# Rewriting of a occupation based Hamiltonian to an spin based Ising Hamiltonian

I run in to the following problem of rewriting a hamiltonian derived in an earlier question to an Ising hamiltonian.

(b)

Identify your result in (a) with the Hamiltonian and the partition function of the Ising model. Be aware that the ni’s take on values of {0, 1}, whereas in the Ising model spins $$s_i$$ take on values{−1, +1}. Introduce quasi-spins $$s_i$$ from the $$n_i$$’s in (a) which take on values of {−1, +1}, i.e. find a simple relation between si in terms of ni , such that $$n_i = 0$$ gives $$s_i = −1$$ and $$n_i = 1$$ gives $$s_i = +1$$. Show that the Hamiltonian you found in (a) can be expressed as an Ising Hamiltonian using a quasi-field $$h = γu/4 + µ/2$$, and interaction parameter $$J = u/4$$. Re-express your results in (a) using thosedefinitions to arrive at an Ising Hamiltonian: $$H_I=−h\sum_i s_i − J \sum_{{(i,j})_{nn}} s_i s_j$$

Start of the excercise and question (a) are as following:

Assume a gas of particles at temperature T, in which all particles exclusively occupy sites on a square lattice. The occupation for a specific site can be $$n_i$$ = {0, 1}. Each occupied site will add an energy −µ to the total system (µ can be interpreted as the chemical potential of the gas). If two particles occupy neighboring sites they interact by an additional energy −u.

(a)

Obtain an expression for the energy per site, $$\epsilon_i$$ . Show that the total lattice gas Hamiltonian is given by:

$$H_{LG}($${$$n_i,n_j$$}$$)=−\sum^N_i n_i (\mu + u/2 \sum_{j_{nn}}n_j)$$

Comment: In general nn stands for nearest neighbours. Question (a) I understand fully and my knowledge on the Ising model is also decent, I just can get this rewriting to work out well.

My thoughts:

The relation between $$n_i$$ and $$s_i$$ is quite simpel, $$s_i=2n_i-1$$ or equivalently $$n_i=1/2(s_i+1)$$. I just started with this subsitution. I see some seemingly good term come up but I cant really tie it together all that well. Cant easily get the h and J in there with the corresponding sum next to it. The $$\gamma$$ factor comes from just counting of the constant in the sum over neighbours. $$\gamma$$ is said to be the number of nearest neighbours in the follow up question (c)