We investigate timelike entanglement measures derived from the spacetime density kernel in the Rosenzweig-Porter model and show that they sharply diagnose both eigenvector ergodicity and spectral chaos. For several Hilbert-space bipartitions, we compute the second Tsallis entropy, the entanglement imagitivity that quantifies non-Hermiticity, and Schatten-norm diagnostics of the kernel. The imagitivity and Frobenius norm exhibit rapid growth and high late-time plateaus in the ergodic regime, are suppressed in the localized regime, and show intermediate behavior in the fractal phase. The real part of the second Tsallis entropy displays a spectral form factor-like dip-ramp-plateau throughout the chaotic window and a suppressed ramp in the localized regime. We further introduce a kernel negativity, defined as the negative spectral weight of the Hermitian part of the kernel. This negativity equals the trace-norm distance to the set of positive semidefinite operators and the maximal witnessable negative quasiprobability, and its time-averaged value decreases across the ergodic-fractal-localized crossover in close correspondence with the fractal dimension.
We investigate timelike entanglement measures derived from the spacetime density kernel in the Rosenzweig-Porter model and show that they sharply diagnose both eigenvector ergodicity and spectral chaos. For several Hilbert-space bipartitions, we compute the second Tsallis entropy, the entanglement imagitivity that quantifies non-Hermiticity, and Schatten-norm diagnostics of the kernel. The imagitivity and Frobenius norm exhibit rapid growth and high late-time plateaus in the ergodic regime, are suppressed in the localized regime, and show intermediate behavior in the fractal phase. The real part of the second Tsallis entropy displays a spectral form factor-like dip-ramp-plateau throughout the chaotic window and a suppressed ramp in the localized regime. We further introduce a kernel negativity, defined as the negative spectral weight of the Hermitian part of the kernel. This negativity equals the trace-norm distance to the set of positive semidefinite operators and the maximal witnessable negative quasiprobability, and its time-averaged value decreases across the ergodic-fractal-localized crossover in close correspondence with the fractal dimension.
A bstract We develop a finite-dimensional formulation of the recently introduced notion of “timelike entanglement”, defined in terms of two-point functions between operators supported on different Cauchy slices. Using a local orthonormal operator basis, we recast this construction in terms of a generalized response tensor . Building on this, we introduce a generalized spacetime density kernel (GSDK) corresponding to higher point correlation functions, including time-ordered as well as out-of-time-ordered correlators. Motivated by the structure of Haar probe averaging, we contract the (2 N )-leg GSDK with cyclic permutation operators and thereby define a kernel form factor . This quantity is computable even when Haar averaging is not feasible, and it reduces to the (2 N )-th moment of the spectral form factor, evaluated at an N -enhanced effective temperature. The correlation functions of the GSDK operators also yield the SFF, with an effective (1/ N )-reduction of the physical time-scales. The GSDK places both early time scrambling diagnostics and late time spectral statistics on a similar footing and clarifies how higher-point correlators and non-trivial time ordering capture fine-grained dynamical information of a quantum system.
We develop a finite-dimensional formulation of the recently introduced notion of ``timelike entanglement'', defined in terms of two-point functions between operators supported on different Cauchy slices. Using a local orthonormal operator basis, we recast this construction in terms of a generalized response tensor. Building on this, we introduce a generalized spacetime density kernel (GSDK) corresponding to higher-point correlation functions, including time-ordered as well as out-of-time-ordered correlators. We show that the Haar-averaged (2N)-point function yields the (2N)-th moment of the spectral form factor (SFF), evaluated at an N-enhanced effective temperature. The correlation functions of the GSDK operators also yield the SFF, with an effective (1/N)-reduction of the physical time-scales. The GSDK places both scrambling diagnostics and spectral statistics on a similar footing and clarifies how higher-point correlators and non-trivial time ordering capture fine-grained dynamical information of a quantum system.
A bstract Utilizing the framework of free probability, we analyze the spectral and operator statistics of the Rosenzweig-Porter random matrix ensembles, which exhibit a rich phase structure encompassing ergodic, fractal, and localized regimes. Leveraging subordination formulae, we develop a perturbative scheme that yields semi-analytic expressions for the density of states up to second order in system size, in good agreement with numerical results. We compute higher-point correlation functions in the ergodic regime using both numerical and suitable analytic approximations. Our analysis of operator statistics for various spin operators across these regimes reveals close agreement with free probability predictions in the ergodic phase, in contrast to persistent deviations observed in the fractal and localized phases, even at late times. Notably, the fractal phase exhibits partial freeness while retaining memory of the initial spectrum, highlighting the importance of non-localized eigenstates and associated with the late-time dynamics of cumulative out-of-time-ordered-correlators (OTOCs). Employing distance measures and statistical tools such as the χ 2 statistic, Kullback-Leibler divergence, and Kolmogorov-Smirnov hypothesis testing, we define a characteristic time scale — the free time — that marks the onset of the validity of free probability predictions for operator spectral statistics in the ergodic phase. Remarkably, our findings demonstrate consistency across these different approaches.
In this work, we study pseudo entropy and spectral dynamics in quantum spin chains (Heisenberg XXZ model and the mixed-field Ising model) to investigate whether they can serve as diagnostics of quantum chaos. First, we derive an exact relation between the real part of the thermal pseudo entropy and the spectral form factor. This provided evidence that pseudo entropy could exhibit the characteristic dip, logarithmic ramp, and plateau, implying the precise manner in which pseudo entropy probes chaos. We found that the imaginary part, while it does not diagnose chaos, can still provide crucial information about Fisher zeros that is not available in the real counterpart. For spatial pseudo entropy in the XXZ chain, we find that the logarithmic critical scaling persists in both integrable and chaotic regimes, and that the non-positivity conjecture holds. Finally, we derive an exact connection between pseudo entropy, relative pseudo entropy, and spectral complexity with a potential connection to holography.
We numerically investigate measurement-induced phase transitions in monitored free fermions through the spectral and eigenstate properties of the entanglement Hamiltonian. By analyzing entanglement scaling, we identify three non-trivial fixed points, for which the entanglement follows exact analytical scaling forms: chaotic unitary dynamics at infinitesimal monitoring, characterized by a Gaussian Page law; a Fermi-liquid fixed point at moderate monitoring, defining a metallic phase with logarithmic entanglement growth and emergent space-time invariance; and a quantum Lifshitz fixed point marking the measurement-induced phase transition into a localized area-law phase. Adopting a random-matrix perspective on the entanglement Hamiltonian, we show that short-range spectral correlations, such as the adjacent gap ratio ⟨r~⟩ and the Kullback-Leibler divergence KL1, sharply detect an ergodic to non-ergodic phase transition at the quantum Lifshitz fixed point, and yield precise estimates of the critical point and correlation length exponent. Long-range probes, including the spectral form factor and the associated Thouless time, corroborate this picture, while the variant KL2 uncovers signatures of a possible non-ergodic extended (multifractal) regime at intermediate monitoring strengths. Together, these results establish the entanglement Hamiltonian as a powerful framework for diagnosing metallic, localized, and multifractal regimes in monitored quantum dynamics, and highlight unexpected scaling structures, Fermi-liquid and Lifshitz criticality, that lie beyond current field-theoretic approaches.
Glasses have the interesting feature of being neither integrable nor fully chaotic. They thermalize quickly within a subspace but thermalize much more slowly across the full space due to high free energy barriers which partition the configuration space into sectors. Past works have examined the Rosenzweig-Porter (RP) model as a minimal quantum model which transitions from localized to chaotic behavior. In this work we generalize the RP model in such a way that it becomes a minimal model which transitions from glassy to chaotic behavior, which we term the “Block Rosenzweig-Porter” (BRP) model. We calculate the spectral form factors of both models at all timescales larger than the inverse spectral width. Whereas the RP model exhibits a crossover from localized to ergodic behavior at the Thouless timescale, the new BRP model instead crosses over from glassy to fully chaotic behavior, as seen by a change in the steepness of the ramp of the spectral form factor.