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Lattice gas ↔ Ising model mapping

Exact change of variables between occupation variables n∈{0,1} and Ising spins σ∈{±1}, relating chemical potential to magnetic field and liquid–gas coexistence to spin-flip symmetry.
Lattice gas ↔ Ising model mapping

Prerequisites and notation

Let Λ\Lambda be a finite graph/lattice with coordination number zz (e.g. z=4z=4 for the square lattice).

Example: nearest-neighbor lattice gas and its Ising representation

Lattice-gas variables and Hamiltonian

A lattice gas uses occupation numbers nx∈{0,1}n_x\in\{0,1\} with Hamiltonian

HΛLG(n)=−ε∑⟨x,y⟩nxny  −  μ∑x∈Λnx, H_{\Lambda}^{\mathrm{LG}}(n) = -\varepsilon\sum_{\langle x,y\rangle} n_x n_y \;-\; \mu\sum_{x\in\Lambda} n_x,

where ε>0\varepsilon>0 is an attractive interaction and μ\mu is the chemical potential.

Change of variables to spins

Define Ising spins σx∈{−1,+1}\sigma_x\in\{-1,+1\} by

nx=1+σx2⟺σx=2nx−1. n_x=\frac{1+\sigma_x}{2} \quad\Longleftrightarrow\quad \sigma_x = 2n_x-1.

Then

nxny=14(1+σx+σy+σxσy). n_x n_y = \frac{1}{4}\bigl(1+\sigma_x+\sigma_y+\sigma_x\sigma_y\bigr).

Resulting Ising Hamiltonian

Substituting into HΛLGH_{\Lambda}^{\mathrm{LG}} and collecting terms yields

HΛLG(n)=−J∑⟨x,y⟩σxσy  −  h∑x∈Λσx  +  constant, H_{\Lambda}^{\mathrm{LG}}(n) = -J\sum_{\langle x,y\rangle}\sigma_x\sigma_y \;-\; h\sum_{x\in\Lambda}\sigma_x \;+\; \text{constant},

with parameters

J=ε4,h=μ2+εz4. J=\frac{\varepsilon}{4}, \qquad h=\frac{\mu}{2}+\frac{\varepsilon z}{4}.

Thus the lattice gas in the grand-canonical ensemble is exactly an in a field.

Dictionary of observables

  • Density ρ\rho vs magnetization mm:

    ρ=⟨nx⟩=1+⟨σx⟩2=1+m2. \rho=\langle n_x\rangle = \frac{1+\langle \sigma_x\rangle}{2} = \frac{1+m}{2}.

  • Compressibility and response correspondences follow similarly by differentiating with respect to μ\mu and hh.

Coexistence line as zero field (particle–hole symmetry)

The Ising spin-flip symmetry corresponds to particle–hole symmetry in the lattice gas. The coexistence curve is the line where the effective field vanishes:

h=0⟺μ=−εz2. h=0 \quad\Longleftrightarrow\quad \mu = -\frac{\varepsilon z}{2}.

On this line, the two Ising phases (for dimensions where symmetry breaking occurs) correspond to high-density “liquid” and low-density “gas” phases. In particular, the 2D lattice gas inherits the phase transition of .

Consequence: universality and order parameters

Because the mapping is exact, critical behavior (exponents, scaling forms) is shared between the lattice gas and the Ising model in the same dimension; the order parameter can be taken as ρ−12\rho-\tfrac12 or equivalently mm (see ).