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Electron-phonon interactions on a single-branch quantum Hall edge

We consider the effect of electron-phonon interactions on edge states in quantum Hall systems with a single edge branch. The presence of electron-phonon interactions modifies the single-particle propagator for general quantum Hall edges, and, in particular, destroys the Fermi liquid even at integer filling. The effect of the electron-phonon interactions may be detected experimentally in the AC conductance or in the tunneling conductance between integer quantum Hall edges.

A two-dimensional electron gas subjected to a strong perpendicular magnetic field may exhibit the quantum Hall (QH) effect [2]. The effect occurs because the electron gas is incompressible at certain densities, which is due to an energy gap to bulk excitations. In the integer QH effect this gap is the kinetic energy gap (the cyclotron energyhω c ) in a magnetic field, and in the fractional QH effect the gap arises because of the electron-electron interactions. As a result of the bulk energy gap, gapless excitations in the system can only exist at the edges. Such edge excitations are density modulations localized at the edges of the system, and all the low-energy properties of a QH system are determined by the edge excitations.
As was first demonstrated by Wen [3], the density operators for edge excitations obey a Kac-Moody algebra, similar in structure to that obeyed by Luttinger liquids [4]. Because time-reversal invariance is broken by the magnetic field, the excitations on one edge can only propagate in one direction [5] corresponding to 'chiral Luttinger liquids' [3]. Wen calculated the single-particle propagator for the electron on the edge of a quantum Hall system at filling factor ν = 1/(2m + 1) and showed that it is given by [3] G(x, t) ∝ (x − v F t) 1/ν . Here, x is a coordinate along the edge, and v F is the edge velocity determined by the details of the potential confining the electron gas at the edge. It is worth noting the contrast between chiral Luttinger liquids and regular Luttinger liquids in one-dimensional (1D) interacting electron systems without magnetic fields. In the regular Luttinger liquid, the Fermi surface is destroyed by repeated backscattering, and the exponent of the single-particle propagator depends on the details of the electron-electron interactions. On the other hand, a chiral Luttinger liquid at a QH edge is caused by the strong correlations in the bulk, and the exponent of the single-particle propagator is fixed by the topological order in the bulk of the system.
The appearance of the anomalous exponent in the single-particle propagator has important experimental implications. It was shown by Wen [6] and by Kane and Fisher [7] that the tunneling conductance between two edges of an FQHE system at bulk filling factor ν = 1/(2m + 1) depends on temperature as T 2(1/ν−1) . The resonant tunneling conductance was calculated by Moon et al. [8], and Fendley et al. [9], and measured by Milliken et al. [10]. The experimentally measured tunneling conductance does indeed exhibit a T 2(1/ν−1)dependence, except for at the very lowest temperatures, where Coulomb interactions between the edges may modify the conductance [11].
In this letter, we consider the effect of electron-phonon interactions on the QH edge states of spin-polarized QH systems with bulk filling factor ν = 1/m, with m an odd integer. Such systems have a single branch of edge excitations on each edge. Electron-phonon interactions in regular Luttinger liquids have been considered previously [12,13]. Martin and Loss [13] showed that coupling the electron system to acoustic phonons destroys the Fermi surface, even in the absence of electron-electron interactions. This effect is only appreciable if the Fermi velocity v F is of the same order as the sound velocity v s , which may be achieved in strongly correlated electron systems where the role of the Fermi velocity v F is played by the charge velocity of the Luttinger liquid. The effect may also be appreciable in QH systems, where v F is determined by the stiffness of the confining potential and the electron density, both of which may be tuned electrostatically by gates. We will show here that the electronphonon interactions will modify the single-particle propagator even in a chiral Luttinger Liquid. In particular, the electron propagator of QH edge state will be shown to have the form where v α is a renormalized edge velocity, and v β,γ are renormalized sound velocities. As a consequence, even the integer quantum Hall edge states will not be Fermi liquids in the presence of electron-phonon interactions.
The modification of the single-particle propagator by the electron-phonon interactions may be detected experimentally, and we will discuss two possibilities. First, the AC conductance will have resonances at (longitudinal) wave-vectors q and frequencies ω related by q = ω/v α , q = ω/v β , and q = −ω/v γ which may in principle be resolved and detected in an experiment. On the other hand, the DC Hall conductance is not modified by the electronphonon interactions. Second, the anomalous exponents in the single-particle propagator will modify the tunneling conductance and can in principle be measured, for example in tunneling between two ν = 1 edges in a bilayer system [14] with an overall filling of ν tot = 2. In the absence of electron-phonon interactions the edges of such a system are (chiral) Fermi liquids which propagate in the same direction. Coulomb interactions alone cannot change the temperature dependence of the tunneling conductance from T 0 . Therefore, any temperature behavior of the tunneling conductance at sufficiently low temperatures must be due to the electron-phonon interactions. The electron-phonon interaction also modifies the temperature dependence of the tunneling conductance between counterpropagating edge states at very low temperatures which has been measured by Milliken et al. [10] The edge excitations in a quantum Hall system with filling fraction ν = 1/(2m + 1) can be described by density modulations of an effectively one-dimensional system [3] where the densities J k obey commutation relations depending on the filling fraction ν: Here, k is the wave-vector along the edge. This theory represents a single U(1) Kac-Moody algebra and it is well known how to "bosonize" it in terms of a chiral boson φ R (x). The current density j(x) is written as Using J k = dx e ikx j(x), we can immediately express the Hamiltonian in terms of the annihilation and creation operators of the boson mode expansion: (the zero modes are omitted). The full (spin-polarized) electron field can also be written in terms of the chiral boson and has been identified as [3] ψ(x) ∝ exp i 1/4νπφ R (this is not to be confused with the quasi-particle field χ which can also be defined in terms of the chiral boson χ ∝ e i √ ν/4πφ R , but carries fractional charge). We now consider the interaction of such a system with phonons One normal mode of the phonons is assumed to be along the quantum Hall edge, so that the crystal displacement d(x) along this edge can be expressed as where ρ is the linear mass density of the crystal. The electron-phonon interaction then becomes where D is the deformation potential constant and the coupling v c = D ν/πρv s is independent of k. For specific values of these parameters, we consider a QH edge in a GaAs heterojunction. We assume that the edge is along one of the cubic axes of GaAs so that the piezoelectric coupling vanishes and can be ignored. For electrostatic confinement by an electrode with potential V g , v F is of the order of ω c ℓ 2 B /ℓ, where ℓ B = [hc/(eB)] 2 is the magnetic length and ℓ = V g ǫ/(4π 2 n 0 e) is the length scale of the electrostatic confining potential [15] (ǫ is the static dielectric constant and n 0 is the two-dimensional electron density). For a magnetic field strength of about 5 T and a density of n 0 = 10 15 m −2 , this gives a Fermi velocity approximately equal to the average sound velocity v s ≈ 5 ×10 3 m/s in GaAs. Thus, it should be possible to optimize the effects of electron-phonon interactions in GaAs heterojunctions under ordinary conditions. The deformation potential constant D is approximately 7.4 eV [16]. Assuming an effective cross-sectional area of 10 −14 m −2 of the GaAs phonon system in the direction perpendicular to the electron propagation, we then arrive at a coupling velocity v c /v s ∼ 0.1.
At this point it is straightforward to diagonalize the complete Hamiltonian by using a generalized Bogliubov transformation T where T is given in terms of three variables φ, θ, η: The Hamiltonian is now written as where the coupling matrix A is given by For the Hamiltonian to become diagonal, the off-diagonal elements of A are required to vanish, which determines the three angles φ, θ, η and in turn also the diagonal elements of A (i.e. the renormalized Fermi and sound velocities). Moreover, the boson representing the electron density now becomes according to Eq. (10) Therefore, the electron field in Eq. (5) must now be expressed in terms of three independent boson fields, namely a right moving "charge" boson α k and one right and one left moving "sound" boson β k , γ k . The electron propagator then becomes a product of three factors according to Eqs. (5) and (14) G where v α = A 11 is the renormalized Fermi velocity and v β = A 22 , v γ = A 33 are the renormalized sound velocities. In Fig. 1 we have plotted the renormalized velocities v i vs. v F (i = α, β, γ) for a coupling of v c /v s = 0.1. The total 'momentum' v α + v β − v γ = v F of the electron is conserved by the transformation T . The right propagating velocities v α and v β show a discontinuity at the resonance v F = v s of equal magnitude which appears to be quadratic in the coupling v c /v s . The left propagating velocity v γ is only slightly modified of the order of 1% from its original value v s . The fact that the electron propagator breaks up into three pieces, corresponding to the normal modes of the Hamiltonian Eq. (12), will have experimental consequences for transport properties. We first calculate the linear response to a scalar potential φ(x, y) = −Ey cos(qx − ωt), where we take q, ω > 0. This potential gives an electric field E(x, y) = E [−qy sin(qx − ωt)x + cos(qx − ωt)ŷ] , where y can be taken to be constant. The perturbed charge density in response to the potential is then obtained as By using the continuity equation ∂ρ/(∂t) + ∂j(x)/(∂x) = 0, we can then obtain the current response function to the applied potential as Experiments dictate that the DC Hall conductance σ H = lim ω→0 lim q→0 σ(q, ω) must not be altered from its quantized value e 2 ν/h, which is indeed the case according to Eq. (17) since the matrix elements obey the sum rule T 2 11 + T 2 12 − T 2 13 = 1. On the other hand, the AC conductance will exhibit resonance structures when ω = v α q, ω = v β q, and ω = −v γ q in response to a potential φ(x, y). Provided at least two of the 'spectral weights' T 2 11 , T 2 12 , and T 3 13 are not too small, and the corresponding renormalized velocities are not too close, these resonances can then in principle be resolved and detected. In Fig. 2 we see that near v F /v s ∼ 1, both T 2 11 and T 2 12 are close to 0.5, while v α and v β are on opposite sides of v s (Fig. 1). With an experimentally reasonable value of q ∼ 10 5 m −1 and v s ∼ v F ∼ 10 3 m/s, this gives a resonance at about 10 8 Hz, well within experimentally accessible range. Figure  2 also shows that the 'total spectral weight' of the electron consists mostly of the forward propagating modes, which contribute almost equally at resonance v F = v s . This means that only very little charge is transported in the counterpropagating direction which was to be expected.
Since the single-particle propagator is changed by the electron-phonon interaction according to Eq. (15), the single-particle density-of-states and properties depending on it such as the tunneling conductance will also be affected by the coupling to the phonons. In particular, we consider the inter-layer tunneling conductance between two edges of a bilayer system with integer filling in each layer, and a total filling factor of ν tot = 2. This can in principle be measured by attaching probes to the different layers separately in a system with large enough separation between the layers that the bulk tunneling probability vanishes. A gate at the edge can then be used to adjust the tunneling probability between the edges. At low enough voltage across the tunneling junction, tunneling through the bulk will be suppressed, and only the edge tunneling current appreciable. Note that in this arrangement, the two edges propagate in the same direction.
The tunneling current is determined by the retarded response function [6] X where ΓA = Γψ 1 (x = 0)ψ 2 † (x = 0)+h.c. is the tunneling operator, with Γ the tunneling amplitude, and ψ i (x), i = 1, 2, the electron field operator on the two edges. From Eq. (15) and following Wen [6] it is a straightforward exercise to determine the temperature dependence of the tunneling differential conductance, with the result that Due to the fact that T 2 13 = 0 in the presence of electron-phonon interactions, the differential tunneling conductance will now depend on temperature, in fundamental contrast to the tunneling between two chiral Fermi liquids propagating in the same direction, which is temperature independent at low temperatures, even in the presence of electron-electron interactions between the two edges. From Fig. 2 we see that the magnitude of the exponent −4T 2 13 for the parameters chosen here is of the order of 10 −2 , which is small (the exponent appears to be quadratic in the coupling constant v c /v s ). However, the main point is that any measured temperature dependence at all will be due to the electron-phonon interactions.
In conclusion we have shown that the electron-phonon interaction modifies the chiral Luttinger Liquid on the quantum Hall edge. The single particle propagator becomes a product of three separate modes, one of which is always counterpropagating. From this we have predicted direct experimental consequences for the AC conductivity of the quantum Hall bar and the temperature dependence of the tunneling conductance.