Can we study an odd number sized lattice model at half filling?

[Luqman Saleem]

So one can numerically study (I am interested in exact diagonalization) any 1D lattice model with $L$ sites and $N$ number of particles. At half filling, $L/N = 2$. My question to a professor was that can we study a system of size $L = 31$ at half filling? He replied yes, there is a way to go from Canonical to Grand Canonical ensembles. He said that system with 31 sites at half filling will be equal to a system of $L = 30$ at half-filling weighted by half + a system of $L = 32$ at half-filling weighted by half. But he said that he's not sure about it.

My question here is, can we really study a system of odd number lattice size at half filling using numerical exact diagonalization? If yes then how?

 

[BigJDubs]

Yes, it is possible to study a system of odd number lattice size at half filling using numerical exact diagonalization. The idea is to use the Grand Canonical ensemble, which allows us to consider systems of different sizes and particle numbers. In this way, we can simulate a system of size L = 31 at half-filling by considering two separate systems of sizes L = 30 and L = 32 with half of the total number of particles in each system. This approach allows us to still keep the total number of particles constant while still capturing the physical behavior of the system at half-filling.