Exact Thermal Eigenstates of Nonintegrable Spin Chains at Infinite Temperature

Introduction— Understanding the mechanism of thermalization in quantum many-body systems has been a pivotal issue in statistical physics [1, 2, 3]. Notably, the eigenstate thermalization hypothesis (ETH) [4, 5, 6] has served as a cornerstone in this field. It posits that all the energy eigenstates in the bulk of the spectrum of quantum many-body systems exhibit thermal properties, thereby giving a plausible explanation of thermalization.

While the ETH is anticipated to hold in most nonintegrable systems, the verification of whether this hypothesis holds in realistic many-body systems relies on numerical calculations, and a theoretical verification has remained elusive [7, 8, 9, 10]. Thus, a significant challenge lies in theoretically addressing the nature of energy eigenstates, particularly in nonintegrable systems. However, it has not been clear whether thermal eigenstates of nonintegrable systems can be treated theoretically. This stems from the difficulty of writing down quantum states whose entanglement entropy obeys a volume law.

One approach to treat quantum many-body states theoretically is to use variational wave functions. Particularly for states that contain a small amount of entanglement, they can be represented via tensor network states such as a matrix product state [11, 12]. Tensor network states are highly tractable, making them not only practical but also significantly contributing to theoretical advancements. Indeed, by utilizing the matrix product state, it has been successful to exactly describe finite-energy-density low-entangled (thus nonthermal) eigenstates even for nonintegrable systems [13, 14, 15], which are examples of many-body scars [16, 17, 18, 19, 20]. However, there has been a lack of variational wave functions suitable for theoretical analysis of volume-law states, which is one of the reasons why thermal eigenstates have not yet been obtained. Hence, there is a craving for a class of volume-law states amenable to the theoretical treatment [21, 22, 23, 24, 25].

In this Letter, we provide pairs of a nonintegrable Hamiltonian and its thermal eigenstate at infinite temperature. We consider a class of volume-law states, which we call the entangled antipodal pair (EAP) states, that are amenable to theoretical calculations. Then we fully characterize Hamiltonians having the EAP state as an eigenstate. It is rigorously shown that some of these Hamiltonians are nonintegrable. In addition, by evolving an EAP state in imaginary time, we construct a thermal pure state at arbitrary temperature, which is locally indistinguishable from the Gibbs state.

Entangled antipodal pair state— We consider quantum spin-$1/2$ systems on a one-dimensional lattice $\Lambda=\{1,2,\cdots,N\}$ with periodic boundary conditions. We assume that the number of lattice sites $N$ is even. Let $\hat{\sigma}_{j}^{\mu}(\mu=x,y,z)$ be the Pauli matrices acting on the $j$-th site, and $\ket{0}_{j}$ and $\ket{1}_{j}$ be the eigenvectors of $\hat{\sigma}_{j}^{z}$.

Our goal is to obtain pairs of a nonintegrable Hamiltonian and its thermal eigenstate. To achieve this, we adopt the following strategy. First, we introduce a class of volume-law states that are theoretically tractable. Next, for each of these volume law states, we search for Hamiltonians that have it as a thermal eigenstate. Finally, we prove that some of these Hamiltonians are nonintegrable. Following this strategy, as the first step, we introduce the entangled antipodal pair (EAP) state as ¹¹1Some special EAP states appear in the study of integrable systems [52, 53].

$\displaystyle\ket{\mathrm{EAP}}=\bigotimes_{j=1}^{N/2}\ket{\Phi_{p_{j}q_{j}}}_{j,j+N/2}.$

(1)

Here $\ket{\Phi_{p_{j}q_{j}}}_{j,j+N/2}(p_{j},q_{j}=0,1)$ are the Bell states between antipodal sites $j$ and $j+N/2$ ²²2It is possible to extend EAP states to higher-dimensional systems. However, unlike the one-dimensional case, there exists arbitrariness in the choice of antipodal sites. defined by

$\displaystyle\ket{\Phi_{pq}}_{j,j^{\prime}}=\frac{\ket{0}_{j}\ket{p}_{j^{\prime}}+(-1)^{q}\ket{1}_{j}\ket{{\bar{p}}}_{j^{\prime}}}{\sqrt{2}}\quad(p,q=0,1),$

(2)

where $\bar{p}$ represents the negation of $p$. It is straightforward to check that

$\displaystyle\hat{\sigma}_{j}^{\mu}\hat{\sigma}_{j+N/2}^{\mu}\ket{\Phi_{p_{j}q_{j}}}_{j,j+N/2}=\omega_{j}^{\mu}\ket{\Phi_{p_{j}q_{j}}}_{j,j+N/2},$

(3)

where $\omega_{j}^{z}=(-1)^{p_{j}}$, $\omega_{j}^{x}=(-1)^{q_{j}}$, and $\omega_{j}^{y}=-\omega_{j}^{x}\omega_{j}^{z}$. Hence, the EAP state is uniquely characterized by $(\omega_{j}^{x})_{j\in\Lambda}$ and $(\omega_{j}^{z})_{j\in\Lambda}$. As a consequence of Eq. (3), the action of $\hat{\sigma}_{j}^{\mu}$ on the EAP state is equivalent to the action of $\hat{\sigma}_{j+N/2}^{\mu}$, except for the factor of $\omega_{j}^{\mu}$, i.e.,

$\displaystyle\hat{\sigma}_{j}^{\mu}\ket{\mathrm{EAP}}=\omega_{j}^{\mu}\hat{\sigma}_{j+N/2}^{\mu}\ket{\mathrm{EAP}}.$

(4)

This plays a key role in the proof of our main results.

Now we explain that EAP states are thermal. Since an EAP state consists of Bell pairs between sites separated by $N/2$ (Fig. 1), for any subsystem $X$ with diameter

$\displaystyle D(X)=\max_{j,j^{\prime}\in X}|j-j^{\prime}|$

(5)

satisfying $D(X)<N/2$, the entanglement entropy obeys a volume law with a maximum coefficient of $\log 2$. Consequently, the reduced density matrix of an EAP state for the subsystem $X$ is the maximally mixed state $\propto\hat{I}_{X}$, which coincides with that of the Gibbs state $\hat{\rho}^{\mathrm{can}}\propto e^{-\beta\hat{H}}$ at the inverse temperature $\beta=0$. Thus, EAP states cannot be distinguished from the thermal equilibrium state at $\beta=0$ by measurements of any local observable, i.e., observable whose support size is independent of $N$. This means that EAP states represent “microscopic thermal equilibrium” (MITE) [28, 29, 3], which is one of the most fundamental definitions of equilibrium states, e.g., in the context of thermalization.

This should be contrasted with the rainbow state [21, 22, 23, 24], which is a product of the Bell states between sites $j$ and $N-j+1$. If we focus only on a subsystem $\{1,2,\cdots,\ell\}$ (with $\ell\leq N/2$), the reduced density matrix of a rainbow state is maximally mixed, and hence coincides with that of the Gibbs state $\hat{\rho}^{\mathrm{can}}\propto e^{-\beta\hat{H}}$ at $\beta=0$. However, for instance, by considering a local observable $\hat{\sigma}^{x}_{N/2}\hat{\sigma}^{x}_{N/2+1}$, rainbow states can be distinguished from the Gibbs state (because the expectation value of $\hat{\sigma}^{x}_{N/2}\hat{\sigma}^{x}_{N/2+1}$ in a rainbow state takes $\pm 1$ while that in the Gibbs state takes $0$). Thus, rainbow states do not represent MITE.

Note that, although EAP states represent MITE, they are very different from typical thermal eigenstates of nonintegrable systems. For instance, the expectation value of a few-body but nonlocal observable $\hat{\sigma}_{j}^{x}\hat{\sigma}_{j+N/2}^{x}$ in an EAP state differs from that in the Gibbs state $\hat{\rho}^{\mathrm{can}}$ at $\beta=0$. (The former takes $\pm 1$, while the latter takes $0$.) In other words, EAP states are examples of states representing MITE but not “few-body thermal equilibrium [3].” Although MITE is more common, some studies [7, 30, 31, 32, 9, 33] have also examined few-body thermal equilibrium and shown that thermal eigenstates of nonintegrable systems often represent it. These facts indicate that EAP states are sort of special thermal eigenstates ³³3 In addition, we can show that EAP states are not thermal in a “deep sense” [54] (even when the moment $k$ of the projected ensemble is not so large, such as $k=2$) in contrast to typical eigenstates of nonintegrable systems [55]..

EAP state as an energy eigenstate— As explained above, in the following, we search for Hamiltonians that have the EAP state as a thermal eigenstate. To this end, we write the Hamiltonian in the most general form as

$\displaystyle\hat{H}$

$\displaystyle=\sum_{X(\subset\Lambda)}\sum_{\vec{\mu}\in\{x,y,z\}^{X}}J_{X}^{\vec{\mu}}\bigotimes_{j\in X}\hat{\sigma}_{j}^{\mu_{j}}.$

(6)

In the first sum, $X$ represents a subset of $\Lambda$. In the second sum, $\vec{\mu}$ represents a combination of $\mu_{j}=x,y,z$ for $j\in X$. Here the origin of the energy is taken such that $\hat{H}$ is traceless. We are interested in the Hamiltonian where some EAP state $\ket{\mathrm{EAP}}$ is an energy eigenstate. Imposing a very mild condition that the locality of interactions in $\hat{H}$ is less than $N/4$, we can characterize such Hamiltonians completely as follows ⁴⁴4See Supplemental Material for proofs of theorems, additional numerical results, and discussions, which includes Refs. [56, 57, 58, 59, 60, 61]:

Note that we can easily check whether a given EAP state is an eigenstate of a given Hamiltonian by testing Eq. (7). This provides a simple explanation to previous results [36, 37] that construct thermal energy eigenstates in certain nonintegrable systems [35].

Note also that Eq. (7) contains only a pair of coefficients $\{J_{X}^{\vec{\mu}},J_{Y}^{\vec{\nu}}\}$, but no other coefficients. This means that any coefficients $J_{X_{1}}^{\vec{\mu}_{1}}$ and $J_{X_{2}}^{\vec{\mu}_{2}}$ belonging to different pairs $\{J_{X_{1}}^{\vec{\mu}_{1}},J_{Y_{1}}^{\vec{\nu}_{1}}\}$ and $\{J_{X_{2}}^{\vec{\mu}_{2}},J_{Y_{2}}^{\vec{\nu}_{2}}\}$ can be taken independently. Thus, for any EAP state, we can construct a Hamiltonian that has the EAP state as an eigenstate by just taking the coefficients such that every pair $\{J_{X}^{\vec{\mu}},J_{Y}^{\vec{\nu}}\}$ satisfies Eq. (7). [In order to restrict the interactions in $\hat{H}$ to a finite range, we can take coefficients $J_{X}^{\vec{\mu}}$ with large $D(X)$ to be zero because $J_{X}^{\vec{\mu}}=J_{Y}^{\vec{\nu}}=0$ trivially satisfies Eq. (7).] However, it is not obvious whether such a Hamiltonian can be translation invariant, even if the EAP state is translation invariant and coefficients are appropriately chosen. Therefore, in the following, we impose translation invariance on the Hamiltonian and show that only several EAP states are allowed as solutions of Eq. (7).

Translation-invariant nonintegrable Hamiltonians— Suppose that $\hat{H}$ defined in Eq. (6) is translation invariant and consists of interactions up to nearest neighbor sites. Then, $\hat{H}$ is characterized by nine nearest-neighbor-interaction coefficients $\{J^{\mu\nu}\}_{\mu,\nu=x,y,z}$ and three magnetic field coefficients $\{h^{\mu}\}_{\mu=x,y,z}$. Since Eq. (7) reduces to

	$\displaystyle J^{\mu_{1}\mu_{2}}(1+\omega_{j}^{\mu_{1}}\omega_{j+1}^{\mu_{2}})=0,\quad h^{\mu_{1}}(1+\omega_{j}^{\mu_{1}})=0,$
	$\displaystyle\quad\text{for all $\mu_{1},\mu_{2}\in\{x,y,z\}$ and $j\in\Lambda$},$		(8)

we can solve them and find all possible choices of $J^{\mu_{1}\mu_{2}},h^{\mu_{1}}\neq 0$ and $(\omega_{j}^{x},\omega_{j}^{z},\omega_{j}^{y}=-\omega_{j}^{x}\omega_{j}^{z})_{j\in\Lambda}$, as will be described in the following Theorem 2. Interestingly, there are some nontrivial solutions whose $(\omega_{j}^{x},\omega_{j}^{z},\omega_{j}^{y})_{j\in\Lambda}$ are not invariant by single-site shift but invariant by $n$-site shift for $n>1$. To express such solutions efficiently, we introduce the following notation for EAP states that are invariant by $n$-site shift ⁵⁵5Note that, for some of states (9), $n$-site shift can cause a phase change, such as $\hat{\mathcal{T}}^{3}\ket{3;01,10,11}=-\ket{3;01,10,11}$, where $\hat{\mathcal{T}}$ is the one-site translation operator.: when $N/2$ is a multiple of $n$,

$\displaystyle\ket{n;p_{1}^{*}q_{1}^{*},\cdots,p_{n}^{*}q_{n}^{*}}$

(9)

denotes the EAP state characterized by $(p_{j},q_{j})_{j\in\Lambda}$ satisfying $p_{mn+j}=p_{j}^{*}$ and $q_{mn+j}=q_{j}^{*}$ for $m=0,1,\cdots,N/2n-1$ and $j=1,2,\cdots,n$. For example, when $N/2$ is even, $\ket{2;10,11}=\bigotimes_{j=1}^{N/4}\ket{\Phi_{10}}_{2j-1,2j-1+N/2}\bigotimes_{j=1}^{N/4}\ket{\Phi_{11}}_{2j,2j+N/2}$. Using this notation, we can list all nontrivial solutions as follows [35]:

The fact that there exist not so many solutions implies that our condition Eq. (7) is strong enough for translation-invariant Hamiltonians.

Considering the importance of these models, we call the models described by $\hat{H}_{1}$, $\hat{H}_{2}$ and $\hat{H}_{3}$ Models 1, 2 and 3, respectively. Note that, since Model 2 is included in Model 1, the EAP state $\ket{1;00}$ is also an eigenstate of $\hat{H}_{2}$.

Furthermore, we can obtain the following theorem regarding the nonintegrability of Models 2 and 3 [35]:

Because it is known that integrable systems have many [$\mathcal{O}(N)$ number of] nontrivial local conserved quantities [40, 41, 42], the above theorem implies that these models are indeed nonintegrable. In addition, since Model 2 (which is $J^{yx}=J^{zy}=h^{y}=0$ case of Model 1) is nonintegrable for any nonzero $J^{xy},J^{yz}$, Model 1 will be nonintegrable, at least for nonaccidental values of the parameters [35].

Combining Theorems 2 and 3 with the fact that EAP states are thermal as explained above, we finally obtain an analytic expression of a thermal energy eigenstate of a nonintegrable system. This highly contrasts with the recent progress made by H. Tasaki [43] in attempts to prove the ETH. He proved, only with respect to special observables, that all energy eigenstates of a certain noninteracting (integrable) system are indistinguishable from the thermal state. However, problems in nonintegrable systems remained elusive in his work. In addition, the restriction on observables is essential in his result because the integrable system has many local observables by which energy eigenstates can be distinguished from the thermal state. By contrast, our approach is to construct, in a nonintegrable system, an energy eigenstate that is indistinguishable from a thermal state with respect to any local observables.

Finite-temperature state— So far, we have only considered states at $\beta=0$. Can we describe thermal states at $\beta\neq 0$ using the EAP state? Here, we provide a method for constructing finite-temperature states using the EAP state. It should be noted that Hamiltonians considered here are not restricted to those having the EAP state as an eigenstate, which we have considered thus far.

Consider a system with translationally invariant finite-range interactions. Suppose that $\hat{H}$ is a real matrix with respect to the products of eigenstates of $\hat{\sigma}_{j}^{z}$, $\{\ket{00\cdots 0},\ket{10\cdots 0},\ket{01\cdots 0},\cdots\}$. This class includes many well studied models in statistical mechanics, such as the Ising model and the Heisenberg model.

Here, we utilize the EAP state characterized by $p_{j}=q_{j}=0$, i.e., $\ket{1;00}$ in the notation introduced in Eq. (9). Then, as will be discussed below, the imaginary-time evolution of the EAP state

$\displaystyle\ket{\beta}\propto e^{-\frac{1}{4}\beta\hat{H}}\ket{1;00}$

(13)

is locally indistinguishable from the Gibbs state $\hat{\rho}^{\mathrm{can}}$ at the inverse temperature $\beta$ with an exponentially small error, although it is not an eigenstate of $\hat{H}$ ⁷⁷7Some readers may wonder, if the EAP state $\ket{1;00}$ is an energy eigenstate, then it is invariant under the imaginary-time evolution, and hence $\ket{\beta}$ cannot describe a finite-temperature state. However, by using Theorem 1, it immediately follows that $\ket{1;00}$ cannot be an energy eigenstate provided that $\hat{H}$ is translation invariant (for single-site translations) and is a real matrix..

Let $\hat{O}$ be a local observable of interest. Since both $\ket{\beta}$ and $\hat{\rho}^{\mathrm{can}}$ are translation invariant, without loss of generality, we can assume that $\hat{O}$ has support around $j=N/4$. We then introduce an approximation $\ket{\tilde{\beta}}$ for $\ket{\beta}$ as

$\displaystyle\ket{\tilde{\beta}}\propto e^{-\frac{1}{4}\beta[\hat{H}-\hat{H}_{\mathrm{int}}]}\ket{1;00}.$

(14)

Here $\hat{H}_{\mathrm{int}}$ represents the interaction between the left half $\{1,2,\cdots,N/2\}$ and the right half $\{N/2+1,N/2+2,\cdots,N\}$ of the whole system, defined as a sum of interaction terms when $\hat{H}$ is decomposed into a linear combination of the Pauli strings as in Eq. (6). The imaginary-time evolution is expected to be stable against local perturbations. Indeed, the Gibbs state, which is proportional to the imaginary-time evolution operator, is not affected by perturbations on infinitely distant points because systems under consideration are one dimensional and do not exhibit the first-order phase transition at finite temperature [35]. Therefore, since $\hat{H}_{\mathrm{int}}$ is a sum of local observables defined around $j=1$ or $N/2$ at a distance of $\mathcal{O}(N)$ from the support of $\hat{O}$, it is expected that $\ket{\tilde{\beta}}$ approximate $\ket{\beta}$ around $j=N/4$ in the limit of $N\to\infty$, i.e.,

$\displaystyle\lim_{N\to\infty}\braket{\tilde{\beta}}{\hat{O}}{\tilde{\beta}}=\lim_{N\to\infty}\braket{{\beta}}{\hat{O}}{{\beta}}.$

(15)

Under this assumption, we have the following [35]:

In other words, $\ket{\beta}$ is a thermal pure state at finite temperature although it is not an energy eigenstate. We should emphasize that this state is a pure state of a closed system, in contrast to a purified Gibbs state [45, 46], which is a pure state of an extended system involving an ancillary system. In the imaginary-time evolved EAP state $\ket{\beta}$, for any local subsystem, its complement plays the role of an ancilla while itself in the correct thermal equilibrium. This property is by no means trivial and relies on the conditions on the Hamiltonian in Theorem 4.

We finally confirm the validity of Eq. (16) by numerically testing our prediction on the quantum Ising chain, defined by the Hamiltonian

$\displaystyle\hat{H}=-\sum_{j=1}^{N}\hat{\sigma}_{j}^{z}\hat{\sigma}_{j+1}^{z}-h^{x}\sum_{j=1}^{N}\hat{\sigma}_{j}^{x}-h^{z}\sum_{j=1}^{N}\hat{\sigma}_{j}^{z}.$

(17)

This model is known to be integrable when $h^{z}=0$ and nonintegrable when $h^{x}\neq 0,h^{z}\neq 0$ [47]. To ensure that the results do not depend on integrability, we investigate both the integrable case ($h^{x}=1,h^{z}=0$) and nonintegrable case ($h^{x}=1,h^{z}=1$). We plot in Fig. 2 the $N$ dependence of the difference in the expectation value of the transverse magnetization $\hat{m}^{x}=\frac{1}{N}\sum_{j=1}^{N}\hat{\sigma}_{j}^{x}$ between the Gibbs state and the imaginary-time evolved EAP state $\ket{\beta}$. For the integrable case, the expectation value in the Gibbs state is evaluated at the thermodynamic limit using the exact solution. For the nonintegrable case, it is evaluated for the same system size as that of $\ket{\beta}$ using the exact diagonalization. In both cases, $\ket{\beta}$ is obtained by applying the imaginary-time evolution operator, expanded using a Taylor series, to the EAP state. We can confirm that the expectation value in $\ket{\beta}$ converges to the correct value. Furthermore, the convergence is exponentially fast. Thus, by utilizing the EAP state evolved in imaginary time, one can calculate the thermal equilibrium values of local observables.

Here, we discuss the relation with the cTPQ state [48], which is also a type of thermal pure states. The cTPQ state is the imaginary-time evolution of a Haar random state. The Haar random state is (almost) maximally entangled and is locally indistinguishable from the maximally mixed state, as is the EAP state. However, while constructing the TPQ state requires an imaginary-time evolution for $\beta/2$, constructing $\ket{\beta}$ requires only half of that, $\beta/4$. In addition, unlike the cTPQ state, $\ket{\beta}$ does not entail statistical uncertainties. This means that generating a single $\ket{\beta}$ suffices, without incurring any sampling costs.

Discussion— We have studied a class of volume-law states, which we call entangled antipodal pair (EAP) states, and thoroughly characterized these states by providing the necessary and sufficient conditions that Hamiltonians must satisfy in order to have an EAP state as an eigenstate. Moreover, we have rigorously shown that some of such Hamiltonians are nonintegrable. Our EAP states are indistinguishable from the Gibbs state at infinite temperature. In other words, we have written down analytic expressions for thermal energy eigenstates of nonintegrable many-body systems. We have also devised a method for constructing finite-temperature thermal states using an EAP state.

Some readers may be concerned that the energy eigenspace containing the EAP eigenstate is excessively large because, with an exponentially large number of states, it can be not so difficult to obtain a highly entangled state as their linear combination, even when each is hardly entangled [49]. However, additional numerical calculations [35] show that in Model 1, the degeneracy at zero energy is only $2$ in the momentum sector ⁸⁸8Furthermore, by extending the locality of interactions from two to three, we have also obtained an example of a pair of the EAP eigenstate and the nonintegrable Hamiltonian that does not have degeneracy in the corresponding momentum sector [35].. In addition, we can also confirm that all states in the degenerate eigenspace have a volume-law entanglement with the maximal coefficient $\log 2$ [35]. Hence, the nontriviality of our findings is not diminished by the degeneracy.

Our result suggests that theoretical analysis may be feasible even for thermal eigenstates of nonintegrable systems and may pave the way for giving a provable example of the ETH. Since EAP states themselves can give only a few of eigenstates at infinite temperature, constructing general eigenstates remains challenging. However, as we have shown, one can obtain thermal states at arbitrary temperature by applying an imaginary-time evolution, a low-complexity operation that can be well approximated via a matrix product operator with small bond dimension [51], to the EAP state. This implies that the expressivity of states derived from EAP states is quite high. Thus, EAP states would be one of the most promising starting points for constructing general eigenstates, including those at finite temperature. Furthermore, it will provide theoretical methodologies not only for thermalization, but also for various areas of quantum statistical mechanics that use thermal pure states, such as the formulation of statistical mechanics and finite temperature simulations.

Acknowledgments— We are grateful to N. Shiraishi, A. Shimizu, H. Katsura, K. Fujii, K. Mizuta, H. Kono, M. Yamaguchi, and Z. Wei for useful discussions. We would like to especially thank H. K. Park and A. N. Ivanov for careful reading and valuable comments on a draft of this Letter. Y.Y. is supported by the Special Postdoctoral Researchers Program at RIKEN. Y.C. is supported by Japan Society for the Promotion of Science KAKENHI Grant No. JP21J14313, the Special Postdoctoral Researchers Program at RIKEN, and JST ERATO Grant No. JPMJER2302, Japan.

Supplemental Material for
“Exact Thermal Eigenstates of Nonintegrable Spin Chains at Infinite Temperature”