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arXiv:cond-mat/9912272 · Journal of the Physical Society of Japan

Exact Dynamics of theSU(K)Haldane-Shastry Model

Takashi Yamamoto, Yasuhiro Saiga, Mitsuhiro Arikawa, Yoshio Kuramoto

Published Mar 15, 2000• 16 citations
arXivHTMLPDF

Abstract

The dynamical structure factor $S(q,ω)$ 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 $S(q,ω)$ 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 $S(q,ω)$ is nonzero, $S(q,ω)$ shows the power-law singularity. It is found that the divergent singularity occurs only in the lowest edges starting from $(q,ω) = (0,0)$ toward positive and negative q. The analytic result is checked numerically for finite systems via exact diagonalization and recursion methods.

DOI
10.1143/jpsj.69.900

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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.
The dynamical structure factor $S(q,\ensuremath{\omega})$ 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 $K\ensuremath{-}1$ 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.
We compute the dynamical spin structure factor $S(k,\ensuremath{\omega})$ of the SU(3) Heisenberg chain variationally using a truncated Hilbert space spanned by the Gutzwiller projected particle-hole excitations of the Fermi sea, introduced by Dalla Piazza et al. [Nat. Phys. 11, 62 (2015)], with a modified importance sampling. We check the reliability of the method by comparing the $S(k,\ensuremath{\omega})$ to exact diagonalization results for 18 sites and to the two-soliton continuum of the Bethe ansatz for 72 sites. We get an excellent agreement in both cases. Detailed analysis of the finite-size effects shows that the method captures the critical Wess-Zumino-Witten SU(3)1{}_{1}1​ behavior and reproduces the correct exponent, with the exception of the size dependence of the weight of the bottom of the conformal tower. We also calculate the single-mode approximation for the SU(N)\mathrm{SU}(N)SU(N) Heisenberg model and determine the velocity of excitations. Finally, we apply the method to the SU(3) Haldane-Shastry model and find that the variational method gives the exact wave function for the lowest excitation at $k=\ifmmode\pm\else\textpm\fi{}2\ensuremath{\pi}/3$.
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.
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 two-spinon part of the dynamic spin structure factor ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ for the one-dimensional s=1/2,s=1/2,s=1/2, XXZ\mathrm{XXZ}XXZ model at T=0T=0T=0 in the antiferromagnetically ordered phase is calculated using recent advances in the algebraic analysis based on (infinite-dimensional) quantum group symmetries of this model and the related vertex models. The two-spinon excitations form a two-parameter continuum consisting of two partly overlapping sheets in $(Q,\ensuremath{\omega})$ space. The spectral threshold has a smooth maximum at the Brillouin zone boundary $(Q=\ensuremath{\pi}/2)$ and a smooth minimum with a gap at the zone center (Q=0).(Q=0).(Q=0). The two-spinon density of states has square-root divergences at the lower and upper continuum boundaries. For the two-spinon transition rates, the two regimes $0<~Q<{Q}_{\ensuremath{\kappa}}$ (near the zone center) and ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2$ (near the zone boundary) must be distinguished, where ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}0$ in the Heisenberg limit and ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}\ensuremath{\pi}/2$ in the Ising limit. In the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2,$ the two-spinon transition rates relevant for ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ are finite at the lower boundary of each sheet, decrease monotonically with increasing \ensuremath{\omega}, and approach zero linearly at the upper boundary. The resulting two-spinon part of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ is then square-root divergent at the spectral threshold and vanishes in a square-root cusp at the upper boundary. In the regime $0<{Q}_{\ensuremath{\kappa}}<~\ensuremath{\pi}/2,$ in contrast, the two-spinon transition rates have a smooth maximum inside the continuum and vanish linearly at either boundary. In the associated two-spinon line shapes of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega}),$ the linear cusps at the continuum boundaries are replaced by square-root cusps. Existing perturbation studies have been unable to capture the physics of the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2.$ However, their line-shape predictions for the regime $0<~Q<{Q}_{\ensuremath{\kappa}}$ are in good agreement with the exact results if the anisotropy is very strong. For weak anisotropies, the exact line shapes are more asymmetric.