グラフ理論に基づいた Lattice fermion とLaplacian operator としての Wilson term
湯本 純
秋田大学大学院 理工学研究科

Lattice fermions as spectral graphs

Background (Species and Wilson term)

格子上の場の理論には，fermion doubling という難問が存在する。
Fermion doubling
QCD に必要な性質を理論に課すことで，複数の lattice fermion (species) が出現するという問題。

J.Y, T.Misumi, JHEP 02, 104 (2022).


これらの species は縮退しているため、区別することができない。
Wilson fermion : species-splitting mass fermion∑ a5
Wilson term : species-splitting term SW = n,µ 2 ψ̄n (2ψn − ψn+µ − ψn−µ)]1 ∑[iγµ sin apµ + (1 − cos apµ)DW (p) =a µPhysical (0, 0, 0, 0) :DW (p) = iγµpµ + O(a)Im λ16Im λRe λNon physical (π/a, 0, 0, 0) :DW (p) = iγµpµ + a2 + O(a)

→

1464

1 Re λµ⊗µwhere ψ = n ψn |n⟩ with |n⟩ ≡ µ |nµ⟩.
[]∑2π(kµ − 1) |k⟩ ⟨k| = 0
D(k) =iγµ sin
Nµ[]2π(kµ − 1) = 0sinNµ⇒k,µ16 species!
Lattice ﬁeld theory
Lattice fermion# of Fermion species=⇒=⇒=⇒

Spectral graph theory
Directed and Weighted spectral graphNullity of spectral matrix

Wilson term as Laplacian operator

A Laplacian operator L of a graph G(V, E) is given bydi if i = j
Lij ≡ −1 if i ̸= j and (i, j) are linked

0otherwise

Maximal # of species depend on the topological invariant of lattice.
Table: Topological invariant and maximum # of the species

latticeTopological invariant

maximal # of species d

4-d torus
Torus T D
Hyperball B D
Sphere S D
T D × Bd

1+4+6+4+1D(1 + 1)1 + 0 + 0 + ···1 + 0 + 0 + ··· + 1
2D + 0162D122D

where di is the number of edges sharing the site i.

e.g. The Lagrangian on T 4 constructed from the Laplacian operator L is∑a5ψ̄Lψ = 2
ψ̄n (2ψn − ψn+µ − ψn−µ) ⇒ SW = 4 ψ̄Lψn,µThe Laplacian operator results in the Wilson term!
Theory (Hodge theory)
The nullity of a Laplacian operator deﬁned on a complex chain coincides with the 0-th Betti
number that is the topological invariant.

数学的な説明を与えるために、graph theory に着目した。

Lattice fermions as graph theory=⇒=⇒∑m,nDeﬁnition (Laplacian operator)

Previous research# of Fermion species
Wilson term

The naive lattice action as spectral graph theory is∑∑∑S=ψ̄mγµDmnψn =ψ̄γµAµψ ≡ ψ̄Dψe.g. The Betti number and nullity of Dirac matrix with the Laplacian on T 4
Nullity of spectral matrix [J.Y, T.MIsumi (2022)]
Laplacian matrix and topological invariant

the r-th Betti number βr of T 4
β0(T 4) = 1,New conjecture on species doubling of lattice fermions!

β1(T 4) = 4,β3(T 4) = 4,

Spectral graph theory

Im λβ2(T 4) = 6,1464

1 Re λβ4(T 4) = 1,The nullity of Dirac matrix with the Laplacian is consistent with
Hodge theory, so # of zero-eigenvalue = β0(T 4) = 1.
The r-th Betti number and each species-splitting number just match!

Deﬁnition (graph)
A graph G is a pair G = (V, E). V is a set of vertices and E is a set of edge.

Wilson term

Deﬁnition (directed graph)
A directed graph is a pair (V, E) of sets of vertices and edges together with two maps
init : E → V and ter : E → V . The two maps are assigned to every edge eij with an initialvertex init(eij ) = vi ∈ V and a terminal vertex ter(eij ) = vj ∈ V . If init(eij ) = ter(eij ), theedge eij is called a loop.

=⇒Laplacian operator
and topological invariant (Betti number)

Summary
Lattice fermions as graph theory

Deﬁnition (weighted graph)# of Fermion species
Wilson term

A weighted graph has a value (weight) for each edge in a graph.

Deﬁnition (adjacency matrix)=⇒=⇒Nullity of spectral matrix
Laplacian matrix and the Betti number

Lattice ﬁeld theory

An adjacency matrix A of graph is the |V | × |V | matrix given by{wij if there is a edge from i to jAij ≡0otherwise

Graph theory

where wij is the weight of an edge from i to j.

Topology

We believe that these are closely related.

e.g. A graph G and an adjacency matrix AG of G.
1/2G12

−1/2

0 1 0 −1
1  −1 0 1 0 −1/2 1/2 AG = 
2  0 −1 0 1 
1 0 −1 0

1/2 −1/24

−1/2Discussion• A new conjecture on the number of fermion species on the discretizedmanifold.
J.Y, T.Misumi [arXiv:2301.09805]• The proof of the new conjecture.
• The Laplacian operator as Persistent homology and Persistent spectral

3graph.

1/2

計算物理 春の学校 2023