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Physical review. B, Condensed matter

Antiferromagnetic clock models in two dimensions

John Cardy

Published Nov 1, 1981• 31 citations
Publisher

Abstract

Antiferromagnetic and asymmetric $p$-state clock models on a square lattice are analyzed with the renormalization group, by studying the effect of $p$-fold symmetry-breaking fields in the planar model. For $p$ odd and \ensuremath{\ge}3 ($p=3$ is the three-state Potts model) these models exhibit two phase transitions with a critical line of continuously varying exponent $\ensuremath{\eta}({\ensuremath{\eta}}_{min}<\ensuremath{\eta}<\frac{1}{4})$ separating them. When the couplings are antiferromagnetic in one direction only, ${\ensuremath{\eta}}_{min}=\frac{2}{{p}^{2}}$, while for the isotropic case ${\ensuremath{\eta}}_{min}=\frac{1}{{p}^{2}}$. In the latter case, the low-temperature phase is characterized by a $2p$-fold degeneracy.

DOI
10.1103/physrevb.24.5128

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We investigate the isotropic-anisotropic phase transition of the twodimensional XY model with six-fold anisotropy, using Monte Carlo renormalization group method. The result indicates difficulty of observing asymptotic critical behavior in Monte Carlo simulations, owing to the marginal flow at the fixed point. PACS numbers: 02.70.Lq, 75.10.Hk Typeset using REVTEX 1 The XY model with Z6 symmetry breaking field represents not only planar spin magnets with Z6 symmetric crystal field [1], but also models with Z2 × Z3 symmetry (which is isomorphic to Z6) such as three-state antiferromagnetic Potts model on the square and the cubic lattice [2] and Ising antiferromagnet on the triangular and the stacked-triangular lattice [3–8]. The critical behavior of the model in two dimension is well understood by scaling argument [1]: The model undergoes two distinct phase transitions, both of which being
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