We propose a simple method of Monte Carlo renormalization group. Thermodynamic quantities of the block-spin system are analyzed phenomenologically on the basis of scaling concept. The field exponent yH is obtained from the square of block-spin magnetization at the fixed point, while the temperature exponent yT is obtained as a consequence of nonlinearity of the scaling fields. In order to check the effectiveness of the method, we perform calculation on the three-dimensional Ising model. The critical temperature and the two critical exponents are estimated precisely. Finite-size effect is shown to be weak, whose origin is the regular part of the free energy density. Statistical errors of the ratios of the block-spin quantities are proved to be surprisingly small compared to those of the block-spin quantities themselves.
We present a real-space renormalization-group (RG) study of the
n-component classical Heisenberg model in the large-
n limit. We obtain exact expressions for correlation functions of block spins, given those of the original spins. We show how to extract the critical behavior of the model from such information. The model provides a good testing ground for evaluating the effectiveness of Monte Carlo RG techniques. We study numerically the implementation of the real-space RG transformation on finite lattices. In two dimensions, we obtain the $\ensuremath{\beta}$ function as a function of temperature. In three dimensions, we calculate the critical temperature and the thermal exponent. We discuss the effects of the use of finite lattices and of the use of different forms for the initial action on the accuracy of the results.
We obtain the critical parameters for the site-percolation problem on the square lattice to a high degree of accuracy (comparable to that of series expansions) by using a Monte Carlo position-space renormalization-group procedure directly on the site-occupation probability. Our method involves calculating recursion relations using progressively larger lattice rescalings,
b. We find smooth sequences for the value of the critical percolation concentration
pc(b) and for the scaling powers
yp(b) and
yh(b). Extrapolating these sequences to the limit $b\ensuremath{\rightarrow}\ensuremath{\infty}$ leads to quite accurate numerical predictions. Further, by considering other weight functions or "rules" which also embody the essential connectivity feature of percolation, we find that the numerical results in the infinite-cell limit are in fact "rule independent." However, the actual fashion in which this limit is approached does depend upon the rule chosen. A connection between extrapolation of our renormalization-group results and finite-size scaling is made. Furthermore, the usual finite-size scaling arguments lead to independent estimates of
pc and
yp. Combining both the large-cell approach and the finite-size scaling results, we obtain ${y}_{p}=0.7385\ifmmode\pm\else\textpm\fi{}0.0080$ and ${y}_{h}=1.898\ifmmode\pm\else\textpm\fi{}0.003$. Thus we find ${\ensuremath{\alpha}}_{p}=\ensuremath{-}0.708\ifmmode\pm\else\textpm\fi{}0.030$, ${\ensuremath{\beta}}_{p}=0.138(+0.006,\ensuremath{-}0.005)$, ${\ensuremath{\gamma}}_{p}=2.432\ifmmode\pm\else\textpm\fi{}0.035$, ${\ensuremath{\delta}}_{p}=18.6\ifmmode\pm\else\textpm\fi{}0.6$, ${\ensuremath{\nu}}_{p}=1.354\ifmmode\pm\else\textpm\fi{}0.015$, and $2\ensuremath{-}{\ensuremath{\eta}}_{p}=1.796\ifmmode\pm\else\textpm\fi{}0.006$. The site-percolation threshold is found for the square lattice at ${p}_{c}=0.5931\ifmmode\pm\else\textpm\fi{}0.0006$. We note that our calculated value of ${\ensuremath{\nu}}_{p}$ is in considerably better agreement with the proposal of Klein et al. that ${\ensuremath{\nu}}_{p}=\frac{\mathrm{ln}\sqrt{3}}{\mathrm{ln}(\frac{3}{2})}\ensuremath{\cong}1.3548$, than with den Nijs' recent conjecture, which predicts ${\ensuremath{\nu}}_{p}=\frac{4}{3}$. However, our results cannot entirely rule out the latter possibility.
The recently developed tensor renormalization-group (TRG) method provides a highly precise technique for deriving thermodynamic and critical properties of lattice Hamiltonians. The TRG is a local coarse-graining transformation, with the elements of the tensor at each lattice site playing the part of the interactions that undergo the renormalization-group flows. These tensor flows are directly related to the phase diagram structure of the infinite system, with each phase flowing to a distinct surface of fixed points. Fixed-point analysis and summation along the flows give the critical exponents, as well as thermodynamic functions along the entire temperature range. Thus, for the ferromagnetic triangular lattice Ising model, the free energy is calculated to better than 10(-5) along the entire temperature range. Unlike previous position-space renormalization-group methods, the truncation (of the tensor index range D) in this general method converges under straightforward and systematic improvements. Our best results are easily obtained with D=24, corresponding to 4624-dimensional renormalization-group flows.
Accurate calculations of the thermodynamic properties of a system near its critical point are now possible using finite-lattice approximations to renormalization groups. In order to investigate some features of such approximations, we introduce a linear-renormalization-group transformation appropriate to a system of continuous spins like the Gaussian model of Berlin and Kac. We solve the renormalization-group equations for the Gaussian model exactly, and then study in detail a finite-lattice approximation to the renormalization group for the three-dimensional case. The renormalization-group transformation contains a parameter
b. The exact transformation has fixed points only for $b={2}^{\ensuremath{-}\frac{5}{2}}$. The approximate transformation has fixed points for a range of values of
b around ${2}^{\ensuremath{-}\frac{5}{2}}$. Eigenvalues of the transformation, which determine critical exponents, depend on the value of
b used in the calculation. We study the problem of identifying the correct value of
b.
The majority-rule, block-spin, renormalization-group transformation for the
d=3 Ising model is studied, with the use of a recently developed technique for calculating renormalized coupling parameters from a single Monte Carlo simulation. Renormalization-group trajectories and the approximate fixed point are calculated. Improvements in the convergence of the Monte Carlo renormalization-group calculations of critical exponents are discussed.