finite-size window in which the energy spectrum looks chaotic while the eigenstates still carry residual structure from integrability. The FESC thus serves as a sharper probe than ⟨r⟩ alone, picking up correlations that standard spectral measures miss.
Out-of-time-ordered correlators (OTOCs) have emerged as a diagnostic of information scrambling and quantum chaos in many-body systems. We investigate the imprints of chaos in the dynamics of OTOCs in the Heisenberg spin-
1/2 chain with random fields. The system is parameterized to exhibit a crossover from integrable to chaotic dynamics. We demonstrate numerically that the approach to saturation of the OTOC can distinguish between integrable and chaotic regimes, with a power-law
(1/t) relaxation for integrable systems and a higher-degree power-law decay
(1/tα;α≥1) followed by an exponential relaxation for the chaotic regime. We further show that long-range spectral statistics, such as the number variance, are more effective in characterizing quantum chaos in the regime near saturation of OTOC. We also demonstrate that the relaxation and initial scrambling regimes exhibit distinct and universal features, with the former being sensitive and the latter being robust against different realizations of random-fields. The long-time saturation of OTOC also fluctuates with different realizations, and its exact expression is derived through the Eigenstate Thermalization Hypothesis.
The Eigenstate Thermalization Hypothesis explains thermalization in isolated quantum systems through the statistical properties of observables in the energy eigenbasis. We investigate the crossover from integrability to chaos in the spin-
1/2 XXZ chain, establishing a direct correspondence between the spectral correlations of the Hamiltonian and local observables expressed in the energy eigenbasis as a signature of ergodicity breaking. By introducing a local perturbation that drives the system from integrability to chaos, we track the standard ETH indicators and the eigenstate entanglement entropy. We introduce a submatrix-based framework for analyzing local observables in the energy eigenbasis. By extracting real-symmetric blocks along the diagonal of the local observables represented in eigenbasis, we show that these submatrices exhibit both the short-range and long-range spectral features of the Hamiltonian. Remarkably, this correspondence persists even in a partially ergodic regime, indicating that the emergence of chaos is already encoded locally within the observables' matrix structure and that small blocks are sufficient to capture the underlying spectral correlations.
We investigate disordered oneand two-dimensional Heisenberg spin lattices across the transition from integrability to quantum chaos from both statistical many-body and quantum-information perspectives. Special emphasis is devoted to quantitatively exploring the interplay between eigenvector statistics, delocalization, and entanglement in the presence of nontrivial symmetries. The implication of the basis dependence of state delocalization indicators such as the number of principal components is addressed, and a measure of relative delocalization is proposed in order to robustly characterize the onset of chaos in the presence of disorder. Both standard multipartite and generalized entanglement are investigated in a wide parameter regime by using a family of spinand fermion-purity measures, their dependence on delocalization and on energy spectrum statistics being examined. A distinctive correlation between entanglement, delocalization, and integrability is uncovered, which may be generic to systems described by the two-body random ensemble and may point to a new diagnostic tool for quantum chaos. Analytical estimates for typical entanglement of random pure states restricted to a proper subspace of the full Hilbert space are also established and compared with random matrix theory predictions.
We study signatures of quantum chaos in finite-loop truncations of the planar dilatation operator in the
su(2) sector of
N=4 super Yang-Mills and its
β-deformation. These truncations define holographically motivated long-range deformations of the nearest-neighbour XXX spin chain. At one-loop the model is integrable, while the all-loop planar theory is expected to again be integrable. Finite-loop truncations therefore provide a natural setting for investigating how chaotic behaviour emerges between these two integrable limits. We analyse this question using spectral statistics, eigenvector diagnostics and spread complexity. We find that the two- and four-loop truncations develop GOE-like level statistics at sufficiently large coupling but with features characteristic of weak integrability breaking. The integrability breaking at four-loops is weaker than at two-loops and the critical coupling at which chaos occurs is larger, at least for long spin chains. The three-loop truncation does not show the same onset of chaos in the range studied. Eigenvector diagnostics show that the corresponding eigenstates remain less random than GOE vectors, indicating weak ergodicity and multifractality. Finally, we can identify signatures of the eigenvalue and eigenvector chaos in the Krylov-space data. Namely, we demonstrate a correlation of the level spacing statistics with the peak of spread complexity and disorder on the Krylov chain. The delocalisation of the initial state in the Hamiltonian eigenbasis is shown to strongly affect the saturation of complexity. Our results suggest that finite-loop dilatation operators are not generic long-range spin chain Hamiltonians, but already display patterns consistent with the restoration of integrability in the all-loop planar theory.
We investigate several distinct spectral crossovers amongst various integrable (Poissonian) and quantum-chaotic (Wigner-Dyson) limits of a
1D disordered quantum spin (
S=1/2) model, by tuning the relative amplitudes of various Hamiltonian parameters to retain or break relevant unitary and antiunitary symmetries. The spin model consists of an isotropic and deterministic Heisenberg term, a random Ising term, an anisotropic and antisymmetric, but deterministic Dzyaloshinskii-Moriya (DM) term, and a Zeeman coupling to a random, inhomogeneous magnetic field. Since we are specially interested in crossovers involving a Gaussian symplectic ensemble (GSE) limit, we carry out all our calculations with an odd number of lattice sites (spins) that naturally results in eigenspectra with Kramers degeneracies (KD's). The various crossovers (viz., the reentrant Poissonian-to-GSE-to-Poissonian, Poissonian-to-GUE, GSE-to-GUE and the reentrant Poissonian-to-GOE-to-Poissonian crossovers) are investigated via detailed studies of both short-range (nearest-neighbour-spacings distribution, NNSD) and long-range [spectral rigidity ${\mathrm{\ensuremath{\Delta}}}_{3}(L)$ and number variance ${\mathrm{\ensuremath{\Sigma}}}^{2}(L)$ ] spectral correlations, where
L is the spectral interval over which the long-range statistic is examined. The short-range studies show excellent agreement with RMT predictions. One of the highlights of this study is the systematic investigation of the consequences of retaining both eigenvalues corresponding to every Kramers doublet, in a crossover involving the GSE limit, and see how it evolves to a limit where the KD is naturally lifted. This is seen most clearly in the NNSD study of the GSE-to-GUE (Gaussian unitary ensemble) transition, achieved by gradually lifting the KD, using the random magnetic field. The NNSD plot in the GSE limit here exhibits a Dirac delta peak at zero splitting and a renormalized GSE hump at finite splitting, whose general analytical form and its asymptotic limit are derived. With an increasing symmetry breaking magnetic field the NNSD shows an interesting, dynamic two-peaked structure that finally converges to the standard GUE lineshape. We explain this trend in terms of a competition between the splittings amongst distinct Kramers doublets (related to unitary symmetries) and the Zeeman-like splittings induced by a breaking of the antiunitary time-reversal symmetry (TRS). This is investigated via the NNSD, the marginal spectral density (MSD) and the densities of states (DOS) for both spin models and RMT crossover matrix models. The first and the final short-range studies involve reentrant Poissonian-to-GOE(GSE)-to-Poissonian crossovers, where the final Poissonian is obtained by a many-body localization of states in the strongly disordered limit, whereas the initial Poissonian regime involve much more delocalized eigenstates. In the long-range spectral correlation studies, we shed light on the extent of agreement between our physical spin systems and RMT predictions. We find that the spin systems depart from the ideal RMT predictions for relatively finite $L\ensuremath{\sim}10\ensuremath{-}15$ at least, for the spectral rigidity and a much smaller $L\ensuremath{\sim}2\ensuremath{-}4$ for the number variance. It is further seen that the departure is usually sooner at the uncorrelated (Poissonian) upper end compared to the correlated (Wigner-Dyson) lower end. We carry out a detailed comparison between the local and the global crossover points, associated with the short-range and the long-range statistics respectively, and find that in most cases they seem to agree reasonably well, but for a few exceptions. Our studies also show that the long-range correlations may serve to distinguish between the two Poissonian limits (nonlocalized and localized) in the reentrant crossovers, which the short-range correlations fail to distinguish.
We investigate disordered one- and two-dimensional Heisenberg spin lattices across the transition from integrability to quantum chaos from both statistical many-body and quantum-information perspectives. Special emphasis is devoted to quantitatively exploring the interplay between eigenvector statistics, delocalization, and entanglement in the presence of nontrivial symmetries. The implication of the basis dependence of state delocalization indicators (such as the number of principal components) is addressed, and a measure of relative delocalization is proposed in order to robustly characterize the onset of chaos in the presence of disorder. Both standard multipartite and generalized entanglement are investigated in a wide parameter regime by using a family of spin- and fermion-purity measures, their dependence on delocalization and on energy spectrum statistics being examined. A distinctive correlation between entanglement, delocalization, and integrability is uncovered, which may be generic to systems described by the two-body random ensemble and may point to a new diagnostic tool for quantum chaos. Analytical estimates for typical entanglement of random pure states restricted to a proper subspace of the full Hilbert space are also established and compared with random matrix theory predictions.