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:1308.1584 · Journal of Statistical Mechanics Theory and Experiment

Integrable deformations of the XXZ spin chain

Niklas Beisert, Lucas Fiévet, Marius de Leeuw, Florian Loebbert

Published Sep 30, 2013• 24 citations
arXivHTMLPDF

Abstract

We consider integrable deformations of the XXZ spin chain for periodic and open boundary conditions. In particular, we classify all long-range deformations and study their impact on the spectrum. As compared to the XXX case, we have the z -spin at our disposal, which induces two additional deformations: the short-range magnetic twist and a new long-range momentum-dependent twist.

DOI
10.1088/1742-5468/2013/09/p09028

Related papers

Integrabilities of the long-range t-J models with twisted boundary conditions

James T. Liu, D. F. Wang

The integrability of the one-dimensional long-range supersymmetric t-J model has previously been established for both open systems and those closed by periodic boundary conditions through explicit construction of its integrals of motion. Recently the system has been extended to include the effect of magnetic flux, which gives rise to a closed chain with twisted boundary conditions. While the t-J model with twisted boundary conditions has been solved for the ground state and full energy spectrum, proof of its integrability has so far been lacking. In this paper we extend the proof of integrability of the long-range supersymmetric t-J model and its SU(m|n) generalization to include the case of twisted boundary conditions.

arXiv:solv-int/9610012•Physical review. B, Condensed matter•Feb 1, 1997•2 citations

Landscapes of integrable long-range spin chains

Rob Klabbers, Jules Lamers

We clarify how the elliptic integrable spin chain recently found by Matushko and Zotov (MZ) relates to various other known long-range spin chains. We evaluate various limits. More precisely, we tweak the MZ chain to allow for a short-range limit, and show it is the XX model with q-deformed antiperiodic boundary conditions. Taking $q\to 1$ gives the elliptic spin chain of Sechin and Zotov (SZ), whose trigonometric case is due to Fukui and Kawakami. It, too, can be adjusted to admit a short-range limit, which we demonstrate to be the antiperiodic XX model. By identifying the translation operator of the MZ chain, which is nontrivial, we show that antiperiodicity is a persistent feature. We compare the resulting (vertex-type) landscape of the MZ chain with the (face-type) landscape containing the Heisenberg XXX and Haldane--Shastry chains. We find that the landscapes only share a single point: the rational Haldane-Shastry chain. Using wrapping we show that the SZ chain is the antiperiodic version of the Inozemtsev chain in a precise sense, and expand both chains around their nearest-neighbour limits to facilitate their interpretations as long-range deformations.

arXiv:2405.09718•ENLIGHTEN (Jurnal Bimbingan dan Konseling Islam)•Jan 1, 2024•1 citation

Generalized Integrable Boundary States in XXZ and XYZ Spin Chains

Xin Qian, Xin Zhang

We investigate integrable boundary states in the anisotropic Heisenberg chain under periodic or twisted boundary conditions, for both even and odd system lengths. Our work demonstrates that the concept of integrable boundary states can be readily generalized. For the XXZ spin chain, we present a set of factorized integrable boundary states using the KT-relation, and these states are also applicable to the XYZ chain. It is shown that a specific set of eigenstates of the transfer matrix can be selected by each boundary state, resulting in an explicit selection rule for the Bethe roots. We also derive the overlaps between integrable boundary states and Bethe states in the U(1)-symmetric cases.

arXiv:2601.16605•Jan 23, 2026•0 citations

Non-Abelian symmetries of the half-infinite XXZ spin chain

Pascal Baseilhac, S. Belliard

The non-Abelian symmetries of the half-infinite XXZ spin chain for all possible types of integrable boundary conditions are classified. For each type of boundary conditions, an analog of the Chevalley-type presentation is given for the corresponding symmetry algebra. In particular, two new algebras arise that are, respectively, generated by the symmetry operators of the model with triangular and special Uq(gl2)-invariant integrable boundary conditions.

arXiv:1611.05390•Nuclear Physics B•Jan 17, 2017•20 citations

Open Perturbatively Long-Range Integrable GL(N) Spin Chains

Niklas Beisert, Florian Loebbert

We construct the most general perturbatively long-range integrable spin chain with spins transforming in the fundamental representation of gl(N) and open boundary conditions. In addition to the previously determined bulk moduli we find a new set of parameters determining the reflection phase shift. We also consider finite-size contributions and comment on their determination.

arXiv:0805.3260•Advanced Science Letters•Jun 1, 2009•16 citations

Recursion relations for long-range integrable spin chains with open boundary conditions

Florian Loebbert

It is well known that integrable charges for short-range (e.g. nearest-neighbor) spin chains with periodic boundary conditions can be recursively generated by a so-called boost operator. In the past, this iterative construction has been generalized to periodic long-range spin chains as they appear in the context of the gauge/gravity correspondence. Here we introduce recursion relations for open long-range spin chain charges converting a short-range into a long-range integrable model.

arXiv:1201.0888•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 25, 2012•13 citations
We consider integrable deformations of the XXZ spin chain for periodic and open boundary conditions. In particular, we classify all long-range deformations and study their impact on the spectrum. As compared to the XXX case, we have the z -spin at our disposal, which induces two additional deformations: the short-range magnetic twist and a new long-range momentum-dependent twist.
The integrability of the one-dimensional long-range supersymmetric t-J model has previously been established for both open systems and those closed by periodic boundary conditions through explicit construction of its integrals of motion. Recently the system has been extended to include the effect of magnetic flux, which gives rise to a closed chain with twisted boundary conditions. While the t-J model with twisted boundary conditions has been solved for the ground state and full energy spectrum, proof of its integrability has so far been lacking. In this paper we extend the proof of integrability of the long-range supersymmetric t-J model and its SU(m|n) generalization to include the case of twisted boundary conditions.
We clarify how the elliptic integrable spin chain recently found by Matushko and Zotov (MZ) relates to various other known long-range spin chains. We evaluate various limits. More precisely, we tweak the MZ chain to allow for a short-range limit, and show it is the XX model with q-deformed antiperiodic boundary conditions. Taking q→1q\to 1q→1 gives the elliptic spin chain of Sechin and Zotov (SZ), whose trigonometric case is due to Fukui and Kawakami. It, too, can be adjusted to admit a short-range limit, which we demonstrate to be the antiperiodic XX model. By identifying the translation operator of the MZ chain, which is nontrivial, we show that antiperiodicity is a persistent feature. We compare the resulting (vertex-type) landscape of the MZ chain with the (face-type) landscape containing the Heisenberg XXX and Haldane--Shastry chains. We find that the landscapes only share a single point: the rational Haldane-Shastry chain. Using wrapping we show that the SZ chain is the antiperiodic version of the Inozemtsev chain in a precise sense, and expand both chains around their nearest-neighbour limits to facilitate their interpretations as long-range deformations.
We investigate integrable boundary states in the anisotropic Heisenberg chain under periodic or twisted boundary conditions, for both even and odd system lengths. Our work demonstrates that the concept of integrable boundary states can be readily generalized. For the XXZ spin chain, we present a set of factorized integrable boundary states using the KT-relation, and these states are also applicable to the XYZ chain. It is shown that a specific set of eigenstates of the transfer matrix can be selected by each boundary state, resulting in an explicit selection rule for the Bethe roots. We also derive the overlaps between integrable boundary states and Bethe states in the U(1)-symmetric cases.