Eigenstate thermalization hypothesis, time operator, and extremely quick relaxation of fidelity

The eigenstate thermalization hypothesis (ETH) insists that for nonintegrable systems each energy eigenstate accurately gives microcanonical expectation values for a class of observables. As a mechanism for ETH to hold, we show that the energy eigenstates are superposition of uncountably many quasi eigenstates of operationally defined"time operator", which are thermal for thermodynamic isolated quantum many-body systems and approximately orthogonal in terms of extremely short relaxation time of the fidelity. In this way, our scenario provides a theoretical explanation of ETH.


Introduction
A considerable research attention has been devoted to the foundation of statistical mechanics on the basis of intrinsic thermal nature of individual pure states. The long-standing fundamental problems involve deriving the principle of equal weight and to explain the mechanism of irreversible thermalization in terms of isolated quantum many-body systems [1][2][3][4][5].
In particular, typicality shows that a pure state uniform randomly sampled with respect to the Haar measure from an appropriate energy shell well represents the microcanonical ensemble [6][7][8][9], and provides a simple scenario to justify the principle of equal weight: Fix a set of observables, then, a majority of the pure states in the Hilbert space are similar to each on another in terms of the expectation values. Thus, we may superpose them with an almost arbitrary weight, and this includes the case of equal weight.
Several different approaches have been studied, including through a restriction on the macroscopic observables [10,11], the general evaluation of relaxation time [12][13][14][15], the Eigenstate thermalization hypothesis (ETH) [1,[16][17][18][19][20][21], and dynamical experiments in autonomous cold atomic systems [22][23][24]. Of these, we focus on the foundation of ETH in terms of the time-energy uncertainty by noting that the energy eigenstates are globally distributed in the basis of suitably defined 'time operator' as detailed below (1) and in section 2. Before this, in the rest of this paragraph, we recall the basic knowledges of ETH. The ETH claims that each energy eigenstate well represents the microcanonical ensemble for nonintegrable systems, i.e. their expectation values for a class of observables agree well with the microcanonical averages. By requiring this property and nondegeneracy condition, arbitrary initial pure states equilibrate for the long time average of expectation values of the fixed observables. The ETH has been discussed in terms of the nonintegrability [1,25,26], partly because the relaxation property is considered to be sensitive to the presence of integrals of motion. On the other hand, [25][26][27] orredhave indicated that most energy eigenstates of integrable systems are thermal. Such an intrinsic thermal nature, shared by most energy eigenstates of integrable systems is often called weak ETH. By considering the observables of a small subsystem, the ETH resembles typicality, although there is still a possibility that the deviations from microcanonical ensemble average of typical states and energy eigenstates are quantitatively different. We will numerically evaluate the deviations later in section 3.
Let us try to understand the mechanism of ETH in terms of typicality. Our starting point is to seek a relevant basis n f ñ {| }that is thermal, and where each energy eigenstate is a superposition of sufficient number of Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. orthonormal states: ] is the dimension of the energy shell, and c n d 1  = ( ) .
In this paper, we present a scenario where the ETH holds by explaining that the quasi eigenstates t Y ñ | ( ) of 'time operator' T form the relevant basis by considering a thermodynamic system where t Y ñ | ( ) thermalizes and stays in equilibrium. Note that the 'time operator' is constructed in (2) via spectral decomposition, and is approximately canonical conjugate to the Hamiltonian up to a constant owing to a long time cutoff. However, proper definitions of 'time operator' remains still controversial in general. Instead of inquiring into the best definition, we explore a foundation of ETH by introducing 'time operator' as (2) and its quasi eigenstates, which are approximately orthogonal through the extremely quick relaxation of the fidelity [12][13][14][15]. Note that such quasi orthogonality is analogous to that of the coherent state, and is used in quantum non demolition measurement [28,29]. In particular, we show that each energy eigenstate can be expressed as a superposition of many mutually almost orthogonal pure states that are considered thermal. Note that [20] quantified the degree of superposition with the use of Shannon entropy, which is basis dependent and maximized to guarantee ETH. Subsequently, [21] addressed the issue to specify a class of observables, such as local and extensive quantities, that satisfy ETH in terms of mutually unbiased basis with respect to the Hamiltonian. Mutually unbiasedness can be regarded as a generalization of the concept of the canonical conjugate, which is significant for our argument, and thus [20,21] are related to the present study. The main difference is that in this article, the quasi eigenstates of the 'time operator' are unbiased with respect to the Hamiltonian. However, we do not attempt to apply ETH to the 'time operator' itself, and instead, we explain that the vast majority of the quasi eigenstates of 'time operator' are regarded as thermal with the use of the typicality [6][7][8][9] by considering observables of a subsystem and the assumption of equilibration. Then, the energy eigenstates are regarded as thermal.
The remainder of this paper is organized as follows. In section 2, we express the energy eigenstates in terms of quasi eigenstates of 'time operator', and explore their orthogonality and thermal nature of quasi eigenstates of the 'time operator'. In section 3, we numerically verify the approximate orthogonality and thermal nature of quasi eigenstates and the ETH. Section 4 is devoted to a summary.  (3) is slightly modified. We set c n as real, since the phase factor at t=0 can be absorbed into the definition of energy eigenstates E n ñ | . Then, we can show that

Time-evolved states
, which is explained later. Let us formally define the 'time operator' as where we consider a large but finite time T (c.f. infinitesimally small cut-off [30,31]). It is well-known that 'time operator', which is canonical conjugate to the Hamiltonian, does not exist as an observable [32][33][34][35], partly because the Hamiltonian is bounded below. On the other contrary, 'time operator' defined by (2) approximately satisfies the commutation relation up to a boundary constant, just as in the case of 'phase operator' [34]:

and a partial integral
| as T  ¥ from the nondegeneracy assumption. In our case, the diagonal basis t Y ñ | ( ) of (2) are approximate eigenstates of T , as the orthogonality holds with time resolution τ, which is in marked contrast to the case of mechanical observables. We explain how the time resolution τ is extremely short for thermodynamic systems.
We can express the energy eigenstates by the inverse Fourier transform: Equation (5) shows that the energy eigenstates are superpositions of continuously many quasi eigenstates of 'time operator', which approximately satisfies the orthogonality(8) [12]. In the next section, we numerically verify how the quasi eigenstate t Y ñ | ( ) typically well represents the microcanonical state, and is considered as a relevant basis to discuss the foundation of ETH. Here, we analytically explore the quasi orthogonality and equilibrium nature of the basis t Y ñ | ( ) . First, we recall the calculation of the fidelity detailed in [12] (see also [14,15]). By using where the discrete sum is evaluated as integral using the density of the states E W ¢ ( ), which renders the spectrum of the eigenenergy continuous, and the dynamics are supposed to be irreversible. At this stage, the recurrence phenomena in case of extremely long time is omitted. Such a continuous approximation accurately holds as shown in figure 1. We expand the density of the states as Here, we evaluate ΔE eff from the condition that the absolute value of the first order term β x is much larger than that of the second order x for x=ΔE eff , which yields C V ?βΔE eff . For thermodynamic density of the states, the energy width ΔE eff is considered to be of the same order as 1 b . As the heat capacity is proportional to the system size, we can accurately calculate the integral up to the first order of x.
, the inner product (8) becomes considerably small [12], |(ˆ|ˆ| ) | , which is compatible to our evaluation of τ. It is also well-known that for long-term regime, the fidelity shows power-law decay by Pailey-Wiener theorem for Fourier-Laplace transformation. Meanwhile, an exponential decay is observed for the time scale of interest to us.
We now discuss some properties of the 'time operator'. The operator | from the quasi orthogonality. Note that the projection operator (9) can be used for measurement of time as projection to t Y ñ | ( ) : Given a state t Y ñ | ( ) with unknown t, such projection determines t with an accuracy τ. By repeating this thought experiment many times with randomly distributed t, we actually obtain the spectral fluctuation. The projection operator (9) also satisfies the completeness

Numerical simulation
First, we explore the quasi orthogonality for quasi eigenstates t Y ñ | ( ) . Then, we also verify the validity of ETH, and investigate the thermal nature of time-evolved states. Regarding the quasi orthogonality, further details of calculation are shown in [12]. For the sake of simplicity and concreteness, we first consider one-dimensional ] as a subspace spanned by eigenstates E n ñ | with (a) 151n200 and (b)251n300. We set the parameters to J=1 and α=1. On the contrary, we also explored various choices of γ j such as uniform case γ j =0.5 (γ=0 corresponds to the integrable case), randomly distributed case γ j ä[0, Δ] with Δ=0.5, 1. The eigenenergies were in increasing order (red broken line) calculated using equation (8), where we set ÿ=1. Note that the relaxation time τ was quite general [12] and was the same order as the Boltzmann time 2πβÿ for macroscopic systems [13], which is extremely short at room temperature s 10 12 t~-. Therefore, we can conclude the bases t Y ñ | ( ) (t0) in the expansion (5) are mutually orthogonal. We then verified the thermalization of the quasi eigenstates of T , i.e. the basis t Y ñ | ( ) well represented the microcanonical state for most t 0, ]. For this purpose, it was necessary to calculate the expectation values of a class of observables Â for t Y ñ | ( ) and compare with those of the microcanonical ensemble. Theoretically, t Y ñ | ( ) describes thermal equilibrium for most t according to the typicality [6,7] and the unitarity of the time evolution. Numerically, we investigated the expectation values of arbitrary observables defined on the left-most m sites A m [8,9,16,25]. Thus, we calculate the Hilbert-Schmidt distance t t m 0 Here, Tr N−m stands for the partial trace for the right-most N−m sites.
In figure 2(a) figure 4, we show the dependences of r Dˆ(blue curve) and t Dˆ(red curve) on the subsystem size m for the case of uniform magnetic field γ j =0.5, and randomly sampled γ j from [0, Δ] with Δ=0.5, 1. The deviations r Dˆand t Dˆagreed well one another all three cases. This means that ETH holds with the same accuracy as the thermal property of t Y ñ | ( ) for these cases. The agreement may get better when the finiteness of the energy width can be negligible for the calculation of t Dˆ.
In the presence of strong spatial disorder, the ETH breaks down [37][38][39]. Exploring the case of non-thermal case including the many-body localization possibly in more than one dimensions is an important future problem.