The phrase "eigenstate thermalization" was first coined by Mark Srednicki in 1994,[1] after similar ideas had been introduced by Josh Deutsch in 1991.
[2] The principal philosophy underlying the eigenstate thermalization hypothesis is that instead of explaining the ergodicity of a thermodynamic system through the mechanism of dynamical chaos, as is done in classical mechanics, one should instead examine the properties of matrix elements of observable quantities in individual energy eigenstates of the system.
[3] If we prepare an isolated, chaotic, classical system in some region of its phase space, then as the system is allowed to evolve in time, it will sample its entire phase space, subject only to a small number of conservation laws (such as conservation of total energy).
[footnote 2] Given the state at time zero in a basis of energy eigenstates the expectation value of any observable
In principle it is thus an open question as to whether an isolated quantum mechanical system, prepared in an arbitrary initial state, will approach a state which resembles thermal equilibrium, in which a handful of observables are adequate to make successful predictions about the system.
However, a variety of experiments in cold atomic gases have indeed observed thermal relaxation in systems which are, to a very good approximation, completely isolated from their environment, and for a wide class of initial states.
[4][5] The task of explaining this experimentally observed applicability of equilibrium statistical mechanics to isolated quantum systems is the primary goal of the eigenstate thermalization hypothesis.
The matrix elements of this operator, as expressed in a basis of energy eigenstates, will be denoted by We now imagine that we prepare our system in an initial state for which the expectation value of
Conversely if a quantum many-body system satisfies the ETH, the matrix representation of any local operator in the energy eigen basis is expected to follow the above ansatz.
This prediction makes absolutely no reference to the initial state of the system, unlike the diagonal ensemble.
Because of this, it is not clear why the microcanonical ensemble should provide such an accurate description of the long-time averages of observables in such a wide variety of physical systems.
This constitutes a foundation for quantum statistical mechanics which is radically different from the one built upon the notions of dynamical ergodicity.
[5] Some analytical results can also be obtained if one makes certain assumptions about the nature of highly excited energy eigenstates.
The original 1994 paper on the ETH by Mark Srednicki studied, in particular, the example of a quantum hard sphere gas in an insulated box.
[1] Currently, it is not well understood how high the energy of an eigenstate of the hard sphere gas must be in order for it to obey the ETH.
[6][5] Notice that this condition allows for the possibility of isolated resurgence times, in which the phases align coherently in order to produce large fluctuations away from the long-time average.
Therefore, while we have been examining this expectation value as the principal object of interest, it is not clear to what extent this represents physically relevant quantities.
As a result of quantum fluctuations, the expectation value of an observable is not typically what will be measured during one experiment on an isolated system.
[6][5] This lends further credence to the idea that the ETH is the underlying mechanism responsible for the thermalization of isolated quantum systems.
[5] However, it has been verified to be true for a wide variety of interacting systems using numerical exact diagonalization techniques, to within the uncertainty of these methods.
It is also known to explicitly fail in certain integrable systems, in which the presence of a large number of constants of motion prevent thermalization.
While one can construct these matrices, it is not clear that they correspond to observables which could be realistically measured in an experiment, or bear any resemblance to physically interesting quantities.
An arbitrary Hermitian operator on the Hilbert space of the system need not correspond to something which is a physically measurable observable.
[11] Typically, the ETH is postulated to hold for "few-body operators,"[4] observables which involve only a small number of particles.
[5] There has also been considerable interest in the case where isolated, non-integrable quantum systems fail to thermalize, despite the predictions of conventional statistical mechanics.
Disordered systems which exhibit many-body localization are candidates for this type of behavior, with the possibility of excited energy eigenstates whose thermodynamic properties more closely resemble those of ground states.
[12][13] It remains an open question as to whether a completely isolated, non-integrable system without static disorder can ever fail to thermalize.
Furthermore, since a classically chaotic system is also ergodic, almost all of its trajectories eventually explore uniformly the entire accessible phase space, which would imply the eigenstates of the quantum chaotic system fill the quantum phase space evenly (up to random fluctuations) in the semiclassical limit
In particular, there is a quantum ergodicity theorem showing that the expectation value of an operator converges to the corresponding microcanonical classical average as
[20] On the other hand, the latter form of scarring has been speculated[24][25] to be the culprit behind the unexpectedly slow thermalization of cold atoms observed experimentally.