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Journal of the Physical Society of Japan

Application of Real-Space Renormalization Group Theory to the Antiferromagnetic Potts Model

Takehiko Oguchi, Takuma Ishikawa

Published Jul 15, 1985• 0 citations
Publisher

Abstract

The real-space renormalization group theory using the first order cumulant is applied to the q -state antiferromagnetic Potts model on 2-d (dimensional) and 3-d lattice. The special features of this paper are that the Potts states are represented by q unit vectors in q -1 dimensional space and the two-sublattice structure is preserved after the renormalization transformation. The critical exponents obtained for q =2 in 3-d are in fairly good agreement with the reliable values and those for q =3 and 4 are obtained for the first time.

DOI
10.1143/jpsj.54.2371

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The real-space renormalization group theory using the first order cumulant is applied to the q -state antiferromagnetic Potts model on 2-d (dimensional) and 3-d lattice. The special features of this paper are that the Potts states are represented by q unit vectors in q -1 dimensional space and the two-sublattice structure is preserved after the renormalization transformation. The critical exponents obtained for q =2 in 3-d are in fairly good agreement with the reliable values and those for q =3 and 4 are obtained for the first time.
A new formulation of the renormalization-group in real space, suitable for quantum spin systems is proposed. The method is applied to the two-dimensional spin-12\frac{1}{2}21​ $X\ensuremath{-}Y$ model on a triangular lattice, the renormalization-group transformation being evaluated up to second order in an appropriate cumulant expansion. To first order an unstable fixed point of the transformation is found, corresponding to a critical temperature and critical indices in satisfactory qualitative agreement with present high-temperature series expansion estimates. In the second-order calculation, however, this fixed point disappears, thus throwing some doubt on the conventional picture of criticality as furnished by high-temperature series. The free energy of the model is also computed. For relatively small values of the nearest-neighbor coupling it is in good agreement with that found by high-temperature series analysis.
Within a real-space renormalization-group framework, we study the qqq-state Potts model in a square lattice, assuming a positive second-neighbor coupling constant and an arbitrary sign for the first-neighbor coupling constant. The qqq evolution of the full phase diagram as well as the thermal and magnetic critical exponents are obtained. Whenever comparison with available results is possible, the agreement is either exact or very satisfactory.
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A previously proposed renormalization-group realization is applied to the qqq-state Potts model for any qqq in (1 + 1)- and (2 + 1)-dimensional lattices of different coordination number. Ferromagnetic and antiferromagnetic critical points are found and the corresponding exponents are computed in a very simple way.
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A generalization of the Niemeijer-van Leeuwen renormalization-group transformation treating disordered cells of spins as vacancies is introduced and applied to the two-dimensional qqq-state Potts model. Finite-lattice approximations yield the changeover from continuous to first-order phase transitions predicted by Baxter but not observed in previous renormalization-group calculations. Exact results are conjectured for the tricritical exponents of dilute Potts models.