Random Matrix Theory, Quantum Chaos, and Eigenstate Thermalization

Hiroki Nakayama Department of Applied Physics, Stanford University, Stanford, CA 94305 (Dated: July 2, 2020) Submitted as coursework for PH470, Stanford University, Spring 2020

In this paper, the main results from Random Matrix Theory (RMT), quantum chaos and Eigen- state Thermalization Hypothesis (ETH) will be surveyed. The theory behind quantum chaos and ETH is deeply rooted in RMT. It will be found that RMT can be used as an indicator for quantum chaos by analyzing the distribution of energy level spacing in quantum systems. ETH is a generalization of RMT in that one can reproduce RMT from ETH in a narrow energy window. The paper explores the connections in RMT, quantum chaos, and ETH. c (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 of the rights, including commercial rights, are reserved to the author.

I. INTRODUCTION focusing on their wave functions, one should instead fo- cus on the observables. In his theorem, the quantum A classical system in which its motions are governed by ergodic theorem, he stated that observable will relax to 9 non-linear equations can exhibit chaotic motions. Chaos a microcanonical distribution at later times. This relax- in the classical sense means that the the system has an ation of observables to the microcanonical distribution exquisite sensitivity to small perturbations in its phase- is called thermalization, a concept deeply connected to space trajectories. A famous example is the ”Butterfly RMT. Since RMT was limited in describing physical ob- Effect”, illustrating the point that any small difference in servables in real systems, in the 1990’s Srednicki gen- the initial condition can cause chaotic behavior. eralized its notion by making an ansatz, known as the 1,4–6 On the other hand, this notion of classical chaos does Eigenstate Thermalization Hypothesis (ETH). not apply directly to a quantum mechanical system, in Following the discussions given by D’Alessio et.al.1, which its dynamics is governed by the linear Schrödinger this paper will survey some of the key concepts in RMT, equation. As in classical chaotic systems, one can use quantum chaos and ETH. The goal of this paper is to phase-space methods to describe quantum mechanical illustrate how these concepts relate to one another. systems; however, the notion of trajectory is meaningless since the uncertainty principle forbids simultaneous measurement of coordinates and momenta. In such case, one may ask what chaos is, in the quantum mechanical setting. II. CLASSICAL CHAOS Although, to this day, there is no precise definition of quantum chaos, it is referred to as chaos in the quan- A. Phase space and Liouville’s Theorem tum mechanical setting. The basis for the understanding of quantum chaos has been formulated by Wigner and Dyson to understand the spectra of complex atomic nu- Suppose we have a system of N point particle systems clei, now called Random Matrix Theory (RMT).1 The in which the position qi and momenta pi of each par- idea of this theory is that rather than finding the exact ticle is known, where i indicates the i-th particle. For eigenspectra of chaotic Hamiltonians, one should instead instance, let there be N particles. In this case, if the probe their statistics. position and momentum for each particle have 3 compo- Following this idea, Bohigas, Giannoni, and Schmit nents, then the subscript i will run from 1 to 3N. A pair has formulated the BGS conjecture stating that quantum of position and momentum (qi, pi) is referred to as the systems with a classical chaotic analog are described by degree of freedom. In the example above, the degree of RMT.3 If the energy level spacing of arbitary quantum freedom is then, 3N. 3N of these pairs are needed to systems follow this he Wigner-Dyson distribution, then describe a dynamical system. it can be said that the system is a quantum chaotic sys- Consider a conserved system, or in other words, a system. RMT acts as a tool to indicate whether a quantum tem without any dissipation. A Hamiltonian of such sys- system is chaotic or not. tem is described as a function of qi and pi that completely RMT also has deep connections with thermalization describes the system. The dynamics of the system can be of quantum mechanical systems. von Neumann’s insight expressed by considering the time evolution of the vari- for quantum mechanical systems was that rather than ables, qi and pi, namely, qi(t) and pi(t). In such case, the 2 following Hamilton’s equations,2 conditions inside a volume in phase space will stay con- dρ 2 stant. This allows us to make the notion that dt = 0. ∂H(q, p) dq 2 = i (1) Applying this to (5), we get the following, ∂pi dt ∂ρ X ∂ρ ∂H ∂ρ ∂H + − = 0. (6) ∂t ∂q ∂p ∂p ∂q ∂H(q, p) dp i i i i i = − i (2) ∂qi dt This is called the Liouville’s theorem. The terms inside the brackets describe how the density ρ varies with re- where i = 1,..., 3N, if as in the above example, each spect to q and p. On the other hand, ∂ρ describes the dy- particle has 3 components for their momenta and posi- ∂t namics of ρ at a fixed point defined in phase space. What tion. It is worth noting that the Hamilton’s equations this theorem implies is that ρ does not evolve chaotically are coupled equations for each degree of freedom. because it is linear in ρ. In other words, if the system The Hamiltonian for the conserved system is depen- starts in a different initial condition, then the probability dent on momenta p, and position q. Let the system distribution of the initial conditions will not evolve ex- not be subject to time dependent forces. The total time ponentially compared to the original distribution. This derivative of the system can be found by the following,2 notion can be extended not only for non-chaotic systems, but also for chaotic systems in which the trajectories are dH(q, p) X ∂H dpi ∂H dqi ∂H = + + . (3) 2 dt ∂p dt ∂q dt ∂t diverging exponentially. i i i Due to the assumption that the system is not subject to ∂H B. Integrable and nonintegrable systems time dependent forces, ∂t = 0. Now, using the coupled differential equations for the position and momenta,2 Consider a conserved system with its trajectory in ∂H ∂H ∂H ∂H phase space start at initial condition (q0, p0). Since the − = 0. (4) system is conserved, the energy of the system will have ∂qi ∂pi ∂pi ∂qi the same value at later times, which leads us to write If (4) holds true for each i in (3), it can be concluded that the Hamiltonian as H(q(t), p(t)) = H(q0, p0). Here, dH(q,p) (q(t), p(t)) is the trajectory in phase space. Recall Hamil- dt = 0 and H itself is a conserved quantity of the system. If H is the total energy, what ever the motion ton’s equation considering momenta in (2). Consider a case where ∂H = − dpi = 0. From this, we can say of the trajectories are in phase space (discussed in this ∂qi dt section later), H is constant. that when H does not have dependence on qi, there is Next, the notion of Phase space of a conserved system no time dependence on pi. Therefore, in this system, will be mentioned. Phase space is a space created by the trajectories are constrained by the constant energy position p and momenta q. Let a system under consider- value H(q0, p0) and pi values. If the system has N de- ation have N degrees of freedom, then its dimensionality grees of freedom, and a total of M conserved quantities, is 2N since each degree of freedom is represented by the then the trajectories will be on a 2N − M dimensional pair q and p. In such conserved systems H(q, p) = E, surface. There are two types of conserved systems: inte- where E is a constant energy value defined by the initial grable and nonintegrable systems. A system is said to be conditions of q and p. The trajectory that the system integrable if the number of conserved quantities is equal will follow in phase space will be confined to this 2N − 1 or greater than the number of degrees of freedom. On the dimensional constant energy surface.2 other hand, a system is considered nonintegrable if the Notice that there are multiple possibilities of initial number of conserved quantities is fewer than the number conditions that the system can take. Therefore, we can of degrees of freedom. In an nonintegrable system there define a distribution of initial conditions in phase space. are less constraints on the trajectories, therefore mak- Let the probability density function of the distribution ing them able able to travel through phase space more be represented as ρ(q, p). In this case, this function is freely. In such system, chaotic motions are possible. In defined so that the probability P (q, p) of finding a tra- an integrable system, on the other hand, the motions of jectory in an infinitesimal volume dV in phase space is the system are constrained by the conserved quantities; defined by P (q, p) = ρ(q, p)dV . The probability density therefore, in such a system, generally, the system will not function ρ(p, q) can be shown to follow the following dy- show chaotic motion. However, there are cases in which namics as the system evolves over time,2 integrable systems display sensitivity to initial conditions leading to chaotic behavior. dρ X ∂ρ ∂H ∂ρ ∂H ∂ρ = − + . (5) In this section, the phase space formulation of con- dt ∂q ∂p ∂p ∂q ∂t served classical systems have been discussed. In such for- i i i i i mulation, it is possible to analyze chaotic behavior and It can be shown that for a conserved system in which classically, it is well understood. One may ask if there its Hamiltonian changes with time, the number of initial is a quantum analogy to these chaotic behaviors, namely 3 quantum chaos. This will be discussed in the following # sections.

III. RANDOM MATRIX THEORY AND QUANTUM CHAOS

As described in the previous section, a system is called chaotic when it has an exquisite sensitivity to its ini- (λ − λ ) tial conditions in its phase-space trajectories. One can- ! " not directly use the phase-space trajectory argument in quantum mechanical systems since it is forbidden by the uncertainty principle to measure both coordinate and momenta simultaneously, making it impossible to define what a trajectory in phase space actually is. The notion of trajectory becomes meaningless in the quantum mechanical setting. Recall Hamilton’s equations in (1) and (2). For chaotic motions to occur in classical mechanics, the Hamiltonian H must be nonlinear. In quan- FIG. 1: Schematic of energy level repulsion in a 2 × 2 Hamil- tum mechanics, the Hamiltonian H is described by the tonian. Schrödinger’sequation, which is linear. It is then obvious that one cannot smoothly apply the concepts in classical chaos directly to chaos in quantum mechanical systems. If the two level system has time reversal symmetry, then Hˆ is a real matrix and hence its off diagonal entries As mentioned in the introduction, chaos in quantum me- ∗ chanical systems, or quantum chaos has its core theory satisfy G = G by hermiticity. In the case where the sys- deeply rooted in Random Matrix Theory (RMT). RMT tem has its time reversal symmetry is broken, then the is also the basis for eigenstate thermalization. This sec- real and the imaginary values of the off diagonal entry G tion will review RMT and its connections to quantum can themselves be treated as random variables, each cho- chaos and necessary concepts to understand the key con- sen from Gaussian distribution. Let the level separation cepts in eigenstate thermalization later developed in this be expressed as ω = E1 − E2. The probability distri- paper. bution of the level separation P (ω) in the case when the system has time reversal symmetry, and in the case when it does not, are given in the following respectively.1 A. Random Matrix Ensembles and Wigner-Dyson ω ω2 Distributions P (ω) = exp − (9) 2σ2 4σ2 Random Matrix Theory (RMT) is a theory to provide insight into the many properties of random matrices in ω2 ω2 P (ω) = √ exp − (10) which its entries are chosen randomly from specific prob- 2 π(σ2)3/2 4σ2 ability distributions called random matrix ensembles. In order to illustrate the characteristics of RMT, first, con- There are two characteristics that can be drawn from sider a simple 2 × 2 Hamiltonian whose entries are taken the statistics. Firstly, in the limit ω → 0, it can be seen from a Gaussian distribution with a mean of zero and that P (ω) → 0, indicating that there is a level repulsion 1,7 with variance σ. between the two eigenvalues of the Hamiltonian E1,E2, " # as mentioned previously and shown schematically in Fig. λ √G 1 2 1. Secondly, P (ω) decays in the limit where ω is large. Hˆ = ∗ (7) √G λ 1 2 2 If we generalize the above distributions, γ 2 The eigenvalues of the Hamiltonian can be found using P (ω) = Nγ ω exp −Mγ ω (11) the general procedure, and it is found that eigenvalues where γ is the variance of the distribution and an indi- E ,E are 1 2 cator of whether the system has time reversal symmetry. p 2 2 γ = 1 is when the system is time reversal symmetric λ1 + λ2 (λ1 − λ2) + 2|G| E = ± . (8) and γ = 2 is when time reversal symmetry is broken. 1,2 2 2 Nγ , Mγ are normalization constants from P (ω). This The plot of E1,E2 with respect to λ1 −λ2 is given in Fig. distribution in (11) is called the Wigner surmise. In par- 1. It is shown that E1 and E2 do not cross each other, ticular, the energy level spacing distribution presented or in other words, the energy levels are repulsive to each here for γ = 1 and γ = 2 are what the Gaussian orthogo- other. This is called energy repulsion. nal ensemble (GOE) and the Gaussian Unitary Ensemble 4

(GUE) follow respectively. The GOE and the GUE will eigenspectra displays RMT statistics, the Wigner-Dyson be explained below. distribution. Suppose there is an ensemble of random matrices of arbitrary size in which the components are drawn from a Gaussian distribution as in the 2 × 2 Hamiltonian ex- B. Matrix elements of observables ample. The distribution of these random matrices are 1,7 expressed as the following, For the convenience of later discussions, let an operator   Oˆ be called the observable of a given system. Since it γ X is an observable, it can be thought of as a Hermitian P (Hˆ ) ∝ exp − H H∗ , (12)  2k2 ij ji operator, which can be represented as,1 ij ˆ X where k represents the energy scale. There are three O = Oj |ji hj| (13) ensembles of matrices that can be extracted from this j distribution. (i) Gaussian orthogonal ensemble (GOE) The ensemble of Hamiltonians with time reversal sym- and that Oˆ |ji = Oj |ji. Suppose a system can be rep- metry, γ = 1, and its components satisfy Hij = Hji. (ii) resented as a Hamiltonian that is a true random matrix Gaussian unitary ensemble (GUE) The Hamiltonians do and that its eigenstates are represented as |ni and |mi. It not have time reversal symmetry, γ = 2, and components can then be said that the entries of the observable under ∗ 1 satisfy Hij = Hji due to the fact that Hamiltonian en- the these eigenstates are, tries are complex. (iii) Gaussian Symplectic Ensemble ˆ X m ∗ n (GSE) The ensemble of Hamiltonians that are invariant Omn = hm| O |ni = Oj(Ψj ) Ψj (14) with respect to symplectic transformations. In this case, j γ = 4.7 m n Note that these are the distribution for the random where Ψj ≡ hj|mi and Ψj ≡ hj|ni are the in- ensembles, not their energy level spacing as was shown ner products of the eigenstates of the observable and for 2 × 2 Hamiltonian. When analyzing a certain com- the Hamiltonian. Before proceeding, it is necessary plex quantum system, one will generate an ensemble of to briefly mention about the eigenvectors of the ran- matrices such as the GOE and GUE that will represent dom matrices. The distribution of the components of the Hamiltonian of the system. In the general case of the eigenvectors corresponding to the GOE and GUE P 2 ensembles of larger matrices, unlike P (ω) in the 2 × 2 can be written as PGOE(Ψ1, Ψ2,...) ∝ δ i Ψi − 1 , P 2 1,7 Hamiltonian system as in the example, the analytic form PGUE(Ψ1, Ψ2,...) ∝ δ i |Ψi| − 1 respectively. is hard to find, but its form is close to the Wigner sur- The probability distribution of the eigenvector compo- mise (11).1,7 Such distribution of level spacing is called nents imply that eigenvectors of random matrices are es- the Wigner-Dyson distribution. Note that ”level spacing” sentially random unit vectors, that is, real or complex here refers to the energy difference between neighboring corresponding to GOE or GUE.1 Now going back to the energy eigenvalues. original discussion, if we let d represent the dimension of As previously stated in the introduction, Bohigas, Gi- the Hilbert space in which it is spanned by the eigenstates annoni, and Schmit discovered that the level spacing dis- of the Hamiltonian, then the average of the product be- tribution of a single particle system followed the Wigner- tween eigenvector components over random eigenstates Dyson distribution in a narrow window of high energies.3 m n 1 1 can be expressed as Ψi Ψj = d δmnδij. This is due to On the same paper, they made a conjecture that in the orthogonality between the eigenvectors that are es- the limit of high energies, if a classical system is highly sentially random unit vectors as previously mentioned. chaotic, then it will have corresponding quantum energy Using this relation, the average value of the observable levels that follow the Wigner-Dyson distribution.3,10 If components can be represented as the following1, the level spacing distribution follows the Wigner-Dyson ( distribution, then it can be said that it is reflecting chaos 1 P O , for m = n O = d i i (15) in the quantum level, or quantum chaos. Studying these mn 0, for m 6= n level spacing statistics from an ensemble of random matrices such as the GOE and the GUE lies at the core of This shows that when evaluating for the average value of RMT. the observable components one finds that only the diag- It is important to note the difference between a physi- onal entries are nonzero, and that it gets suppressed by cal Hamiltonian and the Hamiltonian drawn from RMT. the dimension of the Hilbert space. Most physical Hamiltonians are expressed as sparse ma- The fluctuations of the observable entries can be shown trices that are local, which can be non-random. On the to satisfy the following,1 other hand, the Hamiltonian drawn from RMT is a dense random matrix. It seems RMT cannot be applied to ( 3−γ P 2 physical Hamiltonians; however, under the special condi- 2 2 d2 i Oi , for m = n Omn − Omn = 1 P 2 (16) tion that the physical Hamiltonian is non-integrable, its d2 i Oi , for m 6= n 5

Assuming that the fluctuations are small and independent with respect to the dimension of the Hilbert space, (16) tells us that the fluctuation gets smaller and smaller with increasing dimension. Using these expression for the fluctuations, one can express the observable entries as the following to the leading order 1/d,1 s O2 O ≈ Oδ¯ + R (17) mn mn d mn

1 P 2 1 P 2 where it was defined that O ≡ d j Oj, O ≡ d j Oj and that Rmn is a component of a random matrix chosen from a Gaussian distribution of zero mean and the variance of either γ = 1 for (GOE) and γ = 2 for (GUE), in which it will be real and complex respectively. The FIG. 2: GOE following Wigner-Dyson distribution. The Pois- average expressions given in these set of equations are son distribution is plotted for convenience. averaged over the eigenstates of each of the Hamiltonians in an ensemble. As long as the dimension of the Hilbert space d is large enough, or in other words, the samples then the new probability distribution will not exponen- of the Hamiltonians are large enough, the observable of a tially diverge from the original probability distribution, system governed by a fixed Hamiltonian can be approxi- even if the system itself is chaotic or not.2 The role of mated using equation (17). the probability distribution in phase space in the quantum setting is played by the Wigner-Dyson and Poisson distributions. C. Quantum Integrability To illustrate the above discussions, suppose we generate an ensemble of random matrices. To be more specific, In classical integrable and nonintegrable systems, the let the ensemble be GOE. In Fig.2, the energy level spac- notion of chaos is well defined via trajectories in phase ings for the GOE are shown as the histogram. It can space. However in the quantum setting, the notion of be seen that the energy level spacings follow the Wigner- chaos is nontrivial. To understand quantum chaos, one Dyson distribution. This indicates that the system shows can use RMT and look at many different statistical fea- signs of quantum chaos. For convenience, the Poisson tures to analyze the random matrix eigenspectrum; how- distribution was plotted together in Fig. 2. In the case ever, the common procedure is to probe the distribution where the random matrix ensemble follows the Poisson of energy level spacing of the eigenstates. In the previous distribution, the system is said to be integrable. section, it was stated if the level spacing distributions of To summarize, if one finds that the level spacing dis- the quantum system follows the Wigner-Dyson distribu- tribution follows Poisson or Wigner-Dyson distribution, tion, then it is an indication of quantum chaos. As in then it means that they have found a signature for a classical systems, one may ask if there exists integrabil- quantum integrable or chaotic system respectively. ity in the quantum setting and if there exists its indicator for it. In the quantum setting, integrals of motion and its indicator does exist. As the Wigner-Dyson distribution IV. EIGENSTATE THERMALIZATION was the distribution of energy level spacing for quantum chaotic systems and the indicator for such systems, the A. Classical Thermalization distribution of energy level spacings for quantum integrable systems follow a Poisson distribution,1 Let O represent an observable of an isolated system. P (ω) = exp[−ω]. (18) According to equilibrium statistical mechanics, the time average of such observables over some time interval T , where ω is the energy level spacing and the mean level hOiT are often what gets quantified. This time interval spacing was set to unity. The main difference of this T will be extended to infinity to account for the system to distribution from the Wigner-Dyson distribution, which reach thermal equilibirum, or thermalization. However, again follows a form similar to the Wigner surmise in when the time interval is taken to infinity, it seems impos- (11), is that the Poisson form does not have energy level sible to keep track of the value of the observables for all repulsion. From this view point, it can be said that to times. It is necessary to find a different way to calculate probe the distribution of energy level spacing is to probe this quantity. Recall the phase space formulation in clas- the energy level repulsion. sical mechanics as introduced in section II of this paper. Recall Liouville’s theorem in (6). If the conserved sys- If a system get arbitrarily close to following every possible tem started with a slightly different probability density ρ, point in phase space in its phase space trajectory, Λ(t), 6 then the system is defined to be ergodic.10 In the limit observable O(t) ≡ hψ(t)| Oˆ |ψ(t)i can be expressed as of long times, ergodicity states that the phase space tra- X X i(Em−En)t jectory Λ(t) will follow every point in a constant energy 2 ∗ h¯ O(t) = |Cm| Omm + C Cne Omn (21) 10 m surface. Furthermore, Liouville’s theorem implies that m m6=n this phase space trajectory will end up covering the con- 10 stant energy surface uniformly. This leads to the fact 1 where Omn = hm| Oˆ |ni. Here it can be seen that the that the time average of an observable hOiT can also be time averaged expectation value O(t) is expressed in calculated by averaging over the phase space, which is terms of expectation values of these observables under confined by a constant energy surface ES; therefore,10 energy eigenstates |mi and |ni, Omm and Omn. If we take R the long time average of O(t), the second term in (21) will ES O(Λ)dΛ go to zero (due to exponential decay with system size) hOiT = R . (19) ES dΛ under the assumption that there are no degeneracies. On the other hand, the microcanonical average in the quan- This is the formulation of averages in the microcanonical tum setting involves averaging over all eigenstates within ensemble in the classical setting. Classically, the fact a small energy window ∆E.10 For large systems, this en- that the long time average is equal to the microcanonical ergy window ∆E is very small, and this itself contains average is well defined. This is not so in the quantum many eigenstates. The microcanonical average value is setting. given by averaging over all the energy eigenstates in this window, which is centered around the mean energy of the initial state of the system.12 The microcanonical average 12 B. Quantum Thermalization of the observable O is expressed as the following, 1 X O = Omm (22) Quantum statistical mechanics postulates that a quan- Neigen tum system is in contact with an external reservoir.11 m When a quantum system is not in contact with such reser- where Neigen is the number of eigenstates in the small voir, it is called a closed quantum system. However, it is energy window ∆E. In order for thermalization to occur possible to divide such system into a subsystem, which for this closed quantum system, the microcanonical av- contain a fraction of the total degrees of freedom in the erage O in (22) must equal the long time average value whole system, and the remaining portion of the closed of (21), which is just the first term P |C |2O . This 11 m m mm system as a reservoir. If the rest of the system can act equivalence is the challenge of defining thermalization in a reservoir for the small subsystem, then we can recover closed quantum systems. It is then natural to ask if there the notions of quantum statistical mechanics again and is a way to reconcile this problem. at late times, the subsystem can undergo thermalization. If we draw the Hamiltonian Hˆ out of RMT, we can Before proceeding, note that observables in closed quan- use the notion of observables under RMT (17). This tum systems are local. In a given system, we cannot track states that the diagonal elements of the observable, Omm all the degrees of freedom upon measurement. Instead, is independent of energy eigenstate |mi and that the off- we can only measure a small portion of the entire system, diagonal elements Omn are exponentially small in sys- which what makes the observables very local. tem size.1 Applying these statements from RMT, the Suppose there is a closed quantum system such that P 2 first term in (21) will then become m |Cm| Omm ≈ it is initially in the pure state |ψ0i and its dynamics are O P |C |2 = O. The second term of (21) will van- ˆ m m described by a time independent Hamiltonian H that has ish since the off-diagonal elements become uncorrelated. energy eigenvectors |mi with eigenvalues E , Hˆ |mi = m If we denote the long time average of O(t) as hO(t)iLT , E |mi.1 If we want to find the time-evolving state of m then the above arguments make hO(t)iLT = O, which the system |ψ(t)i, we use the corresponding Schödinger is essentially the long time average of the observable is ∂ ˆ equation, which is expressed as i¯h ∂t |ψ(t)i = H |ψ(t)i = equal to the microcanonical average, a statement of ther- Em |ψ(t)i. The time-evolving state of the closed quantum malization. system can be expressed as the following,1 In practice, RMT is not enough to describe observables in physical systems. As we have seen above, we were only X −iEmt |ψ(t)i = Cme h¯ |mi (20) able to see the microcanonical average equal to the long m time average of the observable when we assumed that the Hamiltonian Hˆ of the closed quantum system was drawn where Cm = hm|ψ(0)i = hm|ψ0i. Now that we have from RMT, meaning that the Hˆ matrix is a dense matrix. an expression for the time evolution of the state of the However, in a physical system, the Hamiltonian is sparse closed quantum system, we can shift directions to ana- and local. A certain generalization of RMT is necessary lyzing the time averaged expectation value of the observ- to account for observables in real systems, which was able O(t) ≡ hψ(t)| Oˆ |ψ(t)i. Using this time dependent done by Srednicki who made an ansatz known as the evolution of the state vector, the time evolution of the Eigenstate Thermalization Hypothesis (ETH).1,4–6. 7

C. Eigenstate Thermalization Hypothesis (ETH) where RMT applies. In this way, RMT and ETH are related to each other with latter being the generalization Srednicki made the following ansatz that will generally of the former. hold for physical observables,1,4–6 −S(E)/2 Omn = O(E)δmn + e f(E, ω)Rmn (23) D. ETH and Thermalization where E ≡ (Em + En)/2, ω ≡ En − Em and S(E) is the thermodynamic entropy at the argument energy E that is As stated in the introduction, von Neumann stated in S(E) P his theorem, the quantum ergodic theorem, that a macro- given by, e = E m δ(E −Em). The functions O(E) and f(E, ω) are smooth functions with respect to their scopic observable will relax to a microcanonical distribution at later times.1 In this section, by following this arguments and Rmn is a real or complex numerical factor that varies randomly with m and n such that R2 = 1 theorem, it will be explicitly shown how the concept of mn thermalization is related to ETH. and |R |2 = 1. Note that the observable represented mn In this theorem,von Neumann considered a quantum in this ansatz is constructed by taking the eigenstates system with N interacting particles confined in a fi- of the Hamiltonian as the basis. If the Hamiltonian of nite volume. Due to the confinement of the system the system obeys time-reversal symmetry, the eigenstate in a closed space, the Hamiltonian Hˆ governing the components are real hence the matrix components of the system will form discrete energy levels E such that observables are real as well. If time reversal symmetry m Hˆ |mi = E |mi where |mi is the corresponding eigen- is broken, the matrix components of the observables are m states or the orthonormal basis of the system’s Hilbert complex values. Without losing generality, we can take space H of dimension d. The theorem focuses in the mi- the function f(E, ω) to be an even function in the ar- crocanonical energy window δE, so it can be expected gument ω.1 Taking the Hermitian conjugate of equation that E ∈ (E − δE/2,E + δE/2). In this window, we (14), the random numerics R and the function f(E, ω), m mn can define an arbitrary state ψ in terms of eigenstates |mi satisfies R = R and f(E, −ω) = f(E, ω) for a sys- mn nm such that |ψi = C |mi where C = hm|ψi. The total tem with time reversal symmetry and R = R∗ and m m mn nm Hilbert space H can be decomposed in terms of mutu- f(E, −ω)∗ = f(E, ω) for a system without time reversal ally orthogonal Hilbert subspaces such that H = L H symmetry.1 ν ν and d = P d where H refers to the Hilbert space of Comparing this ETH ansatz with the RMT prediction ν ν ν microstate ν and d as its dimensionality. What von of the observables given in equation (17), the diagonal ν Neumann meant when he said ”macroscopic” is that elements in the ansatz, O(E), are no longer same val- it is built upon microstates ν. The macroscopic ob- ues, but smooth functions in their arguments E and ω. servable Oˆ = P O Pˆ where Pˆ is the projector to On the other hand, the off-diagonal elements contain a ν ν ν ν Hν . Let O(t) be the observable at time t such that thermodynamic entropy factor to account for the ther- iHtˆ −iHtˆ mal fluctuations in physical systems and a smooth func- O(t) = hψ| e h¯ Oeˆ h¯ |ψi. The microcanonical aver- P ˆ tion f(E, ω). As stated previously, the ansatz is also a age is given as hOiME = m hm| O |mi /d. We then generalization of the RMT prediction of observables in require for Hˆ to not have any nondegenerate energies, 0 0 (17). The ETH ansatz reduces to the RMT prediction Em − En 6= Em0 − En0 unless m = m , n = n or in a narrow energy window called the Thouless energy m = n, m0 = n0. The central statement of the quantum 1,9 ET , a measure of particles’ sensitivity to feel the bound- ergodic theorem is of the following, aries of the system expressed in terms of energy.8 In fact, 2 ET is the smallest possible energy window. It is defined 2 dν hD¯ F = max | hm| Pν |mi | + max hm| Pν |mi − . as ET = L2 where L is the edge length of the system m6=n m d and D is the diffusion constant. If the energy level sep- (24) aration ω satisfies ω < E , then the function f(E, ω) is 2 T If F is exponentially small, then |O(t) − hOîME| < constant in that region because it is smooth in its ar- hOˆ2i where δ is a small number. If we compute for guments. Giving such restriction to ω makes the ETH ME the expectation value of the observable Oˆ, we find the ansatz approximately equal to RMT prediction for the diagonal components as observables shown in equation (17). Note that this re- duction to ETH to RMT occurs only for diffusive systems d hm| Pˆ |mi 2 ˆ X ν X ν with diffusion time t longer than L . Diffusion is in fact a hm| O |mi ≈ Oν = Oν d d result of physical systems with local interactions leading ν m∈H,ν to local Hamiltonians. Once ω < ET is not satisfied, the X hm| Oˆ |mi (25) ETH ansatz will no longer appear as a RMT prediction. = Oν d From the argument made here, it seems RMT is use- m∈H less because of its restrictive necessary condition. Notice = hOiME however, that as the system size increases, energy level P ˆ 1 spacings become closer together. Even in the restriction where m∈H hm| Pν |mi. This states that the diago- that ω < ET , there are exponentially many energy levels nal components are equal to the microcanonical average 8 value, which is indeed what von Neumann stated in the RMT is deeply rooted in the theory behind quantum theorem. For the off-diagonal components,on the other chaos and ETH. Using RMT, one is allowed to find the hand, we find, distributions for energy level spacings. If the the energy level spacing of a quantum system follows the Wigner- X hm| Oˆ |ni = Oν hm| Pˆν |mi . (26) Dyson distribution, then the system is said to be a quan- ν tum chaotic system. If on other hand, the spacing follow Poisson distribution, then the system is said to be quan- ˆ However from (24), it is required that hm| Pν |mi is ex- tum integrable system. Therefore, RMT can be used as ponentially small and Oν is a finite physical observable; an indicator for whether the system is chaotic or not. As therefore, the off-diagonal components are exponentially in quantum chaos, ETH is also deeply related to RMT. 1 ˆ small in system size. Note that Omm = hm| O |mi and ETH was formulated to understand thermalization of ob- Omn = hm| Oˆ |ni. What is significant about this results servables in physical systems. ETH is actually a general- is that this approximately equal to the RMT prediction ization of RMT in a way that ETH reduces to the RMT for observables presented in (17), which in other words, prediction of observables when considering energy level ETH in Thouless energy window δE ∼ ET . spacing within the Thouless energy window. It is re- markable how powerful RMT is in establishing methods to probe quantum chaos and acting as a basis for ETH. V. CONCLUSION

In this paper, the main results from RMT, quantum chaos, and ETH has been surveyed. One can say that

1 Luca D’Alessio, Yariv Kafri, Anatoli Polkovnikov and Mar- 7 Y. Alhassid,“The statistical theory of quantum dots,” Rev. cos Rigol, “From quantum chaos and eigenstate thermal- Mod. Phys. 72 895 (2000). ization to statistical mechanics and thermodynamics,” Ad- 8 E. Akkermans and G. Montambaux, Mesoscopic Physics of vances in Physics, 65:3, 239-362 (2016). Electrons and Photons, Cambridge University Press, New 2 Robert C. Hilborn, Chaos and Nonlinear Dynamics: An York, 2007. Introduction for Scientists and Engineers Oxford Univer- 9 Goldstein, S., Lebowitz, J., Tumulka, R. et al. Long-time sity Press, New York, 2000. behavior of macroscopic quantum systems. EPJ H 35, 3 O. Bohigas, M. J. Giannoni, and C. 173–200 (2010). Schmit,“Characterization of Chaotic Quantum Spec- 10 Joshua M Deutsch, “Eigenstate thermalization hypothe- tra and Universality of Level Fluctuation Laws” Phys. sis,” Rep. Prog. Phys. 81 082001 (2018). Rev. Lett. 52, 1 (1984). 11 R. Nandkishore, D. Huse, “Many-body Localization and 4 Mark Srednicki,“Chaos and quantum thermalization,” Thermalization in Quantum Statistical Mechaincs,” Annu. Phys. Rev. E. 50, 888 (1994). Rev. Condens. Matter Phys. 2015. 6:15-38 (2015). 5 Mark Srednicki,“Thermal ﬂuctuations in quantized chaotic 12 M. Rigol, V. Dunjko and M. Olshanii Nature 452 854 systems,” J. Phys. A. 29.4, L75 (1996). (2008). 6 Mark Srednicki,“The approach to thermal equilibrium in quantized chaotic systems,” J. Phys. A. 32.7, 1163 (1999).