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Journal of Physics A Mathematical and General

Migdal transformations of O(2)-symmetric spin Hamiltonians

Michael N. Barber

Published Dec 1, 1983• 12 citations
Publisher

Abstract

The existence of a massless phase in several different two-dimensional O(2)-symmetric spin models is investigated within the approximate Migdal/Kadanoff renormalisation transformation. The models investigated include the planar rotor model, the truncated model used by Nienhuis (1982) to derive exact results for O(2)-symmetric models, and the step model of Guttmann and Joyce (1973). The truncated model is found to exhibit a 'fixed' line similar to that seen in the planar rotor model, but only for values of the coupling for which the model is unphysical. The step model is found to be always disordered, but a variant exhibits a fixed line. The significance of the results beyond the Migdal approximation is assessed.

DOI
10.1088/0305-4470/16/17/021

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The existence of a massless phase in several different two-dimensional O(2)-symmetric spin models is investigated within the approximate Migdal/Kadanoff renormalisation transformation. The models investigated include the planar rotor model, the truncated model used by Nienhuis (1982) to derive exact results for O(2)-symmetric models, and the step model of Guttmann and Joyce (1973). The truncated model is found to exhibit a 'fixed' line similar to that seen in the planar rotor model, but only for values of the coupling for which the model is unphysical. The step model is found to be always disordered, but a variant exhibits a fixed line. The significance of the results beyond the Migdal approximation is assessed.
The phase diagrams of general qqq-state planar spin models on a square lattice are studied for $q=4\ensuremath{-}7$. The phase diagrams are calculated by a variational method and the infinitesimal Migdal renormalization-group (RG) transformations. Since the Migdal RG commutes with the self-dual transformations of Z(q)Z(q)Z(q) models, one expects reliable phase diagrams (but not accurate critical exponents). In fact all exactly known critical points are reproduced by our calculations together with a fair numerical simulation of lines of critical points. Our results clearly indicate the existence of massless phases for q=6and7q=6 \mathrm{and} 7q=6and7 but are inconclusive for q=5q=5q=5. More accurate methods are needed to determine the phase structure of the Z(5)Z(5)Z(5) model; see, for example, the finite-lattice approximation of Roomany and Wyld (preceding paper).
The quantum spin 1/2 X-Y model in dimensions d = 1, 2 and 3 is studied by means of the extended Migdal approximation for the scale factor of the transformation b = 2. It is found, with the imposed conditions to get the recursion equations, that an unstable fixed point between a T = 0 fixed point and that of T = ∞ exists for d > 1. However, alternate imposed conditions lead to an unusual behavior that the T = 0 fixed point does not appear in any dimension and that a stable fixed point appears at a lower temperature than that of an unstable one for d > 2. The same model on the simple cubic and face-centered cubic lattices is also treated by the two-step decimation transformation.
Two recently proposed methods, the variational cluster wavefunction approximation and the mean-field-like renormalisation group transformation, are used to describe the phase transitions in the Hamiltonian version of the two-dimensional Z(q) symmetric spin models. The convergence of the two schemes to the known results is analysed. It is shown that in their usual form neither of these models can reproduce fine details of the phase diagram, although the renormalisation group method gives in general better convergence.
Abstract We use scale invariant scattering theory to exactly determine the lines of renormalization group fixed points for O(N )-symmetric models with quenched disorder in two dimensions. Random fixed points are characterized by two disorder parameters: a modulus that vanishes when approaching the pure case, and a phase angle. The critical lines fall into three classes depending on the values of the disorder modulus. Besides the class corresponding to the pure case, a second class has maximal value of the disorder modulus and includes Nishimori-like multicritical points as well as zero temperature fixed points. The third class contains critical lines that interpolate, as N varies, between the first two classes. For positive N , it contains a single line of infrared fixed points spanning the values of N from 2−1 \sqrt{2}-1 2​−1 2 − 1 to 1. The symmetry sector of the energy density operator is superuniversal (i.e. N -independent) along this line. For N = 2 a line of fixed points exists only in the pure case, but accounts also for the Berezinskii-Kosterlitz-Thouless phase observed in presence of disorder.
We investigate the controversial issue of the existence of universality classes describing critical phenomena in three-dimensional systems characterized by a matrix order parameter with symmetry $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(N)$ and symmetry-breaking pattern $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(N)\ensuremath{\rightarrow}\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(N\ensuremath{-}2)$. Physical realizations of these systems are, for example, frustrated spin models with noncollinear order. Starting from the field-theoretical Landau-Ginzburg-Wilson Hamiltonian, we consider the massless critical theory and the minimal-subtraction scheme without ϵϵϵ expansion. The three-dimensional analysis of the corresponding five-loop series shows the existence of a stable fixed point for N=2N=2N=2 and N=3N=3N=3, confirming recent field-theoretical results based on a six-loop expansion in the alternative zero-momentum renormalization scheme defined in the massive disordered phase. In addition, we report numerical Monte Carlo simulations of a class of three-dimensional $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(2)$-symmetric lattice models. The results provide further support to the existence of the $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(2)$ universality class predicted by the field-theoretical analyses.
We study the mixed spin-1 and spin-3/2 Blume-Capel model under crystal field in the tridimensional semi-infinite case. This has been done by using the real-space renormalization group approximation and specifically the Migdal-Kadanoff technique. As a function of the ratio R of bulk and surface interactions and the ratios R1 and R2 of bulk and surface crystals fields on the spin-1 and spin-3/2 respectively, we have determined various types of phase diagrams. Besides second- order transition lines, first-order phase transition lines terminating at tricritical points are obtained. We found that there existed nine main types of phase diagram showing a variety of phase transitions associated with the surface, including ordinary, extraordinary, surface and special phase transitions.