Eigenstate thermalization hypothesis


The eigenstate thermalization hypothesis is a set of ideas which purports to explain when and why an isolated quantum mechanical system can be accurately described using equilibrium statistical mechanics. In particular, it is devoted to understanding how systems which are initially prepared in far-from-equilibrium states can evolve in time to a state which appears to be in thermal equilibrium. The phrase "eigenstate thermalization" was first coined by in 1994, after similar ideas had been introduced by Josh Deutsch in 1991. 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.

Motivation

In statistical mechanics, the microcanonical ensemble is a particular statistical ensemble which is used to make predictions about the outcomes of experiments performed on isolated systems that are believed to be in equilibrium with an exactly known energy. The microcanonical ensemble is based upon the assumption that, when such an equilibrated system is probed, the probability for it to be found in any of the microscopic states with the same total energy have equal probability. With this assumption, the ensemble average of an observable quantity is found by averaging the value of that observable over all microstates with the correct total energy:
Importantly, this quantity is independent of everything about the initial state except for its energy.
The assumptions of ergodicity are well-motivated in classical mechanics as a result of dynamical chaos, since a chaotic system will in general spend equal time in equal areas of its phase space. 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. If one can justify the claim that a given physical system is ergodic, then this mechanism will provide an explanation for why statistical mechanics is successful in making accurate predictions. For example, the hard sphere gas has been rigorously proven to be ergodic.
This argument cannot be straightforwardly extended to quantum systems, even ones that are analogous to chaotic classical systems, because time evolution of a quantum system does not uniformly sample all vectors in Hilbert space with a given energy. Given the state at time zero in a basis of energy eigenstates
the expectation value of any observable is
Even if the are incommensurate, so that this expectation value is given for long times by
the expectation value permanently retains knowledge of the initial state in the form of the coefficients.
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. 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.

Statement

Suppose that we are studying an isolated, quantum mechanical many-body system. In this context, "isolated" refers to the fact that the system has no interactions with the environment external to it. If the Hamiltonian of the system is denoted, then a complete set of basis states for the system is given in terms of the eigenstates of the Hamiltonian,
where is the eigenstate of the Hamiltonian with eigenvalue. We will refer to these states simply as "energy eigenstates." For simplicity, we will assume that the system has no degeneracy in its energy eigenvalues, and that it is finite in extent, so that the energy eigenvalues form a discrete, non-degenerate spectrum. This allows us to label the energy eigenstates in order of increasing energy eigenvalue. Additionally, consider some other quantum-mechanical observable, which we wish to make thermal predictions about. 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 is far from its value predicted in a microcanonical ensemble appropriate to the energy scale in question. The eigenstate thermalization hypothesis says that for an arbitrary initial state, the expectation value of will ultimately evolve in time to its value predicted by a microcanonical ensemble, and thereafter will exhibit only small fluctuations around that value, provided that the following two conditions are met:
  1. The diagonal matrix elements vary smoothly as a function of energy, with the difference between neighboring values,, becoming exponentially small in the system size.
  2. The off-diagonal matrix elements, with, are much smaller than the diagonal matrix elements, and in particular are themselves exponentially small in the system size.
These conditions can be written as
where and are smooth functions of energy, is the many-body Hilbert space dimension, and is a random variable with zero mean and unit variance. 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.

Equivalence of the diagonal and microcanonical ensembles

We can define a long-time average of the expectation value of the operator according to the expression
If we use the explicit expression for the time evolution of this expectation value, we can write
The integration in this expression can be performed explicitly, and the result is
Each of the terms in the second sum will become smaller as the limit is taken to infinity. Assuming that the phase coherence between the different exponential terms in the second sum does not ever become large enough to rival this decay, the second sum will go to zero, and we find that the long-time average of the expectation value is given by
This prediction for the time-average of the observable is referred to as its predicted value in the diagonal ensemble, The most important aspect of the diagonal ensemble is that it depends explicitly on the initial state of the system, and so would appear to retain all of the information regarding the preparation of the system. In contrast, the predicted value in the microcanonical ensemble is given by the equally-weighted average over all energy eigenstates within some energy window centered around the mean energy of the system
where is the number of states in the appropriate energy window, and the prime on the sum indices indicates that the summation is restricted to this appropriate microcanonical window. 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.
However, suppose that the matrix elements are effectively constant over the relevant energy window, with fluctuations that are sufficiently small. If this is true, this one constant value A can be effectively pulled out of the sum, and the prediction of the diagonal ensemble is simply equal to this value,
where we have assumed that the initial state is normalized appropriately. Likewise, the prediction of the microcanonical ensemble becomes
The two ensembles are therefore in agreement.
This constancy of the values of over small energy windows is the primary idea underlying the eigenstate thermalization hypothesis. Notice that in particular, it states that the expectation value of ''in a single energy eigenstate is equal to the value predicted by a microcanonical ensemble constructed at that energy scale.'' This constitutes a foundation for quantum statistical mechanics which is radically different from the one built upon the notions of dynamical ergodicity.

Tests

Several numerical studies of small lattice systems appear to tentatively confirm the predictions of the eigenstate thermalization hypothesis in interacting systems which would be expected to thermalize. Likewise, systems which are integrable tend not to obey the eigenstate thermalization hypothesis.
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. This is a system which is known to exhibit chaos classically. For states of sufficiently high energy, Berry's conjecture states that energy eigenfunctions in this many-body system of hard sphere particles will appear to behave as superpositions of plane waves, with the plane waves entering the superposition with random phases and Gaussian-distributed amplitudes. Under this assumption, one can show that, up to corrections which are negligibly small in the thermodynamic limit, the momentum distribution function for each individual, distinguishable particle is equal to the Maxwell–Boltzmann distribution
where is the particle's momentum, m is the mass of the particles, k is the Boltzmann constant, and the "temperature" is related to the energy of the eigenstate according to the usual equation of state for an ideal gas,
where N is the number of particles in the gas. This result is a specific manifestation of the ETH, in that it results in a prediction for the value of an observable in one energy eigenstate which is in agreement with the prediction derived from a microcanonical ensemble. Note that no averaging over initial states whatsoever has been performed, nor has anything resembling the H-theorem been invoked. Additionally, one can also derive the appropriate Bose–Einstein or Fermi–Dirac distributions, if one imposes the appropriate commutation relations for the particles comprising the gas.
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. A rough criterion is that the average thermal wavelength of each particle be sufficiently smaller than the radius of the hard sphere particles, so that the system can probe the features which result in chaos classically. However, it is conceivable that this condition may be able to be relaxed, and perhaps in the thermodynamic limit, energy eigenstates of arbitrarily low energies will satisfy the ETH.