Nature Abhors a Vacuum: A Simple Rigorous Example of Thermalization in an Isolated Macroscopic Quantum System

We show, without relying on any unproven assumptions, that a low-density free fermion chain exhibits thermalization in the following (restricted) sense. We choose the initial state as a pure state drawn randomly from the Hilbert space in which all particles are in half of the chain. This represents a nonequilibrium state such that the half chain containing all particles is in equilibrium at infinite temperature, and the other half chain is a vacuum. We let the system evolve according to the unitary time evolution determined by the Hamiltonian and, at a sufficiently large typical time, measure the particle number in an arbitrary macroscopic region in the chain. In this setup, it is proved that the measured number is close to the equilibrium value with probability very close to one. Our result establishes the presence of thermalization in a concrete model in a mathematically rigorous manner. The key for the proof is a new strategy to show that a randomly generated nonequilibrium initial state typically has a large enough effective dimension by using only mild verifiable assumptions. In the present work, we first give general proof of thermalization based on two assumptions, namely, the absence of degeneracy in energy eigenvalues and a property about the particle distribution in energy eigenstates. We then justify these assumptions in a concrete free-fermion model, where the absence of degeneracy is established by using number-theoretic results. This means that our general result also applies to any lattice gas models in which the above two assumptions are justified. To confirm the potential wide applicability of our theory, we discuss some other models for which the essential assumption about the particle distribution is easily verified, and some non-random initial states whose effective dimensions are sufficiently large.

All data and information relevant to this study are presented in the paper.

1. It is a common misconception that the prediction of equilibrium statistical mechanics should always be compared with an averaged quantity in the corresponding physical system. In fact, the law of large numbers guarantees that the statistical mechanical expectation value accurately predicts the outcome of a single measurement in the equilibrium state, provided that both the system and the quantity to be measured are macroscopic.

2. Researchers who emphasize microscopic mechanisms may not call the process thermalization since everything is governed by free particle dynamics. Our point is to focus only on macroscopically observable phenomena, assuming that the observer has no access to microscopic mechanisms.

3. All the results in Sect. 2 are also valid for a system of hardcore bosons.

4. To be precise this is true only when the final state represents the equilibrium state at infinite temperature (as in our case). In general, if the initial state \(|\Phi (0)\rangle \) has energy close to E then the effective dimension is believed to be close to the dimension of the corresponding energy shell, i.e., the Hilbert space spanned by energy eigenstates whose eigenvalues are close to E. One can argue, although very heuristically, that a large effective dimension is a consequence of (a strong form of) ETH. Consider a system described by a short-ranged translation-invariant Hamiltonian \(\hat{H}\) and assume that ETH is valid. For simplicity, we take the initial state \(|\Phi (0)\rangle \) to be a translation invariant product state. (We assume \(|\Phi (0)\rangle \) is not an eigenstate of \(\hat{H}\).) Then \(|\Phi (0)\rangle \) has energy distribution peaked around some value E. Let \(|\Psi _j\rangle \) be the eigenstate of \(\hat{H}\) with eigenvalue \(E_j\). Since ETH asserts that energy eigenstates with close eigenvalues are similar to each other, it is reasonable to assume that the overlap \(|\langle \Phi (0)|\Psi _j\rangle |^2\) is almost independent of j as long as \(E_j\simeq E\). This implies that \(D_\textrm{eff}\) is almost identical to the dimension of the energy shell around E.

5. We note that the diagonal entropy \(S_\textrm{d}\) studied in these works is believed to be related to the effective dimension as \(D_\textrm{eff}\sim \exp [S_\textrm{d}]\).

6. \({\varDelta }\hat{H}\) may not be small in the class of models considered in Appendix B.

7. The theorem was proved by one of us in [53]. See also Proposition 10.1 in [17] for a similar statement for a slightly complicated model.

8. See [55] for elementary proofs of the two lemmas.

9. We learned the lemma and its proof from Wataru Kai and Kazuaki Miyatani.

10. As is suggested by this conclusion, one can prove, by using essentially the same argument, the absence of degeneracy in certain open fermion chains with a suitable boundary potential.

11. In this trivial model, the energy eigenvalues for a pair of sites (x, 1) and (x, 2) are either zero (when there is no particles), \(\pm s_x+w_x\) (when there is one particle), or \(2w_x+u_x\) (when there are two particles). The total energy eigenvalues are the sums of these eigenvalues and are nondegenerate if we choose \(s_x\), \(w_x\), and \(u_x\) properly.

12. A non-optimal but simple example of a Golomb ruler is obtained by taking N such that \(L=2^N-1\) is a (Mersenne) prime, and setting \(x_j=2^{j-1}\) for \(j=1,\ldots ,N\). In this case, the particle density \(\rho \simeq N/2^N\) is exponentially small in N.

13. In a Golomb ruler, \(x_{j}-x_{Q(j)}-x_{P'(j)}+x_{Q'(j)}=0\mod L\) holds only if (1) \(j=P'(j)\) and \(Q(j)=Q'(j)\), or (2) \(j=Q(j)\) and \(P'(j)=Q'(j)\). Now we decompose a set \(\{1,2,\ldots , N\}\) into two subsets, A and B, where (1) holds in A and (2) holds in B. Then, \(P'\), Q, \(Q'\) can be expressed in the form of \(P'=\textrm{id}^A\oplus \pi ^A\), \(Q=\pi ^B\oplus \textrm{id}^B\), and \(Q'=\pi ^A\oplus \pi ^B\), where \(\pi ^A\) and \(\pi ^B\) represent permutations on A and B, respectively. With the above form of permutations, we easily see \((-1)^{P'QQ'}=1\) if \(x_{j}-x_{Q(j)}-x_{P'(j)}+x_{Q'(j)}=0\mod L\) holds for any j.

14. A derangement is a permutation in which no entry stays at the original position.

15. The floor function \(\lfloor x\rfloor \) is the largest integers less than or equal to x.

It is a pleasure to thank Shelly Goldstein, Takashi Hara, Shu Nakamura, and Marcos Rigol for useful discussions. We also thank Shin Nakano for his patient guidance in number theory, and Wataru Kai and Kazuaki Miyatani for letting us know of Lemma 3.4 and its proof. N.S. was supported by JSPS Grants-in-Aid for Early-Career Scientists No. JP19K14615, and H.T. by JSPS Grants-in-Aid for Scientific Research No. 22K03474.

Authors and Affiliations

1. Faculty of arts and sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, Japan

   Naoto Shiraishi

2. Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo, 171-8588, Japan

   Hal Tasaki

Corresponding author

Correspondence to Hal Tasaki.

Conflict of interest

The authors declare no Conflict of interest.

Communicated by Anatoli Polkovnikov.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

A: Models With Degenerate Energy Eigenvalues

Our general discussion in Sect. 2 is based on the crucial assumption, Assumption 2.1, that all the energy eigenvalues are nondegenerate. Here we shall see how one can treat models in which the degree of degeneracy is at most \(d_\textrm{max}\). We find that our thermalization results remain valid as long as \(d_\textrm{max}\) is not too large. Unfortunately, we do not know of any examples where a nontrivial upper bound for the degree of degeneracy is known.

Let \(E_j\) with \(j=1,\ldots ,N_\textrm{el}\) be the distinct energy eigenvalues. We denote by \(|\Psi _{j,\ell }\rangle \) with \(\ell =1,\ldots ,d_j\) the energy eigenstates corresponding to \(E_j\), where \(d_j\) is the degree of degeneracy of \(E_j\). We assume that the collection of \(|\Psi _{j,\ell }\rangle \) with all j, \(\ell \) forms an orthonormal basis of \(\mathcal{H}_\textrm{tot}\).

We first examine the discussion in Sect. 2.2 about the effective dimension. A straightforward generalization of the definition (2.6) of the effective dimension is

When energy eigenvalues are degenerate, however, it is convenient to employ the definition

where \(\hat{P}_j=\sum _{\ell =1}^{d_j}|\Psi _{j,\ell }\rangle \langle \Psi _{j,\ell }|\) is the projection onto the space corresponding to the energy eigenvalue \(E_j\). Clearly, (A.2) reduces to the original (A.1) when there is no degeneracy. To evaluate (A.2), we note that

where we noted \(ab\le (a^2+b^2)/2\) to get the second line. We thus find

where \(d_\textrm{max}=\max _jd_j\).

Suppose that (2.5) in Assumption 2.2 is valid for the energy eigenstates \(|\Psi _{j,\ell }\rangle \). Then Theorem 2.3 guarantees the crucial lower bound (2.7) for \(D_\textrm{eff}\) defined as (A.1). We thus find from (A.4) that

We see that \(\widetilde{D}_\textrm{eff}\) is large provided that \(d_\textrm{max}\) is not too large. Note that the degeneracy does not essentially change the behavior of the effective dimension if \(d_\textrm{max}\) grows subexponentially in N.

We move onto the discussion in Sect. 2.3 about the time evolution. Taking into account the degeneracy, the expression (2.14) for the time evolution reads

where we defined

Correspondingly, (2.20) is modified as

Therefore the rest of the discussion remains valid if we replace \(D_\textrm{eff}\) with \(\widetilde{D}_\textrm{eff}\).

B: Models Satisfying Assumption 2.2

In this Appendix, we present two classes of models in which we can prove Assumption 2.2 about the particle distribution (under suitable assumptions about nondegeneracy). If we could also justify Assumption 2.1 about the nondegeneracy of energy eigenvalues, we would have further rigorous examples of thermalization. Unfortunately, we still do not know how nondegeneracy can be proved, although we believe it to be highly plausible.

The first class of models is that of interacting fermions on a specific class of lattices, while the second class is that of free fermions on arbitrary lattices with \(\mathbb {Z}_2\) symmetry.

1.1 B.1. Interacting Fermions on Double-Lattice

First, we discuss a class of lattice gas models on a double lattice with special symmetry. In these models, we can easily verify the bound (2.5) for any energy eigenstate corresponding to a nondegenerate energy eigenvalue. See Lemma B.1 below. This means that Assumption 2.2 about the particle distribution is automatically valid if Assumption 2.1 about nondegeneracy of energy eigenvalues is valid. The class of models in fact contains many non-trivial interacting models for which we generally expect that energy eigenvalues are nondegenerate. We thus expect that the present class contains many examples in which our thermalization theorem, Theorem 2.4, is valid. Unfortunately, we are not able to prove nondegeneracy in concrete models, except for trivial decoupled models. See the discussion at the end of the present subsection.

We shall describe the class of models and state the basic observation, i.e., Lemma B.1. Although we here describe models of fermions for notational simplicity, extensions to hardcore bosons or quantum spin systems are trivial.

Let \(\Lambda _0\) be a lattice with L/2 sites, and \(\Lambda _1\) and \(\Lambda _2\) be copies of \(\Lambda _0\). Sites in \(\Lambda _1\) and \(\Lambda _2\) are denoted as (x, 1) and (x, 2), respectively, with \(x\in \Lambda _0\). We consider a model of fermions on the whole lattice \(\Lambda =\Lambda _1\cup \Lambda _2\).

We assume that the Hamiltonian \(\hat{H}\) conserves the total particle number and is invariant under the exchange of two sites (x, 1) and (x, 2) for each \(x\in \Lambda _0\). The latter is a highly nontrivial (and artificial) assumption, which enables us to prove the desired bound (2.5) easily. To be more precise we define for each \(x\in \Lambda _0\) the unitary operator \(\hat{U}_x\) that swaps (x, 1) and (x, 2). It is defined by \(\hat{U}_x|\Phi _\textrm{vac}\rangle =|\Phi _\textrm{vac}\rangle \), \(\hat{U}_x\,\hat{c}_{(x,1)}\hat{U}_x^\dagger =\hat{c}_{(x,2)}\), \(\hat{U}_x\,\hat{c}_{(x,2)}\hat{U}_x^\dagger =\hat{c}_{(x,1)}\), and \(\hat{U}_x\,\hat{c}_{(y,\nu )}\hat{U}_x^\dagger =\hat{c}_{(y,\nu )}\) for \(y\ne x\) and \(\nu =1,2\). Note that \((\hat{U}_x)^2=\hat{1}\). Our symmetry assumption is that \([\hat{U}_x,\hat{H}]=0\) for any \(x\in \Lambda _0\).

If we restrict ourselves to models with standard particle-hopping and two-body interactions, the most general Hamiltonian takes the form

where \(t_{x,y}=(t_{y,x})^*\in \mathbb {C}\), \(v_{x,y}=v_{y,x}\in \mathbb {R}\), and \(s_x,w_x,u_x\in \mathbb {R}\). We defined the number operator by \(\hat{n}_{(x,\sigma )}=\hat{c}^\dagger _{(x,\sigma )}\hat{c}_{(x,\sigma )}\).

Here is the basic observation in the present appendix.

Lemma B.1

Let \(|\Psi \rangle \) be the normalized eigenstate corresponding to a nondgenerate energy eigenvalue of \(\hat{H}\). Then we have

which is the same as (2.5).

Proof

For a fixed particle number N, we define the basis states of the model by

where \(S_1\) and \(S_2\) are arbitrary subsets of \(\Lambda _0\) such that \(|S_1|+|S_2|=N\). Take any normalized eigenstate \(|\Psi \rangle \) of \(\hat{H}\) and expand it in the above basis as

where \(\psi _{S_1,S_2}\in \mathbb {C}\) are coefficients which satisfy \(\sum |\psi _{S_1,S_2}|^2=1\). The symmetry of the Hamiltonian and the nondgeneracy imply \(\hat{U}_x|\Psi \rangle =\pm |\Psi \rangle \) for any \(x\in \Lambda _0\). This means that the expansion coefficients satisfy

for any \(S\subset \Lambda _0\) with \(|S|=N\) and any \(S'\subset S\). We thus have

which, when summed over S, yields

Noting that the left-hand side is \(\langle \Psi |\hat{P}_1|\Psi \rangle \), we get (B.2). \(\square \)

This model considered here is generally non-integrable, and we expect that its energy eigenvalues are nondegenerate. It is desirable to find models in which the absence of degeneracy can be proved rigorously.

Unfortunately, the only case we can prove nondegeneracy is a trivial decoupled model with \(t_{x,y}=v_{x,y}=0\) for all \(x,y\in \Lambda _0\). We readily see that the energy eigenvalues are nondegenerate if \(s_x\), \(w_x\), and \(u_x\) with \(x\in \Lambda _0\) are chosen to be different from each other.^{Footnote 11} Therefore we can fully justify our main theorem, Theorem 2.4, for the model, but we should note that the result is trivial. In the initial state \(|\Phi (0)\rangle \), each pair of sites (x, 1) and (x, 2) is either empty or occupied by one partilce at (x, 1). The time evolution then takes place independently for each pair of sites. If there is a particle in a pair, then a superposition of two states with a particle at (x, 1) and at (x, 2) is generated. This, when viewed macroscopically, results in thermalization. We must say that there is nothing interesting in this observation.

1.2 B.2. Free Fermions With \(\mathbb {Z}_2\) Symmetry

Next, we discuss a class of free fermion models in which the bound (2.5) for the particle distribution, and hence Assumption  2.2 can be justified. We here follow the strategy outlined at the end of Sect. 3.3 and justify the inequality (3.34) for the fermion operators corresponding to single-particle energy eigenstates.

Let \(\Lambda \) be an arbitrary lattice, and consider the most general free fermion Hamiltonian

where the hopping amplitude satisfies \(t_{x,y}=(t_{y,x})^*\in \mathbb {C}\). Note that the diagonal element \(t_{x,x}\in \mathbb {R}\) represents the single-body potential.

We assume that the model has \(\mathbb {Z}_2\) symmetry in the sense that there is a one-to-one map \(p:\Lambda \rightarrow \Lambda \) such that \(p^2=\textrm{id}\), and that the Hamiltonian is invariant under the transformation p, i.e., \(t_{p(x),p(y)}=t_{x,y}\) for any \(x,y\in \Lambda \). We also assume that \(\Lambda \) is disjointly decomposed as \(\Lambda =\Lambda _1\cup \Lambda _2\) and that \(p(\Lambda _1)\subset \Lambda _2\).

As an example, consider the chain \(\Lambda =\{1,\ldots ,L\}\) with odd L, and let p be the inversion \(p(x)=L+1-x\). Then the decomposition with \(\Lambda _1=\{1,\ldots ,(L-1)/2\}\) and \(\Lambda _2=\{(L+1)/2,\ldots ,L\}\) satisfies the above property.

Let \(\varvec{\psi }=(\psi _x)_{x\in \Lambda }\) be a normalized single-particle energy eigenstate. To be precise, it satisfies the Schrödinger equation \(\epsilon \,\psi _x=\sum _{y\in \Lambda }t_{x,y}\psi _x\) for all \(x\in \Lambda \) with the (single-particle) energy eigenvalue \(\epsilon \). Let us further assume that the energy eigenvalue \(\epsilon \) is nondegenerate. Then, with respect to the symmetry transformation p, the corresponding wave function \(\varvec{\psi }\) is either symmetric, i.e., \(\psi _{p(x)}=\psi _x\) for all \(x\in \Lambda \), or antisymmetric, i.e., \(\psi _{p(x)}=-\psi _x\) for all \(x\in \Lambda \). We then see that

where we noted that \(p(\Lambda _1)\subset \Lambda \backslash \Lambda _1\).

Let \(\hat{a}^\dagger _{\varvec{\psi }}=\sum _{x\in \Lambda }\psi _x\,\hat{c}^\dagger _x\) be the creation operator of the state \(\varvec{\psi }\). It can be decomposed as \(\hat{a}^\dagger _{\varvec{\psi }}=\hat{b}^\dagger _{1,\varvec{\psi }}+\hat{b}^\dagger _{2,\varvec{\psi }}\) with \(\hat{b}^\dagger _{1,\varvec{\psi }}=\sum _{x\in \Lambda _1}\psi _x\,\hat{c}^\dagger _x\) and \(\hat{b}^\dagger _{2,\varvec{\psi }}=\sum _{x\in \Lambda _2}\psi _x\,\hat{c}^\dagger _x\). This corresponds to the decomposition (3.30). We also see from (B.9) that \(\hat{b}^\dagger _{1,\varvec{\psi }}\) satisfies \(\Vert \hat{b}_{1,\varvec{\psi }}\,\hat{b}^\dagger _{1,\varvec{\psi }}\Vert =\sum _{x\in \Lambda _1}|\psi _x|^2\le 1/2\), which corresponds to the desired (3.34).

We now assume that single-particle energy eigenvalues \(\epsilon _1,\ldots ,\epsilon _{|\Lambda |}\) are all nondegenerate, and denote by \(\hat{a}^\dagger _j\) the creation operator of the single-particle energy eigenstate corresponding to \(\epsilon _j\). Then the foregoing discussion shows that each \(\hat{a}^\dagger _j\) is decomposed as (3.30), and the operator for the sublattice \(\Lambda _1\) satisfies the bound (3.34). Repeating the derivation in section 3.3, we see an N-body energy eigenstate of the form

satisfies the desired bound (2.5).

Interestingly, it was only necessary to assume the nondegeneracy of single-particle energy eigenvalues to prove the desired bound (2.5) in this model. To ensure the presence of thermalization, we have to assume further that N-body energy eigenvalues are nondegenerate. It is rather likely that degeneracy is absent in a sufficiently complex free fermion model, but we do not know how to justify the claim. We also note that the p-symmetry may not be exact. It can be violated by a small perturbation as long as the bound (2.5) remains valid.

C: Effective Dimensions of Some Initial Particle Configurations in the Free Fermion Chain

In the main text, the initial state \(|\Phi (0)\rangle \) is drawn randomly from the small Hilbert space \(\mathcal{H}_1\). Conceptually speaking, it may be desirable to consider the time evolution starting from a non-random simple initial state. Here we again treat free fermion chains and examine the effective dimensions of some initial states in which particles have definite positions.

In Sect. C.2, we observe that the initial state where particles are arranged in a periodic manner has an effective dimension that is large but not large enough to guarantee thermalization. This observation suggests that a random initial state is mandatory in a free fermion model if we demand the effective dimension to be extremely large. This is very likely to be a common property for integrable models. In a non-integrable model, on the other hand, it is believed that even a regular initial state generally has an effective dimension almost as large as the total dimension.

In Sect. C.3, we consider an artificial class of initial configurations (Golomb ruler configurations) and show that the corresponding effective dimensions are almost as large as the total dimension. This leads to a statement about thermalization with a non-random initial state. In this class of models, however, the particle density inevitably tends to zero according to \(\rho \sim N^{-1}\) as the particle number grows.

1.1 C.1. General formula for \(D_\textrm{eff}\)

We consider the free fermion chain as defined in section 3. Let the initial particle configuration be \(\varvec{x}=(x_1,x_2,\ldots ,x_N)\) with \(x_j\in \{1,\ldots ,L\}\) such that \(x_j<x_{j+1}\) for \(j=1,\ldots ,N-1\), and define the corresponding N fermion state as

We set \(|\Phi _{\varvec{x}}\rangle \) as the initial state \(|\Phi (0)\rangle \). Then we see from (2.6) that the effective dimension is given by

where \(\tilde{\mathcal{K}}_N=\{(k_1,\ldots ,k_N)\,|\,k_j<k_{j+1}\}\subset \mathcal{K}^N\). (The k-space \(\mathcal{K}\) is defined in (3.2).) Noting that (3.3) implies \(\{\hat{c}_x,\hat{a}^\dagger _k\}=e^{ikx}/\sqrt{L}\), we see

where the summation is over all possible N! permutations P of \(\{1,\ldots ,N\}\) and \((-1)^P\) is the signature of P. It is useful to regard \(\varvec{k}\) in the above expression as an element in \(\mathcal{K}^N\) rather than its physical subspace \(\tilde{\mathcal{K}}_N\). This replacement is justified since \(\bigl |\langle \Phi _{\varvec{x}}|\Psi _{\varvec{k}}\rangle \bigr |\) is invariant under any permutations of \(k_1,\ldots ,k_N\) and equals zero if \(k_j=k_{j'}\) for some \(j\ne j'\). We can thus rewrite (C.2) as

This rewriting is useful since one can now sum independently over \(k_1,\ldots ,k_N\in \mathcal{K}\).

From (C.3), we see that

and

with

We shall evaluate the sum (C.4) by using the decomposition (C.6). Clearly

The remaining sums are evaluated by using the standard formula

where \(x\in \mathbb {Z}\). Note that in the expression for \(C_2(\varvec{k})\) in (C.7), one has \(x_{P(j)}-x_{Q(j)}\ne 0\) for at least one j because \(P\ne Q\). We thus see

The sum of \(C_3(\varvec{k})\) is generally nonzero and can be evaluated as

where the characteristic function is defined as \(\chi [\text {true}]=1\) and \(\chi [\text {false}]=0\). Let us write the right-hand side of (C.12) as \(\mathcal{S}_{\varvec{x}}/L^N\). From (C.4), (C.6), (C.9), (C.11), and (C.12), we see that the effective dimension of the initial state \(|\Phi (0)\rangle =|\Phi _{\varvec{x}}\rangle \) is given by

Our main task is to evaluate the sum \(\mathcal{S}_{\varvec{x}}\) defined in (C.12) for a given particle configuration \(\varvec{x}\). For later convenience we sum over P in (C.12) (and write \(P^{-1}Q\), \(P^{-1}P'\), and \(P^{-1}Q'\) as Q, \(P'\), and \(Q'\), respectively) to rewrite the expression as

1.2 C.2. \(D_\textrm{eff}\) in Periodic Configurations

First, we consider regular particle configurations with a period \(p=1,2,\ldots \). Fix p, and choose the chain length L and the particle number N such that \(L=pN\). We consider the initial particle distribution given by

for \(j=1,\ldots ,N\).

Then (C.14) is computed as

which depends only on N and is independent of L and p. Thus, we can evaluate the above sum by employing a useful choice of p. Fortunately, this sum becomes trivial for \(p=1\), and therefore we compute the sum in the case of \(p=1\). A fermion system with \(L=N=1\), which is fully filled, has one-dimensional Hilbert space and hence \(D_\textrm{eff}=1\). We see from (C.13) that \(\mathcal{S}_{\varvec{x}}=N^N-N!\). Recalling the L independence of \(\mathcal{S}_{\varvec{x}}\), we get a remarkably simple result

for any L and N (such that \(L=pN\)), where \(\rho =1/p\) is the particle density. We thus see that the effective dimension grows exponentially with the system size L, as expected. But it turns out that it is not large enough. Combining (C.17) with (2.1), we see

and hence \(D_\textrm{eff}\) is considerably smaller compared with the total dimension \(D_\textrm{tot}\). This conclusion is consistent with the numerical result in [46]. We conclude that our strategy of the proof of Theorem 2.4 is ineffective for this initial state. Interestingly, it was found numerically in [45, 46] that the free fermion chain with similar initial states exhibits thermalization in some sense.

1.3 C.3. \(D_\textrm{eff}\) in Golomb-Ruler Configurations

We next discuss a class of particle configurations for which the effective dimension \(D_\textrm{eff}\) is easily evaluated and turns out to be almost as large as the total dimension \(D_\textrm{tot}\). In these settings, however, the particle density inevitablyapproaches zero as N gets larger.

A sequence of natural numbers \(\varvec{x}=(x_1,\ldots ,x_N)\) is called a Golomb ruler [59] if for any \(j\ne k\), one has \(x_j-x_k=x_\ell -x_m\) only when \(j=\ell \) and \(k=m\). The periodic boundary counterpart (in which one replaces the condition \(x_j-x_k=x_\ell -x_m\) by \(x_j-x_k=x_\ell -x_m \mod L\)) is called a modular Golomb ruler. The optimal (minimum) system size L of a modular Golomb ruler for given N is \(L=N(N-1)\), since \(x_j-x_k\) takes \(N(N-1)\) distinct positive integers. The optimal configuration, if exists, is called a perfect difference set. Interestingly, perfect difference sets are proven to exist if \(N-1\) is a prime power \(p^n\) [60].

We set the configuration of N particles as a modular Golomb ruler \(\varvec{x}=(x_1,\ldots ,x_N)\) (\(x_1<x_2\cdots <x_N\)). By taking \(x_1=1\) and choosing the system size L as a prime such that \(L\ge 2x_N-1\), we see that for any \(j\ne k\), one has \(x_j-x_k=x_\ell -x_m\mod L\) only when \(j=\ell \) and \(k=m\) (i.e., a modular Golomb ruler). A nontrivial and asymptotically optimal example^{Footnote 12} of a Golomb ruler can be found in [61], where the following sequence

for \(j=1,\ldots ,N\) with a prime \(N>2\) is shown to be a Golomb ruler. Since \(1+2N(N-1)\le x_N\le 1+2N(N-1)+N-1\), the aforementioned construction leads to the chain length as \(L\simeq 4N^2\) with the particle density \(\rho \simeq (4N)^{-1}\). Note that the optimal (densest) Golomb ruler has density \(\rho =O(N^{-1})=O(L^{-1/2})\), and thus the above construction is asymptotically optimal.

We shall fix an arbitrary initial particle configuration \(\varvec{x}\) that forms a modular Golomb ruler and evaluate its effective dimension. We first bound the sign factor in (C.14) as \((-1)^{QP'Q'}\le 1\) to get

In fact, it can be shown that this is an equality^{Footnote 13}, but the upper bound is enough for our purpose.

Let us fix a permutation \(Q\ne \textrm{id}\), and examine the conditions for \(\prod _{j=1}^N\chi [\cdots ]=1\), i.e., \(x_{j}-x_{Q(j)}-x_{P'(j)}+x_{Q'(j)}=0\mod L\) for all \(j=1,\ldots ,N\). If j is such that \(Q(j)\ne j\), the condition for \(\varvec{x}\) implies \(P'(j)=j\) and \(Q'(j)=Q(j)\). We see there is no choice for \(P'(j)\) and \(Q'(j)\). If \(j=Q(j)\), on the other hand, the only requirement is \(P'(j)=Q'(j)\). There is some freedom for choosing \(P'(j)\) and \(Q'(j)\).

Let \(n_Q\) be the number of j such that \(Q(j)=j\). Since \(Q\ne \textrm{id}\), we see \(n_Q=0,1,\ldots ,N-2\). From the above consideration, we see that there are \(n_Q!\) choices for \(P'\) (and thus \(Q'\)) for fixed Q. We thus find

where \(\mathcal{N}(n)\) is the number of \(Q\ne \textrm{id}\) such that \(n_Q=n\). The value of \(\mathcal{N}(n)\) is computed explicitly as

where \(d_m\) is the m-th de Montmort number (also known as the m-th derangement number or the subfactorial of m). The de Montmort number counts the number of derangement^{Footnote 14} on n elements. Fortunately, the de Montmort number \(d_m\) is explicitly computed as

with the floor function^{Footnote 15}\(\lfloor \cdot \rfloor \) [62]. This expression, with (C.20) and (C.21), leads to a simple upper bound

Substituting this into (C.13), we can bound the effective dimension from below as

Thus the ratio between the total dimension and the effective dimension is bounded as

Note that \(D_\textrm{tot}=\left( {\begin{array}{c}L\\ N\end{array}}\right) \) is approximated by \((L/N)^N\) when \(N\ll L\), and hence grows super-exponentially in N. (If we take the initial configuration (C.19) then \(D_\textrm{tot}\sim (4N)^N\).) This means that \(D_\textrm{eff}\) satisfying (C.26) is extremely close to \(D_\textrm{tot}\).

As we have stressed, such a large effective dimension is expected in a non-integrable quantum many-body system, but not in an integrable system. Here we have a large \(D_\textrm{eff}\) in a free fermion model because of the artificial Golomb-ruler configuration. But recall that this choice is possible only in the extremely low density \(\rho =O(N^{-1})\).

The above observation about the large effective dimension leads to a statement about thermalization. Take a sufficiently large and arbitrary prime N and a prime L such that \(L\ge 2x_N-1\) with \(x_N\) given by (C.19). We consider the system of N fermions on the chain \(\{1,\ldots ,L\}\) with the Hamiltonian (3.1). We take the phase factor \(\theta \) for which the energy eigenvalues (3.7) are nondegenerate. (See Lemma 3.1.)

For simplicity we restrict our observable only to the particle number in the left half of the chain, i.e.,

The equilibrium value is of course \(\langle \hat{N}_\textrm{left}\rangle _\infty =N/2\). Let the initial state be \(|\Phi (0)\rangle =|\Phi _{\varvec{x}}\rangle =\hat{c}^\dagger _{x_1}\cdots \hat{c}^\dagger _{x_N}|\Phi _\textrm{vac}\rangle \), where the configuration \(x_1,\ldots ,x_N\) is given by (C.19). Since

the initial state is highly nonequilibrium with respect to the quantity \(\hat{N}_\textrm{left}/N\).

Then by using the large deviation type estimate

which follows from (2.22), and the standard argument (as in the proof of Theorem 2.4), we can prove the following.

Theorem C.1

For any \(\varepsilon >0\), there exists a constant \(T>0\) and a subset (a collection of intervals) \(G\subset [0,T]\) with

where \(\mu (G)\) is the total length of the intervals in G. Suppose that one performs a measurement of the number operator \(\hat{N}_\textrm{left}\) in the state \(|\Phi (t)\rangle \) with arbitrary \(t\in G\). Then, with probability larger than \(1-e^{-(\varepsilon ^2/4)N}\), the measurement result \(N_\textrm{left}\) satisfies

i.e., the state is found in thermal equilibrium.

Thus thermalization starting from a deterministic initial state has been established without any unproven assumptions. Here one can choose the precision \(\varepsilon >0\) arbitrarily. But in order to make the factor \(e^{-(\varepsilon ^2/4)N}\) negligibly small, one must take N large enough, which means that the density becomes lower.

D: Possible Extension to the Finite Temperature Situation

Throughout the present paper, we only focused on situations where the initial and the final states correspond to infinite temperature thermal states. See, in particular, Sect. 2.4. We believe that our results can be extended to finite temperature settings with extra technical efforts. Although we do not elaborate on the extension, we here briefly discuss the setting and essential steps in the proof.

We consider the free fermion Hamiltonian (1.1), (3.1). Decompose the Hamiltonian as in (2.30), where we choose \(\Lambda _1\) as the half-chain \(\{1,\ldots ,(L-1)/2\}\). It follows that \(\Vert {\varDelta }\hat{H}\Vert =h_0\) is independent of the system size. Denote by \(|\tilde{\Psi }_j\rangle \in \mathcal{H}_1\) the normalized eigenstate of \(\hat{H}_1\) with eigenvalue \(\tilde{E}_j\). For the energy density \(u\in (-2,0)\) and the energy width \({\varDelta }u>0\), we define the nonequilibrium microcanonical energy shell by

Noting that \(\hat{H}_2|\tilde{\Psi }_j\rangle =0\), we observe

which implies that \(|\tilde{\Psi }_j\rangle \) is (with a minor error when N is large) a superposition of \(|\Psi _k\rangle \) such that \(|E_k-\tilde{E}_j|\lesssim h_0\). We thus find that any state \(|\Phi (0)\rangle \in \mathcal{H}_1^u\) and its time-evolution \(|\Phi (t)\rangle =e^{-i\hat{H}t}|\Phi (0)\rangle \) belongs (again, with minor errors when N is large) to the standard microcanonical energy shell

with \({\varDelta }u'>{\varDelta }u\).

In the finite temperature setting, we choose initial state \(|\Phi (0)\rangle \) randomly and uniformly from the nonequilibrium energy shell \(\mathcal{H}^u_1\). The goal is to show that Theorem 2.4 (with suitable modifications of constants) is valid for the time-evolved state \(|\Phi (0)\rangle \).

Recalling that \(|\Phi (0)\rangle \) (essentially) belongs to \(\mathcal{H}^u_\textrm{tot}\), our strategy for the proof will be to properly replace \(\mathcal{H}_1\) and \(\mathcal{H}_\textrm{tot}\) in the original proof with \(\mathcal{H}^u_1\) and \(\mathcal{H}^u_\textrm{tot}\), respectively. Let us see how the proof of the most important estimate of the effective dimension, Theorem 2.3, is modified. Interestingly, a small modification is sufficient. Denoting by \(\hat{P}^u_1\) the projection onto \(\mathcal{H}^u_1\), and by \(D_1^u\) the dimension of \(\mathcal{H}^u_1\), we find

which is a faithful extension of the key inequality (2.10). The analog of Theorem 2.3 is proved if we properly estimate the ratio \(D^u_\textrm{tot}/D^u_1\). Another nontrivial (but technical) step for the proof of the desired extension of Theorem 2.4 is the derivation of the large-deviation upper bound (2.22) for the microcanonical average.

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