342
R. Boyack and J. Maciejko
former under U(1) chiral transformations ψ α → e −iγ 5 θ/2 ψ α , with a concomitant
rotation of the scalar field φ = (φ 1 , φ 2 ).
If the gauge field is absent, this also establishes the equivalence between the
ungauged NJL model and the chiral O(2) GN model. The latter describes the
semimetal-to-Kekulé-VBS transition for interacting fermions on the honeycomb
lattice [13–15]. The D 6 point-group symmetry of the honeycomb lattice allows for
a term of the form ∝ (φ 1 + iφ 2 ) 3 + c.c. in the critical Lagrangian, which is marginal
in the N f = ∞ limit at the chiral O(2) GN fixed point. However, a renormalizationgroup analysis in the large-N f limit shows that the O(1/N f ) correction renders this
term irrelevant [15]. QMC simulations of the joint probability distribution P (φ 1 , φ 2 )
of the two components of the VBS order parameter also support the emergent SO(2)
symmetry at the critical point [15].
The critical points of the gauged and ungauged NJL models are strongly coupled
(2 + 1)-dimensional conformal field theories characterized by a spectrum of scaling
dimensions that correspond to universal critical exponents. Some of these exponents
have already been computed in the 1/N f expansion in general d spacetime
dimensions [24–27]. The order-parameter anomalous dimension for d = 3 is
chiral O(2) QED 3 -GN : η φ = 1 +
56
3π 2 N f
+ O
1/N
2
f
,
(11)
chiral O(2) GN : η φ = 1 −
8
3π 2 N f
+
544
27π 4 N 2
f
+ O
1/N
3
f
,
(12)
and is related to the scalar-field scaling dimension by Δ φ =
1
2
1 + η φ
. The inverse
correlation length exponent is
chiral O(2) QED 3 -GN : ν
−1
= 1 −
80
3π 2 N f
+ O
1/N
2
f
,
(13)
chiral O(2) GN : ν
−1
= 1 −
16
3π 2 N f
+
8
364 + 27π 2
27π 4 N 2
f
+ O
1/N
3
f
,
(14)
and is related to the scaling dimension of the φ
2 operator by Δ φ 2 = 3 − ν −1 .
3 Fermion Bilinear Scaling Dimensions
The scaling dimension Δ φ characterizes the power-law decay at long distances of
the microscopic VBS correlation function VBS (r)O VBS (r ) ∼ |r − r | −2Δ φ at
the U(1)-VBS critical point of the lattice gauge theory (1), where O VBS (r) can
be chosen as (−1) x
A S A (r)S A (r + ˆ
x) or (−1) y
A S A (r)S A (r + ˆ
y), describing
R. Boyack and J. Maciejko
former under U(1) chiral transformations ψ α → e −iγ 5 θ/2 ψ α , with a concomitant
rotation of the scalar field φ = (φ 1 , φ 2 ).
If the gauge field is absent, this also establishes the equivalence between the
ungauged NJL model and the chiral O(2) GN model. The latter describes the
semimetal-to-Kekulé-VBS transition for interacting fermions on the honeycomb
lattice [13–15]. The D 6 point-group symmetry of the honeycomb lattice allows for
a term of the form ∝ (φ 1 + iφ 2 ) 3 + c.c. in the critical Lagrangian, which is marginal
in the N f = ∞ limit at the chiral O(2) GN fixed point. However, a renormalizationgroup analysis in the large-N f limit shows that the O(1/N f ) correction renders this
term irrelevant [15]. QMC simulations of the joint probability distribution P (φ 1 , φ 2 )
of the two components of the VBS order parameter also support the emergent SO(2)
symmetry at the critical point [15].
The critical points of the gauged and ungauged NJL models are strongly coupled
(2 + 1)-dimensional conformal field theories characterized by a spectrum of scaling
dimensions that correspond to universal critical exponents. Some of these exponents
have already been computed in the 1/N f expansion in general d spacetime
dimensions [24–27]. The order-parameter anomalous dimension for d = 3 is
chiral O(2) QED 3 -GN : η φ = 1 +
56
3π 2 N f
+ O
1/N
2
f
,
(11)
chiral O(2) GN : η φ = 1 −
8
3π 2 N f
+
544
27π 4 N 2
f
+ O
1/N
3
f
,
(12)
and is related to the scalar-field scaling dimension by Δ φ =
1
2
1 + η φ
. The inverse
correlation length exponent is
chiral O(2) QED 3 -GN : ν
−1
= 1 −
80
3π 2 N f
+ O
1/N
2
f
,
(13)
chiral O(2) GN : ν
−1
= 1 −
16
3π 2 N f
+
8
364 + 27π 2
27π 4 N 2
f
+ O
1/N
3
f
,
(14)
and is related to the scaling dimension of the φ
2 operator by Δ φ 2 = 3 − ν −1 .
3 Fermion Bilinear Scaling Dimensions
The scaling dimension Δ φ characterizes the power-law decay at long distances of
the microscopic VBS correlation function VBS (r)O VBS (r ) ∼ |r − r | −2Δ φ at
the U(1)-VBS critical point of the lattice gauge theory (1), where O VBS (r) can
be chosen as (−1) x
A S A (r)S A (r + ˆ
x) or (−1) y
A S A (r)S A (r + ˆ
y), describing