Transfer matrix for a classical spin chain ($Z_N=\mathrm{tr}{W}^N$)

For a nearest-neighbour classical chain with finitely many spin states and dimensionless bond energy $E(s,t)$, its transfer matrix is $W_{st}=e^{-E(s,t)}$. On-site energies are divided between adjacent bonds. Periodic boundary conditions give partition function $Z_N=\mathrm{tr}{W}^N$ by direct multiplication and summation over spin labels. For a strictly positive symmetric matrix, the Perron–Frobenius theorem gives ${\lim }_{N\to \infty }{N}^{-1}\log Z_N=\log {\lambda }_{\max }$. A diagonal local observable $O$ has expectation $\mathrm{tr}(O{W}^N)/Z_N$, tending to $v_{\max }^{T}O{v}_{\max }$ for a normalized dominant eigenvector.

For Ising spins $s,t=\pm 1$ separated by a decorating spin $\tau =0,\pm 1$, summing its Boltzmann weights removes that spin exactly. If its two exchange interactions are $-\tau (J_{1}s+J_{2}t)$, the effective transfer matrix has entries $1+2\cosh [\beta (J_{1}s+J_{2}t)]$. Its eigenvalues are ${\lambda }_{+}=2+4\cosh (\beta J_1)\cosh (\beta J_2)$ and ${\lambda }_{-}=4\sinh (\beta J_1)\sinh (\beta J_2)$. A transfer step contains two physical sites, so the limiting Helmholtz free energy per site is $-(2\beta {)}^{-1}\log {\lambda }_{+}$.

An observable on a spin removed by decimation is inserted before its local sum. For ${\tau }^2$, replace $V_{st}={\sum }_{\tau }{e}^{\beta \tau (J_{1}s+J_{2}t)}$ by $Q_{st}={\sum }_{\tau }{\tau }^2{e}^{\beta \tau (J_{1}s+J_{2}t)}=2\cosh [\beta (J_{1}s+J_{2}t)]$. Contracting the unchanged ring gives the displayed thermal expectation value. In the infinite chain it becomes $v_{+}^{\mathsf{T}}Q{v}_{+}/{\lambda }_{+}$ for normalized dominant eigenvector $v_{+}$.

Hubbard-Stratonovich decoupling of an exponentially coupled Ising chain gives a scalar-field action with a nearest-neighbor gradient term and $U(m)=\tanh (\kappa /2)m^2/J-\log [2\cosh (2m+h)]$. Its continuum partition function is the Euclidean path integral of the displayed quantum particle. At zero field the potential can have two wells, but a finite barrier still permits quantum tunnelling and gives a unique symmetric ground state. The excitation gap produces a finite classical correlation length, so there is no finite-temperature spontaneous magnetization for fixed positive $\kappa$. One classical spatial coordinate becomes imaginary time; a $d$-dimensional classical model corresponds to a quantum model with $d-1$ spatial dimensions.

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 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.