Gross-Witten-Wadia phase transition in induced QCD on the graph

In this paper, we consider an extended Kazakov-Migdal model defined on an arbitrary graph. The partition function of the model, which is expressed as the summation of all Wilson loops on the graph, turns out to be represented by the Bartholdi zeta function weighted by unitary matrices on the edges of the graph. The partition function on the cycle graph at finite N is expressed by the generating function of the generalized Catalan numbers. The partition function on an arbitrary graph can be exactly evaluated at large N which is expressed as an infinite product of a kind of deformed Ihara zeta function. The non-zero area Wilson loops do not contribute to the leading part of the 1/N-expansion of the free energy but to the next leading. The semi-circle distribution of the eigenvalues of the scalar fields is still an exact solution of the model at large N on an arbitrary regular graph, but it reflects only zero-area Wilson loops.

Section: The Fkm Modelmentioning

confidence: 99%

Section: Phase Structure Of the Fkm Model On The Cycle Graphmentioning

confidence: 99%

“…This transition indexed by the eigenvalue distribution is the so-called GWW phase transition [21,22]. For more detail, see [19].…”

Section: Phase Structure Of the Fkm Model On The Cycle Graphmentioning

confidence: 99%

“…From the eigenvalue density (4.14), the free energy of the FKM model on C n can be evaluated as [19]…”

Section: Phase Structure Of the Fkm Model On The Cycle Graphmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Graph zeta functions and Wilson loops in a Kazakov–Migdal model

Matsuura

Ohta

2022

Equivalence of lattice operators and graph matrices

Yumoto,

Misumi

2024

We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix named as “anti-symmetrized adjacency matrix”, specifically tailored for cycle digraphs (T1 lattice) and simple directed paths (B1 lattice). The nontrivial relation between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the anti-symmetrized adjacency matrix, along with its extensions to higher dimensions, are equivalent to naive lattice Dirac operators. Building upon these connections, we provide rigorous proofs for two key assertions: (i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice). (ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the D-dimensional graph results from a cartesian product of cycle digraphs (T1 lattice) and simple directed paths (B1 lattice).

Phases and Duality in the Fundamental Kazakov–Migdal Model on the Graph

Matsuura,

Ohta

2024

We examine the fundamental Kazakov-Migdal (FKM) model on a generic graph, whose partition function is represented by the Ihara zeta function weighted by unitary matrices. The FKM model becomes unstable in the critical strip of the Ihara zeta function. We discover a duality between small and large couplings, associated with the functional equation of the Ihara zeta function for regular graphs. Although the duality is not precise for irregular graphs, we show that the effective action in the large coupling region can be represented by a summation of all possible Wilson loops on the graph similar to that in the small coupling region. We estimate the phase structure of the FKM model both in the small and large coupling regions by comparing it with the Gross-Witten-Wadia (GWW) model. We further validate the theoretical analysis through detailed numerical simulations.