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arXiv:cond-mat/9609262 · Physical Review Letters

Integrable Impurity in the Supersymmetric t - J Model

Gerald Bedürftig, Fabian H. L. Eßler, Holger Frahm

Published Dec 16, 1996• 54 citations
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Abstract

An impurity coupling to both spin and charge degrees of freedom is added to a periodic $t$- $J$ chain such that its interaction with the bulk can be varied continuously without losing integrability. Ground state properties, impurity contributions to the susceptibilities and low temperature specific heat are studied as well as transport properties. The impurity phase shifts are calculated to establish the existence of an impurity bound state in the holon sector.

DOI
10.1103/physrevlett.77.5098

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An impurity coupling to both spin and charge degrees of freedom is added to a periodic ttt- JJJ chain such that its interaction with the bulk can be varied continuously without losing integrability. Ground state properties, impurity contributions to the susceptibilities and low temperature specific heat are studied as well as transport properties. The impurity phase shifts are calculated to establish the existence of an impurity bound state in the holon sector.
We consider an integrable model describing an Anderson-like impurity coupled to an open ttt-JJJ chain. Both the hybridization (i.e., its coupling to bulk chain) and the local spectrum can be controlled without breaking the integrability of the model. As the hybridization is varied, holon and spinon bound states appear in the many body ground state. Based on the exact solution we study the state of the impurity and its contribution to thermodynamic quantities as a function of an applied magnetic field. Kondo behavior in the magnetic response of the impurity can be observed provided that its parameters have been adjusted properly to the energy scales of the holon and spinon excitations of the one-dimensional bulk.
We construct an integrable SU(2)-invariant model consisting of the Heisenberg chain of spin 1 interacting with an impurity of spin S. This generalizes previous results by Andrei and Johannesson for the spin-(1/2 chain. The model Hamiltonian is diagonalized and the thermodynamics is obtained. For ferromagnetic coupling at low temperatures the impurity susceptibility diverges as ${T}^{\mathrm{\ensuremath{-}}2}$ and the impurity specific heat is proportional to T1/2{T}^{1/2}T1/2 for all values of the impurity spin S. For antiferromagnetic coupling and T=0 the impurity susceptibility diverges proportionally to \ensuremath{\Vert}lnH\ensuremath{\Vert} as H\ensuremath{\rightarrow}0 if S=(1/2, while if S>1 the impurity spin is only partially compensated for by the S=1 chain.
We study the zero- and the finite-temperature behavior of the integrable spin-1/2 XXZ periodic chain with an impurity by the algebraic and thermal Bethe ansatz methods. We evaluate the local magnetization on the impurity site at zero temperature analytically and derive the impurity susceptibility exactly from it. In the graphs of the impurity specific heat versus temperature, we show how the impurity spin becomes more liberated from the bulk many-body effect as the exchange coupling between the impurity spin and other spins decreases and that at low temperature it couples strongly to them such as in the Kondo effect. Thus, we observe not only the crossover behavior from the high- to the low-temperature regime, but another from the N -site chain to the ( N − 1)-site chain with a free impurity spin. We also show that the estimate of the Wilson ratio at a given low temperature is independent of the impurity parameter if its absolute value is small enough with respect to the temperature and the universality class is described by the XXZ anisotropy in terms of the dressed charge.
We study exactly the effect of an impurity in the interacting quantum spin chain at low temperature by solving the integrable spin-1/2 XXZ periodic chain with an impurity through the algebraic and thermal Bethe ansatz methods. In particular, we investigate how the crossover temperature for the impurity specific heat depends on the impurity parameter, i.e. the coupling of the impurity to other spins, and show that it is consistent with the analytic expression that is obtained by setting the impurity susceptibility to be proportional to the inverse of the crossover temperature. In the model, two types of crossover behavior appear: one from the high-temperature regime to the low-temperature Kondo regime and another from the N -site homogeneous chain to the ( N − 1)-site chain with a decoupled free impurity spin, with respect to the temperature and the impurity parameter, respectively.
The Kondo problem of two impurities in a 1D strongly correlated electron system within the framework of the open boundary Hubbard chain is solved and the impurities, coupled to the ends of the electron system, are introduced by their scattering matrices with electrons so that the boundary matrices satisfy the reflecting integrability condition. The finite-size correction of the ground-state energy is obtained due to the impurities. Exact expressions for the low-temperature specific heat contributed by the charge and spin parts of the magnetic impurities are derived. The Pauli susceptibility and the Kondo temperature are given explicitly. The Kondo temperature is inversely proportional to the density of electrons.
A Bethe ansatz solution of a (modified) Hubbard chain with a Kondo impurity of arbitrary spin S at a highly symmetric line of parameter space is proposed and explored. Our results confirm the existence of a strong-coupling (line of) fixed-point(s) with ferromagnetic Kondo coupling as first hypothetized by Furusaki and Nagaosa on the basis of perturbative renormalization group calculations. For on-site Hubbard repulsion and ferromagnetic Kondo exchange, the ground state has spin S − 1/2, i.e. , is a singlet when S = 1/2. The contributions of the impurity to the magnetic susceptibility and low-temperature specific heat are discussed. While the Wilson ratio is unity in the half-filled band, it is found to be a function of density and interaction away from half-filling.