The BCS model and the off-shell Bethe ansatz for vertex models (original) (raw)
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Algebraic Bethe ansatz for a discrete-state BCS pairing model
Physical Review B, 2002
We show in detail how Richardson's exact solution of a discrete-state BCS (DBCS) model can be recovered as a special case of an algebraic Bethe Ansatz solution of the inhomogeneous XXX vertex model with twisted boundary conditions: by implementing the twist using Sklyanin's K-matrix construction and taking the quasiclassical limit, one obtains a complete set of conserved quantities, H i , from which the DBCS Hamiltonian can be constructed as a second order polynomial. The eigenvalues and eigenstates of the H i (which reduce to the Gaudin Hamiltonians in the limit of infinitely strong coupling) are exactly known in terms of a set of parameters determined by a set of on-shell Bethe Ansatz equations, which reproduce Richardson's equations for these parameters. We thus clarify that the integrability of the DBCS model is a special case of the integrability of the twisted inhomogeneous XXX vertex model. Furthermore, by considering the twisted inhomogeneous XXZ model and/or choosing a generic polynomial of the H i s as Hamiltonian, more general exactly solvable models can be constructed. -To make the paper accessible to readers that are not Bethe Ansatz experts, the introductory sections include a self-contained review of those of its feature which are needed here. *
Quasi-classical descendants of disordered vertex models with boundaries
2002
We study descendants of inhomogeneous vertex models with boundary reflections when the spin-spin scattering is assumed to be quasi-classical. This corresponds to consider certain power expansion of the boundary-Yang-Baxter equation (or reflection equation). As final product, integrable su(2)-spin chains interacting with a long range with XXZ anisotropy are obtained. The spin-spin couplings are non uniform, and a non uniform tunable external magnetic field is applied; the latter can be obtained when the boundary conditions are assumed to be quasi-classical as well. The exact spectrum is achieved by algebraic Bethe ansatz. Having realized the su(2) operators in terms of fermions, the class of models we found turns out to describe confined fermions with pairing force interactions. The class of models presented in this paper is a one-parameter extension of certain Hamiltonians constructed previously. Extensions to su(n)-spin open chains are discussed.
The six-vertex model eigenvectors as critical limit of the eight-vertex model bethe ansatz
Journal of Statistical Physics, 1989
The critical limit of the eight-vertex model eigenvectors obtained by means of the generalized Bethe Ansatz is shown to give the six-vertex eigenvectors as constructed in a previous paper by two of the authors. Furthermore, an explicit mapping is established between these eigenvectors and the usual Bethe Ansatz eigenvectors of the six-vertex model. This allows one to show that the index v labeling the eight-vertex eigenstates becomes exactly the third component of the total spin in the critical limit.
Analytic Bethe Ansatz and TTT-system in C(1)_2C^{(1)}_2C(1)_2 vertex models
1993
Eigenvalues of the commuting family of transfer matrices are expected to obey the TTT-system, a set of functional relation, proposed recently. Here we obtain the solution to the TTT-system for C(1)_2C^{(1)}_2C(1)_2 vertex models. They are compatible with the analytic Bethe ansatz and Yang-Baxterize the classical characters.
Algebraic Bethe ansatz for thesℓ(2)Gaudin model with boundary
Nuclear Physics B, 2015
Following Sklyanin's proposal in the periodic case, we derive the generating function of the Gaudin Hamiltonians with boundary terms. Our derivation is based on the quasiclassical expansion of the linear combination of the transfer matrix of the XXX Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. By defining the appropriate Bethe vectors which yield strikingly simple off shell action of the generating function, we fully implement the algebraic Bethe ansatz, obtaining the spectrum of the generating function and the corresponding Bethe equations.
RIMS Publications, 2006
These are introductory lectures on application of the free field representation (bosonization) techniques to the solid-on-\S olid (SOS) and eight-vertex models. We start from the very beginnings, including the physical badcground of lattice models and some basic information on quantum integrability. After definitions of the eight-vertex and SOS models, we describe their relation known as the vertex-face correspondence. Then, skipping the Bethe ansatz solution, wc turn to the problem of calculation of correlation functions by mcaiis of the fit.c ficld rcprcscntation. We explain, how the vertex-face correspondence works on the level of vertex operators and bosonization, making it possible to express the correlation functions of the eight-vertex model in terms of the free field representation aimed to describe the SOS model.
Six-vertex model on a finite lattice: Integral representations for nonlocal correlation functions
Nuclear Physics B, 2021
We consider the problem of calculation of correlation functions in the six-vertex model with domain wall boundary conditions. To this aim, we formulate the model as a scalar product of off-shell Bethe states, and, by applying the quantum inverse scattering method, we derive three different integral representations for these states. By suitably combining such representations, and using certain antisymmetrization relation in two sets of variables, it is possible to derive integral representations for various correlation functions. In particular, focusing on the emptiness formation probability, besides reproducing the known result, obtained by other means elsewhere, we provide a new one. By construction, the two representations differ in the number of integrations and their equivalence is related to a hierarchy of highly nontrivial identities. 6.3. Performing integrations 32 Acknowledgments 33 Appendix A. 'Coordinate wavefunction' representation 33 Appendix B. Dual representations for the 'top' and 'bottom' partition functions 35 Appendix C. A remarkable identity 38 References 39
Integrable spin–boson models descending from rational six-vertex models
Nuclear Physics B, 2007
We construct commuting transfer matrices for models describing the interaction between a single quantum spin and a single bosonic mode using the quantum inverse scattering framework. The transfer matrices are obtained from certain inhomogeneous rational vertex models combining bosonic and spin representations of SU (2), subject to non-diagonal toroidal and open boundary conditions. Only open boundary conditions are found to lead to integrable Hamiltonians combining both rotating and counter-rotating terms in the interaction. If the boundary matrices can be brought to triangular form simultaneously, the spectrum of the model can be obtained by means of the algebraic Bethe ansatz after a suitable gauge transformation; the corresponding Hamiltonians are found to be non-hermitian. Alternatively, a certain quasi-classical limit of the transfer matrix is considered where hermitian Hamiltonians are obtained as members of a family of commuting operators; their diagonalization, however, remains an unsolved problem.
The S-matrix of the Faddeev-Reshetikhin model, diagonalizability and PT symmetry
Journal of High Energy Physics, 2007
We study the question of diagonalizability of the Hamiltonian for the Faddeev-Reshetikhin (FR) model in the two particle sector. Although the two particle S-matrix element for the FR model, which may be relevant for the quantization of strings on AdS 5 ×S 5 , has been calculated recently using field theoretic methods, we find that the Hamiltonian for the system in this sector is not diagonalizable. We trace the difficulty to the fact that the interaction term in the Hamiltonian violating Lorentz invariance leads to discontinuity conditions (matching conditions) that cannot be satisfied. We determine the most general quartic interaction Hamiltonian that can be diagonalized. This includes the bosonic Thirring model as well as the bosonic chiral Gross-Neveu model which we find share the same S-matrix. We explain this by showing, through a Fierz transformation, that these two models are in fact equivalent. In addition, we find a general quartic interaction Hamiltonian, violating Lorentz invariance, that can be diagonalized with the same two particle S-matrix element as calculated by Klose and Zarembo for the FR model. This family of generalized interaction Hamiltonians is not Hermitian, but is P T symmetric. We show that the wave functions for this system are also P T symmetric. Thus, the theory is in a P T unbroken phase which guarantees the reality of the energy spectrum as well as the unitarity of the S-matrix.
Properties of linear integral equations related to the six-vertex model with disorder parameter II
Journal of Physics A: Mathematical and Theoretical, 2012
We study certain functions arising in the context of the calculation of correlation functions of the XXZ spin chain and of integrable field theories related with various scaling limits of the underlying six-vertex model. We show that several of these functions that are related to linear integral equations can be obtained by acting with (deformed) difference operators on a master function Φ. The latter is defined in terms of a functional equation and of its asymptotic behavior. Concentrating on the so-called temperature case we show that these conditions uniquely determine the high-temperature series expansions of the master function. This provides an efficient calculation scheme for the high-temperature expansions of the derived functions as well.