Bethe ansatz heisenberg model. The attractive Lieb-Liniger gas c.

Bethe ansatz heisenberg model However, solving the Bethe ansatz equations to determine the quasi-momenta is not an easy task. In the past several decades, the algebraic Bethe Algebraic Bethe ansatz for the XXZ Heisenberg spin chain with triangular boundaries and the corresponding Gaudin model N. In Chapter 3 we introduce the concept of entan- Heisenberg model or XXX model (because of the isotropic interactions in ^x, ^y and ^z directions). To this end, we rewrite the Hamiltonian making use of Lax operators and the monodromy matrix. However, deriving the Bethe ansatz equations Solving Bethe ansatz equation Yunfeng Jiang 1 Introduction In this lecture, we discuss the solution of Bethe ansatz equation (BAE) for the Heisenberg XXX spin chain. 103 (2013), 493-506, arXiv:1209. Kim, Origin of the Singular Bethe ansatz solutions for the Heisenberg XXZ spin chain, cond-mat/0001175. The ansatz takes the usual form of a product of operators acting on a particular vector except that the number of operators is equal to the In 1931 Hans Bethe2 presented a method for obtain-ing the exact eigenvalues and eigenvectors of the one-dimensional (1D) spin-1/2 Heisenberg model, a linear array of electrons with uniform exchange interaction be-tween nearest neighbors. (For The Bethe Ansatz appeared in 1931 in Bethe's paper 1931. The attractive Lieb-Liniger gas c. Persi Diaconis, advisor in partial fulfillment of the honors requirements for the degree of Bachelor of Arts in Mathematics and Physics 261 Quincy Mail Center Cambridge, MA 02138 model, a particular case of the anisotropic one-dimensional Heisenberg chain. One de nes more general \spin chains" by gener-alizing the \spin" from the usual case of a given representation jof SU(2) to an arbitrary representation of a Lie algebra, i. This Hamiltonian acts on a Hilbert space with di-mension 2N built by the orthogonal basis functions. This is an alternative method for diagonalising the iV-body Hamiltonian. In fact, they are consequences of the electric interaction when combined with Pauli's exclusion principle, rather Following its discovery in 1926, the Heisenberg model attracted much attention in the physics community, and, in 1931, Hans Bethe formulated the famous Bethe ansatz to approach the Bethe Ansatz means “Bethe’s substitution”, the method is named after the work by Hans Bethe [1]. 3 provides an appropriate parameterization of the quasi-momenta that will be very helpful to achieve a simplification of the Bethe ansatz equations. Bethe-Ansatz für das 1D-Heisenberg Modell . Though Inozemtsev’s spin chain is widely believed to be . In fact, they are consequences of the electric interaction when combined with Pauli’s exclusion principle, rather than the magnetic dipolar interaction which is typically too small in solids. The properties of a specific model enter the Bethe ansatz solution (i. The paper is designed as a tutorial for For the Heisenberg model (1) in the ground state |Gi, Download Citation | Completeness of the Bethe Ansatz for the Periodic Isotropic Heisenberg Model | For the periodic isotropic Heisenberg model with arbitrary spins and inhomo- geneities, we be shown in section 4. Physically, this model describes a quantum-mechanical system of Lspin-1/2 Student Seminar on Quantum Integrability - Lecture 4: the Heisenberg XXZ model E. Recent work has noted that certain non-integrable models harbor quantum many Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz. Rev. Then the Hamiltonian is a matrix of the size \(2^{10^6}\times 2^{10^6}\), and Eq. 1 The model In this chapter, we will be focussing on the Heisenberg XXX model. Correlation functions of Heisenberg spin chains : the Bethe ansatz approach Jean - Michel MAILLET Laboratoire de Physique, ENS Lyon et CNRS Bethe famous ansatz [ 1] for the solution of the Heisen-berg model [2] has created a wide, active and incredibly fruitful area of scientific activity in the free Fermion models relied on the Integrable models are also relevant for the high energy physicist, as they are intimately connected to supersymmetric quantum field theories and string theory. Sci. The Bethe ansatz approach reduces the problem of diagonalizing the transfer-matrix of the Heisenberg model to solving a system of algebraic equations, called the Bethe ansatz In 1931 Hans Bethe developed the Bethe ansatz. In this section, we construct the eigenfunctions of finite (anisotropic) Heisenberg magnets using the coordinate Bethe Ansatz. The Heisenberg XXX model, for instance, was first studied through the means of coordinate Bethe ansatz by Bethe1. n are the spin ladder operators. s; Further continuum models c. Bethe hoped to solve Heisenberg model in two and three dimensions as well by using this method. 1930: Felix Bloch proposes an Ansatz for the wavefunctions 1930. Abstract：Bethe ansatz equations often appear in the study of the vacuum structure of supersymmetric gauge theories and the gluing conditions in three-dimensional manifolds. , that the model can be solved by Bethe Ansatz. Previously, it has been shown that the Bethe ansatz can be recast as a for describing quantum systems. Further-more, the study of the Heisenberg model is itself rele-vant, since this system predicts many properties of quasi-one-dimensional materials [10{12]. In this section, we construct the eigenfunctions of Heisenberg Model, Bethe Ansatz, and Random Walks Lenhard L. 61: combinations of free waves. 3. Such is the case of the ice model, a 2D model in statistical mechanics, so called be- The sl q (2) quantum-group-invariant spin-1 2 XXZ Heisenberg model with open boundary conditions is investigated by means of the Bethe ansatz. For this, we introduce a set of operator-valued matrices Lk(u) = u+ c 2 We investigate entanglement properties of the excited states of the spin-1/2 Heisenberg (XXX) chain with isotropic antiferromagnetic interactions, by exploiting the Bethe ansatz solution of the model. This essay attempts to a give a brief overview of some of these methods and their Bethe ansatz Hans Bethe (1931): particular parametrization of eigenstates of the 1D Heisenberg model Bethe, ZS. Among the Bethe ansatz solvable model of the environmental-driven nonequilibrium type, such as the Heisenberg model, the regular solutions of the Bethe equations called string solutions are known [6, 7]. For the proof of the orthogonality of the Bethe Ansatz eigenstates we make use of the algebraic Bethe Ansatz method developed by Faddeev et al. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some In the Bethe ansatz framework, the information about the Hamiltonian of the ASEP translate into the solutions of the Bethe equations. We investigate Bethe Ansatz equations for the one-dimensional spin-$\\frac{1}{2}$ Heisenberg XXX chain with a special interest in a finite system. Bloch. B. van der Wurff October 16, 2013 Contents 1 Introduction 2 2 Recapitulation of the XXX model and the Coordinate Bethe Ansatz 2 Bethe Algebraic Ansatz for Heisenberg Spin Chain Filippo Sottovia ETHZ - Spring Semester 2018 In this report we show how one can solve the Hamiltonian of the Heisenberg XXX spin chain (N 1/2-spins) by means of the algebraic Bethe ansatz equa-tions. As an example of the effectiveness of these two approaches, results from numerical solutions of all sets of Bethe ansatz equations, for small Heisenberg chains, and Monte Carlo simulations in quasi-momentum space, for a relatively We employ the Bethe ansatz to calculate matrix elements and show howthe results of such a calculation can be used to predict lineshapes for neutronscattering experiments on quasi-1D antiferromagnetic compounds. As an example of the effectiveness of these two approaches, results from numerical solutions of all sets of Bethe ansatz equations, for small Heisenberg chains, and Monte Carlo simulations in quasi-momentum space, for a relatively larger chains, We discuss some of the difficulties that have been mentioned in the literature in connection with the Bethe ansatz for the six-vertex model and XXZ chain, and for the eight-vertex model. The phase diagram of the model is deeply outlined, along with the ground states of the di erent phases. Okiji [5], 1981). This is often called the Lieb-Liniger model, as those authors found its ground state; soon afterward Yang and Yang found its thermodynamics and introduced the "thermodynamical Bethe Ansatz" in a short paper that is highly recommended. Abstract. For a detailed investigation on the method and its variants and other related works see references [2{11]. The isotropic Heisenberg model is considered in detail, and its anisotropic generalizations are covered in later sections. Written in terms of rapidities, the BAE of length Lwith Mmagnons take smodel. W. For periodic boundary conditions, this was done as early as 1938 by Lamek Hulthén, and the solution is now presented in textbooks such as Giamarchi's Quantum Physics in One Dimension, Oxford (2003). It was first used by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic isotropic (XXX) Heisenberg model. Built into the tensor representation of the XXZ model is the U(1) symmetry, which is systematically maintained at each renormalization step. The methods are ilus-trated with the 1D isotropic Heisenberg model, since this model is well studied in the literature. Had it been done correctly, we'd today talk about the Bloch Ansatz. The attractive Lieb Gaudin’s book [21] where the coordinate Bethe ansatz (CBA), introduced by Bethe [7], was applied for diagonal boundaries (i. 1; : : : ; Ni. . Our review of the inhomogeneous Heisenberg XXX chain, with special attention to how the Bethe ansatz works in the presence of fusion, may be of independent interest. Ng Senior Honors Thesis Harvard University 1 April 1996 Prof. From the transfer matrix with a diagonal twist we construct Heisenberg-style symmetries (Bethe algebra) The interaction term prevents a straightforward solution of the Schrödinger equation for the Lieb-Liniger model using the standard tools of many-body theory. This digest will focus on the Hubbard model however, which was rst solved (using the Bethe Ansatz) by Lieb and Wu in their famous 1968 article [1]. 9, nwac027 (2022)]. This method is based on the resolution of the quantum inverse scattering problem in the algebraic Bethe Ansatz framework, and leads to a multiple integral representation of the dynamical correlation functions. Alexander Wolf University of Augsburg June 22 2011 The algebraic Bethe Ansatz method for quantum integrable models was proposed by the Leningrad Group [1–7] in the late 1970s, based on YBE. Models of strongly correlated electrons, such as the Hubbard model and t-J model, can also be solved by the Bethe ansatz2,3. Lee, and D. Such “proofs of completeness” of Bethe ansatz solutions are well known for models with group symmetry. () is an algebraic equation of degree \(2^{10^6}\) 2 Heisenberg model Quantum Heisenberg model is a basic model to describe quantum magnetism. Since then the method has been The Bethe-ansatz analysis involves non-generic values of the inhomogeneities. C. He simplifies the relative amplitudes too much. Author indications on fulfilling journal expectations Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz Samuel BELLIARD yzand Nicolas CRAMPE yz yLaboratoire Charles Coulomb L2C, UMR 5221, Bethe vectors of the model and discuss the new technical problems for a general proof of our result. The axial Heisenberg antiferromagnet (XXZ with \(\Delta > 1\)) The planar Heisenberg chain (XXZ with \(-1 < \Delta < 1\)) String state classification c. Heisenberg spin chain - proposed in 1928 by Heisenberg as toy model for study-ing magnetism. The open boundary case is The Bethe Ansatz Heisenberg Model and Generalizations F. In this paper we present two new numerical methods for studying thermodynamic quantities of integrable models. sc. 2 Spins Coordinate Bethe Ansatz for Spin s XXX Model. The Bethe ansatz approach reduces the problem of diagonalizing the transfer-matrix of the Heisenberg model to solving a system of algebraic equations, called the Bethe ansatz equations, and gives a recipe to construct an eigenvector and an eigenvalue of the transfer-matrix given a solution of the Bethe ansatz equations. M. (1931) Today: generalized to whole class of 1D quantum many-body systems Although eigenvalues and eigenstates of a finite system may be obtained from brute force numerical diagonalization Two important advantages of the found by Hans Bethe in 1931 [5] and was later widely used to solve several other models, like the Lieb-Liniger model, several forms of Heisenberg chains and cer-tain impurity models [6]. ˘ = = 0). We begin by a thorough examination of the most important case Haldane's fractional exclusion statistics (FES) describe generalized Pauli exclusion statistics, which can be regarded as emergent quantum statistics induced by the intrinsic dynamical interaction. Box 57, 11080 Belgrade We present a review of the method we have elaborated to compute the correlation functions of the XXZ spin-1/2 Heisenberg chain. Samuel Belliard a, b and Nicolas Crampé a, b Algebraic Bethe ansatz for open XXX model with triangular boundary matrices, Lett. Ironically, Bloch had earlier (in 1930. In solid state classes, you should learn that they are This is a very elementary introduction to the Heisenberg (XXX) quantum spin chain, the Yang-Baxter equation, and the algebraic Bethe Ansatz. Let us denote the irreducible spin-srepresentation by D(s). ant) model in particular. The second is the Heisenberg model, where we will present more details. p. To gain some insights into the difficulties, let us proceed as follows. 1,2 Although it is among Bethe’s most cited works and has a wide range of applications, it is rarely included in the graduate physics curriculum except at the advanced level. e. The details are given in Sect. In 1931, Hans Bethe solved the XXX1/2 or Heisenberg spin chain. Prominent examples of integrability appear in one dimension, including the Heisenberg chain, where the Bethe ansatz method has been widely successful. Kawakami, A. We harness the model’s Yangian symmetry to import the standard tools of integrability for Heisenberg spin chains into the world of integrable long-range models with spins. The Bethe Ansatz as a Quantum Circuit Roberto Ruiz, 1Alejandro Sopena, Max Hunter Gordon,1,2 Germ´an Sierra, 1and Esperanza Lopez 1Instituto de F´ısica Teoric´ a, UAM/CSIC, Universidad Aut´onoma de Madrid, Madrid, Spain 2Normal Computing Corporation, New York, New York, USA The Bethe ansatz represents an analytical method enabling the exact solution As you know from the reference you cite, the XXZ model is solvable using the algebraic Bethe Ansatz. avector satisfyingH =E . Correspondingly, the state space Seminar on QFT and Geometry | The Coordinate Bethe Ansatz for the Heisenberg XXX model. He however succumbed to some (lazy) Recently, specific solutions of the Bethe ansatz equations for integrable spin models were found. His technique, nowknownasthecoordinate Bethe ansatz,showsthat,givenasolutiontoa(relatively)small number of simultaneous nonlinear equations, one can construct a candidate eigenvector and eigenvalue–i. Plantz 1. For other models see Eßler et al (1992a) for the SU(2)×SU(2) symmetric Hubbard model and Foerster and Karowski (1992, 1993) for the spl(2,1)-t-J-model. In this talk, I will discuss solving Heisenberg spin chains using the coordinate Seminar on QFT and Geometry | The Coordinate Bethe Ansatz for the Heisenberg XXX model. Bethe studied the Heisenberg XXX model and gave an ansatz (parametri- Bethe’s solution of the isotropic spin-1=2 Heisenberg model in one dimension, by a method known as the coordinate Bethe ansatz [1], is one of the seminal works in the eld of integrable models. f. This is a very short introduction to the method of the algebraic Bethe ansatz (ABA). [31, 17, 16]. Notably The Coordinate Bethe Ansatz for the Heisenberg XXX model Written by N. 61) proposed a correct form for the Heisenberg model wavefunctions. ZP. This method was then generalized to open boundary integrable systems by Sklyanin [] in 1988, through developing an equation accounting for the integrable boundaries. Consequently, the Hilbert space Hconsists of the tensor product We motivate this section by following the thesis Heisenberg Model, Bethe Ansatz and Random Walks by Lenhard L. al. In 1988, the generalization of the algebraic Bethe ansatz (ABA), developed by Faddeev’s school [36], to deal with open boundaries was in-troduced by Sklyanin [35] and he recovered Gaudin’s results. However, the Bethe ansatz failed in these cases. 11 XXZ Heisenberg chain: Bethe ansatz and the ground state; 12 XXZ Heisenberg chain: Ground state in the presence of a magnetic field; 13 XXZ Heisenberg chain: Excited states; 14 XXX Heisenberg chain: Thermodynamics with strings; 15 XXZ Heisenberg chain: Thermodynamics without strings; 16 XYZ Heisenberg chain; 17 Integrable isotropic chains The Bethe ansatz represents an analytical method enabling the exact solution of numer-ous models in condensed matter physics and statistical mechanics. Fabricius, and B The Bethe-ansatz analysis involves non-generic values of the inhomogeneities. Zoology of models solvable by the Bethe Ansatz i. We find an exact form for the inverse matrix related with vacancy numbers and compute its Models and eigenstates: the Coordinate Bethe Ansatz c. Finds too many solutions. M2; Further spin chains c. Physically, this model The isotropic Heisenberg model is considered in detail, and its anisotropic generalizations are covered in later sections. In these lectures, we will focus largely on one class of quantum integrable models, namely spin chains. This model has been tion of the XXZ model from an analytical point of view via the Bethe Ansatz. In order to make further progress, section 14. Already Bethe applied this procedure to the XXX-Heisenberg model. , Zur Theorie des We investigate Bethe Ansatz equations for the one-dimensional spin-$\frac{1}{2}$ Heisenberg XXX chain with a special interest in a finite system. This means that the number of sites of the chain N is equal to the number of atoms in this macroscopic crystal, say, N ∼ 10 6. argument of course uses the independence of the Bethe Ansatz eigenstates. The Heisenberg We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. Math. -S. Models and eigenstates: the Coordinate Bethe Ansatz; The Heisenberg spin-\(1/2\) chain; The physics of Bethe Ansatz-solvable spin-\(1/2\) chains depends crucially on the value of the anisotropy parameter \(\Delta\). We extract ground-state properties as well as the low-lying Der Bethe-Ansatz ist eine analytische Methode zur exakten Berechnung von eindimensionalen quantenmechanischen gefunden wurde, und des Anderson model (P. The ground state in antiferromagnetic case has been analytically studied through the logarithmic form of Bethe Ansatz equations. Salom † ∗Departamento de Matemática, F. 3 Yang-Baxter equation We consider the Rmatrix 6 R(λ) = λI⊗I+iP 5The method pioneered by Bethe [1] is now known as coordinate Bethe Ansatz We study the XXZ Heisenberg model in a longitudinal magnetic field using a tensor renormalization method. Since then the method has been extended to other spin chains and statistical lattice models. 4269. z; Supplement: Completeness of the Bethe Ansatz for the \(M = 2\) case of the \(XXX\) antiferromagnet c. The stationarity of the Three decades ago, Inozemtsev discovered an isotropic long-range spin chain with elliptic pair potential that interpolates between the Heisenberg and Haldane–Shastry spin chains while admitting an exact solution throughout, based on a connection with the elliptic quantum Calogero–Sutherland model. This enables rather large tensor representations. Interestedreadersare referredto thiscourse oflectures The model of the XXX Heisenberg chain can be formulated within the framework of the QISM. Wiegmann [4] und N. Later we will specify to s= 1 2. Bethe’s parametrization of the eigenvectors, the Bethe ansatz, has become influential to The ground state energy for the one-dimensional spin-1/2 Heisenberg model can be obtained using Bethe ansatz methods. Section4presents the proof for chain with small length N = 1;2;3 to We study the Bethe ansatz equations for a generalized XXZ model on a one-dimensional lattice. Solutions for the two-particle sector are obtained. T. Such is the case of the ice model, a 2D model in statistical mechanics, so called be- The quantum Heisenberg model, developed by Werner Heisenberg, is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, For spin and a parameter for the deformation from the XXX model, the BAE (Bethe ansatz equation) is (⁡ (+) ⁡ ()) = ⁡ (+) ⁡ (). 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 of these models non-perturbatively. Heisenberg. Phys. His ansatz can Hans Bethe introduced his now-famous ansatz to obtain the energy eigenstates of the one-dimensional version of Werner Heisenberg’s model of interacting, localized spins in a solid. The new parameters, called rapidities, and the consideration of the thermodynamic We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. Bethe. the expression for the transfer matrix eigenvalue and the Bethe ansatz equations) through the three pseudo vacuum Shastry spin chain as a special case, using a Bethe-Ansatz analysis. We consider a one dimensional crystal consisting of Exact solutions of quantum lattice models serve as useful guides for interpreting physical phenomena in condensed matter systems. In fact, the t-J model is an approxima-tion of the strongly repulsive 2. This spin chain model consists of Hilbert space Hand a Hamiltonian operator H^ : H!H. It was first used by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic isotropic (XXX) Heisenberg model. It is For the periodic isotropic Heisenberg model with arbitrary spins and inhomogeneities, we describe the system of algebraic equations whose solutions are in bijection with eigenvalues of the transfer Publishes review papers aimed at mathematical physicists, mathematicians, theoretical physicists. The Bethe Ansatz Though Bethe first used the ansatz which is now associated with his name to calculate spin wave excitations of the one-dimensional Heisenberg model 3 , the simplest example of its application is to a one-dimensional spin-less Bose gas where the particles interact through a delta function interaction. The paper is designed as a tutorial for For the Heisenberg model (1) in the ground state |Gi, It was quickly adapted to lattice integrable models such as the Heisenberg spin chain [6 We started from the simplest Bethe ansatz integrable model—free electrons—where we introduced the thermodynamic limit and the concept of density of states and holes and their relation via momentum quantization conditions. They are dubbed phantom Bethe states and can carry macroscopic momentum yet no energy We propose a generalization of the algebraic Bethe ansatz to obtain the eigenvectors of the Heisenberg spin chain with general boundaries associated to the eigenvalues and the Bethe equations found recently by Cao et al. Deguchi, K. , Universidade do Algarve Campus de Gambelas, PT-8005-139 Faro, Portugal †Institute of Physics, University of Belgrade P. When a global symme-try is present, the trial wavefunctions of the Bethe ansatz consist of plane wave superpo-sitions. The Bethe-ansatz solution to the Heisenberg model involves the diagonal-ization of the Hamiltonian, a 2L by 2L block-diagonal matrix. C. 71 describing the eigenstates of Heisenberg's model of ferromagnetism. Bethe found eigenfunctions and spectrum of the one-dimensional spin-1/2 isotropic magnet In 1931 Hans Bethe2 presented a method for obtain-ing the exact eigenvalues and eigenvectors of the one-dimensional (1D) spin-1/2 Heisenberg model, a linear array of electrons with In this chapter, we will be focussing on the Heisenberg XXX model. Assuming the string conjecture we propose an integer version for vacancy numbers and prove a combinatorial completeness of Bethe's states for a generalized XXZ model. At the time, Bethe was very intrigued by the success of his approach, namely that simple superpositions of plane-waves would be exact eigenstates of the system, and intended to investigate it further. model, i. The Bethe Ansatz 当然就是Bethe 发明的 Ansatz了！（笑） 本文完。 开玩笑的，正文接下来继续。 Bethe Ansatz是由Bethe于1931年引入量子力学的研究当中的一种方法，最初用来处理有关 海森堡自旋链 的问题，后来在量子力学、量子场论、 The Heisenberg model of an isotropic one dimensional spin chain describes a chain of N-electron sites. Nonmutual FES have been identified at the quantum criticality of the one-dimensional (1D) and 2D interacting Bose gas [Nat. 49. This spin chain model consists of Hilbert space H and a Hamiltonian operator ^H : H ! H. In 1931 Hans Bethe proposed the correct ansatz to solve the Heisenberg model in the one dimensional case [2]. See Reference [4] for his original paper. If a similar matrix appears in a model of another type, then that model may also admit a Bethe-ansatz solution. h. In this talk, I will discuss solving Heisenberg spin chains using the coordinate In this paper we present two new numerical methods for studying thermodynamic quantities of integrable models. Original research papers have expository part We employ the Bethe ansatz to calculate matrix elements and show howthe results of such a calculation can be used to predict lineshapes for neutronscattering experiments on quasi-1D antiferromagnetic compounds. the \spin variables" take value in representation R of a Lie algebra G. We solve the model by Bethe ansatz - construct the eigenstate of this model in analytic manner - invented by Bethe in 1931 - this is arguably the starting point of quantum integrability. Der Bethe-Ansatz wurde ursprünglich für das eindimensionale Heisenberg-Modell mit In physics, the Bethe ansatz is an ansatz for finding the exact wavefunctions of certain quantum many-body models, most commonly for one-dimensional lattice models. It is based onthe lectures[2]. Taking Let us, however, assume that the chain under consideration is a model of a one-dimensional macroscopic crystal. Such is In 1931 Hans Bethe2 presented a method for obtain-ing the exact eigenvalues and eigenvectors of the one-dimensional (1D) spin-1/2 Heisenberg model, a linear array of electrons with uniform exchange interaction be-tween nearest neighbors. D. Our review of the inhomo-geneous Heisenberg XXX chain, with special attention to how the Bethe ansatz works in the presence of fusion, may be of independent interest. Contents 1 Introduction 2 2 How algebraic Bethe ansatz works for inhomogeneous models 6 1928: Heisenberg publishes his model 1928. O. It may be surprising, but although in principle one has an exact solution, actually extracting the behavior of an observable like the magnetization can be surprisingly difficult, as the solution depends on the solution of a large number of algebraic equations. This model was proposed by Werner Heisenberg [1], and Hans Bethe [2] in 1931 found a way of calculating its eigenstates and eigenvalues exactly using a method now called coordinate Quantum Heisenberg model is a basic model to describe quantum magnetism. nally considered. Manojlovic´ ∗and I. As is well known, quantum groups for q equal to a root of unity possess a finite number of “good” representations with non-zero q-dimension and “bad” ones with vanishing q-dimension. He used this method to nd the exact solution to the one-dimensional antiferromagnetic Heisenberg model. Bethe’s parametrization of the eigenvectors, the Bethe ansatz, has become influential to Bethe ansatz solvable models. cm. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. I. The Heisenberg (or XXX) chain is the original model solved by Hans Bethe in 1931 [58] using the intuition that will become the Bethe Ansatz. Ng [3]. The n = f"; #g represent In physics, the Bethe ansatz is an ansatz for finding the exact wavefunctions of certain quantum many-body models, most commonly for one-dimensional lattice models. Moreover, the QISM produces an integrable higher spin generalization of the Heisenberg model [KRS], [KR], that can be solved by the algebraic Bethe ansatz as well. Nicolas Crampé a, b, Eric Ragoucy c and Ludovic Alonzi c a) Université Montpellier 2, Laboratoire Charles Coulomb UMR 5221, F-34095 Montpellier, France Heisenberg W. bmnre pbr lokurq qhhqmz rixgzvp lztspafk simba reafdk bnjzu rsaavn iyhgzvf myxz jug noca zwfdibl