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Physical review. B, Condensed matter

Heisenberg spin chain with many impurities

P. Schlottmann

Published Apr 1, 1994• 36 citations
Publisher

Abstract

We consider the Heisenberg chain of spin 1/2 with a concentration x of impurity spins S. The thermodynamic Bethe ansatz equations are stated for this model. The ground-state equations are analyzed as a function of x and magnetic field. In zero field the ground state is a singlet for finite x, but non-Fermi-liquid-like for x\ensuremath{\rightarrow}0. Two energy bands play a role at T=0. The Van Hove singularities of the empty bands define two critical fields, ${\mathit{H}}_{\mathit{c}1}$(x) and ${\mathit{H}}_{\mathit{c}2}$(x), at which the susceptibility diverges. The non-Fermi-liquid behavior as x\ensuremath{\rightarrow}0 arises from nonanalyticities as ${\mathit{H}}_{\mathit{c}2}$ vanishes.

DOI
10.1103/physrevb.49.9202

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We consider the Heisenberg chain of spin 1/2 with a concentration x of impurity spins S. The thermodynamic Bethe ansatz equations are stated for this model. The ground-state equations are analyzed as a function of x and magnetic field. In zero field the ground state is a singlet for finite x, but non-Fermi-liquid-like for x\ensuremath{\rightarrow}0. Two energy bands play a role at T=0. The Van Hove singularities of the empty bands define two critical fields, Hc1{\mathit{H}}_{\mathit{c}1}Hc1​(x) and Hc2{\mathit{H}}_{\mathit{c}2}Hc2​(x), at which the susceptibility diverges. The non-Fermi-liquid behavior as x\ensuremath{\rightarrow}0 arises from nonanalyticities as Hc2{\mathit{H}}_{\mathit{c}2}Hc2​ vanishes.
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.
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 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'
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.
The thermodynamics of finite open antiferromagnetic XXZ chains is studied using field theory, Bethe ansatz and quantum Monte Carlo methods. For the susceptibility a parameter-free result as a function of the number of sites L and temperature T beyond the scaling limit is derived. The limiting cases ( J being the exchange constant), where the boundary correction shows a logarithmically suppressed Curie behaviour, and , where a crossover to the ground state behaviour takes place, are discussed in detail. On the basis of this analysis we present a simple formula for the averaged susceptibility of a spin-1/2 chain doped with non-magnetic impurities. We show that the effective Curie constant has a highly non-trivial temperature dependence and shows scaling in the low temperature limit. Finally, corrections due to intrachain and interchain couplings and implications for experiments on Sr 2 Cu 1− x Pd x O 3+δ and related compounds are discussed.