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arXiv:2607.03535

Thermalization hierarchy from irreducible degrees of freedom

Pedro Fittipaldi de Castro, Wladimir A. Benalcazar

quant-phcond-mat.stat-mechSubmitted Jul 3, 2026• Announced Jul 7, 2026• 0 citations
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Abstract

The decomposition of the Hilbert space of a quantum many-body system into the irreducible representations of its bond and commutant algebras yields a finer structure of dynamically isolated subspaces than the mere decomposition into symmetry sectors. While it has been recognized that subspaces associated with low bond-irrep dimensions $D_λ$ tend to violate the eigenstate thermalization hypothesis (ETH), here we show that $D_λ$ controls thermalization continuously across the full spectrum of dynamical subspaces. Using SU(2)-symmetric spin-1/2 chains as a paradigmatic example, we demonstrate that $\mathrm{log} \ D_λ$ quantitatively accounts for the average eigenstate entanglement entropy within each sector, establishing a thermalization hierarchy that interpolates from exact quantum many-body scars at $D_λ=1$ to volume-law ergodic states at large $D_λ$. To make this concrete, we introduce the notion of irreducible degrees of freedom (IDOF), defined as the number of independently-varying spatial coordinates parametrizing a many-body state within a given bond-algebra sector, which provides a microscopic interpretation of $D_λ$ and of the resulting thermalization hierarchy. Finally, we show that by selectively breaking symmetries while preserving chosen bond-algebra sectors, one can embed families of nonthermal eigenstates at prescribed entanglement levels into an otherwise ergodic spectrum, generalizing restricted spectrum-generating algebras from towers of individual states to entire dynamical subspaces.

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9 pages, 4 figures

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The decomposition of the Hilbert space of a quantum many-body system into the irreducible representations of its bond and commutant algebras yields a finer structure of dynamically isolated subspaces than the mere decomposition into symmetry sectors. While it has been recognized that subspaces associated with low bond-irrep dimensions DλD_λDλ​ tend to violate the eigenstate thermalization hypothesis (ETH), here we show that DλD_λDλ​ controls thermalization continuously across the full spectrum of dynamical subspaces. Using SU(2)-symmetric spin-1/2 chains as a paradigmatic example, we demonstrate that log Dλ\mathrm{log} \ D_λlog Dλ​ quantitatively accounts for the average eigenstate entanglement entropy within each sector, establishing a thermalization hierarchy that interpolates from exact quantum many-body scars at Dλ=1D_λ=1Dλ​=1 to volume-law ergodic states at large DλD_λDλ​. To make this concrete, we introduce the notion of irreducible degrees of freedom (IDOF), defined as the number of independently-varying spatial coordinates parametrizing a many-body state within a given bond-algebra sector, which provides a microscopic interpretation of DλD_λDλ​ and of the resulting thermalization hierarchy. Finally, we show that by selectively breaking symmetries while preserving chosen bond-algebra sectors, one can embed families of nonthermal eigenstates at prescribed entanglement levels into an otherwise ergodic spectrum, generalizing restricted spectrum-generating algebras from towers of individual states to entire dynamical subspaces.
A quantum many-body scar system usually contains a special nonthermal subspace (approximately) decoupled from the rest of the Hilbert space. In this work, we propose a general structure called deformed symmetric spaces for the decoupled subspaces hosting quantum many-body scars, which are irreducible sectors of simple Lie groups transformed by matrix-product operators (or projected entangled pair operators), of which the entanglement entropies are proved to obey sub-volume-law scaling and thus violate the eigenstate thermalization hypothesis. A deformed symmetric space, in general, is required to have at least a U(1) sub-Lie-group symmetry to allow coherent periodic dynamics from certain low-entangled initial states. We enumerate several possible deforming transformations based on the subgroup symmetry requirement and recover many exact scar eigenstates in the existing scar models, of which the Hamiltonians can be reconstructed using the standard quantum inverse method. In particular, a two-dimensional scar model is proposed, which hosts a periodic dynamical trajectory on which all states are topologically ordered.
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Understanding the evolution towards thermal equilibrium of an isolated quantum system is at the foundation of statistical mechanics and a subject of interest in such diverse areas as cold atom physics or the quantum mechanics of black holes. Since a pure state can never evolve into a thermal density matrix, the Eigenstate Thermalization Hypothesis (ETH) has been put forward by Deutsch and Srednicki as a way to explain this apparent thermalization, similarly to what the ergodic theorem does in classical mechanics. In this paper this hypothesis is tested numerically. First, it is observed that thermalization happens in a subspace of states (the Krylov subspace) with dimension much smaller than that of the total Hilbert space. We check numerically the validity of ETH in such a subspace, for a system of hard core bosons on a two-dimensional lattice. We then discuss how well the eigenstates of the Hamiltonian projected on the Krylov subspace represent the true eigenstates. This discussion is aided by bringing the projected Hamiltonian to the tridiagonal form and interpreting it as an Anderson localization problem for a finite one-dimensional chain. We also consider thermalization of a subsystem and argue that generation of a large entanglement entropy can lead to a thermal density matrix for the subsystem well before the whole system thermalizes. Finally, we comment on possible implications of ETH in quantum gravity.
We study quantum many-body scars (QMBS) in the language of commutant algebras, which are defined as symmetry algebras of of local Hamiltonians. This framework explains the origin of dynamically disconnected subspaces seen in models with exact QMBS, i.e., the large “thermal” subspace and the small “nonthermal” subspace, which are attributed to the existence of unconventional nonlocal conserved quantities in the commutant; hence, it unifies the study of conventional symmetries and weak ergodicity-breaking phenomena into a single framework. Furthermore, this language enables us to use the von Neumann double commutant theorem to formally write down the exhaustive algebra of Hamiltonians with a desired set of QMBS, which demonstrates that QMBS survive under large classes of local perturbations. We illustrate this using several standard examples of QMBS, including the spin- 1 / 2 ferromagnetic, AKLT, spin-1 XY π -bimagnon, and the electronic η -pairing towers of states; in each of these cases, we explicitly write down a set of generators for the full algebra of Hamiltonians with these QMBS. Understanding this hidden structure in QMBS Hamiltonians also allows us to recover results of previous “brute-force” numerical searches for such Hamiltonians. In addition, this language clearly demonstrates the equivalence of several unified formalisms for QMBS proposed in the literature and also illustrates the connection between two apparently distinct classes of QMBS Hamiltonians—those that are captured by the so-called Shiraishi-Mori construction and those that lie beyond. Finally, we show that this framework motivates a precise definition for QMBS that automatically implies that they violate the conventional eigenstate thermalization hypothesis, and we discuss its implications to dynamics. Published by the American Physical Society 2024
Quantum many-body scars (QMBSs) in kinetically constrained systems provide a prominent example of weak ergodicity breaking beyond the eigenstate thermalization hypothesis (ETH). In this work we formulate an open-system perspective on constrained dynamics, where kinetic constraints are interpreted as a balance of information exchange between the system and an auxiliary environment. This viewpoint naturally introduces a quasiparticle number that characterizes the information balance and motivates a revised ETH for constrained systems. Specifically, the eigenstate expectation values of local observables can be approximated by a grand-canonical-like average defined on the energy-quasiparticle-number plane. For the off-diagonal part of local observables, the relevant density of states (DOS) is generalized to the same plane and controls the typical magnitude of off-diagonal matrix elements. The revised ETH accurately describes long-time averages and temporal fluctuations of local observables for both QMBSs and thermal dynamics. Within this framework, scarred dynamics arises from low-DOS regions on the energy-quasiparticle-number plane, where off-diagonal elements are less suppressed and the approximate spectrum-generating algebra becomes more pronounced. Our results provide a unified description of thermalization and QMBSs in kinetically constrained systems.
We investigate thermalization in finite quantum systems with strong long-range interactions. Nearly conserved quantities inherited from the fully connected limit organize the Hilbert space into weakly coupled sectors and split the many-body spectrum into energy bands. Despite the resulting breakdown of global ergodicity, quantum chaos develops within individual energy bands, enabling the definition of microcanonical ensembles within the bands. This supports a band-resolved formulation of thermalization, which we term fragmented eigenstate thermalization hypothesis (fETH). Unlike conventional ETH, finite-size scaling in fETH obeys a symmetry-imposed selection rule that restricts which system sizes can be compared. This band-resolved description has consequences for equilibrium statistical mechanics. While microcanonical ensembles remain confined to a single band, canonical ensembles mix different bands. This mismatch explains ensemble inequivalence without invoking equilibrium phase transitions. Our results apply broadly to Hamiltonians near a fully permutation-symmetric limit.