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Physical Review Letters

Spin-Boson Systems: One-Dimensional Equivalents and the Kondo Problem

Martin Blume, Victor J. Emery, Alan H. Luther

Published Aug 17, 1970• 68 citations
Publisher

Abstract

The partition function for a spin \textonehalf{} in a magnetic field coupled to a set of bosons is shown to be exactly equal to that of a one-dimensional gas and also, in a delicate limit, to that of a one-dimensional Ising model. Both of these are peculiar in that they must provide a free energy which is an intensive quantity. For special values of the parameters in the Hamiltonian, the partition function is equal to that first derived by Anderson and Yuval, and by Hamann, for the Kondo problem.

DOI
10.1103/physrevlett.25.450

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•Sep 28, 2020•0 citations
The partition function for a spin \textonehalf{} in a magnetic field coupled to a set of bosons is shown to be exactly equal to that of a one-dimensional gas and also, in a delicate limit, to that of a one-dimensional Ising model. Both of these are peculiar in that they must provide a free energy which is an intensive quantity. For special values of the parameters in the Hamiltonian, the partition function is equal to that first derived by Anderson and Yuval, and by Hamann, for the Kondo problem.
The equilibrium Kondo problem is solved using a spin-boson representation of the partition function. For antiferromagnetic coupling, a canonical transformation removes infrared singularities in the energy and magnetization series. At T=0T=0T=0 the moment vanishes, the energy is nonanalytic in ${J}_{\ensuremath{\perp}}$, the magnetization saturates at a field HS{H}_{S}HS​ of order ${(\frac{{{J}_{\ensuremath{\perp}}}^{2}}{{J}_{\ensuremath{\parallel}}})}^{\frac{1}{2{J}_{\ensuremath{\parallel}}}}$, and the susceptibility $\ensuremath{\chi}$ is proportional to ${{H}_{S}}^{\ensuremath{-}1}$. ${J}_{\ensuremath{\parallel}}$ and ${J}_{\ensuremath{\perp}}$ are longitudinal and transverse exchange constants.) For ferromagnetic coupling the free moment is only slightly renormalized.
We explicitly evaluate the infinite series of integrals that appear in the "Anderson-Yuval" reformulation of the anisotropic Kondo model in terms of a one-dimensional Coulomb gas. We do this by developing a general approach relating the anisotropic Kondo model of arbitrary spin with the boundary sine-Gordon model, which describes an impurity in a Luttinger liquid and tunneling in the fractional quantum Hall effect. The Kondo solution then follows from the exact perturbative solution of the latter model in terms of Jack polynomials.
Quantum spin systems are shown to be rigorously equivalent to certain Bose systems. As a result the phase transitions occuring in such systems can be regarded as generalized Bose condensation processes. In those processes it is not the number of bosons that is kept constant but a certain function of it. The problem of degeneracy arising in Bosonizing a spin system is solved in a simple fashion. The low-temperature limit of the Bose systems reproduces the magnon gas with correct interactions. The Bloch sum rule becomes exact at all temperatures, when one replaces the magnon number by the number of bosons. The crossover to classical behavior at criticality is discussed and it is shown that at Tc{T}_{c}Tc​ the quantum S=12S=\frac{1}{2}S=21​ spin systems behave like the corresponding classical $S=\ensuremath{\infty}$ systems. A classical effective Hamiltonian whose corresponding partition function is equal to the partition function of the quantum spin systems is derived. Finally, possible application to dynamics are briefly discussed.
Ising’s one-dimensional model is a model whose constituent elements are magnetic moments of spin ±1/2. Spins interact with short-range forces limited only to the first neighbours. The partition function of this model is easy to calculate using transfer matrix (section 4.1). From the known partition function, free energy, magnetization and other thermodynamic functions can be easily obtained. It shows that the magnetization is analytical function of the temperature for all T > 0, which shows that in this Ising model, there is no phase transition for T > 0.