Interaction effects in a multi-channel Fabry-Pérot interferometer in the Aharonov-Bohm regime
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Interaction effects in a multi-channel Fabry-P´erotinterferometer in the Aharonov-Bohm regime
D. Ferraro , E. Sukhorukov D´epartement de Physique Th´eorique, Universit´e de Gen`eve, 24 quai Ernest Ansermet,CH-1211 Geneva, Switzerland Aix Marseille Univ, Universit´e de Toulon, CNRS, CPT, Marseille, France* [email protected] 31, 2018
Abstract
We investigate a Fabry-P´erot interferometer in the integer Hall regime in whichonly one edge channel is transmitted and n channels are trapped into the in-terferometer loop. Addressing recent experimental observations, we assume thatCoulomb blockade effects are completely suppressed due to screening, while keep-ing track of a residual strong short range electron-electron interaction betweenthe co-propagating edge channels trapped into the interferometer loop. Thiskind of interaction can be completely described in the framework of the edge-magnetoplasmon scattering matrix theory allowing us to evaluate the backscat-tering current and the associated differential conductance as a function of the biasvoltage. The remarkable features of these quantities are discussed as a function ofthe number of trapped channels. The developed formalism reveals very generaland provides also a simple way to model the experimentally relevant geometry inwhich some of the trapped channels are absorbed into an Ohmic contact, leadingto energy dissipation. Contents ciPost Physics Submission
A.1 J (1) int and J (2) int W (1) and W (2) Y (1) int and Y (2) int References 18
In the last few years various accurate experimental observations shed new light on the re-markable physics associated to the Fabry-P´erot interferometer (FPI) of integer and fractionalquantum Hall edge channels [1, 2, 3]. Depending on the size of the interference loop two dis-tinct regimes have been achieved. For a small enough loop area, intra-edge interaction playsan essential role and features related to Coulomb blockade occur. In the opposite regimeof large central area, the expected Aharonov-Bohm physics of free electrons is recovered inthe integer quantum Hall regime. A consistent theoretical interpretation of these results aswell as a characterization of the crossover between these two limits has been proposed [4, 5].Various experimental techniques, including gates and ohmic contacts, have been introducedin order to enhance the screening of interaction extending as much as possible the domain ofvalidity of the Aharonov-Bohm regime, where the interaction is negligible and the simple freeparticles picture seems adequate to properly explain the experimental observations.However, very recently, new measurements carried out by Choi et al. [6] have suggestedthat a richer phenomenology occurs also in this apparently trivial case. In particular, whenonly one channel is transmitted throughout the FPI, by increasing the number of channelstrapped in the loop (namely the integer filling factor of the system), one moves from thestandard Aharonov-Bohm effect of electrons (at filling factor 1 ≤ ν ≤
2) to a more puzzlingsituation in which a pair of electrons seems to interfere (at filling factor 3 ≤ ν ≤ ciPost Physics Submission among the channels as the principal responsible of this peculiar pairing effect.The aim of this paper is to provide the proper theoretical background for the descriptionof the experimental setup in Ref. [6] in the framework of the edge-magnetoplasmon scatteringmatrix formalism [7, 8, 9, 10, 11, 12] where the two points electron Green’s function (firstorder electron coherence [13, 14]), crucial ingredient to calculate transport properties likeconductance and noise, explicitly depends on the transmission of the bosonic mode acrossthe interferometer. As simplest possible case we will assume a strong screened Coulombinteraction and we will investigate carefully the functional form of the scattering matrix as afunction of the number of channels trapped into the FPI loop. We will discuss in detail thecase of one trapped channel extrapolating then the behavior in case of more channels into theinterfering loop. The consequences of the form of these scattering matrices on the current andthe conductance will be then investigated. We derive a very powerful and general formalism.On the one hand it allows us to rule out the simplest academic model of short range stronginteraction as a possible way to explain what is observed in experiments, on the other handit appears suitable to extensions towards more realistic models in which finite length of theinteraction and dissipations have been taken into account [15].The paper is organized as follows. In Section 2 we discuss the edge-magnetoplasmonscattering matrix theory for a FPI as a function of the number of integer Hall edge chan-nels trapped into the loop. We focus in particular on a strong short range ( δ -like) screenedCoulomb potential. The classical and quantum contributions to the current and the associ-ated differential conductance are derived in Section 3 by means of the Kubo formula. Theplots of these quantities, as well as the relevant comments concerning the behavior of thesystem as a function of the number of trapped channels are reported in Section 4, also in viewof a possible interpretation of the experimental observations. In Section 5 we investigate therole played by an ohmic contact absorbing some of the channels, analogous to the one usedin realistic setup, in terms of a simple model based only on the energy conservation. Section6 is devoted to the conclusions, while some technical details of the calculation are discussedin Appendix A. Let us start by discussing the physics of two edge channels capacitively coupled along a finiteregion of length L . This problem will be investigated in the framework of the bosonizationformalism [16, 17]. Due to the chirality of the two channels we can identify the incomingregion (1), the interacting region (2) and the outgoing region (3) (see Fig. 1). We will analyzethem in detail in the following. (1)In this region the interaction is absent and the Hamiltonian density can be written in termof the Wen’s hydrodynamical model [18] ( ~ = 1) H (1) = v π ( ∂ x φ ) + v π ( ∂ x φ ) . (1)3 ciPost Physics Submission (1) (2) (3) ˜ φ ,in ( x, ω )˜ φ ,in ( x, ω ) ˜ φ ,out ( x, ω )˜ φ ,out ( x, ω ) ˆ S ( L, ω ) Figure 1: Color on-line. Schematic view of a two integer quantum Hall channels system (fillingfactor ν = 2). According to the chirality one can easily identify the incoming region (1), theinteracting region (2) (shaded area) and the outgoing region (3). In regions (1) and (3) thedynamics of the bosonic fields is well described in terms of free equations of motion, while theoutgoing fields are connected to the incoming ones through the edge-magnetoplasmon scatter-ing matrix ˆ S ( L, ω ) which encodes the information of the inter-channel interaction (assumedas strong and short ranged in the main text).Therefore, one can easily associate a chiral bosonic field to the charge density along eachchannel according to the conventional prescription [16, 17] ρ i = 12 π ∂ x φ i i = 1 , . (2)These bosonic fields propagate freely according to the equations of motion ∂ t φ i ( x, t ) = − v i ∂ x φ i ( x, t ) i = 1 , v and v along the two channels. (2)In this region we assume a density-density short range ( δ -like) interaction in such a way thatthe Hamiltonian density becomes H (2) = v π ( ∂ x φ ) + v π ( ∂ x φ ) + v π ∂ x φ ∂ x φ (4)with v interaction strength. Notice that, in spite of the fact that high frequency measure-ments suggest a relevant role played by a finite range of interaction and dissipation, thisapproximation reveals good at low enough frequencies [15]. According to this, the bosonicfields φ and φ are no longer eigenstates of the Hamiltonian of the system. The equations ofmotion are then decoupled in terms of a charged and a neutral mode, indicated respectivelywith φ ρ and φ σ . They diagonalize the Hamiltonian with associated eigenvelocities v ρ and v σ in such a way that the new equations of motion become ∂ t φ η ( x, t ) = − v η ∂ x φ η ( x, t ) η = ρ, σ. (5)Due to the fact that the incoming fields are co-propagating, the above diagonalizing fields arerelated to φ and φ through a simple rotation of an angle θ in the field space . This becomesmore transparent in the frequency space, namely through a partial Fourier transform withrespect to time. Here, one has˜ φ ρ ( x, ω ) = cos θ ˜ φ ( x, ω ) + sin θ ˜ φ ( x, ω )˜ φ σ ( x, ω ) = − sin θ ˜ φ ( x, ω ) + cos θ ˜ φ ( x, ω ) . (6) Notice that the velocities v ρ and v σ are functions of v , v and v [19]. However, in what follows we arenot interested in their explicit functional form, but only in the fact that typically on has v ρ ≫ v σ [8, 9]. ciPost Physics Submission It is worth to note that the angle θ (0 ≤ θ ≤ π/
4) provides a direct measurement of thestrength of the interaction. In particular, θ = 0 corresponds to the non-interacting case,while for θ = π/ (3)Analogously to region (1), also in this case inter-channel interaction is negligible and theequations of motion write as in Eq. (3) ( H (1) = H (3) ).The general solution of the above systems of equations can be easily found in the frequencydomain and reads ˜ φ α ( x, ω ) = e i ωvα ( x − x ) ˜ φ α ( x , ω ) (7)˜ φ α ( x , ω ) being the (possibly frequency dependent) amplitude at the initial condition x andwhere α = 1 , α = ρ, σ in region (2) respectively.To completely solve the system we need now to impose the continuity of the fields at theboundaries of the three regions, namely at x = 0 and at x = L . Notice that, in the Fourierrepresentation we are considering, this is equivalent to impose the conservation of the currentacross the boundaries. Before investigating the FPI geometry we are interested in, it is useful to recall the ex-pected results in the case of open channels. Here, after some algebra, we obtain the edge-magnetoplasmon scattering matrix representation (cid:18) ˜ φ ( L, ω )˜ φ ( L, ω ) (cid:19) = ˆ S ( L, ω ) (cid:18) ˜ φ (0 , ω )˜ φ (0 , ω ) (cid:19) (8)with ˆ S = (cid:18) cos θe iωτ ρ + sin θe iωτ σ sin θ cos θ (cid:0) e iωτ ρ − e iωτ σ (cid:1) sin θ cos θ (cid:0) e iωτ ρ − e iωτ σ (cid:1) sin θe iωτ ρ + cos θe iωτ σ (cid:19) (9)and where we have introduced the short-hand notation τ α = L/v α ( α = ρ, σ ).Notice that this result is in full agreement with what is discussed in literature [7, 8, 10,11, 12] and satisfies the unitarity condition ˆ S · ˆ S † = I as expected. Thanks to the above results it is now easy to investigate the simplest possible example ofthe geometry described in Fig. 2, where only one edge channel is trapped into the FPI loop( n = 1, k = 0). At filling factor ν = 2 (only blue and red channels in Fig. 2) this systemrepresents the natural starting point to model the Choi’s experiment of Ref. [6], when onlyone channel is trapped into the interferometer loop, while the other is transmitted with atunable amplitude. For sake of generality we will assume two different scattering matricesfor the upper ( ˆ S ( u ) ) and the lower ( ˆ S ( d ) ) part of the interferometer. By properly taking into5 ciPost Physics Submission ˆ Σ ( n ) x Φ x ˜ φ ina ˜ φ in b ˜ φ outb ˜ φ out a Λ Λ Figure 2: Color on-line. Schematic view of a FPI with n trapped channels in the interferingloop. The action of the interferometer in presence of interaction (assumed strong and shortranged in main text) can be described in terms of the edge-magnetoplasmon scattering matrixˆΣ ( n ) . Tunneling of electrons occurs at two quantum point contacts in x and x , respectivelywith tunneling amplitude Λ and Λ . The interferometer is also pierced by a flux Φ of magneticfield which is responsible of the Aharonov-Bohm effect. Moreover, in order to be closer towhat is done in experiments, we can consider the presence of an ohmic contact which on theone hand further enhances the screening of the interaction, on the other absorbs the energyof k ≤ n trapped channels inducing dephasing.account the periodic boundary conditions associated to this closed channel one can derive thescattering matrix ˆΣ (1) for the whole interferometer in the form (cid:18) ˜ φ outa ˜ φ outb (cid:19) = ˆΣ (1) (cid:18) ˜ φ ina ˜ φ inb (cid:19) (10)with ˆΣ (1) = S ( u )11 − S ( d )22 f ( u ) − S ( u )22 S ( d )22 S ( u )12 S ( d )12 − S ( u )22 S ( d )22 S ( u )12 S ( d )12 − S ( u )22 S ( d )22 S ( d )11 − S ( u )22 f ( d ) − S ( u )22 S ( d )22 , (11)where we have introduced the phase factor f ( ω ) = e iω ( τ ρ + τ σ ) . (12)Notice that the frequency dependence has been omitted for notational convenience.It is worth to note that the edge-magnetoplasmon scattering matrix ˆΣ (1) inherits theproperties of matrices ˆ S ( u ) and ˆ S ( d ) and is therefore unitary as expected. In this limit ( θ = π/
4) the expression in Eq. (11) strongly simplifies and one recoversa symmetric configuration ˆ S ( u ) = ˆ S ( d ) = ˆ S . Moreover, we can safely consider the limit v ρ → + ∞ ( τ ρ →
0) for the charged mode.We then obtain the simple matrix elements6 ciPost Physics Submission S = S ≈ (cid:16) e iξ (cid:17) (13) S = S ≈ (cid:16) − e iξ (cid:17) (14)with ξ = ωτ σ .Under this approximation, we can then writeˆΣ (1) ( ξ ) ≈ (cid:18) − h (1) ( ξ ) − h (1) ( ξ ) − h (1) ( ξ ) 2 − h (1) ( ξ ) (cid:19) , (15)where we have defined h (1) ( ξ ) = + ∞ X n =0 (cid:18) − e iξ (cid:19) n = 33 + e iξ . (16)Notice that Eq. (16) calls for the possibility of a simple harmonics expansion of the matrixelements of ˆΣ (1) suitable, as will be clearer in the following, for numerics and useful to easilyderive asymptotic behaviors. Moreover, in the zero frequency limit ( ξ = 0), one directlyobtain ˆΣ (1) (0) = I as required for a purely capacitive coupling. In the strong interaction limit discussed above also the case at filling factor ν = 3, wheretwo channels are trapped in the interferometer loop and the last one is transmitted with atunable amplitude (see Fig. 2), can be easily handled. Without entering into the details ofthe calculation, also in this case we have a charge mode with a velocity v ρ that is greater withrespect to the one of the two neutral modes (assumed v σ for both ).Proceeding exactly on the same way as before, after some quite tedious algebra, oneobtains the edge-magnetoplasmon scattering matrixˆΣ (2) ( ξ ) ≈ (cid:18) − h (2) ( ξ ) − h (2) ( ξ ) − h (2) ( ξ ) 3 − h (2) ( ξ ) (cid:19) (17)with h (2) ( ξ ) = + ∞ X n =0 (cid:18) − e iξ (cid:19) n = 55 + e iξ . (18)Also in this case the zero frequency limit ( ξ = 0) leads to ˆΣ (2) (0) = I . We can now discuss the general expression for the current flowing through the system in theframework of the edge-magnetoplasmon scattering matrix formalism and focusing on the weak Notice that this symmetry is reminiscent of the analogous hidden symmetry observed for the states be-longing to the Jain’s sequence of the fractional quantum Hall effect [20]. ciPost Physics Submission backscattering regime for both the QPCs in Fig. 2. We will consider the first perturbativeorder in the backscattering Hamiltonian H BS = X j =1 , Λ j Ψ † b ( x j )Ψ a ( x j ) + H.c. (19)with Ψ a and Ψ b electronic annihilation operators associated to the two edges and Λ j ( j = 1 , I B = − e X j =1 , i Λ j Ψ † b ( x j )Ψ a ( x j ) + H.c. (20)In order to evaluate the average value of the backscattering current we can use, as usual,the Kubo formula [23] h I B i = − i Z t −∞ dt ′ h (cid:2) I B ( t ) , H BS ( t ′ ) (cid:3) i (21)where operators are written in the interaction picture, namely evolved in time according tothe free edge Hamiltonian only. Notice that all the averages discussed above are taken withrespect to the ground state of the free bosonic systems in absence of backscattering.In order to evaluate the backscattering current in Eq. (21) as a function of the elementsof the edge-magnetoplasmon scattering matrices derived in the previous section we need torecall the fact that the fermionic annihilation operator can be seen as a coherent state ofedge-magnetoplasmon in the form Ψ( x ) = 1 √ πα e iφ ( x ) , (22) α a finite-length cut-off. [16]Because the calculation of the averaged backscattering current naturally involves four-vertex operators it is useful to introduce the general correlator h e − iA e iB e − iC e iD i = exp (cid:26) − (cid:2) h A i + h B i + h C i + h D i (cid:3) + h AB i − h AC i + h AD i + h BC i − h BD i + h CD i} (23)based on the Baker-Campbel-Hausdorff formula and the Wick theorem applied to the bosonicfields and valid for arbitrary gaussian fields A , B , C and D those commutation relations leadto complex functions.By properly taking into account the action of the edge-magnetoplasmon scattering matrixand assuming the same propagation velocity for both free channels ( v = v = v ) we can eval-uate explicitly all the contributions to the current. Notice that the effect of a bias differencebetween the edge b (at voltage V ) and the edge a (grounded) is taken into account through8 ciPost Physics Submission the standard Peierls substitution [24] Ψ b ( x, t ) → e − iω t Ψ b ( x, t ) with ω = eV / ~ and where thefield Ψ a ( x, t ) is left untouched.The averaged current can be naturally separated into a classical contribution, diagonal inthe QPCs action, and a quantum contribution, which is the off diagonal interference term. Inthe following we will discuss these terms in detail mainly focusing on the zero temperaturelimit. For what it concerns the classical contributions to the current, namely the one diagonal inthe QPCs tunneling amplitudes, one obtains I clB = e π α (cid:2) | Λ | + | Λ | (cid:3) Z + ∞−∞ dz sin ( ω z ) ℑ h J ( l ) ( z ) i (24)with J ( l ) ( z ) = exp (cid:26) − Z + ∞ dωω n − ℜ h Σ ( l ) ab ( ω ) io (cid:2) − e − iωz (cid:3)(cid:27) (25)and where ℜ [ ... ] and ℑ [ ... ] indicate respectively the real and the imaginary part. Notice thatthe above formula can be applied to both the case of one ( l = 1) and two ( l = 2) trappedchannels in the FPI. The conventional non interacting case is easily recovered by neglectingthe term associated to the edge-magnetoplasmon scattering matrix (Σ (0) ab = 0). It is worth topoint out the fact that, due to translational invariance in the time domain, the expression inEq. (24) is indeed independent of time.As expected, the above term does not depend on the position of the QPCs and on the fluxof magnetic field piercing the interferometer. It corresponds to the sum of the contributionsassociated to two distinct single QPC geometries, proportional respectively to | Λ | and | Λ | .By replacing the explicit form of the scattering matrix elements respectively for ˆΣ (1) andˆΣ (2) (see Eq. (15) and Eq. (17)) one obtains the factorization J ( l ) ( z ) = J ( l ) int ( z ) J free ( z ) l = 1 , J free ( z ) = exp (cid:26) − Z ∞ dωω (cid:2) − e − iωz (cid:3) e − ω/ω ρ (cid:27) (27)= 1(1 + iω ρ z ) (28)the free fermion contribution and J (1) int ( z ) = exp (cid:26) − Z ∞ dωω (cid:18) cos ωτ σ −
15 + 3 cos ωτ σ (cid:19) (cid:2) − e − iωz (cid:3) e − ω/ω ρ (cid:27) (29) J (2) int ( z ) = exp (cid:26) − Z ∞ dωω (cid:18) cos ωτ σ −
113 + 5 cos ωτ σ (cid:19) (cid:2) − e − iωz (cid:3) e − ω/ω ρ (cid:27) (30)the corrections due to the interaction. Notice that, for further convenience, we haveintroduced the convergence factor e − ω/ω ρ with high frequency cut-off ω ρ = v ρ /α in order tohave well behaved integrals for ω → + ∞ . 9 ciPost Physics Submission Differently from the classical contribution discussed above, this term depends non locally onthe two QPCs amplitudes and encodes information about the Aharonov-Bohm interferenceat the level of the FPI. It is given by I qB = eπ α Z + ∞−∞ dz ℑ h Λ Λ ∗ e iω z W ( l ) i ℑ h Y ( l ) (∆ t − z ) i (31)with ∆ t = ( x − x ) /v , W ( l ) = exp (cid:26)Z + ∞ dωω h Σ ( l ) ab ( ω ) + Σ ( l ) ba ( ω ) i(cid:27) (32)and Y ( l ) ( z ) = exp (cid:26) − Z + ∞ dωω h − (cid:16) Σ ∗ ( l ) aa ( ω ) + Σ ( l ) bb ( ω ) (cid:17) e iωz i(cid:27) . (33)In analogy with what discussed for the classical current, also this contribution does not dependexplicitly on time.In terms of the explicit form of the scattering matrix elements for ˆΣ (1) and ˆΣ (2) (Eq. (15)and Eq. (17)), one has W (1) = exp (cid:26) Z ∞ dωω (cid:18) − e iωτ σ e iωτ σ (cid:19) e − ω/ω ρ (cid:27) (34) W (2) = exp (cid:26) Z ∞ dωω (cid:18) − e iωτ σ e iωτ σ (cid:19) e − ω/ω ρ (cid:27) . (35)Moreover, also in this case we can factorize the interaction contribution in the form Y ( l ) ( z ) = Y ( l ) int ( z ) J free ( − z ) with, Y (1) int ( z ) = exp (cid:26) Z ∞ dωω (cid:18) cos ωτ σ −
15 + 3 cos ωτ σ (cid:19) e iωz e − ω/ω ρ (cid:27) (36) Y (2) int ( z ) = exp (cid:26) Z ∞ dωω (cid:18) cos ωτ σ −
113 + 5 cos ωτ σ (cid:19) e iωz e − ω/ω ρ (cid:27) . (37)According to the Fourier series expansion of the functions h (1) (Eq. (16)) and h (2) (Eq.(18)) it is possible to derive useful asymptotic limits for the functions J ( l ) int , W ( l ) and Y ( l ) int ( l = 1 ,
2) which are discussed in detail in Appendix A.In particular, focusing on the role played by W ( l ) , Eq. (31) can be approximated in a verygood way by ( j = 0 , , I qB (Φ , ω ) ≈ − e | Λ || Λ | π α ( ω ρ τ σ ) a j e − γ j cos ( ω ∆ t − Φ − θ j ) Z + ∞−∞ dz sin ω z ℑ h Y ( j ) int ( z ) J free ( − z ) i (38)where we have included the conventional free fermion case j = 0. In the above expressionwe have introduced the compact notation a = 0, a = 2 / a = 4 / θ = 0, θ = π/ θ = 2 π/ γ = 0, γ ≈ . γ ≈ . ciPost Physics Submission worth to note that in the non interacting case we trivially have Y (0) int = 1. We observe that themore dramatic effects associated to the presence of the trapped channels consists in modifyingthe scaling of the tunneling amplitude through a cut-off dependent contribution and to add anadditional phase shift. Both these contributions crucially depend on the number of trappedchannels. I clB /I G clB /G Figure 3: Color on-line. Classical current in units of I = e | Λ | / ( πv ρ τ σ ) (top) and differentialconductance in units of G = eI (bottom). Every curve is properly further rescaled withrespect to the factor ( ω ρ τ σ ) a j e − γ j in order to keep track of the interaction induced renormal-ization of the tunneling amplitudes for the non interacting case ( j = 0, full black curve), theone trapped channel case ( j = 1, dashed blue curve) and the two trapped channels case ( j = 2,dotted-dashed red curve) as a function of the Josephson frequency ω = eV / ~ . Parametersare: τ σ = 1, ω ρ τ σ = 1000, | Λ | = | Λ | = | Λ | and ~ = 1.In Fig. 3 we show the behavior of the classical contribution to the current and the11 ciPost Physics Submission G qB /G Figure 4: Color on-line. Quantum contribution to the differential conductance in units of G , calculated at ω = 0 as a function of the Aharonov-Bohm phase (Φ). Also in this caseevery curve is properly further rescaled with respect to the factor ( ω ρ τ σ ) a j e − γ j in order tokeep track of the interaction induced renormalization of the tunneling amplitudes for the noninteracting case ( j = 0, full black curve), the one trapped channel case ( j = 1, dashed bluecurve) and the two trapped channels case ( j = 2, dotted-dashed red curve). Parameters are: τ σ = 1, ω ρ τ σ = 1000, | Λ | = | Λ | = | Λ | and ~ = 1.associated differential conductance G clB ∝ ∂ h I clB i ∂ω = e π α (cid:2) | Λ | + | Λ | (cid:3) Z + ∞−∞ dzz cos ( ω z ) ℑ h J ( l ) ( z ) i . (39)In order to take into account the renormalization of the tunneling amplitudes inducedby the presence of the trapped channels and to better compare the results, the curves arenormalized here with respect to the proper cut-off dependent factor ( ω ρ τ σ ) a j e − γ j ( j = 0 , , V ) for ω ≈ ω τ σ ≈ π ) and show a remarkable oscillatingbehavior. This is even more evident for what it concerns the differential conductance (bottompanel of Fig. 3) which is constant (up to a small deviations due to the high frequency cut-offused in the numerics) in the free case, but oscillates and decay quite fast by increasing thenumber of trapped channels. Notice that this phenomenology is reminiscent of what derivedin literature for a FPI in the integer and fractional quantum Hall regime [25] or for thetopological insulators in presence of interaction [26].12 ciPost Physics Submission Concerning the interference contribution, relevant information about the physics of the sys-tems can be extracted form the conductance calculated at zero voltage ( j = 0 , , . ) G qB (Φ , ω = 0) ∝ ∂I qB ∂ω | ω =0 ≈ − e | Λ || Λ | π α ( ω ρ τ σ ) a j e − γ j cos (Φ + θ j ) Z + ∞−∞ dzz ℑ h Y ( j ) int ( z ) J free ( − z ) i (40)The behavior of this quantity as a function of the Aharonov-Bohm phase Φ is representedin Fig. 4, where the phase shifts θ = π/ θ = 2 π/ j = 0), where the integral in Eq. (40) can beevaluated analytically according to the relation Z + ∞−∞ dzz ℑ (cid:20) iω ρ z ) (cid:21) = − πω ρ (41)due to the second order poles in the complex plane and coincide with what obtained numer-ically for the interacting cases ( j = 1 , As discussed above, Choi’s experiment shows a puzzling and extremely interesting evolutionof the effective tunneling charge as a function of the number of trapped channels. In order toshed light on this behavior one can imagine to allow also the tunneling of excitations with m times the charge of an electron ( m ∈ N ) at the QPCs, despite the fact that these multipleexcitations are less relevant with respect to the electrons in the renormalisation group sense[34]. The vertex operator associated to them is given byΨ ( m ) ( x ) ∝ e imϕ ( x ) (42)with the corresponding tunneling Hamiltonian H ( m ) T = X j =1 , Λ ( m ) j e imϕ b ( x j ) e − imϕ a ( x j ) + H.c. (43)In this case, as far as we consider a gaussian model for the edge states dynamics, the ex-pressions for this kind of excitations can be derived directly from the one for the electronsthrough the substitutions
X → [ X ] m , being X = J (1 , , W (1 , or Y (1 , and with a conse-quent replacement of the tunneling amplitudes and the charge associated to the excitations.This leads to a superohmic behavior which reflects on the fact that the conductance at zero Notice that a similar analysis involving the tunneling of multiple fractionally charged excitation has beenproposed in Refs. [27, 28, 29, 30] as a possible explanation for the unexpected evolution of the effective chargein a QPC geometry in the composite edge states of the fractional quantum Hall effect [31, 32, 33]. ciPost Physics Submission bias is identically zero. This fact can be verified analytically in the non interacting case dueto the relation Z + ∞−∞ dzz ℑ (cid:20) iω ρ z ) m (cid:21) = 0 for m = 1 (44)which is a consequence of the presence of higher order poles in the complex plane and numer-ically in the other cases.Therefore, no signature associated to higher charge carrier are expected in the frameworkof the proposed model. According to this, the comparison with experimental observations rules out the simple picture based on short range strong interaction usually considered as avaluable work hypothesis and seems to suggests that a more involved description is required.In particular dissipative effects at the level of the edge-magnetoplasmon scattering matrix[15], finite range interaction and non-gaussianity in the particle injection process [8] couldplay a relevant role. In order to be further closer to what has been done in experiments, we can also consider amodified set-up in which an integer number k ≤ n of the trapped channels is absorbed byan ohmic contact. This configuration has been introduced in order to both enhancing thescreening of the interaction and inducing dephasing among the trapped channels (see Fig.1) [6]. We will study this geometry in the full equilibration limit ( τ σ → + ∞ ) where we canneglect the oscillations appearing in the edge-magnetoplasmon scattering matrices (see Eqs.(15) and (17)). This leads to a semiclassical approximation based only on the conservation ofa unitary energy flow. In full generality it writesˆΣ ( n,k ) ↑ = ( n − k +1)( n +1) n ( k +2)+3 n +1 ( n − k +1) nn ( k +2)+3 n +1( n − k +1) nn ( k +2)+3 n +1 ( n +1) n ( k +2)+3 n +1 ! . (45)The notation ↑ indicates an ohmic contact placed in the upper part of the interferometer loop(see Fig. 2), as opposite to ↓ for an ohmic contact in the lower part whose scattering matrixelements are obtained through the replacements ↑↔↓ , a ↔ b . Notice that the position of theohmic contact does not affect the results concerning the current flowing across the sample.Depending on the incoming arm of injection ( a or b ) one can have two different fractionof the unitary energy flux leaking into the ohmic contact, namely∆ E a, ↑ = 1 − Σ ( n,k ) aa, ↑ − Σ ( n,k ) ba, ↑ = k ( n + 1) n ( k + 2) + 3 n + 1∆ E b, ↑ = 1 − Σ ( n,k ) ab, ↑ − Σ ( n,k ) bb, ↑ = kn ( n + 1) n ( k + 2) + 3 n + 1 . (46)In absence of ohmic contact ( k = 0) the scattering matrix in Eq. (45) reduces toˆΣ ( n, ↑ = ˆΣ ( n, ↓ = ˆΣ ( n, = (cid:18) n +12 n +1 n n +1 n n +1 n +12 n +1 (cid:19) (47)14 ciPost Physics Submission with an obvious zero energy leakage (∆ E a, ↑ = ∆ E b, ↑ = 0). Two important comments are inorder at this point. First of all, we notice that it is possible to obtain the expressions for ˆΣ (1 , and ˆΣ (2 , directly from Eq. (15) and (17) in the semiclassical limit, where all the oscillatingterms are neglected. Eq. (47) represents therefore a generalization of what is done above asfar as equilibration among the channels comes into play. Moreover, this fact also represents avalidation of the renormalisation of the tunneling amplitudes previously derived, due to thefact that the dominant contribution to the integrals comes indeed from a region where theconsidered approximation holds.According to the previous considerations, the classical and quantum contribution to thecurrent can be written, in terms of the elements of the scattering matrix and the energy loss,as I clB = e π α (cid:0) | Λ | + | Λ | (cid:1) R h ω , ω ρ , Σ ( n,k ) aa + Σ ( n,k ) bb + ∆ E a + ∆ E a i (48) I qB = − eπ α ( | Λ || Λ | ) cos ( ω ∆ t − Φ) exp (cid:26) − (∆ E a + ∆ E b ) Z + ∞ dωω e − ω/ω ρ (cid:27) × R h ω , ω ρ , Σ ( n,k ) aa + Σ ( n,k ) bb i (49)with R [ x, y, α ] = 2 π Γ( α ) | x | α − y α e −| x | /y sgn( x ) (50)and Γ( α ) the Euler’s Gamma function. Notice that we have omitted the indication aboutthe placement of the ohmic contact due to the fact that the results are independent of thisbecause of the symmetries relating ˆΣ ( n,k ) ↑ and ˆΣ ( n,k ) ↓ .In absence of ohmic contact (∆ E a = ∆ E b = 0), but still considering the equilibrated limit,one obtains (for | Λ | = | Λ | = | Λ | ) I clB + I qB = e | Λ | π α [1 − cos ( ω ∆ t − Φ)] R h ω , ω ρ , Σ ( n, aa + Σ ( n, bb i (51)showing oscillation as a function of the piercing magnetic flux modulating the overallpower-law behavior ω Σ ( n, aa +Σ ( n, bb − reminiscent of the one observed for the fractional quantumHall effect [25] and topological insulators in presence of interaction [26]. It is worth to notethat the exponents satisfies 1 < Σ ( n, aa + Σ ( n, bb < I clB ∝ ω Σ ( n,k ) aa +Σ ( n,k ) bb +∆ E a +∆ E a − (53) I qB ∝ ω Σ ( n,k ) aa +Σ ( n,k ) bb − . (54)15 ciPost Physics Submission In the present paper we have discussed the physics of a FPI in the integer quantum Hallregime. Motivated by very recent experiments, we focused on a system at filling factor ν = n + 1 ( n ∈ N ) where only one edge channel is transmitted across the sample, whilethe other n are trapped into the interferometer loop. Due to screening, the only residualinteraction effect is given by a strong short range inter-channel interaction that we describedin terms of the edge-magnetoplasmon scattering matrix formalism. The major effects associ-ated to the presence of interacting trapped channels are: a renormalisation of the tunnelingamplitudes affecting in exactly the same way the classical and the quantum contribution tothe current, an evident damped and oscillatory behavior of the differential conductance incontrast to the constant (ohmic) behavoir observed in absence of trapped channels and anadditional phase shift in the Aharonov-Bohm periodicity that is clearly visible in the differen-tial conductance at zero bias. We have also discussed a simple model for a system in which k (0 ≤ k ≤ n ) channels in the loop are absorbed by an ohmic contact, based only on the energyconservation. Here we observed that, as long as the energy is not conserved due to losses intothe contact, the quantum contribution to the conductance is strongly suppressed and onlythe classical contribution survives, showing a non-universal power-law behavior reminiscentof the element of the effective edge-magnetoplasmon scattering matrix. It is worth to notethat the comparison between our analysis and the experiments allows to rule out this simpleand widely accepted model for the inter-edge channel interaction as the proper description ofthe considered set-up. According to this, additional physical effects like dissipation or non-gaussianity could play a major role. However, the developed formalism is general enough toallow estention towards these directions. Acknowledgements
We thanks I. P. Levkivskyi, E. Idrisov, A. Borin and A. Goremykina for useful discussions.We acknowledge the financial support of Swiss National Science Foundation. D. F. wantto acknowledge the support of Grant No. ANR-2010-BLANC-0412 (“1 shot”) and of ANR-2014-BLANC “one shot reloaded” is acknowledged. Part of this work was carried out inthe framework of Labex ARCHIMEDE Grant No. ANR-11-LABX-0033 and of A*MIDEXproject Grant No. ANR- 11-IDEX-0001-02, funded by the “investissements d’avenir” FrenchGovernment program managed by the French National Research Agency (ANR).
A Consideration about the integrals
Aim of this Appendix is to investigate the asymptotic behavior of the functions J ( l ) int , W ( l ) and Y ( l ) int ( l = 1 ,
2) defined in the main text. According to the integral representation f ( η, A ) = Z + ∞ dωω (cid:0) − e − iωη (cid:1) e − ωA = ln (cid:16) i ηA (cid:17) (55)and recalling the Fourier series expansion of h (1) and h (2) in Eqs. (16) and (18) one obtains16 ciPost Physics Submission J (1) int ( z ) = exp ( + ∞ X n =0 ( − n n (cid:20) f (cid:18) z, ω ρ − niτ σ (cid:19) + f (cid:18) z, ω ρ + niτ σ (cid:19)(cid:21) − f (cid:18) z, ω ρ (cid:19)) (56) J (2) int ( z ) = exp ( + ∞ X n =0 ( − n n (cid:20) f (cid:18) z, ω ρ − niτ σ (cid:19) + f (cid:18) z, ω ρ + niτ σ (cid:19)(cid:21) − f (cid:18) z, ω ρ (cid:19)) (57) W (1) = exp ( ∞ X n =0 ( − n n f (cid:18) − τ σ , ω ρ − niτ σ (cid:19)) (58) W (2) = exp ( ∞ X n =0 ( − n n f (cid:18) − τ σ , ω ρ − niτ σ (cid:19)) (59) Y (1) int ( z ) = exp ( + ∞ X n =0 ( − n n (cid:20) f (cid:18) − nτ σ , ω ρ − iz (cid:19) + f (cid:18) nτ σ , ω ρ − iz (cid:19)(cid:21)) (60) Y (2) int ( z ) = exp ( + ∞ X n =0 ( − n n (cid:20) f (cid:18) − nτ σ , ω ρ − iz (cid:19) + f (cid:18) nτ σ , ω ρ − iz (cid:19)(cid:21)) . (61)The above series converge quite fast and are helpful both in the numeric evaluation ofthe integrals for the current and in order to obtain asymptotic expressions under the naturalcondition ω ρ τ σ ≫ A.1 J (1) int and J (2) int In the limit | z | ≫ nτ σ ( n ∈ N ) one directly obtains J (1) int ( | z | → + ∞ ) ≈ ( ω ρ τ σ ) e − γ (62) J (2) int ( | z | → + ∞ ) ≈ ( ω ρ τ σ ) e − γ (63)with γ = 83 X n =1 (cid:18) − (cid:19) n log n ≈ . γ = 245 X n =1 (cid:18) − (cid:19) n log n ≈ . . (65) A.2 W (1) and W (2) In this case there is no dependence on z , but it is still possible to resum the series in anapproximate way obtaining W (1) = ( ω ρ τ σ ) e − i π e − γ (66) W (2) = ( ω ρ τ σ ) e − i π e − γ . (67)Notice that the appearance of a phase factor that will enter directly as a shift of themagnetic flux into the quantum contribution to the current.17 ciPost Physics Submission A.3 Y (1) int and Y (2) int Here, in the limit | z | ≫ nτ σ ( n ∈ N ) these functions reduce to Y (1) int ( | z | → + ∞ ) = Y (2) int ( | z | → + ∞ ) ≈ . (68) References [1] N. Ofek, A. Bid, M. Heiblum, A. Stern, V. Umansky, and D. Nahalu,
Role of interactionsin an electronic Fabry-P´erot interferometer operating in the quantum Hall effect regime ,PNAS , 5276 (2010), doi:10.1073/pnas.0912624107.[2] Yiming Zhang, D. T. McClure, E. M. Levenson-Falk, C. M. Marcus, L. N. Pfeifferand K. W. West,
Distinct signatures for Coulomb blockade and Aharonov-Bohm inter-ference in electronic Fabry-Perot interferometers , Phys. Rev. B , 241304(R) (2009),doi:10.1103/PhysRevB.79.241304.[3] A. Kou, C. Marcus, L. N. Pfeiffer, and K. W. West, Coulomb Oscillations in Antidots inthe Integer and Fractional Quantum Hall Regimes , Phys. Rev. Lett. , 256803 (2012),doi:10.1103/PhysRevLett.108.256803.[4] B. Rosenow and B. I. Halperin,
Influence of Interactions on Flux and Back-Gate Period of Quantum Hall Interferometers , Phys. Rev. Lett. , 106801 (2007),doi:10.1103/PhysRevLett.98.106801.[5] B. I. Halperin, A. Stern, I. Neder, and B. Rosenow, Theory of the Fabry-P´erot quantumHall interferometer , Phys. Rev. B , 155440 (2011), doi:10.1103/PhysRevB.83.155440.[6] H. Choi, I. Sivan, A. Rosenblatt, M. Heiblum, V. Umansky, and D. Mahalu, Robustelectron pairing in the integer quantum hall effect regime , Nature Comm. , 7435 (2015),doi:10.1038/ncomms8435.[7] E. V. Sukhorukov and V. V Cheianov, Resonant Dephasing in the Elec-tronic Mach-Zehnder Interferometer , Phys. Rev. Lett. , 156801 (2007),doi:10.1103/PhysRevLett.99.156801.[8] I. P. Levkivskyi and E. V. Sukhorukov, Dephasing in the electronic Mach-Zehnder interferometer at filling factor ν = 2, Phys. Rev. B , 045322 (2008),doi:10.1103/PhysRevB.78.045322.[9] I. P. Levkivskyi, A. Boyarsky, J. Frohlich, E. V. Sukhorukov, Mach-Zehnder inter-ferometry of fractional quantum Hall edge states , Phys. Rev. B , 045319 (2009),doi:10.1103/PhysRevB.80.045319.[10] P. Degiovanni, Ch. Grenier, G. F`eve, C. Altimiras, H. le Sueur, and F. Pierre, Plasmonscattering approach to energy exchange and high-frequency noise in ν = 2 quantum Halledge channels , Phys. Rev. B , 121302(R) (2010), doi:10.1103/PhysRevB.81.121302.18 ciPost Physics Submission [11] C. Wahl, J. Rech, T. Jonckheere, and T. Martin, Interactions and Charge Fractional-ization in an Electronic Hong-Ou-Mandel Interferometer , Phys. Rev. Lett. , 046802(2014), doi:10.1103/PhysRevLett.112.046802.[12] D. Ferraro, B. Roussel, C. Cabart, E. Thibierge, G. F`eve, C. Grenier, and P. Degiovanni,
Real-Time Decoherence of Landau and Levitov Quasiparticles in Quantum Hall EdgeChannels , Phys. Rev. Lett. , 166403 (2014), doi:10.1103/PhysRevLett.113.166403.[13] C. Grenier, R. Herv´e, E. Bocquillon, F. D. Parmentier, B. Pla¸cais, J.-M. Berroir, G. F`eve,and P. Degiovanni,
Single-electron quantum tomography in quantum Hall edge channels ,New J. Phys. , 093007 (2011), doi:10.1088/1367-2630/13/9/093007.[14] D. Ferraro, A. Feller, A. Ghibaudo, E. Thibierge, E. Bocquillon, G. F`eve, C. Grenier,and P. Degiovanni, Wigner function approach to single electron coherence in quantumHall edge channels , Phys. Rev. B , 205303 (2013), doi:10.1103/PhysRevB.88.205303.[15] E. Bocquillon, V. Freulon, J.-M. Berroir, P. Degiovanni, B. Pla¸cais, A. Cavanna, Y. Jin,and G. F`eve, Separation of neutral and charge modes in one-dimensional chiral edgechannels , Nat. Commun. , 1839 (2013), doi:10.1038/ncomms2788.[16] E. Miranda, Braz. J. Phys. , 3 (2003).[17] T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, Oxford,2004).[18] X. G. Wen,
Topological orders and edge excitations in fractional quantum Hall states ,Adv. Phys. , 405 (1995), doi:10.1080/00018739500101566.[19] A. Braggio, D. Ferraro, M. Carrega, N. Magnoli, and M. Sassetti, Environmental inducedrenormalization effects in quantum Hall edge states due to /f noise and dissipation ,New J. Phys. , 093032 (2012), doi:10.1088/1367-2630/14/9/093032.[20] C. L. Kane and M. P. A. Fisher, Impurity scattering and transport of fractional quantumHall edge states , Phys. Rev. B , 13449 (1995), doi:10.1103/PhysRevB.51.13449.[21] D. Chevallier, J. Rech, T. Jonckheere, C. Wahl, and T. Martin, Poissonian tunnelingthrough an extended impurity in the quantum Hall effect , Phys. Rev. B , 155318 (2010),doi:10.1103/PhysRevB.82.155318.[22] G. Dolcetto, S. Barbarino, D. Ferraro, N. Magnoli, and M. Sassetti, Tunneling be-tween helical edge states through extended contacts , Phys. Rev. B , 195138 (2012),doi:10.1103/PhysRevB.85.195138.[23] G. D. Mahan, Many particle physics (Spingler, New York, 1981).[24] T. Martin,
Les Houches Session LXXXI edited by H. Bouchiat S. Gu´eron, Y. Gefen, G.Montambaux, and J. Dalibard, Elsevier, Amsterdam, 2005.[25] C. de Chamon, D. E. Freed, S. A. Kivelson, S. L. Sondhi, and X. G. Wen,
Twopoint-contact interferometer for quantum Hall systems , Phys. Rev. B , 2331 (1997),doi:10.1103/PhysRevB.55.2331. 19 ciPost Physics Submission [26] D. Ferraro, G. Dolcetto, R. Citro, F. Romeo, and M. Sassetti, Spin current pumping in he-lical Luttinger liquids , Phys. Rev. B , 245419 (2013), doi:10.1103/PhysRevB.87.245419.[27] D. Ferraro, A. Braggio, M. Merlo, N. Magnoli, and M. Sassetti, Relevance of MultipleQuasiparticle Tunneling between Edge States at ν = p/ (2 np + 1), Phys. Rev. Lett. ,166805 (2008), doi:10.1103/PhysRevLett.101.166805.[28] D. Ferraro, A. Braggio, N. Magnoli, and M. Sassetti, Charge tunneling in fractional edgechannels , Phys. Rev. B , 085323 (2010), doi:10.1103/PhysRevB.82.085323.[29] D. Ferraro, A. Braggio, N. Magnoli, and M. Sassetti, Neutral modes’ edge state dynam-ics through quantum point contacts , New J. Phys. , 013012 (2010), doi:10.1088/1367-2630/12/1/013012.[30] M. Carrega, D. Ferraro, A. Braggio, N. Magnoli, and M. Sassetti, Anomalous ChargeTunneling in Fractional Quantum Hall Edge States at a Filling Factor ν = 5 /
2, Phys.Rev. Lett. , 146404 (2011), doi:0.1103/PhysRevLett.107.146404.[31] Y. C. Chung, M. Heiblum, and V. Umansky,
Scattering of Bunched Fractionally ChargedQuasiparticles , Phys. Rev. Lett. , 216804 (2003), doi:10.1103/PhysRevLett.91.216804.[32] A. Bid, N. Ofek, M. Heiblum, V. Umansky, and D. Mahalu, Shot Noise and Charge atthe ν = 2 / Composite Fractional Quantum Hall State , Phys. Rev. Lett. , 236802(2009), doi:10.1103/PhysRevLett.103.236802.[33] M. Dolev, Y. Gross, Y. C. Chung, M. Heiblum, V. Umansky, and D. Mahalu,
De-pendence of the tunneling quasiparticle charge determined via shot noise measure-ments on the tunneling barrier and energetics , Phys. Rev. B , 161303 (2010),doi:10.1103/PhysRevB.81.161303.[34] C. L. Kane and M. P. A. Fisher, Transport in a one-channel Luttinger liquid , Phys. Rev.Lett.68