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The asymmetric simple exclusion process with open boundaries, which is a very simple model of out-of-equilibrium statistical physics, is known to be integrable. In particular, its spectrum can be described in terms of Bethe roots. The large…

Statistical Mechanics · Physics 2009-09-25 Damien Simon

We present a procedure to extract the generalised eigenvectors of a non-diagonalisable matrix by considering a diagonalisable perturbation of it and computing the non-diagonalisable limit of its eigenvectors. As an example of this process,…

High Energy Physics - Theory · Physics 2022-07-13 Juan Miguel Nieto García , Leander Wyss

We describe the Algebraic Bethe Ansatz for the spin-1/2 XXX and XXZ Heisenberg chains with open and periodic boundary conditions in terms of tensor networks. These Bethe eigenstates have the structure of Matrix Product States with a…

Strongly Correlated Electrons · Physics 2012-07-23 Valentin Murg , Vladimir E. Korepin , Frank Verstraete

The task of calculating operator dimensions in the planar limit of N=4 super Yang-Mills theory can be vastly simplified by mapping the dilatation generator to the Hamiltonian of an integrable spin chain. The Bethe ansatz has been used in…

High Energy Physics - Theory · Physics 2012-08-27 Curtis G. Callan, , Jonathan Heckman , Tristan McLoughlin , Ian Swanson

We construct the coordinate Bethe ansatz for all eigenstates of the open spin-$\frac12$ XXZ chain that fulfill the phantom roots criterion (PRC). Under the PRC, the Hilbert space splits into two invariant subspaces and there are two sets of…

Statistical Mechanics · Physics 2022-02-01 Xin Zhang , Andreas Kluemper , Vladislav Popkov

We provide a basis transformation that inverts the coordinate Bethe Ansatz. It is widely believed that the Bethe Ansatz is complete, based on numerical evidence and combinatorial arguments. We present a constructive and comprehensive…

Mathematical Physics · Physics 2025-06-18 Eric I. Corwin , Nikolaus Elsaesser , Axel Saenz

The implementation of the algebraic Bethe ansatz for the XXZ Heisenberg spin chain, of arbitrary spin-$s$, in the case, when both reflection matrices have the upper-triangular form is analyzed. The general form of the Bethe vectors is…

Exactly Solvable and Integrable Systems · Physics 2017-08-21 N. Manojlović , and I. Salom

The Bethe Ansatz is a method for constructing exact eigenstates of quantum-integrable spin chains. Recently, deterministic quantum algorithms, referred to as "algebraic Bethe circuits", have been developed to prepare Bethe states for the…

Quantum Physics · Physics 2025-07-29 Roberto Ruiz , Alejandro Sopena , Esperanza López , Germán Sierra , Balázs Pozsgay

We consider the spectrum of correlation lengths of the spin-$\frac{1}{2}$ XXZ chain in the antiferromagnetic massive regime. These are given as ratios of eigenvalues of the quantum transfer matrix of the model. The eigenvalues are…

Statistical Mechanics · Physics 2017-08-15 Maxime Dugave , Frank Göhmann , Karol K. Kozlowski , Junji Suzuki

The scalar products, form factors and correlation functions of the XXZ spin chain with twisted (or antiperiodic) boundary condition are obtained based on the inhomogeneous $T-Q$ relation and the Bethe states constructed via the off-diagonal…

Mathematical Physics · Physics 2020-03-10 Yi Qiao , Zhirong Xin , Xiaotian Xu , Kun Hao , Tao Yang , Junpeng Cao , Wen-Li Yang

The structure of Bethe vectors for generalised models associated with the XXX- and XXZ-type R-matrix is investigated. The Bethe vectors in terms of two--component and multi--component models are described. Consequently, their structure in…

Mathematical Physics · Physics 2017-08-02 J. Fuksa

We derive exactly scalar products and form factors for integrable higher-spin XXZ chains through the algebraic Bethe-ansatz method. Here spin values are arbitrary and different spins can be mixed. We show the affine quantum-group symmetry,…

Statistical Mechanics · Physics 2011-07-06 Tetsuo Deguchi , Chihiro Matsui

We study the trigonometric quantum spin-Calogero-Sutherland model, and the Haldane-Shastry spin chain as a special case, using a Bethe-ansatz analysis. We harness the model's Yangian symmetry to import the standard tools of integrability…

Mathematical Physics · Physics 2025-03-27 Gwenaël Ferrando , Jules Lamers , Fedor Levkovich-Maslyuk , Didina Serban

In this paper we study the spectrum of the spin-1 Temperley-Lieb spin chain with integrable open boundary conditions. We obtain the eigenvalue expressions as well as its associated Bethe ansatz equations by means of the coordinate Bethe…

Exactly Solvable and Integrable Systems · Physics 2013-06-25 G. A. P. Ribeiro , A. Lima-Santos

The term Bethe Ansatz stands for a multitude of methods in the theory of integrable models in statistical mechanics and quantum field theory that were designed to study the spectra, the thermodynamic properties and the correlation functions…

Mathematical Physics · Physics 2024-12-31 Frank Göhmann

We investigate symmetry properties of the Bethe ansatz wave functions for the Heisenberg $XXZ$ spin chain. The $XXZ$ Hamiltonian commutes simultaneously with the shift operator $T$ and the lattice inversion operator $V$ in the space of…

Statistical Mechanics · Physics 2009-10-31 Jae Dong Noh , Deok-Sun Lee , Doochul Kim

The anomalous dimensions of local single trace gauge invariant operators in N=4 supersymmetric Yang-Mills theory can be computed by diagonalizing a long range integrable Hamiltonian by means of perturbative asymptotic Bethe ansatz. This…

High Energy Physics - Theory · Physics 2009-12-10 Joao Penedones , Pedro Vieira

In this note, we study the eigenvectors and the scalar products the integrable long-range deformation of a XXX spin chain which is solved exactly by algebraic Bethe ansatz, and it coincides in the bulk with the Inozemtsev spin chain. At the…

High Energy Physics - Theory · Physics 2015-06-15 D. Serban

We consider twist $J$ operators with spin $S$ in the $sl(2)$ sector of $\mathcal N=4$ SYM. The small spin expansion of their anomalous dimension defines the so-called slope functions. Much is known about the linear term, but the study of…

High Energy Physics - Theory · Physics 2015-06-19 Matteo Beccaria , Guido Macorini

We present the construction of the full set of eigenvectors of the open ASEP and XXZ models with special constraints on the boundaries. The method combines both recent constructions of coordinate Bethe Ansatz and the old method of matrix…

Statistical Mechanics · Physics 2015-05-28 N. Crampe , E. Ragoucy , D. Simon