Random Matrix Theory, Quantum Chaos, and Eigenstate Thermalization

Fig. 1: The level spacings of GOE follow Wigner-Dyson distribution. As a comparison, a Poisson distribution is plotted as well. (Source: H. Nakayama)

Quantum Chaos and Eigenstate Thermalization Hypothesis (ETH) has its theory deeply rooted in Random Matrix Theory. Here, we will go over the paper by D'Alessio et al. on their discussion on these topics. [1] This article will illustrate how quantum chaos and ETH are bridged with RMT.

Introduction

A classical system in which its motions are governed by non-linear equations can exhibit chaotic motions. Chaos in the classical sense means that the the system has an exquisite sensitivity to small perturbations in its phase-space trajectories. A famous example is the "Butterfly Effect", illustrating the point that any small difference in the initial condition can cause chaotic behavior. However, quantum mechanics is governed by the linear Schrodinger equation. As in classical mechanics, one can use phase-space methods to describe quantum mechanical systems; however, the notion of trajectories in phase-space become meaningless due to the uncertainty principle forbidding simultaneous measurement of coordinates and momenta. If so, one may ask what chaos is, in the quantum mechanical setting, namely, quantum chaos. Although to this day, there is no precise definition of quantum chaos, one can capture the signature of it, using Random Matrix Theory (RMT) to find the distributions of level spacing in a given quantum system. RMT has also great use in explaining thermalization in closed quantum systems, a concept that the long time average of the observable of the system relaxing to its microcanonical expectation value. The limitation of RMT is that it cannot account for the locality of Hamiltonians that govern a physical system. The generalization of RMT, the Eigenstate Thermalization Hypothesis (ETH) has been proposed to overcome this issue.We will bridge together RMT with quantum chaos and ETH in the following discussion below.

Random Matrix Theory and Quantum Chaos

Suppose that we have a Hamiltonian H that governs a quantum mechanical system. If H preserves time reversal symmetry, then the matrix entries are all real numbers. In the case when H does not have time reversal symmetry, the entries are complex numbers. When analyzing complex quantum systems, one can use random matrices in which its components are drawn from a Gaussian distribution, to express the Hamiltonian of a system. By generating an ensemble of random matrices, or in this case, an ensemble of the Hamiltonian of the system, we can study its distribution. If the matrices of the ensemble have time reversal symmetry, the ensemble is called Gaussian orthogonal ensemble (GOE) and if they are instead do not have time reversal symmetry, the ensemble is called Gaussian Unitary Ensemble (GUE). The standard procedure in RMT is to find the distribution of the energy level spacing ω of these ensembles. Such distribution is called the Wigner-Dyson Distribution, which follows the form

where N_γ and M_γ are normalization constants and γ = 1 for GUE and γ = 2 for GOE. [1] When the energy level spacing of the random matrix ensembles follow this distribution, it is a signature that the system is a quantum chaotic system. As in classical chaos there are chaotic and integrable dynamics, and one may ask if there exists quantum integrable systems. In fact there is. RMT can be used to show that the energy spacings behave as a Poisson distribution, given by

where the mean energy spacing was set to unity. To illustrate the arguments given so far, consider Fig. 1. A matrix from GOE was generated and the histogram shown are its distribution of energy level spacings. We can see that the energy level spacings follow the Wigner-Dyson distribution. This indicates that the system shows signs of quantum chaos. For convenience, the Poisson distribution was plotted together in Fig. 1. In the case where the random matrix ensemble follows the Poisson distribution, the system is said to be integrable.

Quantum Chaos and RMT is related in this way that RMT can be used as an indicator to determine if a given quantum system is integrable or chaotic.

Random Matrix Theory and Eigenstate Thermalization Hypothesis

As stated in the introduction, an observable of a system is said to thermalize if its expectation value relaxes to the microcanonical expectation value over time. Consider an isolated quantum system in an initial state |ψ₀> with its dynamics described by the time-independent Hamiltonian H and let O represent its observable. In terms of the eigenstates of the Hamiltonian |m> one can express the time dependent state of the system |ψ> in terms of these eigenstates and hence the expectation value of the time evolution of the observable $O(t)$ as well. This is given by [1]

where E' = E_m - E_n in which E_m and E_n are the energy eigenvalues corresponding to the eigenstates |m> and |n> of the Hamiltonian respectively and C_m is the inner product between the eigenstate |m> and the initial state of the system |ψ₀>. The Planck constant was set to unity. Also note that here, O_mm = < m |O| m > and that O_mn = < m |O| n >. If the system were to thermalize, then the long time average of O(t) must equal to the microcanonical expectation value. When taking the long time average of this expectation value, we can verify that the second term in the expression on the right hand side becomes exponentially small with system size under the assumption that there are no degeneracies; therefore we are left with the first expression. In order for this to equal the microcanonical expectation value, one must use the RMT prediction for observables, in which it states that the diagonal components $O$_mm are independent of the eigenstates m and that the off-diagonal components $O$_mn are exponentially small. [1] It can be said that RMT is necessary in order to formulate the concept of thermalization in quantum systems. However, the problem in this is that this is not enough to describe observables in experimental setups since in reality, we have environmental parameters we must incorporate to the observable such as temperature and the relaxation times that are dependent on observables. To account for this problem, Srednicki made the ansatz for observables in real systems [1-4]

where E" ≡ (E_m + E_n)/2, ω ≡ (E_m - E_n), S(E") is the thermodynamic entropy of the system and O(E") and f₀(E", ω) are smooth functions in their arguments. Often, this ansatz is called Eigenstate Thermalization Hypothesis (ETH). It can be shown that this expression reduces to the RMT prediction of observables in a narrow energy window called the Thouless energy E_T, an energy that is proportional to L² where L is the edge length of the system. ETH is powerful in that it can explore observables outside this narrow energy window where RMT can be applied. In this way, ETH is the generalization of RMT.

© Hiroki Nakayama. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

References

[1] L. D'Alessio et al., "From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics," Adv. Phys. 65, 239 (2016).

[2] M. Srednicki, "Chaos and Quantum Thermalization," Phys. Rev. E 50, 888 (1994).

[3] M. Srednicki, "Thermal Fluctuations in Quantized Chaotic Systems," J. Phys. A: Math. Gen. 29, L75 (1996).

[4] M. Srednicki, "The Approach to Thermal Equilibrium in Quantized Chaotic Systems," J. Phys. A: Math. Gen. 32, 1163 (1999).