Interactions and vertices between fermions and gauge fields . 62

The last part of the Lagrangian density (2.49) to be considered isLIntdescribing the vertices between the fermions and the gauge field fluctuations. In the Fourier space it has the form

LInt(t) =− Z dk

(2π)² dq (2π)²

h

ψ^†(k+q, t)

ea₀(q, t)− e

2m(2k+q)·a(q, t)

ψ(k, t) +

− Z dk⁰

(2π)²ψ^†(k+q, t)e²

2ma(q, t)·a(q⁰, t)

ψ(k+q⁰, t)i

. (2.68)

It can be rewritten as LInt(t) =

Z dk (2π)²

dq (2π)²

h X

µ

v_µ(k,q)ψ^†(k+q, t)a_µ(q, t)ψ(k, t) +

+1 2

X

µ,ν

Z dk⁰

(2π)²w_µν(q,q⁰)ψ^†(k+q, t)a_µ(q, t)a_ν(q⁰, t)ψ(k−q⁰, t)i (2.69)

with the vertices

v_µ(k,q) =

−e, forµ= 0

−m^e ˆz·^k×q^q, forµ= 1 w_µν(q,q⁰) =−e²

m q·q⁰

qq⁰ δµ1δν1. (2.70)

In diagrammatic terms they are represented in Fig (2.3.3).

k⁰−q⁰

q⁰ q

k+q

µ= 1 ν = 1

w_µν k k+q

q v_µ µ= 0,1

Fig. 2.5:The interaction vertices between fermions and gauge fields. Momentum conservation has already been implemented.

Having introduced the free Green’s functions for fermions and gauge fields and their actions we can start considering the perturbation expansion of the propaga-tors in terms of the vertices described by the Lagrangian (2.69).

The exact gauge field propagator is defined as Dµν(q, t) =−ihT

a_µ(q, t)a^†_ν(q,0)

i (2.71)

analogously to the fermionic Green’s function G(k, t) =−ihT

ψ(k, t)ψ^†(k,0)

i, (2.72)

whereT is the time ordering operator. They can also be obtained as Dµν(q, t) =−i

Z Z

Dψ_k^†Dψ_kDa_µ(q)e^iSa_µ(q, t)a^†_ν(q,0) (2.73) and the analogous forG, withS =R

dtdrL(r, t)and the partition function Z=

Z

Dψ^†_kDψ_kDa_µ(q)e^iS . (2.74) The expression (2.73) is particularly suitable for the perturbation treatment. We will in fact expand the exponential of the interaction action contained inSas

e^iSint = X∞

n=0

iⁿ

n!(S^int)ⁿ (2.75)

and then make all the possible contractions of couple of fermions and gauge field operators to produce the corresponding Green’s functions according to (2.71) and (2.72). That is, using Wick’s theorem we will break the average of the product of many operators into contractions of couples of them, with the relations

ha_µ(q, t)a_ν(q⁰, t⁰)i=i(−1)^νD⁽⁰⁾_µν(q, t−t⁰)δ(q+q⁰) (2.76) hψ(k, t)ψ^†(k⁰, t⁰)i=i G⁽⁰⁾(k, t−t⁰)δ(k−k⁰). (2.77) As usual, the total effect of the contractions giving rise to non-connected di-agrams will compensate for the partition function in the denominator due to the interaction action [10, 11, 12, 13, 14]. In the end we will just have to consider con-nected diagrams with free internal Green’s functions.

With the formal apparatus of perturbation theory we can now address the Ran-dom Phase Approximation for the gauge field propagators.

2.4 The RPA for the Gauge field propagator

In order to obtain the dynamics of the gauge field propagators we consider the Dyson equation forD_µν(q,Ω)

D_µν =D_µν⁰ +X

γδ

D⁰_µγΠγδD_δν (2.78)

where momenta and frequency dependencies have been omitted. HereDis the ex-actgauge field propagator andΠis theexact irreduciblepolarization function acting as a selfenergy for the Green’s functionD(see Fig (2.4)).

The Random Phase Approximation (RPA) corresponds to taking the polarization function to lowest order in the vertices and withfreefermionic Green’s functions, the so-calledΠ⁰. In our case it will correspond to have the vertexwat first order andvat the second order, since the average of a single bosonic field vanishes.

= + Π

Fig. 2.6:The Dyson equation for the gauge field propagator. The thick wiggly line is the exact functionDwhile the thin wiggly line is the free Green’s function D⁰. The exact irreducible polarizationΠhas been included.

q,Ω

' q,Ω

+ q,Ω q,Ω k, ω

+ q,Ω k, ω

q,Ω k+q, ω+ Ω

Fig. 2.7:The gauge field propagator at lowest orders in the vertices. Integration over internal momenta and frequencies is implied.

In order to identify Π⁰ we consider the two lowest order corrections to the free gauge field propagatorsiD⁰, represented in Fig (2.4).

These two corrections amount to

−X

γδ

D_µγ⁰ (q,Ω)w_γδ(q,q)Z dk (2π)²

dω

2πG⁰(k, ω)e^iω0+D⁽⁰⁾_δν(q,Ω) + +X

γδ

D_µγ⁰ (q,Ω) Z dk

(2π)² dω

2π v_γ(k,q)G⁰(k+q, ω+ Ω)×

×G⁰(k, ω)v_δ(k,q)D⁽⁰⁾_δν(q,Ω). (2.79) By direct comparison with the Dyson equation (2.78) we deduce the free polariza-tion

Π⁰_µν(q,Ω) =i w_µν(q,q)Z dk (2π)²

dω

2πG⁰(k, ω)e^iω0++ (2.80)

−i Z dk

(2π)² dω

2πv_µ(k,q)G⁰(k+q, ω+ Ω)G⁰(k, ω)v_ν(k,q).

These polarization terms correspond to the diagrams of Fig (2.4).

k, ω

+

k, ω

k+q, ω+ Ω Fig. 2.8:The free polarizationΠ⁰(q,Ω).

The first term, stemming from the vertexw_µν(q,q), is constant while the second is dynamically more interesting. It depends on the momentum of the incoming gauge field and on its frequency.

We can directly see why the RPA is particularly effective for small momenta and energy: in fact, whenever Ωand q tend to zero, the poles of the two fermionic Green’s functions get closer and closer, thereby giving a large contribution to the final integral.

Our purpose in the following is to describe the quasiparticle properties for the CF Fermi Liquid forming at half LL filling [45]. The gauge field propagator will act as the mediator of the effective quasiparticle interaction, keeping trace of both the original Coulomb and CS coupling terms. As in the usual Fermi Liquids we will concentrate on the small energy sector, where the Landau quasiparticles are well defined, their lifetime diverging at the Fermi level. The small energy excitations, moreover, are the relevant ones for the linear transport properties of the system.

Among the interesting quantities to be considered, we will mainly concentrate on the single particle fermionic Green’s function, focusing on the properties of its pole close to the Fermi energy. The issue of the renormalized quasiparticle effective mass will be addressed directly, and we will see how the gauge field fluctuations drastically affect the small energy sector of the spectrum [54].

Being interested in the dominant scattering for fermions close to the Fermi level, we will need the small energy behaviour of the gauge field propagator mediating the coupling. To get this limiting scaling we then consider the polarizationΠ⁰ in the regime|Ω| vFqvFkF. The result (see Appendix A) is

Π⁰(q,Ω)'







−^{m e}2π² 1 +i_v^|Ω_Fq^|

0 0 _24πm^q2e2 −i^2ρe_mv²_F^|Ω_q^|





 . (2.81)

Replacing this result in the exact polarization present in the Dyson equation (2.78) we get the propagator in Random Phase Approximation

D(q,Ω)' 1 χq+iγ_v^|Ω_F^|_q

χ˜q+i˜γq|Ω| −iβq

iβq _2π^m

(2.82)

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

Im Dokument On the role of spin, pairing and statistics for composite fermions in the fractional quantum hall effect (Seite 62-66)