Relaxation in driven integer quantum Hall edge states
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p Relaxation in driven integer quantum Hall edge states
D. L. Kovrizhin and J. T. Chalker Max Planck Institute for the Physics of Complex Systems, N¨othnitzer str. 38, Dresden, D-01187, Germany and Theoretical Physics, Oxford University, 1, Keble Road, Oxford, OX1 3NP, United Kingdom (Dated: November 9, 2018)A highly non-thermal electron distribution is generated when quantum Hall edge states originating fromsources at different potentials meet at a quantum point contact. The relaxation of this distribution to a stationaryform as a function of distance downstream from the contact has been observed in recent experiments [C. Al-timiras et al.
Phys. Rev. Lett. , 056803 (2010)]. Here we present an exact treatment of a minimal model forthe system at filling factor ν =2 , with results that account well for the observations. PACS numbers: 71.10.Pm, 73.23.-b, 73.43.-f, 42.25.Hz
Introduction.
The importance of understanding non-equilibrium dynamics and relaxation in many-body quantumsystems has been recognised since the early years of quan-tum mechanics [1, 2]. Settings in which such problems are ofhigh current interest include, among others, cold atomic gases[2] and nanoscale electronic devices. [3–14] As a particularexample, recent experiments [3] on quantum Hall (QH) edgestates driven out of equilibrium at a quantum point contact(QPC) provide very detailed information on the approach toa steady state in an electron system that appears to be well-isolated from other degrees of freedom. In this paper we de-scribe the exact solution of a simple model for these experi-ments and compare our results with the measurements.In outline, the experiments we are concerned with [3] in-volve two sets of integer QH edge states, which meet at aQPC. When a bias voltage is applied to the QPC, tunnelingbetween the edge states generates a non-equilibrium electrondistribution. The form of this distribution in energy and itsevolution as a function of distance downstream from the QPCare probed by monitoring the tunneling current from a pointon the edge, through a quantum dot that has an isolated levelof controllable energy. Close to the QPC, the measured distri-bution has two steps, reflecting the different energies of Fermisteps in each of the incident edges. With increasing distancefrom the QPC, the distribution relaxes to a single, broad step.The theoretical challenge presented by these observations isto understand and model this relaxation process.The relationship between these edge state experiments andother recent work on many-body quantum dynamics far fromequilibrium has several aspects worth emphasising. First, inthe context of QH edge states, these are the most recent ofa series of striking observations of non-equilibrium effects ininterferometers [6] and in thermal transport [7]. They are alsothe equivalent for a ballistic system of earlier studies [8] of lo-cal distributions in diffusive wires. Second, the measurementsstand apart from earlier theory [13] and experiment [14] onnon-equilibrium transport between fractional QH edge states,because they probe local distribution functions, rather thanthe global non-linear current-voltage characteristic. Third,and more broadly, the system studied is different in impor-tant ways from quantum impurity problems [9–12], as there isno impurity degree of freedom and interactions are not con- fined to an impurity site but instead operate in the ingoingand outgoing channels. Fourth, there is an analogy betweenedge state relaxation and cold atom experiments in the timedomain [2], since distance from the QPC translates roughly astime, using the edge state velocity as a conversion factor. Inthat sense the experiment we consider, probing relaxation aselectrons propagate, is equivalent to a quantum quench [15],in which time evolution is studied following a sudden changein the Hamiltonian. An important question in this context iswhether a system thermalises at long times. Since integrabil-ity is an obstacle to thermalisation, it is noteworthy that theconventional model [16] of a QH edge state as a chiral Lut-tinger liquid is an integrable one.Any attempt to model theoretically the experiments ofRef. 3 starting from a chiral Luttinger liquid description facesan obvious difficulty, since interactions are most naturally de-scribed in terms of collective modes using bosonization, buttunnelling at the QPC is simple only in terms of fermionicvariables. In pioneering work, two alternative approacheshave been developed: one based on a Boltzmann-like equationfor the electron distribution [17]; and the other using a phe-nomenological model for the plasmon distribution generatedat the QPC [18]. From [18], and from subsequent discussionof a quantum quench in an isolated QH edge (with an initialstate chosen to emulate the effects of a QPC) by the presentauthors [19], a physical picture has emerged, in which relax-ation is seen as a consequence of plasmon dispersion or thepresence at ν = 2 of two plasmon modes with distinct veloc-ities. In summary, an electron that tunnels at the QPC can beviewed as a superposition of plasmons. Dispersion or multipleplasmon velocities cause such a wavepacket to broaden as itpropagates. The lengthscale for relaxation of the electron dis-tribution downstream from the QPC is the distance at whichthe width of this wavepacket is comparable to the character-istic separation between tunnelling electrons. The argumentidentifies relevant scales but does not generate a prediction forthe electron distribution and its dependence on distance fromthe QPC. One of our main aims here is to calculate this centralquantity.An overview of our treatment is as follows: we make keyuse of the fact that, since experiments are at Landau levelfilling factor ν =2 , each QH edge carries two co-propagatingchannels. Interactions mix excitations in different channels,and in the minimal model eigenmodes are charactised us-ing two velocities. Our approach combines refermionisation[20, 21] with non-equilibrium bosonization methods [22–26]:by first bosonizing, then recombining bosons to form a newset of fermions, we transform the Hamiltonian for the inter-acting system into one for free particles. Under this trans-formation, the measured distribution (or more accurately, thespectral function probed by the tunnelling conductance) is ex-pressed as a free-fermion determinant. This determinant hasa form similar to that appearing in the the theory [27, 28] offull counting statistics [FCS], and related quantities appear ina variety of other non-equilibrium problems [29]. Its numeri-cal evaluation can be done accurately and efficiently, yieldingthe results we present below. In addition, simple analytical ex-pressions can be found for some quantities, and we give onefor the total energy in each channel as a function of distancefrom the QPC, thus providing a Hamiltonian derivation of aresult obtained previously in the phenomenological treatmentof Ref. 18.The work we set out here is related in a variety of ways toother studies of devices built from integer QH edge states. Inparticular, experiments that revealed striking non-equilibriumeffects in Mach Zehnder interferometers [6] have stimulatedextensive theoretical research, [25, 30–37] including calcu-lations [33] of interferometer dephasing based on the samebosonized model [16] for edge states at ν = 2 with contactinteractions that we adopt in the following. In the contextof edge state relaxation a parallel development to the presentpaper, described in Ref. 26, is based on an approximation (in-volving a factorisation of bosonic correlators) that is expectedto be accurate sufficiently far from the QPC. Our solution re-covers such a factorisation, but only at distances large com-pared to the relaxation length for the model. Model.
The experimental system is illustrated in the upperpanels of Figs. 1 and 2: two alternative geometries arise, ac-cording to whether the tunnelling conductance is measured inthe same or the opposite channel to that coupled by the QPC.We take a Hamiltonian with kinetic, interaction and tun-neling terms, in the form ˆ H = ˆ H kin + ˆ H int + ˆ H tun . Usingthe labels η = 1 , to distinguish edges according to theirsource, and s = ↑ , ↓ to differentiate between the two channelson a given edge, the fermion creation operator at point x forchannel η, s is ˆ ψ † ηs ( x ) . It obeys the standard anticommutationrelation { ˆ ψ † ηs ( x ) ˆ ψ η ′ s ′ ( x ′ ) } = δ ss ′ δ ηη ′ δ ( x − x ′ ) . The densityoperator is ˆ ρ ηs ( x ) = ˆ ψ † ηs ( x ) ˆ ψ ηs ( x ) . Taking all four channels η, s to have the same bare velocity v , and assuming a contactinteraction of strength g between electrons in different chan-nels on the same edge, we have [16] ˆ H kin = − i ~ v X η,s Z ˆ ψ † ηs ( x ) ∂ x ˆ ψ ηs ( x ) d x (1)and ˆ H int = 2 π ~ g X η Z ˆ ρ η ↑ ( x )ˆ ρ η ↓ ( x ) d x . (2) eV d En(E) -0.5 0 0.5V G (mV)-2024 ( d I QD / d V G ) / I QD m a x ( m V - ) µ m2.2 µ m4 µ m10 µ m30 µ m
11 22
FIG. 1: (Color online) Top: sketch of the experimental geometryin which the tunnelling conductance is measured in the same chan-nel as that coupled by the QPC. Bottom: differential conductancecalculated (lines) and measured [3] (symbols) in this geometry at in-dicated distances from QPC. Fits use T = 44 mK , V = 30 . µ V ( βeV = 8) , and p = 0 . . Black lines: calculations (full) and data(dashed) for V =0 . Note that short-range intrachannel interactions can simply beabsorbed into the value of v . Tunneling with amplitude t QPC at the QPC between channels ↓ and ↓ is described by ˆ H tun = t QPC ˆ ψ † ↓ (0) ˆ ψ ↓ (0) + h . c . . (3)A bias voltage V generates a chemical potential difference eV between incident electrons on edge 1 and those on edge 2.The observable of interest is the tunnelling conductance asa function of energy E and distance d > from the QPC, inchannel , s . This is the Fourier transform of the correlator G s ( d, τ ) = h e i ˆ Hτ/ ~ ˆ ψ † s ( d ) e − i ˆ Hτ/ ~ ˆ ψ s ( d ) i , (4)where the average is taken in the non-equilibrium steady state,with s = ↓ in the geometry of Fig. 1 and s = ↑ in that of Fig. 2.The tunnelling conductance is determined by a measurementof the current I QD ( E ) through a quantum dot with a singlelevel at energy E weakly coupled to the channel. With tun-nelling amplitude t D to the dot, this is [19] I QD ( E ) = e | t D | ~ Z G s ( d, τ ) e − iEτ/ ~ dτ. (5) Results.
We show in [38] that the Hamiltonian ˆ H can bebrought into a free particle form by introducing transformedfermion operators ˆΨ α ( x ) with index α = A ± , S ± and eigen-mode velocities v α ≡ v ± = v ± g . Using these new operators ˆ H = − i ~ X α v α Z ˆΨ † α ∂ x ˆΨ α dx − [ t QPC ˆΨ † A + (0) ˆΨ A − (0) + h . c . ] . (6)Moreover the tunnelling conductance is obtained from G s ( d, τ ) ∝ h e − iπ [ ± ˆ N A − ( d,τ )+ ˆ N A + ( d,τ )] i , (7)where upper sign is taken in the first term of the exponent for s = ↑ and the lower sign for s = ↓ . This expectation value istaken in the stationary scattering state of ˆ H specified by thetemperature and chemical potentials of incident channels, andthe operators ˆ N α ( d, τ ) = Z d + v ± τd ˆΨ † α ( y ) ˆΨ α ( y ) dy (8)count electrons within the spatial intervals d ≤ y ≤ d + v ± τ on the channels A ± .Equations (6) - (8) form the central results of this paper:they express the observable of interest in an interacting, non-equilibrium system, in terms of the expectation value of asingle-particle operator, evaluated as an average in a station-ary scattering state of a single-particle Hamiltonian. We nowuse these equations to discuss the physics of relaxation in thissystem and to calculate the tunnelling conductance.A simple physical picture of the relaxation process, and anidentification of relevant length scales, follows from the formof Eq. (7). This picture has the same content as our discussion(above and in Ref. 19) of relaxation arising from two plasmonvelocities, but is phrased in terms of the transformed fermions.Note first that the operators ˆ N ± ( d, τ ) count fermions that passthrough the QPC in two time windows, both of duration τ , butwith a relative delay of d/v eff , where /v eff = 1 /v − − /v + = 2 g/ ( v + g ) . (9)At d = 0 the two windows exactly overlap. Then, for ex-ample, G ↑ (0 , τ ) acquires the same contribution from eachfermion inside the window, regardless of tunneling. In con-sequence, and as expected, the electron distribution at d = 0 in the ↑ channel is unaffected by tunneling, since the QPCcouples ↓ channels. (Conversely, it is clear that G ↓ ( d, τ ) isaffected by tunnelling even for d → , since it depends onthe difference and not the sum of the fermion numbers inthe two windows). Far downstream from the QPC, by con-trast, d/v eff ≫ τ , the two windows are widely separated intime, and the fermion numbers in each are uncorrelated. Inthis case, the right side of Eq. (7) is the product of indepen-dent factors, each with the form of a FCS generating function[27, 28]. At large d the tunnelling conductance is the samein both channels. It matches exactly the result obtained previ-ously [19] by considering a quantum quench, with the impor-tant consequence that even the limiting distribution far fromthe QPC is non-thermal, although deviations from a Fermi-Dirac form are small. [19] The velocity v eff can be combinedwith voltage V or temperature T to define two lengths: l V = ~ v eff /e | V | and l T = ~ v eff / πk B T. (10)These set the scale for relaxation, and diverge in the non-interacting limit. eV d En(E) -0.5 0 0.5V G (mV)-4-202 ( d I QD / d V G ) / I QD m a x ( m V - ) µ m4 µ m10 µ m FIG. 2: (Color online) Top: sketch of the experimental geometry inwhich the tunneling conductance is measured in the opposite channelto that coupled by the QPC. Bottom: differential conductance calcu-lated (lines) and measured [3] (symbols) in this geometry at indicateddistances from the QPC. Fits use T = 44 mK , V = 45 . µ V , and p = 0 . . Black lines: calculations (full) and data (dashed) for V = 0 . Detailed comparison of our results with the measurementsof Ref. 3 requires an evaluation of the tunnelling conductanceat general d . For this we compute G s ( d, τ ) from Eq. (7) assummarised in [38]. The outcome is shown for the two al-ternative geometries in Figs. 1 and 2. The theoretical curvesmatch the data well, except for the smallest value of d inFig. 1. Four parameters enter the calculations: v eff , T , V andthe tunneling probability p at the QPC. In addition, one furtherparameter is needed to set the energy scale for the data: thegate voltage-to-energy lever arm η G (defined in Ref. 3, sup-plemental material). Of these parameters, only v eff is uncon-strained by independent measurements. We arrive at the fits inFigs. 1 and 2 as follows. We fix η G = 0 . and p ≃ . (con-sistent with the values determined in Refs. 3 and 4) and obtainthe bath temperature T = 44 mK (close to the experimentalvalue of T = 40 mK) by fitting data at zero bias voltage to athermal Fermi-Dirac distribution. We then compare numeri-cal results at large d with data measured in the ↓ channel at themaximum distance from the QPC ( µ m ). We obtain a goodfit at βeV = 8 , and use this value for computations shownin Fig. 1 at all other d . Finally, we fix the interaction scale v eff by fitting data at one intermediate distance ( µ m ). Thetheoretical predictions at other distances are then fully deter-mined. (Our specific value for p [ . rather than . ] issignificant for the fit only at intermediate d and small energy.)Data for the geometry of Fig. 2 were taken using a differentbias voltage from that in Fig. 1 and we calculate the theoreticalcurves shown in Fig. 2 using the same values of η G , T and v eff as for Fig. 1, but scaling the value of βeV by the ratio of ex-perimental voltages. Both values of βeV are 25% lower thanthe experimental ones, and this discrepancy is consistent withthe ‘missing energy’ reported in Ref. 3. Possible interpreta-tions of this discrepancy include loss of energy at the QPCor the existence of additional, neutral edge modes,[3, 17, 18]or invoke [19, 26] the distinction in an interacting system be-tween the measured spectral function and the electron distri-bution in energy. The fact that we obtain good fits to all dataexcept that in Fig. 1 for the smallest d suggests that, if energyis lost from the edge modes in the experimental system, thisprobably happens very close to the QPC. The most significantoutcome from this fitting process is a value for the interactionparameter: v eff = 6 . × ms − . A roughly comparableresult ( v eff = 10 ms − ) was obtained from the same data inRef.18, but we believe our fitting procedure is more precise.Both determinations are in the range measured for edge statevelocities in gated samples.[3]An additional quantity of interest is the energy density ε s ( d ) in channel s : it characterises with a single value theelectron distribution [3]. Moreover, we can evaluate it ex-plicitly from Eq. (7) because (see Ref. 19) it is proportionalto ∂ τ G ± ( d, τ ) | τ =0 and G ± ( d, τ ) has an expansion for small τ in cumulants of ˆ N ± ( d, τ ) . Interactions cause energy ex-change between channels, so that ε s ( d ) evolves with d . Weobtain ε ± ( d ) = ε T + ε V (cid:20) ∓ l l sin d/ l V sinh d/ l T (cid:21) . (11)Here ε V ≡ p (1 − p ) eV / π ~ v is the excess energy in a singlechannel due to a double-step electron distribution generatedby a QPC with tunneling probability p . Eq. (11), derived usinga different approach in Ref. 18, shows that the energy densitiesinterpolate between values just after the QPC of ε − (0) = ε T + ε V and ε + (0) = ε T , and values at large d of ε ± ( d ) = ε T + ε V / . The relaxation length is set by the smaller of l T and l V . This relaxation is oscillatory, but oscillations are stronglysuppressed if eV ≪ k B T .Details of calculations are given in Supplementary Material[38]. We thank F. Pierre for extensive discussions and for pro-viding access to the data of Ref. 3. The work was supportedin part by EPSRC grant EP/I032487/1. [1] J. von Neumann, Z. Phys. , 30 (1929), available in a translationby R. Tumulka as: European Phys. J. H , 201 (2010).[2] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore,Rev. Mod. Phys. , 863 (2011).[3] H. le Sueur, C. Altimiras, U. Gennser, A. Cavanna, D. Mailly,F. Pierre, Phys. Rev. Lett. , 056803 (2010).[4] C. Altimiras, H. le Sueur, U. Gennser, A. Cavanna, D. Maillyand F. Pierre, Nature Physics , 34 (2009).[5] C. Altimiras, H. le Sueur, U. Gennser, A. Cavanna, D. Mailly,F. Pierre, Phys. Rev. Lett. , 226804 (2010). [6] I. Neder, M. Heiblum, Y. Levinson, D. Mahalu, and V. Umansky,Phys. Rev. Lett. , 016804, (2006).[7] G. Granger, J. P. Eisenstein, and J. L. Reno, Phys. Rev. Lett. ,086803 (2009).[8] H. Pothier, S. Guˇzron, N. O. Birge, D. Esteve, and M. H. Devoret,Phys. Rev. Lett. , 3490-3493 (1997).[9] Y. Meir, N. S. Wingreen, and P. A. Lee, Phys. Rev. Lett. , 2601(1993).[10] R. M. Konik, H. Saleur, and A. Ludwig, Phys. Rev. B ,125304 (2002).[11] P. Mehta and N. Andrei, Phys. Rev. Lett. , 216802 (2006).[12] S. De Franceschi et al., Phys. Rev. Lett. , 156801 (2002).[13] P. Fendley, A. W. W. Ludwig, and H. Saleur Phys. Rev. Lett. , 3005 (1995).[14] A. M. Chang, Rev. Mod. Phys. , 1449 (2003).[15] P. Calabrese and J. Cardy, Phys. Rev. Lett. , 136801 (2006).[16] X. G. Wen, Phys. Rev. Lett. , 2206 (1990); Phys. Rev. B ,11025 (1991).[17] A. M. Lunde, S. E. Nigg, and M. B¨uttiker, Phys. Rev. B ,041311(R) (2010).[18] P. Degiovanni, C. Grenier, G. F`eve, C. Altimiras, H. le Sueur,F. Pierre, Phys. Rev. B , 121302(R) (2010).[19] D. L. Kovrizhin and J. T. Chalker, Phys. Rev. B , 085105(2011).[20] M. Fabrizio and A. Parola, Phys. Rev. Lett. , 226 (1993).[21] J. von Delft, G. Zarand and M. Fabrizio, Phys Rev. Lett, ,196 (1998); G. Zarand and J. von Delft, Phys. Rev. B, , 6918(2000); arXiv:cond-mat/9812182.[22] M.A. Cazalilla, Phys. Rev. Lett. , 156403 (2006).[23] D. B. Gutman, Y. Gefen, and A. D. Mirlin, Phys. Rev. Lett. ,126802 (2008); Phys. Rev. B , 085436 (2010).[24] I. P. Levkivskyi and E. V. Sukhorukov, Phys. Rev. Lett. ,036801 (2009).[25] D. L. Kovrizhin and J. T. Chalker, Phys. Rev. B , 161306(R)(2009); Phys. Rev. B , 155318 (2010).[26] I. P. Levkivskyi and E. V. Sukhorukov, Phys. Rev. B , 075309(2012).[27] L. S. Levitov and G. B. Lesovik, JETP Lett. , 230 (1993); L.S. Levitov in Quantum Noise in Mesoscopic Systems , ed. Yu. V.Nazarov (Kluwer, Amsterdam, 2003); I. Klich ibid. [28] D. A. Abanin and L. S. Levitov, Phys. Rev. Lett. , 186803(2005).[29] D. B. Gutman, Yuval Gefen and A. D. Mirlin, J. Phys. A , 165003 (2011); A. Abanov, D. A. Ivanov, and Y. Qian,arXiv:1108.1355.[30] I. Neder, F. Marquardt, M. Heiblum, D. Mahalu and V. Uman-sky, Nature Physics, , 534 (2007).[31] J. T. Chalker, Y. Gefen, and M. Y. Veillette, Phys. Rev. B ,085320 (2007).[32] E. V. Sukhorukov and V. V. Cheianov, Phys. Rev. Lett. ,156801 (2007).[33] I. P. Levkivskyi and E. V. Sukhorukov, Phys. Rev. B 78, 045322(2008).[34] I. Neder and E. Ginossar, Phys. Rev. Lett. , 196806 (2008).[35] S.-C. Youn, H.-W. Lee, and H.-S. Sim, Phys. Rev. Lett. ,075401 (2011).[37] I. Neder, Phys. Rev. Lett. , 186404 (2012).[38] Supplementary Material. SUPPLEMENTARY MATERIALMODEL
We take a Hamiltonian with kinetic, interaction and tun-nelling terms, in the form ˆ H = ˆ H kin + ˆ H int + ˆ H tun . Us-ing the labels η = 1 , to distinguish edges according totheir source, and s = ↑ , ↓ to differentiate between the twochannels on a given edge, the fermion creation operator atpoint x for channel η, s is ˆ ψ † ηs ( x ) . The density operator is ˆ ρ ηs ( x ) = ˆ ψ † ηs ( x ) ˆ ψ ηs ( x ) . Taking all four channels η, s to havethe same bare velocity v , and assuming a contact interactionof strength g between electrons in different channels on thesame edge, we have [1] ˆ H kin = − i ~ v X η,s Z ˆ ψ † ηs ( x ) ∂ x ˆ ψ ηs ( x ) d x (12)and ˆ H int = 2 π ~ g X η Z ˆ ρ η ↑ ( x )ˆ ρ η ↓ ( x ) d x . (13)Tunneling with amplitude t QPC at the QPC between channels ↓ and ↓ is described by ˆ H tun = t QPC ˆ ψ † ↓ (0) ˆ ψ ↓ (0) + h . c . . (14)A bias voltage V generates a chemical potential difference eV between incident electrons on edge 1 and those on edge 2.The observable of interest is the tunnelling conductance asa function of energy E and distance d > from the QPC, inchannel , s . This is the Fourier transform of the correlator G s ( d, τ ) = h e i ˆ Hτ/ ~ ˆ ψ † s ( d ) e − i ˆ Hτ/ ~ ˆ ψ s ( d ) i , (15)where the average is taken in the non-equilibrium steady state.The tunnelling conductance is determined by a measurementof the current I QD ( E ) through a quantum dot with a singlelevel at energy E weakly coupled to the channel. With tun-nelling amplitude t D to the dot, this is [2] I QD ( E ) = e | t D | ~ Z G s ( d, τ ) e − iEτ/ ~ dτ. (16) REFERMIONIZATION
The steps required to derive Eqns. (6) - (8) of the main textare as follows. First we bosonize the Hamiltonian in the stan-dard way [3], introducing bosonic fields ˆ φ ηs ( x ) with com-mutation relations [ ˆ φ ηs ( x ) , ∂ y ˆ φ ηs ′ ( x ′ )] = − πiδ ηη ′ δ ss ′ δ ( x − x ′ ) , and Klein factors ˆ F ηs , and representing the fermion oper-ators via the relation ˆ ψ ηs ( x ) = (2 πa ) − / ˆ F ηs e i πL ˆ N ηs e − i ˆ φ ηs ( x ) , (17) where, as usual, a is a short-distance cut-off, ˆ N ηs is thefermion number operator, and L is the length of the edge.After bosonization, the combination ˆ H kin + ˆ H int is diag-onalized by a rotation to new bosonic fields ˆ χ α , where ( ˆ χ S + ˆ χ A − ˆ χ A + ˆ χ S − ) T = U ( ˆ φ ↑ ˆ φ ↓ ˆ φ ↓ ˆ φ ↑ ) T and U = 12 − −
11 1 − − − − . (18)Next we use the fields ˆ χ α to define new fermion operators[4,5] ˆΨ α . Crucially, besides diagonalising ˆ H kin + ˆ H int , the trans-formations ensure that ˆ H tun remains a single-particle operatorwhen expressed in terms of ˆΨ α . Specifically, after bosoniza-tion ˆ H tun = t QPC ˆ F † ↓ ˆ F ↓ e i [ ˆ φ ↓ (0) − ˆ φ ↓ (0)] + h . c ., (19)while under rotation ˆ φ ↓ (0) − ˆ φ ↓ (0) = ˆ χ A + (0) − ˆ χ A − (0) .Since ˆ χ A ± (0) appear in this expression with unit coefficients,we can introduce new Klein factors ˆ F α and new fermion fields Ψ α ∼ ˆ F α e − i ˆ χ α to obtain the expression for ˆ H displayedin Eq. (6) of the main text. The required transformation ofKlein factors has been given previously in the context of atwo-channel Kondo model:[5] since fermion number opera-tors should transform following Eq. (18), we require ˆ F † S − ˆ F † A − = ˆ F † ↑ ˆ F ↓ , ˆ F S − ˆ F † A − = ˆ F † ↓ ˆ F ↑ , ˆ F † S − ˆ F † A + = ˆ F † ↑ ˆ F ↓ , ˆ F † S + ˆ F † A − = ˆ F † ↑ ˆ F † ↓ . (20)This implies that the combination appearing in ˆ H tun has thetransformation ˆ F † ↓ ˆ F ↓ = − ˆ F † A + ˆ F A − . Note that a unit changein the occupation number of one of the new fermions resultsin changes of one half for the occupation numbers of the orig-inal fermions. Physical states in the new basis must thereforesatisfy certain selection rules, to ensure that fermion occupa-tion numbers in the original basis are integer. These selectionrules are set out in Ref. 5. For our purposes the key point isthat ˆ H tun does not connect the physical and unphysical sec-tors, because the new fermion operators appear in it in pairs.We can also transform in this way the operators that are re-quired to generate an incident state with a density differencebetween channels. Since ˆ F † ↑ ˆ F † ↓ = ˆ F † A + ˆ F † S + and ˆ F † ↓ ˆ F † ↑ =ˆ F A + ˆ F † S + , a bias voltage V between channels and is rep-resented by setting the chemical potential to be eV in channel A + and zero in channels A − and S ± .The task now is to evaluate the correlation function G s ( d, t ) . For this it is convenient to work in the interac-tion representation, with ˆ H tun as the ‘interaction’ and ˆ H ≡ ˆ H kin + ˆ H int , using a superscript I to indicate operators in thisrepresentation: A I ( t ) = e i ˆ H t Ae − i ˆ H t . Time evolution ofthe new bosonic and fermionic fields fields is simple in thispicture: we have ˆ χ IA ± ( x, t ) = ˆ χ A ± ( x − v ± t ) , ˆΨ IA ± ( x, t ) = ˆΨ A ± ( x − v ± t ) , and similarly for ˆ χ IS ± ( x, t ) and ˆ ψ IS ± ( x, t ) .The time evolution operator in the interaction representa-tion is ˆ S I ( t ) = exp (cid:20) − i ~ Z t −∞ ˆ H I tun ( τ )d τ (cid:21) , (21)(the usual time-ordering is not required here because [ ˆ H I tun ( t ) , ˆ H I tun ( t )] = 0 for all t and t ). Scattering statesare generated by the action of ˆ S I ( t ) : Eq. (15) takes the form G s ( d, τ ) = h Q i , where Q ≡ [ ˆ S I ( τ )] † [ ˆ ψ I s ( d, τ )] † ˆ S I ( τ )[ ˆ S I (0)] † ˆ ψ I s ( d,
0) ˆ S I (0) and h . . . i denotes a conventional thermal average, withHamiltonian ˆ H and chemical potentials µ and µ on the twoedges, defined by h . . . i = Z − Tr n e − β ( ˆ H − µ ˆ N − µ ˆ N ) . . . o , (22)where Z = Tr { e − β ( ˆ H − µ ˆ N − µ ˆ N ) } and ˆ N ≡ ˆ N ↑ + ˆ N ↓ ,the number operator for edge 1 (and correspondingly for ˆ N ).We now describe how the quantity Q may be simplified.First, by a straightforward though lengthy calculation one canshow that [ ˆ H I tun ( t ) , ˆ ψ I s ( d, t )] = 0 for t / ∈ [ t − d/v + , t − d/v − ] . Next, we insert [ ˆ S I ( ∞ )] † ˆ S I ( ∞ ) between the factors of ˆ S I ( τ ) and [ ˆ S I (0)] † in our expression for Q . We then commute ˆ S I ( τ )[ ˆ S I ( ∞ )] † to the left, and ˆ S I ( ∞ )[ ˆ S I (0)] † to the right,to obtain Q = [ ˆ S I ( ∞ )] † [ ˆ ψ I s ( d, τ )] † ˆ ψ I s ( d,
0) ˆ S I ( ∞ ) . (23)Bosonizing using Eq. (17) we have [ ˆ ψ I s ( d, τ )] † ˆ ψ I s ( d,
0) = (2 πa ) − e i ˆ φ I s ( d,τ ) e − i πL ˆ N I s ( τ ) × [ ˆ F I s ( τ )] † ˆ F I s (0) e i πL ˆ N I s (0) e − iφ I s ( d, . Rotating to the new fields and using ˆ χ Iα ( d, − ˆ χ Iα ( d, τ ) + 2 πL [ ˆ N Iα ( τ ) − ˆ N Iα (0)] = 2 π ˆ N α ( d, τ ) with ˆ N α ( d, τ ) defined in Eq. (8) of the main text, and setting µ = 0 so that ˆ F I s ( τ ) = ˆ F s , we find [ ˆ ψ I s ( d, τ )] † ˆ ψ I s ( d,
0) =(2 πa ) − e − iπ [( N A + ( d,τ ) ±N A − ( d,τ )) − ( N S + ( d,τ ) ±N S − ( d,τ )] where the signs ± correspond to the choices s = ↑ , ↓ . Since ˆ S I ( t ) transforms only the channels A ± and not S ± , we canextract a normalisation factor to arrive at G s ( d, τ ) = G ( τ ) h e − iπ [ N A + ( d,τ ) ±N A − ( d,τ )] ih e − iπ [ N A + ( d,τ ) ±N A − ( d,τ )] i (24)where h . . . i ≡ h [ ˆ S I ( ∞ )] † . . . ˆ S I ( ∞ ) i and G ( τ ) ≡ h ˆ ψ † s ( d, τ ) ˆ ψ s ( d, i = i β ~ ( v + v − ) × [ πβ ~ v + ( − v + τ + ia )] × [ πβ ~ v − ( − v − τ + ia )] . (25)A final step is to consider the effect of the evolution opera-tors [ ˆ S I ( ∞ )] † and ˆ S I ( ∞ ) on the basis states in which the ex-pectation value is calculated. As these states are generated bythe action of the operators ˆΨ A ± ( x ) on the vacuum, we mustfind how these operators transform. We do this by solving theSchr¨odinger equation with the Hamiltonian of Eq. (6) of themain text. The solution involves the scattering amplitudes atthe QPC: introducing θ = t QPC / ~ √ v + v − , the tunnelling andreflection probabilities are sin θ and cos θ , while [ ˆ S I ( ∞ )] † ˆΨ A + ( x ) ˆ S I ( ∞ ) =cos θ ˆΨ A + ( x ) − i sin θ [ v − /v + ] / ˆΨ † A − ( v − x/v + ) (26)and [ ˆ S I ( ∞ )] † ˆΨ A − ( x ) ˆ S I ( ∞ ) =cos θ ˆΨ A − ( x ) − i sin θ [ v + /v − ] / ˆΨ † A + ( v + x/v − ) . (27)Hence the average h . . . i in a scattering state is evaluated bycombining the rotation between channels defined in Eqns. (26)and (27) with the thermal average h . . . i . NUMERICAL EVALUATION OF THE TUNNELLINGCONDUCTANCE
In the following we outline the methods we use for numer-ical evaluation of the correlation function defined in Eq. (7)of the main text, and hence the differential tunnelling conduc-tance shown in Figs. 1 and 2 of the main text. The centralproblem is to evaluate the normalised expectation value ap-pearing on the right hand side of Eq. (24), which has the form h X i norm ≡ h X i / h X i (28)with X = e − iπ [ N A + ( d,τ ) ±N A − ( d,τ )] . In the limit of large dis-tance d from the QPC, contributions to the two time windowsappearing in the exponent in X are independent and it is con-venient to use the approach described in Ref. 6. At finite d there is no such factorisation and we resort instead to a methodsimilar to one described in Ref.2, appropriately modified totake account of finite temperature.(i) Long distance limit.
At large d the expectation valuefactorizes into a product of two functions, each of which countnumber of particles in a fixed time window, so that h e − iπ [ N A + ( d,τ ) ±N A − ( d,τ )] i norm = χ − ( ∓ π, τ ) χ + ( − π, τ ) with χ ± ( δ, τ ) = h e iδ ˆ N ± ( d,τ ) i norm . (29)The functions χ ± ( δ, τ ) have a FCS form and can be writtenin terms of determinants for non-interacting fermions. Onehas[7] h e iδ ˆ N ± ( d,τ ) i = det { [1 − ˆ P ( e − iδ − n ± ( ε ) ˆ P ] } . (30)Here n ± ( ε ) is the corresponding electron energy distributionin a given channel at finite temperature and bias voltage afterthe action of ˆ S I ( ∞ ) (i.e. a double-step) while ˆ P is a projec-tion operator that is diagonal in the time domain, having theaction on a time-dependent function y ( t ) : ˆ P y ( t ) = y ( t ) if t ∈ [0 , τ ] and ˆ P y ( t ) = 0 otherwise. Using the regularizationprocedure proposed in Ref. 6 we can write the determinant(30) in the form h e iδ ˆ N ± ( τ ) i = det[ f ( t i − t j )] , (31)where the function f ( t ) is defined as a Fourier transform of ˜ f ( ε ) = [1 − n ± ( ε )( e − iδ − e − i δ ε Λ , (32)which is periodic in the domain [ − Λ , Λ] , with Λ a high energycutoff on single-particle states and t j = jπ/ Λ . Note that thephase factor e − i δ ε Λ appearing here is crucial, since without itone would wrongly obtain a result periodic in δ .The normalisation h e iδ ˆ N ± ( d,τ ) i is obtained from similarexpressions in which the non-equilibrium distribution n ± ( ε ) is replaced by a thermal one, and in practice we evaluate di-rectly the ratio χ ± ( δ, τ ) . We check numerical convergenceby changing the value of the cutoff Λ . The size of the matri-ces required grows linearly with τ , the largest necessary being × . Some results obtained using this approach areshown in Figs. 3 and 4. The numerical calculations can betested at large τ by comparison with the asymptotic analyticresults derived in Ref. 6: as shown in Fig. 3 the agreementis excellent. An incidental by-product of our calculations isthe discovery of an interesting new feature of the asymptoticbehaviour. According to Ref. 6 this is exponential in τ , witha rate that diverges for δ = π and p = 1 / . In fact we findnumerically that decay at these parameter values is not expo-nential but Gaussian in τ . It would be interesting to search foran analytical derivation of that form.(ii) Finite distance.
At finite distance d the correlation func-tion G s ( d, τ ) is not simply a product of independent contribu-tions from the two channels A ± , and so cannot be expressed eV τ/ π (cid:22) hv | χ ± ( π , τ ) | FIG. 3: The quantity | χ ± ( π, τ ) | [see Eq. (29)] as a function of time τ for different values of parameter βeV at tunneling probability p =0 . Thick solid line: βeV = 2 ; dashed line: βeV = 5 ; dot-dashedline: βeV = 10 ; circles: βeV = 100 . Thin solid line correspondsto asymptotic behaviour at large βeV , which is represented withinnumerical errors by the function exp[ − π τ ( k B T ds ) / ~ ] . eV τ/ π (cid:22) hv L o g | χ ± ( π , τ ) | FIG. 4: (Color online). Comparison of the asymptotic behaviour of | χ ± ( π, τ ) | [see Eq. (29)] from Ref.6, (solid lines) with results ob-tained numerically using Eq. (31) (dashed lines), at tunneling proba-bility p = 0 . for different values of δ . From top right to bottom left:red lines δ = π/ , green lines ( δ = 3 π/ ), blue lines ( δ = π − . )and violet lines ( δ = π − . ). in terms of quantities familiar from the theory of FCS. We re-quire instead a different numerical approach. Using Levitov’sdeterminant formula [7], we have h [ ˆ S I ( ∞ )] † e − iπ [ N A + ( d,τ ) ±N A − ( d,τ )] ˆ S I ( ∞ ) i = det[1 − n ( ε )[[ ˆ S ( ∞ )] † e − iπ [ ± ˆ N A − ( d,τ )+ ˆ N A + ( d,τ )] ˆ S I ( ∞ ) − . Here the determinant is in the two-channel Fock space and n ( ε ) is a Fermi-Dirac distribution in each channel, but withtwo distinct chemical potentials. To evaluate this determinantwe express ˆ N A ± ( d, τ ) as bilinears in the fermion creationand annihilation operators ˆΨ † A ± ( x ) and ˆΨ A ± ( x ) . We eval-uate this determinant in a basis of eigenstates of ˆ H kin , con- eV τ/ π (cid:22) hv h e − i π [ N A + ( d , τ ) − N A − ( d , τ ) ] i n o r m FIG. 5: Time dependence of the normalised correlator h e − iπ [ N A + ( d,τ ) −N A − ( d,τ )] i norm for the same channel as thatcoupled by the QPC. Curves are for tunnelling probability p = 1 / , eV /k B T = 5 and different distances d from the QPC: (thin solidline) d/l v = 0 ; (dashed line) d/l v = 2 . ; (dotted line) d/l v = 5 ;(dot-dashed line) d/l v = 10 ; and (thick solid line) x/l v = 20 . sidering edges of finite length with periodic boundary condi-tions and imposing energy cut-offs to obtain a matrix of finitesize. For adequate convergence with increasing system size,we find that it is necessary to scale the lengths L ± of the twochannels according to the velocity, setting v + /L + = v − /L − .With this choice, a basis of a few thousand states is sufficientto obtain the results we present here.We show representative results for the normalised correla-tor h e − iπ [ N A + ( d,τ ) ±N A − ( d,τ )] i norm at finite d in Figs. 5 and6. The most distinctive feature is an oscillatory time depen-dence of the correlator in the channel that is coupled by theQPC, which at d = 0 and T = 0 can be obtained exactly as h e − iπ [ N A + (0 ,τ ) ±N A − (0 ,τ )] i norm = cos( eV τ / π ~ ) . h e − i π [ N A + ( d , τ ) + N A − ( d , τ ) ] i n o r m eV τ/ π (cid:22) hv FIG. 6: Time dependence of the normalised correlator h e − iπ [ N A + ( d,τ )+ N A − ( d,τ )] i norm for the channel not coupledby the QPC. Other parameters as for Fig. 5. [1] X. G. Wen, Phys. Rev. Lett. , 2206 (1990); Phys. Rev. B ,11025 (1991).[2] D. L. Kovrizhin and J. T. Chalker, Phys. Rev. B , 085105(2011).[3] See: J. von Delft and H. Schoeller, Annalen Phys.
225 (1998);T. Giamarchi,
Quantum Physics in One Dimension (OxfordUniv. Press, Oxford, 2004).[4] M. Fabrizio and A. Parola, Phys. Rev. Lett. , 226 (1993).[5] J. von Delft, G. Zarand and M. Fabrizio, Phys Rev. Lett, ,196 (1998); G. Zarand and J. von Delft, Phys. Rev. B, , 6918(2000); arXiv:cond-mat/9812182.[6] D. B. Gutman, Yuval Gefen and A. D. Mirlin, J. Phys. A , 165003 (2011); A. Abanov, D. A. Ivanov, and Y. Qian,arXiv:1108.1355.[7] L. S. Levitov and G. B. Lesovik, JETP Lett. , 230 (1993); L.S. Levitov in Quantum Noise in Mesoscopic Systems , ed. Yu. V.Nazarov (Kluwer, Amsterdam, 2003); I. Klich, ed. Yu. V.Nazarov (Kluwer, Amsterdam, 2003); I. Klich