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arXiv:cond-mat/9706017 · Physical Review Letters

Low-Temperature Behavior of a Magnetic Impurity in a Heisenberg Chain

Yu-Liang Liu

Published Jul 14, 1997• 15 citations

Abstract

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 ${g}_{c}<g<1$. 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 ${g}_{c}<g<1$, and $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}1$, for $g\ensuremath{\le}{g}_{c}$, where ${g}_{c}$ satisfies ${4g}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({1\ensuremath{-}g}_{c}{)}^{2}$, and $g$ is a dimensionless coupling strength parameter.

DOI: 10.1103/physrevlett.79.293

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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

{g}_{c}<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

{g}_{c}<g<1

, and

\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}1

, for

g\ensuremath{\le}{g}_{c}

, where

{g}_{c}

satisfies

{4g}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({1\ensuremath{-}g}_{c}{)}^{2}

, and

g

is a dimensionless coupling strength parameter.

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.

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 calculate the impurity susceptibility for a spin-1/2 magnetic impurity symmetrically coupled to its two nearest-neighbor spins in an antiferromagnetic Heisenberg spin-1/2 chain. This is done by extending White's density-matrix renormalization group (DMRG) technique to finite temperatures. It is found that the impurity susceptibility satisfies Curie law at low temperatures, and that the Curie constant decreases rapidly with increasing coupling constant between the impurity spin and the spins in the chain. For strong couplings, a logarithmic dependence of the susceptibility on temperature emerges. The Curie law behavior agrees with the DMRG analysis at zero temperature but deviates from the prediction of conformal field theory.

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.

Low-temperature thermodynamic properties of the spin-1/2 Heisenberg ladder system with non-magnetic impurities are discussed using an effective Hamiltonian for the impurity-induced spins in the background of a spin liquid with a gap. It is shown that the uniform susceptibility shows Curie-like behavior in high-, intermediate-, and low-temperature regimes, with three different Curie constants. The ratio of specific heat to temperature, C/T, is divergent in the low-temperature limit, reflecting a singular distribution of the effective couplings among the impurity spins.

We investigate the low-temperature thermodynamics of the spin-1/2 Heisenberg chain with open ends. On the basis of boundary conformal field theory arguments and numerical density matrix renormalization group calculations, it is established that in the isotropic case the impurity susceptibility exhibits a Curie-like divergent behavior as the temperature decreases, even in the absence of magnetic impurities. A similar singular temperature dependence is also found in the boundary contributions of the specific heat coefficient. In the anisotropic case, for 1/2<Delta<1, these boundary quantities still show a singular temperature dependence obeying a power law with an anomalous dimension. Experimental consequences will be discussed.