In Sec.3.2.3we argued that the impact of interactions on transport through the system is closely related to the LDOS at the chemical potential in the barrier center,Aµ( ˜Vc). Here we want to further elaborate on this point by discussing the effect of many-body processes to first order in the interaction parameter U. In other words, we consider the first diagram in the self-energy series (Eq. (3.19)), which describes Hartree type physics only. This provides intuition as to why the effective strength of interactions, i.e. the measure how strongly the system is influenced by electron-electron processes, can be written as product of the interaction parameter and the local density of states:
Ueff(Vc) =U × Aµ(Vc). (3.22) Once again, we can interpret this result in a (somewhat loose but nevertheless instructive) semiclassical picture in terms of the electron’s velocity: Slow electrons feel interactions particularly strong.
Let us make some considerations how interactions influence the propagation of a charge carrier in the QPC: An electron that enters the QPC region sees not only the effective 1D barrierV(x)but in addi- tion feels the presence of other electrons in the constriction. To first order in the interaction this amounts to an enhancement of the barrier by the Hartree self-energyΣh(x) =n
3.3 Physics of a parabolic barrier in the presence of interactions
37
the noninteracting electronic densityn0(x)(note that in similar manner the electrostatic QPC potential is created by the electronic density in the metallic gates above the 2DES). The resulting potentialVh(x) =V(x) + Σh(x) (3.23) is called the effective Hartree barrier (depicted in Fig.3.4(a)). To first order transport is determined by Vh(x) rather than by the bare potentialV(x). Let us calculate how the barrier height of the Hartree potential,Vh
c =Vh(0), evolves with the barrier height of the bare potential,Vc. Since an increase inVc decreases the noninteracting density in the barrier center (with growing barrier height more and more electrons are pushed into the leads), we know thatVh
c increases less thanVc: dVh c dVc = d dVc Vc+ Σh(0)= 1 +Udn0(0) dVc <1. (3.24)
We evaluate the change in noninteracting density in the barrier center due to a potential shift, i.e. the derivative on the r.h.s. of Eq. (3.24). Since the density is given by the energy integral over the LDOS (see Eq. (3.14)) up to the chemical potential (for simplicity evaluated at temperatureT= 0),
n0(0) =
Z −V˜c
−∞
d˜εLDOS(˜ε,˜x= 0), (3.25)
its derivative w.r.t. Vc is just equal to the negative of theLDOSat the chemical potential in the barrier
sub-o p en regime 1 0 0 -2 2 0 1 2 3 -1 conductance g barrier height ˜Vc position ˜x energ y ˜ ε V(˜x)
(a)
(b)
U/(Ωxlx) 0 2 3 Hartre e heig h t ˜V h c ˜ Vc= ˜Vch ˜ Vh(˜x) µ Σh(˜x)Figure 3.4:
The Hartree barrier:(a)Illustration of the bare (U= 0) potentialV(˜x)and the Hartree
potentialV
h(˜x)in the vicinity of the barrier top. The combination of finite densityn
0
(˜x)and
finite interaction strengthUcauses an enhancement w. r. t. the bare barrier given by the Hartree
self-energyΣ
h(˜x) =n
0(˜x)U.(b)The barrier heightV
chof the effective Hartree potential depends
nonlinear on the original barrier height
V
c: Beyond pinch-off (V˜
c0), where all density is
pushed away from the QPCs center, both coincide independent from the value ofU. Yet, in the
sub-open regime the dependence of
V
chon
V
cdescreases with increasingU. The plot shows
center. Hence, we find (from Eq. (3.24)) dVh c dVc = 1−U·A µ(Vc) = 1 −Ueff(Vc). (3.26) This is a logical, but nevertheless remarkable result which we illustrate in Fig.3.4(b). A change in the actual barrier height of the interacting system (to first order given byVh
c ) with the bare barrier height is strongly dependent onVcvia the LDOS,Aµ(Vc). This implies thatVh
c changes least quickly withVc˜ in the sub-open regime. Note that this indirectly supports the picture of a weaker dependence of conductance onVc in the sub-open regime, in other words provides a possible mechanism for the formation of an interaction-induced plateau-like structure even atB=T=Vsd= 0.
It is important to note though, that not only the height of the potential, but of course also its over- all geometry is influenced by interactions. Surely this includes a change in the barrier curvature Ωx, which subsequently changes transmission and conductance. From this perspective the above calcula- tion is insufficient to draw definite conclusions about the shape of the conductance in the presence of interactions. Rather, we present it here to provide the reader with intuition, why the local density of statesAµ( ˜Vc)determines the interacting physics of a QPC viaU
eff( ˜Vc). Still, Hartree contributions are of great importance for the physics of the0.7 anomaly. In order to produce reliable results for the shape of the conductance step in the presence of interactions, it is necessary to include them to all orders: The densityn0(x)within the QPC gives rise to an enhanced Hartree barrier, which in turn generates a modified densityn1(x)etc. This provides an infinite feedback loop and a self-consistency condition for the Hartree density,ni+1(x) =ni(x). The number of iterations needed to fulfill this condition with due accuracy depends sensitively on the value ofAµ( ˜Vc)and thus sensitively on the barrier height. In addition, successive orders usually show oscillatory behavior, meaning an alternating overestimate and underestimate of interactions before convergence. Hence, truncation at low (or too low) orders leads to an inconsistent description of interactions along the conductance step. Since Hartree contribution are of particular interest for the physics at zero excitation-energy T = Vsd= 0, where inelastic scattering (covered by Fock contributions) is forbidden, we refrain from drawing conclusions about the detailed shaping of the zero temperature linear conductance curve from our SOPT results (in which Hartree con- tributions are restricted to second order). In contrast, fRG provides a generic way to sum up Hartree contributions to infinite order, hence allowing for an interpretation of the shaping of the T=Vsd= 0 conductance step.
After having presented a simple Hartree argument, which makes anomalous conductance bahavior in the sub-open regime plausible, we now present fRG calculation for the linear conductance, which show that our model is indeed capable of reproducing the features of the 0.7 anomaly.