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Register a new userDegenerate observables and the many Eigenstate Thermalization Hypotheses
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Under unitary time evolution, expectation values of physically reasonable observables often evolve towards the predictions of equilibrium statistical mechanics. The eigenstate thermalization hypothesis (ETH) states that this is also true already for individual energy eigenstates. Here we aim at elucidating the emergence of ETH for observables that can realistically be measured due to their high degeneracy, such as local, extensive or macroscopic observables. We bisect this problem into two parts, a condition on the relative overlaps and one on the relative phases between the eigenbases of the observable and Hamiltonian.
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We study the matrix elements of few-body observables, focusing on the off-diagonal ones, in the eigenstates of the two-dimensional transverse field Ising model. By resolving all symmetries, we relate the onset of quantum chaos to the structure of the matrix elements. In particular, we show that a general result of the theory of random matrices, namely, the value 2 of the ratio of variances (diagonal to off-diagonal) of the matrix elements of Hermitian operators, occurs in the quantum chaotic regime. Furthermore, we explore the behavior of the off-diagonal matrix elements of observables as a function of the eigenstate energy differences, and show that it is in accordance with the eigenstate thermalization hypothesis ansatz.
We use exact diagonalization to study the eigenstate thermalization hypothesis (ETH) in the quantum dimer model on the square and triangular lattices. Due to the nonergodicity of the local plaquette-flip dynamics, the Hilbert space, which consists of highly constrained close-packed dimer configurations, splits into sectors characterized by topological invariants. We show that this has important consequences for ETH: We find that ETH is clearly satisfied only when each topological sector is treated separately, and only for moderate ratios of the potential and kinetic terms in the Hamiltonian. By contrast, when the spectrum is treated as a whole, ETH breaks down on the square lattice, and apparently also on the triangular lattice. These results demonstrate that quantum dimer models have interesting thermalization dynamics.
In a recent Letter [PhysRevLett.119.030601 (2017), arXiv:1702.08227], Shiraishi and Mori claim to provide a general method for constructing local Hamiltonians that do not exhibit eigenstate thermalization. We argue that the claim is based on a misunderstanding of the eigenstate thermalization hypothesis (ETH). More specifically, on the assumption that ETH is valid for the entire Hamiltonian matrix instead of each symmetry sector independently. We discuss what happens if one mixes symmetry sectors in the two-dimensional transverse field Ising model.
Boltzmanns ergodic hypothesis furnishes a possible explanation for the emergence of statistical mechanics in the framework of classical physics. In quantum mechanics, the Eigenstate Thermalization Hypothesis (ETH) is instead generally considered as a possible route to thermalization. This is because the notion of ergodicity itself is vague in the quantum world and it is often simply taken as a synonym for thermalization. Here we show, in an elementary way, that when quantum ergodicity is properly defined, it is, in fact, equivalent to ETH. In turn, ergodicity is equivalent to thermalization, thus implying the equivalence of thermalization and ETH. This result previously appeared in [De Palma et al., Phys. Rev. Lett. 115, 220401 (2015)], but becomes particularly clear in the present context. We also show that it is possible to define a classical analogue of ETH which is implicitly assumed to be satisfied when constructing classical statistical mechanics. Classical and quantum statistical mechanics are built according to the familiar standard prescription. This prescription, however, is ontologically justified only in the quantum world.
We discuss eigenstate correlations for ergodic, spatially extended many-body quantum systems, in terms of the statistical properties of matrix elements of local observables. While the eigenstate thermalization hypothesis (ETH) is known to give an excellent description of these quantities, the butterfly effect implies structure beyond ETH. We determine the universal form of this structure at long distances and small eigenvalue separations for Floquet systems. We use numerical studies of a Floquet quantum circuit to illustrate both the accuracy of ETH and the existence of our predicted additional correlations.
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