Abstract
We investigate the spectral characteristics of the naive discrete Dirac operator on a Face-Centered Cubic
(FCC) lattice in D = 3 spatial dimensions. While standard hypercubic lattices are notorious for generating
2
3
= 8 fermion doublers at the Brillouin Zone corners, the FCC lattice offers a richer, non-trivial spectral
geometry shaped by its Truncated Octahedron Brillouin zone. By explicitly deriving the dispersion relation,
we map the complete set of zero modes. Our analysis reveals a distinct topological structure: exactly four
isolated zero modes reside in the zone interior (at the L-points), while a continuous family of zeros forms
nodal lines along the boundary connecting the X and W points. Calculating the Jacobian determinant
allows us to assign chiralities to the isolated modes: the fundamental Γ-mode (χ = +1) is balanced by the
four L-modes (χ = −1 each). To satisfy the Nielsen-Ninomiya theorem, the boundary nodal lines must
therefore carry a net chirality of +3. We conclude by discussing the decomposition of the L-point quartet
under the tetrahedral group S
4
, showing how it naturally splits into a singlet and a triplet (4 → 1 ⊕ 3)—a
geometric feature that suggests a novel mechanism for generating flavor structures in lattice gauge theories.
I. INTRODUCTION
Putting fermions on a lattice has always been a messy business. The naive discretization of
the Dirac equation runs headlong into the doubling problem: on a simple hypercubic grid in D
dimensions, replacing derivatives with finite differences generates 2
D
fermion species [1, 2]. In three
spatial dimensions, this means getting eight particles for the price of one, crowding the corners of
the cubic Brillouin zone.
Physicists have developed clever ways to suppress these artifacts—Wilson fermions [3] break
chiral symmetry to give the doublers infinite mass, while Staggered fermions [4, 5] accept a smaller
number of doublers in exchange for remnant chiral symmetry. Yet, these solutions usually involve
tweaking the operator or the fields, rather than rethinking the geometry of the space itself.
Ideally, one should look at non-hypercubic lattices. The Face-Centered Cubic (FCC) lattice is
particularly interesting because it represents the tightest possible way to pack spheres in 3D (the
Kepler conjecture) and boasts the highest possible point group symmetry (O
h
). While gauge fields
on Body-Centered Cubic (BCC) lattices have seen some study [8], the behavior of fermions on the
reciprocal FCC lattice remains largely unexplored.
In this Letter, we map out the naive Dirac operator on an FCC lattice. We find that the
geometry itself breaks the degeneracy typical of hypercubic doublers. Instead of clustering at
∗
raghu@idrive.com
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identical corners, the zero modes separate into three distinct topological classes:
1. A single fundamental mode at the center (Γ).
2. A quartet of isolated chiral partners at the face centers (L).
3. Continuous nodal lines running along the zone boundary (X − W).
This natural separation implies that the FCC lattice might provide a purely geometric solution
to the generation problem in the Standard Model.
II. LATTICE DEFINITIONS
A. Real Space Basis
The FCC lattice is defined by the 12 vectors connecting a point to its nearest neighbors. If we
take the cubic lattice constant to be a, these vectors are permutations of (±1, ±1, 0)×a/2. To keep
the momentum space algebra clean, we define our hopping vectors in integer units. Let N be the
set of 12 unnormalized integer vectors:
N =
(±1, ±1, 0),
(±1, 0, ±1),
(0, ±1, ±1)
(1)
B. Reciprocal Space
The reciprocal of an FCC lattice is a Body-Centered Cubic (BCC) lattice. Consequently, the
First Brillouin Zone (FBZ) takes the shape of a Truncated Octahedron. We focus on the high-
symmetry points in the FBZ (using the integer basis defined above) [12]:
• Γ (Gamma): The Zone Center, k = (0, 0, 0).
• L-Points: The centers of the 8 hexagonal faces. Because k and −k are identified, we have 4
independent points:
L = (±π/2, ±π/2, ±π/2) (2)
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• X-Points: The centers of the 6 square faces (3 independent points):
X = {(π, 0, 0), (0, π, 0), (0, 0, π)} (3)
• W-Points: The corners of the zone, e.g., (π, π/2, 0).
III. THE NAIVE DIRAC OPERATOR
We construct the naive Dirac operator in momentum space by summing over the 12 nearest
neighbors. We use the spatial components (µ = 1, 2, 3) of the standard 4D Euclidean gamma
matrices to handle the spatial hopping:
D(k) =
1
a
X
n∈N
iγ
µ
n
µ
sin(k · n) (4)
where {γ
µ
, γ
ν
} = 2δ
µν
.
A. Dispersion Relation
The operator’s eigenvalues depend on the magnitude of the kinetic vector field f
µ
(k):
f
µ
(k) =
1
a
X
n∈N
n
µ
sin(k · n) (5)
The dispersion relation is simply |f(k)|
2
=
P
µ
f
µ
(k)
2
. Zero modes appear wherever |f(k)| vanishes.
B. Zero Mode Analysis
We ran a numerical minimization of |f(k)|
2
across the entire Brillouin Zone. The findings are
summarized in Table I.
1. The Fundamental Mode (Γ) At k = (0, 0, 0), sin(k · n) is zero for every neighbor vector
n. This is the physical fermion we expect.
2. The Isolated Doublers (L-Points) Things get interesting at the L-point k = (π/2, π/2, π/2).
Here, the 12 scalar products L · n land in the set {0, ±π}. Specifically, six neighbors give L · n = 0
while the other six give L · n = ±π. Since sine vanishes at both 0 and π, every term drops out.
This holds for all 4 independent L-points, meaning we have four exact zero modes sitting isolated
in the bulk of the zone.
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TABLE I. Spectral characteristics of high-symmetry points in the FCC Brillouin Zone. M
W ilson
denotes the
mass acquired from a standard Wilson term (r/a)
P
(1 − cos k · n) with r = 1.
Point Coordinates (k) |f(k)|
2
M
W ilson
Chirality (χ)
Γ (0, 0, 0) 0 0 +1
L (
π
2
,
π
2
,
π
2
) 0 12/a −1
X (π, 0, 0) 0 16/a N/A (Line)
W (π,
π
2
, 0) 0 16/a N/A (Line)
K (
3π
4
,
3π
4
, 0) = 0 ∼ 15.7/a N/A
3. Boundary Nodal Lines Unlike the hypercubic lattice, where doublers are discrete points,
the FCC operator generates continuous lines of zeros. Take the line connecting an X-point (π, 0, 0)
to a W -point (π, π/2, 0), parametrized by k(t) = (π, t, 0). Plugging this into Eq. (4), we find the
vector sum vanishes identically. This happens because of the inversion symmetry of the FCC set:
for every n, there is a −n. When one component of k is π, the terms from n and −n cancel each
other out pairwise (since sin(π + x) = − sin(x) = sin(−x)).
IV. CHIRALITY AND NIELSEN-NINOMIYA
The Nielsen-Ninomiya theorem [1, 6] provides a strict consistency check: the total chirality of
all zero modes on the lattice must sum to zero. Let’s see how the FCC lattice satisfies this. We
calculate the chirality χ
i
at a zero k
i
by taking the sign of the Jacobian determinant of the kinetic
field f
µ
(k):
J
µν
=
∂f
µ
∂k
ν
=
1
a
X
n∈N
n
µ
n
ν
cos(k · n) (6)
A. Chirality of Γ
At the origin, cos(0) = 1, so the Jacobian is just the sum of the outer products of the neighbor
vectors. For our 12 vectors:
J
µν
(Γ) =
1
a
X
n
n
µ
n
ν
=
8
a
δ
µν
(7)
This is a positive definite matrix with eigenvalues {8/a, 8/a, 8/a}.
det J(Γ) > 0 ⇒ χ
Γ
= +1 (8)
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We define this as the Right-Handed sector.
B. Chirality of L-Points
At L = (π/2, π/2, π/2), the cosine terms flip signs. We get −1 for neighbors where L · n = ±π
and +1 where L · n = 0. Computing the Jacobian matrix explicitly gives an indefinite matrix with
eigenvalues:
λ
L
=
1
a
{+4, +4, −8} (9)
The determinant is negative (det J = −128/a
3
):
det J(L) < 0 ⇒ χ
L
= −1 (10)
With 4 independent L-points, the total contribution is:
X
L
χ = 4 × (−1) = −4 (11)
These represent the Left-Handed sector.
C. The Summation
So far, the isolated modes give us 1 + (−4) = −3. Since the total chirality must be zero, the
“missing” +3 chirality must reside on the continuous nodal lines at the zone boundary.
χ
boundary
= −(χ
Γ
+ χ
L
) = −(1 − 4) = +3 (12)
This confirms that the boundary lines aren’t just artifacts; they are topological defects carrying
the necessary compensating charge [9].
V. DISCUSSION: THE TETRAHEDRAL QUARTET
The spectral structure we’ve derived has some intriguing implications for model building. In
standard hypercubic lattice theory, the 2
3
= 8 doublers are all geometrically equivalent—there is
no obvious way to pick out a specific number of generations.
But on the FCC lattice, the spectrum is naturally stratified:
• The Γ-point is a unique Singlet.
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• The boundary lines are continuous (and thus easily gapped).
• The L-points form a distinct, isolated Quartet deep in the bulk.
Crucially, the four L-points transform under the tetrahedral group S
4
. This 4-dimensional
representation is reducible:
4 → 1 ⊕ 3 (13)
This decomposition hints at a geometric origin for the “3 + 1” flavor structure seen in nature (three
active generations plus one sterile state). If the vacuum were to develop a condensate that selects
a preferred time direction [11], the S
4
symmetry would break, splitting the quartet into a singlet
and a triplet.
This suggests that N
g
= 3 might not be an arbitrary input, but a consequence of the intrinsic
geometry of the lattice itself. We leave the construction of such a mass term for future work.
VI. CONCLUSION
We have mapped the spectral landscape of the naive Dirac operator on the FCC lattice. The
picture that emerges is one of a fundamental mode at Γ, four isolated chiral partners at L, and
continuous nodal lines at the zone boundary. The geometry satisfies the Nielsen-Ninomiya theorem
in a non-trivial way (χ
total
= +1 − 4 + 3 = 0), providing a consistent topological starting point.
Unlike the hypercubic lattice, which gives us an unmanageable swarm of identical fermions, the
FCC lattice offers a structured spectrum that naturally supports a 3-generation model via the
L-point triplet.
ACKNOWLEDGMENTS
Numerical results presented in this work were verified using publicly available code [13].
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