Spin space

A spin space is the set $S$ of allowed values for a single spin variable at one lattice site. In a lattice model on the integer lattice (see integer lattice ), each site $i$ carries a spin $\sigma_i \in S$.

For probabilistic and Gibbsian formulations, one typically equips $S$ with a $\sigma$-algebra $\mathcal{S}$ (see sigma-algebra ) so that local observables are measurable functions of $\sigma_i$. Often one also fixes an a priori measure $\rho$ on $(S,\mathcal{S})$ (see measure ), which is counting measure for discrete spins and a reference (e.g. Haar/surface) measure for continuous spins.

The spin space determines what a spin configuration is and underlies the full configuration space .

Common examples

• Ising spins: $S=\{-1,+1\}$ (see Ising model ).
• Potts spins: $S=\{1,2,\dots,q\}$ (see Potts model ).
• Lattice gas/occupancy: $S=\{0,1\}$ (see lattice gas ).
• XY spins: $S$ is the unit circle (see XY model ).
• Heisenberg spins: $S$ is a unit sphere in $\mathbb{R}^n$ (see Heisenberg model ).

Key properties

• Discrete vs continuous: If $S$ is finite/countable, sums replace integrals in the partition function ; if $S$ is continuous, the choice of a priori measure $\rho$ is part of the model.
• Compactness/topology (often used): When $S$ is finite or compact, the product configuration space can inherit useful compactness properties (see configuration space ).
• Symmetries: Many models have a symmetry group acting on $S$ (e.g. spin-flip for Ising, rotations for XY/Heisenberg). Symmetries on $S$ induce symmetries of the Hamiltonian and Gibbs measures, relevant for spontaneous symmetry breaking .

Physical interpretation

The spin space represents the local microscopic degrees of freedom:

• orientations of a magnetic moment (continuous $S$),
• discrete “up/down” magnetic moments (Ising),
• internal states/colors (Potts),
• presence/absence of a particle (lattice gas).

Changing the choice of $S$ changes the nature of fluctuations and possible ordered phases (e.g. continuous symmetries vs discrete symmetries), influencing whether and how phase transitions can occur.