2008/02/21 by I. A. Campbell, P. Butera · 12 citations
Mathematics · Physics and Astronomy · #Condensed matter physics #Critical point (mathematics) #Ferromagnetism #Geometry #Ising model #Lattice (music) #Mathematical analysis #Mathematical physics #Mathematics #Physics #Quantum many-body systems #Scaling #Scaling limit #Square lattice #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physrevb.78.024435
published in Physical Review B 78(2) (American Physical Society) · 6 pages, 6 figures
arxiv created 2008/02/21 · openalex publication_date 2008/07/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
If scaling variables and scaling expressions are chosen judiciously, ``critical'' scaling analyses do not need to be restricted to a narrow range of temperatures near the critical point. The standard scaling variable (T\ensuremath-Tc)/Tc is inadapted to wide ranges of temperature because it diverges at high temperatures. With the variable \ensuremathτ=(T\ensuremath-Tc)/T, in the high-dimension (mean-field) limit, the reduced susceptibility and the second-moment correlation length of the Ising ferromagnet depend on temperature as \ensuremathχ(T)=\ensuremathτ^\ensuremath-1 and \ensuremathξ(T)=T^\ensuremath-1/2\ensuremathτ^\ensuremath-1/2 exactly over the entire temperature range above the critical temperature Tc. For the canonical two-dimensional square lattice near-neighbor Ising ferromagnet, it is shown that compact ``extended scaling'' expressions analogous to the high-dimensional limit form but with the appropriate exponents give accurate approximations to the true temperature dependencies, again over the entire temperature range from Tc to infinity. Within this approach, for near-neighbor interaction systems, there is no crossover temperature above which mean-field-like behavior sets in.