Mean Field and Random Phase Approximation

b(x, t) =B− B =B−ϕφ˜ ₀ρ(x, t) (2.3.14) and we can divide the total electromagnetic field or Chern Simons field in a mean field and a dynamical field respecting the fluctuations:

aµ=Aµ− hAµi. (2.3.15)

The averagehAµiis not a dynamical field and gives no interesting contribution to the equation of motions. The only effect is that it reduces the external field and in some cases, at even fraction, it completely eliminates the external field. Since ∂_ihAji = 0 only the dynamical part a_µ contributes to the equation of motions and this leads to the relation:

ρ(x, t) = 1

˜

ϕφ₀ ε^ij∂_ia_j(x, t). (2.3.16) We may now calculate the free propagator of the low energy free massive charge carriers.

The spin-less fields are given by ψ(x, t) =

Z dkdt

(2π)³ a⁺(k)e^{−i(kx−ωt)}+a(k)e^{−i(kx−ωt)}

. The free part of the action is

S_F = Z

dxdt ψ⁺(x, t)

− p²

2m +i∂_t+µ

ψ(x, t). (2.3.17)

In Fourier space this is exactly SF=

Z

dkdt ψ⁺(k, t)

ω− k² 2m −µ]

| {z }

[G⁰(k,ω)]⁻1

ψ(k, t) (2.3.18)

and the low energy Fermionic Greens function is (ǫ >0):

G⁰(k, ω) = [ω− k²

2m−µ+iǫsign(ω)]⁻¹. (2.3.19) Random Phase Approximation

In the random phase approximation (RPA) the two-point function of the propagator of the gauge field is calculated up to second order time dependent perturbation theory. Therefore the Chern Simons fields have to be quantized first. Here we face the problem that the Chern Simons Theory is a gauge theory, usually a U(1) but also U(1)⊗SU(2) or U(N) fields are discussed. In gauge theories we have to incorporate gauge invariance and the uniqueness of the Cauchy data. To satisfy the Cauchy problem we have to fix the gauge. So far the quantization

in the Chern Simons composite Fermion picture is performed in Coulomb gauge fixing. From usual quantum electrodynamics it is well known that the advantage of Coulomb gauge is, that there are only physical degrees of freedom left in the theory, only the transverse modes are included. As an effect of Coulomb gauge the zero component of the gauge fieldA₀ becomes a Lagrangian multiplier resulting in a constrained equation, the Poisson equation. However, there is also a disadvantage namely there appear terms violating intrinsically causality and this terms have to be eliminated in the propagator by counter-terms. For this reason the Gupta Bleuler Method in Lorentz gauge is used to circumvent the handling with counter-terms. Furthermore in the case of more complicated gauge theories especially nonabelian gauge theories, but also gauge theories on noncommutative spaces, the method of Becchi, Rouet, Stora [BRS76] and Tyutin [Tyu] called BRST quantization is preferred. It can be viewed as a generalization of the Gubta Bleuler method and is a central aspect of the Batalin Vitkovsky formalism [BV81] in the geometric quantization procedure [BV81], [AKSZ97] and for an introduction see [Fio03]. Now that we have a low energy theory, which violates causality at a fundamental level we may think that this fact might be ignored but this we should not do.

The theory should always be thought as a low energy limit of a relativistic theory, like in the situation of the hydrogen atom. Then we have to perform in the same way as in the covariant formalism. More concrete this means the counter-terms required in the Coulomb gauge have to be included also in a low energy theory. The impact for the Chern Simons gauge theory is similar. Either we choose Coulomb gauge and evaluate suitable counter-terms or we choose Lorentz gauge and the Goupta Bleuler method or the BRST method respectively. This will be discussed in the next chapter in this section we perform the Coulomb gauge method.

Let us now turn to the free gauge field propagator. The quantization procedure requires a unique Cauchy problem so we fix the gauge. The Coulomb gauge ∂ⁱa_i(x) = 0 is also called transverse gauge since only transverse modes, the physical modes are left and the longitudinal modes are eliminated from the beginning. This can be seen best in momentum space. The Coulomb gauge condition is here kⁱa_i(k) = 0. In the two dimensional plane it is clear that we can only have one longitudinal mode and one transverse mode. If we choose for example the x-direction as the direction of the momentum then clearly a_x =a_L = 0 anda_y =a_T is the physical mode. Without fixing a coordinate system we have

a= (a_T, a_L) =:a₁iσ₂k k +a₂k

k.

We now quantize the theory and define the operators to evaluate the free gauge field prop-agator and the S-matrix defined by the interaction Hamiltonian. The quantum fields in momentum space obey the following relations:

a⁺₀(k) =a₀(−k), a⁺₁(k) =−a₁(−k). (2.3.20) The part of the action with Chern Simon fields can be written as

S_G = ε^ij

˜ ϕφ0

Z

dxdt a₀∂_ia_j (2.3.21)

+ ε^ijε^mn 2( ˜ϕφ₀)²

Z

dy ∂iaj(x, t) V(x−y) ∂man(y, t).

Instead of the Coulomb potential we may consider here also a screened potential, for example a Yukawa like potential. Then in Fourier space the action is given by

S_G=

Z dkdΩ (2π)³

iek

˜

ϕφ₀a₀(k,Ω)a⁺₁(k,Ω) + k²V(k)

2( ˜ϕφ₀)²a₁(k,Ω)a⁺₁(k,Ω). (2.3.22) We rewrite it in terms of the free Greens function

D⁰_µν(k,Ω)

=

V(k) e²

iek

˜ ϕφ0

−_ϕφ^iek_{˜ 0} −^k_{( ˜ϕφ}^2V₀^(k)₎²

!

(2.3.23) and obtain:

SG=

Z dkdΩ

(2π)³ a⁺_µ(k,Ω)[D_µν⁰ (k,Ω)]⁻¹aν(k,Ω). (2.3.24) The Fermionic fiels interact with the Chern Simons gauge fields through the interaction part of the Lagrangian (2.3.7)

S_Int = − Z

dt

Z dkdq (2π)⁴

X

µν

v_µ(kq)ψ⁺(k+q, t)a_µ(q, t)ψ(k, t) (2.3.25) +1

2

Z dq^′

(2π)²w_µν(q,q^′)ψ⁺(k+q, t)a_µ(q, t)a_ν(q^′, t)ψ(k−q^′, t)

Were v_µ and w_µν are the vertices with contribution vµ =

(−e, µ= 0

−_mq^e ε^ijk_iq_j, µ= 1 (2.3.26) w_µν = − e²

mqq^′qⁱq_i^′δ_µ1δ_ν1 (2.3.27) The propagators are given by the expectation value of the time ordered product:

hT^[t,0][a_µ(q, t)a⁺_ν(q^′,0)]i = iD_µν(q, t) (2.3.28) hT^[t,0][ψ(k, t)ψ⁺(k^′,0)]i = iG(k, t) (2.3.29) T^[t1,...,tn] denotes the time ordering with respect to the times t₁, . . . , t_n. The interaction term can be treated in a perturbative S-matrix formulation. The formal S-matrix is defined through the Dyson series

S_[λ] = Texp{i Z

dt Z

dx δ(x⁰−t)λ(x)LInt(x)} (2.3.30)

= I+ X∞

N=1

(−i)^N

N! S_[λ]^(N) (2.3.31)

T denotes the time ordering andI is the identity. We introduced for the time being formally an adiabatic switching function from Schwartz space λ(x) ∈ S(R³) to get a well defined expression. The N-th coefficientS^(N_[λ]⁾ of the Dyson series is given by

S^(N_[λ]⁾= Z

dx1. . . dxnλ(x1):LInt(x1):. . . λ(xn):LInt(xn): (2.3.32) We may remark that aS-matrix is formally unitary S_[λ]⁺S_[λ]=S_[λ]S_[λ]⁺ =¹ if the interaction Hamiltonian

H_Int^λ (t) = Z

dx δ(x⁰−t)λ(x)LInt(x) (2.3.33) is symmetric: H_Int^λ (t) =H_Int^λ (t)⁺. Then we only have to show that the interaction Hamilto-nian is symmetric but this is given by construction.

The Dyson equation for the gauge field propagators D(q,Ω) is formally given by

D_µν =D_µν⁰ +D_µρ⁰ Π^ρσD_σν (2.3.34) The self-energy part Π is called the exact irreducible polarization function andD⁰ is the free propagator

ha_µ(q, t)a_ν(q^′, t)i=i(−1)^νD⁽⁰⁾(q, t−t^′)δ(q−q^′). (2.3.35) We will call the lowest order approximation Π⁰ of the polarization function Π the random phase approximation together with the free Fermionic Green’s functions. In diagrammatic terms the Dyson equation can be drawn as:

= + . (2.3.36)

The thick wiggled line corresponds to the exact propagator, the thin wiggled line to the free propagator and the shaded area represents the exact polarization. The lowest order corrections to the free gauge field propagator is given by (ǫ >0)

S¹ = −D_µρ⁰ (q,Ω)w^ρσ(q,q)

Z dkdω

(2π)³ G⁰(k, ω) e^iωǫ D⁰_σν(q,Ω) (2.3.37) +D_µρ⁰ (q,Ω)

Z dkdω

(2π)³ v^ρ(k,q) G⁰(k+q, ω+ Ω)G⁰(k, ω) v^σ(k,q)D_σν⁰ (q,Ω).

The self-energy part gives the free polarization function to lowest order perturbation theory Π⁰_µν = −iw^ρσ(q,q)

Z dkdω

(2π)³ G⁰(k, ω) e^iωǫ (2.3.38)

+ i

Z dkdω

(2π)³ v^ρ(k,q) G⁰(k+q, ω+ Ω)G⁰(k, ω) v^σ(k,q).

In term of Feynman graphs this corresponds to the approximation to lowest order in the vertices:

≈ + + (2.3.39)

wherein the contribution of the free polarization function corresponds to the loops:

Π⁰_µν = + (2.3.40)

The polarization function is analyzed in [HLR93] and [SH95] and it renormalizes the com-posite Fermions effective mass in the small energy sector in the Fermi Liquids theory. The relevant regime is defined by |Ω| ≪vFq ≪vFkF. The polarization function in this regime is calculated to

Π⁰(q,Ω)≈ −^me_2π²(1 +i_v^|Ω|

Fq) 0

0 _24πm^q2e2 −i^2ρe_mv^2|Ω|

Fq

!

(2.3.41) Then the approximated gauge field propagator is

D(q,Ω)≈ 1 ξ_q+iγ_v^|Ω|

Fq

−ξ˜_q+i˜γ_q|Ω| −iβq iβq _2π^m

(2.3.42) with the abbreviations β = e/φΦ˜ ₀ and the potential like terms ˜ξ_q = −q²V(q)/φΦ˜ ² and ξ_q = ˜ξ_qm/2π and ˜γ_q = 2ρ/k_Fq and γ = ρ/π. The dominant component is D₁₁. The imaginary pole

Ω =−i 2πe²

k_FεφΦ˜ ²₀q² (2.3.43)

gives a slowly decaying mode where the decay time diverges for very small momenta.

Self-energy Correction for the Fermionic Two Point Function The Dyson equation for the Fermion propagator is given by

G(k, ω) =G⁰(k,Ω) +G⁰(k,Ω)Σ(k, ω)G(k, ω) (2.3.44) and can also be understood in terms of irreducible Feynman graphs:

= + . (2.3.45)

The thick line corresponds to the exact propagator, the thin line to the free propagator and the exact self-energy part is denoted by the shaded loop. The renormalized spectrum can by calculated via the self-consistent equation

ε^∗(k) =ε(k) + Re(Σ(k, ε(k))) (2.3.46) with the freeε(k) and renormalizedε^∗ Fermionic dispersion. Then the quasi-particle lifetime is

τ = Im(Σ(k, ε^∗(k)))⁻¹. (2.3.47)

The effective quasi-particle massm^∗ is defined via the relation ε^∗(k) := k²

2m^∗ −µ. (2.3.48)

If we take the derivative of (2.3.46) with respect tok at the Fermi level we obtain k_F

m^∗ = k_F

m +∂_εΣ|k=kF

k_F

m +∂_ωΣ|ω=ε^∗

k_F

m^∗ (2.3.49)

which gives the relation

m^∗

m = 1−∂_ωΣ|ω=ε^∗

1 +∂εΣ|k=kF

. (2.3.50)

The Fermionic propagator in the random phase approximation is considered only up to the first order in the gauge field propagator:

iG(k, ω) ≈ iG⁰(k, ω) (2.3.51)

−iG⁰(k, ω)

Z dk^′dΩ

(2π)³ v1(k,k^′)G⁰(k^′.ω−Ω)D11(k−k^′,Ω)iG⁰(k, ω).

The Hartree correction and the first order contribution are neglected since they are only uninteresting constants. In diagrammatic terms this approximation corresponds to

≈ + , (2.3.52)

wherein the gauge field loop corresponds to the Fock type self-energy contribution. The self-energy part in terms of the propagators is

Σ(k, ω)≈i

Z dk^′dΩ

(2π)³ v²₁(k,k^′)G⁰(k^′, ω−Ω)D₁₁(k−k^′,Ω) (2.3.53) and it is convenient to introduce the difference

δΣ(kb, ω) = Σ(k, ω)−Σ(k,0) (2.3.54)

explicitly given by:

δΣ(k, ω)≈i

Z dk^′dΩ

(2π)³ v²₁(k,k^′)D11(k−k^′,Ω)(G⁰(k^′.ω−ω)−G⁰(k^′.ω−Ω)). (2.3.55) We introduce spherical coordinates q=|k−k^′|sov₁ becomes

v₁²= e²k²k^′2

m²q² sin²θ and Z

dk^′ = 2 Z _∞

0

Z _k+k^′

|k−k^′|

dk^′dq q

ksinθ. (2.3.56) The angle θis between the two momenta and

sinθ=

1− k²+k^′2−q² 2kk^′

21/2

. (2.3.57)

At the Fermi level q≪k_F q ∈[0,2k_F] this gives sinθ≈q/kandv₁²≈e²k²_F/m². The measure can be approximated to one and the self-energy difference is

δΣ(k, ω)≈i 2e²k_F² (2π)³m²

Z _∞

−∞

dΩ Z _2kF

0

Z _∞

0

D₁₁(q,Ω)(G⁰(k^′.ω−ω)−G⁰(k^′.ω−Ω)).(2.3.58) The evaluation of these integrals is rather technical and is done only approximately. We start with the integral over k^′ with smallω, which we can approximate

Z _∞

0

dk^′ (G⁰(k^′.ω−ω)−G⁰(k^′.ω−Ω))≈ (2.3.59)

≈ 1 2

Z _∞

−∞

dk^′ ω

(ω−Ω−η+iǫsign(ω−Ω))(Ω +ε_k^′+iǫsign(Ω))

Near the Fermi energy the dispersion relation may be approximated byε_k^′ ≈v_F =η and this simplifies the integral to

ω 2vF

Z _∞

−∞

dη ω

(ω−Ω−η+iǫsign(ω−Ω))(Ω +η+iǫsign(Ω)) = (2.3.60)

= −iπ vF

Θ(Ω)Θ(ω−Ω).

For positive ω (quasi-particle properties) the integral is calculated by closing the contour in the upper half-space on the complex plane. Next we calculate

Z 2kF

0

dq D₁₁ =

Z 2kF

0

dq − qe²

εϕ˜²φ²₀ +i2ρ|Ω| kFq

(2.3.61)

= εϕ˜²φ²₀

2e² ln i2ρ|Ω| k_F

−ln i2ρ|Ω|

k_F −4k²_F e² εϕ˜²φ²₀

.

Being in the regime ω ≪vFq, we are allowed to expand the integral with respect to Ω/vFq and this gives

εϕ˜²φ²₀

2e² ln i2ρ|Ω| k_F

−ln −4k_F² e² εϕ˜²φ²₀

(2.3.62)

and the self-energy difference is calculated to δΣ≈ k_Fεϕ˜²φ²₀

8π²m ωln εϕ˜²φ²₀ 8πe²kF

ω

−ik_Fεϕ˜²φ²₀

16πm ω. (2.3.63)

Having calculated the self-energy we can perform a mass renormalization m^∗∝ εϕ˜²φ²₀

8πe²mlnω. (2.3.64)

This quasi-particle mass diverges due to the 1/q singularity leading to infrared divergences.

It seems that the correction δΣ is exact in the limit ω ≪ v_Fq, which is shown in [SH95] by evaluating Ward identities. These singularities of the single particle Green’s function of the composite Fermions allows for a derivation of an energy gap [SH95]:

∆(p)≈ e²k_Fπ εϕ( ˜˜ ϕ+ 1)

1

ln(2p+ 1) (2.3.65)

withpbeing the composite Fermion filling. In the regime wherepis large the approximations ln(2p+ 1)≈lnp and k_F ≈p

2/p/l leads to the energy gap:

∆(p)≈ r2

˜ ϕ

e²π

εlϕ˜ln(2p). (2.3.66)

The interaction with the gauge field leads to incompressibility, which scales in the large p limit with the suggested potential, interacting between the charge densities. It is in this case the Coulomb potential. This scaling is expected also in terms of dimensional estimations [HLR93]. Some experiments [DST⁺94][PXS⁺99] [YST⁺99] near half and quarter filling factor suggests a large effective composite Fermion mass.

Im Dokument Quantum Field Theory and Composite Fermions in the Fractional Quantum Hall Effect (Seite 32-39)