Eigenstate thermalization: Deutsch's approach and beyond

Following Deutsch [44] we consider Hamiltonians H of the form

$$\begin{eqnarray}&&H={{H}_{0}}+V,\end{eqnarray} \tag{ 8 }$$

consisting of an 'unperturbed' part H₀ and a 'perturbation' V. As before, eigenvectors and eigenvalues of H are denoted as $|n\rangle$ and E_n with ${{E}_{n+1}}\geqslant {{E}_{n}}$. Likewise, those of H₀ are denoted as $|n{{\rangle }_{0}}$ and E_n⁰ with $E_{n+1}^{0}\geqslant E_{n}^{0}$.

Typical examples one has in mind [44] are H₀ which describe a non-interacting many-body system, e.g. an ideal gas in a box, while V accounts for the particle–particle interactions. Further examples are so-called quantum quenches: H₀ describes the system for times $t\lt 0$, while H applies to $t\geqslant 0$. In other words, some external condition or some system property suddenly changes at time t = 0.

In various such examples, the perturbation matrix

$$\begin{eqnarray}&&V_{mn}^{0}:={{}_{0}}{{\langle m|V|n\rangle }_{0}}\end{eqnarray} \tag{ 9 }$$

is often expected or numerically found to be a banded matrix [21, 44, 59, 60], i.e., the typical magnitude of V⁰_mn decreases with increasing $|m-n|$ towards zero. Furthermore, in the above mentioned example where H₀ describes a non-interacting many-body system, the perturbation matrix V⁰_mn is usually very sparse, i.e., only a small fraction of all matrix elements is non-zero [60–62].

In any case, the perturbation V is required to be sufficiently weak so that the two systems H and H₀ still exhibit similar thermodynamic properties at the considered system energy E, in particular similar densities of the energy levels, see above equation (4).

As a next step, the common lore of random matrix theory is adopted [44, 47, 61]: one samples matrices V⁰_mn from a certain random matrix ensemble with statistical properties which imitate reasonably well the main features of the 'true' perturbation V (band structure, sparsity etc), and it is assumed that if a certain property can be shown to apply to the overwhelming majority of such randomly sampled V-matrices, then it will also apply to the actual (non-random) V in (8). A priori, such a random matrix approach may appear 'unreasonable' since most of those randomly sampled perturbations V amount to systems which are physically very different from the one actually modelled in (8). Yet, in practice such a random matrix approach turned out to be surprisingly successful in a large variety of specific examples [61], and hence, as in Deutsch's work [44], will be tacitly taken for granted from now on.

The randomness of V entails via in (8) a randomization of the eigenstates $|n\rangle$ of H and hence of the basis-transformation matrix

$$\begin{eqnarray}&&{{U}_{mn}}:={{\langle m|n\rangle }_{0}}.\end{eqnarray} \tag{ 10 }$$

Likewise, any given observable A and its matrix elements in the unperturbed basis

$$\begin{eqnarray}&&A_{mn}^{0}:={{}_{0}}{{\langle m|A|n\rangle }_{0}}\end{eqnarray} \tag{ 11 }$$

are non-random quantities, while

$$\begin{eqnarray}&&{{A}_{mn}}:=\langle m|A|n\rangle \end{eqnarray} \tag{ 12 }$$

will be the elements of a random matrix, inheriting the randomness of the U-matrix via

$$\begin{eqnarray}&&{{A}_{mn}}=\mathop{\sum }\limits_{jk}{{U}_{mj}}A_{jk}^{0}U_{nk}^{*}.\end{eqnarray} \tag{ 13 }$$

Demonstrating ETH thus amounts to showing that ${{A}_{mm}}-{{A}_{nn}}$ is small for most V's and sufficiently close m and n. Formally, this will be achieved by considering the variances

$$\begin{eqnarray}&&\sigma _{n}^{2}:={{\langle {{({{A}_{nn}}-{{\langle {{A}_{nn}}\rangle }_{V}})}^{2}}\rangle }_{V}}={{\langle {{({{A}_{nn}})}^{2}}\rangle }_{V}}-\langle {{A}_{nn}}\rangle _{V}^{2},\end{eqnarray} \tag{ 14 }$$

where ${{\langle ...\rangle }_{V}}$ indicates the average over the random perturbations V. In a first step (section 5.4), we will show that the mean values ${{\langle {{A}_{nn}}\rangle }_{V}}$ for sufficiently close n's differ very little in comparison to the experimental resolution limit $\delta A$ introduced in section 2. In a second step (section 5.5), we will show that ${{\sigma }_{n}}\ll \delta A$, implying that A_nn differs very little from ${{\langle {{A}_{nn}}\rangle }_{V}}$ for most V. Altogether, this will imply the desired result that for most V's the A_nn's change very little upon changing n (section 5.6).

To simplify the algebra, we henceforth assume that the largest and smallest eigenvalues of A in (1) satisfy

$$\begin{eqnarray}&&{{a}_{{\rm min} }}=-{{a}_{{\rm max} }}.\end{eqnarray} \tag{ 15 }$$

As a consequence, ${{a}_{{\rm max} }}={{\Delta }_{A}}/2$ according to (1). Note that the assumption (15) does not imply any loss of generality, since adding an arbitrary constant (times the identity operator) to the observable A, and hence to all its eigenvalues, does not entail any non-trivial physical consequences. In particular, the above-mentioned changes of ${{\langle {{A}_{nn}}\rangle }_{V}}$ upon variation of n and the variances (14) remain exactly the same. For later use, we thus can conclude that

$$\begin{eqnarray}&&|\langle \psi |{{A}^{\nu }}|\psi \rangle |\leqslant {{({{\Delta }_{A}}/2)}^{\nu }}\end{eqnarray} \tag{ 16 }$$

for any $\nu \in \mathbb{N}$.

For the sake of simplicity, we furthermore assume that all matrix elements V⁰_mn from (9) and U_mn from (10) are real numbers. For example, for systems without spins and magnetic fields, H₀ and V in (8) are both purely real operators in position representation and hence the eigenstates $|n{{\rangle }_{0}}$ and $|n\rangle$ can be chosen so that all V⁰_mn and U_mn become real. So, it is natural to assume that also the corresponding random matrix ensembles only involve real matrix elements. In particular, this implies with equations (13) and (14) that

$$\begin{eqnarray}&&{{\langle {{A}_{nn}}\rangle }_{V}}=\mathop{\sum }\limits_{jk}A_{jk}^{0}{{\langle {{U}_{nj}}{{U}_{nk}}\rangle }_{V}}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&\sigma _{n}^{2}=\mathop{\sum }\limits_{ijkl}A_{ij}^{0}A_{kl}^{0}{{\langle {{U}_{ni}}{{U}_{nj}}{{U}_{nk}}{{U}_{nl}}\rangle }_{V}}-\langle {{A}_{nn}}\rangle _{V}^{2}.\end{eqnarray} \tag{ 18 }$$

It may well be that our subsequent calculations can be readily extended to systems for which such a transformation to purely real matrices V⁰_mn and U_mn is no longer possible. However, the so far available knowledge, e.g. from random matrix theory or numerical investigations, regarding the statistical properties of the U_mn's is only sufficient for our purposes for real matrices (see next section). Since the subject of our paper is not the exploration of such statistical random matrix properties but rather their implications with respect to ETH, we confine ourselves to the case of real matrices.

In view of (17), (18), some basic statistical properties of the matrix elements U_mn are needed in order to make any further progress.

At this point it is crucial to note that the Hamiltonian H in (8) gives rise to a very special type of random matrix. Namely, the matrix $_{\,0}{{\langle m|H|n\rangle }_{0}}$ is the sum of the above discussed random perturbation V⁰_mn and of the non-random diagonal matrix $_{\,0}{{\langle m|{{H}_{0}}|n\rangle }_{0}}={{\delta }_{mn}}E_{n}^{0}$, whose diagonal elements E_n⁰ grow approximately linearly with n (at least within a sufficiently small vicinity of the preset system energy E, onto which we tacitly restrict ourselves, see also sections 3 and 5.6). Out of the huge literature on random matrix theory, only a relatively small number of works pertains to this special case, see e.g. [46, 60, 62, 63] and further references therein. Strictly speaking, they are obtained for infinitely large matrices V⁰_mn, whose statistical properties do not depend on m and n separately, but only on the difference $m-n$. Likewise, the unperturbed matrix $_{\,0}{{\langle m|{{H}_{0}}|n\rangle }_{0}}$ is assumed to be infinitely large and of the form ${{\delta }_{mn}}E_{n}^{0}$ with equally spaced energy gaps $E_{n+1}^{0}-E_{n}^{0}$. Intuitively, these seem quite plausible approximations, at least for not too strong perturbations V in (8). They can be readily justified by numerical examples, but somewhat more rigorous analytical results do not seem to exist. Here, we adopt the widely accepted viewpoint that, for out present purposes, they can be taken for granted [44, 60, 61].

As a consequence, also the statistical properties of the U_mn only depend on $m-n$, e.g. the νth moments are of the form

$$\begin{eqnarray}&&{{\langle {{({{U}_{mn}})}^{\nu }}\rangle }_{V}}={{u}_{\nu }}(m-n),\end{eqnarray} \tag{ 19 }$$

where the ${{u}_{\nu }}(n)$ are real (but not necessarily even [63]) functions of n, and are furthermore non-negative for even ν.

Known analytical results mainly concern the second moment ${{u}_{2}}(n)$ for various ensembles of possibly banded and/or sparse random V-matrices, see e.g. [46, 60, 62, 63] and references therein. In all cases, it is found that ${{u}_{2}}(n)$ is monotonically decreasing for $n\geqslant 0$ and monotonically increasing for $n\leqslant 0$, hence exhibiting a global maximum at n = 0:

$$\begin{eqnarray}&&\mathop{{\rm max} }\limits_{n}\{{{u}_{2}}(n)\}={{u}_{2}}(0).\end{eqnarray} \tag{ 20 }$$

Since ${{\sum }_{k}}{{U}_{mk}}{{U}_{nk}}={{\delta }_{mn}}$ we can conclude that

$$\begin{eqnarray}&&\mathop{\sum }\limits_{n}{{u}_{2}}(n)=1,\end{eqnarray} \tag{ 21 }$$

implying that ${{u}_{2}}(n)$ must approach zero for large $|n|$.

In all those analytical results, the mean values ${{\langle V_{mn}^{0}\rangle }_{V}}$ are tacitly assumed to vanish and it is found that ${{u}_{2}}(n)$ then only depends on the second moments ${{\langle {{(V_{mn}^{0})}^{2}}\rangle }_{V}}$. Since we are not aware of any justification for this assumption ${{\langle V_{mn}^{0}\rangle }_{V}}=0$, we have numerically investigated various examples and found that, indeed, the statistical properties of the U_mn's seem to be independent of the first moments ${{\langle V_{mn}^{0}\rangle }_{V}}$ (while keeping all other cumulants fixed). Furthermore, a simple physical argument is as follows: replacing an eigenstate $|n{{\rangle }_{0}}$ of H₀ by $-|n{{\rangle }_{0}}$ is supposed not to change any physically relevant properties of the given (non-random) model in (8). Note that this argument applies separately to any single n. Hence, it is quite plausible that upon randomly flipping the signs for half of all n's, the resulting 'new' V⁰_mn's will be 'unbiased' for $m \ne n$. (More precisely: if a random matrix description works at all, then an ensemble with ${{\langle V_{mn}^{0}\rangle }_{V}}=0$ seems most appropriate). Finally, a possibly remaining systematic 'bias' of the diagonal elements V⁰_nn can be removed by adding an irrelevant constant to V. (The typical magnitude of the V⁰_mn's is also estimated in Appendix B of [45], however, not taking into account the possibility that their average may be zero.)

For the rest, the detailed properties of ${{u}_{2}}(n)$ are found—as expected—to still depend on the quantitative details of ${{\langle {{(V_{mn}^{0})}^{2}}\rangle }_{V}}$. Since no general statement about the latter seems possible for the general class of systems we have in mind with (8), we will focus on conclusions which do not depend on the corresponding details of ${{u}_{2}}(n)$. Rather, we will only exploit the following very crude common denominator of all so far explored particular classes of random matrices V_mn, see e.g. [46, 60, 62, 63] and further references therein:

$$\begin{eqnarray}&&{{u}_{2}}(0)={{10}^{-{\rm O}(N)}}.\end{eqnarray} \tag{ 22 }$$

The basic physical reason is that exceedingly 'weak' perturbations V in (8) are tacitly ignored so that the smallest relevant energy scale is the mean level spacing ${{E}_{n+1}}-{{E}_{n}}$, being of the order of ${{10}^{-{\rm O}(N)}}\;$ J according to section 2. Moreover, the ratio between this energy scale and any other relevant energy scale of the system can be very roughly estimated by ${{10}^{-{\rm O}(N)}}$, independently of any further details of the specific model system in (8). As a consequence, the very crude estimate (22) is also independent of these details.

Further statistical properties of the U_mn, which we will, similarly as in [44, 45, 60, 62, 63], take for granted later on, are:

• (i)  
  Their average is zero, i.e.

  $$\begin{eqnarray}&&{{u}_{1}}(n)=0\ \ {\rm for\ all}n.\end{eqnarray} \tag{ 23 }$$
• (ii)  
  They are statistically independent of each other, i.e.

  $$\begin{eqnarray}&&{{U}_{mn}}\;{\rm is\ independent\ of}{{U}_{jk}}\;{\rm if}\;m \ne j\;{\rm or}\;n \ne k.\end{eqnarray} \tag{ 24 }$$
• (iii)  
  Their distribution does not exhibit long tails, i.e.

  $$\begin{eqnarray}&&{{u}_{4}}(n)\leqslant c\;{{u}_{2}}(n){{u}_{2}}(0)\ \ {\rm for\ all}n\end{eqnarray} \tag{ 25 }$$

with an n-independent constant c, which may possibly be very large but which is required not to be so large that it can compete in order of magnitude with $1/{{u}_{2}}(0)$ from (22). For instance for a system with $N={{10}^{23}}$ particles, it would be sufficient that $c\leqslant {{10}^{{{10}^{22}}}}$. In other words, we adopt the very weak assumption

$$\begin{eqnarray}&&c\;{{u}_{2}}(0)={{10}^{-{\rm O}(N)}}.\end{eqnarray} \tag{ 26 }$$

For example, for a Gaussian distribution (with zero mean, cf (23)) one finds that ${{u}_{4}}(n)=3u_{2}^{2}(n)$ and hence (25) is satisfied for c = 3. Though a Gaussian distribution is often taken for granted [22, 44, 45], non-Gaussian distributions have been actually observed e.g. in [59] and also in our own numerical explorations (unpublished), but the more general condition (25) was still satisfied in all cases. We also note that since ${{u}_{2}}(n)/{{u}_{2}}(0)$ approaches zero for large $|n|$, the condition in (25) becomes weaker and weaker with increasing $|n|$.

Evaluating (17) by means of (19), (23), and (24) yields

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{\langle {{A}_{nn}}\rangle }_{V}} & = & \mathop{\sum }\limits_{j}A_{jj}^{0}{{\langle {{U}_{nj}}{{U}_{nj}}\rangle }_{V}}+\mathop{\sum }\limits_{j \ne k}A_{jk}^{0}{{\langle {{U}_{nj}}\rangle }_{V}}{{\langle {{U}_{nk}}\rangle }_{V}} \\ {} & = & \mathop{\sum }\limits_{j}A_{jj}^{0}{{u}_{2}}(n-j). \\ \end{array}\end{eqnarray} \tag{ 27 }$$

It follows that

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{\Delta }_{n}} & := & {{\langle {{A}_{n+1,n+1}}\rangle }_{V}}-{{\langle {{A}_{nn}}\rangle }_{V}} \\ {} & = & \mathop{\sum }\limits_{j}A_{jj}^{0}[{{u}_{2}}(n+1-j)-{{u}_{2}}(n-j)] \\ \end{array}\end{eqnarray} \tag{ 28 }$$

and hence that

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} |{{\Delta }_{n}}| & \leqslant & \mathop{\sum }\limits_{j}|A_{jj}^{0}||{{u}_{2}}(n+1-j)-{{u}_{2}}(n-j)| \\ {} & \leqslant & \mathop{{\rm max} }\limits_{j}|A_{jj}^{0}|\mathop{\sum }\limits_{k}|{{u}_{2}}(k+1)-{{u}_{2}}(k)|. \\ \end{array}\end{eqnarray} \tag{ 29 }$$

The maximum over j can be estimated from above by ${{\Delta }_{A}}/2$ according to (16). Recalling that ${{u}_{2}}(k)$ is monotonically decreasing for $k\geqslant 0$ and monotonically increasing for $k\leqslant 0$ (see above (20)), the sum over k amounts to $2\;{{u}_{2}}(0)$ (more generally, this sum amounts to the total variation of ${{u}_{2}}(k)$; hence, if ${{u}_{2}}(k)$ exhibits M local maxima, it can be estimated from above by $2M{{{\rm max} }_{k}}{{u}_{2}}(k)$). Altogether, we thus can conclude that

$$\begin{eqnarray}&&|{{\langle {{A}_{n+1,n+1}}\rangle }_{V}}-{{\langle {{A}_{nn}}\rangle }_{V}}|\leqslant {{\Delta }_{A}}\;{{u}_{2}}(0).\end{eqnarray} \tag{ 30 }$$

This upper bound is tight: one can readily find examples A for which (30) becomes an equality. Moreover, it follows that

$$\begin{eqnarray}&&|{{\langle {{A}_{mm}}\rangle }_{V}}-{{\langle {{A}_{nn}}\rangle }_{V}}|\leqslant |m-n|\;{{\Delta }_{A}}\;{{u}_{2}}(0).\end{eqnarray} \tag{ 31 }$$

By generalizing the line of reasoning in (29), (30) one can also show that

$$\begin{eqnarray}&&|{{\langle {{A}_{mm}}\rangle }_{V}}-{{\langle {{A}_{nn}}\rangle }_{V}}|\leqslant {{\Delta }_{A}}\mathop{\sum }\limits_{k=-\kappa }^{\kappa }{{u}_{2}}(k)\end{eqnarray} \tag{ 32 }$$

with $\kappa := |m-n|-1$. Under certain conditions, the bound (31) may be better than (32), but never by more than a factor of 2. For sufficiently large $|m-n|$, (32) is always better since the sum on the right-hand side is bounded by unity (see (21)), while the right-hand side of (31) is unbounded. (The relevance of large $|m-n|$-values will become apparent in section 5.6 below). In any case, (32) is a rather tight bound in the sense that one can find examples for A_jj⁰ so that the left-hand side is larger than the right-hand side divided by 2 for a set of suitably chosen pairs $(m,n)$ so that the differences $m-n$ may still take any integer value.

We rewrite the variance from (18) as

$$\begin{eqnarray}&&\sigma _{n}^{2}+\langle {{A}_{nn}}\rangle _{V}^{2}=\mathop{\sum }\limits_{ijkl}A_{ij}^{0}A_{kl}^{0}{{\langle {{U}_{ni}}{{U}_{nj}}{{U}_{nk}}{{U}_{nl}}\rangle }_{V}}\end{eqnarray} \tag{ 33 }$$

and evaluate the four-fold sum by distinguishing four possible cases. Case 1: $i \ne k$ and i = j. In this case, we only have to keep summands with l = k: otherwise the factor U_nk on the right-hand side of (33) would be independent of the remaining three factors according to (24), and the corresponding summand would vanish according to (19) and (23). Case 2: $i \ne k$ and $i \ne j$. As before, we can conclude that only summands with l = i and j = k give rise to non-vanishing terms. Case 3: i = k and $i \ne l$, implying, as before, that only j = l contribute. Case 4: i = k and i = l, implying j = i. Consequently, we can rewrite (33) as

$$\begin{eqnarray}&&\sigma _{n}^{2}+\langle {{A}_{nn}}\rangle _{V}^{2}={{\Sigma }_{1}}+{{\Sigma }_{2}}+{{\Sigma }_{3}}+{{\Sigma }_{4}},\end{eqnarray} \tag{ 34 }$$

where the four summands correspond to the above four cases and can be rewritten as:

$$\begin{eqnarray}&&{{\Sigma }_{1}}=\mathop{\sum }\limits_{i \ne k}A_{ii}^{0}A_{kk}^{0}{{\langle {{U}_{ni}}{{U}_{ni}}{{U}_{nk}}{{U}_{nk}}\rangle }_{V}},\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&{{\Sigma }_{2}}=\mathop{\sum }\limits_{i \ne k}A_{ik}^{0}A_{ki}^{0}{{\langle {{U}_{ni}}{{U}_{nk}}{{U}_{nk}}{{U}_{ni}}\rangle }_{V}},\end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray}&&{{\Sigma }_{3}}=\mathop{\sum }\limits_{i \ne l}A_{il}^{0}A_{il}^{0}{{\langle {{U}_{ni}}{{U}_{nl}}{{U}_{ni}}{{U}_{nl}}\rangle }_{V}},\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}&&{{\Sigma }_{4}}=\mathop{\sum }\limits_{i}A_{ii}^{0}A_{ii}^{0}{{\langle {{U}_{ni}}{{U}_{ni}}{{U}_{ni}}{{U}_{ni}}\rangle }_{V}}.\end{eqnarray} \tag{ 38 }$$

With the help of (24) and (19) we can rewrite (35) as

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{\Sigma }_{1}} & = & \mathop{\sum }\limits_{i \ne k}A_{ii}^{0}A_{kk}^{0}{{u}_{2}}(n-i){{u}_{2}}(n-k) \\ {} & = & \mathop{\sum }\limits_{ik}A_{ii}^{0}A_{kk}^{0}{{u}_{2}}(n-i){{u}_{2}}(n-k)-\mathop{\sum }\limits_{i}{{(A_{ii}^{0})}^{2}}u_{2}^{2}(n-i) \\ {} & = & \langle {{A}_{nn}}\rangle _{V}^{2}-\mathop{\sum }\limits_{i}{{(A_{ii}^{0})}^{2}}u_{2}^{2}(n-i), \\ \end{array}\end{eqnarray} \tag{ 39 }$$

where we exploited (27) in the last equation. Likewise, one finds that

$$\begin{eqnarray}&&{{\Sigma }_{2}}=\mathop{\sum }\limits_{ik}|A_{ik}^{0}{{|}^{2}}{{u}_{2}}(n-i){{u}_{2}}(n-k)-\mathop{\sum }\limits_{i}{{(A_{ii}^{0})}^{2}}u_{2}^{2}(n-i),\end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray}&&{{\Sigma }_{3}}=\mathop{\sum }\limits_{il}{{(A_{il}^{0})}^{2}}{{u}_{2}}(n-i){{u}_{2}}(n-l)-\mathop{\sum }\limits_{i}{{(A_{ii}^{0})}^{2}}u_{2}^{2}(n-i),\end{eqnarray} \tag{ 41 }$$

$$\begin{eqnarray}&&{{\Sigma }_{4}}=\mathop{\sum }\limits_{i}{{(A_{ii}^{0})}^{2}}{{u}_{4}}(n-i).\end{eqnarray} \tag{ 42 }$$

Introducing these results into (34) thus yields

$$\begin{eqnarray}&&\sigma _{n}^{2}={{S}_{1}}+{{S}_{2}}+{{S}_{3}},\end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray}&&{{S}_{1}} := \mathop{\sum }\limits_{ik}|A_{ik}^{0}{{|}^{2}}{{u}_{2}}(n-i){{u}_{2}}(n-k),\end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray}&&{{S}_{2}} := \mathop{\sum }\limits_{il}{{(A_{il}^{0})}^{2}}{{u}_{2}}(n-i){{u}_{2}}(n-l),\end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray}&&{{S}_{3}} := \mathop{\sum }\limits_{i}{{(A_{ii}^{0})}^{2}}[{{u}_{4}}(n-i)-3u_{2}^{2}(n-i)].\end{eqnarray} \tag{ 46 }$$

The three factors on the right-hand side of (44) are all non-negative and hence

$$\begin{eqnarray}&&{{S}_{1}}=|{{S}_{1}}|\leqslant \mathop{{\rm max} }\limits_{k}\{{{u}_{2}}(k)\}\mathop{\sum }\limits_{i}{{u}_{2}}(n-i)\mathop{\sum }\limits_{k}A_{ik}^{0}A_{ki}^{0}.\end{eqnarray} \tag{ 47 }$$

With (11) one readily sees that the last sum over k amounts to $_{\,0}{{\langle i|{{A}^{2}}|i\rangle }_{0}}({{A}^{2}})_{ii}^{0}$. Exploiting (20) we thus obtain

$$\begin{eqnarray}&&{{S}_{1}}\leqslant {{u}_{2}}(0)\mathop{\sum }\limits_{i}({{A}^{2}})_{ii}^{0}\;{{u}_{2}}(n-i).\end{eqnarray} \tag{ 48 }$$

Likewise, since ${{u}_{2}}(i)\geqslant 0$ for all i, the modulus of (45) can be estimated as

$$\begin{eqnarray}&&|{{S}_{2}}|\leqslant \mathop{\sum }\limits_{il}|A_{il}^{0}{{|}^{2}}{{u}_{2}}(n-i){{u}_{2}}(n-l)={{S}_{1}}.\end{eqnarray} \tag{ 49 }$$

Turning to (46), we first note that the last factor ${{\kappa }_{4}}:={{u}_{4}}(n-i)-3u_{2}^{2}(n-i)$ represents the 4th cumulant of the random variable U_ni. For a Gaussian distribution (the case considered by Deutsch [44, 45]), this cumulant vanishes, but for more general distributions it may be finite and of either sign. We thus estimate $|{{\kappa }_{4}}|$ from above by ${{u}_{4}}(n-i)+3u_{2}^{2}(n-i)$. Observing (26) and $u_{2}^{2}(n-i)\leqslant {{u}_{2}}(0){{u}_{2}}(n-i)$ (see (20)), we thus can bound (46) by

$$\begin{eqnarray}&&|{{S}_{3}}|\leqslant {{u}_{2}}(0)(c+3)\mathop{\sum }\limits_{i}{{(A_{ii}^{0})}^{2}}{{u}_{2}}(n-i).\end{eqnarray} \tag{ 50 }$$

Next, we invoke the Cauchy–Schwarz inequality to conclude

$$\begin{eqnarray}&&{{\langle \psi |B|\psi \rangle }^{2}}\leqslant \langle \psi |\psi \rangle \langle \psi |{{B}^{2}}|\psi \rangle \end{eqnarray} \tag{ 51 }$$

for arbitrary Hermitian operators B and vectors $|\psi \rangle$. In particular, it follows that ${{(A_{ii}^{0})}^{2}}\leqslant ({{A}^{2}})_{ii}^{0}$. Altogether, we thus arrive at

$$\begin{eqnarray}&&\sigma _{n}^{2}\leqslant (c+5){{u}_{2}}(0)\mathop{\sum }\limits_{i}({{A}^{2}})_{ii}^{0}\;{{u}_{2}}(n-i).\end{eqnarray} \tag{ 52 }$$

Finally, we exploit (16) and (21), resulting in

$$\begin{eqnarray}&&\sigma _{n}^{2}\leqslant (c+5){{u}_{2}}(0)\;{{({{\Delta }_{A}}/2)}^{2}}={{10}^{-{\rm O}(N)}}\;\Delta _{A}^{2},\end{eqnarray} \tag{ 53 }$$

where we used (26) in the last step.

From (14), (53), and Markov's inequality it follows that

$$\begin{eqnarray}&&{\rm Prob}\left( |{{A}_{nn}}-{{\langle {{A}_{nn}}\rangle }_{V}}|\geqslant \epsilon \right)\leqslant {{({{\Delta }_{A}}/\epsilon )}^{2}}\;{{10}^{-{\rm O}(N)}}\end{eqnarray} \tag{ 54 }$$

for any $\epsilon \gt 0$, where Prob $(X)$ denotes the probability that a randomly sampled V in (8) entails property X. For instance, if in the last term ${\rm O}(N)={{10}^{23}}$ and $\epsilon ={{\Delta }_{A}}\;{{10}^{-{{10}^{22}}}}$ then the right-hand side of (54) is still ${{10}^{-{\rm O}({{10}^{23}})}}$. Consequently, the joint probability that every A_nn is practically indistinguishable from ${{\langle {{A}_{nn}}\rangle }_{V}}$ simultaneously for all $n\in \{{{n}_{0}},..,{{n}_{0}}+\Delta n\}$ still remains negligibly small if $0\leqslant \Delta n\ll {{10}^{{\rm O}(N)}}$.

On the other hand, (22) and (31) imply that the difference ${{\langle {{A}_{mm}}\rangle }_{V}}-{{\langle {{A}_{nn}}\rangle }_{V}}$ remains below the experimental resolution limit $\delta A$ of A (cf section 2) even for quite large range-to-resolution ratios ${{\Delta }_{A}}/\delta A$, provided $|m-n|$ remains much smaller than of the order of ${{10}^{{\rm O}(N)}}$. In other words, the variations of the ${{\langle {{A}_{nn}}\rangle }_{V}}$ also remain negligibly small within the 'window' of n-values $\{{{n}_{0}},..,{{n}_{0}}+\Delta n\}$ if $0\leqslant \Delta n\ll {{10}^{{\rm O}(N)}}$.

Altogether, we thus arrive at the conclusion that for the vast majority of randomly sampled perturbations V in (8), the A_nn's remain practically constant (below the experimental resolution limit) as long as n varies by much less than ${{10}^{{\rm O}(N)}}$.

The latter property is sometimes referred to as the strong ETH [33, 38]. It immediately implies the practical indistinguishability of the two expectation values (6) and (7), and hence thermalization, provided both the ${{\rho }_{nn}}(0)$ and the $\rho _{nn}^{{\rm mic}}$ are negligibly small outside a window of n-values much smaller than ${{10}^{{\rm O}(N)}}$, but otherwise without any further restriction on the initial condition $\rho (0)$.

If the range $\Delta n$ of admitted n-values is not any more much smaller than ${{10}^{{\rm O}(N)}}$, then we can no longer conclude from (53) that with high probability all the A_nn's remain simultaneously close to the ${{\langle {{A}_{nn}}\rangle }_{V}}$'s. However, we still can conclude that for the vast majority of n's, those differences remain negligibly small² . If, in addition, the variations of the ${{\langle {{A}_{nn}}\rangle }_{V}}$'s would remain small, we still could conclude that 'most' A_nn's are practically equal (for the overwhelming majority of V's), i.e. the so-called weak ETH is satisfied [33, 38]. As a consequence, the expectation values (6) and (7) would again be practically equal under certain additional conditions on the initial condition $\rho (0)$. For instance, the total weight of all ${{\rho }_{nn}}(0)^{\prime} s$ corresponding to exceptionally large differences ${{A}_{nn}}-{{\langle {{A}_{nn}}\rangle }_{V}}$ should remain sufficiently small. For example (4) would obviously be a sufficient condition.

However, this line of reasoning contains a problem: if the variations $\Delta n$ of n are not any more much smaller than ${{10}^{{\rm O}(N)}}$, then equations (31) and (22) no longer imply that the variations of ${{\langle {{A}_{nn}}\rangle }_{V}}$ remain negligible. The same conclusion follows from the bound (32). Since the latter bound is already rather tight (see below (32)), we can conclude that the restriction to windows of n-values much smaller than ${{10}^{{\rm O}(N)}}$ is not merely a technical problem but rather an indispensable prerequisite of the random matrix model from section 5.1. In particular, this restriction also concerns the original findings by Deutsch [44].

Note that $\Delta n$ from above is identical to the number D of energy eigenvalues E_n contained in the microcanonical energy window ${{I}_{{\rm mic}}} :=[E-\Delta E,E]$ from section 3. The above discussed restriction thus amounts to

$$\begin{eqnarray}&&D\ll {{10}^{{\rm O}(N)}}\end{eqnarray} \tag{ 55 }$$

and implies that $\Delta E$ must remain very much smaller than any macroscopically resolvable energy difference. (This follows from the fact that the energy eigenstates are exponentially dense in the system size N, see section 2).

We consider a pure energy eigenstate of the unperturbed Hamiltonian H₀ in (8), i.e.

$$\begin{eqnarray}&&\rho =|m{{\rangle }_{0\,0}}\langle m|\end{eqnarray} \tag{ 61 }$$

with an arbitrary but fixed m. Its eigenvalues are either zero or one, hence the range from (1) is ${{\Delta }_{\rho }}=1$. Observing that $\rho _{ik}^{0}={{\delta }_{im}}{{\delta }_{km}}$ it follows from (13) and (17) to (19) that

$$\begin{eqnarray}&&{{\rho }_{nn}}=|{{U}_{mn}}{{|}^{2}},\end{eqnarray} \tag{ 62 }$$

$$\begin{eqnarray}&&{{\langle {{\rho }_{nn}}\rangle }_{V}}={{u}_{2}}(n-m),\end{eqnarray} \tag{ 63 }$$

$$\begin{eqnarray}&&\sigma _{n}^{2}={{u}_{4}}(n-m)-u_{2}^{2}(n-m),\end{eqnarray} \tag{ 64 }$$

where the variance from (14) is given for $A=\rho$ by

$$\begin{eqnarray}&&\sigma _{n}^{2}:={{\langle {{({{\rho }_{nn}})}^{2}}\rangle }_{V}}-\langle {{\rho }_{nn}}\rangle _{V}^{2}.\end{eqnarray} \tag{ 65 }$$

As a concrete example, we may focus on Gaussian distributed U_mn's (see below equation (26)), so that ${{u}_{4}}(n)=3u_{2}^{2}(n)$ and hence

$$\begin{eqnarray}&&\sigma _{n}^{2}=2\;u_{2}^{2}(n-m).\end{eqnarray} \tag{ 66 }$$

Altogether, the standard deviation ${{\sigma }_{n}}$ of the random variable ${{\rho }_{nn}}$ from (62) is thus comparable to its mean value ${{\langle {{\rho }_{nn}}\rangle }_{V}}$, and both are, according to (20) and (22), extremely small compared to the range ${{\Delta }_{\rho }}=1$ of the considered observable ρ. Within any reasonable resolution limit $\delta \rho$ of this observable we thus can conclude that, for the vast majority of random perturbations V in (8), all ${{\rho }_{nn}}$'s are practically equal (namely zero), in agreement with the general validation of ETH from section 5. But for our present purposes, this usual resolution limit $\delta \rho$ is still way too large. On the actual scale of interest, the ${{\rho }_{nn}}$'s from (62) are not at all a slowly varying function of n, but rather exhibit very significant random fluctuations (see also (24)). In particular, it would be wrong to argue that the ${{\rho }_{nn}}(0)$'s in (7) are now practically constant and hence, upon comparison with (6), thermalization follows.

We return to general density operators ρ, i.e., we only assume that ρ is a Hermitian, non-negative operator of unit trace and purity

$$\begin{eqnarray}&&{\rm Tr}\{{{\rho }^{2}}\}\leqslant 1.\end{eqnarray} \tag{ 67 }$$

In the following, we will exploit these properties of ρ, which, however, would be lost after adding a constant to ρ so that (15) is satisfied. Hence we only can employ those previous results which were obtained without the help of (15). Along these lines, one finds exactly as in (27) and (52) that

$$\begin{eqnarray}&&{{\langle {{\rho }_{nn}}\rangle }_{V}}=\mathop{\sum }\limits_{j}\rho _{jj}^{0}{{u}_{2}}(n-j),\end{eqnarray} \tag{ 68 }$$

$$\begin{eqnarray}&&\sigma _{n}^{2}\leqslant (c+5){{u}_{2}}(0)\mathop{\sum }\limits_{i}({{\rho }^{2}})_{ii}^{0}\;{{u}_{2}}(n-i),\end{eqnarray} \tag{ 69 }$$

where $\sigma _{n}^{2}$ is defined in (65) and $({{\rho }^{2}})_{ii}^{0}:={{}_{0}}{{\langle i|{{\rho }^{2}}|i\rangle }_{0}}$. Likewise, (57) and (59) yield

$$\begin{eqnarray}&&{{\langle {{\rho }_{mn}}\rangle }_{V}}=0,\end{eqnarray} \tag{ 70 }$$

$$\begin{eqnarray}&&{{\langle |{{\rho }_{mn}}{{|}^{2}}\rangle }_{V}}=\mathop{\sum }\limits_{ik}|\rho _{ik}^{0}{{|}^{2}}{{u}_{2}}(m-i){{u}_{2}}(n-k)\end{eqnarray} \tag{ 71 }$$

for all $m \ne n$.

Introducing (20) into (68) implies

$$\begin{eqnarray}&&{{\langle {{\rho }_{nn}}\rangle }_{V}}\leqslant \mathop{{\rm max} }\limits_{j}\{{{u}_{2}}(j)\}\mathop{\sum }\limits_{j}\rho _{jj}^{0}={{u}_{2}}(0),\end{eqnarray} \tag{ 72 }$$

where we exploited that the sum over j equals ${\rm Tr}\rho =1$. Likewise, (69) yields

$$\begin{eqnarray}&&\sigma _{n}^{2}\leqslant (c+5)u_{2}^{2}(0)\;{\rm Tr}\{{{\rho }^{2}}\}.\end{eqnarray} \tag{ 73 }$$

Rewriting the definition from (28) as

$$\begin{eqnarray}&&{{\Delta }_{n}}:={{\langle {{\rho }_{n+1,n+1}}\rangle }_{V}}-{{\langle {{\rho }_{nn}}\rangle }_{V}}\end{eqnarray} \tag{ 74 }$$

we find along similar lines of reasoning that

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} |{{\Delta }_{n}}| & \leqslant & \mathop{\sum }\limits_{j}\rho _{jj}^{0}|{{u}_{2}}(n+1-j)-{{u}_{2}}(n-j)| \\ {} & \leqslant & \mathop{{\rm max} }\limits_{n}|{{u}_{2}}(n+1)-{{u}_{2}}(n)|. \\ \end{array}\end{eqnarray} \tag{ 75 }$$

Moreover, we can conclude that

$$\begin{eqnarray}&&\mathop{\sum }\limits_{n}|{{\Delta }_{n}}|\leqslant \mathop{\sum }\limits_{j}\rho _{jj}^{0}\mathop{\sum }\limits_{n}|{{u}_{2}}(n+1-j)-{{u}_{2}}(n-j)|.\end{eqnarray} \tag{ 76 }$$

Similarly as in (30), the last sum is seen to be equal to $2\;{{u}_{2}}(0)$. The remaining sum over j equals ${\rm Tr}\rho =1$ and hence

$$\begin{eqnarray}&&\mathop{\sum }\limits_{n}|{{\Delta }_{n}}|\leqslant 2\;{{u}_{2}}(0).\end{eqnarray} \tag{ 77 }$$

Equations (72), (73) indicate that, in contrast to pure states in (61), for mixed states of small purity ${\rm Tr}\{{{\rho }^{2}}\}$, the random fluctuations of the ${{\rho }_{nn}}$'s about their mean values may become negligible. Moreover, the right-hand side of (75) usually turns out [44, 45, 60, 62, 63] to be of the order of $u_{2}^{2}(0)$. The same conclusion is also suggested by (77). Consequently, also the variations of ${{\langle {{\rho }_{nn}}\rangle }_{V}}$ as a function of n become small. Unlike for pure states we thus can now conclude that (6) and (7) are approximately equal, implying thermalization. However, assuming a small purity of ρ represents a quite strong restriction in the first place.

Returning to general ρ, we can deduce from (20) and (68) that

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} \mathop{\sum }\limits_{n}\langle {{\rho }_{nn}}\rangle _{V}^{2} & \leqslant & \mathop{\sum }\limits_{n}\mathop{{\rm max} }\limits_{j}\{{{u}_{2}}(j)\}\mathop{\sum }\limits_{j}\rho _{jj}^{0}\mathop{\sum }\limits_{k}\rho _{kk}^{0}{{u}_{2}}(n-k) \\ {} & \leqslant & {{u}_{2}}(0)\mathop{\sum }\limits_{j}\rho _{jj}^{0}\mathop{\sum }\limits_{k}\rho _{kk}^{0}\mathop{\sum }\limits_{n}{{u}_{2}}(n-k). \\ \end{array}\end{eqnarray} \tag{ 78 }$$

The sum over n equals one according to (21) and the two remaining sums are equal to ${\rm Tr}\rho =1$, i.e.

$$\begin{eqnarray}&&\mathop{\sum }\limits_{n}\langle {{\rho }_{nn}}\rangle _{V}^{2}\leqslant {{u}_{2}}(0).\end{eqnarray} \tag{ 79 }$$

Likewise, equations (67) and (69) imply

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} \sum \sigma _{n}^{2} & \leqslant & (c+5){{u}_{2}}(0)\mathop{\sum }\limits_{i}({{\rho }^{2}})_{ii}^{0}\;\mathop{\sum }\limits_{n}{{u}_{2}}(n-i) \\ {} & = & (c+5){{u}_{2}}(0){\rm Tr}\{{{\rho }^{2}}\}\leqslant (c+5){{u}_{2}}(0). \\ \end{array}\end{eqnarray} \tag{ 80 }$$

Altogether, this yields

$$\begin{eqnarray}&&\mathop{\sum }\limits_{n}{{\langle {{({{\rho }_{nn}})}^{2}}\rangle }_{V}}\leqslant (c+6){{u}_{2}}(0)\end{eqnarray} \tag{ 81 }$$

and with (25)

$$\begin{eqnarray}&&{{\langle \mathop{\sum }\limits_{n}{{({{\rho }_{nn}})}^{2}}\rangle }_{V}}\leqslant {{10}^{-{\rm O}(N)}}.\end{eqnarray} \tag{ 82 }$$

Since ${{({{{\rm max} }_{n}}\;{{\rho }_{nn}})}^{2}}={{{\rm max} }_{n}}{{({{\rho }_{nn}})}^{2}}\leqslant {{\sum }_{n}}{{({{\rho }_{nn}})}^{2}}$ it follows that

$$\begin{eqnarray}&&{{\langle {{(\mathop{{\rm max} }\limits_{n}\{{{\rho }_{nn}}\})}^{2}}\rangle }_{V}}\leqslant {{10}^{-{\rm O}(N)}}\end{eqnarray} \tag{ 83 }$$

and hence by Markov's inequality that

$$\begin{eqnarray}&&{\rm Prob}\left( \mathop{{\rm max} }\limits_{n}\{{{\rho }_{nn}}\}\geqslant \epsilon \right)\leqslant {{\epsilon }^{-2}}\;{{10}^{-{\rm O}(N)}},\end{eqnarray} \tag{ 84 }$$

see also the explanations below (54).

We focus on the off-diagonal matrix elements ${{\rho }_{mn}}$ with $m \ne n$. Their mean values are zero according to (70). Their variance can be readily bounded by the Cauchy–Schwarz inequality

$$\begin{eqnarray}&&|{{\rho }_{mn}}{{|}^{2}}\leqslant {{\rho }_{mm}}\;{{\rho }_{nn}}.\end{eqnarray} \tag{ 85 }$$

Likewise, introducing $|\rho _{ik}^{0}{{|}^{2}}\leqslant \rho _{ii}^{0}\rho _{kk}^{0}$ into (71) yields with (68) the estimate

$$\begin{eqnarray}&&{{\langle |{{\rho }_{mn}}{{|}^{2}}\rangle }_{V}}\leqslant {{\langle {{\rho }_{mm}}\rangle }_{V}}\;{{\langle {{\rho }_{nn}}\rangle }_{V}}.\end{eqnarray} \tag{ 86 }$$

Similarly as below (47) one sees that ${{\sum }_{mn}}|{{\rho }_{mn}}{{|}^{2}}={\rm Tr}\{{{\rho }^{2}}\}$ and hence

$$\begin{eqnarray}&&\mathop{\sum }\limits_{m \ne n}{{\langle |{{\rho }_{mn}}{{|}^{2}}\rangle }_{V}}={\rm Tr}\{{{\rho }^{2}}\}-{{\langle \mathop{\sum }\limits_{n}\rho _{nn}^{2}\rangle }_{V}}\leqslant {\rm Tr}\{{{\rho }^{2}}\}.\end{eqnarray} \tag{ 87 }$$

According to (82), the last inequality in (87) is in fact a very tight upper bound. The same estimate follows by summing in (71) over m and n. Neither of these results indicate that the off-diagonal matrix elements are typically much smaller than the diagonal elements. We thus conjecture that typical off-diagonal matrix elements will in fact not be small compared to the diagonal elements. Trivial exceptions are pure states (61). Non-trivial exceptions may be mixed states of low purity, similarly as below (77).

The main result of this section is (84): it implies that for the overwhelming majority of randomly sampled perturbations V in (8) the last term in (3) is unimaginably small (essentially in agreement with (4)). In other words, equilibration in the sense of section 2 is verified. We emphasize that all these conclusions do not depend an any further details of the actual initial condition $\rho (0)$, except that it is assumed to be fixed, i.e. independent of the randomly sampled V.

The physical interpretation is as follows: one specific, usually not very well known, but nevertheless well-defined initial state $\rho (0)$ (pure or mixed) is 'given' to us, and then evolves further in time according to one particular, randomly picked system Hamiltonian (8). Our results guarantee that the vast majority of those randomly sampled Hamiltonians gives rise to equilibration in the sense of equation (3). As the only unproven part remains the assumption that the actual (in detail not exactly known) system does not correspond to one of the rare, untypical Hamiltonians of the considered random ensemble.

Since ${\rm Tr}\{\rho \}={{\sum }_{n}}{{\rho }_{nn}}=1$ we can conclude from (84) that, for most V's, the number of non-negligible ${{\rho }_{nn}}$'s cannot be much smaller than ${{10}^{{\rm O}(N)}}$. As a consequence, the strong ETH scenario from section 5.6 does not apply. In turn, to apply this scenario, a different physical set up is required, with a different physical view of how the initial condition ρ arises. Namely, one particular, but 'typical' V in (8) is considered to have been randomly sampled but now is held fixed. Since we assumed the system is typical, strong ETH as specified in section 5.6 can and will be taken for granted. In a next step, the initial state $\rho =\rho (0)$ for this particular system is specified, arising, e.g., as the result of an experimental preparation procedure for this very system (with respect to other systems H of the ensemble, this preparation procedure may not be physically meaningful or not even well defined). Finally, there must be good reasons (e.g. a very careful experimentalist) to assume that this preparation process yields level populations ${{\rho }_{nn}}$ which are negligible outside a window of n-values much smaller than ${{10}^{{\rm O}(N)}}$, see equation (55).

In conclusion, our present formalism is able to validate either equilibration or thermalization, but not both of them simultaneously for one and the same physical model system.