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Journal of the Physical Society of Japan

Validity of the Finite-Size-Correction Method Based on Conformal Field Theories

Tohru Koma

Published Oct 15, 1993• 2 citations
Publisher

Abstract

We question the validity of the finite-size-correction method based on conformal field theories. It is shown that the method leads to incorrect critical exponents on some of the correlation functions for a one-dimensional free fermion model. The model exhibits the same critical behaviour as in the well-known square-lattice Ising model that is believed to be conformally invariant.

DOI
10.1143/jpsj.62.3402

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We question the validity of the finite-size-correction method based on conformal field theories. It is shown that the method leads to incorrect critical exponents on some of the correlation functions for a one-dimensional free fermion model. The model exhibits the same critical behaviour as in the well-known square-lattice Ising model that is believed to be conformally invariant.
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A method to calculate perturbative corrections to the spectrum of a one-dimensional quantum chain in the scaling variable z is developed from the conformal invariance at its fixed point. Then, from the knowledge of certain correlation functions, one can calculate the first coefficients of the scaling functions. The method is applied successfully to the Ising model, where the corrections are also known from the exact solution. The first coefficients of the magnetic scaling functions of the Ising model are also obtained.
(received; accepted) PACS. 75.10Hk – Classical spin models. PACS. 75.40Mg – Numerical simulation studies. PACS. 11.25Hf – Conformal field theory, algebraic structures. Abstract. – Conformal field theory predicts finite-size scaling amplitudes of correlation lengths universally related to critical exponents on sphere-like, semi-finite systems S d−1 ×R of arbitrary dimensionality d. Numerical studies have up to now been unable to validate this result due to the intricacies of lattice discretisation of such curved spaces. We present a cluster-update Monte Carlo study of the Ising model on a three-dimensional geometry using slightly irregular lattices that confirms the validity of a linear amplitude-exponent relation to high precision. Introduction. – The observation that the symmetry of systems at a critical point goes beyond scale invariance emerging from the divergence of correlation lengths, and additionally comprises rotational and translational as well as inversional invariance has greatly enlarged the scope of exact results that can be derived for such systems on field-theory grounds. While customary scaling theory and the real-space renormalisation ansatz [1, 2] give rise to general
Conformal field theory predicts finite-size scaling amplitudes of correlation lengths universally related to critical exponents on sphere-like, semi-finite systems S d − 1 × of arbitrary dimensionality d . Numerical studies have up to now been unable to validate this result due to the intricacies of lattice discretisation of such curved spaces. We present a cluster-update Monte Carlo study of the Ising model on a three-dimensional geometry using slightly irregular lattices that confirms the validity of a linear amplitude-exponent relation to high precision.
Finite-size scaling and transfer-matrix techniques are used to determine the conformal anomaly and critical exponents of O(n) models on the square lattice. These calculations were performed on five branches of critical points parametrised by n. The results for two of the branches agree with the known universal properties of the O(n) model as derived for the honeycomb lattice. Finite-size convergence is poor for most of the third branch, so that in this case the classification in terms of known models is uncertain. The authors' data show that the fourth branch can be interpreted in terms of a combination of a low-temperature O(n) and a critical Ising model. Finally, the numerical results for the O(n) model on the remaining branch agree accurately with the known results for the low-temperature O(n+1) model. They prove this equivalence by means of an exact mapping.