Normalize the ring modes as

φ_{j} (x) = L^{- 1/2} e^{i k_{j} x}

, with distinct

k_{j} = 2 π ℓ_{j} / L

and integers

ℓ_{j}

. Inserting

\hat{ψ} (x) = φ_{1} (x) a_{1} + φ_{2} (x) a_{2}

into the contact interaction gives

H_{int} = \frac{g}{2} \sum_{i, j, k, l = 1}^{2} I_{ijk l} a_{i}^{†} a_{j}^{†} a_{k} a_{l}, I_{ijk l} = \int_{0}^{L} φ_{i}^{*} φ_{j}^{*} φ_{k} φ_{l} d x .

(1)

Orthogonality makes

I_{ijk l} = 1/ L

when

k_{i} + k_{j} = k_{k} + k_{l}

, and zero otherwise: this is momentum conservation in the projected Hamiltonian operator. There is one all-mode-1 term, one all-mode-2 term, and four mixed terms with one particle in each mode. A pair conversion between distinct ring modes would require

2 k_{1} = 2 k_{2}

and therefore vanishes.

Using

a_{j}^{† 2} a_{j}^{2} = n_{j} (n_{j} - 1)

and

a_{1}^{†} a_{2}^{†} a_{1} a_{2} = n_{1} n_{2}

yields the contact interaction between two ring modes,

H_{int} = \frac{g}{2 L} [n_{1} (n_{1} - 1) + n_{2} (n_{2} - 1) + 4 n_{1} n_{2}], α = β = \frac{g}{2 L}, γ = \frac{2 g}{L} .

(2)

The factor four comes from two independent choices of ordering for the creation operators and annihilation operators. This is the two-mode projected model; a full ring contact interaction can also scatter particles into other modes while conserving total momentum.