The partition function of the Ising model of ferromagnetism is examined in the limit of high density in the anticipation that in the limit of infinite density one recovers the Weiss molecular field. The formal parameter of expansion is
z1 where
z is the number of spins in the range of the exchange potential (not restricted to nearest neighbor interactions). In the absence of long-range order, only ring diagrams in the cluster expansion contribute. These give a divergence in the specific heat at $k{T}_{c}=\ensuremath{\Sigma}{j\ensuremath{\ne}i}^{}{v}_{\mathrm{ij}}$ where
vij is the exchange potential. This is the molecular field value for the Curie point
Tc. In the presence of a magnetic field the partition function is evaluated for fixed magnetic moment
M in the same approximation,
M being determined by minimization. This results in a susceptibility differing from the molecular field theory and hence an inconsistency in the theory.The inconsistency is traced back to the observation that the acceptance of ring diagrams is equivalent to the gaussian model of Kac and Berlin which violates the sum rule ${{\ensuremath{\Sigma}}_{i=1}}^{N}{{\ensuremath{\mu}}_{i}}^{2}=N$. Here ${\ensuremath{\mu}}_{i}$ is the "spin" per particle and
N is the total number of particles. This condition is reinstated by insuring the sum rule. The result leads to the spherical model. Thus, a consistent high density approximation to the Ising model is the spherical model. Below the Curie point or for fixed magnetic field,
M is again held fixed and only the Fourier components of the spin density with nonvanishing wave vector are "sphericalized." The result leads to a physically acceptable model which becomes the molecular field theory at low temperatures or high fields and deviates in
O(z1) in general. Formally, the results are simply expressed in terms of a temperature dependent Weiss field. These results differ from the ordinary spherical model which is physically unacceptable below the Curie point. However, a molecular field modification of the spherical model due to M. Lax yields the same result when properly interpreted.It is shown that the above results are also valid (to the same approximation) in the quantum mechanical Heisenberg model, for temperatures above the Curie point.
A relation between O ( n ) models and Ising models has been recently conjectured (Casetti et al 2011 Phys. Rev. Lett. 106 057208 ). Such a relation, inspired by an energy landscape analysis, implies that the microcanonical density of states of an O ( n ) spin model on a lattice can be effectively approximated in terms of the density of states of an Ising model defined on the same lattice and with the same interactions. Were this relation exact, it would imply that the critical energy densities of all the O ( n ) models (i.e. the average values per spin of the O ( n ) Hamiltonians at their respective critical temperatures) should be equal to that of the corresponding Ising model. It is therefore worth investigating how different the critical energies are and how this difference depends on n . We compare the critical energy densities of O ( n ) models in three dimensions in some specific cases: the O (1) or Ising model, the O (2) or XY model, the O (3) or Heisenberg model, the O (4) model and the O (∞) or spherical model, all defined on regular cubic lattices and with ferromagnetic nearest-neighbor interactions. The values of the critical energy density in the n = 2, n = 3 and n = 4 cases are derived through a finite-size scaling analysis of data produced by means of Monte Carlo simulations on lattices with up to 128 3 sites. For n = 2 and n = 3 the accuracy of previously known results has been improved. We finally derive an interpolation formula showing that the difference between the critical energy densities of O ( n ) models and that of the Ising model is smaller than 1% if n < 8 and never exceeds 3% for any n .
The behavior of the spherical model of a ferromagnet with an interaction energy between the magnetic spins which varies with distance as $\frac{1}{{r}^{d+\ensuremath{\sigma}}}$ (where
d is the dimensionality of the lattice and $\ensuremath{\sigma}>0$) is analyzed. It is shown that the model exhibits a ferromagnetic transition in one and two dimensions, providing $0<\ensuremath{\sigma}<d$. (The usual spherical model with nearest-neighbor interactions does not have a transition in one and two dimensions.) The critical-point behavior is investigated. It is found that the singularities in the specific heat and susceptibility are dependent on $\ensuremath{\sigma}$ and
d, but the behavior of the magnetization is independent of $\ensuremath{\sigma}$ and
d. In three dimensions the susceptibility diverges as ${(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}$, where $\ensuremath{\gamma}=1$ for $0<\ensuremath{\sigma}<\frac{3}{2}$, $\ensuremath{\gamma}=\frac{\ensuremath{\sigma}}{(3\ensuremath{-}\ensuremath{\sigma})}$ for $\frac{3}{2}<\ensuremath{\sigma}<2$ and $\ensuremath{\gamma}=2$ for $\ensuremath{\sigma}>2$. The asymptotic form of the spin-spin correlation function $\ensuremath{\Gamma}(\mathrm{r})$ is studied in the neighborhood of the critical temperature
Tc. At
T=Tc, $\ensuremath{\Gamma}(\mathrm{r})$ decays for large
r as $\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}$. Several two-dimensional models with long-range interactions falling off as
r21 in certain directions only are also investigated.
A new and simple diagrammatic expansion is developed for the free energy of the Ising model with arbitrary spin and range of interaction. The contribution of each diagram is a product of (1) the reciprocal of the order of the symmetry group of the diagram, (2) a product of semi-invariants, with a factor
Mn for each spin in the diagram
n being the number of bonds joined to that spin) and (3) a sum of products $\ensuremath{\Sigma}{i,j,k,\ensuremath{\cdots}}^{}(2\ensuremath{\beta}{J}_{\mathrm{ij}})(2\ensuremath{\beta}{J}_{\mathrm{kl}})\ensuremath{\cdots}$, where $2\ensuremath{\beta}{J}_{\mathrm{ij}}$ corresponds to a bond between spins
i and
j, and where the summation is carried out with no restrictions on the summation indices (i.e., no "excluded volume" corrections). The expansion variables are the spin deviation operators ${\ensuremath{\sigma}}_{i}\ensuremath{\equiv}\overline{S}\ensuremath{-}{S}_{\mathrm{iz}}$. The quantity
S is chosen to eliminate a large class of diagrams; this choice also minimizes the corresponding free-energy contribution and implies
S=〈Siz〉. By further renormalization of the vertices all reducible (i.e., simply articulated) diagrams are eliminated. To zero order the molecular field solution is obtained. The next approximation consists of the summation of renormalized simple loop diagrams. The justification of this choice of diagrams rests, at low temperature, on the decrease of the value of the higher order semi-invariants, on the relationship of these loop diagrams to the random phase approximation, and on the agreement with the rigorous low-temperature series result. At the Curie temperature the same choice of diagrams is justified by a modification of Brout's
z1 criterion, so that the expansion can be viewed as an expansion in
z1, where
z is the effective number of spins interacting with a given spin. Finally the choice of loop diagrams is justified at high temperature by exact agreement to second order in
TTc (and by a very close agreement to fourth order) with the rigorous high-temperature series expansion. Thus the theory agrees with rigorous results in the low-temperature and paramagnetic regions and has a well-defined criterion of validity in the intermediate temperature region. For nearest-neighbor interactions and spin \textonehalf{} the specific heat is continuous through the Curie temperature, with an infinite slope on the low-temperature side of the transition.
The Ising model with ferromagnetic interactions that decay as $\frac{1}{{r}^{\ensuremath{\alpha}}}$ is analyzed in the nonextensive regime $0<~\ensuremath{\alpha}<~d$, where the thermodynamic limit is not defined. In order to study the asymptotic properties of the model in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit (
N being the number of spins) we propose a generalization of the Curie-Weiss model, for which the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit is well defined for all $\ensuremath{\alpha}>~0$. We conjecture that mean-field theory is exact in the last model for all $0<~\ensuremath{\alpha}<~d$. This conjecture is supported by Monte Carlo heat-bath simulations in the
d=1 case. Moreover, we confirm a recently conjectured scaling by Tsallis that allows for a unification of extensive ($\ensuremath{\alpha}>d$) and nonextensive ($0<~\ensuremath{\alpha}<~d$) regimes.
The matching of leading singular contributions to the different mean-field or critical behaviors of a system with long-but-finite-ranged spatially anisotropic interactions leads to "scaling laws" connecting criticial exponents for different dimensions. The results are exact for mean-field theory (MFT) and spherical model exponents and predict MFT exponents for
d=4. From the known
d=2 Ising-model exponents, the scaling laws give $\ensuremath{\alpha}=0$, $\ensuremath{\beta}=\frac{4}{11}$, $\ensuremath{\gamma}=\frac{14}{11}$, $\ensuremath{\nu}=\frac{2}{3}$ for
d=3. Agreement with previous results for the
d=2,3 excluded volume problem is also quite good. For the $X\ensuremath{-}Y$ and Heisenberg models the results predict $\ensuremath{\gamma}\ensuremath{\approx}2.0 \mathrm{and} 2.4$, respectively, for
d=2.