Exact Mirror-Parity Selection Rule for Electron-Phonon Coupling in Two-Dimensional Crystals
Abstract
The electron--phonon (e-ph) interaction underlies conventional superconductivity and plays a central role in transport and other many-body phenomena in solids. Here, we derive a symmetry selection rule for e-ph scattering in two-dimensional crystals possessing a horizontal mirror plane, σh. For electronic states of mirror parity sn(k) and sm(k+q), and a phonon mode of parity χν(q), we show that the e-ph matrix element gmn,ν(k,q) vanishes identically unless sn(k) χν(q) sm(k+q)=+1. The rule follows from the covariance of the phonon-induced deformation potential under σh, constitutes the Cs case of the general group-theoretical selection rule for matrix elements, and partitions the e-ph interaction throughout the Brillouin zone into symmetry-allowed and symmetry-forbidden scattering channels. Consequently, mirror-even phonons couple electronic states of the same mirror parity, either even--even or odd--odd, whereas mirror-odd phonons couple only states of opposite mirror parity, for both intra- and interband processes. We illustrate the selection rule through first-principles calculations for the two-dimensional superconductor Sc2BCH4, where mirror parity is well defined throughout the entire Brillouin zone and symmetry-forbidden matrix elements vanish up to numerical precision. We then extend the analysis to bulk MgB2, where the rule applies exactly only on mirror-invariant planes of the three-dimensional Brillouin zone. Moving away from these planes removes the exact mirror constraint and activates scattering channels that are symmetry forbidden when the mirror eigenvalues are well defined. Our results establish horizontal mirror symmetry as an exact and practical filter of e-ph scattering processes.
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BibTeX
@misc{barrera2026exact,
title = {{Exact Mirror-Parity Selection Rule for Electron-Phonon Coupling in Two-Dimensional Crystals}},
author = {Nicolás F. Barrera and Javiera Cabezas-Escares and Francisco Munoz and Carlos Cardenas and Julia Contreras-García},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05710},
eprint = {2611.05710},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05710},
}
RIS
TY - UNPB
TI - Exact Mirror-Parity Selection Rule for Electron-Phonon Coupling in Two-Dimensional Crystals
AU - Nicolás F. Barrera
AU - Javiera Cabezas-Escares
AU - Francisco Munoz
AU - Carlos Cardenas
AU - Julia Contreras-García
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05710
UR - https://arxiv.org/abs/2611.05710
AB - The electron--phonon (e-ph) interaction underlies conventional superconductivity and plays a central role in transport and other many-body phenomena in solids. Here, we derive a symmetry selection rule for e-ph scattering in two-dimensional crystals possessing a horizontal mirror plane, σ_h. For electronic states of mirror parity s_n(k) and s_m(k+q), and a phonon mode of parity χ_ν(q), we show that the e-ph matrix element g_mn,ν(k,q) vanishes identically unless s_n(k) χ_ν(q) s_m(k+q)=+1. The rule follows from the covariance of the phonon-induced deformation potential under σ_h, constitutes the C_s case of the general group-theoretical selection rule for matrix elements, and partitions the e-ph interaction throughout the Brillouin zone into symmetry-allowed and symmetry-forbidden scattering channels. Consequently, mirror-even phonons couple electronic states of the same mirror parity, either even--even or odd--odd, whereas mirror-odd phonons couple only states of opposite mirror parity, for both intra- and interband processes. We illustrate the selection rule through first-principles calculations for the two-dimensional superconductor Sc_2BCH_4, where mirror parity is well defined throughout the entire Brillouin zone and symmetry-forbidden matrix elements vanish up to numerical precision. We then extend the analysis to bulk MgB_2, where the rule applies exactly only on mirror-invariant planes of the three-dimensional Brillouin zone. Moving away from these planes removes the exact mirror constraint and activates scattering channels that are symmetry forbidden when the mirror eigenvalues are well defined. Our results establish horizontal mirror symmetry as an exact and practical filter of e-ph scattering processes.
ER -