Appendices

Evaluation of the Fermion Free Energy

There are \(2^N\) possible configurations of the spins in the LRFK model. In the language of ions and electrons (immobile and mobile species), we define \(n^k_i\) to be the occupation of the \(i\)th site of the \(k\)th configuration. The quantum part of the free energy can then be defined through the quantum partition function \(\mathcal{Z}^k\) associated with each state \(n^k_i\):

\[\begin{aligned} F^k &= -1/\beta \ln{\mathcal{Z}^k}, \\ \end{aligned}\]

such that the overall partition function is:

\[\begin{aligned} \mathcal{Z} &= \sum_k e^{- \beta H^k} Z^k \\ &= \sum_k e^{-\beta (H^k + F^k)}. \\ \end{aligned}\]

Fermions are limited to occupation numbers of 0 or 1, so \(Z^k\) simplifies nicely. If \(m^j_i = \{0,1\}\) is defined as the occupation of the level with energy \(\epsilon^k_i\) then the partition function is a sum over all the occupation states labelled by \(j\):

\[\begin{aligned} Z^k &= \mathrm{Tr} e^{-\beta F^k} = \sum_j e^{-\beta \sum_i m^j_i \epsilon^k_i}\\ &= \sum_j \prod_i e^{- \beta m^j_i \epsilon^k_i}= \prod_i \sum_j e^{- \beta m^j_i \epsilon^k_i}\\ &= \prod_i (1 + e^{- \beta \epsilon^k_i})\\ F^k &= -1/\beta \sum_k \ln{(1 + e^{- \beta \epsilon^k_i})}. \end{aligned}\]

Observables can then be calculated from the partition function, for examples the occupation numbers:

\[\begin{aligned} \langle N \rangle &= \frac{1}{\beta} \frac{1}{Z} \frac{\partial Z}{\partial \mu} = - \frac{\partial F}{\partial \mu}\\ &= \frac{1}{\beta} \frac{1}{Z} \frac{\partial}{\partial \mu} \sum_k e^{-\beta (H^k + F^k)}\\ &= 1/Z \sum_k (N^k_{\mathrm{ion}} + N^k_{\mathrm{electron}}) e^{-\beta (H^k + F^k)},\\ \end{aligned}\]

with the definitions:

\[\begin{aligned} N^k_{\mathrm{ion}} &= - \frac{\partial H^k}{\partial \mu} = \sum_i n^k_i\\ N^k_{\mathrm{electron}} &= - \frac{\partial F^k}{\partial \mu} = \sum_i \left(1 + e^{\beta \epsilon^k_i}\right)^{-1}.\\ \end{aligned}\]

Next Section: Particle-Hole Symmetry