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Physical review. B, Condensed matter

Group structure of block transformations: Matching conditions for the critical properties of lattice spin systems

Attilio L. Stella, Raf Dekeyser, Joseph O. Indekeu, Flavio Toigo

Published Nov 1, 1979• 8 citations
Publisher

Abstract

Matching procedures, related to the group structure of block-spin renormalization, are proposed for an optimal choice of parameter-dependent transformations. The method also allows to determine the parameter's dependence on the reduced coupling near the fixed point. Applications to linear transformations for two- and three-dimensional Ising models and two-dimensional three-, four-, and five-state Potts systems are reported. With simple approximations good estimates of critical couplings and exponents are obtained. Results for the two-dimensional spin-\textonehalf{} $\mathrm{XY}$ system and their possible implications, as far as an eventual transition is concerned, are discussed. Some applications to nonlinear transformations are also considered.

DOI
10.1103/physrevb.20.3799

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Physical review. B, Solid state•Apr 1, 1976•24 citations
Matching procedures, related to the group structure of block-spin renormalization, are proposed for an optimal choice of parameter-dependent transformations. The method also allows to determine the parameter's dependence on the reduced coupling near the fixed point. Applications to linear transformations for two- and three-dimensional Ising models and two-dimensional three-, four-, and five-state Potts systems are reported. With simple approximations good estimates of critical couplings and exponents are obtained. Results for the two-dimensional spin-\textonehalf{} XY\mathrm{XY}XY system and their possible implications, as far as an eventual transition is concerned, are discussed. Some applications to nonlinear transformations are also considered.
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 bbb. 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 bbb around ${2}^{\ensuremath{-}\frac{5}{2}}$. Eigenvalues of the transformation, which determine critical exponents, depend on the value of bbb used in the calculation. We study the problem of identifying the correct value of bbb.
Most renormalization groups studied heretofore are linear, in that the block-spin variables are linearly related to the old spin variables. These renormalization groups all require an adjustable parameter that must be properly fixed in order that the renormalization-group transformation have a useful fixed point. Niemeijer and van Leeuwen, however, have investigated the critical behavior of the two-dimensional Ising model with a nonlinear renormalization group, and find a fixed point without introducing such a parameter. We introduce here another nonlinear renormalization group and study it in detail for the Gaussian model. We find that (i) within certain limits, no such parameter need be adjusted in order to reach a fixed point; (ii) an eigenvalue of the linearized transformation with no physical significance depends on the nonlinearity of the transformation; and (iii) a physically significant eigenvalue is unchanged (to the order examined).
The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (JJJ) and biquadratic (KKK) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in JJJ, KKK, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (HHH) and another odd interaction (LLL) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.
The Monte Carlo renormalization group is applied to the three-dimensional Ising model on simple cubic lattices with 83{8}^{3}83, 163{16}^{3}163, 323{32}^{3}323, and 643{64}^{3}643 sites. The comparison of block-spin correlation functions from the largest lattices yields the nearest-neighbor critical coupling K1c=0.221654(6){K}_{1}^{c}=0.221654(6)K1c​=0.221654(6). After allowing for (i) interpolation to this best estimate for K1c{K}_{1}^{c}K1c​, (ii) an apparent finite-size effect in the renormalization-group transformation due to the measurement of correlation functions of too few (seven) operators, and (iii) the extrapolation for the effect of a slow transient towards the fixed point, the values $\ensuremath{\nu}=0.629(4)$ and $\ensuremath{\eta}=0.031(5)$ are obtained for the thermal and magnetic exponents. The correction-to-scaling exponent $\ensuremath{\omega}$ is estimated to be around 1; to obtain an accuracy competitive with other methods requires measurements with more than seven operators. We briefly review the problem of redundant operators and indicate the future prospects for this kind of calculation.
The fixed-point topology of Kadanoff's lower-bound renormalization transformation is discussed for the d=2d=2d=2 square-lattice Ising model in the three-dimensional space of coupling constants for the nearest-neighbor, next-nearest neighbor, and four-spin interactions. An obstacle one encounters in trying to calculate a critical surface is pointed out. In the three-dimensional space of coupling constants Kadanoff's fixed point has two relevant vectors and can only be reached from a critical line, not from a critical surface. Another fixed point is reported which does have only one relevant vector and can be reached from a critical surface. Another fixed point is reported which does have only one relevant vector and can be reached from a critical surface. The critical exponents of this fixed point are close to the exact values, though not nearly so close as for Kadanoff's fixed point. The properties of both fixed points as a function of the variational parameter ppp are described. Critical values of the nearest-neighbor couplings leading to the fixed points are compared.