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Physical Review Letters

Microcanonical Renormalization Group

Michael J. Creutz, Andreas Gocksch, Michael C. Ogilvie, Masanori Okawa

Published Aug 27, 1984• 48 citations
Publisher

Abstract

We argue that microcanonical Monte Carlo techniques can provide direct information on the values of renormalized coupling constants in numerical renormalization-group studies. The method is tested on SU(2) lattice gauge theory with fundamental and adjoint couplings and on the two-dimensional O(3) Heisenberg model.
DOI
10.1103/physrevlett.53.875

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We report the results of a microcanonical simulation of the two- and three-dimensional Ising models at criticality. We also present a microcanonical algorithm that allows a simultaneous simulation of a lattice spin system at different energies, in our case the number of different energies is 32. The critical behavior of the systems was studied via a recently proposed microcanonical renormalization-group technique that yields independent estimates for the critical energy and the critical temperature, as well as for three critical exponents providing a direct test of hyperscaling. Our results in two dimensions are in good agreement with exact results. In the three dimensions our quoted values are consistent with Monte Carlo estimates recently reported in the literature. We obtain uc{\mathit{u}}_{\mathit{c}}uc​=-0.991(1), Tc{\mathit{T}}_{\mathit{c}}Tc​=4.5112(3), \ensuremath{\beta}/\ensuremath{\nu}=0.541(1), \ensuremath{\nu}=0.630(3), and \ensuremath{\alpha}=0.109(1).
We implement a momentum-space renormalization-group procedure into the Monte Carlo renormalization-group (MCRG) method. Usually the symmetries of the system must be known and incorporated into the renormalization group, but here we let the renormalization group be given by the Monte Carlo simulation itself. This allows the MCRG method to be applicable to systems where the underlying symmetries may not be known, such as in Monte Carlo studies of quantum systems. As a test of this method, we apply it to the study of the two-dimensional square-lattice q-state Potts model, both for integer values of q and for noninteger values of q near q=4.
Monte Carlo renormalization-group methods were applied to the classical three-component Heisenberg model on a two-dimensional lattice. Expectation values of local correlations of spins and various sized block spins were computed using traditional Monte Carlo methods. By matching quantities at different length scales generated by different Monte Carlo Hamiltonians we directly determined the renormalization of the nearest-neighbor coupling. Using these data and the results of high-temperature expansions and low-temperature renormalization-group calculations we have determined that this model does not have a phase transition. We have also obtained the amplitude for the low-temperature divergence of the susceptibility and the correlation length.