1. Introduction

Recent progress in the field of thermalization in quantum systems has provided a better understanding of the related questions (see the reviews [4, 5, 8, 13, 17, 24]). Now one of the main ideas is known as the eigenstate thermalization hypothesis (ETH).

This idea goes back to von Neumann’s seminal paper [15] (discussed in [11]) of 1929, where the main notions of statistical mechanics were re-interpreted in a quantum-mechanical way, as well as the ergodic theorem and the H-theorem were formulated and proved. Von Neumann also noted that when discussing thermalization in isolated quantum systems, one should focus on physical observables as opposed to wave functions or density matrices describing the entire system.

The next important step was made in 1991 by Deutsch. In his paper [6] based on the random matrix theory (RMT), the relationship between diagonal matrix elements of an observable and microcanonical averages was revealed.

A few years later, Srednicki provided a generalization of the RMT prediction. In 1994, for a gas of hard spheres at high energy and low density, he showed [18] that each energy eigenstate which satisfies Berry’s conjecture [2] predicts a thermal distribution for the momentum of a single constituent particle. He referred to this remarkable phenomenon as eigenstate thermalization. One of the well-known formulations of the ETH (see (2.8) below) was given by Srednicki later in [19, 20]. A slightly different formulation was suggested by Rigol et al. [21].

At present there are several different formulations of the ETH (see, for example, the discussion in [1]).

Some current progress on the ETH, its applications, and related topics can be found in the recent papers [9, 12, 14, 22].

The subject of our interest in the present paper is the common formulation (2.8) of the ETH [20] (hereafter the ansatz ). We show that this formulation does not necessarily imply thermalization (in the sense described below) of an observable of isolated many-body quantum systems. In the ETH framework, to get such thermalization, one has to postulate the canonical or microcanonical distribution in the ansatz (2.8). More generally, any other average can also be postulated in the ansatz (2.8), which leads to a corresponding equilibration condition. We hope that this consideration will help to clarify some questions related to the formulation of the ETH.

In the next section we recall the common formulation of the ETH given by Srednicki in [20] and introduce some notation. In Section 3 we consider the ansatz (2.8) as a sufficient condition for thermalization and discuss how the ansatz should be written in this context. In Section 4 we discuss how the ansatz is related to the canonical thermal average. In Section 5 we answer the question of whether the ansatz (2.8) indeed implies thermalization of observables of isolated quantum systems. This consideration suggests a natural formulation of the thermalization condition (TC) which implies such thermalization (Section 6). In Section 7, there is a generalization of TC which leads to equilibration with any density matrix.

2. On one formulation of the eigenstate thermalization hypothesis

Throughout the paper we will consider a bounded isolated \(N\)-body quantum system with \(N\gg1\). Let a Hilbert space \( {\mathcal H} \) correspond to one particle, and let \( {\mathcal H} _N = {\mathcal H} ^{\otimes N}\) correspond to the entire quantum system. Denote a Hamiltonian of the system by \( \widehat{H}{} \), the energy eigenvalues by \(E_n\), where \(0 < E_0 \leq E_1 \leq E_2 \leq \dots\), and the corresponding eigenstates by \( |n\rangle \), so that

$$ \widehat{H}{} |n\rangle = E_n |n\rangle , \qquad n=0,1,2,\dots.$$
(2.1)

A hermitian operator \( \widehat{A}{} \) corresponds to an observable \(A\) of this system, and

$$ A_{mn} := \langle m| \widehat{A}{} \kern1pt |n\rangle$$
(2.2)

are the matrix elements of the operator \( \widehat{A}{} \). The pure state \( |\psi(t)\rangle \) of the system at any time \(t\) is

$$ |\psi(t)\rangle = \sum_n c_n \kern1pt e^{-iE_n t} |n\rangle , \qquad \sum_n |c_n|^2 = 1,$$
(2.3)

and

$$ |\psi\rangle \equiv |\psi(0)\rangle := \sum_n c_n |n\rangle .$$
(2.4)

We will refer to this system as the quantum system \( {\mathfrak S} \). For a more precise mathematical definition of a quantum system see, for example, [16, 23].

The thermodynamic entropy \(S(E)\) of continuous energy \(E\) is defined by

$$ e^{S(E)} = E \sum_m \delta_\varepsilon(E-E_m),$$
(2.5)

where \(\delta_\varepsilon(x)\) is a Dirac delta function that has been smeared just enough to make \(S(E)\) monotonic. Suppose that the mean energy

$$ E := \sum_m |c_m|^2E_m$$
(2.6)

and entropy of the system are extensive, i.e., as \(N\to\infty\)

$$ E = \varepsilon N + o(N) \qquad\text{and}\qquad S(E) = sN + o(N).$$
(2.7)

In [20] the following formula for the matrix elements of the observable \(A\) in the energy eigenstate basis was suggested:

$$ A_{mn} = {\mathcal A} (E_{mn})\delta_{mn} + e^{-S(E_{mn})/2}f(E_{mn},\omega)R_{mn},$$
(2.8)

where \(E_{mn} := (E_m+E_n)/2\), \(\omega := E_m-E_n\), \( {\mathcal A} (E)\) is a smooth function, \(S(E)\) is the thermodynamic entropy at energy \(E\), \(f(E,\omega)>0\) is an even function of \(\omega\) and a smooth function of both arguments, and \(R_{mn}\in {\mathbb C}\) is a random variable which varies erratically with \(m\) and \(n\), with

$$\mathbb{E}[\operatorname{Re}(R_{mn})] = \mathbb{E}[\operatorname{Im}(R_{mn})] = 0 \qquad\text{and}\qquad \mathrm{Var} [\operatorname{Re}(R_{mn})] = \mathrm{Var} [\operatorname{Im}(R_{mn})] = 1.$$

We will refer to formula (2.8) as the ansatz .

According to [20], the ansatz (2.8) implies thermalization in many-body quantum systems in the sense that

  1. •

    the time average of the expectation value of an observable is approximately equal to its canonical thermal average at the appropriate temperature, and

  2. •

    the fluctuations of this expectation value about its time average are small.

In the present paper thermalization will be understood precisely in this sense.

This significant statement about thermalization is quite surprising because the ansatz (2.8) has nothing like the canonical distribution, but nevertheless the observable allegedly equilibrates to the canonical thermal average.

Two questions arise:

  1. (1)

    Does the ansatz (2.8) indeed imply thermalization in the above sense?

  2. (2)

    If not, how should this ansatz be modified to give a sufficient condition for thermalization?

For further use, let us introduce some more notation. The quantum uncertainty of the mean energy is

$$ \varDelta^2 := \sum_m |c_m|^2 (E_m-E)^2.$$
(2.9)

For any \(E>0\) the temperature \(T>0\) is defined by

$$ \frac{1}{T} \equiv \beta := \frac{dS}{dE}.$$
(2.10)

The canonical thermal average of an observable \(A\) is

$$ \langle \widehat{A}{} \kern1pt \rangle _\beta^{\textrm{can}} := \frac{ \operatorname{Tr} (e^{-\beta H} \widehat{A}{} \,)} { \operatorname{Tr} e^{-\beta H}}.$$
(2.11)

The microcanonical thermal average of an observable \(A\) at energy \(E\) is

$$ \langle \widehat{A}{} \kern1pt \rangle _E^{\textrm{mic}} := \frac{1}{ {\mathcal N} _{E,\Delta E}} \sum_{|E-E_n|<\Delta E} A_{nn},$$
(2.12)

where \([E-\Delta E,E+\Delta E]\), \(\Delta E>0\), is an appropriately chosen energy window (see [21]) and \( {\mathcal N} _{E,\Delta E}\) is the number of energy eigenstates in the energy window.

From (2.1) and (2.2) we get a surjective map \(E_n \mapsto A_{nn}\). Let a piecewise linear continuous function \( {\mathcal L} (E)\) be the linear interpolation for the countable set of points

$$(E_0,A_{0,0}),\ (E_1,A_{1,1}),\ (E_2,A_{2,2}),\ \dots.$$

It is obvious that \(A_{nn} = {\mathcal L} (E_n)\) for all \(n\in{\mathbb N}_0 = \{0,1,2,\dots\}\).

3. Ansatz as a condition for thermalization

Let us make some comments on the ansatz (2.8).

1. Random variable \(R_{mn}\) .

The ansatz (2.8) has a random variable on the right-hand side, whereas its left-hand side is deterministic. This is contradictory. In this sense equality (2.8) is not quite correct.

In [6, 7, 17] Hamiltonians are taken in the form \( \widehat{H}{} = \widehat{H}{} _0 + \widehat{V}{} \), where \( \widehat{H}{} _0\) is an “unperturbed” part and \( \widehat{V}{} \) is a weak “perturbation.” One can think of \( \widehat{H}{} _0\) as describing an ideal gas in a box and of \( \widehat{V}{} \) as describing two-particle interactions. However, instead of adding in these interactions explicitly, the authors model \( \widehat{V}{} \) by a random matrix from a certain random matrix ensemble with statistical properties which imitate well the main features of the perturbation \( \widehat{V}{} \); that is, the many-body Hamiltonian \( \widehat{H}{} \) resembles a random matrix. Hence, its eigenstates and matrix elements are random as well.

However, for a deterministic Hamiltonian, formula (2.8) is not correct since there is a deterministic value on the left-hand side, while the right-hand side includes the random variable \(R_{mn}\).

2. Exponentially small off-diagonal matrix elements.

From the ansatz (2.8) we see that the off-diagonal matrix elements \(A_{mn}\) are exponentially small in \(N\). But from the proof of Proposition 2 about thermalization (see Appendix B), one can see that the off-diagonal matrix elements must be small but not necessarily exponentially small in \(N\).

3. Smooth function \( {\mathcal A} (E)\) .

In the ansatz (2.8), \( {\mathcal A} (E)\) is postulated to be a smooth function. Then in [20] some arguments are given in favor of the fact that \( {\mathcal A} (E)\) is approximately equal to the canonical thermal average. From the proof of Proposition 1 (see Appendix A), one can see that \( {\mathcal A} (E)\) may be just a continuous piecewise linear interpolating function \( {\mathcal L} (E)\) for the countable set of points \((E_0,A_{0,0}),(E_1,A_{1,1}),(E_2,A_{2,2}),\dots\).

Taking this consideration into account, we can conjecture the following form of a sufficient condition for thermalization:

$$ A_{mn} = {\mathcal L} (E)\delta_{mn} + \alpha_{mn}(N), \qquad \alpha_{mn}(N)\xrightarrow[N\to\infty]{} 0.$$
(3.1)

However, below we will suggest a better formula.

4. Quantum uncertainty \(\varDelta\) .

In [20] there is an additional assumption

$$ \varDelta^2\frac{ {\mathcal A} ''(E)}{ {\mathcal A} (E)} \ll 1$$
(3.2)

(\(\varDelta\) is defined in (2.9)), which is used in showing the thermalization of quantum systems obeying the ETH. To avoid this assumption, one should postulate a slightly different thing. Namely, separating the diagonal matrix elements \(A_{nn}\) and off-diagonal matrix elements \(A_{mn}\), we get the conjectural thermalization condition

$$ \begin{cases} \displaystyle \sum_n |c_n|^2A_{nn} = {\mathcal L} \Biggl(\sum_n |c_n|^2E_n\Biggr) + o(1), \\ \displaystyle A_{mn} = A_{mn}(N) \xrightarrow[N\to\infty]{} 0, \qquad m\neq n. \end{cases}$$
(3.3)

The first equality can be written in a short and nice form, so we would have

$$ \begin{cases} \displaystyle \langle \psi| \widehat{A}{} \kern1pt |\psi\rangle = {\mathcal L} (E) + o(1), \\ \displaystyle A_{mn} = A_{mn}(N) \xrightarrow[N\to\infty]{} 0, \qquad m\neq n. \end{cases}$$
(3.4)

However, this is not the end of the story.

4. Ansatz and canonical average

In [20] there is a statement that the function \( {\mathcal A} (E)\) can be related to a standard expression in statistical mechanics, the equilibrium value of \(A\), as given by the canonical thermal average (see [20, (2.10)]). From the arguments presented in [20], we see that the precise statement should actually be as follows.

Proposition 1 (see [20]).

Consider the quantum system \( {\mathfrak S} \) (see (2.1)–(2.3) ). Then for any \(\beta>0\) there exists an \(E_\beta>0\) such that

$$ \langle \widehat{A}{} \kern1pt \rangle _\beta^{\textrm{can}} = {\mathcal A} (E_\beta) + O(N^{-1}),$$
(4.1)

where

$$ \beta = \frac{\partial S(E)}{\partial E}\biggr|_{E=E_\beta}.$$
(4.2)

Remark 1.

This proposition states that \( \langle \widehat{A}{} \kern1pt \rangle _\beta^{\textrm{can}} \) can be written for any \(\beta>0\) in the form (4.1) and does not state that this equality holds for an arbitrary \(E>0\).

We reproduce a proof of this proposition in more detail in Appendix A (but it is still not rigorous).

5. Does the ansatz imply thermalization?

Define the expectation value of an observable \(A\) as

$$ A_t := \langle \psi(t)| \widehat{A}{} \kern1pt |\psi(t)\rangle = \sum_{m,n} c_m^* c_n \kern1pt e^{i(E_m-E_n)t} A_{mn},$$
(5.1)

where \(\psi(t)\) is defined in (2.3), and its infinite time average (Cesàro mean) as

$$ \overline{A} := \lim_{\tau\to\infty} \frac{1}{\tau} \intop_0^\tau\! A_t\,dt.$$
(5.2)

In [20, (3.6), (3.7)] there is a statement about thermalization of quantum systems obeying the ETH, which can be written as follows:

Consider the quantum system \( {\mathfrak S} \) (see (2.1)–(2.3)). If the ETH (2.8) holds for an observable \(A,\) then for \(N\to\infty\)

$$ \overline{A} = \langle \widehat{A}{} \kern1pt \rangle _\beta^{\textrm{can}} + O(\varDelta^2) + O(N^{-1}) + O(e^{-S/2}), $$
(5.3)
$$ \overline{(A_t-\overline{A} \kern1pt )^2} = O(e^{-S}). $$
(5.4)

In short, this statement is justified in [20] as follows (up to some big O’s; \(N = \mathrm{const}\gg 1\)):

$$ \begin{aligned} \, \overline{A} &= \sum_n |c_n|^2 A_{nn} \stackrel{\textrm{ETH}}{=} \sum_n |c_n|^2 {\mathcal A} (E_n) = {\mathcal A} \Biggl(\sum_n |c_n|^2 E_n\Biggr) \equiv {\mathcal A} (E) \stackrel{(4.1)}{=} \langle \widehat{A}{} \kern1pt \rangle _\beta^{\textrm{can}} \\[4pt] &\equiv \frac{\sum_n [ \exp(-\beta E_n) A_{nn} ]}{\sum_n \exp(-\beta E_n)}. \end{aligned}$$
(5.5)

Here we see that Proposition 1 is used to turn from \( {\mathcal A} (E)\) to \( \langle \widehat{A}{} \kern1pt \rangle _\beta^{\textrm{can}} \). However, one cannot guarantee that (4.1) from Proposition 1 necessarily holds for \(E_\beta=E\) for some \(\beta\) (see Remark 1). Hence, it is not proved that the ansatz (2.8) implies thermalization.

Moreover, if equalities (5.5) held, then we would have

$$ \sum_n |c_n|^2 A_{nn} = \sum_n \frac{e^{-\beta E_n}}{Z_\beta} A_{nn}.$$
(5.6)

On the left-hand side there is a weighted arithmetic mean of \(A_{nn}\) in a general form, and on the right-hand side there is also a weighted arithmetic mean but in a special form, the canonical thermal average. Although this equality may hold for some \(c_n\), \(E_n\), and \(A_{nn}\), \(n\in{\mathbb N}_0\) (see Example 1 below), one can easily find \(c_n\), \(E_n\), and \(A_{nn}\), \(n\in{\mathbb N}_0\), such that the two means differ significantly.

So, the answer to the question in the title of this section is negative: the ansatz (2.8) does not necessarily imply thermalization.

6. Thermalization condition

The above consideration shows that the ansatz (2.8) does not necessarily imply thermalization. Obviously, the same is true for the conjectural thermalization condition (3.3), (3.4) based on the ansatz (2.8).

If we want to get thermalization in the above sense, we should postulate the canonical thermal average in the ETH ansatz. Then from (3.3) we get the following thermalization condition (TC).

Thermalization condition.

A set of states \( |\psi\rangle \) of the form (2.4) and observables \(A\) satisfies the thermalization condition if as \(N\to\infty\) the following equalities hold:

$$\sum_n |c_n|^2A_{nn} = \langle \widehat{A}{} \kern1pt \rangle _\beta^{\textrm{can}} + o(1),$$
(6.1)
$$A_{mn}=A_{mn}(N) \to 0, \qquad m\neq n.$$
(6.2)

Using (2.3) and (6.2), we can write (6.1) in the form

$$ \langle \psi| \widehat{A}{} \kern1pt |\psi\rangle = \langle \widehat{A}{} \kern1pt \rangle _\beta^{\textrm{can}} + o(1).$$
(6.3)

It is easy to show that the TC implies thermalization of an observable of isolated many-body quantum systems.

Proposition 2 (thermalization).

Consider the quantum system \( {\mathfrak S} \) (see (2.1)–(2.3) ). If the TC holds for an observable \(A,\) then for \(N\to\infty\)

  1. (1)

    the infinite time average of \(A_t\) is approximately equal to its equilibrium value (2.11) at the appropriate temperature :

    $$ \overline{A} = \langle \widehat{A}{} \kern1pt \rangle _\beta^{\textrm{can}} + o(1);$$
    (6.4)
  2. (2)

    the fluctuations of \(A_t\) about \(\overline{A}\) are small :

    $$ \overline{(A_t-\overline{A})^2} = o(1).$$
    (6.5)

The proof is given in Appendix B.

This proof shows that with the TC there is no need in assumption (3.2) on quantum uncertainty.

Example 1.

An observable of any quantum system in the state \( |\psi\rangle \) of the form (2.4) with

$$ c_n = \frac{e^{-\beta E_n/2}}{\sqrt{Z_\beta}}, \qquad |\psi\rangle = \sum_n \frac{e^{-\beta E_n/2}}{\sqrt{Z_\beta}} \, |n\rangle$$
(6.6)

obviously satisfies the TC.

7. Generalization of the thermalization condition

Note that in [5, 21] the ETH is formulated as

$$ \langle n| \widehat{A}{} \kern1pt |n\rangle = \langle \widehat{A}{} \kern1pt \rangle ^{\textrm{mic}}(E_n).$$
(7.1)

Here we see that the microcanonical average is postulated in the ETH ansatz. Based on numerical simulations, the authors claim that the microcanonical predictions are robust with respect to the choice of the width of the energy window. This formulation implies the TC in the case of equality of the canonical and microcanonical ensembles.

Similarly, in [13] the validity of the ETH is investigated through the relation

$$ I_{\textrm{ETH}}[ \widehat{A}{} \kern1pt ] := \max_{\phi_n\in {\mathcal H} _{E,\Lambda}} \bigl| \langle \phi_n| \widehat{A}{} \kern1pt |\phi_n\rangle - \langle \widehat{A}{} \kern1pt \rangle ^{\textrm{mic}}\bigr| \xrightarrow[N\to\infty]{} 0.$$
(7.2)

Actually, the TC can be written not only with the canonical thermal average \( \langle \widehat{A}{} \kern1pt \rangle _\beta^{\textrm{can}} \) for some \(\beta\), but also with the microcanonical average \( \langle \widehat{A}{} \kern1pt \rangle _E^{\textrm{mic}}\) at energy \(E\), and even with any other average \( \langle \widehat{A}{} \kern1pt \rangle = \operatorname{Tr} (\rho \widehat{A}{} \kern1pt )\), where \(\rho\) is an arbitrary density matrix. So, we come to the equilibration condition.

Equilibration condition.

A set of states \( |\psi\rangle \) in the form of (2.4) and observables \(A\) satisfies the equilibration condition with the mean \( \langle \widehat{A}{} \kern1pt \rangle = \operatorname{Tr} (\rho \widehat{A}{} \kern1pt )\) if the equalities

$$\sum_n |c_n|^2A_{nn} = \langle \widehat{A}{} \kern1pt \rangle + o(1) $$
(7.3)

and (6.2) hold as \(N\to\infty\).

Obviously, since the TC implies thermalization, the equilibration condition implies equilibration. Proposition 2 holds with the first statement replaced by the following:

$$ \overline{A} = \langle \widehat{A}{} \kern1pt \rangle + o(1).$$
(7.4)

If the density matrix commutes with the Hamiltonian, i.e., \([\rho, \widehat{H}{} \kern1pt ] = 0\), then in the same basis we can write

$$\widehat{H}{} = \sum_n E_n |n\rangle\langle n|$$

and

$$\rho = \sum_n \lambda_n |n\rangle\langle n| , \qquad \sum_n \lambda_n = 1, \qquad \lambda_n \geq 0 \quad \forall n.$$

From (7.3) we get

$$\sum_n |c_n|^2A_{nn} = \operatorname{Tr} (\rho \widehat{A}{} \kern1pt ) = \sum_{m,n} \langle m| \rho |n\rangle \langle n| \widehat{A}{} \kern1pt |m\rangle = \sum_{m,n} \lambda_n \delta_{mn}A_{mn} = \sum_n \lambda_n A_{nn}.$$

From this equality we see that in the simplest case \(c_n=\sqrt{\lambda_n}\). In addition,

$$\dot{\rho} =i[\rho,H] = 0$$

implies that \(\rho\) corresponds to a stationary state, according to general expectations that \(\rho(t)\) is close to \(\rho^{\textrm{st}}\) when \(t\gg 1\).

Example 2.

An observable of any quantum system in a state \( |\psi\rangle \) of the form (2.4) with

$$ c_n = ( {\mathcal N} _{E,\Delta E})^{-1/2}, \qquad |\psi\rangle = ( {\mathcal N} _{E,\Delta E})^{-1/2} \sum_{|E-E_n|<\Delta E} |n\rangle$$
(7.5)

(see (2.12)) satisfies the equilibration condition. This observable equilibrates to the microcanonical average.

Example 3.

Let \(\zeta(s)\) be the Riemann zeta function, \(s>1\). Any observable of a quantum system in a state \( |\psi\rangle \) of the form (2.4) with

$$ c_n = \frac{1}{\sqrt{\zeta(2s)}\,n^s}, \qquad \sum_{n=1}^\infty |c_n|^2 = \frac{1}{\zeta(2s)} \sum_{n=1}^\infty \frac{1}{n^{2s}} = 1, \qquad |\psi\rangle = \frac{1}{\sqrt{\zeta(2s)}} \sum_n \frac{1}{n^s} |n\rangle$$
(7.6)

(see (2.12)) satisfies the equilibration condition. This observable equilibrates to the mean value

$$ \langle \widehat{A}{} \kern1pt \rangle := \frac{1}{\zeta(2s)} \sum_{n=1}^\infty \frac{1}{n^{2s}}A_{nn},$$
(7.7)

which differs from the canonical and microcanonical ones.

8. Conclusions

The eigenstate thermalization hypothesis in the form of the ansatz (2.8) as a sufficient condition for thermalization is discussed. We have shown that observables of bounded isolated many-body quantum systems satisfying the ETH ansatz (2.8) do not necessarily thermalize in the above sense.

In the ETH framework, to get such thermalization, one should in fact postulate it in the ETH ansatz. Then the thermalization condition (6.1), (6.2) for bounded isolated many-body quantum systems is readily formulated. We have shown that this condition implies the needed thermalization (Proposition 2).

More generally, any other average can also be postulated in the ETH ansatz, which leads to a corresponding equilibration condition (7.3), (6.2).

Further insight is required in this area to better understand the thermalization phenomenon of isolated many-body quantum systems.