ABSTRACT - DRUM - University of Maryland

onding basis. Hamiltonian in the bonding sector is just a theory of free bosons. In
the anti-bonding sector, it reads
Ĥ= v 2
[K(∂ x θ) 2 + 1 K (∂ xϕ) 2 ]
+ 2t ⊥
πa 0
cos √ 2πϕ cos √ 2πθ. (7.20)
The bosonic fields ϕ and θ are in the anti-bonding basis. The perturbation (t ⊥ )
term has nonzero conformal spin which implies that two-particle processes are automatically
generated by RG flow even when they are absent in the bare Hamiltonian.
Therefore, one has to include two-particle perturbations in the RG flow
Ĥ 2 = g 1
(πa 0 ) 2 cos √ 8πϕ + g 2
(πa 0 ) 2 cos √ 8πθ. (7.21)
The RG flow equations for weak couplings have been derived by Yakovenko [169]
and Nersesyan et al. [170]. Here we cite their results [159]:
dz
(
dl = 2 − K + )
K−1
z
2
dy 1
= (2 − 2K)y 1 + (K − K −1 )z 2
dl
, (7.22)
dy 2
= (2 − 2K −1 )y 2 + (K −1 − K)z 2
dl
dK
= 1 dl 2 (y2 2 − y1K 2 2 )
where the dimensionless couplings are defined as z = t ⊥a
2πv and y 1,2 = g 1,2
πv .
Since we are interested in the phase where the pair tunneling dominates at low
energy, we assume K > 1 so y 1 is irrelevant and can be put to 0. Also we neglect
renormalization of K.
Integrating the RG flow equations with initial conditions
z(0) = z 0 ≪ 1, y 2 (0) = 0 we obtain
y 2 (l) = z 2 0
K −1 − K [
e 2(1−α)l − e ] 2(1−K−1 )l , (7.23)
2α
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