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Physical Review Letters

Ordering and Criticality in Spin-1 Chains

Rajiv R. P. Singh, Martin P. Gelfand

Published Sep 25, 1989• 2 citations
Publisher

Abstract

work' which also agrees very well with field theory pre- dictions once logarithmic corrections are taken into account. Our statement summarizing previous numerical work pertinentto critical behavior at the Bethe-Ansatz- solvable point was not correct.Critical exponents at this point, along with the possibility of logarithmic correc- tions, had been studied previously by Blote, Bonner, and co-workers, ' and, more recently, by Alcaraz and Martins.We are grateful to J. C. Bonner for bringing this to our attention.We have also discovered that, due to a coding error, the expansions in the dimerization parameter X for an- isotropy h, W1 were incorrect.The estimated critical points are only affected significantly for A& 1: The point C in Fig. 1 should be moved from k =0 to

DOI
10.1103/physrevlett.63.1442

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work' which also agrees very well with field theory pre- dictions once logarithmic corrections are taken into account. Our statement summarizing previous numerical work pertinentto critical behavior at the Bethe-Ansatz- solvable point was not correct.Critical exponents at this point, along with the possibility of logarithmic correc- tions, had been studied previously by Blote, Bonner, and co-workers, ' and, more recently, by Alcaraz and Martins.We are grateful to J. C. Bonner for bringing this to our attention.We have also discovered that, due to a coding error, the expansions in the dimerization parameter X for an- isotropy h, W1 were incorrect.The estimated critical points are only affected significantly for A& 1: The point C in Fig. 1 should be moved from k =0 to
work' which also agrees very well with field theory pre- dictions once logarithmic corrections are taken into account. Our statement summarizing previous numerical work pertinentto critical behavior at the Bethe-Ansatz- solvable point was not correct.Critical exponents at this point, along with the possibility of logarithmic correc- tions, had been studied previously by Blote, Bonner, and co-workers, ' and, more recently, by Alcaraz and Martins.We are grateful to J. C. Bonner for bringing this to our attention.We have also discovered that, due to a coding error, the expansions in the dimerization parameter X for an- isotropy h, W1 were incorrect.The estimated critical points are only affected significantly for A& 1: The point C in Fig. 1 should be moved from k =0 to
There has recently been considerable interest in the numerical calculation of critical exponents for quantum spin chains with general spin s, stimulated by a remarkable prediction by Haldane and conformally invariant field theoretic calculations. In several instances numerical calculations have differed considerably from theoretical predictions, with misleading results. These discrepancies have generally been attributed to the presence of marginal operators resulting in logarithmic corrections which slow numerical convergence. Such phenomena seem to be ubiquitous in the area of quantum spin chains. However, until recently, no calculation was available for estimating quantitatively the expected shifts in exponent values obtained by numerical means. The purpose of this work is to call attention to such a calculation1 for the integrable Takhtajan–Babujian family of quantum spin chains, and make specific comparisons with a variety of spin chain exponents obtained numerically.2,3 Agreement with the analytic formulas is extremely good in all cases. This is an indication that finite-size scaling is a powerful tool for extracting critical exponents even in models with logarithmic corrections, provided that these corrections are taken into account in lowest order using the method of Ref. 1.
We present an analysis based on the extrapolation of finite-size results to the infinite one-dimensional spin-(1/2) dimerized isotropic Heisenberg system for the whole range of the dimerization parameter \ensuremath{\delta} (\ensuremath{\Vert}\ensuremath{\delta}\ensuremath{\Vert}\ensuremath{\le}1) at zero temperature. This system undergoes a transition at \ensuremath{\delta}=0, and a gap opens in the spectrum of elementary excitations. The exponent \ensuremath{\nu}, which characterizes the opening of the gap, is estimated with use of the finite-size results (with size up to N=18). We investigate two finite-size-scaling hypotheses, assuming a pure power-law behavior [case (1)], or taking into account logarithmic corrections [case (2)]. To estimate \ensuremath{\nu}, we use the derivative of the reciprocal of the gap, the derivative of the gap, and the Callan-Symanzik function. We show that the first of these is less affected by finite-size corrections than are the other two. Using it, we have obtained, in case (1), \ensuremath{\nu}=0.71\ifmmode\pm\else\textpm\fi{}0.01, in agreement with previous estimates, and in case (2), \ensuremath{\nu}=0.668\ifmmode\pm\else\textpm\fi{}0.001, in very good agreement with the value \ensuremath{\nu}=(2/3) conjectured by den Nijs. We also show that, far from criticality, the ground-state energy per site may be described by the form \ensuremath{\Vert}\ensuremath{\delta}${\ensuremath{\Vert}}^{x}$ with x=1.34\ifmmode\pm\else\textpm\fi{}0.02. However, results for the derivative of this quantity show a different functional dependence upon \ensuremath{\delta}, at least for \ensuremath{\delta}\ensuremath{\gtrsim}0.4. In fact, both the values of the ground-state energy and its derivative agree with the third-order perturbation theory of Harris, with agreement improving as \ensuremath{\delta} approaches 1.