# Fermionic two-level system

Selveste
Homework Statement:
a
Relevant Equations:
a
Problem statement:

Consider a fermionic system with two states $1,2$ with energy levels $\epsilon_i, i=1,2$. Moreover, the number of particles in state $i$ is $n_i = 0,1$. Let the Hamiltonian of the system be
$$H = \sum_{i=1}^2 \epsilon_i n_i + \sum_{i \neq j} U n_i n_j$$
Here, $U > 0$ is a Coulomb-repulsion present in the system if both states $i$ and $j$ are occupied.

a) Compute the grand canonical partition function $Z_g$ of the system by direct summation.
b) Let $\epsilon_i = \epsilon; (i=1,2)$, and compute $<N>$.
c) Let $\beta U \gg 1$ and find $<U>$ in this limit. Finally, for $\beta U \gg 1$, set $\mu = \epsilon$ and compute $<U>$ in this case.

Attempt at solution:

Just to make it clear:
$n_k$ is the number of particles in the state with wave number $k$.
$\mu$ is the chemical potential - the energy required to remove one particle from the system.

a)
$$Z_g = \sum_{[n_k]} e^{-\beta \sum_{k}(\epsilon_k-\mu)n_k} = \prod_k \sum_{n_k}e^{-\beta (\epsilon_k-\mu)n_k} = \prod_k \sum_{n_k=0}^{1}e^{-\beta (\epsilon_k-\mu)n_k}$$
$$=\prod_k \Big( 1 + e^{-\beta(\epsilon_k-\mu)}\Big) = \Big(1+e^{-\beta (\epsilon_1 -\mu)}\Big)\Big( 1+e^{-\beta (\epsilon_2 -\mu)} \Big)$$
$$= 1 + e^{-\beta (\epsilon_1 - \mu)} + e^{-\beta (\epsilon_2 - \mu)} + e^{-\beta (\epsilon_1 + \epsilon_2 - 2\mu)}$$

b)
$$<N> = \frac{\partial ln Z_g}{\partial (\beta \mu)} = \frac{2e^{-\beta (\epsilon - \mu)}+2e^{-2\beta(\epsilon - \mu)}}{1 +2e^{-\beta(\epsilon - \mu)}+e^{-2\beta(\epsilon - \mu)}}$$

c)
Here I don't know what to do as there is no $U$ is the expressions I have found.

$$Z_g = \sum_{[n_k]} e^{-\beta \sum_{k}(\epsilon_k-\mu)n_k} = \prod_k \sum_{n_k}e^{-\beta (\epsilon_k-\mu)n_k} = \prod_k \sum_{n_k=0}^{1}e^{-\beta (\epsilon_k-\mu)n_k}$$
A basic way to express ##Z_g## is as a sum over all possible microstates, ##s##, $$Z_g = \sum_s e^{-\beta (E_s - \mu N_s)}$$ where ##E_s## is the total energy of the system for the microstate ##s## and ##N_s## is the total number of particles in the microstate ##s##. The sum over all microstates includes states for all possible values of the total number of particles.