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arXiv:2601.16177

Exact Thermal Stabilizer Eigenstates at Infinite Temperature

Akihiro Hokkyo

quant-phcond-mat.stat-mechSubmitted Jan 22, 2026• Announced Jan 23, 2026• 0 citations• Revised Jan 29, 2026 (v2)
v1 · Jan 22, 2026v2 · Jan 29, 2026
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Abstract

Understanding how microscopic few-body interactions give rise to thermal behavior in isolated quantum many-body systems remains a central challenge in nonequilibrium statistical mechanics. While individual energy eigenstates are expected to reproduce thermal equilibrium values, analytic access to highly entangled thermal eigenstates of nonintegrable Hamiltonians remains scarce. In this Letter, we construct exact infinite-temperature eigenstates of generically nonintegrable two-body Hamiltonians using stabilizer states. These states can fully reproduce thermal expectation values for all spatially local observables, extending previously known Bell-pair-based constructions to a broader class. At the same time, we prove a sharp no-go theorem: stabilizer eigenstates of two-body Hamiltonians cannot satisfy microscopic thermal equilibrium for all four-body observables. This bound is tight, as we explicitly construct a translationally invariant Hamiltonian whose stabilizer eigenstate is thermal for all two-body and three-body observables as well as all spatially local observables. Our results suggest that reproducing higher-order thermal correlations requires nonstabilizer degrees of freedom, providing analytic insight into the interplay between interaction locality, microscopic thermal equilibrium, and quantum computational complexity.

Comments
6+2 pages, 3 figures; Figure 3 (numerical calculation of energy-level spacings) has been added

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Understanding how microscopic few-body interactions give rise to thermal behavior in isolated quantum many-body systems remains a central challenge in nonequilibrium statistical mechanics. While individual energy eigenstates are expected to reproduce thermal equilibrium values, analytic access to highly entangled thermal eigenstates of nonintegrable Hamiltonians remains scarce. In this Letter, we construct exact infinite-temperature eigenstates of generically nonintegrable two-body Hamiltonians using stabilizer states. These states can fully reproduce thermal expectation values for all spatially local observables, extending previously known Bell-pair-based constructions to a broader class. At the same time, we prove a sharp no-go theorem: stabilizer eigenstates of two-body Hamiltonians cannot satisfy microscopic thermal equilibrium for all four-body observables. This bound is tight, as we explicitly construct a translationally invariant Hamiltonian whose stabilizer eigenstate is thermal for all two-body and three-body observables as well as all spatially local observables. Our results suggest that reproducing higher-order thermal correlations requires nonstabilizer degrees of freedom, providing analytic insight into the interplay between interaction locality, microscopic thermal equilibrium, and quantum computational complexity.
The eigenstate thermalization hypothesis (ETH) plays a major role in explaining thermalization of isolated quantum many-body systems. However, there has been no proof of the ETH in realistic systems due to the difficulty in the theoretical treatment of thermal energy eigenstates of nonintegrable systems. Here, we write down analytically thermal eigenstates of nonintegrable spin chains. We consider a class of theoretically tractable volume-law states, which we call entangled antipodal pair (EAP) states. These states are thermal, in the most fundamental sense that they are indistinguishable from the Gibbs state with respect to all local observables, with infinite temperature. We then identify Hamiltonians having the EAP state as an eigenstate and rigorously show that some of these Hamiltonians are nonintegrable. Furthermore, a thermal pure state at an arbitrary temperature is obtained by the imaginary time evolution of an EAP state. Our results offer a potential avenue for providing a provable example of the ETH.
Nonintegrable many-body quantum systems typically thermalize at long times through the mechanism of quantum chaos. However, some exceptional systems, such as those harboring quantum scars, break thermalization, serving as testbeds for foundational problems of quantum statistical physics. Here, we investigate a class of nonintegrable bond-staggered models that is endowed with a large number of zero-energy eigenstates and possesses a non-Abelian internal symmetry. We use character theory to give a lower bound on the zero-energy degeneracy, which matches exact diagonalization results and is found to grow exponentially with the system size. We also show that few-magnon zero-energy states have an exact analytical description, allowing us to build a basis of low-entangled fixed-separation states, which is stable to most perturbations found in experiments. This remarkable dynamical stability of special states elucidates our understanding of nonequilibrium processes in non-Abelian chaotic quantum models.
Deriving conditions under which a macroscopic system thermalizes directly from the underlying quantum many-body dynamics of its microscopic constituents is a long-standing challenge in theoretical physics. The well-known eigenstate thermalization hypothesis (ETH) is presumed to be a key mechanism, but has defied rigorous verification for generic systems thus far. A weaker variant (weak ETH), by contrast, is provably true for a large variety of systems, including even many integrable models, but its implications with respect to the problem of thermalization are still largely unexplored. Here we analytically demonstrate that systems satisfying the weak ETH exhibit thermalization for two very natural classes of far-from-equilibrium initial conditions: the overwhelming majority of all pure states with a preset nonequilibrium expectation value of some given local observable, and the Gibbs states of a Hamiltonian which subsequently is subject to a quantum quench in the form of a sudden change of some local system properties.
The eigenstate thermalization hypothesis (ETH) posits how isolated quantum many-body systems thermalize, assuming that individual eigenstates at the same energy density have identical expectation values of local observables in the limit of large systems. While the ETH apparently holds across a wide range of interacting quantum systems, in this work we show that it requires generalization in the presence of thermal first-order phase transitions. We introduce a class of all-to-all spin models, featuring first-order thermal phase transitions that stem from two distinct mean-field solutions (two ``branches'') that exchange dominance in the many-body density of states as the energy is varied. We argue that for energies in the vicinity of the thermal phase transition, eigenstate expectation values do not need to converge to the same thermal value. The system has a regime with coexistence of two classes of eigenstates corresponding to the two branches with distinct expectation values at the same energy density, and another regime with Schrodinger-cat-like eigenstates that are inter-branch superpositions; these two regimes are separated by an eigenstate phase transition. We support our results by semiclassical calculations and an exact diagonalization study of a microscopic spin model, and argue that the structure of eigenstates in the vicinity of thermal first-order phase transitions can be experimentally probed via non-equilibrium dynamics.
The eigenstate thermalization hypothesis (ETH), which asserts that every eigenstate of a many-body quantum system is indistinguishable from a thermal ensemble, plays a pivotal role in understanding thermalization of isolated quantum systems. Yet, no evidence has been obtained as to whether the ETH holds for all few-body operators in a chaotic system; such few-body operators include key quantities in statistical mechanics, such as the total magnetization, the momentum distributions, and their low-order thermal and quantum fluctuations. Here, we formulate a conjecture that for a generic nonintegrable system the ETH holds simultaneously for all mmm-body operators with m<α∗Nm < α_{\ast} Nm<α∗​N in the thermodynamic limit for some nonzero constant α∗>0α_{\ast} > 0α∗​>0. We first show the existence of such nontrivial constants for idealized (pseudo) random-matrix descriptions of many-body eigenstates. We then verify the conjecture for generic spin, Bose, and Fermi systems with local and few-body interactions by large-scale numerical calculations. Our results imply that generic systems satisfy the ETH simultaneously for all few-body operators, including their thermal and quantum fluctuations.
Thermalization conventionally describes local properties of isolated many-body systems at equilibrium. Magic, or nonstabilizerness $\unicode{x2013}$ the resource enabling universal quantum computation $\unicode{x2013}$ is by contrast encoded in the global structure of the many-body wavefunction. We show that, despite its global nature, the magic of equilibrium pure states of chaotic many-body systems, including late-time evolved states and energy eigenstates, is universal: it is captured by the thermal Scrooge ensemble, the minimally informative ensemble of pure states consistent with the Gibbs state at the same effective temperature. Therefore, for systems with no conserved quantities other than the total energy, equilibrium magic is a function of temperature alone, independent of the initial state and other microscopic features of the equilibrium state. This yields concrete universal predictions for the stabilizer Rényi entropies (SREs). At infinite temperature, the SRE is set by Haar-like fluctuations of the Pauli spectrum, while at finite temperature energy conservation induces a volume-law thermodynamic correction controlled by the thermal Pauli spectrum. We support these predictions with analytical arguments and extensive numerical simulations. We further show that chaotic many-body systems at high temperatures possess long-range magic and entanglement that cannot be removed by finite-depth local quantum circuits. Our results establish magic as a thermodynamic property of chaotic many-body systems and suggest that Scrooge ensembles may provide a unified framework for quantum many-body resources.