Integrable models for confined fermions: applications to metallic grains

We study integrable models for electrons in metals when the single particle spectrum is discrete. The electron–electron interactions are BCS-like pairing, Coulomb repulsion, and spin-exchange coupling. These couplings are, in general, nonuniform in the sense that they depend on the levels occupied by the interacting electrons. By using the realization of spin-1/2 operators in terms of electrons the models describe spin-1/2 models with nonuniform long range interactions and external magnetic field. The integrability and the exact solution arise since the model Hamiltonians can be constructed in terms of Gaudin models. Uniform pairing and the resulting orthodox model correspond to an isotropic limit of the Gaudin Hamiltonians. We discuss possible applications of this model to a single grain and to a system of few interacting grains.     

Exact solution of the XXX Gaudin model with generic open boundaries

The XXX Gaudin model with generic integrable open boundaries specified by the most general non-diagonal reflecting matrices is studied. Besides the inhomogeneous parameters, the associated Gaudin operators have six free parameters which break the U(1)$U\left(1\right)$-symmetry. With the help of the off-diagonal Bethe ansatz, we successfully obtained the eigenvalues of these Gaudin operators and the corresponding Bethe ansatz equations.     

On the theory of superconductivity in the extended Hubbard model

A microscopic theory of superconductivity in the extended Hubbard model which takes into account the intersite Coulomb repulsion and electron-phonon interaction is developed in the limit of strong correlations. The Dyson equation for normal and pair Green functions expressed in terms of the Hubbard operators is derived. The self-energy is obtained in the noncrossing approximation. In the normal state, antiferromagnetic short-range correlations result in the electronic spectrum with a narrow bandwidth. We calculate superconducting T _c by taking into account the pairing mediated by charge and spin fluctuations and phonons. We found the d-wave pairing with high-T _c mediated by spin fluctuations induced by the strong kinematic interaction for the Hubbard operators. Contributions to the d-wave pairing coming from the intersite Coulomb repulsion and phonons turned out to be small.     

Integrable spin–boson models descending from rational six-vertex models

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)$\mathit{SU}(2)$, 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.     

Bethe Subalgebras in Hecke Algebra and Gaudin models

The generating function for elements of the Bethe subalgebra of the Hecke algebra is constructed as Sklyanin’s transfer-matrix operator for the Hecke chain. We show that in a special classical limit ${q \to 1}$ the Hamiltonians of the Gaudin model can be derived from the transfer-matrix operator of the Hecke chain. We construct a non-local analog of the Gaudin Hamiltonians in the case of the Hecke algebras.     

Exactly-solvable models derived from a generalized Gaudin algebra

We introduce a generalized Gaudin Lie algebra and a complete set of mutually commuting quantum invariants allowing the derivation of several families of exactly solvable Hamiltonians. Different Hamiltonians correspond to different representations of the generators of the algebra. The derived exactly-solvable generalized Gaudin models include the Hamiltonians of Bardeen–Cooper–Schrieffer, Suhl–Matthias–Walker, Lipkin–Meshkov–Glick, the generalized Dicke and atom–molecule, the nuclear interacting boson model, a new exactly-solvable Kondo-like impurity model, and many more that have not been exploited in the physics literature yet.     

Gaudin Model,KZ Equation and an Isomonodromic Problem on the Tours

This Letter presents a construction of isospectral problems on the torus. The construction starts from an SU(n) version of the XYZ Gaudin model recently studied by Kuroki and Takebe within the context of a twisted WZW model. In the classical limit, the quantum Hamiltonians of the generalized Gaudin model turn into classical Hamiltonians with a natural r-matrix structure. These Hamiltonians are used to build a nonautonomous multi-time Hamiltonian system, which is eventually shown to be an isomonodromic problem on the torus. This isomonodromic problem can also be reproduced from an elliptic analogue of the KZ equation for the twisted WZW model. Finally, a geometric interpretation of this isomonodromic problem is discussed in the language of a moduli space of meromorphic connections.     

Bethe Vectors of the osp(1|2) Gaudin Model

The eigenvectors of the osp(1|2) invariant Gaudin Hamiltonians are found using explicitly constructed creation operators. Commutation relations between the creation operators and the generators of the loop superalgebra are calculated. The coordinate representation of the Bethe states is presented. The relation between the Bethe vectors and solutions to the Knizhnik–Zamolodchikov equation yields the norm of the eigenvectors.     

Rational SU(3) Gaudin Model

The Hamiltonians of the SU(3) Gaudin model are constructed based on the nonrelativistic limit of the SU(3) chain.After the quantum determinant being well defined,the eigenvectors and eigenvalues of the Hamiltonians of the SU(3) Geudin model are given.These results can be generalized to any number of constituting spins (SU(N)).     

Elliptic Gaudin models and elliptic KZ equations

The Gaudin models based on the face-type elliptic quantum groups and the XYZ Gaudin models are studied. The Gaudin model Hamiltonians are constructed and are diagonalized by using the algebraic Bethe ansatz method. The corresponding face-type Knizhnik–Zamolodchikov equations and their solutions are given.     

Integrability of the pairing hamiltonian

We show that a many-body Hamiltonian that corresponds to a system of fermions interacting through a pairing force is an integrable problem, i.e. it has as many constants of the motion as degrees of freedom. At the classical level this implies that the time-dependent Hartree-Fock-Bogoliubov dynamics is integrable and at the quantum level that there are conserved operators of two-body character which reduce to the number operators when the pairing strength vanishes. We display these operators explicitly and study in detail the three-level example.     

An Integrable Discretization of the Rational {\mathfrak su(2)} Gaudin Model and Related Systems

The first part of the present paper is devoted to a systematic construction of continuous-time finite-dimensional integrable systems arising from the rational Gaudin model through certain contraction procedures. In the second part, we derive an explicit integrable Poisson map discretizing a particular Hamiltonian flow of the rational Gaudin model. Then, the contraction procedures enable us to construct explicit integrable discretizations of the continuous systems derived in the first part of the paper.     

Off-shell Bethe ansatz equation for osp(21) Gaudin magnets

The semi-classical limit of the algebraic Bethe ansatz method is used to solve the theory of Gaudin models. Via off-shell Bethe ansatz method we find the spectra and eigenvectors of the N−1 independents Gaudin Hamiltonians with symmetry osp(21). We also show how the off-shell Gaudin equation solves the trigonometric Knizhnik–Zamolodchikov equation.     

Generating Function of Correlators in the sl2 Gaudin Model

An exponential generating function of correlators is calculated explicitly for the sl2 Gaudin model (degenerated quantum integrable XXX spin chain). The calculation relies on the Gauss decomposition for the SL(2) loop group. A new explicit expression for the correlators is derived from the generating function, from which the determinant formulas for the norms of Bethe eigenfunctions due to Richardson and Gaudin are obtained.     

Application of a diagram technique for Hamiltonians with Coulomb interactions to the Hubbard model

We give a general statistical Wick theorem for Hamiltonians containing the Coulomb interaction. The diagram technique developed for some class of interactions is applied to the calculation of the transverse Green function and effective interaction within the Hubbard model.     

Exact correlation functions of the BCS model in the canonical ensemble

We evaluate correlation functions of the BCS model for a finite number of particles. The integrability of the Hamiltonian relates it with the Gaudin algebra G[sl(2)]. Therefore, a theorem that Sklyanin proved for the Gaudin model, can be applied. Several diagonal and off-diagonal correlators are calculated. The finite-size scaling behavior of the pairing correlation function is studied.     

Electrostatic Analogy for Integrable Pairing Force Hamiltonians

For the exactly solved reduced BCS model an electrostatic analogy exists; in particular it served to obtain the exact thermodynamic limit of the model from the Richardson Bethe ansatz equations. We present an electrostatic analogy for a wider class of integrable Hamiltonians with pairing force interactions. We apply it to obtain the exact thermodynamic limit of this class of models. To verify the analytical results, we compare them with numerical solutions of the Bethe ansatz equations for finite systems at half-filling for the ground state.     

Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes

Searches for possible new quantum phases and classifications of quantum phases have been central problems in physics. Yet, they are indeed challenging problems due to the computational difficulties in analyzing quantum many-body systems and the lack of a general framework for classifications. While frustration-free Hamiltonians, which appear as fixed point Hamiltonians of renormalization group transformations, may serve as representatives of quantum phases, it is still difficult to analyze and classify quantum phases of arbitrary frustration-free Hamiltonians exhaustively. Here, we address these problems by sharpening our considerations to a certain subclass of frustration-free Hamiltonians, called stabilizer Hamiltonians, which have been actively studied in quantum information science. We propose a model of frustration-free Hamiltonians which covers a large class of physically realistic stabilizer Hamiltonians, constrained to only three physical conditions; the locality of interaction terms, translation symmetries and scale symmetries, meaning that the number of ground states does not grow with the system size. We show that quantum phases arising in two-dimensional models can be classified exactly through certain quantum coding theoretical operators, called logical operators, by proving that two models with topologically distinct shapes of logical operators are always separated by quantum phase transitions.     

The Open Boundary Dynamical Elliptic Quantum Gaudin Model and Its Solution

We construct the Hamiltonians of open elliptic quantum Gaudin model and show its relation with the open boundary elliptic quantum group. We define eigenstates of the model to be Bethe vectors with η=0 of the boundary elliptic quantum group. Then, the Hamiltonian is exactly diagonalized by using the algebraic Bethe ansatz method.     

The SYM integrable super spin chain

Recently it was established that the one-loop planar dilatation generator of super-Yang–Mills theory may be identified, in some restricted cases, with the Hamiltonians of various integrable quantum spin chains. In particular Minahan and Zarembo established that the restriction to scalar operators leads to an integrable vector chain, while recent work in QCD suggested that restricting to twist operators, containing mostly covariant derivatives, yields certain integrable Heisenberg XXX chains with non-compact spin symmetry . Here we unify and generalize these insights and argue that the complete one-loop planar dilatation generator of is described by an integrable super spin chain. We also write down various forms of the associated Bethe ansatz equations, whose solutions are in one-to-one correspondence with the complete set of all one-loop planar anomalous dimensions in the gauge theory. We finally speculate on the non-perturbative extension of these integrable structures, which appears to involve non-local deformations of the conserved charges.