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Low Temperature Physics

Thermal conductivity of a quantum spin-1/2 antiferromagnetic chain with magnetic impurities

Andrei A. Zvyagin

Published Mar 1, 2008• 0 citations
Publisher

Abstract

We present an exact theory that describes how magnetic impurities change the behavior of the thermal conductivity for the integrable Heisenberg antiferromagnetic quantum spin-1/2 chain. Single magnetic impurities and a large concentration of impurities with similar values of the couplings to the host chain (a weak disorder) do not change the linear-in-temperature low-T behavior of the thermal conductivity: Only the slope of that behavior becomes smaller, compared to the homogeneous case. The strong disorder in the distribution of the impurity-host couplings produces more rapid temperature growth of the thermal conductivity, compared to the linear-in-T dependence of the homogeneous chain and the chain with weak disorder. Recent experiments on the thermal conductivity in inhomogeneous quasi-one-dimensional quantum spin systems manifest qualitative agreement with our results.

DOI
10.1063/1.2889411

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We present an exact theory that describes how magnetic impurities change the behavior of the thermal conductivity for the integrable Heisenberg antiferromagnetic quantum spin-1/2 chain. Single magnetic impurities and a large concentration of impurities with similar values of the couplings to the host chain (a weak disorder) do not change the linear-in-temperature low-T behavior of the thermal conductivity: Only the slope of that behavior becomes smaller, compared to the homogeneous case. The strong disorder in the distribution of the impurity-host couplings produces more rapid temperature growth of the thermal conductivity, compared to the linear-in-T dependence of the homogeneous chain and the chain with weak disorder. Recent experiments on the thermal conductivity in inhomogeneous quasi-one-dimensional quantum spin systems manifest qualitative agreement with our results.
In the framework of the exactly solvable model, using nonperturbative methods, the temperature and magnetic field behavior of the NMR relaxation rate (and low-energy dynamical structure factor) for the antiferromagnetic Heisenberg spin-1/2 chain with magnetic impurities is calculated. For a single magnetic impurity, totally screened at low temperatures by spinons of the chain, the critical exponents, which determine the low-temperature and magnetic field behavior, get renormalized. For the finite concentration of impurities with the random distribution of the impurity-chain coupling the weak disorder in the distribution does not produce essential changes to the dynamical structure factor. Contrary, the strong disorder yields drastically different behavior of the low-energy dynamical characteristics, comparing to the homogeneous chain. Our results are generic, and they qualitatively agree in limiting cases with the results of perturbative calculations for disordered spin chains and with the data of experiments on quasi-one-dimensional spin compounds.
We explore the effect of a (non-)magnetic impurity on the thermal transport of the spin-12\frac{1}{2}21​ Heisenberg chain model. This unique system allows to probe Kondo-type phenomena in a prototype strongly correlated system. Using numerical diagonalization techniques we study the scaling of the frequency dependent thermal conductivity with system size and host-impurity coupling strength as well as the dependence on temperature. We focus in particular on the analysis of ``cutting healing'' of weak links or a magnetic impurity by the host chain via Kondo-like screening as the temperature is lowered.
We study the magnetic field-dependence of the thermal conductivity due to magnetic excitations in frustrated spin-1/2 Heisenberg chains. Near the saturation field, the system is described by a dilute gas of weakly interacting fermions (free-fermion fixed point). We show that in this regime the thermal conductivity exhibits a nonmonotonic behavior as a function of the ratio $\ensuremath{\alpha}={J}_{2}/{J}_{1}$ between second- and first-nearest-neighbor antiferromagnetic exchange interactions. This result is a direct consequence of the splitting of the single-particle dispersion minimum into two minima that takes place at the Lifshitz point $\ensuremath{\alpha}=1/4$. Upon increasing $\ensuremath{\alpha}$ from zero, the inverse mass vanishes at $\ensuremath{\alpha}=1/4$ and it increases monotonically from zero for $\ensuremath{\alpha}\ensuremath{\ge}1/4$. By deriving an effective low-energy theory of the dilute gas of fermions, we demonstrate that the Drude weight Kth{K}_{\mathrm{th}}Kth​ of the thermal conductivity exhibits a similar dependence on $\ensuremath{\alpha}$ near the saturation field. Moreover, this theory predicts a transition between a two-component Tomonaga-Luttinger liquid and a vector-chiral phase at a critical value $\ensuremath{\alpha}={\ensuremath{\alpha}}_{c}$ that agrees very well with previous density matrix renormalization group results. We also show that the resulting curve ${K}_{\mathrm{th}}(\ensuremath{\alpha})$ is in excellent agreement with exact diagonalization (ED) results. For the low-magnetic field regime, our ED results show that ${K}_{\mathrm{th}}(\ensuremath{\alpha})$ has a pronounced minimum at $\ensuremath{\alpha}\ensuremath{\simeq}0.7$. We also demonstrate that the thermal conductivity is significantly affected by the presence of magnetothermal coupling.
Exact results for the thermodynamic properties of ensembles of magnetic impurities with randomly distributed host-impurity couplings in the quantum antiferromagnetic Heisenberg model are presented. Exact calculations are done for arbitrary values of temperature and external magnetic field. We have shown that for strong disorder the quenching of the impurity moments is absent. For weak disorder the screening persists, but with the critical non-Fermi-liquid behaviors of the magnetic susceptibility and specific heat. A comparison with disordered Kondo effect experiments in dirty metallic alloys is performed.
We calculate exactly the thermodynamics of ensembles of magnetic impurities with randomly distributed host-impurity couplings in the critical region of the uniaxial spin-1/2 quantum chain. We derive a finite set of integral equations describing the complete temperature and magnetic field dependence of the system. Exact numerical results for arbitrary values of temperature and external magnetic field are obtained. We show that for strong disorder the quenching of the impurity moments is absent. For weak disorder the screening persists, but with a critical non-Fermi-liquid behaviour of the magnetic susceptibility and specific heat. We point out that the `easy-plane' magnetic anisotropy does not qualitatively change the behaviour of the thermodynamic characteristics of the disordered impurities in the antiferromagnetic chain. On the other hand, anisotropy of ferromagnetic type can enhance the tendency to quenching of the impurities for strong disorder.
We study thermal transport in one-dimensional spin systems both in the presence and absence of impurities. In the absence of disorder, all these spin systems display in the continuum limit a temperature-dependent Drude peak in the thermal conductivity. In gapless systems, the low-temperature Drude weight is proportional to temperature and to the central charge, which characterizes the conformal field theory that describes the system at low energies. On the other hand, the low-temperature Drude weight of spin gap systems shows an activated behavior modulated by a power law. For temperatures higher than the spin gap, one recovers the linear T behavior akin to gapless systems. For temperatures larger than the exchange coupling, the Drude weight decays as ${T}^{\ensuremath{-}2}.$ We argue that this behavior is a generic feature of quasi-one-dimensional spin gap systems with a relativistic like low-energy dispersion. We also consider the effect of a magnetic field on the Drude weight with emphasis on the commensurate-incommensurate transition induced by it. We then study the effect of nonmagnetic impurities on the thermal conductivity of the dimerized XY\mathrm{XY}XY chain and the spin-12\frac{1}{2}21​ two-leg ladder. Impurities destroy the Drude peak and the thermal conductivity exhibits a purely activated behavior at low temperature, with an activation gap renormalized by disorder. The relevance of these results for experiments is briefly discussed.