Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2501.14017 · Physical review. B./Physical review. B

Exact few-magnon Bethe scars of the nonintegrable alternating Heisenberg chain

Ronald Melendrez, Bhaskar Mukherjee, Marcin Szyniszewski, Christopher J. Turner, Arijeet Pal, Hitesh J. Changlani

Published Oct 16, 2026• 0 citations
arXivHTMLPDF

Abstract

Exact solutions of quantum lattice models serve as useful guides for interpreting physical phenomena in condensed matter systems. Prominent examples of integrability appear in one dimension, including the Heisenberg chain, where the Bethe ansatz method has been widely successful. Recent work has noted that certain non-integrable models harbor quantum many-body scar states, which form a superspin of regular states hidden in an otherwise chaotic spectrum. Here we consider one of the simplest examples of a non-integrable model, the alternating ferromagnetic-antiferromagnetic (bond-staggered) Heisenberg chain, a close cousin of the spin-1 Haldane chain and a spin analog of the Su-Schrieffer-Heeger model, and show the presence of exponentially many zero-energy states. We highlight features of the alternating chain that allow treatment with the coordinate Bethe ansatz (with important modifications), and surprisingly, for a non-integrable system, we find simple compact expressions for zero-energy eigenfunctions for a few magnons, including solutions with fractionalized particle momentum. These states exhibit wavefunction scarring similar to that of single-particle states in chaotic billiards. We discuss a general numerical recipe to diagnose the existence of such generalized Bethe ansatz (GBA) scar states and also provide exact analytic expressions for their entanglement. We conclude by conjecturing a picture of magnon pairing which may generalize to multiple magnons. Our work opens the avenue to describe certain eigenstates of partially integrable systems using the GBA.

DOI
10.1103/2g33-l3y6

Related papers

Periodic orbits and quantum many-body scars in integrable spin chains

Elena Petrova, Maksym Serbyn, Marko Ljubotina

Quantum many-body scars provide an exception to generic thermalising dynamics, but their relation to integrability remains unclear. Here we apply the periodic-orbit framework to the integrable XYZ/XXZ spin-1/2 chain, developing an energy-resolved approach that enables us to locate and track periodic orbits across the spectrum. We identify families of scarred orbits and resolve their supporting eigenstates in terms of Bethe-ansatz quantum numbers, revealing a common string core dressed by zero-momentum magnon pairs. This allows us to reconstruct both the scarred eigenstates and the associated towers directly within the Bethe ansatz, and explain the equidistant tower spacing analytically from the decoupling of zero-momentum magnon pairs in the Bethe equations, providing a microscopic realisation of scar phenomenology in an integrable setting. We further show that this structure persists upon breaking integrability with a transverse field, with the periodic orbits continuing to govern the dynamics. Our results establish a direct connection between the periodic-orbit picture of quantum scarring and the algebraic structure of integrable models, showing that key features of scar dynamics can be understood analytically within the Bethe ansatz framework.

quant-pharXiv:2607.15132•Jul 17, 2026•0 citations

Exact Valence-Bond Solid Scars in the Square-Lattice Heisenberg Model

David D. Dai

We show that the spin-s square-lattice Heisenberg model has exact many-body scars. These scars are simple valence-bond solids with exactly zero energy, and they exist in even-by-even systems and ladders of width 2. Ladders have additional scars corresponding to injecting one or two magnons on top of a parent valence-bond solid scar. These scars have a remarkably simple physical origin based only the angular momentum algebra and cancellations from spin-antialignment within a valence bond. Our comprehensive exact diagonalization calculations suggest that our valence-bond solids exhaust all exact eigenstates in the Heisenberg model except for few-magnon states near the top of the spectrum. Our scars are interesting because they are not part of a tower, have area-law entanglement, break translation symmetry, and exist for Heisenberg models of all spin.

arXiv:2412.08874•Dec 12, 2024•0 citations

Catching Bethe phantoms and quantum many-body scars: Long-lived spin-helix states in Heisenberg magnets

Niklas Jepsen, Yoo Kyung Lee, Hanzhen Lin +4 more

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.

arXiv:2110.12043•Oct 22, 2021•2 citations

Bethe ansatz for two-magnon scattering states in 2D and 3D Heisenberg–Ising ferromagnets †

P. N. Bibikov

Abstract Two different versions of Bethe ansatz are suggested for evaluation of scattering two-magnon states in 2D and 3D Heisenberg–Ising ferromagnets on square and simple cubic lattices. It is shown that the two-magnon sector is subdivided on two subsectors related to non-interacting and scattering magnons. The former subsector possess an integrable regular dynamics and may be described by a natural modification of the usual Bethe Ansatz. The latter one is characterized by a non-integrable chaotic dynamics and may be treated only within discrete degenerative version of Bethe Ansatz previously suggested by the author. Some of these results are generalized for multi-magnon states of the Heisenberg–Ising ferromagnet on a D dimensional hyper cubic lattice.

arXiv:1705.04117•Journal of Statistical Mechanics Theory and Experiment•Apr 1, 2018•4 citations

Bethe ansatz solvable multi-chain quantum systems

Andrei A. Zvyagin

In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example (`basic' model) of this class of models: the antiferromagnetic two-chain spin-1/2 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate-incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the `quantum transfer matrix') for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behaviour of non-integrable (less constrained) multi-chain quantum models is discussed.

Journal of Physics A Mathematical and General•Oct 8, 2001•40 citations

Phantom Bethe roots in the integrable open spin- 12 XXZ chain

Xin Zhang, Andreas Klümper, Vladislav Popkov

We investigate special solutions to the Bethe Ansatz equations (BAE) for open integrable XXZ Heisenberg spin chains containing phantom (infinite) Bethe roots. The phantom Bethe roots do not contribute to the energy of the Bethe state, so the energy is determined exclusively by the remaining regular excitations. We derive the phantom Bethe roots criterion and focus on BAE solutions for mixtures of phantom roots and regular (finite) Bethe roots. We prove that in the presence of phantom Bethe roots, all eigenstates are split between two invariant subspaces, spanned by chiral shock states. Bethe eigenstates are described by two complementary sets of Bethe ansatz equations for regular roots, one for each invariant subspace. The respective ``semiphantom'' Bethe vectors are states of chiral nature, with chirality properties getting less pronounced when more regular Bethe roots are added. For the easy plane case, ``semiphantom'' Bethe states carry nonzero magnetic current and are characterized by quasiperiodic modulation of the magnetization profile, the most prominent example being the spin helix states (SHS). We illustrate our results investigating ``semiphantom'' Bethe states generated by one regular Bethe root (the other Bethe roots being phantom), with a simple structure of the invariant subspace, in all details. We obtain the explicit expressions for Bethe vectors and calculate the simplest correlation functions, including the spin-current for all the states in the single-particle multiplet.

arXiv:2102.03299•Physical review. B./Physical review. B•Mar 22, 2021•19 citations
Exact solutions of quantum lattice models serve as useful guides for interpreting physical phenomena in condensed matter systems. Prominent examples of integrability appear in one dimension, including the Heisenberg chain, where the Bethe ansatz method has been widely successful. Recent work has noted that certain non-integrable models harbor quantum many-body scar states, which form a superspin of regular states hidden in an otherwise chaotic spectrum. Here we consider one of the simplest examples of a non-integrable model, the alternating ferromagnetic-antiferromagnetic (bond-staggered) Heisenberg chain, a close cousin of the spin-1 Haldane chain and a spin analog of the Su-Schrieffer-Heeger model, and show the presence of exponentially many zero-energy states. We highlight features of the alternating chain that allow treatment with the coordinate Bethe ansatz (with important modifications), and surprisingly, for a non-integrable system, we find simple compact expressions for zero-energy eigenfunctions for a few magnons, including solutions with fractionalized particle momentum. These states exhibit wavefunction scarring similar to that of single-particle states in chaotic billiards. We discuss a general numerical recipe to diagnose the existence of such generalized Bethe ansatz (GBA) scar states and also provide exact analytic expressions for their entanglement. We conclude by conjecturing a picture of magnon pairing which may generalize to multiple magnons. Our work opens the avenue to describe certain eigenstates of partially integrable systems using the GBA.
Quantum many-body scars provide an exception to generic thermalising dynamics, but their relation to integrability remains unclear. Here we apply the periodic-orbit framework to the integrable XYZ/XXZ spin-1/2 chain, developing an energy-resolved approach that enables us to locate and track periodic orbits across the spectrum. We identify families of scarred orbits and resolve their supporting eigenstates in terms of Bethe-ansatz quantum numbers, revealing a common string core dressed by zero-momentum magnon pairs. This allows us to reconstruct both the scarred eigenstates and the associated towers directly within the Bethe ansatz, and explain the equidistant tower spacing analytically from the decoupling of zero-momentum magnon pairs in the Bethe equations, providing a microscopic realisation of scar phenomenology in an integrable setting. We further show that this structure persists upon breaking integrability with a transverse field, with the periodic orbits continuing to govern the dynamics. Our results establish a direct connection between the periodic-orbit picture of quantum scarring and the algebraic structure of integrable models, showing that key features of scar dynamics can be understood analytically within the Bethe ansatz framework.
We show that the spin-s square-lattice Heisenberg model has exact many-body scars. These scars are simple valence-bond solids with exactly zero energy, and they exist in even-by-even systems and ladders of width 2. Ladders have additional scars corresponding to injecting one or two magnons on top of a parent valence-bond solid scar. These scars have a remarkably simple physical origin based only the angular momentum algebra and cancellations from spin-antialignment within a valence bond. Our comprehensive exact diagonalization calculations suggest that our valence-bond solids exhaust all exact eigenstates in the Heisenberg model except for few-magnon states near the top of the spectrum. Our scars are interesting because they are not part of a tower, have area-law entanglement, break translation symmetry, and exist for Heisenberg models of all spin.
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 Two different versions of Bethe ansatz are suggested for evaluation of scattering two-magnon states in 2D and 3D Heisenberg–Ising ferromagnets on square and simple cubic lattices. It is shown that the two-magnon sector is subdivided on two subsectors related to non-interacting and scattering magnons. The former subsector possess an integrable regular dynamics and may be described by a natural modification of the usual Bethe Ansatz. The latter one is characterized by a non-integrable chaotic dynamics and may be treated only within discrete degenerative version of Bethe Ansatz previously suggested by the author. Some of these results are generalized for multi-magnon states of the Heisenberg–Ising ferromagnet on a D dimensional hyper cubic lattice.
In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example (`basic' model) of this class of models: the antiferromagnetic two-chain spin-1/2 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate-incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the `quantum transfer matrix') for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behaviour of non-integrable (less constrained) multi-chain quantum models is discussed.
We investigate special solutions to the Bethe Ansatz equations (BAE) for open integrable XXZ Heisenberg spin chains containing phantom (infinite) Bethe roots. The phantom Bethe roots do not contribute to the energy of the Bethe state, so the energy is determined exclusively by the remaining regular excitations. We derive the phantom Bethe roots criterion and focus on BAE solutions for mixtures of phantom roots and regular (finite) Bethe roots. We prove that in the presence of phantom Bethe roots, all eigenstates are split between two invariant subspaces, spanned by chiral shock states. Bethe eigenstates are described by two complementary sets of Bethe ansatz equations for regular roots, one for each invariant subspace. The respective ``semiphantom'' Bethe vectors are states of chiral nature, with chirality properties getting less pronounced when more regular Bethe roots are added. For the easy plane case, ``semiphantom'' Bethe states carry nonzero magnetic current and are characterized by quasiperiodic modulation of the magnetization profile, the most prominent example being the spin helix states (SHS). We illustrate our results investigating ``semiphantom'' Bethe states generated by one regular Bethe root (the other Bethe roots being phantom), with a simple structure of the invariant subspace, in all details. We obtain the explicit expressions for Bethe vectors and calculate the simplest correlation functions, including the spin-current for all the states in the single-particle multiplet.