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Physical Review Letters

Comment on "Ordering and Criticality in Spin-1 Chains."

Ian K. Affleck

Published Sep 25, 1989• 0 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.63.1442.2

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Ordering and Criticality in Spin-1 Chains

Rajiv R. P. Singh, Martin P. Gelfand

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

Physical Review Letters•Sep 25, 1989•2 citations

Logarithmic corrections and antiferromagnetic chains (abstract)

Ian K. Affleck, J. C. Bonner

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.

Journal of Applied Physics•May 1, 1990•0 citations

Logarithmic corrections in antiferromagnetic chains

Ian K. Affleck, Jill C. Bonner

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.

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Critical Exponents from Other Expansions

It is useful to compare the results obtained so far with other approaches to the critical exponents. One is a similar field-theoretic approach based on perturbation expansions of φ 4-theories. But instead of working in D = 4−ε dimensions and continuing the results to ε = 1 to obtain critical exponents in the physical dimension D = 3, perturbation expansions can be derived directly in three dimensions, and studied in the regime of small masses. Fundamentally, this approach is less satisfactory than the previous one since there is no small parameter analogous to ε which permits us to prove the existence of a scaling limit at least for small ε. Another disadvantage is that it is impossible to calculate the expansion coefficients analytically, except for one- and two loop-diagrams. In one respect the three-dimensional approach has, however, an advantage: the power series have long been available up to six loops for ω, η, and ν, and there exist recent results up to seven loops for the critical exponents η and ν. Although numerical expansions are not as esthetic as the exact expansions derived in this book, their increased order leads to a higher accuracy in the critical exponents, as we shall see below in Section 20.4. Another approach is based on high-temperature expansions for lattice models of the O(N)symmetric classical Heisenberg model with the energy of Eq. (1.52). The critical exponents will be derived from such expansions in Section 20.8. Finally we take advantage of the fact that the O(N)-symmetric classical Heisenberg model is exactly solvable in the limit of large N, where it reduces to the spherical model [recall the remark after Eq. (1.69)]. It is therefore possible to expand the φ 4-theory around this limit in powers of 1/N. The results will be compared with those from the perturbation expansions in

WORLD SCIENTIFIC eBooks•Jul 1, 2001•0 citations

Comment on "Ordering and Criticality in Spin-1 Chains"

Ian K. Affleck

Received 31 October 1988DOI:https://doi.org/10.1103/PhysRevLett.62.839©1989 American Physical Society

Physical Review Letters•Feb 13, 1989•17 citations

Finite-size study of the one-dimensional spin-(1/2) dimerized Heisenberg chain

G. Spronken, B. Fourcade, Yves Lépine

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.

Physical review. B, Condensed matter•Feb 1, 1986•50 citations
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.
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.
It is useful to compare the results obtained so far with other approaches to the critical exponents. One is a similar field-theoretic approach based on perturbation expansions of φ 4-theories. But instead of working in D = 4−ε dimensions and continuing the results to ε = 1 to obtain critical exponents in the physical dimension D = 3, perturbation expansions can be derived directly in three dimensions, and studied in the regime of small masses. Fundamentally, this approach is less satisfactory than the previous one since there is no small parameter analogous to ε which permits us to prove the existence of a scaling limit at least for small ε. Another disadvantage is that it is impossible to calculate the expansion coefficients analytically, except for one- and two loop-diagrams. In one respect the three-dimensional approach has, however, an advantage: the power series have long been available up to six loops for ω, η, and ν, and there exist recent results up to seven loops for the critical exponents η and ν. Although numerical expansions are not as esthetic as the exact expansions derived in this book, their increased order leads to a higher accuracy in the critical exponents, as we shall see below in Section 20.4. Another approach is based on high-temperature expansions for lattice models of the O(N)symmetric classical Heisenberg model with the energy of Eq. (1.52). The critical exponents will be derived from such expansions in Section 20.8. Finally we take advantage of the fact that the O(N)-symmetric classical Heisenberg model is exactly solvable in the limit of large N, where it reduces to the spherical model [recall the remark after Eq. (1.69)]. It is therefore possible to expand the φ 4-theory around this limit in powers of 1/N. The results will be compared with those from the perturbation expansions in
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.