Fermions and Zeta Function on the Graph

TL;DR

This work constructs a fermionic model on graphs where the Dirac operator is built from deformed incidence matrices and the partition function is given by the inverse of a graph zeta function, linking cycle counting to fermionic dynamics. By employing the Bartholdi zeta function and its determinant representations, the authors derive finite polynomial expansions in the coupling and interpret coefficients as counts of fermionic cycles; they also extend to grid graphs via covering graphs and the Artin-Ihara L-function, showing the absence of fermion doubling and enabling overlap fermions. The framework connects to statistical models through winding numbers, with the poles of the zeta function encoding phase-transition data (e.g., Ising model correspondences via r = -1). A covering-graph perspective unifies gauge theories on graphs with L-functions, suggesting broad applications in lattice gauge theory on graphs, condensed matter, and beyond.

Abstract

We propose a novel fermionic model on the graphs. The Dirac operator of the model consists of deformed incidence matrices on the graph and the partition function is given by the inverse of the graph zeta function. We find that the coefficients of the inverse of the graph zeta function, which is a polynomial of finite degree in the coupling constant, count the number of fermionic cycles on the graph. We also construct the model on grid graphs by using the concept of the covering graph and the Artin-Ihara $L$-function. In connection with this, we show that the fermion doubling is absent, and the overlap fermions can be constructed on a general graph. Furthermore, we relate our model to statistical models by introducing the winding number around cycles, where the distribution of the poles of the graph zeta function (the zeros of the partition function) plays a crucial role. Finally, we formulate gauge theory including fermions on the graph from the viewpoint of the covering graph derived from the gauge group in a unified way.

Figures (12)

• Figure 1: The graph of the cycle graph $C_3$, which has three vertices and three edges. There are three fermions $\xi^v$ on each vertex $v$ and three pairs of fermions $(\psi^e,\tilde{\psi}^e)$ on each edge $e$.
• Figure 2: The graph of the double triangle graph DT, which has four vertices and five edges. There are four fermions $\xi^v$ on each vertex $v$ and five pairs of fermions $(\psi^e,\tilde{\psi}^e)$ on each edge $e$.
• Figure 3: An example of the fermionic cycle $\Psi_{12\bar{2}\bar{1}}$ on the cycle graph $C_3$. This cycle is a single primitive cycle with length four, two bumps and $F=1$.
• Figure 4: Examples of the fermionic cycles on the double triangle graph (DT). The left figure (a) shows the fermionic cycle $\Psi_{1234}\Psi_{\bar{5}\bar{2}\bar{1}}$ with length 7, which is a product of two primitive cycles, then $F=2$. The right figure (b) shows the fermionic cycle $\Psi_{\bar{4}\bar{3}51234\bar{5}\bar{2}\bar{1}}$ with maximal length 10, which is a single primitive cycle of $F=1$.
• Figure 5: The cycle graph $C_N$ as the covering graphs over $C_1$ graph. The voltage assignment is given by the representation of $\mathbb{Z}_N$. It is not shown in the figure, but the vertex at $(v,\rho_{N-1})$ is again connected to the vertex at $(v,\rho_0)$ owing to the periodic boundary condition.