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Annalen der Physik

Numerical tests of conjectures of conformal field theory for three‐dimensional systems

Martin Weigel, Wolfhard Janke E

Published Nov 1, 1998• 0 citations
Publisher

Abstract

Abstract The concept of conformal field theory provides a general classification of statistical systems on two‐dimensional geometries at the point of a continuous phase transition. Considering the finite‐size scaling of certain special observables, one thus obtains not only the critical exponents but even the corresponding amplitudes of the divergences analytically. A first numerical analysis brought up the question whether analogous results can be obtained for those systems on three‐dimensional manifolds. Using Monte Carlo simulations based on the Wolff single‐cluster update algorithm we investigate the scaling properties of O( n ) symmetric classical spin models on a three‐dimensional, hyper‐cylindrical geometry with a toroidal cross‐section considering both periodic and antiperiodic boundary conditions. Studying the correlation lengths of the Ising, the XY, and the Heisenberg model, we find strong evidence for a scaling relation analogous to the two‐dimensional case, but in contrast here for the systems with antiperiodic boundary conditions.

DOI
10.1002/andp.199851005-628

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Abstract The concept of conformal field theory provides a general classification of statistical systems on two‐dimensional geometries at the point of a continuous phase transition. Considering the finite‐size scaling of certain special observables, one thus obtains not only the critical exponents but even the corresponding amplitudes of the divergences analytically. A first numerical analysis brought up the question whether analogous results can be obtained for those systems on three‐dimensional manifolds. Using Monte Carlo simulations based on the Wolff single‐cluster update algorithm we investigate the scaling properties of O( n ) symmetric classical spin models on a three‐dimensional, hyper‐cylindrical geometry with a toroidal cross‐section considering both periodic and antiperiodic boundary conditions. Studying the correlation lengths of the Ising, the XY, and the Heisenberg model, we find strong evidence for a scaling relation analogous to the two‐dimensional case, but in contrast here for the systems with antiperiodic boundary conditions.
The concept of conformal field theory provides a general classification of statistical systems on two-dimensional geometries at the point of a continuous phase transition. Considering the finite-size scaling of certain special observables, one thus obtains not only the critical exponents but even the corresponding amplitudes of the divergences analytically. A first numerical analysis brought up the question whether analogous results can be obtained for those systems on three-dimensional manifolds. Using Monte Carlo simulations based on the Wolff single-cluster update algorithm we investigate the scaling properties of O(n) symmetric classical spin models on a three-dimensional, hyper-cylindrical geometry with a toroidal cross-section considering both periodic and antiperiodic boundary conditions. Studying the correlation lengths of the Ising, the XY, and the Heisenberg model, we find strong evidence for a scaling relation analogous to the two-dimensional case, but in contrast here for the systems with antiperiodic boundary conditions.
The concept of conformal field theory provides a general classification of statistical\nsystems on two-dimensional geometries at the point of a continuous phase transition.\nConsidering the finite-size scaling of certain special observables, one thus obtains not only\nthe critical exponents but even the corresponding amplitudes of the divergences analytically.\nA first numerical analysis brought up the question whether analogous results can be obtained\nfor those systems on three-dimensional manifolds.\nUsing Monte Carlo simulations based on the Wolff single-cluster update algorithm we investigate\nthe scaling properties of O(n) symmetric classical spin models on a three-dimensional,\nhyper-cylindrical geometry with a toroidal cross-section considering both periodic and antiperiodic\nboundary conditions. Studying the correlation lengths of the Ising, the XY, and\nthe Heisenberg model, we find strong evidence for a scaling relation analogous to the twodimensional\ncase, but in contrast here for the systems with antiperiodic boundary conditions.
In a Monte Carlo study using a cluster update algorithm we investigate finite-size scaling (FSS) of the correlation lengths of several representatives of the class of three-dimensional classical O(n)\mathrm{O}(n)O(n) symmetric spin models on the geometry ${T}^{2}\ifmmode\times\else\texttimes\fi{}R$. For all the models we find strong evidence of a linear relation between FSS amplitudes and scaling dimensions when applying antiperiodic instead of periodic boundary conditions across the torus. This type of scaling relation can be proven analytically for systems on two-dimensional strips with periodic boundary conditions using conformal field theory.
(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.
Using an elaborate set of simulational tools and statistically optimized methods of data analysis, we investigate the scaling behavior of the correlation lengths of three-dimensional classical O(n)O(n)O(n) spin models. Considering three-dimensional slabs ${S}^{1}\ifmmode\times\else\texttimes\fi{}{S}^{1}\ifmmode\times\else\texttimes\fi{}R,$ the results over a wide range of n indicate the validity of special scaling relations involving universal amplitude ratios that are analogous to results of conformal field theory for two-dimensional systems. A striking mismatch of the $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}$ extrapolation of these simulations against analytical calculations is traced back to a breakdown of the identification of this limit with the spherical model.