Eigenstate Thermalization for Degenerate Observables

Abstract

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 the 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. We show that the relative overlaps are unbiased for highly degenerate observables and demonstrate that unless relative phases conspire to cumulative effects, this makes such observables verify the ETH. Through this we elucidate potential pathways towards proofs of thermalization.

Eigenstate thermalization hypothesis, time operator, and extremely quick relaxation of fidelity

“…Several different approaches have been studied, including through a restriction on the macroscopic observables [10,11], the general evaluation of relaxation time [12][13][14][15], the Eigenstate thermalization hypothesis (ETH) [1,[16][17][18][19][20][21], and dynamical experiments in autonomous cold atomic systems [22][23][24]. Of these, we focus on the foundation of ETH in terms of the time-energy uncertainty by noting that the energy eigenstates are globally distributed in the basis of suitably defined 'time operator' as detailed below (1) and in section 2.…”

“…Note that [20] quantified the degree of superposition with the use of Shannon entropy, which is basis dependent and maximized to guarantee ETH. Subsequently, [21] addressed the issue to specify a class of observables, such as local and extensive quantities, that satisfy ETH in terms of mutually unbiased basis with respect to the Hamiltonian. Mutually unbiasedness can be regarded as a generalization of the concept of the canonical conjugate, which is significant for our argument, and thus [20,21] are related to the present study.…”

“…Subsequently, [21] addressed the issue to specify a class of observables, such as local and extensive quantities, that satisfy ETH in terms of mutually unbiased basis with respect to the Hamiltonian. Mutually unbiasedness can be regarded as a generalization of the concept of the canonical conjugate, which is significant for our argument, and thus [20,21] are related to the present study. The main difference is that in this article, the quasi eigenstates of the 'time operator' are unbiased with respect to the Hamiltonian.…”

Anomalous Behavior of Magnetic Susceptibility Obtained by Quench Experiments in Isolated Quantum Systems

“…if and only if condition (10), which is similar to but weaker than the ordinary 'off-diagonal' ETH [44,[49][50][51], is satisfied.…”

“…This is similar to but different from the ordinary two forms of ETH in the following points. The ordinary strong ETH [48][49][50][51] requires that all ν|σ z 0 |ν behave like a smooth function of E ν /N , which is often satisfied in nonintegrable systems [60]. Since a smooth function of E ν /N can be regarded as linear within the narrow region |δE ν | < ∼ T √ c h N , any system satisfying the strong ETH also satisfies condition (8), while notice that the converse is not necessarily true.…”