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 for Heisenberg spin chains into the world of integrable long-range models with spins. From the transfer matrix with a diagonal twist we construct Heisenberg-style symmetries (Bethe algebra) that refine the usual hierarchy of commuting Hamiltonians (quantum determinant) of the spin-Calogero–Sutherland model. We compute the first few of these new conserved charges explicitly, and diagonalise them by Bethe Ansatz inside each irreducible Yangian representation. This yields a new eigenbasis for the spin-Calogero–Sutherland model that generalises the Yangian Gelfand–Tsetlin basis of Takemura–Uglov. The Bethe-Ansatz analysis involves non-generic values of the inhomogeneities. 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.
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 for Heisenberg spin chains into the world of integrable long-range models with spins. From the transfer matrix with a diagonal twist we construct Heisenberg-style symmetries (Bethe algebra) that refine the usual hierarchy of commuting Hamiltonians (quantum determinant) of the spin-Calogero–Sutherland model. We compute the first few of these new conserved charges explicitly, and diagonalise them by Bethe Ansatz inside each irreducible Yangian representation. This yields a new eigenbasis for the spin-Calogero–Sutherland model that generalises the Yangian Gelfand–Tsetlin basis of Takemura–Uglov. The Bethe-Ansatz analysis involves non-generic values of the inhomogeneities. 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.
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 for Heisenberg spin chains into the world of integrable long-range models with spins. From the transfer matrix with a diagonal twist we construct Heisenberg-style symmetries (Bethe algebra) that refine the usual hierarchy of commuting Hamiltonians (quantum determinant) of the spin-Calogero-Sutherland model. We compute the first few of these new conserved charges explicitly, and diagonalise them by Bethe ansatz inside each irreducible Yangian representation. This yields a new eigenbasis for the spin-Calogero-Sutherland model that generalises the Yangian Gelfand-Tsetlin basis of Takemura-Uglov. The Bethe-ansatz analysis involves non-generic values of the inhomogeneities. 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.
The Haldane-Shastry spin chain is a prototypical integrable model with long-range interactions, notable for hosting quasiparticles with fractional statistics and serving as a discrete analogue of a conformal field theory. Its remarkable simplicity is closely tied to a full Yangian spin symmetry. While the highest-weight states for this symmetry are known explicitly, a systematic treatment of the descendant states, needed for the computation of various physical quantities, has remained incomplete. In this work, we provide a detailed construction of these descendants in terms of the algebraic Bethe ansatz following recent work of Ferrando et al. In the limit of extreme twist, it includes the Gelfand-Tsetlin basis. As an application, we derive product and determinant formulae for norms and overlaps of these states.
Integrable models are physical models for which some quantities can be exactly obtained, without use of perturbation theory. Those very special models are source of an increasing interest in theoretical physics. The direct applications in condensed matter physics and the subtle links evidenced more recently with some supersymmetric gauges theories motivated the development of complex mathematical tools. Among these, Bethe ansatz played an important role, and provides an efficient approach for diagonalizing a lot of models of various nature. The first chapter of this thesis is devoted to the introduction to the two approaches of the Bethe ansatz, said “coordinate” and “algebraic”, in the context of the XXX Heisenberg spin chain and a continuous spin generalization of the Totally Asymmetric Simple Exclusion Process, the so called Zero-range Chipping model with factorized steady state (ZCM). The second chapter is devoted to the Modified Algebraic Bethe Ansatz in the context of the periodic XXX chain. This modified ansatz is proposed for solving the spectral problem of the open spin chain, for which the usual ansatz fails. The scalar product of the obtained modified Bethe states is studied. The third chapter concerns the resolution of the identity and the inverse functional problem. An expression for the spin states in terms of Bethe states est presented for the ZCM, and an expression for the resolution of the identity in term of Bethe states for the infinite XXZ chain is proved, involving in both cases the contribution of bound states. At last, the fourth chapter concerns determinant representations in the Bethe ansatz. An expression for the “matrix elements of the particle number operator” for the delta-Bose gas in terms of a determinant is proved, and some integral representations for the Izergin-Korepin and Slavnov determinants are investigated, then establishing a new formal link between these two determinant representations.
In this thesis we theoretically investigate the close relation between spin chains and Yangians. First, we investigate the Haldane-Shastry model, which symmetry algebra is a representation of the Yangian. We show this in full detail for the spin-1/2 model. We also give an argument why a spin-1 chain with Yangian symmetry cannot exist. \nAfter that, we turn to integrable spin chains which are constructed using the quantum inverse scattering method. Using this method, an integrable spin chain can be constructed from each so called R-matrix, which is a solution of the Yang-Baxter equation. Our main result is the construction of an R-matrix that is invariant under the adjoint representation of SU(n), which is found using an intertwining operator for the corresponding Yangian representation. The integrable\nspin chain that corresponds to this R-matrix is non-Hermitian, making its physical interpretation unclear. These results have been published recently as a preprint under arXiv:1606.02516. \nFinally, integrable spin chains with defects are studied. We present the Hamiltonian for the Heisenberg model and the Babujian-Takhtajan model with defects, such that these models keep there integrability while staying Lie algebra invariant.
This chapter addresses the long-standing challenge of constructing the Bethe wave function in large but finite volumes via the diagonalization of the transfer matrix within the spin representation. While the coordinate Bethe ansatz provides effective tools for specific models, it becomes increasingly cumbersome in systems with internal degrees of freedom, such as the Lieb–Liniger model. The algebraic Bethe ansatz offers a more general and systematic framework, particularly well suited for handling nested structures. Spin chains serve as a natural setting for this formalism, offering both mathematical richness and physical relevance as models of interacting quantum spins. Starting from the Heisenberg spin chain, the chapter introduces the algebraic structure underpinning the method, including the R -matrix formulation, quantum group symmetries, and the construction of the transfer matrix. To connect the algebraic formalism with thermodynamic behavior, the chapter explores the string hypothesis, which organizes solutions to the Bethe equations into regular complex patterns. This leads naturally to the Bethe–Takahashi equations, which govern the thermodynamic limit of integrable spin chains. These tools enable a tractable analysis of excited states and physical observables, establishing a foundation for applying the algebraic Bethe ansatz to a broader class of quantum integrable systems.
In this thesis the nested algebraic Bethe ansatz technique is applied to various orthogonal and symplectic closed and open spin chain models. Each spin chain considered is regarded as a representation of an underlying quantum group algebra, and expressions for eigenvectors of transfer matrices associated to these models are constructed using the algebra relations, reducing the problem to a set of Bethe equations. The specific models considered are the Ol'shanskii twisted Yangian spin chain, where gl_n bulk symmetry is broken to orthogonal or symplectic symmetry; the MacKay twisted Yangian spin chain, an open spin chain with bulk orthogonal or symplectic symmetry and various boundary types; and the q-deformed orthogonal or symplectic closed spin chain. For the first and third cases, a closed 'trace formula' expression for the eigenvector is also provided.