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Physical review. B, Condensed matter

Logarithmic corrections in antiferromagnetic chains

Ian K. Affleck, Jill C. Bonner

Published Jul 1, 1990• 8 citations
Publisher

Abstract

Various numerical estimates of critical exponents for antiferromagnetic chains are compared with conformal field-theory predictions. Field theory allows the explicit calculation of logarithmic corrections, and hence the value of the effective exponent that is expected to be observed numerically. Comparison is made with numerically obtained exponent values, and agreement is found to be very good, even when logarithmic corrections are as large as 70%. Logarithmic corrections can lead to misleading results, for example an incorrect phase diagram, and examples in the literature where this seems to have occurred are discussed.

DOI
10.1103/physrevb.42.954

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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

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Various numerical estimates of critical exponents for antiferromagnetic chains are compared with conformal field-theory predictions. Field theory allows the explicit calculation of logarithmic corrections, and hence the value of the effective exponent that is expected to be observed numerically. Comparison is made with numerically obtained exponent values, and agreement is found to be very good, even when logarithmic corrections are as large as 70%. Logarithmic corrections can lead to misleading results, for example an incorrect phase diagram, and examples in the literature where this seems to have occurred are discussed.
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.
Affleck has made predictions for the critical exponents of a class of integrable spin chains of general spin sss. The expressions are sss dependent. Our interest here is a finite-size scaling analysis of spin-1 systems to determine the validity of the prediction for the alternation critical exponent. We discuss these results in the light of our numerical results for the conformal anomaly ccc, the exponent $\ensuremath{\eta}$, and the mass-gap exponent associated with biquadratic exchange. The numerical results are not generally in good agreement with theory, and hence the possible presence of logarithmic corrections is discussed.
Field theory calculations predict multiplicative logarithmic corrections to correlation functions from marginally irrelevant operators. However, for the numerically most suitable model, the spin-\textonehalf{} chain, these corrections have been controversial. In this paper, the spin-spin correlation function of the antiferromagnetic spin-\textonehalf{} chain is calculated numerically in the presence of a next-nearest-neighbor coupling J2{J}_{2}J2​ for chains of up to 32 sites. By varying the coupling strength J2{J}_{2}J2​ we can control the effect of the marginal operator, and our results unambiguously confirm the field theory predictions. The critical value at which the marginal operator vanishes has been determined to be at $\frac{{J}_{2}}{J}=0.241167\ifmmode\pm\else\textpm\fi{}0.000005$.
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