We have extended the series for the zero-field susceptibility of the spin-
s Ising model through tenth order in the reduced inverse temperature
K on the square, triangular, simple-cubic, body-centered-cubic, and face-centered-cubic lattices. The series coefficients
hn(s) are expressed as simple polynomials in
X=s(s+1). Using extended methods of analysis we have estimated the nature of the leading singularities on the face-centered lattice and conclude with good confidence that the susceptibility exponent ${\ensuremath{\gamma}}_{1}$ equals 1.25, independent of
s. The exponent of the leading correction term is estimated to be ${\ensuremath{\gamma}}_{2}\ensuremath{\simeq}0.75\ifmmode\pm\else\textpm\fi{}0.08$ in good agreement with renormalization-group theory. For
s=21 only, the amplitude of the confluent correction apparently vanishes. We have also studied the leading singularities on the triangular lattice and conclude that ${\ensuremath{\gamma}}_{1}=\frac{7}{4}$, independent of
s, with, however, much stronger corrections than in three dimensions. These results provide a very strong corroboration of the universality hypothesis.
Our aim is to investigate the critical behaviour of lattice spin models such as the three-dimensional Ising model in the thermodynamic limit. The exact partition functions (typically summed over the order of 1075 states) for finite simple cubic Ising lattices are computed using a transfer matrix approach. Q-state Potts model partition functions on two- and three-dimensional lattices are also computed and analysed. Our results are analysed as distributions of zeros of the partition function in the complex-temperature plane. We then look at sequences of such distributions for sequences of lattices approaching the thermodynamic limit. For a controlled comparison, we show how a sequence of zero distributions for finite 2d Ising lattices tends to Onsager’s thermodynamic solution. Via such comparisons, we find evidence to suggest, for example, a thermodynamic limit singular point in the behaviour of the specific heat of the 3d Ising model.
The three-dimensional Ising model in a zero external field is exactly solved by operator algebras, similar to the Onsager’s approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along two directions and the screw boundary condition along the third direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. The critical temperatures, at which order–disorder transitions in the infinite crystal occur along three axis directions, are determined. The analytical expressions for the internal energy and the specific heat are also presented.
We have extended the series for the zero-field susceptibility of the spin-
S Ising model to eighth order in the reduced temperature
K, on the triangular, simple cubic, body-centered-cubic, and face-centered-cubic lattices. The coefficients of these series
hn(S) are expressed as simple polynomials in
X=S(S+1). For the face-centered-cubic lattice, an accurate polynomial fit to the critical point
Kc(S) is presented; and the apparent spin dependence of the critical exponent $\ensuremath{\gamma}$ is briefly discussed. The series are quite well behaved for all
S. However, the large-
S series seems to exhibit more rapid apparent convergence.
We construct a single reduced free-energy generator for the simple-cubic three-dimensional Ising model by finite gluing of complementary local relations. With the convention
Φ=+logZ/V, the final object is
Φ3(K,H,L)=Φ0(K)+ΔΦH(K,H)+ΦFSS(K,H,L), from which
M=∂HΦ3,
χ=∂H2Φ3, and
UK=∂KΦ3 are generated. The construction covers the high- and low-temperature regimes, the critical region, finite magnetic field, strong-field saturation, periodic-film finite size, and simultaneous finite
H and finite
L. The materials and operations are separated explicitly. A diagnostic dimensional stage uses
d=1,2,4 anchors in
μc=−logtanhKc to transport critical coordinates and positive film amplitudes toward
d=3. The practical 3D generator then protects established 3D values of
Kc, critical exponents, selected amplitude ratios, and scalar zero-field critical data. The zero-field term joins truncated simple-cubic high- and low-temperature series through a singular critical patch with
C2 matching. The inner magnetic increment joins an
O(K2) high-temperature field expansion, a minimal 3D parametric critical free energy fixed only by the basic 3D anchors, and a low-temperature/cluster continuation using fixed quintic switches in
K. A recovered
z4 strong-field expansion is integrated to an absolute free energy. The final inner-to-strong crossover is performed by constructing a monotone beta-CDF magnetization whose endpoint values and integral reproduce the two endpoint free energies, and then integrating that derivative; thus convexity is enforced at the free-energy level rather than by clipping observables. The periodic-film excess is built from dimensionally weighted 2D and leading 4D film surfaces in common correlation-length coordinates, with an
H-independent residual stabilizer that preserves the
H=0 axis. A previously reconstructed full 3D parametric equation-of-state benchmark and the full 3D finite-size scaling function are withheld from construction. The pure dimensional stage gives
Kc(3)=0.22301549 and sub-percent estimates of
β,
δ, and
ν. After 3D calibration, the final implementation passes 29/29 stated structural and numerical checks. In the withheld low-field window
∣K−Kc∣≤0.01,
H≤0.1, the magnetization has RMS relative error
1.21% and maximum error
4.29%. The result is not an exact lattice solution; it is a finite, auditable reconstruction in which the input anchors, switch intervals, matching conditions, numerical tolerances, and validation data are kept distinct.
We test the Gluing Method on the square-lattice two-dimensional Ising model by reconstructing global free-energy generators from complementary local relations. The emphasis is kept explicit throughout: the inputs used as \emph{gluing material} are separated from quantities obtained \emph{after gluing} and from data reserved for validation. At zero magnetic field, Kramers--Wannier duality is absorbed into the coordinate
τ=logsinh2K, so that the reduced free energy decomposes as
Φ(τ)=τ/2+G(∣τ∣). The gluing materials are then only a critical logarithmic local expansion and a low-order high-temperature outer expansion; the low-temperature sector is generated by duality rather than fitted independently. Positive flat weights preserve the prescribed critical asymptotics to every finite algebraic order. With one crossover parameter fixed from three scalar anchors, the resulting bulk free energy agrees with the exact Onsager benchmark at a maximum relative error of
1.915×10−5 on the tested interval. Internal energy and specific heat are derived from the same glued generator. Zero-field finite-size corrections are organized separately as one-variable scaling functions multiplying
L−2,L−4,…. At nonzero field in the thermodynamic limit, the gluing materials are the established high-temperature, critical, and low-temperature local expansions of the universal Ising magnetic scaling free energy
Φmag(η),
η=m/∣h∣8/15. The low-temperature input contains the
∣h∣ cusp associated with spontaneous symmetry breaking. Mapping the glued scaling free energy back to the square lattice through nonlinear scaling fields yields a single generator for the magnetic free-energy increment, magnetization, and susceptibility. The construction reproduces high-temperature linear response, the critical isotherm
M∝∣H∣1/15, and the two one-sided limits
M(H→0±)=±M0 without separately fitting magnetization branches. Finally, finite size and finite field are combined in the scaling variables
xt=Lm and
xh=L15/8h. The two protected one-dimensional axes, a central mixed-response expansion, a low-temperature spectral-rounding relation, and a massive outer description are glued into a two-variable scaling surface
Yglue(xt,xh). The resulting lattice free energy closes in the compact form
ΦLglue(K,H)=Φlatglue(K,H)+L2Yglue(Lm,L15/8h)−Ybulk(Lm,L15/8h)+O(L−4) where
m=m(r,H) and
h=h(r,H) are the nonlinear scaling fields and
Ybulk is the thermodynamic singular scaling surface expressed in the same variables. Thus all finite-size/finite-field information is carried by the excess
Yglue−Ybulk, while the thermodynamic lattice free energy remains the previously constructed
Φlatglue. On eleven transfer-matrix points withheld from the construction, the blind free-energy test gives RMS absolute error
3.90×10−4 and maximum absolute error
9.55×10−4 in
Y. Differentiation of the same free energy produces finite-size magnetization and susceptibility; magnetization is especially stable on the low-temperature side, while the second derivative is more sensitive in crossover regions.