Spin-Spin Correlations in an Ising Model for Which Scaling is Exact

Abstract

with the requirement of conservation of energy, this constitutes a selection rule so restrictive that no currents would be observable in our experimental range. In tunneling from axially symmetric vortex states into the plane-wave states of a normal metal, the selection rules, according to our preliminary calculations, are much less restrictive. Again there are difficulties in applying such a rule to a vortex in dirty material in which /x is not well defined, but we expect that an equivalent rule will apply, limiting the amount of tunneling, at least within a mean free path of the vortex center.In conclusion, we believe that the vortex cores in our films are well aligned across our tunneling barriers, that the # 2 -dependent tunnel currents in our S^I-Sz junctions arise from those regions outside the smaller vortex cores whose properties are not independent of the vortex packing density, and, finally, that although the tunneling from a vortex core to a normal metal may show the usual probability, the tunneling between two vortex cores is reduced to negligible proportions by the requirement that angular momentum be conserved. 1 M. Tinkham, Phys. Rev.In their recent work using the renormalization-group approach, Fisher, Ma, and Nickel derived results for 17 when there is a long-range interaction, r~d~°, which were seemingly at variance with the numerical results of Nagle and Bonner. The problem is the presence of nonthermodynamic terms in the spin-spin correlation function which nevertheless contribute to the thermodynamics. We also find that the scaling relation between 6 and v\ fails to hold for d >2a.Recently Fisher, Ma, and Nickel 1 have computed theoretically the value of the critical index 77 for a system with long-range spin-spin interactions in d dimensions of the form l/r d+°. They employed the renormalization-group approach introduced by Wilson 2,3 and Fisher, 4 which has been rather successful in this general area. Consequently it is somewhat disturbing that these results (d=l, cr <0.3) do not agree with the careful numerical estimates of Nagle and Bonner. 5 Baker 6 showed that there is a model, with long-range forces (but not translationally invariant), in which the approximate recursion relations, derived by Wilson 2 using the renormalization-group approach, are exact. Since this model is exactly solvable using renormalization-group methods, we have investigated its solution in order to attempt to resolve the discrepancy. We found that it is necessary for cr/d < §, when considering spinspin correlations in this model, to distinguish between long long-range order and short longrange order. That is to say, we may have in the thermodynamic limit, when the system size tends to infinity, a different behavior when the length considered is long compared to the lattice spacing but short compared to the system size than we get when it is comparable to the system size. The quantity studied numerically by Nagle and Bonner, 5 called rj.by them, is a characteristic of long long-range order, while the usual definiti...

The renormalization group and critical phenomena

“…More general multicritica1 points have been analyzed (see, for example, Fisher, 1974). Effects of dipolar forces Aharony and Fisher, 1973;Aharony, 1973aAharony, , 1973b, other long-range forces (Suzuki, 1972(Suzuki, , 1973aBaker and Golner, 1973;Suzuki et al , 1972;Sak, 1973), cubic perturbations, and anisotropies (Pfeuty and Fisher, 1972;Wegner, 1972b;Wallace, 1973;Ketley and Wallace, 1973;Aharony, 1973cAharony, , 1973dSuzuki, 1973b;Liu, 1973;Grover, 1973;Chang and Stanley, 1973"a recent reference is Blankschtein and Makamel, 1982) were pursued. The problems of dynamics of critical behavior were extensively studied.…”

Scaling and Finite-Size Scaling above the Upper Critical Dimension

“…Baker and Golner also determined a negative η in an analytical study of an Ising model for which scaling is exact. 68 Their explanation was that there are two long-range distance scales: long long range and short long range. They posited that long long-range order is controlled by a new, different anomalous dimension, while "short long-range order" is controlled by the usual η.…”

Ensemble renormalization group for the random-field hierarchical model

“…Even for modest sizes (i.e. n = 5) the phase diagram that we obtain is quite good, and the critical line gives the expected values for h c and T c , apart from finite size corrections [10,15] (see [20]). …”

“…The variable ρ controls the strength of the interaction: for ρ ∈ [1; 2) the model has a ferromagnetic phase transition. When 1 < ρ < 3/2 the transition is mean field like whereas when 3/2 < ρ < 1 it is non-mean field [10]. Thus, HM can be used to study phase transitions in nonmean field systems, alike to D−dimensional short range models, with D smaller than the upper critical dimension (i.e.…”

“…It was later understood that the topology of this model could be used to implement an exact real-space RG transformation [9]. Therefore, analytical and numerical studies were pursued in this direction [10], [11], [12][13][14][15]. Among the numerical works, an approximate real space transformation was suggested as a new approach to deal with the RG in disordered systems.…”