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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Eigenstate thermalization hypothesis | 5/5 | https://en.wikipedia.org/wiki/Eigenstate_thermalization_hypothesis | reference | science, encyclopedia | 2026-05-05T09:59:23.990681+00:00 | kb-cron |
== General validity == Currently, there is no known analytical derivation of the eigenstate thermalization hypothesis for general interacting systems. 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 has also been proven to be true in certain special cases in the semi-classical limit, where the validity of the ETH rests on the validity of Shnirelman's theorem, which states that in a system which is classically chaotic, the expectation value of an operator
A
^
{\displaystyle {\hat {A}}}
in an energy eigenstate is equal to its classical, microcanonical average at the appropriate energy. Whether or not it can be shown to be true more generally in interacting quantum systems remains an open question. 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. It is also important to note that the ETH makes statements about specific observables on a case-by-case basis - it does not make any claims about whether every observable in a system will obey ETH. In fact, this certainly cannot be true. Given a basis of energy eigenstates, one can always explicitly construct an operator which violates the ETH, simply by writing down the operator as a matrix in this basis whose elements explicitly do not obey the conditions imposed by the ETH. Conversely, it is always trivially possible to find operators which do satisfy ETH, by writing down a matrix whose elements are specifically chosen to obey ETH. In light of this, one may be led to believe that the ETH is somewhat trivial in its usefulness. However, the important consideration to bear in mind is that these operators thus constructed may not have any physical relevance. 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. Typically, the ETH is postulated to hold for "few-body operators," observables which involve only a small number of particles. Examples of this would include the occupation of a given momentum in a gas of particles, or the occupation of a particular site in a lattice system of particles. Notice that while the ETH is typically applied to "simple" few-body operators such as these, these observables need not be local in space - the momentum number operator in the above example does not represent a local quantity. 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. It remains an open question as to whether a completely isolated, non-integrable system without static disorder can ever fail to thermalize. One intriguing possibility is the realization of "Quantum Disentangled Liquids." It also an open question whether all eigenstates must obey the ETH in a thermalizing system. The eigenstate thermalization hypothesis is closely connected to the quantum nature of chaos (see quantum chaos). 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
ℏ
→
0
{\displaystyle \hbar \rightarrow 0}
. In particular, there is a quantum ergodicity theorem showing that the expectation value of an operator converges to the corresponding microcanonical classical average as
ℏ
→
0
{\displaystyle \hbar \rightarrow 0}
. However, the quantum ergodicity theorem leaves open the possibility of non-ergodic states such as quantum scars. In addition to the conventional scarring, there are two other types of quantum scarring, which further illustrate the weak-ergodicity breaking in quantum chaotic systems: perturbation-induced and many-body quantum scars. Since the former arise a combined effect of special nearly-degenerate unperturbed states and the localized nature of the perturbation (potential bums), the scarring can slow down the thermalization process in disordered quantum dots and wells, which is further illustrated by the fact that these quantum scars can be used to propagate quantum wave packets in a disordered nanostructure with high fidelity. On the other hand, the latter form of scarring has been speculated to be the culprit behind the unexpectedly slow thermalization of cold atoms observed experimentally.
== See also ==
== Footnotes ==
== References ==
== External links == "Overview of Eigenstate Thermalization Hypothesis" by Mark Srednicki, UCSB, KITP Program: Quantum Dynamics in Far from Equilibrium Thermally Isolated Systems "The Eigenstate Thermalization Hypothesis" by Mark Srednicki, UCSB, KITP Rapid Response Workshop: Black Holes: Complementarity, Fuzz, or Fire? "Quantum Disentangled Liquids" by Matthew P. A. Fisher, UCSB, KITP Conference: From the Renormalization Group to Quantum Gravity Celebrating the science of Joe Polchinski