Emergent Chern-Simons excitations due to electron-phonon interaction

Abstract

We address the problem of Dirac fermions interacting with transversal optical phonons. A gap in the spectrum of fermions leads to the emergence of the Chern-Simons excitations in the spectrum of phonons. We study the effect of those excitations on observable quantities: the phonon dispersion, the phonon spectral density, and the Hall conductivity.

Spontaneous mass generation due to phonons in a two-dimensional Dirac fermion system

“…It turned out that the transport properties for a Fermi energy near the Dirac node are very robust against disorder [12,13]. This has also been supported by theoretical studies [3,5] and was recently attributed to an attractive fixed point in the renormalization-group flow [14]. Lattice distortions in the form of phonons are expected to have a more severe effect on transport because they may open an electronic gap for sufficiently strong electron-phonon coupling [14][15][16][17][18][19].…”

“…For details of this coupling we also refer to our earlier work Ref. [14]. The 2 + 1-dimensional action of this model then reads…”

“…Thus, it is sufficient to study the Hall conductivity at a single massive Dirac fermion and to assume that the other Dirac fermion appears with the opposite mass sign. This idea was employed to study the effect of phonons on the Hall conductivity in the case of weak electron-phonon interaction [14]. The effective Chern-Simons term gives the correct values of the two Hall plateaux, but contributions from other quantum fluctuations generate small corrections.…”

Quantum Hall effect induced by electron-phonon interaction

“…This resembles the popular in electrodynamics "light cone" gauge fixing n • A = 0, with n denoting a unit vector pointing into the time direction [2]. It was shown in [31] that in such system phonon Chern-Simons excitations can in principle be generated.…”

“…The spectrum of these modes reveals a pronounced weak alteration over the entire Brillouin zone [30,33,34], which makes it possible to model them in the form of dispersionless monochromatic lattice vibrations. Neglecting the slow dynamics of heavy lattice ions we effectively end up with the following Euclidean action of an interacting electron-phonon system: is related to the inverse frequency of monochromatic phonons [29,31]. The correlation functions of currents can be obtained from the generating functional W[A] = log exp{−S[A]} by repeating variations with respect to the fields A = { α, β}, S[A] is the action (5) augmented by an auxiliary source term…”