Searching the Null Space: A novel approach for calculating partition functions

Knowing the partition function of a particle system is immensely valuable, as it can be used to determine various important thermodynamic properties, including entropy, free energy, chemical potential, magnetization, and specific heat. However, due to the exponential growth rate of microstates, finding exact, analytic solutions is possible only for a few simple models. Numerical approaches have traditionally focused on estimating the density of energy states using Monte Carlo (MC) techniques, but the energy space is rough and difficult to traverse. In this talk, I will present an alternative framework for calculating the partition function of lattice systems. By constructing a two-dimensional binary matrix from the lattice, the problem can be reduced to finding the number of specific vectors in the null space of that matrix, which can be numerically approximated using a modified version of the Wang-Landau algorithm. I will discuss results obtained using this methodology for multidimensional Ising and Baxter-Wu models, compared against the traditional energy space approach, and highlight domains where this approach offers improved partition function estimates.