2.3 Summary
3.1.1 Bosonized Hamiltonian
We start from the Hamiltonian containing two-particle interactions for spin-polarized electrons moving in both directions
Hˆ10 = Hˆ00 + ˆHint0
= −
ˆ
dxψˆ†(x) 1
2m∂x2
ψ(x) + ˆHint0 Hˆint0 = 1
2 ˆ
dx ˆ
dx0ψˆ†(x) ˆψ†(x0)U(x−x0) ˆψ(x0) ˆψ(x), (3.1) whereU(x) denotes the real interaction potential. The fermionic fields are defined as
ψ(x)ˆ = 1
√ L
X
k
eikxˆck, (3.2)
where ˆc†kapplied to the vacuum state creates an electron with momentumk. Here, we assume a finite interaction range, i.e. the Fourier transform ofU, Uq =´
dx e−iqxU(x), is cut off for q qc with
3.1 Bosonization 19 qc kF. Linearizing the electronic dispersion relation about the Fermi momentum|kF|
Hˆ0 = X
k
n vF
k: ˆc†k,Rˆck,R :−k: ˆc†k,Lˆck,L :
−vFkF(δNˆR+δNˆL)o
, (3.3)
wherevF denotes the Fermi velocity (in the Mach-Zehnder interferometer the edge channel velocity), we introduce the so called Luttinger model. For this, the electrons are separated into two species, i.e., into right- and left moving electrons, ˆck,R and ˆck,L respectively and their linear spectrum is extended down to−∞[14]. The Fermi sea where all the states below the chemical potential are filled is replaced by a ’Dirac sea’ where the infinite number of states with negative energy are assumed to be filled (cf. Fig. 3.1a). Nevertheless, the low-energy properties of ˆH0 and ˆH00 are similar, as the interaction potential Uq is cut off forq qc (where qc kF) and the deep lying electrons do not contribute to the low-energetic excitations of the system. Formally, the Dirac sea is filled by an infinite number of electrons, which is why we normal order the Hamiltonian ˆH0 with respect to the filled, non-interacting Dirac sea (denoted as|vaci). As usual, we label normal ordered operators with : · · · :, where equivalently we could write : A := ˆA−D
vac|A|vacˆ E
. For example, in Eq.(3.3) δNˆR≡P
k: ˆc†k,Rcˆk,R: is the normal ordered number operator. As we are only interested in a chiral electron system, the following considerations are restricted to right-moving electrons. The chiral, interacting part of the Hamiltonian in the momentum representation yields
Hˆint = 1 2L
X
q,k,k0
Uq : ˆc†k,Rcˆ†k0,Rˆck0−q,Rˆck+q,R:. (3.4)
In the next step we re-express Eq. (3.4) in terms of density fluctuations.
Density operators The normal ordered density operator ˆρR(x) is introduced for the chiral system consisting of right-moving electrons as
ˆ
ρR(x) ≡ : ˆψR†(x) ˆψR(x) : ˆ
ρq,R = X
k
ˆ
c†k,Rˆck+q,R≈X
k>0
ˆ
c†k,Rˆck+q,R, q6= 0. (3.5)
In the second line we safely neglect the low-lying electrons (with formal momentum k < 0) as due to the momentum cutoffqckF in the interaction potentialUq they do not contribute to the low-energy properties of the system. We can now re-express the interacting part of the Hamiltonian ˆHint
energy
0 1 3
1 3
(a) (b)
left moving electrons
right moving electrons
filled Dirac sea
plasmonic spectrum
Figure 3.1: (a) Pictorial plot of the dispersion relation of non-interacting one-dimensional electrons (solid gray line). As mentioned in Subsection 3.1.1, the Luttinger theory assumes a spectrum lin-earized about the Fermi momentumkF (solid blue lines), which is extended down to−∞. Thereby, the right and left moving electrons are separated into two species. The filled Fermi sea is substi-tuted by the filled “Dirac sea”. Due to the linearization procedure, the theory is only valid at low energies, where only electrons in the close vicinity to the Fermi edge are involved in excitations (the finite bandwith, related to the cutoff parametera(see main text) is indicated by red boxes). Inset:
Applying the density operator ˆρ†q to the vacuum state creates a superpostion of electron-hole pairs.
Basically, these are the relevant low-energy excitations. (b) Dispersion relation of the bosonic modes ˆbq for an interaction potential Uq = 2παvFe−(q/qc)2 with 2πα= 5. For qqc , the velocity of the plasmons is renormalized to ¯v = vF(1 +α), while due to the finite interaction range in the limit qqc the bare velocityvF is reproduced, i.e., there are two distinct energy regimes.
3.1 Bosonization 21
The emerging of the Hartree-Fock terms in Eq. (3.6) has to be noted in particular. The Fock correction naturally cancels out contributions stemming from the unphysical interaction of the electrons with itself, as the Pauli principle does not allow for two particles to be exactly at the same position in space. For small changes in the particle number we can linearize the Hartree-Fock term inδNˆR. As a consequence, we can simply incorporate the Hartree-Fock contribution into a re-definition of the chemical potential. The main point is that it is obviously possible to reformulate the interacting part of the Hamiltonian only in terms of the density operators ˆρq,R.
Bosonic operators{bˆq,bˆ†q} The crucial step behind the bosonization of the Hamiltonian Eq. (3.3) is that it is possible to find operators{ˆbq,ˆb†q}which diagonalize the Hamiltonian and fulfill the bosonic commutation relations. It can be shown [21, 14, 4] that for the Tomonaga-Luttinger model (where the linearized dispersion relation is is extended down to−∞) the operators
(q >0) ˆbq =
represent well defined bosonic excitations, i.e., they fulfill
[ˆbq,ˆb†q0] =δq,q0. (3.8) In the remainder of this work we will refer to these modes as ’plasmons’, i.e., modulations in the electron density. Furthermore, it turns out [21, 4, 14] that the free part of ˆH0can be written in terms of those operators:
Hˆ0 ∼ X
q>0
vFqˆb†qˆbq. (3.9)
Formula Eq. (3.9) lies at the root of the bosonization method; however, we still have to fix the constant in Eq. (3.9). As the bosonic operators{ˆbq,ˆb†q}only describe the low-energetic excitations of the chiral interacting electron system, we have to add the energy related to a change in the electron number.
Addingnelectrons to the filled Fermi sea increases the energy by 2πvF
L
N¯R+n
X
j= ¯NR
j = πvF
L
n+ ¯NR
n+ ¯NR+ 1
−N¯R N¯R+ 1 . (3.10)
where ¯NR is the mean electron number in the channel. Therefore, up to a constant we have to add the contribution πvLF( ˆNR+ ¯NR)( ˆNR+ ¯NR+ 1).
Bosonized Hamiltonian Finally, we can sum up the different terms. Linearizing the part depend-ing on the normal ordered electron number δNˆR, i.e., keeping only terms linear in δNˆR yields the bosonized Hamiltonian (omitting a further constant)
Hˆ1 = X
q>0
ωqˆb†qˆbq+µδNˆR, (3.11)
where µ=u+ 2πvFρ¯R (with ¯ρR = ¯NR/L) anduis a constant which fixes the chemical potential of the channel. For example, u contains the energy shift due to the Hartree-Fock contribution in Eq. (3.6) and to any external applied gate voltage (here we set u = 0). The plasmonic dispersion relation (see Fig. 3.1b) is given by
ωq = vFq
1 + Uq 2πvF