Ising transfer matrix in one dimension

Take a ring with N spins, ferromagnetic coupling J, and field energy h per spin. Introduce dimensionless couplings $K=J/(k_BT)$ and $H=h/(k_BT)$. Assign half of each endpoint's field energy to each bond so every site field is counted exactly once around the ring. Matrix indices are classical spin values, not quantum states whose amplitudes interfere. Multiplication sums positive statistical weights over intermediate spin signs.

Periodic closure makes the partition sum the trace of the Nth matrix power. An open chain instead needs boundary vectors, which changes finite-size factors even though the bulk limit is the same. Diagonalizing the matrix gives two eigenvalues. At any finite positive temperature its entries are positive, and the largest eigenvalue is unique and smooth for finite real field. An eigenvalue crossing that could create a finite-temperature bulk singularity is absent in this nearest-neighbour chain.

At zero field the eigenvectors are symmetric and antisymmetric under reversal, with eigenvalues $2\cosh K$ and $2\sinh K$. In the infinite ferromagnetic chain, the spin correlation at a separation of j bonds is $(\tanh K)^j$. Writing that decay as an exponential defines a correlation length measured in lattice spacings. At finite K, the ratio lies strictly between zero and one, so the correlation length is finite even when it is large.

The dominant eigenvalue alone determines the thermodynamic free energy per site, but it does not contain all finite-size information. The omitted contribution relative to the leading partition term is $(\lambda_-/\lambda_+)^N$. That correction can matter in a cold chain shorter than its correlation length. At infinite temperature the zero-field ratio vanishes and distinct spins are independent; as temperature tends to zero the ratio approaches one and the correlation length diverges.

An isolated reversal boundary costs a finite bond energy rather than an energy proportional to the length of the reversed region. At every positive temperature there is a nonzero density of such domain walls in an infinite chain, breaking it into finite correlated domains. Their density becomes exponentially small at low temperature, so a short experimental chain can look ordered over its whole length without representing an infinite-system ordered phase.

The connected-correlation sum gives the zero-field susceptibility per site to h as $\chi=(k_BT)^{-1}(1+q)/(1-q)$ with $q=\tanh K$, hence $\chi=e^{2K}/(k_BT)$. It grows sharply as temperature falls but remains finite at each positive temperature. This exact result exposes the failure of mean-field independence assumptions in one dimension. Long-range couplings, infinitely many local states, or a different model can evade the simple nearest-neighbour conclusion, so its assumptions belong to the result.