Low-temperature behavior of a Magnetic Impurity in a Heisenberg Chain

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: Cim(T ) ∼ T μ, μ = 4− 1 g (1− g)2, for gc < g < 1. The impurity spin susceptibility has the following temperature dependence: χim(T ) ∼ T ν , ν = 3− 1g (1−g)2, for gc < g < 1, and ν = −1, for g ≤ gc, where gc satisfies: 4gc = (1 − gc), and g is a dimensionless coupling strength parameter. 78.70.Dm, 79.60.Jv, 72.10.Fk Typeset using REVTEX 1 Recently, the quantum impurity scattering of the Tomonaga-Luttinger(TL) liquid has been extensively studied by using different techniques [1–19]. There is some controversy on the treatment of backward scattering of the conduction electrons on a quantum impurity or impurity-like hole in the valence band. One thinks that including backward scattering drastically changes the properties of a TL-liquid. In principle, this problem can be formulated by means of the Bethe-Ansatz to obtain exact results for static and thermodynamic quantities. However to obtain frequency dependent quantities one has to use perturbative methods. Due to strong correlations between the conduction electrons and impurity, perturbation-theory may fail. In Ref. [19], we have given a clear expression and some exact results within the validity of the bosonization method for a quantum impurity scattering of a general one-dimensional(1D) interacting electronic system, and shown that the system has two independent collective modes: one is drastically influenced by electronelectron interaction, and another is independent of the electron-electron interaction which is clearly observed in the numerical calculations [18], while it is missed in the renormalization group calculations [3]. Here we use the same method as that in Ref. [19] to study a magnetic impurity in the Heisenberg chain which can be easily treated by numerical methods. The impurity susceptibility of this system shows an unusual temperature dependence: χim(T ) ∼ T , ν = 3 − 1g (1 − g), for gc < g < 1, and ν = −1, for g ≤ gc, where gc is defined as that: 4gc = (1 − gc), and g is a dimensionless coupling strength parameter. For an antiferromagnetic Heisenberg chain, the dimensionless coupling strength parameter g takes the value [21] g = 1 2 , and the temperature dependence of the impurity susceptibility χim(T ) is T . The impurity specific heat Cim(T ) has the temperature dependence: Cim(T ) ∼ T , μ = 4 − 1 g (1 − g), for gc < g < 1. Although the impurity specific heat as well as the spin susceptibility depend on the dimensionless coupling strength parameter g, we can still define a temperature independent Wilson-ratio in this system. We consider the following one-dimensional Heisenberg model H = − ∑