Relaxation and edge reconstruction in integer quantum Hall systems

The interplay between the confinement potential and electron-electron interactions causes reconstructions of Quantum Hall edges. We study the consequences of this edge reconstruction for the relaxation of hot electrons injected into integer quantum Hall edge states. In translationally invariant edges, the relaxation of hot electrons is governed by three-body collisions which are sensitive to the electron dispersion and thus to reconstruction effects. We show that the relaxation rates are significantly altered in different reconstruction scenarios.


I. INTRODUCTION
The kinetic properties of one-dimensional (1D) quantum systems are an active area of current research. 1,2 What makes the field exciting is that many-particle physics is drastically different in one spatial dimension. This difference is already evidenced in basic nonequilibrium properties such as the microscopic mechanisms of relaxation. Within the scope of Fermi-liquid theory, relaxation processes in higher dimensions proceed by pair collisions of electrons which provide an efficient mechanism for relaxation of initial nonequilibrium states. In contrast, conservation of energy and momentum strongly restricts scattering in one spatial dimension so that pair collisions necessarily result in zeromomentum exchange or an interchange of the momenta of the colliding particles. Neither process causes relaxation. This poses the fundamental question of the microscopic origin of relaxation in 1D systems. Notably, the absence of relaxation by pair collisions, which holds regardless of the strength of interaction, has received experimental support. 3 The question of equilibration emerges in a diverse set of 1D many-body systems. These include energy and momentum-resolved tunneling experiments with nanoscale quantum wires, 3,4 quench dynamics of cold atomic gases, 5,6 as well as energy-spectroscopy experiments on quantum Hall edge states driven out of equilibrium. [7][8][9][10] The present paper is motivated by the latter experiments which are carried out in a highmobility two-dimensional electron system at Landau level filling factor ν = 2. This system hosts two copropagating edge states which can be driven out of equilibrium by inter-edge tunneling in the vicinity of quantum point contacts. This generates a nonequilibrium distribution of electron energy (in the sense of electronic edge transport) downstream from the contact which is monitored as a function of the propagation distance by means of a quantum-dot-based energy spectrometer. The experiments show that the initial nonequilibrium distribution relaxes to a stationary form which is close to the ther-mal distribution but with an effective temperature and chemical potential.
Edge-state equilibration was also probed in experiments at Landau-level filling factor ν = 1. 11 Heat is carried unidirectionally by the single chiral edge mode as confirmed by thermopower measurements along the edge. These experiments found that hot electrons injected locally into the edge cool down while propagating along the edge. It is worth emphasizing that the standard chiral-Luttinger-liquid model for quantum Hall edge states 12 does not account for equilibration effects. Indeed, this model is exactly solvable and as usual, its integrability is an obstacle to thermalization. In early works 13 this apparent difficulty was overcome by assuming a disordered edge where impurity mediated scattering allows for interchannel equilibration.
These experimental discoveries led to a flurry of theoretical activity. We briefly summarize these contributions and place our work into their context. Two initial publications 14,15 used entirely different concepts. Ref. 14 was based on a Boltzmann kinetic equation for a disordered edge. Since translation invariance is broken, momentum is no longer a good quantum number and relaxation becomes possible even by two-particle collisions. Ref. 15 adopted a bosonization approach and combined it with a phenomenological model for the plasmon distribution generated at the quantum point contact. Within this model, thermalization was interpreted as a consequence of plasmon dispersion which causes the electron wave packets to broaden as they propagate with different group velocities. This picture was elegantly elaborated and extended in Refs. [16][17][18]. A third mechanism was proposed in the context of electronic Mach-Zehnder interferometers 19,20 based on electron-plasmon scattering. 21 This mechanism relies on scattering of high-energy electrons by low-energy plasmons enabled by the curvature of the fermionic spectrum.
Despite the insight provided by these theories, important issues need to be sorted out. First, these works do not give a definitive answer whether relaxation is possible in translationally-invariant clean edges. Specifically, the dispersion of plasmon modes may lead to a steady state but does not constitute true relaxation as the energy in each plasmon mode is conserved. Second, the edge of quantum Hall systems can be reconstructed due to Coulomb interactions. The precise nature of reconstruction depends on the steepness of the confinement potential, ranging from no reconstruction for very sharp confinement potentials 22 to alternating compressible and incompressible stripes for very smooth edges. 23 Indeed, experiments 24,25 point towards an important role of reconstruction effects in energy transfer along the edge.
The purpose of the present study is to address these issues within minimal models of unreconstructed and reconstructed edges. Specifically, we consider energy relaxation of a hot particle injected into translationally invariant quantum Hall edges at Landau level filling factors ν = 1, 2. With the assumption that the velocity v 1 of the injected particle differs sufficiently from the Fermi velocity v F , we treat the Coulomb interaction perturbatively. 26,27 In this limit, relaxation processes are dominated by three-body collisions which depend sensitively on the electron dispersion and hence on the edge reconstruction. We begin with a discussion of energy relaxation for the unreconstructed edge in Sec. II. We then discuss two simple models of reconstructed quantum Hall edges. In Sec. III, we discuss relaxation processes for a spin-reconstructed edge for filling factor ν = 2. In Sec. IV, we turn to a minimal model of charge reconstruction of a ν = 1 edge which provides the simplest realization of counter-propagating edge modes. We conclude in Sec. V.

II. UNRECONSTRUCTED EDGE
A confinement potential V c (x) that is sharp on the scale of the Coulomb interaction (i.e., V c e 2 /(κl 2 B ), where κ is the dielectric constant and l B denotes the magnetic length) remains stable against interactioninduced reconstructions and the electron dispersion ε(k) can be obtained approximately from the noninteracting Schrödinger equation. 22 A generic electronic dispersion of an unreconstructed edge is sketched in Fig. 1a, exhibiting a confinement-induced bending of the Landau levels near the edge of the sample.
In the limit of high magnetic fields (V c ω c /l B ), the electron states near the edge can be described by the lowest Landau level wave functions in the Landau gauge. Here, k = X/l 2 B and L denotes the length of the sample edge (taken along the y direction). The defining feature of the unreconstructed edge is the sharp zero-temperature occupation function ν σ (X) = Θ(−X) of Landau level states with guiding center X when the Zeeman splitting ε Z is negligible [see Fig.  1b].
The single particle dispersion near the Fermi energy (corresponding to momentum k F ) is controlled by the confinement potential and can be approximated as (2) The dispersion is parametrized through the edge velocity v F = V c l 2 B at the Fermi energy and the curvature 1/m c = V c l 4 B . Note that these parameters become maximal for an infinitely sharp edge for which v F ∼ ω c l B and 1/m c ∼ 1/m. 22 Note however, that a description in terms of the wave functions in Eq. (1) is no longer valid in this extreme limit.
The finite curvature of the dispersion implies that at least three particles are required for an energy and momentum conserving relaxation process. Relaxation of a high-energy electron (labeled by i = 1 in Fig. 1a) is possible by scattering two electrons (labeled i = 2, 3 in Fig.  1a) near the Fermi energy. Indeed, due to the curvature of the dispersion near the Fermi energy, exciting electron i = 2 from the Fermi energy requires more energy than scattering electron i = 3 deeper into the Fermi sea. Clearly, this relaxation process relies on finite temperature and typical energy transfers for electrons i = 2, 3 at the Fermi energy are of the order of T . Quantitatively, this process can relax the hot particle with excess energy is the momentum transferred to particle i in the collision. Note that q 1 q 3 so that relaxation occurs in many small steps v F q 1 ∼ T 2 /ε.
For Landau-level filling factor ν = 2, these considerations apply when the Zeeman splitting is small compared to temperature. In the opposite limit ε Z T , the curvature of the dispersion implies that the Fermi momenta and hence the Fermi velocities differ for the two spin directions. In this case, relaxation is dominated by processes in which the electrons i = 1 and i = 2 have opposite spins, and thus different Fermi momenta k F j and Fermi velocities v j with j = 1, 2. To include a finite Zeeman splitting at Landau level filling factor ν = 2 as well as for later convenience, it is thus beneficial to consider a modified dispersion which is linearized in the vicinity of each of the three particles, including the hot particle with velocity v 1 and momentum k 1 . This captures the behavior in the regime of strong Zeeman splitting ε Z T on which we will focus in the following. Nevertheless, we can also recover the results for the quadratic dispersion and weak Zeeman splitting ε Z T by identifying v 2 − v 3 with the typical velocity difference T /(v F m c ) due to the curvature of the dispersion.
Using the dispersion in Eq. (3), energy and momentum conservation leads to is controlled by the Zeeman splitting which we assume to be small compared to the excitation energy ε such that

A. Three-body scattering formalism
Energy relaxation by processes of the kind shown in Fig. 1a was already discussed in the context of quantum wires in Ref. 26. While our calculation here follows the same outline, there are characteristic differences related to the nature of the interaction matrix elements. The energy relaxation rate via three-body collisions is again given by (5) where n i is the Fermi-Dirac distribution function at k i . The factor involving q 1 weights the out-scattering rate with the relative relaxed energy, accounting for the fact that the hot particle relaxes only a fraction of its energy in a single collision. The three-body matrix element can be evaluated by the generalized golden rule Here, G 0 is the free Green's function, is the generic two-body interaction potential, and the subscript c emphasizes that only connected processes contribute which involve all three particles.
The calculation for quantum Hall edges differs from that for quantum wires in the form of the Coulomb matrix element V q (k 1 −k 2 ) which now has to be evaluated using the Landau level wave functions in Eq. (1). For quantum Hall systems, the Coulomb matrix element is exponentially suppressed by a factor of exp(−q 2 l 2 B /2) for large momentum transfers. This is especially relevant because large momentum transfers yield the leading contribution to relaxation in quantum wires. 26 Moreover, V q (k 1 −k 2 ) does not only depend on the momentum transfer but also on the initial momentum difference which controls the distance between the guiding centers of the interacting electrons. Focusing on the remaining low momentum transfer processes (q 1/l B ), one obtains (see Appendix-A for details) with the understanding that at small q, the matrix elements will be eventually cut off by a large length scale λ l B which is given by the distance to a screening gate. For k 1 − k 2 1/l B the Coulomb matrix element is that of a quantum wire of width l B . For k 1 − k 2 1/l B , the interaction is that of electrons in two quantum wires separated by a distance of (k 1 − k 2 + q)l 2 B which equals the average of the guiding center distances of the electrons before and after the collision.
With the absence of large momentum transfer processes the three-body scattering is dominated by the direct matrix element. The importance of the (k 1 − k 2 ) dependence of the Coulomb matrix element can be seen from the fact that the linearized dispersion of Eq. (3) leads to a vanishing direct matrix element for a quantumwire-like Coulomb interaction V q (0) (see Appendix B). In contrast, when reiterating the derivation 28 of the direct matrix element T 123 1 2 3 including the dependence of V q (k 1 − k 2 ) on the initial momenta, the result does not vanish and takes the form Here we used k 1 − k 2 ≈ k 1 − k F 2 = ∆k. This expression is applicable under the assumption that all initial momentum-differences are large compared to 1/l B to also suit the reconstruction effects that will be discussed later. For the unreconstructed edge, it is however more reasonable to assume k 2 − k 3 1/l B (for typical ε Z , T e 2 /κl B ) in which case the last term of Eq. (8), involving V q1 , does not show up (see Appendix B).

B. Results for the unreconstructed edge
For the unreconstructed edge, the momentum and velocity differences are linked by the curvature of the confinement potential via v 2 − v 3 = (k 2 − k 3 )/m c and v 1 − v 2 = ∆k/m c . The direct matrix element then takes the form Since for large Zeeman energy the particles at k 2 and k 3 have opposite spins, there is no exchange contribution (remember that exchange is appreciable for small momentum transfers only) and Eq. (9) fully determines the three-body matrix element. The corresponding energy relaxation rate can then be obtained by power counting which yields Here and we also used ∆k = m c (v 1 − v 2 ) = ε/v 1 . In obtaining Eq. (10), a factor of L/(v 1 − v 2 ) emerges from eliminating the energy δ-function in Eq. (5), each summation over the remaining k 2 , k 3 , q 3 contributes a phase space factor of ∼ T /v 2 and the weighting factor v 1 q 1 /ε takes the form Finally we have to account for the competition between excitation (q 1 > 0) and relaxation (q 1 < 0) of the hot particle. The latter is slightly favored because the momentum transfer working against the Fermi distribution is reduced by a fraction Equation (10), valid at ε m c v 1 e 2 /κ and ε Z T implies that the relaxation rate is strongly temperature dependent and can be enhanced by increasing the magnetic field.
As mentioned above, the relaxation rate in the opposite limit of weak Zeeman splitting ε Z T can be obtained up to prefactors by replacing (v 2 −v 3 ) ∼ T /(v 2 m c ). Note that this regime allows for a low momentum transfer exchange term T 123 1 3 2 because the particles 2 and 3 are no longer necessarily of opposite spin. T 123 1 3 2 can then be obtained from Eq. (9) by replacing q 3 → k 2 − k 3 , which does not change the power counting argument.
It is therefore possible to combine both cases by setting v 2 − v 3 = max{ε Z , T }/(v 2 m c ). In the case v 1 ≈ v 2 = V c l 2 B it is then possible to rewrite Eq. (10) as which applies in the regime V c l 2 B ε(V c l 2 B /V c ) e 2 /κ. For the later comparison of the relaxation rates before and after edge reconstruction it will be useful to consider the unreconstructed case as the v 1 v 2 ∼ e 2 /κ limit of Eq. (10) [which can be applied for ε (e 2 /κl B ) 2 /(V c l 2 B )]. Formally, this regime leaves the condition of applicability for the Taylor expansion of the confinement potential that defines 1/m c = V c (µ)l 4 B and would lead to another inverse mass V c (µ+ε)l 4 B for curvature effects at energies of the order of ε. Distinguishing these different masses does however not lead to qualitative changes of the results and for brevity of the presentation we assume a quadratic confinement potential where we used that in this regime m c v 2 1 ∼ . The crossover between Eqs. (11) and (12) can be obtained at their limits of applicability by setting ε = e 2 V c /(κl 2 B V c ) and V c = e 2 /(κl 2 B ). Note that for a spin polarized edge, Eqs. (10)- (12) only apply if the Coulomb interaction is not screened for momenta of the order of T /v 2 . For a screened short range interaction (T /v 2 1/λ), the Pauli principle then leads to a suppression of the energy relaxation rate by an additional factor of (T λ/v 2 ) 4 1. 27

III. SPIN RECONSTRUCTION
Edge reconstruction in quantum Hall systems results from the competition between the Coulomb interaction and the confinement potential. Spin reconstruction at ν = 2 takes place when the confinement potential V c varies sufficiently slowly so that V c < e 2 /κl 2 B and can be understood at the level of the Hartree-Fock approximation. [29][30][31][32] Once the slope of the confinement potential becomes weaker than that of the repulsive Hartree potential V H , it is favorable to deposit charges outside the edge. This can be done without paying exchange energy by a relative shift of the Fermi momenta of spin up and spin down particles, as depicted in Fig. 2. In the absence of a Zeeman splitting, ε Z = 0, this is a second order phase transition with spontaneous breaking of the spin symmetry. Then, the distance of the two Fermi momenta varies as k F 2 −k F 3 ∝ (|V H |−V c ) 1/2 , eventually saturating at ∼ 1/l B . 29 For finite Zeeman splitting ε Z , the spin symmetry is lifted by the Zeeman field and the transition is smeared on the scale of k F 2 − k F 3 ∼ ε Z /v 2 .
Spin reconstruction leads to characteristic changes in the single particle dispersion that develops an "eye structure"[cf. Fig. 2a]. Important for the relaxation dynamics is the increase of v 2 − v 3 = (k F 2 − k F 3 )/m c , which enhances the typical energy transferred per step of relaxation [cf. Eq. (4)].
For truly long range interactions, the Hartree-Fock approximation predicts a logarithmic singularity ∼ e 2 /κ ln(|k − k F |l B ) of the particle velocity at the Fermi energy, which is however cut off in the presence of screening, say by a nearby gate electrode. The Fermi velocity is thus still of the order of v 2 , v 3 ∼ e 2 /κ for typical choices of the screening length.
Even with spin reconstruction, the relaxation of hot particles can be described within the model dispersion of Eq. (3). We consider the case where the hot particle (not shown in Fig. 2) is injected well outside the energy window e 2 /(κl B ) of the reconstructed region. This is compatible with the condition for the validity of a perturbative expansion, which reduces to v 1 v 2 for the case of the Fermi velocity determined by the interaction.
The energy relaxation rate 1/τ (s) E can now be derived in the same way as for the unreconstructed edge and consequently, Eq. (10) also applies to spin reconstructed edges. The crucial difference is that the velocity difference v 2 −v 3 is now strongly enhanced by the spin reconstruction, taking values up to v 2 −v 3 ∼ 1/(m c l B ). Comparing the rates before [v 2 − v 3 ∼ max{ε Z , T }/(m c v 2 )] and well after spin reconstruction, we find an enhancement of the relaxation rates given by (13)

IV. CHARGE RECONSTRUCTION
For confinement potentials that vary even more smoothly, changing by e 2 /κl B over a region w > l B , charge reconstruction may occur such that part of the electrons at the edge are pushed away from the bulk by a length of the order of l B . 31,32 It leads to a nonmonotonic behavior of the dispersion with momentum and the creation of two additional counter-propagating 33 edge modes, as depicted in Fig. 3.
A minimal model for charge reconstruction considers filling factor ν = 1 within the Hartree-Fock approximation. 31 It is convenient to formally model the confinement potential by a positive background charge which is distributed spatially as if it was occupying lowest Landau level wave functions ψ X with occupation numbers ν c (X) = Θ(−X). The advantage of this model is that such a confinement potential exactly cancels the Hartree potential of the electrons for an unreconstructed edge. In this case, the electron occupation of the unreconstructed edge is stabilized by the (attractive) exchange potential. The reconstruction transition can then be modeled by changing the abrupt drop of ν c (X) into a linear decrease over a length w. For the unreconstructed electron occupations, this leads to negative (at X < 0) and positive (at X > 0) excess charges, causing a dipole field that favors separating electrons from the bulk. Once this dipole field overcomes the exchange potential, a charge reconstruction transition takes place. Within the Hartree-Fock approximation, this happens for w ∼ 8l B . Due to the particle-hole symmetric choice of the confinement potential around X = 0, the width and the distance of the additional stripe from the bulk electron droplet both take the same value b. Moreover, the transition is of first order in the sense that b changes abruptly at the transition from zero to a value of the order of l B .
Note that the same mechanism induces new (weaker) effective dipole fields at each of the three Fermi points as the edge becomes yet smoother. Thus, increasing w even further causes additional stripes to appear, eventually approaching the limit of a compressible edge which is expected for w l B . 23 In the following we will focus on w > ∼ l B , remaining well outside the compressible limit. Energy relaxation in the charge reconstructed case can also be captured by the dispersion (3) when setting v 3 < 0 and choosing the particle i = 2 to lie in one of the co-propagating branches. 34 The three Fermi velocities of the charge reconstructed edge are essentially determined by the variation of the exchange potential, which is short ranged such that b > ∼ l B already approximates the bulk edge (b → ∞) behavior. Consequently, the magnitudes of the Fermi velocities are equal to that of the unreconstructed edge and ∼ e 2 /κ. In line with the discussions above, we consider the relaxation of a hot particle injected well outside the reconstructed region with v 1 e 2 /κ. The nonmonotonic behavior of the dispersion introduces a new relaxation process which relaxes the hot particle by exciting two counter-propagating electron-hole pairs [see Fig. 3]. This eliminates the restriction that the energy transfers at the Fermi energy cannot exceed the temperature and makes the relaxation process similar to that for non-chiral quantum wires. 26 Unlike for quantum wires, however, the momentum transfers at the Fermi en-ergy of the co-and counter-propagating branch are of the same order.
The three-body matrix element of Eq. (8) still applies in the presence of charge reconstruction because its derivation did not require a specific sign of v 3 . Note however that for the charge reconstructed case v 2 −v 3 ∼ v 2 ∼ e 2 /κ and is therefore not connected to k F 2 − k F 3 ∼ 1/l B by the curvature of the confinement potential. Assuming that there are no substantial curvature effects on the scale of the reconstructed region V c l 2 B e 2 /(κl B ) the last term of Eq. (8) dominates the three-body matrix element and Eq. (9) modifies to The crucial difference for the energy relaxation rates compared to the unreconstructed case arises from the large allowed momentum q 3 ∼ 1/l B , which is limited only by the size of the reconstructed region for which the linearized dispersion applies. This increases both the momentum phase space to (L/l B ) 3 and the typical relaxed momentum to Moreover, excitation and relaxation processes no longer need to be balanced when e 2 /κl B T , and we find which applies for ε (e 2 /κl B ) 2 /(V l 2 B ) and allows for relaxation even at T = 0. Equation (15) implies that the increased phase space and the energy relaxation step size leads to a dramatic enhancement of the relaxation rate compared to the unreconstructed case [see Eq. (12)] as where we used the limit T ε Z .

V. CONCLUSIONS
We studied three-body processes as an intrinsic mechanism for relaxation of hot electrons in clean integer quantum Hall edges at Landau level filling factors ν = 1 and ν = 2. These processes rely crucially on the form of the electron dispersion and are thus susceptible to edge reconstruction effects. For an unreconstructed edge, energy relaxation requires a finite temperature which determines the phase space for the relaxation processes. The energy given up by the hot electron in a single three-body collision is controlled by curvature effects on the scale of temperature or Zeeman energy so that the relaxation rate can be tuned by a magnetic field once ε Z T . While unreconstructed edges are expected for steep confinement potentials, smoother confinement potentials with V c < ∼ e 2 /(κl 2 B ) may lead to an interaction-induced spin reconstruction, which causes a relative shift of the Fermi momenta of the two spin species by ∼ 1/l B . The three-body processes are then controlled by curvature effects on the scale of the interaction energy e 2 /(κl B ) which causes a strong increase of the relaxation rate [see Eq. (13)]. Even softer confinement may cause charge reconstruction which introduces additional co-and counterpropagating edge modes. The presence of counterpropagating modes allows for relaxation even at T = 0. Consequently, the phase space for three-body collisions is no longer controlled by temperature but by the size of the reconstructed region ∼ e 2 /(κl B ) which ensues an additional dramatic enhancement of the relaxation rate [see Eq. (16)].
Experimental studies of interaction-induced reconstruction transitions in high magnetic fields have been performed. 24 Our study suggests that it would be rewarding to experimentally investigate relaxation processes in such systems.
Within this section we provide all essential details needed for the derivation of Eq. (7) presented in the main text. We assume that the edge is smooth enough that we can approximate the electron wave functions by those of the bulk. We start from the interaction matrix element in real space (A1) In the following we will measure all lengths scales in units of magnetic length l B . In this units the guiding center coordinate directly translates to momenta. With the lowest Landau level wave functions of Eq. (1) we then find where we used the screened Coulomb potential which carries an extra factor e − √ ∆x 2 +∆y 2 /λ with λ being the distance to a screening gate. The integration over ∆y gives 2K 0 (|∆xdX|) in the case when dX 1/λ, where K 0 is the Bessel function of imaginary argument. If, however, dX 1/λ the integral is cut off and the result changes to 2K 0 (|∆x/λ|). We will derive results for the dX 1/λ case and keep in mind appropriate changes for the other limit. After y integration, that gives a factor of L we obtain the intermediate step Performing now the Gaussian integral over x, that gives a factor of π/2, followed by using the Landau gauge to replace guiding center coordinates by momenta one arrives at where we used short-hand notation X − X = kl 2 B = k (with l B = 1). Note that V q (k) = V q (−k − 2q) and is therefore not symmetric, which plays an important role. We see immediately that scattering processes with large momentum transfer q 1 are exponentially suppressed. We therefore concentrate on the opposite limit of q 1 when the exponential prefactor e −q 2 /2 can be set to unity.
Let us study limiting cases of Eq. (A4). In the case when k 1 one can approximate the exponential under the integral by the delta-function √ 2πδ(k + q + ξ), and thus obtains Using the asymptotic form of the Bessel function and restoring units of l B one recovers the second limit in Eq. (7).
In the other limiting case when k 1 one can approximate the exponential under the integral of Eq. (A4) by e −ξ 2 /2 and then complete integration exactly with the result With the logarithmic accuracy at small q this translates into the first limit of Eq. (7).
Appendix B: Calculation of the three-body matrix element T 123 1 2 3 In general the three-particle scattering amplitude 1 2 3 |V G 0 V |123 c contains six terms: one direct and five exchange contributions. 28 As explained in the text we need only the former one which reads explicitly 28 where the spin structure is δ Σ,Σ = δ σ1,σ 1 δ σ2,σ 2 δ σ3,σ 3 and the Coulomb matrix element V q (k) was derived in the preceding section. Now using the dispersion relation from Eq. (3), and constrain on momentum transfers from Eq. (4), imposed by the conservation laws, one can simplify T 123 1 2 3 to where we used the property V q (k) = V q (−k − 2q). It is important to stress that the above expression would vanish by ignoring the dependence of the Coulomb matrix element on initial momenta, namely for V q (k) = V q . To proceed further we make use of the assumption that injected particle is of high energy, such that v 1 v 2,3 and k 1 k 2, 3 . In this case we expand V qi (∆k + q i ) in q i . For the interaction V q (k) = − 2e 2 κ ln(|kq|l 2 B ) we obtain after the expansion . (B3) Note that if we are in the regime when k 2 − k 3 l −1 B we have to use the interaction potential V q (k) = − 2e 2 κ ln(|q|l B ), which has a vanishing derivative with respect to k. This can be accounted for by removing the two terms with V q1 (. . .) in the above formula for T 123 1 2 3 . Finally, to leading logarithmic order we can set V q1 (k 1 − k 2 ) = V q1 (k 1 − k 3 ) as well as V q2 (k 2 − k 1 ) = V q3 (k 3 −k 1 ) = V q3 (k 1 −k 2 ) and V q3 (k 3 −k 2 ) = V q2 (k 2 −k 3 ) to obtain Eq. (8) since the spin summation is equal to unity.