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arXiv:2505.14994

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

Mingchen Zheng, Liang, Chenguang, Shu Chen, Xin Zhang

Published May 21, 2025• 0 citations
arXivHTMLPDF

Abstract

Spin helix states-characterized by their 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 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.

DOI
10.48550/arxiv.2505.14994

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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.
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.
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 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.
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.
Quantum many-body scars are highly excited eigenstates of non-integrable Hamiltonians which violate the eigenstate thermalization hypothesis and are embedded in a sea of thermal eigenstates. We provide a general mechanism to construct partially integrable models with arbitrarily large local Hilbert space dimensions, which host exact many-body scars. We introduce designed integrability-breaking terms to several exactly solvable spin chains, whose integrable Hamiltonians are composed of the generators of the Temperley-Lieb algebra. In the non-integrable subspace of these models, we identify a special kind of product states -- the generalized spin helix states as exact quantum many-body scars, which lie in the common null space of the non-Hermitian generators of the Temperley-Lieb algebra and are annihilated by the integrability-breaking terms. Our constructions establish an intriguing connection between integrability and quantum many-body scars, meanwhile provide a systematic understanding of scarred Hamiltonians from the perspective of non-Hermitian projectors.
We demonstrate the existence of a special chiral ``phantom'' mode with some analogy to a Goldstone mode in the anisotropic quantum XXZXXZXXZ Heisenberg spin chain. The phantom excitations contribute zero energy to the eigenstate, but a finite fixed quantum of momentum k0{k}_{0}k0​. The mode exists not due to symmetry principles, but results from nontrivial scattering properties of magnons with momentum k0{k}_{0}k0​ given by the anisotropy via $cos{k}_{0}=\mathrm{\ensuremath{\Delta}}$. Different occupations of the phantom mode lead to energetical degeneracies between different magnetization sectors in the periodic case. This mode originates from special string-type solutions of the Bethe ansatz equations with unbounded rapidities, the phantom Bethe roots (PBRs). We derive criteria under which the spectrum contains eigenstates with PBRs, both in open and periodically closed integrable systems, for spin-1/21/21/2 and higher spins, and discuss the respective chiral eigenstates. The simplest of such eigenstates, the spin helix state, which is a periodically modulated state of chiral nature, is built up from the phantom excitations exclusively. Implications of our results for experiments are discussed.