Lattice partition function

Fix a finite region $\Lambda$ of the lattice (see finite boxes in a lattice ) and a single-site spin space $S$ with a reference (a priori) measure $\lambda$ on $S$. A spin configuration on $\Lambda$ is an element $\sigma_\Lambda \in S^\Lambda$, and a boundary condition is a configuration $\tau$ on the complement $\Lambda^c$.

Given an inverse temperature $\beta$ (see inverse temperature ) and a finite-volume lattice Hamiltonian $H_\Lambda(\sigma_\Lambda \mid \tau)$, the finite-volume lattice partition function is

where $\lambda_\Lambda$ is the product measure on $S^\Lambda$ induced by $\lambda$ (see product measure ).

If $S$ is finite (e.g. the Ising single-spin set $\{\pm 1\}$), the integral reduces to a finite sum over $\sigma_\Lambda \in S^\Lambda$.

This object is the lattice analogue of the canonical partition function .

Key properties

• Normalization for Gibbs weights. The partition function is the normalizing constant for the finite-volume Gibbs measure :

  $$\mu_\Lambda^{\beta,\tau}(d\sigma_\Lambda) ={} \frac{1}{Z_\Lambda(\beta,\tau)} \exp\!\bigl(-\beta\,H_\Lambda(\sigma_\Lambda\mid\tau)\bigr)\, \lambda_\Lambda(d\sigma_\Lambda).$$

• Boundary dependence. In general $Z_\Lambda(\beta,\tau)$ depends on the boundary condition $\tau$; for finite-range interactions the dependence is through spins near the boundary of $\Lambda$.

• Generates thermodynamics. Derivatives of $\log Z_\Lambda$ with respect to parameters in $H_\Lambda$ (e.g. a magnetic field coupling ) produce finite-volume expectations and fluctuations (see ensemble averages ).

Physical interpretation

$Z_\Lambda(\beta,\tau)$ is the weighted “count” of microstates in $\Lambda$ compatible with an environment $\tau$, with each configuration weighted by the Boltzmann factor $\exp(-\beta H_\Lambda)$. Its logarithm controls the finite-volume free energy and leads directly to the lattice pressure and its thermodynamic limit.