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With the convention $H=J{\sum }_{⟨ij⟩}{\sigma }_i{\sigma }_j-h{\sum }_i{\sigma }_i$, positive $J$ favors opposite neighboring Ising spins. On a bipartite graph, the Néel order parameter measures opposite sublattice magnetizations; it is distinct from the uniform magnetization. Changing the signs of spins on one sublattice maps the zero-field model to the ferromagnetic Ising model.

For a nearest-neighbor auxiliary-field representation of an Ising model, sharing each local potential between adjacent bonds gives the displayed symmetric integral operator. Its periodic partition function is the trace of its $N$th power, up to Gaussian normalization constants. In the slowly varying continuum limit, a slice of length $ϵ$ has kernel $[M/(2\pi ϵ){]}^{1/2}\exp [-M(m^{\prime }-m{)}^2/(2ϵ)-ϵ\{U(m^{\prime })+U(m)\}/2]$. Expanding its action on a smooth function gives $1-ϵ\hat{H}+O({ϵ}^2)$, with $\hat{H}=-(2M{)}^{-1}{\partial }_m^2+U$. Thus fixed endpoints give an imaginary-time transition kernel and periodic endpoints give a Euclidean path integral trace. The finite-spacing integral operator and the continuum differential operator need not have identical high-energy spectra.

The Blume–Capel model is a spin model with ${\sigma }_i\in \{-1,0,1\}$ and HamiltonianCompetition between the Ising interaction and the single-ion anisotropy $D$ produces a tricritical point in its phase diagram.

Checkerboard spin decimation retains one parity sublattice of a square-lattice Ising model and sums over the other. The retained primitive vectors are $a(1,1)$ and $a(1,-1)$, so the coarse lattice is again square with spacing $\sqrt{2}a$. A rotation and length rescaling restore the original geometry. Every original nearest-neighbour bond joins the two parity classes, whereas a diagonal next-nearest-neighbour bond stays within one class.

The quartic interaction $g_1{\varphi }_1^4+g_2{\varphi }_2^4+\lambda {\varphi }_1^2{\varphi }_2^2$ includes the Gaussian fixed point, single and double Ising model Wilson-Fisher fixed points, and an O(N) model fixed point with $N=2$. A rotation of two decoupled identical Ising models produces a second coordinate representation of their renormalization-group fixed point.

The zero-field ferromagnetic Curie–Weiss model is an Ising model on $n$ spins with an equal interaction between every pair. For $s_i\in \{-1,1\}$ and $\beta >0$, its normalized probability mass function is $Z_{n,\beta }^{-1}\exp [\beta ({\sum }_i{s}_i{)}^2/(2n)]$. The partition function $Z_{n,\beta }$ sums these weights over all configurations. The square includes constant diagonal terms; these affect only normalization. Spin inversion symmetry makes every spin's expected value zero, while grouping configurations by their spin magnetization givesHere $m=-1+2k/n$, $0\le k\le n$.

The zero-field Hubbard–Stratonovich transformation of an exponentially coupled Ising model produces $U(m)=A{m}^2-\log [2\cosh (2m)]$, where $A=\tanh (\kappa /2)/J>0$. Since $U^{\prime \prime }(0)=2A-4$ and $U$ tends to positive infinity, a symmetric double well exists exactly when $A<2$. Its nonzero stationary points solve $Am=\tanh (2m)$; the ratio $\tanh (2m)/m$ decreases from 2 to 0 for $m>0$, proving that there is exactly one positive minimum. At $A=2$ the minimum is a single quartic minimum. A small conjugate field adds $-h\tanh (2m)+O(h^2)$ and tilts the wells, while a sufficiently large field can remove the metastable well. A finite barrier permits quantum tunnelling; two classical minima alone do not establish spontaneous symmetry breaking.

The Edwards-Sokal coupling assigns one common random spin to every random-cluster component. For $q=2$ it couples the random-cluster model to the Ising model.

For the ferromagnetic binary Ising model, all-zero boundary measures increase and all-one boundary measures decrease as boxes exhaust the lattice, in stochastic domination. The nearest-neighbor spatial conditional probability distribution reduces each domain comparison to a mixture of boundary conditions for the smaller box. Probabilities that a specified finite set is all one consequently converge; inclusion-exclusion gives convergence of every finite Ising spin pattern. Their consistent limits define the two extreme infinite-volume measures. Existence does not assert that the two limits coincide.

For $H(\sigma )={\sum }_{xy}{J}_{xy}{\sigma }_x{\sigma }_y+{\sum }_x{h}_x{\sigma }_x$ on a finite graph, assume $J_{xy}\ge 0$. Let $U=\{x:{\sigma }_x=1,{\tau }_x=-1\}$ and $W=\{x:{\sigma }_x=-1,{\tau }_x=1\}$. The displayed difference is $4{\sum }_{xy:x\in U,y\in W\text{ or conversely}}{J}_{xy}$: all other edge terms and all field terms cancel. Exponentiating proves the FKG lattice condition and hence positive association of random variables for this Ising model.

The orthogonal transformation ${\psi }_{\pm }=({\varphi }_1\pm {\varphi }_2)/\sqrt{2}$ takes $g({\varphi }_1^4+{\varphi }_2^4+6{\varphi }_1^2{\varphi }_2^2)$ to $2g({\psi }_{+}^4+{\psi }_{-}^4)$. These are the same two decoupled Ising models expressed in rotated fields.

For finitely many microstates with finite energies $E_{\alpha }(h)$ that are real analytic functions of a real control $h$, the canonical partition function at $T>0$ is a finite positive sum:It and the resulting Helmholtz free energy are locally real analytic functions of $T$ and $h$. Hence such a finite Ising model has no literal nonanalytic equilibrium phase transition. A limit of these real analytic functions can nevertheless fail to be a real analytic function; the narrow but smooth crossover can become a sharp phase transition.^[ref]

Finite particle number alone is not the hypothesis of a finite sum. Classical phase space is continuous, and a system of quantum states can have infinitely many energy eigenvalues: its partition function requires convergence and local bounds that permit differentiation or an analytic extension of the integral or thermal trace. A divergent partition function, a singularly changed statistical Hamiltonian, or the zero-temperature limit falls outside the finite-positive-sum argument. Observing a steep change in a finite sample therefore motivates a limiting model; it does not measure an actual infinity.

A fixed Ising boundary condition pins the Ising spins on a specified boundary, leaving the interior distributed with the original interaction weights. The boundary can be the inner graph vertex boundary of a finite box, or an exterior neighboring set if that convention is explicitly chosen. Conditioning on pinned Ising spins preserves the FKG lattice condition of a ferromagnetic Ising model. The FKG inequality then makes interior expectations of increasing observables nondecreasing in the pinned boundary Ising spins.

An Ising Ising spin variable is a classical two-state degree of freedom. Its thermal mean can lie anywhere in $[-1,1]$ even though each microscopic value is $-1$ or $1$. It should not be confused with an arbitrary quantum-spin operator. Products of neighbouring Ising spins supply the pair interactions of the Ising model, and their average defines spin magnetization.

For a ferromagnetic Ising model with coordination number of a lattice $z$, its mean-field self-consistency equation is $M=\tanh (zJM/(k_{B}T))$. Put $A=T_c/T$. For $A\le 1$, the only solution is zero. For $A>1$, strict concavity of $\tanh (AM)-M$ on $M>0$ gives one positive solution and its spin-reversed partner. Expanding $\tanh (AM)=AM-(AM{)}^3/3+O(M^5)$ gives $M^2=3(A-1)/A^3+O((A-1{)}^2)$, proving a continuous onset with order-parameter critical exponent $\beta =1/2$. The Bragg-Williams free energy of the Ising model has a positive quartic coefficient at $T_c$, so these are the stable ordered minima rather than a first-order jump.

Mean-field theory replaces fluctuating interactions with a self-consistent average field. For example, an Ising spin feels a field determined by the average neighboring spin magnetization; a self-consistency equation approximates the interacting Ising model. Neglecting correlated fluctuations can fail near a continuous phase transition. The mean-field approximation in Landau theory is a particular use of this general framework.

For $N\ge 2$, the open-chain Ising model coupling $[J_N{]}_{ij}=J{\rho }^{|i-j|}$, $0<\rho <1$, has a tridiagonal inverse. Its endpoint diagonal entries are $1/[J(1-{\rho }^2)]$, its interior diagonal entries are $(1+{\rho }^2)/[J(1-{\rho }^2)]$, and adjacent off-diagonal entries are $-\rho /[J(1-{\rho }^2)]$. Direct matrix multiplication verifies the inverse because the geometric sequences cancel in every interior row. The displayed quadratic form follows with $D=\rho /[J(1-{\rho }^2)]$, $A=(1-\rho )/[J(1+\rho )]$ and $B=\rho /[J(1+\rho )]$. For $\rho =e^{-\kappa }$, $D=(2J\sinh \kappa {)}^{-1}$ and $A=\tanh (\kappa /2)/J$. Omitting the endpoint term requires an infinite chain, a correctly periodized coupling, or a bulk thermodynamic limit.

Write the interaction matrix as $\mathsf{J}$ to distinguish it from its scalar strength. The printed pair sum contains both orientations and has no factor $1/2$. For a positive-definite matrix $\mathsf{J}$, completion of the square gives the Hubbard–Stratonovich transformationsince $m^T{\mathsf{J}}^{-1}m-2m^T\sigma =(m-\mathsf{J}\sigma {)}^T{\mathsf{J}}^{-1}(m-\mathsf{J}\sigma )-{\sigma }^T\mathsf{J}\sigma$. For the exponential kernel with $J>0$ and $\kappa >0$, positivity follows from the positive Fourier symbol derived in part (b), or from its finite open-chain covariance matrix.

Insert this identity into the Ising model partition sum and interchange the finite spin sum with the convergent Gaussian integral. The spin variables now factor independently:It follows thatwith $C=({\pi }^N\det \mathsf{J}{)}^{-1/2}$ for the measure used here. The factor two in the spin-field coupling is required by the double pair sum; replacing it by one would decouple a statistical Hamiltonian with a different normalization. The diagonal $J_{ii}{\sigma }_i^2=J$ is a spin-independent energy. Including it as printed makes the positive Gaussian identity directly applicable; if diagonal self-couplings are removed they should first be restored as an irrelevant constant before this real auxiliary-field decoupling.

With lattice spacing set to one and a smoothly varying auxiliary field, the discrete action becomesThis is precisely the Euclidean path integral of a quantum particle with $\hslash =1$, mass $M$ and potential $U$. With fixed endpoints it gives the transition amplitude $⟨m_f|e^{-\mathcal{L}{\hat{H}}_Q}|m_i⟩$, while periodic classical boundary conditions give its trace. The corresponding operator isThis is classical Ising chain as a quantum particle. It is the continuum approximation to the exact positive transfer kernel, so its numerical spectrum is accurate when the relevant auxiliary configurations vary slowly. The transfer-kernel reasoning and the domain-wall argument below also support the qualitative conclusion without treating a continuum approximation as an exact finite-lattice identity.

At zero field write $a=\tanh (\kappa /2)/J>0$. Expansion givesA double well exists when $a<2$; the stationary points satisfy $am=\tanh (2m)$. For $a\ge 2$ there is a single minimum at zero, with the quadratic curvature marginal at equality. Thus a double well is the low-effective-temperature regime of the model, not an unconditional property for every $J$. The exact potential remains confining because $U(m)=a{m}^2-2|m|+O(1)$ at large $|m|$.

Even when two wells exist, a quantum particle with finite mass and a finite barrier tunnels between them. Its one-dimensional confining Schrödinger operator has a unique, strictly positive ground-state wavefunction. At $h=0$ reflection symmetry makes that state even, so every odd observable has zero expectation. An antisymmetric first excited state is split from it by a positive gap. A spectral expansion of an odd-observable correlation has factors $e^{-x(E_n-E_0)}$; the smallest contributing gap therefore gives a finite classical correlation length. The original spin observable, after decoupling, is $\tanh (2m+h)$, which is odd at zero field. HenceA deep well can give an exponentially small quantum tunnelling gap and an extremely long correlation length, which can mimic order in a finite sample, but it does not produce true long-range order.

The classical counterpart is the finite cost of a domain wall. Because the pair sum is doubled, a sharp wall has dimensionless energyAt any finite temperature such walls have nonzero fugacity, divide a long chain into domains and destroy spontaneous magnetization. The range is large for $\kappa \ll 1$ but finite. The zero-temperature limit and an infinite-range limit are singular and must not be interchanged casually with the thermodynamic limit; an infinite-range interaction also requires an extensive normalization before comparing models.

A nonzero $h$ tilts the potential: to first order $U_h(m)=U_0(m)-h\tanh (2m)+O(h^2)$. A positive field favors the positive well and produces positive magnetization. For fixed finite parameters the ground state and classical free energy vary smoothly, so there is no zero-field discontinuity due merely to a double-well bare potential. A discrete broken-symmetry state appears only in an appropriate singular limit, not by ignoring quantum tunnelling.

More generally a local $d$-dimensional classical statistical model can be sliced along one coordinate and represented by a transfer operator. That coordinate becomes imaginary time; the other $d-1$ coordinates become the spatial coordinates of a quantum system. Infinite extent in the transfer direction corresponds to its ground-state, or zero-quantum-temperature, limit. The classical Ising model is thus related to a quantum Ising system in $d-1$ spatial dimensions, with transverse dynamics generated by the transfer operator. For $d=1$ this is quantum mechanics of finitely many local variables, where quantum tunnelling restores symmetry. For $d\ge 2$ the quantum system has infinitely many spatial degrees of freedom and can have distinct symmetry-broken ground states, allowing a classical finite-temperature ordering transition. Locality and anisotropy of the mapping determine the quantum couplings; the dimension correspondence alone does not fix them.