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Journal of Physics A Mathematical and General

Fisher renormalisation and the application of RSRG methods to annealed dilute systems

Heber R. da Cruz, Robin B. Stinchcombe

Published Aug 21, 1986• 2 citations
Publisher

Abstract

The application of RSRG methods to annealed random systems is studied with special reference to the violation of Fisher renormalisation by approximate techniques. An RSRG cluster approximation is first applied to the annealed dilute Ising model to obtain flowlines and fixed points. Other possible forms of scaling are considered, and how they may be tested against the scaling of equivalent pure models. It is argued that to satisfy Fisher renormalisation it is necessary to allow the probability distribution of the annealed system to evolve, under length scaling, from its initial simple form. The evolution is investigated for the annealed dilute q-state Potts model on the diamond Berker lattice (which can have either sign for the pure specific heat exponent alpha ). By using the width of the distribution as a crucial additional scaling parameter, it is shown how the flow in the increased parameter space can satisfy the Fisher renormalisation criterion and allow the proper appearance of the random fixed point. The treatment is less complete than that for the quenched case because of the need to approximate a relation between indicator functions and the probability distribution.

DOI
10.1088/0305-4470/19/12/022

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Scale-invariant quenched disorder and its stability criterion at random critical points

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Physical review. B, Condensed matter•Mar 1, 1984•104 citations
The application of RSRG methods to annealed random systems is studied with special reference to the violation of Fisher renormalisation by approximate techniques. An RSRG cluster approximation is first applied to the annealed dilute Ising model to obtain flowlines and fixed points. Other possible forms of scaling are considered, and how they may be tested against the scaling of equivalent pure models. It is argued that to satisfy Fisher renormalisation it is necessary to allow the probability distribution of the annealed system to evolve, under length scaling, from its initial simple form. The evolution is investigated for the annealed dilute q-state Potts model on the diamond Berker lattice (which can have either sign for the pure specific heat exponent alpha ). By using the width of the distribution as a crucial additional scaling parameter, it is shown how the flow in the increased parameter space can satisfy the Fisher renormalisation criterion and allow the proper appearance of the random fixed point. The treatment is less complete than that for the quenched case because of the need to approximate a relation between indicator functions and the probability distribution.
The authors elucidate the violation of the Harris criterion that often occurs in real-space renormalization group (RSRG) treatments of quenched random systems. The system studied is the bond-diluted q-state Potts model on the diamond Berker lattice, since it allows either sign for the pure specific heat exponent alpha which is calculated exactly and satisfies a generalised hyperscaling relation involving the fractal dimension. It is argued that it is necessary to allow the bond probability distribution to evolve, under length scaling, from its initial binary form. The width of the distribution (excluding the weight of the missing bonds) is found to play a crucial role and is used as an additional parameter. These conclusions are confirmed by actual calculation for this system of the RG flow in a generalised parameter space, in which the random fixed point is located. The associated fixed-point distributions are discussed, and it is demonstrated that the generalised view allows the Harris criterion to be satisfied.
"Renorma1ization-group methods have been applied in the study of quenched random magnetic systems in recent years. We begin with a brief review of second-order phase transitions in pure, homogeneous systems and also of the .. renorma1ization group framework. Then we provide an introduction to quenched random magnetic systems. Next, momentum-space methods and position-space techniques as applied to quenched random magnets are outlined and compared. Grinstein and Luther applied the Wilson-Fisher E-expansion to random n-vector models; Khme1'nitsky discovered that the random Ising model (n = 1) possessed a ""random"" fixed point of 0(squareroot(e)). This fixed point was found to have one marginal and one irrelevant operator. We have investigated the stability of this fixed point using Ca11an-Symanzik equations and renorma1ized perturbation theory. We find the fixed point stable in the next order; we have also obtained critical exponents to one higher order. Next, position-space techniques are used to study some simple model systems. In addition to critical exponents, global thermodynamic properties are determined. These calculations are based on the Migda1-Kadanoff approximate recursion relations suitably generalized to the inhomogeneous case. Firstly we study the randomly bond-dilute two-dimensional nearest- neighbor Ising model on a square lattice. Calculations give both thermal and magnetic exponents associated with the percolative fixed point. Differential recursion relations yield a phase diagram which is in quantitative agreement with all known results. Curves for the specific heat, percolation probability, and magnetization are displayed. The critical region of the specific heat becomes unobservably narrow well above the percolation threshold Pc. This provides a possible explanation for the apparent specific-heat rounding in certain experiments. We then study the Edwards-Anderson model of a spin glass. The current theoretical situation, which is far from satisfactory at present, is briefly reviewed. We treat the spin-l/2 Ising model with independently random nearestneighbor interactions in dimensionalities d = 2, 3, and 4. The phase diagram, which is in qualitative agreement with mean-field results, exhibits paramagnetic, ferromagnetic, antiferromagnetic, and spin-glass phases. The spinglass and paramagnetic phases meet along an extended second-order phase boundary, which terminates in two tricritical points. Critical and tricritical exponents are calculated. The spin-glass specific-heat exponent turns out to be large and negative, compatibly with recent experiments which show a rounded specific heat anomaly. Global specific-heat curves are also displayed for d = 2."
A zero-temperature real space renormalisation group block method is applied to the random quantum Ising model with a transverse field on the planar honeycomb and square lattices. For the bound diluted system the magnetisation and the separation of the ground-state energy level (in the paramagnetic phase) are presented for several bond concentrations p. The critical exponents extracted both from the fixed points and from direct numerical computations preserve some scaling relations, and the critical curve displays a characteristic discontinuity at the percolation concentration. For the McCoy and Wu distribution (1968) the random fields and bonds are found to introduce a strong relevant disorder. The order parameter still falls off continuously to zero for well defined values of the parameters but a new fixed point yields a slight change in the critical exponents.
A position space renormalisation group technique (decimation) is used to treat site- and bond-diluted Ising ferromagnets on square and triangular lattices. In every case the renormalisation equations lead to two fixed points, one corresponding to the pure Ising system and one to percolation. Linearising around the fixed points, eigenvalues and hence the critical exponents nu T , nu P (correlation length), alpha (specific heat) and phi (crossover) are obtained. Numerical iteration of the recursion relations provides the critical curves. Values of critical temperatures, critical concentrations, exponents, and of limiting slopes of critical curves are in very satisfactory agreement with all known exact or series results. Points on the critical curve flow into the Ising fixed point under the transformation. The sign of the pure Ising exponent alpha is consistent with this flow for the site-diluted cases but not for the bond-diluted cases.
The critical properties of systems with quenched bond disorder are determined from a fixed distribution, under renormalization group, of the random bonds. Full fixed distributions with all moments are obtained numerically by histograms and, to a good approximation, in terms of $\ensuremath{\Gamma}$ distributions. For such systems, the specific-heat exponent $\ensuremath{\alpha}$ does not equal the crossover exponent $\ensuremath{\varphi}$ at random criticality. We derive a new relation between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$, which invokes characteristics of the fixed distribution. The difference between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$ is noted for nnn-vector models in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and for Potts models on hierarchical lattices solved exactly. In general, stable random critical behavior with positive $\ensuremath{\alpha}$ appears to be possible. We develop a general treatment of quenched disorder and illustrate it by calculating specific-heat curves. It is suggested that the critical exponents of the three- and four-state random-bond Potts models in two dimensions are $\ensuremath{\nu}\ensuremath{\simeq}1.06 \mathrm{and} 1.19$.