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Journal of the Physical Society of Japan

Critical Properties ofS=1/2 XXZ Chain

Satoshi Takada, Kenn Kubo

Published May 15, 1986• 32 citations
Publisher

Abstract

The S x S x correlation function of S =1/2 XXZ chain is numericaly obtained at low temperatures by means of the transfer matrix with use of the Trotter approximation. The inplane susceptibility and the correlation length are extrapolated to N =∞ where N is the number of the Trotter slicings. It is shown that the extrapolation works quite effectively and makes the estimation of the critical indices ν, γ and η possible for the anisotropy parameter -0.75≤Δ≤0.5. The obtained values of ν, γ and η satisfy the scaling relation and agree quite well with the analytic expression derived by Luther and Peschel.

DOI
10.1143/jpsj.55.1671

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The S x S x correlation function of S =1/2 XXZ chain is numericaly obtained at low temperatures by means of the transfer matrix with use of the Trotter approximation. The inplane susceptibility and the correlation length are extrapolated to N =∞ where N is the number of the Trotter slicings. It is shown that the extrapolation works quite effectively and makes the estimation of the critical indices ν, γ and η possible for the anisotropy parameter -0.75≤Δ≤0.5. The obtained values of ν, γ and η satisfy the scaling relation and agree quite well with the analytic expression derived by Luther and Peschel.
The S x - S x correlation function of S =1 XXZ chain is calculated at low temperatures by means of the transfer matrix with use of the Trotter formula. Critical properties of the correlation function, such as critical indices γ, η and ν and the mass identified by the inverse correlation length at T =0, are obtained. We conclude that a groundstate with an excitation gap appears for the anisotropy parameter Δ<Δ 1 ≃ -0.3 including the isotropic antiferromagnetic point (Δ=-1). We have obtained η≃1/4 at Δ=Δ 1 in accord with the theoretical prediction by Haldane and den Nijs. Our result, however, shows the monotonic increase of the excitation gap with decreasing Δ for Δ<Δ 1 contrary to Haldane's prediction.
Using the quantum transfer-matrix method we have calculated the spin correlation functions of the S =3/2 XXZ chain and examined the temperature dependence of the mass (≡ the inverse of the correlation length), susceptibility χ x ( Q ) and the magnetic structure factor S z ( Q ). The result shows that the mass vanishes at T =0 for the planar and isotropic cases (-1≦ Δ ≦1) in agreement with the theoretical prediction. At a low temperature the peak of S z ( Q ) at Q =π has been observed to persist even in the ferromagnetic region (0< Δ <0.8). The susceptibility χ z ( Q =0) at T =0 has been obtained with use of the extrapolation. The critical indices have been also examined.
A set of equations that allows one to calculate the correlation length and the free energy of the S=1/2 XYZ chain at a given temperature is obtained. It contains infinite unknown numbers and is derived by the largest and the second-largest eigenvalues of the quantum transfer matrix in the limit of infinite Trotter number. The numerical solution of this set of equations gives very accurate values of the free energy and the correlation length at arbitrary temperature. The energy gaps that appear in the correlation length and the free energy at low temperature are discussed.
We study the criticality at the isotropic limit of the S=(1/2, antiferromagnetic, XXZ chain on approach from the Ising side. Long series are developed for the longitudinal and transverse correlation sums or structure factors Szz{S}_{\mathrm{zz}}Szz​ and ${S}_{+\mathrm{\ensuremath{-}}}$ around the Ising limit. Their extrapolation indicates a divergence as the critical point is approached, with two different powers, a result quite unexpected if one employs a naive scaling argument since the critical correlations are isotropic and there is only one correlation length, \ensuremath{\xi}. An analytical calculation is presented, in which this paradox is resolved and the numerical results are confirmed.
The Suzuki-Trotter formula has been used to get the mmm th approximant to the classical representation of the partition function of the NNN-spin S=12S=\frac{1}{2}S=21​ XY\mathrm{XY}XY chain. The equivalent two-dimensional ($N\ifmmode\times\else\texttimes\fi{}2m$) Ising model with four-spin interactions has been studied in detail by use of the numerically exact transfer-matrix method for T> 0.05T>~0.05T> 0.05 and m< 8m<~8m< 8. The limiting $m\ensuremath{\rightarrow}\ensuremath{\infty}$ behavior has been estimated from the extrapolation formula of the form $E(m)=E(\ensuremath{\infty})+\frac{a}{{m}^{2}}$. The limiting value of the energy agrees with the exact solution of Katsura.