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Journal of Physics Condensed Matter

The open spin chain with impurity: an exact solution

Holger Frahm, Andrei A. Zvyagin

Published Nov 10, 1997• 83 citations
Publisher

Abstract

Using a model of an open spin chain that is exactly soluble by means of a Bethe ansatz , we study the effects of a boundary magnetic field and an impurity spin coupled to the chain. An impurity spin only scatters forward, while the boundary is purely a back-scatterer. Two parameters for the impurity and one for the boundary permit us to mimic the effect of real magnetic impurity, with both forward and backward scattering.

DOI
10.1088/0953-8984/9/45/021

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Using a model of an open spin chain that is exactly soluble by means of a Bethe ansatz , we study the effects of a boundary magnetic field and an impurity spin coupled to the chain. An impurity spin only scatters forward, while the boundary is purely a back-scatterer. Two parameters for the impurity and one for the boundary permit us to mimic the effect of real magnetic impurity, with both forward and backward scattering.
We propose an integrable model of the spin-1/2 Heisenberg chain coupled to two impurity moments. With the open boundary conditions at the impurity sites, the model can be exactly solved for arbitrary impurity spin and arbitrary exchange constants between the bulk and the impurities. The absence of redundant terms in the Hamiltonian makes the model very reasonable. The Hamiltonian is diagonalized via algebraic Bethe ansatz. It is found that the impurity spins can only be screened (partially for S>1/2S>1/2S>1/2) for antiferromagnetic coupling between the impurity and the bulk. Otherwise the impurity spins cannot be screened. The residual entropy of the ground state and the Kondo temperature are also derived explicitly based on the Bethe ansatz and the local Fermi-liquid theory.
In this paper, the Kondo magnetic effect is studied for the XXZ\mathrm{XXZ}XXZ spin chain where the impurities are coupled to the edges of the system. The Hamiltonian of our model can be constructed from the transfer matrix. It is exactly solvable and the exchange constants between the bulk and the boundary impurities are arbitrary. The finite-size corrections to the ground-state energy are obtained for the magnetic impurities and the boundary condition. The specific heat and the susceptibility contributed by the impurities are derived and the Kondo temperature is given explicitly by the use of the Luttinger-Fermi liquid picture and the Bethe ansatz method.
The Bethe ansatz solution for the Kondo problem with open boundary conditions is presented. We show that one of the features of the generic Kondo effect---the finite magnetic susceptibility at low energies (present in the Bethe ansatz solution for periodic boundary conditions for the same model)---is absent due to the effect of open edges, if one considers the total effect of the impurity and the open edge together. However, we show that the behavior of the dynamical magnetic impurity itself is similar for periodic and open boundaries. It is shown that a real static boundary local potential (magnetic field) cannot model all features of the behavior of the dynamical magnetic impurity in the generic Kondo situation.
The spin-1/2 Heisenberg chain coupled to a spin-S impurity moment with anti-periodic boundary condition is studied via the off-diagonal Bethe ansatz method. The twisted boundary breaks the U(1) symmetry of the system, which leads to that the spin ring with impurity can not be solved by the conventional Bethe ansatz methods. By combining the properties of the R-matrix, the transfer matrix, and the quantum determinant, we derive the T–Q relation and the corresponding Bethe ansatz equations. The residual magnetizations of the ground states and the impurity specific heat are investigated. It is found that the residual magnetizations in this model strongly depend on the constraint of the topological boundary condition, the inhomogeneity of the impurity comparing with the hosts could depress the impurity specific heat in the thermodynamic limit. This method can be expand to other integrable impurity models without U(1) symmetry.
We develop a general Bethe ansatz formalism for diagonalizing an integrable model of a magnetic impurity of arbitrary spin coupled ferro- or antiferromagnetically to a chain of interacting electrons. The method is applied to an open chain, with the exact solution revealing the existence of a ``hidden'' Kondo effect driven by forward electron scattering off the impurity. We argue that the so-called ``operator reflection matrices'' proposed in recent Bethe ansatz studies of related models emulate only forward electron-impurity scattering, which may explain the absence of complete Kondo screening for certain values of the impurity-electron coupling in these models.