Non-equilibrium noise in transport across a tunneling contact between $\nu = 2/3$ fractional quantum Hall edges

In a recent experimental paper [1] a qualitative confirmation of the existence of upstream neutral modes at $\nu = 2/3$ quantum Hall edge was reported. Using the chiral Luttinger liquid theory of quantum Hall edge we develop a quantitative model of the experiment [1]. A good quantitative agreement of our theory with the experimental data reinforces the conclusion of existence of the upstream neutral mode. Our model also enables us to extract important quantitative information about non-equilibrium processes in Ohmic and tunneling contacts from the experimental data. In particular, for $\nu = 2/3$, we find a power-law dependence of the neutral mode temperature on the charge current injected from the Ohmic contact.


INTRODUCTION
The quasi-one-dimensional edge channels supported by fractional quantum Hall (FQH) states have for a long time attracted attention of both theorists and experimentalists. During the 1980s models emphasizing role of edge states for FQH transport developed into a powerful field-theoretical framework of a chiral Luttinger liquid (CLL) [1]. A very rigid mathematical structure of the latter led to a number of nontrivial predictions such as fractionally charged quasiparticles, and excitations with anyonic or even non-Abelian statistics. Some of these predictions have been tested experimentally while others still pose a challenge to experimentalists.
One of the milestones in the experimental studies of the FQH effect is the recently reported observation [2] of the neutral current (a transport channel which does not carry electric charge) at the edge of the ν = 2/3 FQH state. The ν = 2/3 state is one of the simplest for which the CLL theory is not consistent without a neutral current. Moreover, the predicted flow direction of this current is opposite to the electrons' drift velocity [3,4] and thus contradicts intuition based on the magnetic hydrodynamics [5].
Apart from the detection of the upstream neutral mode, the design of the experiment, Ref. [2], gave access to a significant amount of quantitative data characterizing the system [2,6]. This motivates the present work, where a detailed quantitative description of the experiment is developed basing on the minimal ν = 2/3 edge model worked out in [3,4] and supported by numerical simulations of small systems [7,8]. Within the developed framework we analyze the data of [2] in order to (a) check its consistence with the minimal ν = 2/3 edge model quantitatively and (b) extract new information about ν = 2/3 edge physics.
While we find excellent agreement of our theory with the experimental data of Ref. [2], we would like to remark that a number of alternative theories have been proposed recently in order to explain other experimental results such as those of Ref. [9]. These theories extend the minimal ν = 2/3 edge model by introducing new physics, such as edge reconstruction [10] or bandwidth cutoffs [11], at some intermediate energy scale. Such extensions can be incorporated into our framework. However, as they contain additional unknown parameters, their comparison against experiment can only be insightful with more independent experimental data.

DESCRIPTION OF THE EXPERIMENT [2]
Here we briefly discuss the experiment [2] where the upstream neutral currents at the quantum Hall (QH) edge were investigated. Figure 1 shows a sketch of the experimental device. The green (color online) region is the AlGaAs heterostructure with the light-green showing where the 2DEG (two-dimensional electron gas) is actually present. The sample is in the transverse magnetic field so that the filling factor is 2/3 and the corresponding quantization of the Hall conductivity is observed. Green arrows show the direction of the electrons' drift velocity which coincides with the flow direction of the charge transporting channel (charged mode). Yellow patches represent Ohmic contacts. The purple rectangular pads on top of the sample are the gates which allow one to make a constriction which plays the role of tunneling junction (denoted as QPC -quantum point contact -in the figure). Contacts Ground 1 and Ground 2 are grounded. Source N and Source S are used to inject electric current into the device. Measurements of electric current and its noise are performed at Voltage probe.
The idea of the experiment is as follows. Suppose a current I n is injected into Source N. If the edge supports only one chirality (counterclockwise) then anything injected into Source N will be absorbed by Ground 1 and have no effect on Voltage probe. However, if we assume that there is a neutral mode flowing clockwise, informa- tion about the events in Source N carried by the neutral mode may reach QPC. In QPC such information may be transmitted to the opposite edge and then transported to Voltage probe by the charged mode. In particular, let us assume that the injection of the current I n excites the neutral mode flowing out of Source N towards QPC. Due to the tunneling across QPC of quasiparticles having both charged and neutral degrees of freedom the neutral mode excitations will be converted into the current noise at Voltage probe. Thus, the presence of the counterpropagating neutral mode implies that the noise observed at Voltage probe should depend on the current I n . Such a dependence was reported in [2].
The observation of the theoretically predicted upstream neutral mode is a very important qualitative result. However, experimental techniques and numerical data reported in [2] go far beyond this achievement providing a lot of implicit quantitative information about current fractionalization in Ohmic contacts, transport along the QH edges and quasiparticle tunneling across the QPC. In order to effectively utilize this quantitative information one needs an analytical theory of the experiment based on the modern understanding of the FQH edge. The goal of this work is to discuss the results of [2] within such a theoretical framework.

THEORETICAL PICTURE OF THE EXPERIMENT
Our theoretical description of the experiment has three key ingredients: the effective theory of the quantum Hall edge, a model of the QPC, and phenomenological assumptions about the interaction of the Ohmic contacts with the QH edge. The former two are based on the standard theoretical framework which we briefly review in the next section. In this section we focus on the general pic-ture of the experiment, paying particular attention to the assumptions regarding the Ohmic contacts.
Our theoretical model of the experiment is illustrated in Fig. 2. Each edge supports one counterclockwise charged mode and one clockwise neutral mode. The two edges approach each other in the QPC region where the tunneling of the quasiparticles between the edges occurs. Our quantitative theory is developed for the case of weak quasiparticle tunneling. The Ohmic contacts are shown as rectangles. We assume that any excitations of neutral and charged modes are fully absorbed by the Ohmic contacts they flow into. We further assume strong equilibration mechanisms at the edge so that the hydrodynamic description can be used. That is, each edge can be characterized by local point-dependent thermodynamic variables including the charged mode chemical potential µ (c) , the charged mode temperature T and the neutral mode temperature T , and any other thermodynamic variables arising due to the existence of extra conserved quantities. We assume that in the absence of currents (I n = 0 and I s = 0) the edges are in equilibrium with the environment so that all modes' temperatures are equal to the base temperature T 0 and the chemical potentials are equal to zero. Away from this state the temperatures and chemical potentials are unknown functions of I n and I s , and other thermodynamic variables are assumed to be unaffected by the injection of currents I n , I s . The functions µ (c) (I n , I s ), T (I n , I s ), T (I n , I s ) for each edge are defined by the interaction of the Ohmic contact with the edge, however no predictive theoretical model of such interaction is known today. As we show, these functions can be inferred from the experimental data under some plausible phenomenological assumptions.
We assume that there is a strong heat exchange between the modes at each edge. In this approximation the local temperatures of the two modes coincide at each point along the edge: T n = T n , T s = T s . Moreover, following [2] we assume that the lower edge temperature is equal to the base temperature (T s = T s = T 0 ); that is, the electric current I s injected by the Ohmic contact Source S does not induce any nonequilibrium noise to the lower edge charged mode.

FORMALISM OF THE EDGE FIELD THEORY
In this section we give a brief overview of the CLL formalism [1,12,13] which is believed to provide the effective theoretical description of a fractional QH edge. We then focus on a particular edge model relevant to the experiment [2]. We conclude this section by a discussion of the model Hamiltonian describing the tunneling of quasiparticles between the QH edges. Tunneling of the quasiparticles at the constriction induces extra noise in the charged mode of the lower edge which is detected at the Voltage probe. Injection of the current Is only changes the chemical potential of the charged mode of the lower edge.

General formalism[14]
Abelian QH edge theories are usually formulated in terms of bosonic fields ϕ i (x, t), where t is time and x is the spatial coordinate along the edge. Each field ϕ i represents an edge mode. Suppose that we have N edge modes and correspondingly N fields ϕ i with i = 1, ..., N . Then the dynamics of the edge is described by the effective action [15] where v i ∈ R + are the propagation velocities, χ i = ±1 represent chiralities of the modes (plus for the clockwise and minus for the counterclockwise direction), and a µ (x, t) is the electromagnetic field potential at the edge. Covariant derivatives are defined as D µ ϕ i = ∂ µ ϕ i − χ i q i a µ . The coupling constants q i provide information on how the electric charge is distributed between the modes. The symbol ε µν denotes the fully antisymmetric tensor with µ, ν taking values t and x (or 0 and 1 respectively) and ε tx = ε 01 = 1.
Conservation of total electric current in the whole volume of a 2D sample leads to the condition [12,13] The electric current at the edge is In the presence of the electric field it is not conserved: which is a manifestation of the inflow of the Hall current from the bulk.
In the absence of the electric field a µ (x, t) = 0 the current is conserved and has the form In the rest of this section we assume that a µ (x, t) = 0.
Beyond the electric current one can also define neutral currents with vector p = (p 1 , ..., p N ) being linearly independent of vector q = (q 1 , ..., q N ). [16] The quantized fields ϕ i can be presented as follows where X i = −χ i x + v i t, k = 2πn/L, n ∈ N; L → ∞ is the system size, a i (k) and a † i (k) are the annihilation and the creation operators respectively, and ϕ 0 and π 0 are the zero modes: The fields ϕ i obey the commutation relation of chiral bosons: The edge supports quasiparticles of the form which are important for the processes of tunneling at the QPC. The notation : ... : stands for the normal ordering, g = (g 1 , ..., g N ), and g i ∈ R are the quasiparticle quantum numbers.
Among the quasiparticle fields there has to be a field representing an electron which is the fundamental constituent particle: Minimal models of the QH states of Jain series ν = N/(2N ± 1) have N electron operators each representing a composite fermion Landau level: The electron fields have to satisfy the following constraints: where J 0 is charge density operator defined in Eq. (5), {...} denotes the anti-commutator, and a plus or minus sign in the second equation can be chosen independently for each pair (α, β); α, β = 1, ..., N .
For the parameters e αi in Eq. (12) these constraints imply e α · e α ∈ 2Z + 1, e α · e β ∈ Z, q · e α = −1 where we defined e α = (e α1 , ..., e αN ) and q = (q 1 , ..., q N ) with q i being the coupling constants from the action (1), and the operation Equations (14) have many inequivalent solutions each defining a topological QH class. It is convenient to parametrize these classes with the help of the K-matrix: Consider now a QH fluid corresponding to a particular solution {e 1 , ..., e N } of Eqs. (14). The spectrum of the quasiparticles (10) present in the model is determined from the requirement of mutual locality with all the electron operators: where s is either +1 or −1 depending on the particular quasiparticles. This leads to the following restrictions on the parameters g i in Eq. (10): where g = (g 1 , ..., g N ). The set of numbers n α completely defines the properties of a quasiparticle operator.
For the following considerations two quantum numbers of the quasiparticle operator (10) are of particular importance: the electric charge Q and the scaling dimension δ. They are given by The minimal model the ν = 2/3 QH edge Here we use the general principles discussed above to obtain the minimal model of the ν = 2/3 QH edge. This model emerges from different semi-phenomenological theoretical approaches to the QH edge [17,18] and is the most likely candidate to describe this fraction [8].
First, we note that it is impossible to satisfy the constraints (2) and (14) assuming that N = 1. For N = 2 we choose the charge vector [19] q = ( 2/3, 0) (20) and chiralities Then equations (14) lead to an infinite one-parameter family of solutions: where m = −1, 0, 1, 2, ... The electron operators have the smallest scaling dimension for m = −1, which gives and the K-matrix This defines the minimal model of the ν = 2/3 QH edge.
The quasiparticle spectrum of the model is defined by Eq. (17). The parameters of the three excitations which are most relevant for tunneling across the QPC are given in Table I.
Tunneling of quasiparticles across the QPC Wherever the two QH edges approach each other at a distance on the order of the magnetic length processes of quasiparticle exchange between the edges are possible. It is widely accepted [12,20,21] that such processes can be described by adding the following term to the Hamiltonian: where the superscripts (u), (l) label quantities relating to the upper and the lower edge respectively; for simplicity we assume that tunneling occurs at the origin. In the case of weak tunneling across the bulk of the QH state the sum runs over all quasiparticles in the model. However, at small energies quasiparticles with the smallest scaling dimension δ(g) have the largest tunneling amplitude η g , thus giving the most important contribution.

CALCULATION OF OBSERVABLE QUANTITIES
In this section we derive analytical expressions for two observable quantities as functions of the experimentally variable parameters. These quantities include the tunneling rate that is the ratio of the current tunneling across the QPC to the Source S current I s and the excess noise in the Voltage probe which is the noise in the Voltage probe in the presence of currents I n , I s less the equilibrium noise at I n = I s = 0. We further demonstrate that it is advantageous to consider the ratio of these quantities rather than each separate one. This way the influence of non-universal physics of the tunneling contact can be reduced.

Tunneling rate
As it was mentioned in the previous section, the most important contribution to the tunneling processes is due to the most relevant excitations. Such excitations are listed in Table I, and we restrict our considerations to these excitations only. To this end we introduce the following notation ψ i (x, t) = V gi (x, t) where g i , i = 1, 2, 3 are the three most relevant quasiparticle vectors given in Table I.
The tunneling Hamiltonian can then be written as where the superscripts (u), (l) label quantities relating to the upper and the lower edge respectively and η i are unknown complex phenomenological parameters.
We calculate the tunneling current within the secondorder perturbation theory in the tunneling Hamiltonian. The detailed derivation can be found in Appendix B. The resulting tunneling rate is given by the Kubo formula: where I T is the tunneling current and I s is the current originating from Source S. Apart from r, paper [2] uses t = 1 − r.

Excess noise
Noise spectral density of the electric current flowing into the Voltage probe (see Fig. 2) can be calculated as the Fourier transform of the two-point correlation function of the current operator I, where {. . . } denotes the anti-commutator, and It is convenient to separate the operator I of the full current flowing to the Voltage probe into I 0 + I T with I 0 = J µ(l) (x = −0, t = 0)| µ=1=x being the spatial component of the operator J µ (x, t) defined in Eq. (5) which represents the electric current flowing along the lower edge just before the tunneling point, and I T being the tunneling current operator. Then the noise can be represented as follows: with ∆I a = I a − I a , indices a, b take values 0 and T .
We are interested in the low-frequency component measured in the experiment. To a good approximation this can be replaced by the zero-frequency component S(ω = 0). Within the second order perturbation theory we find We remind the reader that T s is the lower edge temperature in the neighborhood of the QPC. These formulas are derived in Appendix C.
The contribution S 00 is the Johnson-Nyquist noise of the lower edge. If we restore e, , and k B we see that S 00 (0) = k B T s /R, R = 2π /(νe 2 ) = h/(νe 2 ). Since the Voltage probe contact not only absorbs the lower edge charged mode but also emits another charged mode which flows to the right of it, the actual Nyquist noise measured in the contact will be S JN (0) = 2k B T s /R, in agreement with general theory of Johnson-Nyquist noise. The factor of 2 difference from the Nyquist noise expression used in [2] is due to the noise spectral density definition as discussed in footnote [23].
Following [2] we define the excess noisẽ where S eq is the equilibrium noise spectral density (i.e. the noise when I s = 0 and I n = 0, meaning that the edge temperatures are equal to the base temperature: T s = T n = T 0 ). It turns out that S eq (0) = S 00 (0) resulting iñ This fact is proven in Appendix D using the explicit formulas for S 0T (0), S T T (0) obtained in Appendices C1 and C2.
Noise to tunneling rate ratio [24] Expressions (30), (35), and (36) depend on the tunneling constants η i . It is well known (see e.g. [25][26][27] and references therein) that the tunneling amplitudes η i in electrostatically confined QPCs strongly depend on the applied bias voltage in a non-universal way, probably due to charging effects. Therefore one would like to exclude this dependence from the quantities used for comparison with experiment.
Consider the ratio of the excess noise to the tunneling rate: , v c and v n are the propagation velocities of the charged and the neutral mode respectively, and e is the elementary charge. The number (g i ) 2 is presented in the column g 2 of Table I for each of the three excitations enumerated by i. Functions F i and G i (see Appendices B1, C1, C2 and E) represent contribution of different excitations to the excess noise and tunneling current respectively. In particular, the excess noise is given bỹ and the tunneling rate is equal to The explicit form of these functions is presented below. Note that F 1 = F 2 and G 1 = G 2 .
where λ = T n /T s , T n is the local upper edge temperature at QPC, T s = T 0 is the local lower edge temperature at QPC, e is the elementary charge, h = 2π is the Planck constant, k B is the Boltzmann constant, ν = 2/3 is the filling factor, and the scaling dimension δ and the quasiparticle charges in the units of the elementary charge Q i can be found in Table I.
Remarks on non-universality in the noise to tunneling rate ratio It is easy to see that if any one quasiparticle dominates tunneling (for example, if θ → ∞) then the unwanted non-universal dependence of the tunneling amplitudes on the applied bias voltage does not enter the expression (39). If we assume that the SU(2) symmetry of the edge [3,4] is for some reason preserved at the tunneling contact so that |η 1 | 2 = |η 2 | 2 = |η 3 | 2 , then again the non-universal behavior of the tunneling amplitudes does not enter the expression X; moreover, in this case finding θ allows us to determine the v c /v n ratio. In general, though, θ may exhibit some non-universal behavior. Anticipating results, we can say that, surprisingly, θ does not seem to exhibit any strong dependence on I s or I n .
For the following considerations we also give the large-I s asymptotic behavior of the noise to tunneling rate ratio (39) which we derive using Eqs. (42)-(46): This asymptotic expression can give the reader an idea as to the effect introduced by the non-universal function θ(I n , I s ). One can see, for example, that the gradient of the asymptote increases by a factor of 2 as θ increases from zero to infinity.

COMPARISON WITH THE EXPERIMENT
In this section we compare our analytical results with the experimental data.
The following data are available from the paper [2]: the transmission rate t = 1 − r dependence on the currents I n and I s (Fig. 3(a) of [2]), the dependence of the excess noise at zero frequency on the currents I n and I s (Fig. 3(a) of [2]) and the dependence of the excess noise at zero frequency on the current I n for I s = 0 (Fig. 2 of  [2]).
It is a well-known problem (see, e.g., [25][26][27] and references therein) that the dependence of the transmission rate t on the current I s does not have the form predicted by the minimal model of tunneling defined in Eqs. (28), (29). A possible explanation is the non-universal dependence of the tunneling amplitudes η i on I s due to electrostatic effects. As discussed in the previous section this problem can be avoided in simple cases by considering the ratio of the excess noise to the tunneling rate r = 1−t. However, in the present case a certain degree of non-universality remains due to the non-universal function θ(I n , I s ). The theoretical expression for the noise to tunneling rate ratio X λ,θ (I s ) is given by Eq. (39), where λ = T n /T s is the ratio of the two edges' temperatures. Neither θ, nor λ can be calculated theoretically and we will deduce them from fits of the experimental data. We assume that λ depends on I n but not on I s ; we also assume that the non-universal behavior of the tunneling amplitudes does not lead to any significant dependence of θ on the currents I n , I s . While the former assumption is physically plausible in the weak tunneling regime, the latter one is motivated by our intention to reduce the number of fitting parameters as much as we can.
In Fig. 3 the results of fitting X λ,θ (I s ) to the experimental data taken from Fig. 3(a) of [2] are shown. Optimal fits are found for each set of points corresponding to a given value of I n with θ and λ being fitting parameters. The corresponding values and standard deviations of fitting parameters are shown in Table II. For I n = 0 we have set λ = 1 by definition. As one can see from the Table II, the values of θ do not vary significantly. Thus we repeat the fitting procedure with θ equal to the mean of the five values and λ being the only fitting parameter. The resulting fits and values of λ are presented in Fig. 4 and Table III. As one can see the fits remain good, thus we cannot reliably find the extent of deviation of θ from a constant value with the available experimental data.  Table III gives us some data on the dependence of T n = λT s = λT 0 on the current I n . We further investigate this by fitting it with the following function: where C and a are fitting parameters. The resulting fit is shown in Fig. 5. The corresponding values of fitting parameters are a = 0.54(5), C = 5.05(13) nA −a . This disagrees with the claim of Ref. [22] that the experimental data are consistent with a linear T n dependence on I n . We cannot analyze the source of this discrepancy because Ref. [22] does not contain sufficient detail as to how the comparison with the experiment was done.
Using the phenomenological dependence (48) it is possible to predict the noise to tunneling rate ratio at any I s , I n without any further fitting procedures (we still take θ = θ mean = 0.39). So we can test the formula (48) by comparing the theoretical prediction of X λ,θ (I s ) to another data set. We take the experimental data for the excess noiseS(0) dependence on I n for I s = 0 from Fig. 2 of the paper [2] for t = 1 − r = 80%. The resulting comparison of the noise to tunneling rate ratio X is shown in Fig. 6. An excellent agreement of theoretical curves and experimental points gives an independent confirmation of the result (48).

DISCUSSION
In this section we discuss the results of the comparison of theoretical predictions with the experimental data, and emphasize some important aspects of our analysis.   Table III with the fit of these points by formula (48) is shown.
The good quality of the fits shown in Fig. 4 suggests that the minimal model of the ν = 2/3 quantum Hall edge is consistent with the experimental data. Note, that the existence of good fits is not trivial because of the following reasons. The number of fitting parameters is small; namely, two fitting parameters are used to get Fig. 3, only one is used for Fig. 4 and no fitting parameters are involved in obtaining Fig. 6. Moreover, our theory imposes strong constraints on the shape of the function X λ,θ (I s ) in the whole region of parameters λ, θ. For example, as can be seen from Eq. (47), the gradient of the large I s asymptote of the curve X λ,θ (I s ) varies between e/3 and 2e/3 as θ increases from zero to infinity. The fact that the experimental curve lies between these bounds is non-trivial. The fact that the gradient of the large I s asymptote of the curve X λ,θ (I s ) does not coincide with the limiting values of e/3 and 2e/3 provides an indirect confirmation of the presence of more than one quasiparticle species taking part in tunneling. Indeed, in the case of a single quasiparticle species of charge Q, the asymptote gradient would equal Q. From considerations similar to the flux insertion argument [28,Section 7.5] one can deduce that the natural ν = 2/3 fractional charges are integer multiples of e/3. Interpreting the asymptotic behavior of the noise to tunneling rate ratio in terms of a single quasiparticle tunneling, one would get an unnatural value of Q lying between e/3 and 2e/3. The minimal model of the ν = 2/3 edge explains this contradiction in a natural way: the experimentally observed "charge" is a weight average of the charges of two equally relevant quasiparticles with weights defined by non-universal tunneling amplitudes' ratio θ. This gives an extra argument in favor of the minimal model of the ν = 2/3 edge with the K-matrix (25). A similar point was made in paper [11] in relation to the experiment [9].
Note, that the minimal model analyzed here can also be regarded as the low-energy limit of the extended models proposed in Refs. [10,11]. At higher energies both extended models predict tunneling contact physics to be dominated by a quasiparticle with charge e/3. Since we do not see this in our analysis, we conclude that either the extended physics is not present in the system or occurs above the energies probed in the experiment of Ref. [2].
It should, however, be emphasized that the minimal ν = 2/3 edge model alone is not sufficient to describe the present experiment. Extra assumptions are needed to model the non-universal physics of Ohmic contacts, edge equilibration mechanisms, and the tunneling con-tact. Such assumptions have been discussed throughout the text, and here we summarize them: • injection of electric current into an Ohmic contact induces non-equilibrium noise in the neutral mode but not in the charged mode; • injection of electric current into an Ohmic contact does not induce a shift in the neutral mode chemical potential (that is the thermodynamic potential dual to the neutral charge defined through Eqs. (6) and (26)); • strong equilibration of the charged and the neutral modes takes place along the edge resulting in some current-dependent local temperature of the edge; • the tunneling contanct can be modeled by the minimal tunneling Hamiltonian (28) with tunneling amplitudes depending on the edge chemical potential in some non-universal way.
While these phenomenological assumptions are plausible, they may not be accurate. Moreover, their validity may depend on the experimental conditions. The theoretical framework presented here enables a more detailed experimental investigation and refinement of our understanding of non-equilibrium processes at the edge. For example, in the present work we use experimental data to establish a phenomenological law (48) describing the dependence of the neutral mode temperature at the QPC on the current I n (see Figs. 5 and Fig. 6). Recently there has been some theoretical progress in understanding of the interaction of Ohmic contacts with the quantum Hall edge [29]. However, at present a complete theoretical predictive model of Ohmic contacts is still missing, and the information on the neutral mode heating may contribute to its development.
It is also interesting to note that we do not find any significant dependence of the ratio of the tunneling amplitudes of different species of quasiparticles on the currents I n , I s (see discussion of Figs. 3 and 4). This is surprising since the tunneling amplitudes themselves appear to vary significantly to explain the tunneling rate dependence on I s observed in [2]. This fact suggests the existence of a mechanism which ensures roughly equal participation of all three quasiparticles species in the tunneling. It is known [3,4] that disorder scattering at the edge enforces the SU(2) symmetry between the quasiparticle species. A similar mechanism might be responsible for the discussed phenomenon.
We emphasize that our theoretical predictions are derived in the limit of perturbatively weak tunneling of the quasiparticles. Therefore, the tunneling rate at which the comparison with the experimental data is made should be small enough so that our theory remains valid, but large enough in order to minimize statistical errors of the noise to tunneling rate ratio.

CONCLUSIONS
Using the chiral Luttinger liquid theory of the quantum Hall edge we develop a quantitative model of the experiment reported in [2]. This model enables us to extract important quantitative information about nonequilibrium processes in Ohmic and tunneling contacts from the experimental data. In particular, for ν = 2/3, we find a power-law dependence of the neutral mode temperature on the charge current injected from the Ohmic contact. We also find a surprising behavior of quasiparticle tunneling amplitudes which may be a signature of the SU(2) symmetry in the quasiparticle tunneling across the QPC.
Here we give explicit expressions for the correlation functions at a single edge without tunneling (described by the minimal model for ν = 2/3 defined in the main text) which are used to calculate the quantities of experimental interest. In all the correlation functions of this appendix we assume the infinite system size limit L → ∞.
The two-point correlation function of quasi-particle operators is equal to t) is a quasiparticle excitation operator defined in Eq. (10), g = (g 1 , g 2 ) = (g c , g n ) is the excitation vector, p enumerates charged (c or 1) and neutral (n or 2) modes, X p = −χ p x + v p t, χ p and v p are the mode chirality and velocity respectively which enter the action (1) (in our case χ 1 = −χ 2 = 1), and T is the temperature of the edge. The electric charge Q (c) = Q = g 1 √ ν, ν = 2/3, and µ (c) = µ is the chemical potential of the charged mode at the edge. It coincides with the chemical potential of the Ohmic contact where the charged mode originates. The neutral charge Q (n) = g 2 and the chemical potential µ (n) do not enter the formulas in the other sections as we assume µ (n) = 0, though, in principle, injection of the current from an Ohmic contact could shift the neutral mode chemical potential. We have also introduced an infinitesimally small positive number ε → +0.
The electric current along the edge in equilibrium is given by the average of the current operator J µ=1 defined in Eq. (5): in agreement with the quantization law of Hall conductance [30]. The two-point correlation function of quasi-particle operators with the current operator inserted is given by where Y i = t − χ c x/v c + iκ i . κ 0 = 0, κ 1 = κ → +0 is an infinitesimally small positive number, κ 2 = κ 1 + ε, and ε is the same as in the two-particle correlation function. Finally, the current-current correlation function is J 1 (x 0 , t 0 )J 1 (x 1 , t 1 ) = J 1 (x 0 , t 0 ) J 1 (x 1 , t 1 ) + where Y i = t − χ c x/v c + iκ i , κ 0 = 0, κ 1 = κ → +0 is an infinitesimally small positive number.
where Q i are the quasiparticle charges Q in Table I and A i are the operators defined in Eq. (29). This is the tunneling current operator in the interaction picture with interaction H T (which is emphasized by the subscript "int"). We calculate the expression for the tunneling current operator in the Heisenberg picture within the perturbation theory in H T : The observed tunneling current is then We have used the relationships A i (t) = A j (τ )A i (t) = 0, A † j (τ )A i (t) ∝ δ ij . It can be checked with explicit correlation functions (50) that the integral of each of the summands in the formula (58) is convergent. Thus, one can split them and manipulate separately. Using time translational invariance of the correlation functions in both summands and changing sign of the integration variable in the second one we finally get which leads to the expression (30) for the tunneling rate r.

ture.
The noise spectral density S(ω) defined in Eq. (31) then separates into four terms, see Eqs. (32) and (33), where the identity S ab (ω) = S ba (−ω) following from the time translational invariance of the correlation functions has been used.
Using Eq. (53) one obtains where T (l) is the lower edge temperature, and ε → +0 is an infinitesimally small positive number. This is the identity (34).
Since Moving to S 0T (ω), we find up to the corrections O(|η i | 3 ) that In analogy with the calculation of the tunneling current expectation value, the integral of each of the two summands in the last formula is convergent, thus we can manipulate the two summand integrals separately. Changing the order of integration and renaming τ ↔ τ in the second summand we arrive at the expression (36) for ω = 0.