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arXiv:cond-mat/0011069 · Physical Review Letters

Spinon Attraction in Spin- 1/2 Antiferromagnetic Chains

Bogdan Andrei Bernevig, Domenico Giuliano, R. B. Laughlin

Published Apr 9, 2001• 26 citations
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Abstract

We derive the representation of the two-spinon wave function for the Haldane-Shastry model in terms of the spinon coordinates. This result allows us to rigorously analyze spinon interaction and its physical effects. We show that spinons attract one another. The attraction gets stronger as the size of the system is increased and, in the thermodynamic limit, determines the power law with which the susceptibility diverges.

DOI
10.1103/physrevlett.86.3392

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•Jan 1, 2003•0 citations
We derive the representation of the two-spinon wave function for the Haldane-Shastry model in terms of the spinon coordinates. This result allows us to rigorously analyze spinon interaction and its physical effects. We show that spinons attract one another. The attraction gets stronger as the size of the system is increased and, in the thermodynamic limit, determines the power law with which the susceptibility diverges.
By deriving and studying the coordinate representation for the two-spinon wave function, we show that spinon excitations in the Haldane-Shastry model interact. The interaction is given by a short-range attraction and causes an enhancement in the two-spinon wave function at short separations between the spinons. We express the spin susceptibility for a finite lattice in terms of the enhancement, given by the two-spinon wave function at zero separation. In the thermodynamic limit, the spinon attraction turns into the square-root divergence in the dynamical spin susceptibility.
The mapping between the fermion and spinon compositions of eigenstates in the one-dimensional spin-1/2 XX model on a lattice with N sites is used to describe the spinon interaction from two different perspectives: (i) For finite N the energy of all eigenstates is expressed as a function of spinon momenta and spinon spins, which, in turn, are solutions of a set of Bethe ansatz equations. The latter are the basis of an exact thermodynamic analysis in the spinon representation of the XX model. (ii) For N -> infinity the energy per site of spinon configurations involving any number of spinon orbitals is expressed as a function of reduced variables representing momentum, filling, and magnetization of each orbital. The spins of spinons in a single orbital are found to be coupled in a manner well described by an Ising-like equivalent-neighbor interaction, switching from ferromagnetic to antiferromagnetic as the filling exceeds a critical level. Comparisons are made with results for the Haldane-Shastry model.
By deriving and studying the coordinate representation for the one-spinon one-holon wave function we show that spinons and holons in the supersymmetric $t\ensuremath{-}J$ model with 1/r2{1/r}^{2}1/r2 interaction attract each other. The interaction causes a probability enhancement in the one-spinon one-holon wave function at short separation between the particles. We express the hole spectral function for a finite lattice in terms of the probability enhancement, given by the one-spinon one-holon wave function at zero separation. In the thermodynamic limit, the spinon-holon attraction turns into the square-root divergence in the hole spectral function.
In this chapter, we discuss the static and dynamic properties of the spin chain which is often referred to as the Haldane–Shastry model [77, 161]. In contrast with more familiar spin chains with the nearest-neighbor exchange interaction, the Haldane–Shastry model has a particular form of the longrange exchange interaction. In spite of its peculiar shape, it has turned out that the Haldane–Shastry model is the most fundamental one-dimensional spin system. We shall start with a discussion of how the model is solved for the ground state in Sections 4.1 to 4.3, and proceed to static correlation functions in Sections 4.4 and 4.5. The excitation spectrum is interpreted by magnons in Section 4.6, and by spinons in Section 4.7. The spinon picture gives the correct degeneracy of the energy levels in Section 4.8, and leads to the derivation of thermodynamics in Section 2.4, and the dynamical correlation function in Section 4.11. Most results have been obtained in an analytic form without any approximation. There is a close connection between the spectrum of the spin chain and that of the Sutherland model. The most remarkable fact is that the spin chain has a correspondence with two different values of the coupling parameter λ in the Sutherland model. Namely, the spectrum is mapped to the case with either λ = 2 or λ = ∞. In the latter case, particles crystallize with equal spacing, and a small oscillation from the equilibrium, as well as exchange interactions, make up the spectrum. In both cases of λ, however, the degeneracy of each level in the spin chain is not reproduced. Hence, the mapping as it stands cannot be used for thermodynamics.
The elementary excitations of strongly correlated one-dimensional electronic systems - spinons and holons - are discussed in an exact solution of the Haldane-Shastry and Kuramoto-Yokoyama model. We derive and exactly solve the equation of motion both for the two-spinon wavefunction and for the one-spinon, one-holon wavefunction. By solving the equations of motion we find the spinon-spinon and spinon-holon interaction to be a short-range attraction. The physical consequences of such an attraction on the spin-form factor and on the hole-spectral function are also worked out.