A one-dimensional metal containing interacting electrons moving in both directions and magnetic impurities is studied. The model is shown to be integrable by the Bethe ansatz provided that an algebraic relation is satisfied between the coupling constants of the electron-electron and electron-impurity interactions.
A model describing the Anderson impurity in the Bardeen-Cooper-Schriffer superconductor is proven to exhibit hidden integrability and is diagonalized exactly by the Bethe ansatz. Typeset using REVTEX 1 The basic theoretical models describing magnetic impurities in nonmagnetic normal metals, such as the s-d (Kondo) model, the Anderson model, etc., are integrable under two additional conditions [1]: (i) an electron-impurity coupling is assumed to be energy independent, and (ii) a band electron dispersion ε(k) can be linearized around the Fermi level, ε(k) ≃ vF(k − kF), where kF and vF are the Fermi momentum and velocity, respectively. Only under these conditions, both an electron-impurity scattering and an effective electronelectron coupling are described in terms of discontinuous jumps in the Bethe ansatz wave functions. Therefore a linear dispersion of particles and a pointlike particle-impurity coupling are considered now as the necessary mathematical conditions for integrability of the “impurity ” models. Because a carrier dispersion in superconducting metals cannot be linearized,
We consider the one-dimensional t-J model, which consists of spin-1/2 electrons on a lattice with nearest neighbor hopping t constrained by the excluded multiple occupancy of the lattice sites and spin-exchange J between neighboring sites. The model is integrable at the supersymmetric point, J=2t. We extend the model by introducing an impurity of arbitrary spin S that interacts with the electrons on the neighboring sites without destroying the integrability. The lattice model is defined by the scattering matrices via the quantum inverse scattering method. The interaction Hamiltonian between the impurity and the itinerant electrons is only explicitly constructed in the continuum limit. The discrete Bethe ansatz equations diagonalizing the model are derived and the solutions are classified according to the string hypothesis. The thermodynamic Bethe ansatz equations are derived and the impurity free energy is obtained for arbitrary bandfilling as a function of temperature and external magnetic field. The properties of the impurity depend on one coupling parameter. The impurity can localize up to one itinerant electron and has in general mixed valent properties. The integer valent low T small H fixed point of the impurity corresponds to an asymptotically free spin S, while if either T or H (or both) become large the impurity behaves like an asymptotically free spin (S+).
We propose an integrable Kondo problem in a one-dimensional $t\ensuremath{-}J$ model. With the open boundary condition of the wave functions at the impurity sites, the model can be exactly solved via Bethe ansatz for a set of JL,R (Kondo coupling constants) and VL,R (impurity potentials) parametrized by a single parameter c. The integrable value of JL,R runs from negative infinity to positive infinity, which allows us to study both the ferromagnetic Kondo problem and the antiferromagnetic Kondo problem in a strongly correlated electron system. Generally, there is a residual entropy for the ground state, which indicates a typical non-Fermi liquid behavior.