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Frontiers in Physics

Minimalistic real-space renormalization of Ising and Potts Models in two dimensions

Gary Willis, Gunnar Pruessner, Jonathan Keelan

Published Jun 23, 2015• 0 citations
Publisher

Abstract

We introduce and discuss a real-space renormalization group (RSRG) procedure on very small lattices, which in principle does not require any of the usual approximations, e.g. a cut-off in the expansion of the Hamiltonian in powers of the field. The procedure is carried out numerically on very small lattices (4x4 to 2x2) and implemented for the Ising Model and the q=3,4,5 Potts Models. Nevertheless, the resulting estimates of the correlation length exponent and the magnetization exponent are typically within 3% to 7% of the exact values. The 4-state Potts Model generates a third magnetic exponent which seems to be unknown in the literature. A number of questions about the meaning of certain exponents and the procedure itself arise from its use of symmetry principles and its application to the q=5 Potts Model.

DOI
10.3389/fphy.2015.00046

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We introduce and discuss a real-space renormalization group (RSRG) procedure on very small lattices, which in principle does not require any of the usual approximations, e.g. a cut-off in the expansion of the Hamiltonian in powers of the field. The procedure is carried out numerically on very small lattices (4x4 to 2x2) and implemented for the Ising Model and the q=3,4,5 Potts Models. Nevertheless, the resulting estimates of the correlation length exponent and the magnetization exponent are typically within 3% to 7% of the exact values. The 4-state Potts Model generates a third magnetic exponent which seems to be unknown in the literature. A number of questions about the meaning of certain exponents and the procedure itself arise from its use of symmetry principles and its application to the q=5 Potts Model.
We develop a novel real-space renormalization group (RG) scheme which accurately estimates correlation length exponent ννν near criticality of higher-dimensional quantum Ising and Potts models in a transverse field. Our method is remarkably simple (often analytical), grouping only a few spins into a block spin so that renormalized Hamiltonian has a closed form. A previous difficulty of spatial anisotropy and unwanted terms is avoided by incorporating rotational invariance and internal Zq\mathbb{Z}_qZq​ symmetries of the Hamiltonian. By applying this scheme to the (2+1)-dim Ising model on a triangular lattice and solving an analytical RG equation, we obtain ν≈0.6300ν\approx 0.6300ν≈0.6300. This value is within statistical errors of the current best Monte-Carlo result, 25th-order high-temperature series expansions, ϕ4ϕ^4ϕ4-theory estimation which considers up to seven-loop corrections and experiments performed in low-Earth orbits. We also apply the scheme to higher-dimensional Potts models for which ordinary Monte-Carlo methods are not effective due to strong hysteresis and suppression of quantum fluctuation in a weak first-order phase transition.
The two- and three-dimensional transverse-field Ising models with ferromagnetic exchange interactions are analyzed by means of the real-space renormalization-group method. The basic strategy is a generalization of a method developed for the one-dimensional case, which exploits the exact invariance of the model under renormalization and is known to give the exact values of the critical point and critical exponent ν. The resulting values of the critical exponent ν in two and three dimensions are in good agreement with those for the classical Ising model in three and four dimensions. To the best of our knowledge, this is the first example in which a real-space renormalization group on (2+1)- and (3+1)-dimensional Bravais lattices yields accurate estimates of the critical exponents.
Real Space Renormalization Group (RSRG) treatment of Ising model for square and simple cubic lattice is investigated and critical coupling strengths of these lattices are obtained. The mathematical complications, which appear inevitable in the decimated partition function due to Block-spin transformation, is treated with a relevant approximation. The approximation is based on the approximate equivalence of [Formula: see text] for small [Formula: see text], where K is the nearest neighbor coupling strength and [Formula: see text] is the nearest neighbor spins degrees of freedom around a central spin. The values of the critical coupling strengths are obtained as 0.4830 for square lattice and 0.2225 for simple cubic (SC) lattice. The corresponding critical exponents values [Formula: see text] and [Formula: see text] are also calculated within very acceptable agreement with those values obtained from numerical works.
We explain the recent numerical successes obtained by Tao Xiang's group, who developed and applied tensor renormalization group methods for the Ising model on square and cubic lattices, by the fact that their new truncation method sharply singles out a surprisingly small subspace of dimension two. We show that in the two-state approximation, their transformation can be handled analytically, yielding a value of 0.964 for the critical exponent $\ensuremath{\nu}$ much closer to the exact value 1 than the 1.338 value obtained in the Migdal-Kadanoff approximation. We propose two alternative blocking procedures that preserve the isotropy and improve the accuracy to $\ensuremath{\nu}=0.987$ and 0.993, respectively. We discuss applications to other classical lattice models, including models with fermions, and suggest that it could become a competitor for Monte Carlo methods suitable for accurate calculations of critical exponents, taking continuum limits, and the study of near-conformal systems in arbitrarily large volumes.
We describe how to evaluate approximately various physical interesting quantities in random Ising systems by direct renormalization of a finite system. The renormalization procedure is used to reduce the number of degrees of freedom to a number that is small enough, enabling direct summing over the surviving spins. This procedure can be used to obtain averages of functions of the surviving spins. We show how to evaluate averages that involve spins that do not survive the renormalization procedure. We show, for the random field Ising model, how to obtain $\overline{\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{r})}=\overline{〈\ensuremath{\sigma}(0)\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{r})〉\ensuremath{-}〈\ensuremath{\sigma}(0)〉〈\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{r})〉},$ the ``connected'' correlation function, and $\overline{S(\stackrel{\ensuremath{\rightarrow}}{r})}=\overline{〈\ensuremath{\sigma}(0)\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{r})〉},$ the ``disconnected'' correlation function. Consequently, we show how to obtain the average susceptibility and the average energy. For an Ising system with random bonds and random fields, we show how to obtain the average specific heat. We conclude by presenting our numerical results for the average susceptibility and the function $\overline{\ensuremath{\Gamma}(r)}$ along one of the principal axes. (In this work, the full three-dimensional (3D) correlation is calculated and not just parameters such $\ensuremath{\nu}$ or $\ensuremath{\eta}).$ The results for the average susceptibility are used to extract the critical temperature and critical exponents of the 3D random field Ising system.
The one-dimensional q-state Potts model with ferromagnetic pair interactions which decay with the distance r as is considered. We calculate, through a real-space renormalization group technique using Kadanoff blocks of length b, the critical temperature and the correlation length critical exponent as a function of for different values of q. Some of the very few known rigorous results for general q are reproduced by our approach. Several asymptotic behaviours are derived analytically for q = 2, 3 in the limit. We also obtain extrapolated critical temperatures for arbitrary values of and for q = 2, 3, 4, which we believe approximate the exact ones well, except in the region near . Furthermore, the use of another extrapolation procedure suitable only in the vicinity of led us to conjecture that the exact critical temperature is the same for any value of q . We also verify that , which is consistent with a recent conjecture of Tsallis.