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# Triviality problem and high-temperature expansions of higher susceptibilities for the Ising and scalar-field models in four-, five-, and six-dimensional lattices

Dipartimento di Fisica Universita' di Milano-Bicocca and Istituto Nazionale di Fisica Nucleare Sezione di Milano-Bicocca 3 Piazza della Scienza, I-20126 Milano, Italy.

Physical Review E (Impact Factor: 2.29). 02/2012; 85(2 Pt 1):021105. DOI: 10.1103/PhysRevE.85.021105 Source: PubMed

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**ABSTRACT:**We have extended, in most cases through 24th order, the series expansions of the dimer density in powers of the activity in the case of bipartite [(hyper)-simple-cubic and (hyper)-body-centered-cubic] lattices of dimensionalities 2⩽d⩽7. A numerical analysis of these data yields estimates of the exponents characterizing the Yang-Lee edge singularities for lattice ferromagnetic spin models as d varies between the lower and the upper critical dimensionalities. Our results are consistent with, but more extensive and sometimes more accurate than, those obtained from the existing dimer series or from the estimates of related exponents for lattice animals, branched polymers, and fluids. We mention also that it is possible to obtain estimates of the dimer constants from our series for the various lattices. - [Show abstract] [Hide abstract]

**ABSTRACT:**The high-temperature expansion coefficients of the ordinary and the higher susceptibilities of the spin-1/2 nearest-neighbor Ising model are calculated exactly up to the 20th order for a general d-dimensional (hyper)-simple-cubical lattice. These series are analyzed to study the dependence of critical parameters on the lattice dimensionality. Using the general $d$ expression of the ordinary susceptibility, we have more than doubled the length of the existing series expansion of the critical temperature in powers of 1/d. - [Show abstract] [Hide abstract]

**ABSTRACT:**Above the upper critical dimension, the breakdown of hyperscaling is associated with dangerous irrelevant variables in the renormalization group formalism at least for systems with periodic boundary conditions. While these have been extensively studied, there have been only a few analyses of finite-size scaling with free boundary conditions. The conventional expectation there is that, in contrast to periodic geometries, finite-size scaling is Gaussian, governed by a correlation length commensurate with the lattice extent. Here, detailed numerical studies of the five-dimensional Ising model indicate that this expectation is unsupported, both at the infinite-volume critical point and at the pseudocritical point where the finite-size susceptibility peaks. Instead the evidence indicates that finite-size scaling at the pseudocritical point is similar to that in the periodic case. An analytic explanation is offered which allows hyperscaling to be extended beyond the upper critical dimension.