It is well-known that thermalization breaks down in presence of an extensive number of symmetries or conservation laws that are mutually compatible or commuting with each other. We ask here how an extensive number of mutually incompatible local symmetries govern eigenstate thermalization (ETH). We will investigate this through models within the class of bond-dependent $\mathbb{Z}_2$-symmetric quantum spin-$\frac{1}{2}$ Hamiltonians that naturally admit this structure. This leads to exponentially large degeneracies in the spectrum governed by the incompatible symmetry structure. Here, using spectral diagnostics, we show that non-integrability can survive despite extensively many conserved quantities when they are incompatible. Furthermore, we show that ETH behavior persists deep into the many-body spectral edge due to this structure. In other words, incompatible local symmetries enable a mechanism to obtain enough thermodynamic entropy at low excitation energy densities to make ETH operational or predictive near the ground state.
It is well-known that thermalization breaks down in presence of an extensive number of symmetries or conservation laws that are mutually compatible or commuting with each other. We ask here how an extensive number of mutually incompatible local symmetries govern eigenstate thermalization (ETH). We will investigate this through models within the class of bond-dependent Z2-symmetric quantum spin-21 Hamiltonians that naturally admit this structure. This leads to exponentially large degeneracies in the spectrum governed by the incompatible symmetry structure. Here, using spectral diagnostics, we show that non-integrability can survive despite extensively many conserved quantities when they are incompatible. Furthermore, we show that ETH behavior persists deep into the many-body spectral edge due to this structure. In other words, incompatible local symmetries enable a mechanism to obtain enough thermodynamic entropy at low excitation energy densities to make ETH operational or predictive near the ground state.
The eigenstate thermalization hypothesis (ETH) explains how generic quantum many-body systems thermalize internally. It implies that local operators' time-averaged expectation values approximately equal their thermal expectation values, regardless of microscopic details. The ETH's range of applicability therefore impacts theory and experiments. Murthy et al. [Phys. Rev. Lett. 130, 140402 (2023)0031-900710.1103/PhysRevLett.130.140402] recently showed that non-Abelian symmetries conflict with the ETH. Such symmetries have excited interest in quantum thermodynamics lately, as they are equivalent to conserved quantities that fail to commute with each other and noncommutation is a quintessentially quantum phenomenon. Murthy et al. proposed a non-Abelian ETH, which we support numerically. The numerics model a one-dimensional next-nearest-neighbor Heisenberg chain of 18 qubits. We represent local operators with matrices relative to an energy eigenbasis. The matrices bear out seven predictions of the non-Abelian ETH. We also prove analytically that the non-Abelian ETH exhibits a self-consistency property. The proof relies on a thermodynamic-entropy definition different from that of Murthy et al. This work initiates the observation and application of the non-Abelian ETH.
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 Dλ tend to violate the eigenstate thermalization hypothesis (ETH), here we show that Dλ 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 logDλ 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=1 to volume-law ergodic states at large Dλ. 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 Dλ 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.
The eigenstate thermalization hypothesis (ETH) explains why nonintegrable quantum many-body systems thermalize internally if the Hamiltonian lacks symmetries. If the Hamiltonian conserves one quantity ("charge"), the ETH implies thermalization within a charge sector-in a microcanonical subspace. But quantum systems can have charges that fail to commute with each other and so share no eigenbasis; microcanonical subspaces may not exist. Furthermore, the Hamiltonian will have degeneracies, so the ETH need not imply thermalization. We adapt the ETH to noncommuting charges by positing a non-Abelian ETH and invoking the approximate microcanonical subspace introduced in quantum thermodynamics. Illustrating with SU(2) symmetry, we apply the non-Abelian ETH in calculating local operators' time-averaged and thermal expectation values. In many cases, we prove, the time average thermalizes. However, we find cases in which, under a physically reasonable assumption, the time average converges to the thermal average unusually slowly as a function of the global-system size. This work extends the ETH, a cornerstone of many-body physics, to noncommuting charges, recently a subject of intense activity in quantum thermodynamics.
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.
We investigate the eigenstate thermalization hypothesis (ETH) in integrable models, focusing on the $\text{spin}\ensuremath{-}\frac{1}{2}$ isotropic Heisenberg (XXX) chain. We provide numerical evidence that the ETH holds for typical eigenstates (weak ETH scenario). Specifically, using a numerical implementation of state-of-the-art Bethe ansatz results, we study the finite-size scaling of the eigenstate-to-eigenstate fluctuations of the reduced density matrix. We find that fluctuations are normally distributed, and their standard deviation decays in the thermodynamic limit as ${L}^{\ensuremath{-}1/2}$, with L the size of the chain. This is in contrast with the exponential decay that is found in generic nonintegrable systems. Based on our results, it is natural to expect that this scenario holds in other integrable spin models and for typical local observables. Finally, we investigate the entanglement properties of the excited states of the XXX chain. We numerically verify that typical midspectrum eigenstates exhibit extensive entanglement entropy (i.e., volume-law scaling).
There is a dichotomy in the nonequilibrium dynamics of quantum many-body systems. In the presence of integrability, expectation values of local operators equilibrate to values described by a generalized Gibbs ensemble, which retains extensive memory about the initial state of the system. On the other hand, in generic systems such expectation values relax to stationary values described by the thermal ensemble, fixed solely by the energy of the state. At the heart of understanding, this dichotomy is the eigenstate thermalization hypothesis (ETH): individual eigenstates in nonintegrable systems are thermal, in the sense that expectation values agree with the thermal prediction at a temperature set by the energy of the eigenstate. In systems where ETH is violated, thermalization can be avoided. Thus, establishing the range of validity of ETH is crucial in understanding whether a given quantum system thermalizes. Here, we study a simple model with confinement, the quantum Ising chain with a longitudinal field, in which ETH is violated. Despite an absence of integrability, there exist rare (nonthermal) states that persist far into the spectrum. These arise as a direct consequence of confinement: pairs of particles are confined, forming new ``meson'' excitations whose energy can be extensive in the system size. We show that such states are nonthermal in both the continuum and in the low-energy spectrum of the corresponding lattice model. We highlight that the presence of such states within the spectrum has important consequences, with certain quenches leading to an absence of thermalization and local observables evolving anomalously.