Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

�
p
�
dθp
= g
2π
RG for Interacting Fermions 19
we obtain a convolution of the two Fermi liquid functions
�
u(θk − θp)u(θp − θk ′)=g
�
∞�
u 2 m cos m(θ − θ ′ �
)
p
u 2 0 + 1
2
m=1
(38)
(39)
where we have reverted to the notation θ = θk,θ ′ = θk ′. In the second term of
(37), the δp,−p ′ turns out to be subleading, while the other allows independent
sums over p, p ′ . This means that only u0 contributes to this term, (other
avrerage to zero upon summing over all angles) which produces
− �
(40)
pp ′
u(θk − θp)u(θp ′ − θk ′)=g2 u 2 0
Feeding this into full expression for this contribution to the particle-hole
diagram, we find it to be
dVαβγδ
dt Leading
g′ �
= ∆ ln 2
g
kk ′
�
∞�
u
m=1
2 m cos m(θ − θ ′ �
)
φ ∗ α(k)φ ∗ β(k ′ )φγ(k ′ )φδ(k) (41)
Notice that the result is still of the Fermi liquid form. In other words the couplings
Vαβγδ which were written in terms of Landau parameters um, flow into
renormalized coupling once again expressible in terms of renormalized Landau
parameters. By comparing the two sides, we see each um flows independently
of the others as per
dum
dt = −e−t (ln 2)u 2 m m �= 0 (42)
The above equation can be written in a more physically transparent form
by using a rescaled variable (for m �= 0 only)
ũm = e −t um
in terms of which the flow equation becomes
(43)
dũm
dt = −ũm − (ln 2)ũ 2 m ≡ β(ũm) (44)
where the last is a definition of the β-function.
The reason uo does not flow is that the corresponding interaction commutes
with the one-body “kinetic” part, and therefore does not suffer quantum
fluctuations.
This is the answer at large g. We have dropped subleading contributions
of the following type: