Zero modes of non-abelian Dirac operator in topologically non-trivial band insulator
Abstract
We show that the local gauge-invariance of the quantum geometric tensor (QGT) defined in the two-dimensional Block-momentum (BM) space of a generic N-band (arising out of sub-lattice degrees of freedom) insulator represented by a tight-binding hamiltonian, leads to the existence of zero modes of non-abelian Dirac operator in such BM space. These zero modes in the two-dimensional Brillouin zone torus T2, determines the probability amplitudes associated with the N-component ground state wave-function under adiabatic approximation. We analytically derive the expansion coefficients of maximally localized Wannier function in one-dimension from the solutions of these zero mode equations, and obtain momentum-space generalization of celebrated Laughlin gauge condition as a by-product.The commutator of these non-abelian Dirac operators define the algebra of the guiding center co-ordinates which provide a more general frame-work to study such systems than one based on the Landau-level analogues. In the ideal quantum geometric limit, these zero-mode solutions become self-dual momentum-space instantons, and can be linked to the non-abelian generalization of the vortexability criterion of Chern bands in momentum-space. We derive some exact solutions of such zero-mode on the complex torus geometry, and discuss implications of this framework for realistic systems such as two-dimensional topological insulators, rhombohedral NL-layer graphene.
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BibTeX
@misc{kumari2026zero,
title = {{Zero modes of non-abelian Dirac operator in topologically non-trivial band insulator}},
author = {Neha Kumari and Sankalpa Ghosh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.27608},
eprint = {2605.27608},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.27608},
}
RIS
TY - UNPB
TI - Zero modes of non-abelian Dirac operator in topologically non-trivial band insulator
AU - Neha Kumari
AU - Sankalpa Ghosh
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2605.27608
UR - https://arxiv.org/abs/2605.27608
AB - We show that the local gauge-invariance of the quantum geometric tensor (QGT) defined in the two-dimensional Block-momentum (BM) space of a generic N-band (arising out of sub-lattice degrees of freedom) insulator represented by a tight-binding hamiltonian, leads to the existence of zero modes of non-abelian Dirac operator in such BM space. These zero modes in the two-dimensional Brillouin zone torus T^2, determines the probability amplitudes associated with the N-component ground state wave-function under adiabatic approximation. We analytically derive the expansion coefficients of maximally localized Wannier function in one-dimension from the solutions of these zero mode equations, and obtain momentum-space generalization of celebrated Laughlin gauge condition as a by-product.The commutator of these non-abelian Dirac operators define the algebra of the guiding center co-ordinates which provide a more general frame-work to study such systems than one based on the Landau-level analogues. In the ideal quantum geometric limit, these zero-mode solutions become self-dual momentum-space instantons, and can be linked to the non-abelian generalization of the vortexability criterion of Chern bands in momentum-space. We derive some exact solutions of such zero-mode on the complex torus geometry, and discuss implications of this framework for realistic systems such as two-dimensional topological insulators, rhombohedral N_L-layer graphene.
ER -