Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics C Solid State Physics

Critical phenomena with many body interactions: an exactly soluble model system

A. McKerrell, Roger Bowers

Published Jan 17, 1972• 21 citations
Publisher

Abstract

A model Hamiltonian with 2-point, 4-point,..., 2n-point infinite-range interactions chosen in a particular way is introduced and the complete thermodynamics is obtained exactly in an elementary manner. The critical exponents alpha '=1-1/n, beta =1/2n, gamma = gamma '=1, delta =2n+1, Delta '=1+1/2n are derived and a minor violation of scaling is found.

DOI
10.1088/0022-3719/5/1/003

Related papers

First order and critical point phase transitions in cooperative assemblies with many-body interactions: some exact results

Roger Bowers, A. McKerrell

The complete thermodynamics of a model hamiltonian with general-spin infinite range Ising 2-point, 4-point,... interaction terms is found exactly in two different statistical mechanical ensembles which prove to be essentially equivalent. The coefficients of the above terms may be chosen to give critical behaviour which is largely consistent with the idea of scaling and which has delta equal to any odd positive integer and gamma =1; alternatively a first order phase transition may be obtained, a finite spontaneous magnetization appearing as the temperature is decreased through that of the transition. The Heisenberg case is also considered and found to be closely similar to the Ising case.

Journal of Physics C Solid State Physics•Sep 5, 1972•12 citations

The critical exponents of Ising model systems with pure three-body interactions. III

David W. Wood, H P Griffiths

For pt. II, see abstr. A64455 of 1973. The critical-point exponents of two Ising models with pure three-spin coupling terms are examined using low-temperature (T<T c ) series expansions for the thermodynamic functions. In particular the critical isotherm exponent delta is examined by applying numerical extrapolation methods to the critical isotherm. The two models are (a) a two-dimensional triangular lattice and (b) a three-dimensional face-centred cubic lattice. Direct numerical estimates of alpha ', beta and delta for (b) show that the scaling relations hold for this model and it is found that alpha '=0.655, beta =0.0538, gamma =1.24, nu =0.448, eta =-0.76, and delta =24 on the basis of the scaling relations. The numerical accuracy of the exponents beta and delta for model (a) does not permit an independent verification of the scaling relations; assuming scaling to hold here also, and using the exact result alpha '= 2 / 3 obtained by Baxter and Wu, it is found that gamma '=1.187+or-0.006, nu = 2 / 3 , eta =0.219+or-0.009, beta =0.073+or-0.003, and delta =17.3+or-0.8.

Journal of Physics C Solid State Physics•Apr 21, 1974•19 citations

Critical Exponents and Scaling Relations in 1/nExpansion. II

S. Hikami

The critical exponents δ and Δ are derived in powers of 1/n to first order for the n-vector model with short range interactions. Scaling relations α=2+γ-Δ, δ=(d+2-η)/(d-2+η) and δ=Δ/(Δ-2γ) are confirmed exactly up to order 1/n. By considering successive derivatives of magnetization with respect to external field and temperature, the equations of state above the critical point M=∑∞m=0cmε-γ-(m-1)ΔH2m-1 and M=∑∞m=0amH1-2(γ+m)/δεm are confirmed up to order 1/n.

Progress of Theoretical Physics•Apr 1, 1973•8 citations

Critical Exponents for Long-Range Interactions. III

Masuo Suzuki

The extended scaled equation of state (namely, the scaled form of the wave-number dependent nonlinear susceptibility) is confirmed up to the first order of 1/n, in the classical n-vector model with general range of interactions. Critical exponents γ, δ, η, ν and Δ are evaluated systematically to order 1/n, both for short-range and long-range interactions to check universality and scaling relations.

Progress of Theoretical Physics•May 1, 1973•17 citations

Critical Exponents and Scaling Relations in 1/nExpansion

Ryuzo Abe, S. Hikami

The critical exponents ν, η and α are derived in powers of 1/n to first order for the n-vector model (or n-component system) with short-range interactions. Scaling relations χ(q)∼ε-γf(q/εν), dν=d-α and γ=ν(2-η) are confirmed up to the order of 1/n.

Progress of Theoretical Physics•Feb 1, 1973•69 citations

Phase boundaries near critical end points. III. Corrections to scaling and spherical models

Márcia C. Barbosa

Thermodynamics in the vicinity of a critical end point is studied. Phenomenological arguments are used to show the presence of further critical nonanalyticities beyond the leading-order singularities found in papers I and II. These contributions are related to the universal correction-to-scaling features of the bulk thermodynamics on the critical \ensuremath{\lambda} line. The predictions are checked on exactly soluble spherical models with short- and long-range interactions.

Physical review. B, Condensed matter•Mar 1, 1992•21 citations
A model Hamiltonian with 2-point, 4-point,..., 2n-point infinite-range interactions chosen in a particular way is introduced and the complete thermodynamics is obtained exactly in an elementary manner. The critical exponents alpha '=1-1/n, beta =1/2n, gamma = gamma '=1, delta =2n+1, Delta '=1+1/2n are derived and a minor violation of scaling is found.
The complete thermodynamics of a model hamiltonian with general-spin infinite range Ising 2-point, 4-point,... interaction terms is found exactly in two different statistical mechanical ensembles which prove to be essentially equivalent. The coefficients of the above terms may be chosen to give critical behaviour which is largely consistent with the idea of scaling and which has delta equal to any odd positive integer and gamma =1; alternatively a first order phase transition may be obtained, a finite spontaneous magnetization appearing as the temperature is decreased through that of the transition. The Heisenberg case is also considered and found to be closely similar to the Ising case.
For pt. II, see abstr. A64455 of 1973. The critical-point exponents of two Ising models with pure three-spin coupling terms are examined using low-temperature (T<T c ) series expansions for the thermodynamic functions. In particular the critical isotherm exponent delta is examined by applying numerical extrapolation methods to the critical isotherm. The two models are (a) a two-dimensional triangular lattice and (b) a three-dimensional face-centred cubic lattice. Direct numerical estimates of alpha ', beta and delta for (b) show that the scaling relations hold for this model and it is found that alpha '=0.655, beta =0.0538, gamma =1.24, nu =0.448, eta =-0.76, and delta =24 on the basis of the scaling relations. The numerical accuracy of the exponents beta and delta for model (a) does not permit an independent verification of the scaling relations; assuming scaling to hold here also, and using the exact result alpha '= 2 / 3 obtained by Baxter and Wu, it is found that gamma '=1.187+or-0.006, nu = 2 / 3 , eta =0.219+or-0.009, beta =0.073+or-0.003, and delta =17.3+or-0.8.