Introduction. The emergence of thermodynamics from unitary dynamics in isolated quantum many-body systems [1], [2] provides a fascinating puzzle in contemporary physics. At the heart of this enigma lies the Eigenstate Thermalization Hypothesis (ETH) [3]–[7], which posits that all the eigenstates of a quantum many-body system exhibit thermal properties for macroscopic observables. Thus, a comprehensive understanding of thermalization necessitates venturing beyond the traditional focus on the ground- and low-lying-excited- state properties of quantum many-body systems, and developing a thorough understanding of the structure of states in the bulk of their energy spectrum. Although the special class of integrable systems [8]–[11] offers a pathway to obtain solutions for excited states owing to the existence of extensive conserved quantities in them, they paradoxically serve as prime examples wherein the ETH is violated.

Obtaining analytical expressions for excited eigenstates remains a formidable challenge for generic non-integrable systems. This daunting task has spurred a plethora of research to identify tractable non-integrable models where a description beyond the low-energy states can be given [12]–[25]. This quest was further fueled by the fact that many of these exact eigenstates violate the ETH leading to anomalous non-thermal dynamics similar to those seen in experiments realizing the so-called PXP model [26], [27]. Hence, these eigenstates were dubbed as quantum many-body scars (QMBS) [28]–[32]. Examples of exact QMBS include towers of quasiparticle states in the Affleck-Kennedy-Lieb-Tasaki spin chain [33]–[36], \(\pi{-}\)bimagnon states in the spin-\(1\) \(XY\) model [20], [37], \(\eta{-}\)pairing states in the Hubbard model [38], [39], isolated area-law states in a variety of models [16] including the PXP [19] and others [40]–[50]. Multiple constructions that produce models hosting exact QMBS eigenstates have also been proposed [16], [38], [39], [51]–[57].

However, most of these QMBS states exhibit sub-volume entanglement entropy in contrast to the volume-law predicted by ETH. For some time, the main exception was a class of QMBS called rainbow scars [58], which do exhibit volume entanglement for some bipartitions, but the expectation value of appropriate local observables in it are still non-thermal. However, recent results have showcased exact volume-law entangled states hosted in various PXP-type models [59] and several spin-\(1/2\) chains with nearest-neighbor interactions [60], [61], which are “thermal" for local observables. These examples are few and far between and have been demonstrated in specific spin models, and the precise conditions for the existence of such solvable states are still unclear.

In this work, we provide a generic approach and explicit sufficient conditions for identifying a class of exact excited eigenstates in a large class of spin models. We show that any translation-invariant Hamiltonian whose every term formally anti-commutes with an on-site invertible operator times complex conjugation possesses these eigenstates. We start by constructing an exact zero energy eigenstate for spin-\(S\) chains with periodic boundary conditions (PBC) and then generalize it to higher dimensions. We further establish that these states show a volume-law scaling of entanglement entropy as predicted by ETH for all contiguous bipartitions. Moreover, these states exhibit thermal properties for any local observable, although they are atypical and athermal for non-local (but few-body) observables. Our construction recovers previously proposed volume-entangled states in various spin-\(1/2\) models [59]–[61]. Additionally, we construct analytical expressions of many new exact zero-energy states for a diverse set of spin-\(S\) Hamiltonians in arbitrary dimensions for both non-integrable and integrable models.

General construction in spin chains. We are interested in the class of spin-\(S\) chains of size \(L\) with a translation invariant interaction (with PBC) of the form \[H(L){=}\sum_{\alpha,i{=}1}^{L} \hat{H}^{\alpha}_{i{+}j_{1},\cdots,i{+}j_{n}}{=}\sum_{\alpha,i{=}1}^{L}\prod_{k{=}1}^{n} (S_{i{+}j_{k}}^{\mu})^{q}, \label{eq:32generic95Hamiltonian}\tag{1}\] where \(S_{j}^{\mu}(\mu{=}x,y,z)\) is the spin-\(S\) operator at site \(j\) and \(q\) is a non-negative integer. The term \(\hat{H}^{\alpha}_{i+j_{1},{\cdots},i{+}j_{n}}\) represents interaction between any \(n{-}\)spins at sites \(\{i{+}j_{1},{\cdots},i{+}j_{n}\}\) with \(0{\leq}j_{k}{<}L\). The Hamiltonian is defined in the Hilbert space \(\mathcal{H}_{L}\) of dimension \(\mathcal{D}_{L}{=}(2S{+}1)^{L}\) on the global computational basis \(|\vec{S}\rangle{=}{\bigotimes}_{i{=}1}^{L}|s_{i}\rangle\) \((\mathcal{H}_{L}{\equiv}\text{span}\{|\vec{S}\rangle\})\), where \(|s_{i}\rangle\) denotes the \((2S{+}1)\) dimensional local Hilbert space on site \(i\). In this setting, we establish the following theorem.

The anti-commutation relation in Eq. ?? implies \[\mathcal{C}(\hat{H}^{\alpha}_{i{+}j_{1}, \cdots, i{+}j_{n}})^{\ast}{=}-\hat{H}^{\alpha}_{i{+}j_{1}, \cdots, i{+}j_{n}}\mathcal{C}, \label{eq:32condition}\tag{2}\] where the complex conjugation is defined with respect to the computational basis \(|\vec{S}\rangle\). The proof of the theorem proceeds by using the mapping \(\overline{i}{\equiv}i{+}L\), and expressing the Hamiltonian for the \(2L\)-site system with PBC as \(H(2L){=}\sum_{\alpha,i{=}1}^{L} (\hat{H}^{\alpha}_{i{+}j_{1},{\cdots},i{+}j_{n}}{+}\hat{H}^{\alpha}_{\overline{i{+}j_{1}},{\cdots},\overline{i{+}j_{n}}})\), and then showing that for the state \(|\Lambda\rangle_{\mathcal{C}}\), the action of the term \(\hat{H}^{\alpha}_{i{+}j_{1}, {\cdots}, i{+}j_{n}}\) is canceled by that of \(\hat{H}^{\alpha}_{\overline{i{+}j_{1}},{\cdots},\overline{i{+}j_{n}}}\) for all \(i\) [see 1 of supplemental material (SM) [62] for details], i.e., \[\hat{H}^{\alpha}_{i{+}j_{1},\cdots,i{+}j_{n}}|\Lambda\rangle_{\mathcal{C}}{=}{-}\hat{H}^{\alpha}_{\overline{i{+}j_{1}},\cdots,\overline{i{+}j_{n}}}|\Lambda\rangle_{\mathcal{C}}. \label{eq:32cancellation}\tag{3}\] By noting that Eq. 2 always holds for \(\mathcal{C} {=} \mathbb{I}\), we obtain the following corollary to Theorem 1.

Next, we show that the state \(|\Lambda\rangle_{\mathcal{C}}\) is volume-law entangled for an arbitrary contiguous bipartition. By definition, the state \(\ket{\Lambda}_{\mC}\) is a product of “dimers" between sites \(i\) and \(\overline{i}\). The entanglement between the spins on the sites \(i\) and \(\overline{i}\) is therefore strictly greater than \(0\). It can be as high as \(\text{log}(2S{+}1)\) depending upon the nature of \(\mathcal{C}_{i}\) (the reduced density matrix of each dimer is just \(\mC_i \mC_i^\dagger\), whose eigenvalues are just the singular values of \(\mC_i\)), and is maximally entangled if and only if \(\mathcal{C}_{i}\) is unitary.

An arbitrary contiguous subsystem of size \(N{\leq}L\) [shaded grey in Fig. 1\((a)\)] cuts exactly \(N\) bonds between the correlated spins leading to the entanglement entropy of the subsystem \(\mathcal{S}_{N}{\propto} N\) thereby obeying volume-law scaling. Moreover, when \(\mC_i\) is unitary, the reduced density matrix over any contiguous subsystem is just proportional to the identity matrix, which is just the infinite-temperature Gibbs density matrix, hence the state is thermal for strictly local observables [59], [61]. However, for tailored subsystems chosen to consist solely of sites \(\{i,\overline{i}\}\), the entanglement entropy is strictly zero, and non-local observables with support over these sub-systems exhibit highly athermal properties.

We now illustrate the power of our general construction by illustrating exact excited zero-energy states for several integrable and non-integrable spin chains. We begin by considering spin-\(1/2\) systems for which any Hamiltonian can be recast to the form given in Eq. 1 . The spin operators at site \(i\) can be represented by Pauli matrices \(\sigma_{i}^{\mu} ({\mu}{=}x,y,z)\) and the local Hilbert space is defined by the \(\sigma^{z}_{i}\)-eigenvectors \(|0\rangle_{i}\) and \(|1\rangle_{i}\). In what follows, the state \(|\Lambda\rangle_{\mathcal{C}}\) can be expressed in terms of Bell states between sites \(i,j\) defined as \[|\Psi^{\pm}\rangle_{i, j}{=}\frac{(|00\rangle {\pm} |11\rangle)_{i,j}}{\sqrt{2}};~ |\Phi^{\pm}\rangle_{i,j} {=} \frac{(|01\rangle {\pm} |10\rangle)_{i,j}}{\sqrt{2}}. \label{eq:32bell32states}\tag{4}\] For instance, consider the following non-integrable Hamiltonian discussed in Ref. [61] \[\begin{align} H_{1}(L){=}\sum_{i{=}1}^{L}\big(J_{1}\sigma_{i}^{x}\sigma_{i{+}1}^{y}+J_{2}\sigma_{i}^{y}\sigma_{i+1}^{z}\big), \end{align} \label{eq:32spin-147232example2}\tag{5}\] with arbitrary values of the coupling constants \(J_{1}\) and \(J_{2}\). Here \(n{=}2,j_{k}{=}k{-}1\) and each \(\hat{H}^{\alpha}_{i, i{+}1}\) (i.e., \(\sigma_{i}^{x}\sigma_{i{+}1}^{y}\) and \(\sigma_{i}^{y}\sigma_{i+1}^{z}\)) is purely imaginary in the computational basis. From Corollary 1, the state \(|\Lambda\rangle_{\mathbb{I}}\) given in Eq. ?? is a zero-energy eigenstate of \(H_{1}(2L)\) and takes a simple form \(|\Lambda(S{=}1/2)\rangle_{\mathbb{I}}{=}\bigotimes_{i{=}1}^{L}|\Psi^{+}\rangle_{i,\overline{i}}\). This state is depicted in Fig. 1\((a)\) where all the dotted lines represent the Bell state \(|\Psi^{+}\rangle\). Since the Hamiltonian \(H_{1}(2L)\) has global spectral-reflection symmetry generated by the operator \(\mathcal{M}_{1}{=}{\prod}_{i{=}1}^{2L}\sigma_{i}^{y}\), the zero-energy state mentioned above is a mid-spectrum excited state. Note that \(|\Lambda(S{=}1/2)\rangle_{\mathbb{I}}\) is a zero-energy eigenstate of any spin-\(1/2\) Hamiltonian with even number of sites of form Eq. 1 that has an odd number of \(\sigma_{y}\)’s in each \(\hat{H}^{\alpha}_{i{+}j_{1}, {\cdots}, i{+}j_{n}}\).

Furthermore, when the half-chain length \(L\) is a multiple of three, the operator \(\mathcal{C}_{1}{=}{\prod}_{p{=}1}^{L/3}\sigma_{3p{-}2}^{x} \sigma_{3p{-}1}^{y}\sigma_{3p}^{z}\) satisfies the conditions given in Eq. 2 . This allows us to identify another zero-energy eigenstate (and its translations) of \(H_{1}(2L)\) of the form \(|\Lambda\rangle_{\mathcal{C}_{1}}{=}\bigotimes_{i{=}1}^{L{-}2}|\Phi^{+}\rangle_{i,\overline{i}}|\Phi^{-}\rangle_{i{+}1,\overline{i{+}1}}|\Psi^{-}\rangle_{i{+}2,\overline{i{+}2}}\). The state \(|\Lambda\rangle_{\mathcal{C}_{1}}\) is precisely the exact zero-energy eigenstate dubbed as Entangled Antipodal Pair (EAP) state presented in Ref. [61]. While Ref. [61] derives the Hamiltonian \(H_{1}(2L)\) via an exhaustive search of all possible nearest-neighbor spin-\(1/2\) interactions for which the predefined EAP state has zero-energy, we show that \(H_1(2L)\) satisfies a very general sufficient condition for the existence of such an EAP state. Similarly, other zero energy eigenstates found in different spin-\(1/2\) Hamiltonians of Ref [60], [61] including the PXP chain [59] can be understood using our construction (see 3 of SM [62] for details with other examples).

Next, we investigate a spin-\(S\) \(XY\) chain, described by the Hamiltonian \[H_{XY}(L){=} \sum_{i=1}^{L} (J_{1}S_{i}^{x}S_{i+1}^{x}+J_{2}S_{i}^{y}S_{i+1}^{y}+hS_{i}^{z}), \label{eq:32spin-S32XY}\tag{6}\] where \(J_{1},J_{2}\) and \(h\) are arbitrary and \(S_{i}^{\mu}\)(\(\mu{=}x,y,z\)) are spin\({-}S\) operators at site \(i\). The spin components define site parity operators \(\mathcal{P}_{i}^{\mu}{=}e^{i\pi S_{i}^{\mu}}\). One can show that \(\mathcal{P}_{i}^{\mu}\) commutes with \(S_{i}^{\mu}\) but anti-commutes with \(S_{i}^{\mu'} (\mu{\neq}\mu')\). Hence when \(L\) is even, in the orthonormal basis \(|m\rangle_{i}\) that are eigenstates of \(S_{i}^{z}\) with eigenvalues \(m{=}{-}S, {-}S{+}1, {\cdots}, S\), the operator defined by \(\mathcal{C}_{XY}{=}\prod_{p{=}1}^{L/2}\mathcal{P}_{2p{-}1}^{x}\mathcal{P}_{2p}^{y}\), satisfies Eq. 2 . As a result of Theorem 1, the Hamiltonian \(H_{XY}(2L)\) hosts an exact zero-energy eigenstate \(|\Lambda\rangle_{\mathcal{C}_{XY}}\) (and its translations). Using the action of the on-site parity operators on the basis states \(\mathcal{P}_{i}^{x}|m\rangle_{i}{=}({-}1)^{S}|{-}m\rangle_{i}\) and \(\mathcal{P}_{i}^{y}|m\rangle_{i}{=}({-}1)^{S{+}m}|{-}m\rangle_{i}\), we can express the state as \[\begin{align} &|\Lambda\rangle_{\mathcal{C}_{XY}}=\bigotimes_{i{=}1}^{L{-}1}|\kappa_{S}^{-}\rangle_{i,\overline{i}}|\kappa_{S}^{+}\rangle_{i{+}1,\overline{i{+}1}},\text{where}\\ &|\kappa_{S}^{\pm}\rangle_{i,j}{=}\frac{1}{\sqrt{2S{+}1}}\sum_{p{=}0}^{2S}({\pm}1)^{p}(S^{-})^{p}|S\rangle_{i}{\otimes}(S^{+})^{p}|{-}S\rangle_{j}. \end{align} \label{eq:32Lambda95S95XY32state}\tag{7}\] For \(S{>}1/2\), \(H_{XY}\) is non-integrable [20] for generic values of the couplings, while for \(S{=}1/2\), \(H_{XY}\) is integrable (reduces to the transverse field Ising model for \(J_{2}{=}0\)) [63]. For \(S{=}1/2\), this state reads \(|\Lambda\rangle_{\mathcal{C}_{XY}}{=}\bigotimes_{i{=}1}^{L{-}1}|\Phi^{-}\rangle_{i,\overline{i}}|\Phi^{+}\rangle_{i{+}1,\overline{i{+}1}}\) and shows that volume-law eigenstates exist even in integrable models.

The sufficient conditions we derive allow us to construct a large family of Hamiltonians that will inherently host the specific volume-law entangled eigenstate of the particular form of Eq. ?? . For instance, the zero-energy state \(|\Lambda_{S}\rangle_{\mathcal{C}_{XY}}\) described in Eq. 7 is also a zero-energy eigenstate of the spin-\(S\) Kitaev chain [64]–[66] given by the Hamiltonian \(H_{K}(2L){=}\sum_{i{=}1}^{L}(J_{1}S_{2i{-}1}^xS_{2i}^x {+}J_{2} S_{2i}^y S_{2i{+}1}^y)\) for arbitrary values of \(J_{1}\) and \(J_{2}\). Furthermore, Hamiltonians involving longer-range (beyond nearest neighbors) interactions between spins, e.g. \(H_{\rm NNNN}(2L){=}\sum_{i{=}1}^{2L} (J_{1}S_{i}^{x}S_{i{+}3}^{x}{+}J_{2}S_{i}^{y}S_{i{+}3}^{y})\) or models with more than two-spin interactions, e.g. \(H_{3{-}4}(2L){=}\sum_{i{=}1}^{2L} (J_{1}S_{i}^{z}S_{i{+}1}^{z}S_{i{+}2}^{z}{+}J_{2}S_{i}^{x}S_{i{+}1}^{z}S_{i{+}2}^{y}S_{i{+}3}^{z})\) host the same zero-energy state \(|\Lambda_{S}\rangle_{\mathcal{C}_{XY}}\) in their spectrum as they satisfy Eq. 2 . Many such Hamiltonians exist physically or can be engineered experimentally.

Generalization to arbitrary dimensions. To investigate higher-dimensional spin systems, it is convenient to revisit the one-dimensional case and rearrange the \(2L\) sites labeled by indices \({1}, \overline{1}, {\cdots},{L}, \overline{L}\) into the configuration shown in Fig. 1\((b)\). Our goal is to determine local Hamiltonians that host the state \(|\Lambda\rangle_\mathcal{C}\) defined in Eq. ?? as a zero energy eigenstate provided the operator \(\mathcal{C}\) satisfies conditions analogous to those in Eq. 2 .

Similar to the one-dimensional case, for all the interaction terms \(\hat{H}_{j_{1},{\cdots}, j_{n}}^{\alpha}\) between any \(n{-}\)sites \(j_{1},{\cdots}, j_{n}\) (where \(j_k{\in}\{1,\overline{1},{\cdots}, L,\overline{L}\}\) and \(j_{k}{\neq} \overline{j}_{k'}\) for all \(k,k'\)), restricted to the form \(\hat{H}_{j_{1},{\cdots}, j_{n}}^{\alpha}{=}\prod_{k{=}1}^{n}(S_{j_{k}}^{\mu})^{q}\) and satisfying \(\{\hat{H}^{\alpha}_{j_{1},{\cdots},j_{n}},\mathcal{C}\mathcal{K}\}{=}0\), we find that \[(\hat{H}^{\alpha}_{j_{1},\cdots,j_{n}} + \hat{H}^{\alpha}_{\overline{j}_{1},\cdots,\overline{j}_{n}})|\Lambda\rangle_{\mathcal{C}}=0, \label{eq:32cancelation95in95arb95dimension}\tag{8}\] where we have considered the identification \(\overline{\overline{j}}_{k}{=}j_{k}\). Consequently, all Hamiltonians of the form \(H{=}\sum_{\alpha,\{j\}} J_{\alpha}(\hat{H}^{\alpha}_{j_{1},{\cdots},j_{n}} {+} \hat{H}^{\alpha}_{\overline{j}_{1},{\cdots},\overline{j}_{n}})\) with arbitrary coefficients \(J_{\alpha}\), host \(|\Lambda\rangle_{\mathcal{C}}\) as a zero energy eigenstate.

For instance, consider the Hamiltonian of the nearest-neighbor spin-\(S\) \(XY\) model with a transverse field defined on a \(2L{\times}2L\) square lattice with PBC in both directions, \[\begin{align} H_{XY}^{2D}{=}\sum_{\mu,i,j}J_{\mu}(S_{i,j}^{\mu}S_{i{+}1,j}^{\mu}{+}S_{i,j}^{\mu}S_{i,j{+}1}^{\mu}){+}h\sum_{i,j}S_{i,j}^{z}, \end{align} \label{eq:322D32XY32model32Hamiltonian}\tag{9}\] where \(S_{i,j}^{\mu}(\mu{=}x,y,z)\) are the spin operators at site \((i,j)\) with \(i,j{\in}\{1, {\cdots}, 2L\}\) with \(i\) and \(j\) labeling the horizontal and vertical directions, respectively, with \((1,1)\) at the bottom-left corner, and \(J_{\mu}\) and \(h\) are arbitrary couplings.

Now relabeling the \(4L^{2}\) sites with a single index \(r_{i,j}{\in}\{1, \overline{1}, {\cdots}, 2L^{2}, \overline{2L^{2}}\}\) as shown in Fig. 2\((a)\), where \[r_{i,j}{\equiv} \begin{cases} i{+}2L{\times}(j{-}1)&\text{for} j{\leq}L \\ \overline{1{+}[(i{-}1{+}L){\rm~mod~}2L]{+}2L{\times}(j{-}1{-}L)}&\text{for} j{>}L, \end{cases} \label{eq:322D32XY-relabelling}\tag{10}\] the Hamiltonian \(H_{XY}^{2D}\) reduces to the convenient form \[\begin{align} H&_{XY}^{2D}{=}\sum_{\mu,i,j}^{j{<}L}\Big[J_{\mu}(S_{r_{i,j}}^{\mu}S_{r_{i{+}1,j}}^{\mu}{+}S_{\overline{r_{i,j}}}^{\mu}S_{\overline{r_{i{+}1,j}}}^{\mu})\\ &{+}J_{\mu}(S_{r_{i,j}}^{\mu}S_{r_{i,j{+}1}}^{\mu}{+}S_{\overline{r_{i,j}}}^{\mu}S_{\overline{r_{i,j{+}1}}}^{\mu}){+}h(S_{r_{i,j}}^{z}{+}S_{\overline{r_{i,j}}}^{z})\Big]. \end{align} \label{eq:322D32XY32model32Hamiltonian95relabelled}\tag{11}\] In the computational basis \(|m\rangle_{r_{i,j}}\) that are eigenstates of \(S_{r_{i,j}}^{z}\), each \(\hat{H}_{\{r\}}^{\alpha}\) of the Hamiltonian, i.e., \(S_{r_{i,j}}^{x}S_{r_{i{+}1,j}}^{x},S_{r_{i,j}}^{y}S_{r_{i{+}1,j}}^{y}, S_{r_{i,j}}^{x}S_{r_{i,j{+}1}}^{x}\), \(S_{r_{i,j}}^{y}S_{r_{i,j{+}1}}^{y}\) and \(S_{r_{i,j}}^{z}\) are real and anticommute with the operator \(\mathcal{C}_{XY}^{2D}{=}\prod_{i,j}(\mathcal{P}_{r_{i,j}}^{x})^{\delta_{(i{+}j){\rm~mod~}2,0}}(\mathcal{P}_{r_{i,j}}^{y})^{\delta_{(i{+}j){\rm~mod~}2,1}}\) [thus satisfy Eq. 2 ]. Thus, using Theorem 1 and Eq. 8 , the Hamiltonian \(H_{XY}^{2D}\) hosts a zero energy eigenstate of the form \(|\Lambda\rangle_{C_{XY}}^{2D}\) which can be expressed as \[|\Lambda\rangle_{C_{XY}}^{2D}{=}\bigotimes_{i,j}^{j{<}L}(|\kappa_{S}^{-}\rangle_{r,\overline{r}})^{\delta_{(i{+}j){\rm~mod~}2,0}} (|\kappa_{S}^{+}\rangle_{r,\overline{r}})^{\delta_{(i{+}j){\rm~mod~}2,1}}, \label{eq:32zeroenergy95eigstate952DXY}\tag{12}\] where \(\ket{\kappa_S^\pm}\) are defined in Eq. 7 . The state \(|\Lambda\rangle_{C_{XY}}^{2D}\) is shown in Fig. 2\((a)\), where the spins at sites \(r_{i,j}\) and \(\overline{r_{i,j}}\) are in the state \(|\kappa_{S}^{-}\rangle\) when \(i{+}j\) is even (marked in red) and in the state \(|\kappa_{S}^{+}\rangle\) when \(i{+}j\) is odd (marked in blue). Similarly, with different choices of labeling of the sites [e.g., see Fig. 2\((b)\) and more in the 2 of SM [62]] we can find many more zero-energy eigenstates \(|\Lambda\rangle_{\mathcal{C}}\) for \(H_{XY}^{2D}\) with different pairing patterns. Using the same strategy, we can also find zero-energy eigenstates of spin-\(S\) \(XY\) model in different geometries as shown in Figs. 2\((c)\),\((d)\) (and more in the SM [62]). Many such zero-energy eigenstates can be found for other spin Hamiltonians in arbitrary dimensions using our construction. Unlike the one-dimensional case, in higher dimensions, there can be special contiguous bipartitions for which the entanglement is not a volume-law, although it is for most of them.

Discussion. In this work, we have introduced a general method for constructing a specific class of exact zero-energy excited eigenstates in spin-\(S\) models that satisfy certain conditions. This construction only uses the existence of an operator, which, along with complex conjugation, anti-commutes with each term of the Hamiltonian. The existence of such an operator is easy to guess and check, and this directly allows us to construct exact eigenstates without analyzing the complex microscopic details of the model. We demonstrated the broad applicability of our construction by finding new exact eigenstates for various natural examples of spin-\(S\) Hamiltonians. We also recovered some previously known results for spin-\(1/2\) models [59], [61] as special cases of our construction.

In one dimension, the states we identified exhibit volume-law scaling of entanglement entropy for general contiguous bipartitions. In cases when the operator \(\mC\) is unitary, they also exhibit infinite-temperature thermal properties for all strictly local observables. This phenomenology mostly also extends to higher dimensions, although a few contiguous bipartitions might not display a volume-law scaling of entanglement. However, in any dimension, these states are always athermal for non-local but few-body observables and hence can be considered examples of highly entangled QMBS. Indeed, by construction, they are simultaneous eigenstates of non-commuting local operators, consistent with the proposed definition of QMBS based on commutant algebras [56]. It would be interesting to explore whether other examples of QMBS that exhibit superficially similar structures, e.g., the exponentially many exact eigenstates in the 2D PXP model [22], fit into this construction.

Building on this foundation, it is natural to wonder if extensions of these ideas to models beyond spins such as fermionic and bosonic systems would work, or generalizations such as going beyond pairwise entanglement to entangled clusters of \(l\)-spins, with \(l{\geq}3\) can be engineered. Along with the theoretical interest in states close to completely satisfying ETH, the existence of such states can also aid in the experimental measurement of Out-of-Time-Ordered Correlators (OTOC), as discussed in [59].

Finally, on a different note, while we focused on constructing non-integrable models with these states, there are also integrable models, such as the one-dimensional transverse-field Ising model, which host such states. While these states are athermal and violate the ETH [3] in non-integrable models (at least with respect to some non-local but few-body observables), it is worth exploring the implications of their existence in integrable models to the generalized ETH [10], [11], [67], the natural analog of ETH for integrable models.

Acknowledgments. We thank Andrew Ivanov, Maurizio Fagotti, and Diptiman Sen for useful discussions. This work was undertaken on the Nandadevi supercomputer maintained and supported by the Institute of Mathematical Sciences’ High-Performance Computing Center, India. This research was supported by the Munich Center for Quantum Science and Technology (MCQST), and in part by the International Centre for Theoretical Sciences (ICTS) for participating in the program - Stability of Quantum Matter in and out of Equilibrium at Various Scales (code: ICTS/SQMVS2024/01). S. Moudgalya acknowledges the hospitality of the Institute of Mathematical Sciences (IMSc), Chennai, where this collaboration was initiated. S. Mohapatra is grateful to the School of Natural Sciences, Technical University of Munich (TUM) for its hospitality in the summer of 2024 where a part of this work was completed.

1 Proof of Theorem 1 of main text: \(|\Lambda\rangle_\mathcal{C}\) is a zero energy eigenstate of \(H(2L)\)↩︎

For convenience, let us restate the form of the Hamiltonian and Theorem 1 of the main text here. We are considering the class of spin-\(S\) chains of size \(L\) with a translation invariant interaction [with periodic boundary condition (PBC)] of the form \[H(L){=}\sum_{\alpha,i{=}1}^{L} \hat{H}^{\alpha}_{i{+}j_{1},\cdots,i{+}j_{n}}{=}\sum_{\alpha,i{=}1}^{L}\prod_{k{=}1}^{n} (S_{i{+}j_{k}}^{\mu})^{q}, \label{suppeq:32generic95Hamiltonian}\tag{13}\] where \(S_{j}^{\mu}(\mu{=}x,y,z)\) is the spin-\(S\) operator at site \(j\) and \(q\) is a non-negative integer. The term \(\hat{H}^{\alpha}_{i+j_{1},\cdots,i+j_{n}}\) represents interaction between any \(n{-}\)spins at sites \(\{i{+}j_{1},{\cdots},i{+}j_{n}\}\) with \(0{\leq}j_{k}{<}L\). The Hamiltonian is defined in the Hilbert space \(\mathcal{H}_{L}\) of dimension \(\mathcal{D}_{L}{=}(2S{+}1)^{L}\) on the global computational basis \(|\vec{S}\rangle{=}{\bigotimes}_{i{=}1}^{L}|s_{i}\rangle\) \((\mathcal{H}_{L}{\equiv}\text{span}\{|\vec{S}\rangle\})\), where \(|s_{i}\rangle\) denotes the \((2S{+}1)\) dimensional local Hilbert space on site \(i\). For this setting, we stated the following theorem in the main text:

2 Some additional zero energy eigenstates for nearest neighbor spin-\(S\) \(XY\) model↩︎

In this section, we provide additional examples of exact zero energy eigenstates of the spin-\(S\) \(XY\) model with a transverse field in geometries different from those we considered in the main text. The states in Fig. 3\((a{-}c)\) are different zero energy states on a \(2L{\times}2L\) square lattice with PBC, while states in Fig. 3\((d{-}f)\) depict distinct zero energy states on a \(2L{\times}2L{\times}M\) cuboid with PBC. The spins at sites \(r\) and \(\overline{r}\) share the state \(|\kappa_{S}^{-}\rangle\) (when marked in red) and the state \(|\kappa_{S}^{+}\rangle\) (when marked in blue), where \[|\kappa_{S}^{\pm}\rangle_{i,j}{=}\frac{1}{\sqrt{2S{+}1}}\sum_{p{=}0}^{2S}({\pm}1)^{p}(S^{-})^{p}|S\rangle_{i}{\otimes}(S^{+})^{p}|{-}S\rangle_{j}. \label{suppeq:32kappa}\tag{31}\] Note that by interchanging the dimers (i.e., changing the red color to blue color and vice-versa), we get a different zero energy state (which can be verified to be orthogonal to the previous ones in this case) in each case. The examples presented in Fig. 3 serve as representatives of classes of zero energy eigenstates in the spin-\(S\) \(XY\) model in higher dimensions. A systematic investigation of the various classes, in particular, enumerating the number of distinct classes and the corresponding number of states within each class, is an interesting direction worth exploring in the future.

3 Reproducing previously constructed eigenstates of various spin models↩︎

In this section, we apply our formalism to previously studied spin-\(1/2\) Hamiltonians in the literature to demonstrate that it can reproduce the same zero energy states as reported in those works. Note that for all of the following examples, we consider the Hilbert space defined by computational basis \(|\vec{b}\rangle{=}{\bigotimes}_{i{=}1}^{L}|b_{i}\rangle\) \((\mathcal{H}_{L}{\equiv}\text{span}\{|\vec{b}\rangle\})\), where \(\vec{b}{=}(b_{1}, b_{2}, {\cdots}, b_{L})\) is either a string of spins, i.e., \(b_j \in \{\uparrow, \downarrow\}\), or a “bit-string" of length \(L\), i.e., \(b_j \in \{0, 1\}\).

3.1 A model with three-spin interaction and transverse magnetic field [60]↩︎

First, we consider the following Hamiltonian with a three-spin interaction and a transverse magnetic field for a system of size \(L\) (with PBC) that was studied in Ref. [60] \[\begin{align} H_{1}(L){=}\sum_{i{=}1}^{L}\big(h\sigma_{i}^{z}+J\sigma_{i}^{x}\sigma_{i+1}^{x}\sigma_{i+2}^{x}\big), \end{align} \label{suppeq:32three95spin95ising}\tag{32}\] where \(h\) and \(J\) are arbitrary constants. Here each \(\hat{H}^{\alpha}_{i+j_{1},\cdots,i+j_{n}}\), i.e., \(\sigma_{i}^{x}\sigma_{i+1}^{x}\sigma_{i+2}^{x}\) and \(\sigma_{i}^{z}\) are real in the chosen basis and anti-commute with the operator \(\mathcal{C}_{1}{=}\prod_{i}^{L} \sigma_{i}^{y}\) [hence they satisfy Eq. 14 ]. Thus using Theorem 1, a zero energy eigenstate of Hamiltonian \(H_{1}(2L)\) is given by \[|\Lambda\rangle_{\mathcal{C}_{1}}{=}\bigotimes_{i{=}1}^{L}|\Phi^{-}\rangle_{i,\overline{i}}, \label{suppeq:32Lambda95state1}\tag{33}\] where \(|\Phi^{\pm}\rangle_{i,j}{=}(|01\rangle {\pm} |10\rangle)_{i,j}/\sqrt{2}\). Similarly, when the half-chain length \(L\) is a multiple of three, the operator \(\mathcal{C}_{1}'{=}\prod_{p{=}1}^{\frac{L}{3}} \sigma_{3p{-}2}^{x}\sigma_{3p{-}1}^{y}\sigma_{3p}^{x}\) anti-commutes with each \(\hat{H}_{i{+}j_{1}, {\cdots}, i{+}j_{n}}^{\alpha}\) [thus satisfying Eq. 14 ] resulting in another zero energy eigenstate of the form \[|\Lambda\rangle_{\mC_{1}'}{=}\bigotimes_{i{=}1}^{L{-}2}|\Phi^{+}\rangle_{i,\overline{i}}|\Phi^{-}\rangle_{i{+}1,\overline{i{+}1}}|\Phi^{+}\rangle_{i{+}2,\overline{i{+}2}}. \label{suppeq:32Lambda95state1p}\tag{34}\] Note that due to PBC the translations of the state \(|\Lambda\rangle_{\mC_{1}'}\) by one site, i.e., \(|\Lambda^{\mathcal{T}_1}\rangle_{\mC_{1}'}{=}\bigotimes_{i{=}1}^{L{-}2}|\Phi^{-}\rangle_{i,\overline{i}}|\Phi^{+}\rangle_{i{+}1,\overline{i{+}1}}|\Phi^{+}\rangle_{i{+}2,\overline{i{+}2}}\) and two sites i.e. \(|\Lambda^{\mathcal{T}_2}\rangle_{\mC_{1}'}{=}\bigotimes_{i{=}1}^{L{-}2}|\Phi^{+}\rangle_{i,\overline{i}}|\Phi^{+}\rangle_{i{+}1,\overline{i{+}1}}|\Phi^{-}\rangle_{i{+}2,\overline{i{+}2}}\) are also distinct zero energy eigenstates of \(H_{1}(2L)\). The states of Eqs. 33 and 34 are exactly the volume-law entangled states reported in Ref. [60].

3.2 PXP model [59]↩︎

Next, we reproduce the volume-law state constructed in the PXP model in Ref. [59] using our framework. The PXP model for a spin chain of size \(L\) with PBC is described by the Hamiltonian [29] \[H_{\rm PXP}(L)=\sum_{i{=}1}^{L} \hat{P}_{i{-}1}\sigma_{i}^{x}\hat{P}_{i{+}1}=\sum_{i{=}1}^{L} \Big[\sigma_{i}^{x}{+}\sigma_{i}^{x}\sigma_{i{+}1}^{z}{+}\sigma_{i{-}1}^{z}\sigma_{i}^{x}{+}\sigma_{i{-}1}^{z}\sigma_{i}^{x}\sigma_{i{+}1}^{z}\Big]. \label{suppeq:32PXP95Hamiltonian}\tag{35}\] Here, we used \(\hat{P}_{i}{=}(\mathbb{I}{+}\sigma_{i}^{z})/2\) to write the Hamiltonian in the form given in Eq. 13 . Now in our chosen computational basis the global particle-hole symmetry operator \(\mathcal{C}_{\rm PXP}{=}\prod_{i{=}1}^{L}\sigma_{i}^{z}\) satisfies the conditions stated in Eq. 14 . Thus, using Theorem 1, a zero energy eigenstate of \(H_{\rm PXP}(2L)\) is \[|\Lambda\rangle_{\rm PXP}{=}\bigotimes_{i{=}1}^{L}|\Psi^{-}\rangle_{i,\overline{i}}{=} \sum_{|\vec{b}\rangle\in\mathcal{H}_{L}} ({-}1)^{|b|}|\vec{b}\rangle_{1,\cdots,L} \otimes |\vec{b}\rangle_{\overline{1},\cdots,\overline{L}}, \label{suppeq:32Lambda95PXP}\tag{36}\] where we have used \(\mC_{\rm PXP}|\vec{b}\rangle{=}({-}1)^{|b|}|\vec{b}\rangle\) with \(|b|\) denoting the parity of the bit string \(\vec{b}\) and \(|\Psi^{-}\rangle_{i,j}{=}(|00\rangle {-} |11\rangle)_{i,j}/\sqrt{2}\)

While \(\ket{\Lambda}_{\rm PXP}\) is an eigenstate of the PXP model in the full \(2^L\)-dimensional Hilbert space, the Hilbert space of the PXP model is usually truncated to the so-called nearest-neighbor Rydberg blockade Hilbert space [29]. This truncated blockade subspace \(\mathcal{F}_{L}\) for chains of \(L\) spins with PBC is spanned by basis vectors \(|\vec{f}\rangle\) whose bit-strings \(\vec{f}\) do not contain the sub-bit-string “11". The projector on this subspace defined by \(\mathcal{P}_{\text{Ryd}}{=}\prod_{i}(\mathbb{I}{-}\hat{Q}_{i}\hat{Q}_{i{+}1})\) (i.e., \(\mathcal{P}_{\text{Ryd}}\mH_{L}{=}\mathcal{F}_{L})\), where \(\hat{Q}_{i}{=}\mathbb{I}{-}\hat{P}_{i}\), commutes with the Hamiltonian \(H_{\rm PXP}\), which enables the truncation of this Hilbert space for the PXP Hamiltonian. Thus, the projection of the state \(|\Lambda\rangle_{\rm PXP}\) into the Rydberg blockade sector should also be an eigenstate of the PXP model truncated into this Hilbert space, and this state reads \[|\Lambda\rangle_{\rm Ryd}{=}\mathcal{P}_{\text{Ryd}}|\Lambda\rangle_{\rm PXP}{=}\sum_{|\vec{f}\rangle\in \mathcal{F}_{L}}({-}1)^{|f|}|\vec{f}\rangle_{1,\cdots, L}\otimes |\vec{f}\rangle_{L{+}1,\cdots, 2L}. \label{suppeq:32PXP95eigenstate95Rydberg95subspace}\tag{37}\] This is precisely the volume-law eigenstate reported in Ref. [59] for the PXP chain. We note that this construction also works for the PXP model in arbitrary dimensions and in a wide variety of geometries, which then reproduce the results of Ref. [59].

3.3 Entangled antipodal paired (EAP) state [61]↩︎

Finally, we apply our formalism to a next-nearest-neighbor interacting Hamiltonian for a system of size \(L\) with PBC considered in Ref. [61] given by \[\begin{align} H_{\rm NNN}(L){=}\sum_{i{=}1}^{L} \Big( J^{xzx}\sigma_{i}^{x}\sigma_{i{+}1}^{z}\sigma_{i{+}2}^{x}{+}J^{yzy}\sigma_{i}^{y}\sigma_{i{+}1}^{z}\sigma_{i{+}2}^{y}{+}J^{xzy}\sigma_{i}^{x}\sigma_{i{+}1}^{z}\sigma_{i{+}2}^{y}{+}J^{yzx}\sigma_{i}^{y}\sigma_{i{+}1}^{z}\sigma_{i{+}2}^{x}{+}\\ J^{zzz}\sigma_{i}^{z}\sigma_{i{+}1}^{z}\sigma_{i{+}2}^{z}{+} J^{xx}\sigma_{i}^{x}\sigma_{i{+}1}^{x}{+} J^{yy}\sigma_{i}^{y}\sigma_{i{+}1}^{y}{+} J^{xy}\sigma_{i}^{x}\sigma_{i{+}1}^{y}{+} J^{yx}\sigma_{i}^{y}\sigma_{i{+}1}^{x}{+} J^{z}\sigma_{i}^{z}\Big), \end{align}\] where all \(J\)’s are arbitrary. In our chosen computational basis, the Hamiltonian consists of terms \(\hat{H}^{\alpha}_{i+j_{1},\cdots,i+j_{n}}\) that are either purely real (i.e., \(\sigma_{i}^{x}\sigma_{i{+}1}^{z}\sigma_{i{+}2}^{x}\), \(\sigma_{i}^{y}\sigma_{i{+}1}^{z}\sigma_{i{+}2}^{y}\), \(\sigma_{i}^{z}\sigma_{i{+}1}^{z}\sigma_{i{+}2}^{z}\), \(\sigma_{i}^{x}\sigma_{i{+}1}^{x}\), \(\sigma_{i}^{y}\sigma_{i{+}1}^{y}\) and \(\sigma_{i}^{z}\)) or purely imaginary (i.e., \(\sigma_{i}^{x}\sigma_{i{+}1}^{z}\sigma_{i{+}2}^{y}\), \(\sigma_{i}^{y}\sigma_{i{+}1}^{z}\sigma_{i{+}2}^{x}\), \(\sigma_{i}^{x}\sigma_{i{+}1}^{y}\) and \(\sigma_{i}^{y}\sigma_{i{+}1}^{x}\)). When \(L\) is even, all real \(\hat{H}^{\alpha}_{i+j_{1},\cdots,i+j_{n}}\) anticommute with an operator \(\mathcal{C}_{\rm NNN}{=}\prod_{p{=}1}^{\frac{L}{2}}\sigma_{2p{-}1}^{x}\sigma_{2p}^{y}\) while all purely imaginary \(\hat{H}^{\alpha}_{i+j_{1},\cdots,i+j_{n}}\) commute with \(\mathcal{C}_{\rm NNN}\) satisfying the condition given in Eq. 14 . Thus, as a result of Theorem 1, the Hamiltonian \(H_{\rm NNN}(2L)\) hosts a zero energy eigenstate given by \[|\Lambda\rangle_{\rm NNN}{=}\bigotimes_{i{=}1}^{L{-}1}|\Phi^{-}\rangle_{i,\overline{i}}|\Phi^{+}\rangle_{i{+}1,\overline{i{+}1}}, \label{suppeq:32EAP95eigenstate}\tag{38}\] where \(|\Phi^{\pm}\rangle_{i,j}{=}(|01\rangle {\pm} |10\rangle)_{i,j}/\sqrt{2}\). Note that due to PBC the translation of the state by one site given by \[|\Lambda^{\mathcal{T}_1}\rangle_{\rm NNN}{=}\bigotimes_{i{=}1}^{L{-}1}|\Phi^{+}\rangle_{i,\overline{i}}|\Phi^{-}\rangle_{i{+}1,\overline{i{+}1}}\] produces another distinct zero energy eigenstate (orthogonal to \(|\Lambda\rangle_{\rm NNN}\)). These are precisely the entangled antipodal paired (EAP) states [named so since there are Bell-pairs between antipodal points when the lattice sites are arranged uniformly on a circle] identified in Ref. [61].

References↩︎