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Physical review. B./Physical review. B

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

Mingchen Zheng, Chenguang Liang, Shu Chen, Xin Zhang

Published Sep 3, 2025• 2 citations
Publisher

Abstract

Spin helix states with spatially modulated spin textures are exact eigenstates of the one-dimensional anisotropic spin-$\frac{1}{2}$ Heisenberg model under specific parameter conditions. In this work, we extend this framework by constructing exact eigenstates of the fully anisotropic XYZ Heisenberg model, with spin helix structures modulated by elliptic functions, across all spatial dimensions and spin quantum numbers. The XXZ and XY cases are also analyzed to clarify the conditions for the emergence of spin helix eigenstates and their key properties. Within the subspace spanned by these elliptic spin helix states, a family of analytically tractable states can be identified. Our results broaden the class of exactly solvable eigenstates in nonintegrable systems and deepen the understanding of spin modulation states in many-body systems.

DOI
10.1103/6dbn-n6rv

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Spin helix states with spatially modulated spin textures are exact eigenstates of the one-dimensional anisotropic spin-12\frac{1}{2}21​ Heisenberg model under specific parameter conditions. In this work, we extend this framework by constructing exact eigenstates of the fully anisotropic XYZ Heisenberg model, with spin helix structures modulated by elliptic functions, across all spatial dimensions and spin quantum numbers. The XXZ and XY cases are also analyzed to clarify the conditions for the emergence of spin helix eigenstates and their key properties. Within the subspace spanned by these elliptic spin helix states, a family of analytically tractable states can be identified. Our results broaden the class of exactly solvable eigenstates in nonintegrable systems and deepen the understanding of spin modulation states in many-body systems.
Exact spin helix eigenstates in the anisotropic spin-sss Heisenberg model with arbitrary dimensions
Spin helix states-characterized by their spatially modulated spin textures-are exact eigenstates of the one-dimensional anisotropic spin-12\frac{1}{2}21​ Heisenberg model under specific parameter conditions. In this work, we extend this framework by constructing exact spin helix eigenstates for the fully anisotropic XYZ Heisenberg model with arbitrary spatial dimensions and arbitrary spin quantum numbers. Our results demonstrate that some non-trivial exact eigenstates can persist beyond the integrable regime. The XXZ and XY cases are also analyzed to clarify the conditions for the emergence of spin helix eigenstates and their key properties. Our results broaden the class of analytically tractable exact eigenstates in non-integrable systems and deepen understanding of spin modulation states in many-body systems.
Abstract In general, the summation of a set of sub-Hamiltonians cannot share a common eigenstate of each one, only if it is an unentangled product state, such as a spin helix state in quantum spin system. Here we present a method, referred to as the building block method (BBM), for constructing possible spin-1/2 XXZ Heisenberg lattice systems possessing spin helix states. We focus on two types of XXZ dimers as basic elements, with a non-Hermitian parity-time ( P T ) magnetic field and Hermitian Dzyaloshinskii-Moriya interaction (DMI), which share the same degenerate eigenstates. Based on these two building blocks, one can construct a variety of Heisenberg quantum spin systems, which support helix states with zero energy. Furthermore, we show that such systems act as quantum spin graphs since they obey the analogs of Kirchhoff’s laws for sets of spin helix states when the non-Hermitian P T terms cancel each other out. In addition, the dynamic response of the helix states for three types of perturbations is also investigated analytically and numerically. Our findings provide a way to study quantum spin systems on irregular lattices.
In general, the summation of a set of sub-Hamiltonians cannot share a common eigenstate of each one, only if it is an unentangled product state, such as a phantom helix state in quantum spin system. Here we present a method, referred to as the building block method (BBM), for constructing possible spin-1/2 XXZ Heisenberg lattice systems possessing phantom helix states. We focus on two types of XXZ dimers as basic elements, with a non-Hermitian parity-time (PT ) field and Hermitian Dzyaloshinskii-Moriya interaction (DMI), which share the same degenerate eigenstates. Based on these two building blocks, one can construct a variety of Heisenberg quantum spin systems, which support helix states with zero energy. The underlying mechanism is the existence of a set of degenerate eigenstates. Furthermore, we show that such systems act as quantum spin graphs since they obey the analogs of Kirchhoff's laws for sets of spin helix states when the non-Hermitian PT fields cancel each other out. In addition, the dynamic response of the helix states for three types of perturbations is also investigated analytically and numerically. Our findings provide a way to study quantum spin systems with irregular geometries beyond the Bethe ansatz approach.
Exact solutions for excited states in nonintegrable quantum Hamiltonians have revealed novel dynamical phenomena that can occur in quantum many-body systems. This work proposes a method to analytically construct a specific set of volume-law entangled zero-energy exact excited eigenstates in a large class of spin Hamiltonians. In particular, we show that all spin chains that satisfy a simple set of conditions host exact volume law entangled zero-energy eigenstates in the middle of their spectra. Examples of physically relevant spin chains of this type include the transverse-field Ising model, PXP model, spin-S XY model, and spin-S Kitaev chain. Although these eigenstates are highly atypical in their structure, they are thermal with respect to local observables. Our framework also unifies many recent constructions of volume-law entangled eigenstates in the literature. Finally, we show that a similar construction also generalizes to spin models on graphs in arbitrary dimensions.
Exact solutions for quantum many-body systems are rare and provide valuable insight to universal phenomena. Here we show experimentally in anisotropic Heisenberg chains that special helical spin patterns can have very long lifetimes. This finding confirms the recent prediction of phantom Bethe states, exact many-body eigenstates carrying finite momenta yet no energy. We theoretically find analogous stable spin helices in higher dimensions and in other non-integrable systems, where they imply non-thermalizing dynamics associated with quantum many-body scars. We use phantom spin helices to directly measure the interaction anisotropy which has a major contribution from short-range off-site interactions that have not been observed before. Phantom helix states open new opportunities for quantum simulations of spin physics and studies of many-body dynamics.
We derive a criterion under which splitting of all eigenstates of an open XYZ\mathrm{XYZ}XYZ Hamiltonian with boundary fields into two invariant subspaces, spanned by chiral shock states, occurs. The splitting is governed by an integer number, which has the geometrical meaning of the maximal number of kinks in the basis states. We describe the generic structure of the respective Bethe vectors. We obtain explicit expressions for Bethe vectors, in the absence of Bethe roots, and those generated by one Bethe root, and we investigate the single-particle subspace. We also describe in detail an elliptic analog of the spin-helix state, appearing in both the periodic and the open XYZ\mathrm{XYZ}XYZ model, and we derive the eigenstate condition. The elliptic analog of the spin-helix state is characterized by a quasiperiodic modulation of the magnetization profile, governed by Jacobi elliptic functions.