ABSTRACT - DRUM - University of Maryland

The standard Abelian bosonization reads
ˆψ r,a =
ˆη r,a
√ 2πa0
e i√ π(θ a+rϕ a)
(7.3)
where a 0 is the short-distance cutoff, r = +/− for R/L movers and ˆη r,a are Majorana
operators which keep track of the anti-commuting character of the fermionic
operators. We follow the constructive bosonization as being thoroughly reviewed in
[160]. The two bosonic fields ϕ a and θ a satisfy the canonical commutation relation:
[∂ x ϕ a (x), θ a (x ′ )] = iδ(x − x ′ ). (7.4)
The ϕ a field is related to the charge density on chain a by ρ a = 1 √ π
∂ x ϕ a , and θ a is
its conjugate field, which can be interpreted as the phase of the pair field.
It is convenient to work in the bonding and anti-bonding basis:
ϕ ± = 1 √
2
(ϕ 1 ± ϕ 2 ), θ ± = 1 √
2
(θ 1 ± θ 2 ). (7.5)
The resulting bosonized Hamiltonian decouples as Ĥ = Ĥ+ + Ĥ−:
Ĥ + = v +
2
Ĥ − = v −
2
[
K+ (∂ x θ + ) 2 + K −1
+ (∂ x ϕ + ) 2] ,
[
K− (∂ x θ − ) 2 + K −1
− (∂ x ϕ − ) 2]
+ g p
2(πa 0 ) cos √ 8πθ 2 − +
g bs
2(πa 0 ) cos √ 8πϕ 2 − .
(7.6)
Here a 0 is the short-distance cutoff. This decoupling of the bonding and the antibonding
degrees of freedom is analogous to the spin-charge separation of electrons in
one dimension. Without any inter-chain forward scattering, we have K ± = K, v ± =
v.
The bonding sector is simply a theory of free bosons.
The Hamiltonian
in the anti-bonding sector can be analyzed by the perturbative Renormalization
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