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Physical review. B, Condensed matter

Integrable spin-1 Heisenberg chain with impurity

Kong-Ju-Bock Lee, P. Schlottmann

Published Jan 1, 1988• 85 citations
Publisher

Abstract

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 ${T}^{1/2}$ 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.

DOI
10.1103/physrevb.37.379

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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\ensuremath−2{T}^{\mathrm{\ensuremath{-}}2}T\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.
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.
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'
By extending White's density-matrix renormalization group (DMRG) technique to finite temperatures, we calculate the impurity susceptibility for the following systems: (a) a spin-1/2 impurity coupled to a single site of an antiferromagnetic Heisenberg spin-1/2 chain and (b) an impurity coupled symmetrically to the two spins at the open ends of the chain. We also perform a Monte Carlo simulation and show that the DMRG adaptation gives reliable results. The calculated impurity susceptibility shows Pauli-like behavior at low temperatures in the former case while a ln T divergence is found in the latter case, in agreement with predictions based on conformal field theory.
Using the bosonization technique, we have studied a spin-1/2 magnetic impurity in Heisenberg chain, and shown that the impurity specific heat and spin susceptibility have an anomalous temperature dependence. The temperature dependence of the impurity specific heat is ${C}_{\mathrm{im}}(T)\ensuremath{\sim}{T}^{\ensuremath{\mu}}$, $\ensuremath{\mu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4\ensuremath{-}\frac{1}{g}(1\ensuremath{-}g{)}^{2}$, for gc<g<1{g}_{c}<g<1gc​<g<1. The impurity spin susceptibility has the following temperature dependence: ${\ensuremath{\chi}}_{\mathrm{im}}(T)\ensuremath{\sim}{T}^{\ensuremath{\nu}}$, $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3\ensuremath{-}\frac{1}{g}(1\ensuremath{-}g{)}^{2}$, for gc<g<1{g}_{c}<g<1gc​<g<1, and $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}1$, for $g\ensuremath{\le}{g}_{c}$, where gc{g}_{c}gc​ satisfies ${4g}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({1\ensuremath{-}g}_{c}{)}^{2}$, and ggg is a dimensionless coupling strength parameter.
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.