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Physical review. B, Condensed matter

Privman-Fisher hypothesis on finite systems: Verification in the case of the spherical model of ferromagnetism

Surjit Singh, Raj K. Pathria

Published Apr 1, 1985• 69 citations
Publisher

Abstract

The Privman-Fisher hypothesis on the singular part of the free-energy density of a finite system, near the bulk critical point T=${T}_{c}$, is examined in the context of the spherical model of ferromagnetism. A d-dimensional hypercubical lattice (of size ${N}_{1}$a\ifmmode\times\else\texttimes\fi{}${N}_{2}$a\ifmmode\times\else\texttimes\fi{}...\ifmmode\times\else\texttimes\fi{}${N}_{d}$a) is considered and, subject to periodic boundary conditions, explicit expressions are derived for the free energy, the specific heat, and the magnetic susceptibility of the system at temperatures close to ${T}_{c}$. The relevant scaling functions governing the critical behavior of the system are obtained and, with the use of the asymptotic properties of these functions, various predictions of the Privman-Fisher hypothesis are verified. By implication, the passage of the given system towards standard bulk behavior, as ${N}_{j}$\ensuremath{\rightarrow}\ensuremath{\infty}, is also elucidated.

DOI
10.1103/physrevb.31.4483

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Physical Review Letters•Jul 22, 1985•35 citations
The Privman-Fisher hypothesis on the singular part of the free-energy density of a finite system, near the bulk critical point T=Tc{T}_{c}Tc​, is examined in the context of the spherical model of ferromagnetism. A d-dimensional hypercubical lattice (of size N1{N}_{1}N1​a\ifmmode\times\else\texttimes\fi{}N2{N}_{2}N2​a\ifmmode\times\else\texttimes\fi{}...\ifmmode\times\else\texttimes\fi{}Nd{N}_{d}Nd​a) is considered and, subject to periodic boundary conditions, explicit expressions are derived for the free energy, the specific heat, and the magnetic susceptibility of the system at temperatures close to Tc{T}_{c}Tc​. The relevant scaling functions governing the critical behavior of the system are obtained and, with the use of the asymptotic properties of these functions, various predictions of the Privman-Fisher hypothesis are verified. By implication, the passage of the given system towards standard bulk behavior, as Nj{N}_{j}Nj​\ensuremath{\rightarrow}\ensuremath{\infty}, is also elucidated.
Explicit expressions are derived for the free energy, the specific heat, and the magnetic susceptibility of a spherical model of spins on a d-dimensional hypercubical lattice, of size L1{L}_{1}L1​\ifmmode\times\else\texttimes\fi{}L2{L}_{2}L2​\ifmmode\times\else\texttimes\fi{}...\ifmmode\times\else\texttimes\fi{}Ld{L}_{d}Ld​, under antiperiodic boundary conditions. The relevant scaling functions that govern the critical behavior of the system are obtained and, with the use of the asymptotic properties of these functions, various predictions of the Privman-Fisher hypothesis [Phys. Rev. B 30, 322 (1984)] on the hyperuniversality of finite systems are verified. Approach towards standard critical behavior, both for TTc{T}_{c}Tc​(\ensuremath{\infty}) and T>Tc{T}_{c}Tc​(\ensuremath{\infty}), is examined. In the former case, the approach generally takes place through a power law; only in some special situations does one obtain an exponential law instead. In the latter case, the approach is generally exponential, except for the susceptibility of the system which (somewhat surprisingly) displays a finite-size effect dominantly determined by the surface-to-volume ratio of the lattice.
The finite-size scaling hypothesis of Privman and Fisher (Phys. Rev. B, 30, 322 (1984)) is applied to systems with O(n) symmetry [Formula: see text], confined to geometry Ld − d′ × ∞d′ (where d and d′ are continuous variables such that 2 < d′ < d < 4) and subjected to periodic boundary conditions. Predictions, involving amplitudes as well as exponents, are made on the singular part of the specific heat c(s), spontaneous magnetization m0, magnetic susceptibility χ0, and correlation length ξ0 in the region of the second-order phase transition [Formula: see text]. Analytical verification of these predictions is carried out in the case of the spherical model of ferromagnetism (n = ∞), which includes evaluation of the shift in the critical temperature of the system.
The Privman-Fisher hypothesis on the singular part of the free-energy density of a finite system, near the bulk critical point T=Tc{T}_{c}Tc​, is examined in the context of an ideal relativistic Bose gas confined to a cuboidal enclosure (L1{L}_{1}L1​\ifmmode\times\else\texttimes\fi{}L2{L}_{2}L2​\ifmmode\times\else\texttimes\fi{}L3{L}_{3}L3​) under periodic boundary conditions. Taking into account the possibility of particle-antiparticle pair production in the system, explicit expressions are derived for the free energy, the specific heat, and the condensate density at temperatures close to Tc{T}_{c}Tc​, and the special cases of a cube, a square channel and a film are investigated at length. The various predictions of the Privman-Fisher hypothesis are fully borne out and the scaling functions governing the critical behavior of the system are found to be universal---irrespective of the severity of the relativistic effects. The influence of the latter enters only through the nonuniversal scale factors, C1{C}_{1}C1​ and C2{C}_{2}C2​, which depend on the particle mass m and density \ensuremath{\rho} as well.
We propose a finite-size scaling hypothesis for the ``singular'' part of the free-energy density, f(s){f}^{(s)}f(s)(T,H;L), and the correlation function, G(R,T,H;L), of a system with O(n) symmetry (n\ensuremath{\ge}2), confined to geometry ${L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\infty}}^{d\mathcal{'}}$ (d>4, d'\ensuremath{\le}2) and subjected to periodic boundary conditions. On the basis of this hypothesis, we predict finite-size effects in the zero-field susceptibility, \ensuremath{\chi}(T;L), the correlation length, \ensuremath{\xi}(T;L), and the ``singular'' part of the specific heat, c(s){c}^{(s)}c(s)(T;L), in the region of first-order phase transition (TTc{T}_{c}Tc​) as well as the region of second-order phase transition (T\ensuremath{\simeq}Tc{T}_{c}Tc​). It turns out that in both these regions the system in zero field can be described by a single, unifying variable v\ensuremath{\sim}${L}^{2(d\mathrm{\ensuremath{-}}d\mathcal{'})/(4\mathrm{\ensuremath{-}}d\mathcal{'})}$t\ifmmode \tilde{}\else \~{}\fi{}, where t\ifmmode \tilde{}\else \~{}\fi{} is an appropriate temperature parameter of the system. To test the various predictions, a detailed analytical study is carried out in the case of the spherical model of ferromagnetism (n=\ensuremath{\infty}), and all predictions are seen to be fully borne out. For general n, we compare our findings with those of the previous authors, wherever possible, and in each case the comparison turns out to be perfect.
Finite-size effects on the first-order phase transition in a three-dimensional classical Heisenberg model with nearest-neighbor ferromagnetic interactions and periodic boundary conditions are studied theoretically and numerically in this paper. We find that the current Fisher-Privman theory fails at temperatures close to criticality and/or for small systems. We rederive explicit scaling functions for the longitudinal magnetization and susceptibility under another phenomenological assumption. The results are in very good agreement with Monte Carlo simulations.
We predict the manner in which a physical system, of size ${\mathrm{L}}^{\mathrm{d}\mathrm{\ensuremath{-}}\mathrm{d}\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathrm{d}\mathcal{'}}$, subject to periodic boundary conditions, approaches bulk critical behavior as L\ensuremath{\rightarrow}\ensuremath{\infty}. While for T>Tc{\mathrm{T}}_{\mathrm{c}}Tc​(\ensuremath{\infty}) the approach is exponential, for TTc{\mathrm{T}}_{\mathrm{c}}Tc​(\ensuremath{\infty}) it is generally governed by power laws whose indices are determined by the bulk exponents for the corresponding d- and d'-dimensional systems. Specific predictions on the spherical model of ferromagnetism and the relativistic Bose gas with pair production are verified by analytical results.