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Journal of Applied Physics

Logarithmic corrections and antiferromagnetic chains (abstract)

Ian K. Affleck, J. C. Bonner

Published May 1, 1990• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.345903

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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.
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.
The authors are concerned with the critical properties of antiferromagnetic Takhtajan-Babujian models with spin S=1, 3/2 and 2. The leading eigenenergies of this Hamiltonian, in a finite chain, are calculated by investigating numerically and analytically the Bethe ansatz equations for the finite system. The critical exponents and the conformal anomaly are obtained from their relations with the eigenspectrum of the finite Hamiltonian. The appearance of logarithmic corrections produces poor estimates. However, a combination of analytical and numerical methods produces very good estimates. Their results strongly support the conjecture that the Wess-Zumino-Witten-Novikov non-linear sigma models with topological charge k=2S are the underlying field theories for these spin-S statistical mechanics models.
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 $\frac{{J}_{2}}{J}=0.241167\ifmmode\pm\else\textpm\fi{}0.000005$.
Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.