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arXiv:1510.02548 · Journal of Physics Conference Series

Crossover temperature of the spin-1/2 XXZ chain with an impurity

Ryoko Yahagi, Jun Sato, Tetsuo Deguchi

Published Jan 25, 2016• 24 citations
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Abstract

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.

DOI
10.1088/1742-6596/670/1/012054

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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.
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.
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.
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.
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.
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.