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arXiv:2608.29810

Pseudo Entropy in Quantum Spin Chains: from Integrability to Chaos

Tara Bahadur Rana, Yadav Raj Dahal, Kiran Adhikari

Published Aug 30, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2608.29810

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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.
A bstract We explore spread and spectral complexity in quantum systems that exhibit a transition from integrability to chaos, namely the mixed-field Ising model and the next-to-nearest-neighbor deformation of the Heisenberg XXZ spin chain. We corroborate the observation that the presence of a peak in spread complexity before its saturation, is a characteristic feature in chaotic systems. We find that, in general, the saturation value of spread complexity post-peak depends not only on the spectral statistics of the Hamiltonian, but also on the specific state. However, there appears to be a maximal universal bound determined by the symmetries and dimension of the Hamiltonian, which is realized by the thermofield double state (TFD) at infinite temperature. We also find that the time scales at which the spread complexity and spectral form factor change their behaviour agree with each other and are independent of the chaotic properties of the systems. In the case of spectral complexity, we identify that the key factor determining its saturation value and timescale in chaotic systems is given by minimum energy difference in the theory’s spectrum. This explains observations made in the literature regarding its earlier saturation in chaotic systems compared to their integrable counterparts. We conclude by discussing the properties of the TFD which, we conjecture, make it suitable for probing signatures of chaos in quantum many-body systems.
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A bstract Quantum informatic quantities such as entanglement entropy are useful in detecting quantum phase transitions. Recently, a new entanglement measure called pseudo-entropy was proposed which is a generalization of the more well-known entanglement entropy. It has many nice properties and is useful in the study of post-selection measurements. In this paper, one of our goals is to explore the properties of pseudo-entropy and study its effectiveness as a quantum chaos diagnostic, i.e., as a tool to distinguish between chaotic and integrable systems. Using various variants of the SYK model, we study the signal of quantum chaos captured in the pseudo-entropy and relate it to the spectral form factor (SFF) and local operator entanglement (LOE).
We investigate the integrability-to-chaos transition and information scrambling in Ising spin networks via a graph-theoretic formulation. Modeling spins as vertices and interactions as edges encoded by adjacency matrices across path, Erdos-Renyi, and Watts-Strogatz topologies, we demonstrate that long-range couplings and heterogeneous degree distributions markedly accelerate quantum information propagation. The Hamiltonian comprises local and normalized non-local interactions; tuning the non-local coupling and field heterogeneity drives integrability breaking. To quantify scrambling, we employ bipartite mutual and tripartite information. Increasing non-local interactions drives tripartite information to large negative values, signaling deep information scrambling. The squared commutators constructed from out-of-time-order correlators (OTOCs) exhibit early-time exponential growth; equivalently, the OTOC four-point functions themselves decay exponentially, yielding quantum Lyapunov exponents that scale systematically with parameters governing the chaotic regime. Complementing this, Krylov complexity reveals rapid operator growth in the chaotic phase, synchronizing with OTOC and mutual information dynamics. Spectrally, the transition manifests as a shift from Poissonian to Wigner-Dyson level spacing statistics. The spectral form factor (SFF) exhibits the characteristic slope-dip-ramp-plateau structure, enabling the extraction of Thouless and Heisenberg times. Crucially, a reduced Thouless time strongly correlates with accelerated information and operator scrambling. Ultimately, this work establishes a unified framework bridging network topology with information-theoretic, operator, and spectral diagnostics, offering insights into thermalization and non-equilibrium dynamics in quantum many-body systems.
Entanglement entropy has proven to be a powerful probe of phenomena such as quantum chaos and phase transitions. Pseudo entropy is a recently proposed time-like generalization of an entanglement measure, motivated by de Sitter holography. In this work, we find that pseudo entropy can also serve as a novel probe for distinguishing topological phases of matter. For this, we consider the Su--Schrieffer--Heeger model as a representative example and investigate the averaged excess entropy ΔS12ΔS_{12}ΔS12​, defined as the difference between pseudo entropy and the average entanglement entropy, across the topological-to-trivial and trivial-to-topological phase transitions. When the two states are in the same phase, we find that $ ΔS_{12}$ is non-positive under periodic boundary conditions, while for open boundary conditions, it is non-positive only when the system is sufficiently large. Moreover, we analyze ground-state quench protocols for topology-crossing quenches and find that the imaginary pseudo entropy tracks the critical times predicted by the Fisher zeros.