Exact spin helix eigenstates in anisotropic spin-$s$ Heisenberg model in arbitrary dimensions

Exact eigenstates in quantum many-body systems have attracted growing interest, as they offer rare analytic access to the structure of strongly correlated quantum matter Sutherland ; Amico et al. (2008). These states naturally arise in integrable systems Baxter (2016); Korepin et al. (1997), where an extensive set of conserved quantities enables exact solutions and leads to constrained, nonergodic dynamics describable by generalized Gibbs ensembles Ilievski et al. (2015, 2016); Essler and Fagotti (2016); Vidmar and Rigol (2016); Cecile et al. (2024) and generalized hydrodynamics Castro-Alvaredo et al. (2016); Bertini et al. (2016); Bastianello et al. (2022); Alba et al. (2021); De Nardis et al. (2022); Doyon et al. (2025). Remarkably, exact eigenstates can also emerge in nonintegrable systems, most notably as quantum many-body scar states (QMBS), in which a set of atypical, long-lived coherent states coexist with an otherwise thermalizing spectrum Serbyn et al. (2021); Moudgalya et al. (2022); Chandran et al. (2023); Turner et al. (2018a, b); Lin and Motrunich (2019); Mark et al. (2020); Moudgalya et al. (2018a, b); Schecter and Iadecola (2019); Chattopadhyay et al. (2020); Shibata et al. (2020); O’Dea et al. (2020); Pakrouski et al. (2020); Ren et al. (2021); Ivanov and Motrunich . Such states offer valuable benchmarks for understanding entanglement structure Amico et al. (2008); Eisert et al. (2010), coherence and ergodicity breaking D’Alessio et al. (2016); Nandkishore and Huse (2015); Abanin et al. (2019); Sierant et al. (2025), and serve as experimentally accessible targets for testing quantum dynamics and controlling many-body evolution in programmable quantum simulators Bernien et al. (2017); Bluvstein et al. (2021).

A particularly striking class of exact eigenstates emerges in the paradigmatic spin-1/2 XXZ Heisenberg chain with periodic boundary conditions, where the so-called spin-helix states (SHSs) arise under fine-tuned conditions (the anisotropy parameter $\eta$ takes root-of-unity values) Popkov et al. (2021); Jepsen et al. (2022a). These eigenstates exhibit spatially rotating spin textures and provide rare examples of highly structured, nonthermal excited eigenstates in an interacting many-body system. Their origin can be traced back to Baxter’s seminal work on the eight-vertex (XYZ) model in the 1970s Baxter (1973a, b, c), which revealed spectral degeneracies at special parameter values—later understood to stem from an enhanced quantum group symmetry $U_{q}(\mathfrak{sl}_{2})$ where $q$ is a root of unity Pasquier and Saleur (1990); Deguchi et al. (2001). More recently, SHSs have been reinterpreted via the Bethe ansatz as arising from “phantom” Bethe roots, in which all quasiparticles share identical momentum and do not scatter Popkov et al. (2021). Importantly, SHSs have been experimentally observed in ultracold atom systems simulating the XXZ chain, where long-lived helical spin patterns were detected Jepsen et al. (2021, 2022a). Moreover, SHSs persist beyond the integrable setting, appearing in higher-dimensional and higher-spin extensions of the XXZ model where they represent robust examples of QMBS Jepsen et al. (2022a), suggesting a unifying framework for nonthermal states across integrable and nonintegrable regimes.

Despite this progress, the status of SHSs in more general settings—such as the fully anisotropic XYZ Heisenberg model—remains poorly understood. The XYZ model features unequal couplings along all spin directions and lacks continuous symmetries, unlike the $U(1)$-symmetric XXZ chain. This more intricate parameter landscape raises fundamental questions about the existence and structure of exact spin helix eigenstates beyond the integrable regime. In this work, we investigate SHSs in the XYZ model and demonstrate that, under specific parameter conditions, it supports eigenstates with helical structure. We further extend the analysis to higher spatial dimensions $d$ and higher spin $s$, and offer a unified framework for understanding spin helix eigenstates across different parameter regimes. We also examine the XXZ and XY limits in detail, clarifying the characteristics and parameter constraints of spin helix eigenstates in these cases. Our results provide a comprehensive perspective on analytically tractable, spatially inhomogeneous eigenstates in interacting quantum spin systems.

This paper is organized as follows. In Section II, we introduce the notation and conventions for elliptic theta functions, which are essential for describing the modulated structure of spin helix eigenstates in the XYZ model. Section III presents the construction and characterization of exact spin helix eigenstates in the $d$-dimensional hypercubic XYZ spin model. In Section IV, we analyze the degenerate limits of the XYZ model—such as the XXZ and XY cases—and demonstrate how spin helix eigenstates persist and transform in these regimes. In Section V, we extend our discussion to other classes of spin models that support spin helix eigenstates, including systems with long-range interactions and more complex lattice geometries. Finally, Section VI is the summary and outlook.

We introduce the following Jacobi theta functions $\vartheta_{\alpha}(u,q)$ Whittaker and Watson (1950)

$\displaystyle\begin{aligned} &\vartheta_{1}(u,q)=2\sum_{n=0}^{\infty}(-1)^{n}q^{(n+\frac{1}{2})^{2}}\sin[(2n+1)u],\\ &\vartheta_{2}(u,q)=2\sum_{n=0}^{\infty}q^{(n+\frac{1}{2})^{2}}\cos[(2n+1)u],\\ &\vartheta_{3}(u,q)=1+2\sum_{n=1}^{\infty}q^{n^{2}}\cos(2nu),\\ &\vartheta_{4}(u,q)=1+2\sum_{n=1}^{\infty}(-1)^{n}q^{n^{2}}\cos(2nu).\end{aligned}$

(1)

For convenience, we use the following shorthand notations $\theta_{\alpha}(u),\,\tilde{\theta}_{\alpha}(u),\,\alpha=1,2,3,4$

$\displaystyle\theta_{\alpha}(u)\equiv\vartheta_{\alpha}(\pi u,{\mathrm{e}}^{{\mathrm{i}}\pi\tau}),\quad\tilde{\theta}_{\alpha}(u)\equiv\vartheta_{\alpha}(\pi u,{\mathrm{e}}^{2{\mathrm{i}}\pi\tau}),$

(2)

where $\tau$ is a complex with a positive imaginary part. Further details on theta functions can be found in Appendix A.

The Hamiltonian of the spin-$s$ XYZ Heisenberg model is

$\displaystyle H=\sum_{\langle i,j\rangle}H_{i,j},\quad$

(3)

where $H_{i,j}$ is the two-site XYZ interaction term

$$H_{i,j}=J_{x}{S}_{i}^{x}S_{j}^{x}+J_{y}S_{i}^{y}S_{j}^{y}+J_{z}S_{i}^{z}S_{j}^{z},$$

(4)

Here, ${S}_{j}^{\alpha}$ $(\alpha=x,\ y,\ z)$ denote the spin-$s$ operators at lattice site $j$ of a $d$-dimensional hypercubic lattice with volume $V=\prod_{\alpha=1}^{d}L_{\alpha}$, where $L_{\alpha}$ is the number of sites along the $\alpha$-th spatial direction. The notation ${\langle i,j\rangle}$ indicates summation over nearest-neighbor site pairs, and the exchange coefficients are parameterized by theta functions as follows Wang et al. (2016)

$\displaystyle J_{x}=\frac{\theta_{4}(\eta)}{\theta_{4}(0)},\quad J_{y}=\frac{\theta_{3}(\eta)}{\theta_{3}(0)},\quad J_{z}=\frac{\theta_{2}(\eta)}{\theta_{2}(0)}.$

(5)

Obviously, the Hamiltonian depends on two parameters: $\eta$ and $\tau$. From the identities

$\displaystyle J_{\alpha}|_{\eta\to\eta+2}=J_{\alpha},\quad J_{\alpha}|_{\eta\to\eta+2\tau}={\mathrm{e}}^{-4{\mathrm{i}}\pi(\eta+\tau)}J_{\alpha},$

(6)

we can thus restrict parameter $\eta$ to the rectangle $0\leq(\eta)<2,\,0\leq{\rm Im}(\eta)<2{\rm Im}(\tau)$ without loss of generality.

When $\tau$ is purely imaginary and $\eta$ is real or also purely imaginary, the Hamiltonian in Eq. (3) is Hermitian, as follows

• •

  when ${\rm Im}(\eta)=(\tau)=0$,   $|J_{x}|\geq|J_{y}|\geq|J_{z}|$,

• •

  when $(\eta)=(\tau)=0$,   $|J_{x}|\leq|J_{y}|\leq|J_{z}|$.

Let us first introduce the following local vector

	$\displaystyle\psi^{(s)}(u)$	$\displaystyle=\frac{1}{\mathcal{N}(u)}\sum_{n=0}^{2s}\kappa_{n}\left[\tilde{\theta}_{1}(u)\right]^{2s-n}\left[\tilde{\theta}_{4}(u)\right]^{n}\ket{s-n},$		(7)
	$\displaystyle\kappa_{n}$	$\displaystyle=\sqrt{(2s)!/(n!(2s-n)!)},$
	$\displaystyle\mathcal{N}(u)$	$\displaystyle=\left[\theta_{4}((u))\theta_{3}({\mathrm{i}}{\rm Im}(u))\right]^{s}$

where $u$ is a free parameter, $\{\ket{m}\}$ are the $S^{z}$ basis with $S^{z}\ket{m}=m\ket{m}$. The state $\psi^{(s)}(u)$ in (7) is a spin coherent state which can be obtained by rotating the highest-weight state $\ket{s}$ first around the $y$-axis and then around the $z$-axis, specifically as follows

$\displaystyle\psi^{(s)}(u)\propto{\mathrm{e}}^{-{\mathrm{i}}\beta(u)S_{z}}{\mathrm{e}}^{-{\mathrm{i}}\gamma(u)S_{y}}\ket{s},$

(8)

where

$\displaystyle\gamma(u)=2\arctan\left|\frac{\tilde{\theta}_{4}(u)}{\tilde{\theta}_{1}(u)}\right|,\quad\beta(u)=\arg\left(\frac{\tilde{\theta}_{4}(u)}{\tilde{\theta}_{1}(u)}\right).$

(9)

Thus we can also use $\gamma(u)$ and $\beta(u)$ to represent the state $\psi^{(s)}(u)$, and the expectation value of $S^{\alpha}$ in the state $\psi^{(s)}(u)$ reads

$\displaystyle\begin{aligned} \langle S_{x}\rangle&=s\sin(\gamma(u))\cos(\beta(u)),\\ \langle S_{y}\rangle&=s\sin(\gamma(u))\sin(\beta(u)),\\ \langle S_{z}\rangle&=s\cos(\gamma(u)).\end{aligned}$

(10)

The proof of Eqs. (8) and (10) is presented in Appendix B.

The local vector $\psi^{(s)}_{i}(u)$ in (7) satisfies the following divergence condition:

	$\displaystyle H_{i,j}\,\psi^{(s)}_{i}(u)\psi^{(s)}_{j}(u\pm\eta)=s^{2}b(\pm u)\psi^{(s)}_{i}(u)\psi^{(s)}_{j}(u\pm\eta)$
	$\displaystyle\quad\pm s\left[a(u)S_{i}^{z}-a(u\pm\eta)S_{j}^{z}\right]\,\psi^{(s)}_{i}(u)\psi^{(s)}_{j}(u\pm\eta),$		(11)

where

	$\displaystyle a(u)=\frac{\theta_{1}(\eta)\theta_{2}(u)}{\theta_{2}(0)\theta_{1}(u)},\quad b(u)=g(\eta)+g(u)-g(u+\eta),$
	$\displaystyle g(u)=\frac{\theta_{1}(\eta)\theta_{1}^{\prime}(u)}{\theta_{1}^{\prime}(0)\theta_{1}(u)},\quad\theta^{\prime}_{j}(u)=\frac{\partial\theta_{j}(u)}{\partial u}.$		(12)

Equation (11) implies that only two non-eigen terms arise when $H_{i,j}$ acts on the product state $\psi^{(s)}_{i}(u)\psi^{(s)}_{j}(v)$ and the “phase” factors $u$ and $v$ in $\psi^{(s)}_{i}(u)$ and $\psi^{(s)}_{j}(v)$ has a fixed difference $\pm\eta$. The foundational identities presented in Eq. (11) inspire the construction of specific tensor product states in Theorem 1. When $s=1/2$, Eq. (11) was established in Refs. Popkov et al. (2022); Zhang et al. (2022, 2024). In this paper, we generalize it to the case of an arbitrary $s$. The proof of Eq. (11) involves numerous elliptic function identities. Due to its inherent complexity, we present complete proof details in Appendix C.

We can construct the following tensor product state

$\displaystyle\ket{\Psi^{(s)}(u,\bm{\epsilon})}=\bigotimes_{j}\psi^{(s)}_{j}\left(u+\eta\,\bm{\epsilon}\cdot\bm{n}_{j}\right),\quad u\in\mathbb{C},$

(13)

where $\bm{n}_{j}=(n_{j,1},\ldots,n_{j,d})$ is the coordinate of site $j$ in the $d$-dimensional hypercubic lattice, $\bm{\epsilon}=(\epsilon_{1},\ldots,\epsilon_{d})$, $\epsilon_{\alpha}=\pm 1$ is a $d$-dimensional vector, and $\bm{\epsilon}\cdot\bm{n}_{j}=\epsilon_{1}n_{j,1}+\dots+\epsilon_{d}\,n_{j,d}$ describes a phase that exhibits linear spatial variation along each lattice direction. A visualization of the phase $\bm{\epsilon}\cdot\bm{n}_{j}$ is demonstrated in Fig. 1. Such an inhomogeneous state was first proposed by Baxter in the pioneering study of the one-dimensional spin-$1/2$ XYZ model Baxter (1973a, c). In Eq. (13), we generalize Baxter’s result to $d$-dimensional spin-$s$ Heisenberg chain.

In the state $\ket{\Psi^{(s)}(u,\bm{\epsilon})}$, the polar and azimuthal angles $\gamma_{j}(u)$ and $\beta_{j}(u)$ defined in Eq. (9) vary systematically with site index $j$, forming a spatially modulated spin texture characteristic of a spin helix. The winding direction is set by the vector $\bm{\epsilon}$, while the pitch and phase of the helix depend sensitively on the parameters $u$, $\eta$, and the modular parameter $\tau$, as illustrated in Figs. 2 and 3. Owing to the factorized structure of the state, its spin configuration admits a direct classical interpretation in the large-$s$ limit, where quantum fluctuations are suppressed and the local spin orientation approaches a classical vector field.

The eigenstate $\ket{\Psi^{(s)}(u,\bm{\epsilon})}$ depends on a free phase parameter $u$, whereas the energy $E$ remains independent of this parameter. This implies that the spin helix eigenstates form a degenerate manifold. $\ket{\Psi^{(s)}(u,\bm{\epsilon})}$ is an exact nonthermal eigenstate that exists in higher dimension XYZ models, which are generally believed to be non-integrable Shiraishi and Tasaki . As such, $\ket{\Psi^{(s)}(u,\bm{\epsilon})}$ constitutes a QMBS.

For finite systems, Eq. (15) implies the parameter $\eta$ can only take several discrete values

$$\eta=\frac{2p_{\alpha}\tau+2q_{\alpha}}{L_{\alpha}},\,\,p_{\alpha},q_{\alpha}\in\mathbb{Z}.$$

(20)

In the thermodynamic limit, these discrete points become densely distributed, allowing $\eta$ to approach any value within the complex plane. This demonstrates the universality of spin helix eigenstates in Heisenberg models.

The XYZ model constitutes the most general form of the Heisenberg model class. In specific parameter regimes, it reduces to partially anisotropic models, such as the XXZ and XY models.

In the previous sections, we study the Heisenberg model with only nearest-neighbor interactions in a hypercubic lattice. In this section, we will demonstrate that the Heisenberg model with other settings can still have spin helix eigenstates, provided the exchange coefficients are appropriately modulated.

Let us consider the following periodic spin-$s$ Heisenberg model

	$\displaystyle H=\sum_{k}F_{k}\sum_{\langle i,j\rangle_{k}}H_{i,j}^{(k)},$		(45)
	$\displaystyle H_{i,j}^{(k)}=J_{x}^{(k)}{S}_{i}^{x}S_{j}^{x}+J_{y}^{(k)}S_{i}^{y}S_{j}^{y}+J_{z}^{(k)}S_{i}^{z}S_{j}^{z},$		(46)

where $\langle i,j\rangle_{k}$ represents the $k$-th nearest-neighbor pairs along the axial lattice directions, $\{F_{k}\}$ are free parameters and the exchange coefficients $\{J_{\alpha}^{(k)}\}$ are

$\displaystyle\left\{J_{x}^{(k)},J_{y}^{(k)},J_{z}^{(k)}\right\}=\left\{\frac{\theta_{4}(k\eta)}{\theta_{4}(0)},\frac{\theta_{3}(k\eta)}{\theta_{3}(0)},\frac{\theta_{2}(k\eta)}{\theta_{2}(0)}\right\}.$

(47)