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.
A functional-integral formulation of the Ising model is used as a link between the usual approach in terms of summation over spin variables and field-theory-like formulas. The latter take the form of Feynman-diagram expansions, Dyson integral equations, or Schwinger functional-derivative field equations. Basic to the theory is a spinless-nucleon-like Green's function, related to the Ising spin-spin correlation function. By means of an infinite renormalization technique, a mesonlike propagator, related to local energy-energy correlations, is bootstrapped. Finally, a vertex function, associated with energy-spin-spin correlations, is introduced. To zeroth order this theory bears similarity to the spherical model, but vital differences are also noted. A brief discussion is presented of the relation of methods employed in field theory for the treatment of the infrared divergence, and approximations which might be of value for critical correlations.
Starting with the previously derived vacuum-state expectation representation of the Ising partition function, an integral transform is applied to part of the Bose field operator arising from pair interactions, generating thereby the effect of random external fields. The remaining pair interaction is treated as a perturbation, whose effect is small in both the high- and the low-temperature extremes. The transformed part of the potential is selected first to generate a Markoff process, and the perturbation equations may then be regarded as conditions that the perturbation have no effect on either the partition function or the long-range order below the transition. In two dimensions, it is shown how the characteristic integral equation of the Markoff process reduces to the matrix eigenvalue problem solved by Onsager, in the limit of just nearest-neighbor interactions.
We show the way perturbation procedures originated by Brout and Hemmer and further developed by Lebowitz and co-workers can be used to study the critical behavior of lattice systems with a potential
v(r) having a weak, long-ranged tail that approaches ${\ensuremath{\gamma}}^{d}\ensuremath{\phi}(\ensuremath{\gamma}\mathrm{r})$ as $\ensuremath{\gamma}$ approaches zero, where
d is dimensionality. We consider both the Ising and the spherical models, and note that the results for the latter model are also those that follow from the Ornstein-Zernike assumption that the direct correlation function behaves like $\ensuremath{-}\frac{v(\mathrm{r})}{\mathrm{kT}}$, when $v(\mathrm{r})\ensuremath{\ll}\mathrm{kT}$. We recover by a graph technique the previously known result that in one dimension, quantities of interest such as the inverse correlation length $\ensuremath{\kappa}$ cannot be expanded in $\ensuremath{\gamma}$ at the mean-field critical temperature
T0 and density ${\ensuremath{\rho}}_{0}$ that characterize the critical point in the $\ensuremath{\gamma}\ensuremath{\rightarrow}0$ limit. Instead, at (${T}_{0}, {\ensuremath{\rho}}_{0}$), $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}^{\frac{4}{3}}$ as $\ensuremath{\gamma}\ensuremath{\rightarrow}0$. This is true for both the Ising and the spherical models. In the two-dimensional spherical model, we find $\ensuremath{\kappa}$ to vary as ${\ensuremath{\gamma}}^{2}{[\mathrm{ln}(\frac{1}{\ensuremath{\gamma}})]}^{\frac{1}{2}}$ when $\ensuremath{\gamma}\ensuremath{\rightarrow}0$, while in three dimensions $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}^{\frac{5}{2}}$. In the Ising case for
d=1, we characterize topologically the infinite sum of graphs that contribute to the $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}^{\frac{4}{3}}$ term in the expansion of the pair correlation function. (In the spherical model, a chain graph gives the entire contribution to the pair correlation function for all
d). For
d=3 we do not attempt to deal with the correlation length in the Ising case, but instead consider the shift in the critical temperature as $\ensuremath{\gamma}\ensuremath{\rightarrow}0$. We find that the spherical-model result that gives a shift of order ${\ensuremath{\gamma}}^{3}$ is exact to that order in $\ensuremath{\gamma}$ for the Ising model. We also find that the lowest-order correction to the spherical-model result is of the form ($\mathrm{const}{\ensuremath{\gamma}}^{6}\mathrm{ln}(\mathrm{const}\ensuremath{\gamma})$, in agreement with recent work by Thouless; but in general we expect to find terms of order ${\ensuremath{\gamma}}^{4}$ and ${\ensuremath{\gamma}}^{5}$ from the spherical-model result dominating this correction.
A previously developed method for diagrammatic expansion of the Ising-model partition function and pair distribution function is applied to calculation of the field-free quadruplet spin averages $〈{\ensuremath{\mu}}_{1}{\ensuremath{\mu}}_{2}{\ensuremath{\mu}}_{3}{\ensuremath{\mu}}_{4}〉$. The central four-vertex function
Q generated by these averages is topologically analyzed in standard fashion in terms of an irreducible four-vertex quantity
e. Several rigorously necessary conditions that must be satisfied by
Q are listed. It is furthermore pointed out that the existence of a logarithmic specific-heat anomaly puts an additional constraint on quadruplet averages (which is verified in an Appendix by direct calculation on the Ising-Onsager two-dimensional square lattice). Upon making a simplifying functional assumption for
Q, this latter quantity may be entirely determined above
Tc by one of the necessary conditions. This leads at
Tc to a
k2d spectrum (
d=dimensionality) and above
Tc to a logarithmic specific heat, both of which were adduced by Abe's Ising-model version of the
He4 speculative analysis due to Patashinskii and Pokrovskii (but for different reasons from those in the present analysis). Sinec the Abe-Patashinskii-Pokrovskii spectrum almost certainly exhibits an incorrect exponent, an alternative and more powerful functional assumption for
Q is suggested which still yields a soluble theory in principle, but construction of the solution is not attempted here.