2. Composite Fermions and the Chern-Simons theory of the FQHE
2.5 Selfenergy correction to the fermionic Green’s function: CF effective
whereβ=e/ϕΦ˜ 0,χ˜q =−q2V(q)/ϕ˜2Φ20=−q e2/ϕ˜2Φ20,χq = ˜χqm/2π,γ˜q = 2ρ/kFq andγ=ρ/π.
Clearly the D11 component is the dominant one, being the only non-vanishing small energy-small momentum matrix element. It comes out to be
D11(q,Ω)≈
− q e2
ϕ˜2Φ20 +i2ρ|Ω| kFq
−1
. (2.83)
This is the leading gauge field channel mediating effective interactions between the fermions. By inspection we see thatD(q,Ω)has a pole (for the retarded propagator) at
Ω =−i 2πe2
kFϕ˜2Φ20q2. (2.84) Such an imaginary pole means that the dominant coupling is mediated by a slowly decaying channel rather than a conventional stable mode. However the decay time diverges for very small momenta asq−2.
Having obtained the RPA gauge field propagator we will now concentrate on the Fermi Liquid corrections to the single quasiparticle Green’s function. Among the physical properties to be extracted from it we will observe a peculiar diverging CF effective mass close to the Fermi level [54].
2.5 Selfenergy correction to the fermionic Green’s function: CF
In terms ofΣ(k, ω), we can deduce the renormalized spectrum of the single particle excitations out of the pole ofG(k, ω)by solving the selfconsistent equation
ε∗(k) =ε(k) + Re [Σ (k, ε∗(k))] (2.86) whereε(k)andε∗(k)are the free and renormalized fermionic dispersions, respec-tively. On the same footing the quasiparticle lifetime is
τ = 1
Im [Σ (k, ε∗(k))]. (2.87) It is possible to define an effective quasiparticle massm∗ from (2.86) by insisting that the renormalized dispersion has the form
ε∗(k)≡ k2
2m∗ −µ . (2.88)
Thus, taking the derivative of (2.86) with respect tokat the Fermi level we get kF
m∗ = kF m+ ∂Σ
∂ε(k)
k=kF·kF m +∂Σ
∂ω
ω=ε∗(k)· kF
m∗ (2.89)
whence the equation for the renormalized effective mass
m∗ m =
1−∂Σ∂ω
ω=ε∗(k) 1 +∂ε(k)∂Σ
k=kF
. (2.90)
In order to analyze the fermionic selfenergy and deduce the outcoming quasi-particle effective mass we start considering the Green’s function up to first order in the gauge field propagator (see Fig. (2.5))
iG(k, ω)'iG0(k, ω)−G0(k, ω)× (2.91)
×
Z dk0 (2π)2
dΩ
2πv1(k,k0)2G0(k0, ω−Ω)D11(k−k0,Ω)
G0(k, ω).
Here we neglected the first order contribution coming from the vertexwand from the Hartree correction since they are just constants, at most renormalizing the chem-ical potential.
By direct comparison with the Dyson equation (2.85) we extract the first order self-energy correction
Σ(k, ω)'i Z dk0
(2π)2 dΩ
2πv1(k,k0)2G0(k0, ω−Ω)D11(k−k0,Ω), (2.92)
k, ω ' k, ω
+ k, ω k, ω
k−k,Ω
k0, ω−Ω
Fig. 2.10:The fermionic Green’s function up to first order in the gauge field prop-agator. The first order contributions coming from the vertex wµν and the Hartree correction have been neglected since they are just constants renormalizing the chemical potential. The Fock-type selfenergy insertion is clearly visible.
k−k0,Ω
k0, ω−Ω
Fig. 2.11:The Fock type selfenergy correction to the single particle fermionic Green’s function G0(k, ω). Integration over internal momenta and fre-quency is implied.
represented by the Fock-type diagram of Fig (2.5).
Having a dynamical bosonic mode as mediator of the interaction, as in the theory of superconductivity for example, it is often more convenient to evaluateδΣ(k, ω) = Σ(k, ω)−Σ(k,0)[55]. Thus our task is to calculate
δΣ(k, ω)'i Z dk0
(2π)2 dΩ
2π v1(k,k0)2D11(k−k0,Ω)
G0(k0, ω−Ω)−G0(k0,−Ω) . (2.93) It is convenient to introduceq≡ |k−k0|and replace
v1(k,k0)2= e2 m2
k2k02
q2 sin2θ (2.94)
whereθis the angle betweenkandk0. The measure is then changed into Z
dk0= Z ∞
0
k0dk0 Z 2π
0
dθ= 2 Z ∞
0
dk0 Z k+k0
|k−k0|
dq q
ksinθ (2.95) with
sinθ= r
1−hk2+k02−q2 2kk0
i2
. (2.96)
For quasiparticle scattering at the Fermi level andqkFwe havesinθ'q/ksuch that the measure can be approximated to be 1, with q ∈ [0,2kF] andv1(k,k0)2 ' kF2e2/m2. The selfenergy (2.93) becomes
δΣ(k, ω)'i 2e2k2F (2π)3m2
Z ∞
−∞
dΩ Z 2kF
0
dq D11(q,Ω) Z ∞
0
dk0
G0(k0, ω−Ω)−G0(k0,−Ω) (2.97) where we used the fact that the propagators depend only on the modulus of their momentum.
Let us consider first the integral overk0. For smallωwe can approximate Z ∞
0
dk0
G0(k0, ω−Ω)−G0(k0,−Ω)
' (2.98)
'1 2
Z ∞
−∞
dk0 ω
[ω−Ω−εk0 +i0+sign(ω−Ω)] [Ω +εk0+i0+sign(Ω)] . Linearizing the dispersion close toEF,εk0 'vF(k0−kF)≡η, we get
ω 2vF
Z ∞
−∞
dη 1
[ω−Ω−η+i0+sign(ω−Ω)] [Ω +η+i0+sign(Ω)] . (2.99) If we concentrate on positiveω(meaning to investigate quasi-particleproperties) we can perform the integration onη by closing the contour in the positive imaginary complex plane with the result that (2.99) becomes
−iπ vF
Θ(Ω) Θ(ω−Ω), (2.100)
withΘ(x)the step function equal to1forx >0and0elsewhere.
The integration overq, Z 2kF
0
dq D11(q,Ω) = Z 2kF
0
dq 1
−ϕq e˜2Φ220 +i2ρk|Ω|
Fq
, (2.101)
can be easily carried out to give ϕ˜2Φ20
2e2 log
i2ρk|Ω|
F
i2ρk|Ω|
F −4k2Fϕ˜e22Φ20
. (2.102)
Having shown thatΩ∈[0, ω](see (2.100)) and beingω vFq, we can expand the result inΩ/vFqto be
ϕ˜2Φ20 2e2 log
"
− i2ρk|FΩ| 4kF2ϕ˜e22Φ20
#
. (2.103)
Finally the integration overΩbrings δΣ(k, ω)' 2e2k2F
(2π)3m2 π vF
ϕ˜2Φ20 2e2
Z ω 0
dΩ log
"
− i2ρk|FΩ| 4kF2ϕ˜e22Φ20
#
= ' kFϕ˜2Φ20
8π2m ω log
ϕ˜2Φ20 8πkFe2ω
−ikFϕ˜2Φ20
16πm ω . (2.104)
We can see directly that the imaginary part of the selfenergy is correctly producing a pole for the single particle Green’s function in the negative imaginary plane for positive frequencies, as requested from its analytical properties. Moreover, the ratio between imaginary and real part of the selfenergy vanishes for small frequencies, indicating that the concept of quasiparticle is well defined close to the Fermi level.
Now let us discuss the physical consequencies of (2.104) for the CF effective mass. First of all the selfenergy is essentially frequency dependent and it is domi-nated by its real part for small frequencies. Inserting (2.104) into (2.90) we get the dominant scaling
m∗∼ −mkFϕ˜2Φ20
8π2m logω , (2.105)
that is, a logarithmic divergence at small energy.
Thus CF are well defined quasiparticles in the Landau Fermi-liquid sense but their renormalized parameters can show ”anomalies”. The origin of the diverging effec-tive mass is the1/qsingularity of the gauge field propagator in the static limit. This gives rise to many infrared divergencies found while calculating response functions out of the original Lagrangian (2.48).
It is natural to ask whether the singular behaviour is just an artifact of the lowest or-der approximation we kept, and is removed by higher oror-der corrections. This issue has been investigated in detail by Stern and Halperin [54] who showed the remark-able fact that the result (2.104) isexactat low energies. More specifically they show that, due to Ward identities valid in the limitω vFq, the corrections to the in-ternal fermionic Green’s function leading to the exactGin the selfenergy (2.92) are cancelled by the vertex corrections. Thus, surprisingly, the first order calculation we performed gives theexactresult, in the low energy sector.
Stern and Halperin considered also the case when a small residual effective magnetic field B∗ exists, i.e. the limit of very large CF filling factor, p 1. In particular the states at fillingsν = 1/ϕ˜can be looked at aslimp→∞p/( ˜ϕp±1). By studying the pole of the single particle Green’s function in the CFLL basis they were able to extract the FQH energy gap∆(p), and they found
∆(p)' e2kFπ ϕ( ˜˜ ϕp+ 1)
1
ln(2p+ 1) . (2.106)
The origin of the logarithmic correction is the same as the one for vanishing residual field. In the limitp1we can estimatekF =`−1p
2/ϕ˜andln(2p+ 1) 'lnp, so that
∆(p)'e2
` π
˜ ϕ2plnp
r2
˜
ϕ . (2.107)
This is a very important result, since it shows how the gauge field mediated interac-tion induces incompressibility with the correct scaling determined by the Coulomb coupling. At least in the largeplimit it is then possible to obtain the scaling pre-dicted by HLR via dimensional analysis arguments (see (2.25)). Moreover the gaps are predicted to show a logarithmic dependence on the filling factor, which is linked to the diverging CF effective mass at the Fermi level forν = 1/ϕ.˜
Thus, an accurate measurement of the energy gaps close to even denominator states
would allow a direct test of the theoretically expected logarithmic divergence. Of course, since the gaps become smaller and smaller for increasing p, it is experi-mentally very difficult to obtain precise measurements, since the disorder tends to
”close the gap” before the logarithmic corrections show up (see Fig (2.4)). There are claims [56] of the observation of a very large effective mass close toν= 1/2, but the issue is still experimentally not fixed.