340
R. Boyack and J. Maciejko
gauge field as noncompact; the effects of compactness due to the original lattice
formulation will be discussed in Sect. 4.
For J larger than some critical value J c , a VBS phase with unbroken global
SU (N f )/Z N f symmetry is found for N f = 4, 6, 8 [6, 7]. Columnar VBS order
doubles the unit cell of the square lattice and spontaneously breaks the latter’s
D 4 point-group symmetry to a D 2 subgroup; it is represented by a time-reversalinvariant vector order parameter V = (V x , V y ) transforming in the two-dimensional
E irreducible representation of D 4 . Using the projective symmetry group approach,
one can determine how gauge-invariant operators in the low-energy QED 3 theory
transform under the microscopic lattice symmetries [10]. Defining the two 4 × 4
Hermitian matrices
Γ 3 =
0 −i
i 0
,
Γ 5 = Γ 0 Γ 1 Γ 2 Γ 3 =
0 1
1 0
,
(7)
which square to the identity and anticommute with each other and with the
Dirac matrices (6), one finds that the pair of time-reversal-invariant and flavorsymmetric Dirac bilinears
α iΨ α Γ 5 Ψ α ,
α iΨ α Γ 3 Ψ α
transform precisely
in the E irreducible representation of D 4 . Furthermore,
α iΨ α Γ 5 Ψ α is odd
under x-reflections and lattice x-translations and even under y-translations, while
α iΨ α Γ 3 Ψ α transforms oppositely. Thus one can identify V x ∼
α iΨ α Γ 5 Ψ α
and V y ∼
α iΨ α Γ 3 Ψ α . A nonzero expectation value of V corresponds to a
nonzero fermion mass, in accordance with the gapped spectrum observed in the
VBS phase [7]. Note that V is Lorentz invariant since Γ 3 and Γ 5 commute with the
Euclidean transformations exp
−
i
2 ω μν σ μν
where σ μν =
i
4 [Γ μ , Γ ν ].
The occurrence of a VBS phase for J > J c can be understood as arising
from a short-ranged four-fermion interaction term ∼(g 2 /N f )V 2 generated by gauge
fluctuations at the lattice scale. Such interactions are perturbatively irrelevant at the
conformal QED 3 fixed point, but if sufficiently strong can give rise to dynamical
fermion mass generation via a quantum critical point. Decoupling this interaction
term with a pair φ = (φ 1 , φ 2 ) of scalar fields and tuning to the quantum critical
point, we obtain the chiral O(2) QED 3 -GN model,
L O(2)QED 3 −GN
=
N f
α=1
Ψ α Γ μ
∂ μ +
e
N f
iA μ
Ψ α +
g
N f
iφ · Ψ α MΨ α
+ . . . ,
(8)
where M = (Γ 3 , Γ 5 ), and . . . includes Maxwell and gauge-fixing terms for the
gauge field, and symmetry-allowed kinetic and self-interaction terms for the scalar
field φ. At the free-field fixed point, the gauge coupling e 2 and the Yukawa coupling
g 2 have units of mass and are thus relevant. However, the fields have been rescaled to
make explicit the fact that e and g appear with a suppressing factor of 1/
N f . In the
large-N f limit, the physics at momenta |q| | e 2 , g 2 is dominated by the coupling
R. Boyack and J. Maciejko
gauge field as noncompact; the effects of compactness due to the original lattice
formulation will be discussed in Sect. 4.
For J larger than some critical value J c , a VBS phase with unbroken global
SU (N f )/Z N f symmetry is found for N f = 4, 6, 8 [6, 7]. Columnar VBS order
doubles the unit cell of the square lattice and spontaneously breaks the latter’s
D 4 point-group symmetry to a D 2 subgroup; it is represented by a time-reversalinvariant vector order parameter V = (V x , V y ) transforming in the two-dimensional
E irreducible representation of D 4 . Using the projective symmetry group approach,
one can determine how gauge-invariant operators in the low-energy QED 3 theory
transform under the microscopic lattice symmetries [10]. Defining the two 4 × 4
Hermitian matrices
Γ 3 =
0 −i
i 0
,
Γ 5 = Γ 0 Γ 1 Γ 2 Γ 3 =
0 1
1 0
,
(7)
which square to the identity and anticommute with each other and with the
Dirac matrices (6), one finds that the pair of time-reversal-invariant and flavorsymmetric Dirac bilinears
α iΨ α Γ 5 Ψ α ,
α iΨ α Γ 3 Ψ α
transform precisely
in the E irreducible representation of D 4 . Furthermore,
α iΨ α Γ 5 Ψ α is odd
under x-reflections and lattice x-translations and even under y-translations, while
α iΨ α Γ 3 Ψ α transforms oppositely. Thus one can identify V x ∼
α iΨ α Γ 5 Ψ α
and V y ∼
α iΨ α Γ 3 Ψ α . A nonzero expectation value of V corresponds to a
nonzero fermion mass, in accordance with the gapped spectrum observed in the
VBS phase [7]. Note that V is Lorentz invariant since Γ 3 and Γ 5 commute with the
Euclidean transformations exp
−
i
2 ω μν σ μν
where σ μν =
i
4 [Γ μ , Γ ν ].
The occurrence of a VBS phase for J > J c can be understood as arising
from a short-ranged four-fermion interaction term ∼(g 2 /N f )V 2 generated by gauge
fluctuations at the lattice scale. Such interactions are perturbatively irrelevant at the
conformal QED 3 fixed point, but if sufficiently strong can give rise to dynamical
fermion mass generation via a quantum critical point. Decoupling this interaction
term with a pair φ = (φ 1 , φ 2 ) of scalar fields and tuning to the quantum critical
point, we obtain the chiral O(2) QED 3 -GN model,
L O(2)QED 3 −GN
=
N f
α=1
Ψ α Γ μ
∂ μ +
e
N f
iA μ
Ψ α +
g
N f
iφ · Ψ α MΨ α
+ . . . ,
(8)
where M = (Γ 3 , Γ 5 ), and . . . includes Maxwell and gauge-fixing terms for the
gauge field, and symmetry-allowed kinetic and self-interaction terms for the scalar
field φ. At the free-field fixed point, the gauge coupling e 2 and the Yukawa coupling
g 2 have units of mass and are thus relevant. However, the fields have been rescaled to
make explicit the fact that e and g appear with a suppressing factor of 1/
N f . In the
large-N f limit, the physics at momenta |q| | e 2 , g 2 is dominated by the coupling