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arXiv:2305.04984 · Physical Review Letters

Violation of Eigenstate Thermalization Hypothesis in Quantum Field Theories with Higher-Form Symmetry

Osamu Fukushima, Ryusuke Hamazaki

Published Sep 29, 2023• 10 citations
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Abstract

We elucidate how the presence of higher-form symmetries affects the dynamics of thermalization in isolated quantum systems. Under reasonable assumptions, we analytically show that a p-form symmetry in a (d+1)-dimensional quantum field theory leads to the breakdown of the eigenstate thermalization hypothesis for many nontrivial (d-p)-dimensional observables. For discrete higher-form (i.e., p≥1) symmetry, this indicates the absence of thermalization for observables that are nonlocal but much smaller than the whole system size without any local conserved quantities. We numerically demonstrate this argument for the (2+1)-dimensional Z_{2} lattice gauge theory. While local observables such as the plaquette operator thermalize even for mixed symmetry sectors, the nonlocal observable exciting a magnetic dipole instead relaxes to the generalized Gibbs ensemble that takes account of the Z_{2} one-form symmetry.

DOI
10.1103/physrevlett.131.131602

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We elucidate how the presence of higher-form symmetries affects the dynamics of thermalization in isolated quantum systems. Under reasonable assumptions, we analytically show that a p-form symmetry in a (d+1)-dimensional quantum field theory leads to the breakdown of the eigenstate thermalization hypothesis for many nontrivial (d-p)-dimensional observables. For discrete higher-form (i.e., p≥1) symmetry, this indicates the absence of thermalization for observables that are nonlocal but much smaller than the whole system size without any local conserved quantities. We numerically demonstrate this argument for the (2+1)-dimensional Z_{2} lattice gauge theory. While local observables such as the plaquette operator thermalize even for mixed symmetry sectors, the nonlocal observable exciting a magnetic dipole instead relaxes to the generalized Gibbs ensemble that takes account of the Z_{2} one-form symmetry.
Abstract The existence of p-form symmetry in a (d + 1)-dimensional quantum field is known to always lead to the breakdown of the eigenstate thermalization hypothesis for certain (d − p)-dimensional operators other than symmetry operators under some assumptions. The assumptions include the mixing of symmetry sectors within a given energy shell, which is rather challenging to verify because it requires information on the eigenstates in the middle of the spectrum. We reconsider this assumption from the viewpoint of projective representations to avoid this difficulty. In the case of ZN\mathbb {Z}_NZN​ symmetries, we can circumvent the difficulty by considering ZN×ZN\mathbb {Z}_N\times \mathbb {Z}_NZN​×ZN​-symmetric theories with nontrivial projective phases, and perturbing the Hamiltonian while preserving one of the ZN\mathbb {Z}_NZN​ symmetries of our interest. We also perform numerical analyses for (1 + 1)-dimensional spin chains and the (2 + 1)-dimensional Z2\mathbb {Z}_2Z2​ lattice gauge theory.
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.
We test the eigenstate thermalization hypothesis (ETH) in 1 + 1 -dimensional SU(2) lattice gauge theory (LGT) with one flavor of dynamical fermions. Using the loop-string-hadron framework of the LGT with a bosonic cutoff, we exactly diagonalize the Hamiltonian for finite size systems and calculate matrix elements (MEs) in the eigenbasis for both local and nonlocal operators. We analyze different indicators to identify the parameter space for quantum chaos at finite lattice sizes and investigate how the ETH behavior emerges in both the diagonal and off-diagonal MEs. Our investigations allow us to study various timescales of thermalization and the emergence of random matrix behavior, and highlight the interplays of the several diagnostics with each other. Furthermore, from the off-diagonal MEs, we extract a smooth function that is closely related to the spectral function for both local and nonlocal operators. We find numerical evidence of the spectral gap and the memory peak in the nonlocal operator case. Finally, we investigate aspects of subsystem ETH in the lattice gauge theory and identify certain features in the subsystem reduced density matrix that are unique to gauge theories.
We review exact approaches and recent results related to the relaxation dynamics and description after relaxation of various one-dimensional lattice systems of hard-core bosons after a sudden quench. We first analyze the integrable case, where the combination of analytical insights and computational techniques enable one to study large system sizes. Thermalization does not occur in this regime. However, after relaxation, observables can be described by a generalization of the Gibbs ensemble. We then utilize full exact diagonalization to study what happens as integrability is broken. We show that thermalization does occur in finite nonintegrable systems provided they are sufficiently far away from the integrable point. We argue that the onset of thermalization can be understood in terms of the eigenstate thermalization hypothesis.
We test the eigenstate thermalization hypothesis (ETH) for 2+12+12+1 dimensional SU(2) lattice gauge theory. By considering the theory on a chain of plaquettes and truncating basis states for link variables at j=1/2j=1/2j=1/2, we can map it onto a quantum spin chain with local interactions and numerically exactly diagonalize the Hamiltonian for reasonably large lattice sizes. We find energy level repulsion in momentum sectors with no remaining discrete symmetry. We study two local observables made up of Wilson loops and calculate their matrix elements in the energy eigenbasis, which are shown consistent with the ETH.
A bstract The Eigenstate Thermalization Hypothesis (ETH) provides a sufficient condition for thermalization of isolated quantum systems. While the standard ETH is formulated in the absence of degeneracy, physical systems often possess symmetries that induce degenerate energy eigenstates. In this paper, we investigate ETH in the presence of nontrivial projective representations of Abelian symmetries, which arise naturally from ’t Hooft anomalies. We argue that such projective structures can lead to degenerate excited states, and how the ETH can be formulated under such degeneracies. In the presence of projective charges supplied by symmetry operators, our projective-representation ETH indicates that the stationary values of the operators are described by the generalized Gibbs ensemble instead of the standard Gibbs ensemble. Our findings elucidate the role of symmetry and degeneracy in quantum thermalization and pave the way for further exploration of the ETH in anomalous symmetry settings.