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Physical review. B, Condensed matter

Symmetry-breaking fixed points in the two-dimensional planar model

Ganpathy N. Murthy

Published Jan 1, 1990• 0 citations
Publisher

Abstract

Perturbative fixed points and critical behavior are found for the planar model with symmetry-breaking interactions. These interactions favor p directions for the planar spin. The expansion parameter is \ensuremath{\surd}\ensuremath{\epsilon} , where \ensuremath{\epsilon}=(p-4)/4. .AE
DOI
10.1103/physrevb.41.776

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Landau-Ginzburg-Wilson symmetry analyses and Monte Carlo calculations for the classical antiferromagnetic planar (XY\mathrm{XY}XY) model on a triangular lattice reveal a wealth of interesting critical phenomena. From this simple model arise a zero-field transition to a state of long-range order, a new mechanism for spin disordering, and a critical point associated with a possible new universality class.
A renormalization-group technique is used to study the critical behavior of spin models in which each interaction has a small independent random width about its average value. The cluster approximation of Niemeyer and Van Leeuwen indicates that the two-dimensional Ising model has the same critical behavior as the homogeneous system. The $\ensuremath{\epsilon}$ expansion for nnn-component continuous spins shows that this behavior holds to first order in $\ensuremath{\epsilon}$ for n>4n>4n>4. For n<4n<4n<4, there is a new stable fixed point with $2\ensuremath{\nu}=1+[\frac{3n}{16(n\ensuremath{-}1)}]\ensuremath{\epsilon}$.
A renormalization-group approach is used for investigating the effects of the Parisi replica symmetry breaking (``spin-glass effects'') on critical properties of d-dimensional systems with extended quenched disorder along ${\ensuremath{\varepsilon}}_{d}$ directions. Within a double expansion scheme to first order in $(\ensuremath{\varepsilon}=4\ensuremath{-}d, {\ensuremath{\varepsilon}}_{d})$ it is found that for $d<~4+{\ensuremath{\varepsilon}}_{d}$ and order-parameter dimensionalities n< 4,n<~4,n< 4, a full fixed-point instability with a consequent runaway towards a strong-coupling regime occurs in the parameter space while, for d<4d<4d<4 and n>4,n>4,n>4, compatibly with the modified Harris criterion, an additional marginally stable fixed point exists which might control a new anisotropic random critical behavior. In any case, the traditional replica-symmetric Dorogovtsev-like solution for classical and quantum systems is unstable against the replica symmetry-breaking modes.
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