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arXiv:cond-mat/9406059 · Physical Review Letters

Elementary Excitations of One-Dimensional t−J Model with Inverse-Square Exchange

Zou Ha, F. D. M. Haldane

Published Nov 21, 1994• 38 citations
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Abstract

We identify exact excitation content of the intermediate states for the one-particle Green's functions, spin-spin, and (charge) density-density correlation functions of the periodic one-dimensional $t\ensuremath{-}J$ model with inverse-square exchange. The excitations consist of neutral $S=\frac{1}{2}$ spinons and spinless (charge $\ensuremath{-}e$ holons with semionic fractional statistics and bosonic (charge $+2e$) "antiholons" which are excitations of the holon condensate. We find a set of selection rules and the regions of nonvanishing spectral weight in the energy-momentum space for the various correlation functions.

DOI
10.1103/physrevlett.73.2887

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We identify exact excitation content of the intermediate states for the one-particle Green's functions, spin-spin, and (charge) density-density correlation functions of the periodic one-dimensional t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model with inverse-square exchange. The excitations consist of neutral S=12S=\frac{1}{2}S=21​ spinons and spinless (charge \ensuremath−e\ensuremath{-}e\ensuremath−e holons with semionic fractional statistics and bosonic (charge +2e+2e+2e) "antiholons" which are excitations of the holon condensate. We find a set of selection rules and the regions of nonvanishing spectral weight in the energy-momentum space for the various correlation functions.
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.
Quasi-particle picture in a magnetic field is pursued for dynamical spin and charge correlation functions of the one-dimensional supersymmetric t-J model with inverse-square interaction. With use of exact diagonalization and the asymptotic Bethe-ansatz equations for finite systems, excitation contents of relevant excited states are identified which are valid in the thermodynamic limit. The excitation contents are composed of spinons, antispinons, holons and antiholons obeying fractional statistics. Both longitudinal and transverse components of the dynamical spin structure factor are independent of the electron density in the region where only quasi-particles with spin degrees of freedom (spinons and antispinons) contribute. The dynamical charge structure factor does not depend on the spin-polarization density in the region where only quasi-particles with charge (holons and antiholons) are excited. These features indicate the strong spin-charge separation in dynamics, reflecting the high symmetry of the model.
Following the Bethe ansatz we determine the dynamical spectra of the one-dimensional supersymmetric t-J model. A series of fractionalized excitations are identified through two sets of Bethe numbers. Typical patterns in each set are found to yield wavefunctions containing elementary spin and charge carriers, manifested as distinct boundaries of the collective excitations in the spectra of single electron Green functions. In spin channels, gapless excitations fractionalized into two spin and a pair of postive and negative charge carriers, extending to finite energy as multiple continua. These patterns connect to the half-filling limit where only fractionalized spinons survive. In particle density channel, apart from spin-charge fractionalization, excitations involving only charge fluctuations are observed. Furthermore, nontrivial Bethe strings encoding bound state structure appear in channels of reducing or conserving magnetization, where spin and charge constituents can also be identified. These string states contribute significantly even to the low-energy sector in the limit of vanishing magnetization.
In this work, we present results for the dynamics of the doped one-dimensional t-J model for very large systems and in a wide parameter range. These results are obtained by quantum Monte Carlo simulation based on a newly developed algorithm, the hybrid-loop Monte Carlo. Within this algorithm the loop-algorithm is used to update the spin degrees of freedom, assuring optimal efficiency, whereas the charge degrees of freedom are evolved exactly for a given spin background in the frame of the determinantal algorithm, so that both static and dynamical observables can be measured. The study of the one-particle spectral function in the whole Luttinger-liquid regime of the phase diagram shows a splitting of the spectral weight into two branches on the inverse photoemission side, in contradiction with what is expected for an ordinary metal. This feature can be understood if we assume that the elementary excitation content of the nearest-neighbor t-J model and their dispersion is the same as the one analytically obtained at J/t=2 from a t-J model with both hopping and exchange interaction decaying as 1/r². In this case the elementary excitations are spinons with charge Q=0 and spin S=1/2, holons with charge Q=-e and spin S=0, and antiholon with charge Q=2e and spin S=0. Since the antiholon has twice the mass of the holon and charge with opposite sign, holon and antiholon are not charge conjugate but two independent elementary excitations. The agreement between the analytic results derived from our assumptions and the Monte Carlo data for a wide range of parameters and for different values of doping strongly indicate, that antiholons are generic excitations in the nearest-neighbor t-J model. Since the quantum numbers of the antiholons are the same as those of for a Cooper pair, the question arises whether these new excitations are related to superconductivity, that is indeed one of the possible phases in the one-dimensional nearest-neighbor t-J model. A further study of the system in the phase-separated region shows that the spectral function is very discontinuous in the photoemission side. This feature reminds an interference scenario generated by superposition of a continuous dispersion with a regular pattern. The nature of a phase-separated regime in a simulation with periodic boundary condition can be the origin of the regular pattern.
We construct thermodynamics of the one-dimensional supersymmetric t-J model with the 1/sin 2 interaction and hopping. The thermodynamics is described exactly in terms of free spinons and holons obeying Haldane's fractional exclusion statistics at all temperatures. Moreover, at low temperatures the semionic spinons and holons decouple resulting in the spin-charge separation in thermodynamic properties. We obtain explicit results for the spin and charge susceptibilities and specific heat, and interpret them in terms of the fractional exclusion statistics. Extension to the multi-component t-J model shows that the excitations obey either fractional statistics for g-ons with partial polarization of components, or the parafermionic one without polarization.
We present a selection rule for matrix elements of local spin operators in the S=1/2 Haldane-Shastry model. Based on this rule we extend a recent exact calculation by Haldane and Zirnbauer of the ground-state dynamical spin correlation function Sab{\mathit{S}}^{\mathit{a}\mathit{b}}Sab(n,t)=〈0\ensuremath{\Vert}Sa{\mathit{S}}^{\mathit{a}}Sa(n,t)Sb{\mathit{S}}^{\mathit{b}}Sb(0,0)\ensuremath{\Vert}0〉 and its Fourier transform Sab{\mathit{S}}^{\mathit{a}\mathit{b}}Sab(Q,E) of this model to a finite magnetic field. In zero field, only two-spinon excitations contribute to the spec tral function; in the (positively) partially spin-polarized case, there are two types of elementary excitations: spinons (\ensuremath{\Delta}Sz{\mathit{S}}^{\mathit{z}}Sz=\ifmmode\pm\else\textpm\fi{}1/2) and magnons (\ensuremath{\Delta}Sz{\mathit{S}}^{\mathit{z}}Sz=-1). The magnons are divided into left- or right-moving branches. The only classes of excited states contributing to the spectral functions are (I) two spinons, (II) two spinons+one magnon, (IIIa) two spinons+two magnons (moving in opposite directions), and (IIIb) one magnon. The contributions to the various correlations are ${\mathit{S}}^{\mathrm{\ensuremath{-}}+}$: (I); Szz{\mathit{S}}^{\mathit{z}\mathit{z}}Szz: (I)+(II); ${\mathit{S}}^{+\mathrm{\ensuremath{-}}}$: (I)+(II)+(III). In the zero-field limit there are no magnons, while in the fully polarized case, there are no spinons. We discuss the relation of the spectral functions to correlations of the Calogero-Sutherland model at coupling \ensuremath{\lambda}=2.