Inverse Monte Carlo Renormalization Group Transformations for Critical Phenomena

We introduce a computationally stable inverse Monte Carlo renormalization group transformation method that provides a number of advantages for the calculation of critical properties. We are able to simulate the fixed point of a renormalization group for arbitrarily large lattices without critical slowing down. The log-log scaling plots obtained with this method show remarkable linearity, leading to accurate estimates for critical exponents. We illustrate this method with calculations in two- and three-dimensional Ising models for a variety of renormalization group transformations.

Figure 1

A log-log plot of the linear size of the lattice $L$ versus the average of the squared magnetization for the $d=2$ and $d=3$ Ising models. The line connecting the ▽ is obtained for the $d=2$ fixed point using the seven-coupling Hamiltonian calculated by Swendsen. The line connecting the ○ is obtained for the 17-coupling Hamiltonian calculated by Swendsen to approximate the $d=3$ fixed point. For both cases the RG transformation used is the majority rule for blocks of $2^d$ spins.

Figure 2

A log-log plot of the spin-spin correlation function of distance $r$ versus $r$, for the $d=2$ Ising model with the majority rule, for lattice sizes $L=4,8,\dots ,512$ where $r=1,2,\dots ,L/2$ using the seven-coupling Hamiltonian calculated by Swendsen. The line connecting the □ connects all points for which $r=L/2$, and the $*$ are for $r=L/4$. The dashed line has the exact slope of $-0.25$.