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arXiv:cond-mat/9912244 · Physical Review Letters

Exact Dynamical Structure Factor of the Degenerate Haldane-Shastry Model

Takashi Yamamoto, Yasuhiro Saiga, Mitsuhiro Arikawa, Yoshio Kuramoto

Published Feb 7, 2000• 21 citations
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Abstract

The dynamical structure factor $S(q,\ensuremath{\omega})$ of the $K$-component ( $K\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$, $3$, $4$) spin chain with a $1/{r}^{2}$ interaction is derived exactly at zero temperature for the arbitrary size of the system. The result is interpreted in terms of a free quasiparticle picture which is a generalization of the spinon picture in the $\mathrm{SU}(2)$ case. The excited states consist of $K$ quasiparticles each of which is characterized by a set of $K\ensuremath{-}1$ quantum numbers. Divergent singularities of $S(q,\ensuremath{\omega})$ at the spectral edges are derived analytically. The analytic result is checked numerically for finite systems.

DOI
10.1103/physrevlett.84.1308

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The dynamical structure factor S(q,\ensuremathω)S(q,\ensuremath{\omega})S(q,\ensuremathω) of the KKK-component ( K=2K\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2K=2, 333, 444) spin chain with a 1/r21/{r}^{2}1/r2 interaction is derived exactly at zero temperature for the arbitrary size of the system. The result is interpreted in terms of a free quasiparticle picture which is a generalization of the spinon picture in the SU(2)\mathrm{SU}(2)SU(2) case. The excited states consist of KKK quasiparticles each of which is characterized by a set of K\ensuremath−1K\ensuremath{-}1K\ensuremath−1 quantum numbers. Divergent singularities of S(q,\ensuremathω)S(q,\ensuremath{\omega})S(q,\ensuremathω) at the spectral edges are derived analytically. The analytic result is checked numerically for finite systems.
The dynamical structure factor S(q,ω)S(q,ω)S(q,ω) of the SU(K) (K=2,3,4) Haldane-Shastry model is derived exactly at zero temperature for arbitrary size of the system. The result is interpreted in terms of free quasi-particles which are generalization of spinons in the SU(2) case; the excited states relevant to S(q,ω)S(q,ω)S(q,ω) consist of K quasi-particles each of which is characterized by a set of K-1 quantum numbers. Near the boundaries of the region where S(q,ω)S(q,ω)S(q,ω) is nonzero, S(q,ω)S(q,ω)S(q,ω) shows the power-law singularity. It is found that the divergent singularity occurs only in the lowest edges starting from (q,ω)=(0,0)(q,ω) = (0,0)(q,ω)=(0,0) toward positive and negative q. The analytic result is checked numerically for finite systems via exact diagonalization and recursion methods.
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 Haldane–Shastry spin chain can be mapped to the infinite coupling limit of the SU (2) spin Calogero–Sutherland model. We use the 2 Jack polynomial technology to compute the form factors of the spin operator on the multi-spinon spectrum. The spin structure factor is obtained through a form factor expansion. The expansion is proved to converge in the small momentum limit. Numerics based on two- and four-spinon contributions give an approximate result for the infinite temperature static and dynamic spin structure factor.
The exact expression derived by Bougourzi, Couture, and Kacir for the two-spinon contribution to the dynamic spin structure factor S(q,\ensuremath{\omega}) of the one-dimensional s=1/2 Heisenberg antiferromagnet at T=0 is evaluated for direct comparison with finite-chain transition rates (N\ensuremath{\leqslant}28) and an approximate analytical result previously inferred from finite-N data, sum rules, and Bethe ansatz calculations. The two-spinon excitations account for 72.89% of the total intensity in S(q,\ensuremath{\omega}). The singularity structure of the exact result is determined analytically and its spectral-weight distribution evaluated numerically over the entire range of the two-spinon continuum. The leading singularities of the frequency-dependent spin autocorrelation function, static spin structure factor, and q dependent susceptibility are determined via sum rules. The impact of the non-two-spinon excitations on the integrated intensity, the susceptibility, the frequency moments, and the Euclidian time representation of S(q,\ensuremath{\omega}) is studied on the basis of finite-size data.
Dynamical properties, such as dynamical spin and charge structure factors and single-particle spectral functions, are studied for the one-dimensional supersymmetric t-J model with inverse-square interaction. Exact diagonalization and the recursion method are used for finite systems up to 16 sites. A simple rule is proposed to understand the supermultiplet structure in the excitation spectrum. Numerical calculations show that the dynamical spin structure factor is independent of the electron density in the region where only two spinons contribute. In the electron removal spectrum from half-filling, close correspondence is found between the dominant contribution from three-spinon plus one-holon states in the t-J model and the two-spinon excitations in the Haldane-Shastry model. Comparison is made with dynamics of the nearest-neighbor supersymmetric t-J model.