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Journal of Physics Condensed Matter

Impurity-induced critical behaviour in antiferromagnetic Heisenberg chains

P. Schlottmann

Published Aug 26, 1991• 65 citations
Publisher

Abstract

The authors consider an integrable SU(2)-invariant model consisting of the Heisenberg chain of arbitrary spin S (Takhtajan-Babujian model) interacting with an impurity of spin S'. The impurity is assumed to be located on the mth link of the chain and interacts only with both neighbouring sites. The starting point is a set of commuting transfer matrices, whose local weights satisfy triangular Yang-Baxter relations. The diagonalization of the transfer matrices leads to the Bethe ansatz equations for the model. The thermodynamics of the system is studied. Three situations have to be distinguished: (i) If S'=S the impurity just corresponds to one more site in the chain. (ii) If S'>S the impurity spin is only partially compensated at T=0, leaving an effective spin of (S'-S). (iii) S'

DOI
10.1088/0953-8984/3/34/008

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The authors consider an integrable SU(2)-invariant model consisting of the Heisenberg chain of arbitrary spin S (Takhtajan-Babujian model) interacting with an impurity of spin S'. The impurity is assumed to be located on the mth link of the chain and interacts only with both neighbouring sites. The starting point is a set of commuting transfer matrices, whose local weights satisfy triangular Yang-Baxter relations. The diagonalization of the transfer matrices leads to the Bethe ansatz equations for the model. The thermodynamics of the system is studied. Three situations have to be distinguished: (i) If S'=S the impurity just corresponds to one more site in the chain. (ii) If S'>S the impurity spin is only partially compensated at T=0, leaving an effective spin of (S'-S). (iii) S'
The numerical solution of the Bethe ansatz equations of an integrable SU(2)-invariant generalization of the spin-S antiferromagnetic Heisenberg chain (Takhtajian-Babujian model) with a spin-S' impurity in zero magnetic field is presented. The entropy, specific heat and susceptibility of an impurity of spin S' are obtained numerically as a function of the spin S of the chain. Three situations have to be distinguished: (i) if S'=S the impurity just corresponds to one more site in the chain; (ii) if S'>S the impurity spin is only partially compensated (undercompensated) at T=0, leaving an effective spin of (S'-S); (iii) if S'
An impurity of spin S′=1/2 is introduced in the Babujian-Takhtajian Heisenberg model of spin S in one dimension. The impurity is located on the mth link of the chain and interacts with both neighboring sites. The interaction between impurity and Heisenberg chain is of a special type so that the integrability is preserved. The diagonalization of the transfer matrices leads to the Bethe ansatz equations of the model. The thermodynamics of the system is studied. For ferromagnetic coupling in the Heisenberg chain, the impurity is locked into the critical behavior of the lattice, i.e., at low temperatures the specific heat is proportional to T1/2 (ferromagnetic magnons) and the susceptibility diverges as T−2 (with logarithmic corrections). For antiferromagnetic coupling and S =1/2 the impurity is just one more site in the chain. If S ≳1/2 the collective properties of impurity and lattice give rise to critical behavior in complete analogy to the overcompensated n-channel Kondo problem. The zero-temperature entropy is finite in zero-field, but zero if the field is nonzero, giving rise to an essential singularity at T=H=0. As a consequence a two-peak structure arises in the specific heat in a small but finite field.
We consider an integrable SU(2)-invariant model consisting of a Heisenberg chain of spins S (the Takhtajan-Babujian model) interacting with a finite concentration c of impurity spins S ´. The thermodynamic Bethe- ansatz equations are stated for this model. The ground-state equations are analysed as a function of c , the magnetic field and the coupling parameter (impurity rapidity p 0 ) of the impurities to the lattice. In zero field the ground state is a singlet for finite c , but becomes non-Fermi-liquid-like as c 0 for S ´< S . Two rapidity bands play a role at T = 0 corresponding to strings of length 2 S and 2 S ´, respectively. The van Hove singularities of the empty bands define two critical fields, H c ( c , p 0 ) and H s ( c , p 0 ), at which the susceptibility diverges. H c tends to zero as c 0 giving rise to a crossover from non-Fermi-liquid behaviour for H > H c to Fermi-liquid-like behaviour for H < H c . The spectrum of elementary excitations and the finite-size corrections to the ground-state energy are calculated, and used to discuss the asymptotic behaviour of spin-spin correlation functions for long times and large distances.
We study the impurity-induced critical behavior in an integrable SU(2)\mathrm{SU}(2)SU(2)-invariant model consisting of an open spin chain of arbitrary spin SSS (Takhatajian-Babujian model) interacting with an impurity of spin ${S}^{\ensuremath{'}}$ located at one of the boundaries. For S=1/2S=1/2S=1/2 or ${S}^{\ensuremath{'}}=1/2,$ the impurity interaction takes a very simple form $J{S}_{1}\ensuremath{\cdot}{S}^{\ensuremath{'}}$ that describes the deformed boundary bond between the impurity ${S}^{\ensuremath{'}}$ and the first bulk spin S1{S}_{1}S1​ with an arbitrary coupling strength J.J.J. For a weak coupling $0<J<{J}_{0}/[{(S+S}^{\ensuremath{'}}{)}^{2}\ensuremath{-}1/4],$ the impurity is completely compensated, undercompensated, and overcompensated for ${S}^{\ensuremath{'}}=S,$ ${S}^{\ensuremath{'}}>S,$ and ${S}^{\ensuremath{'}}<S$ as in the usual Kondo problem. While for strong coupling $J>~{J}_{0}/[{(S+S}^{\ensuremath{'}}{)}^{2}\ensuremath{-}1/4],$ the impurity spin is split into two ghost spins. Their cooperative effect leads to a variety of new critical behaviors with different values of $|{S}^{\ensuremath{'}}\ensuremath{-}S|.$
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 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.