Bloch states of an infinite alternating-charge lattice under a transverse electric field
Abstract
We consider an infinite one dimensional array of alternating charges embedded in a two-dimensional configuration space and subjected to a weak static electric field transverse to the lattice axis. Expansion of the Coulomb interaction about a lattice site gives a transverse restoring stiffness κ∝ ζ(3)/L3, defining the oscillator length ay, while Bloch periodicity is retained along the lattice direction and the transverse motion is represented in a harmonic basis. The Coulomb matrix elements are governed by the dimensionless scale ηr−s, which couples the reciprocal-lattice transfer to the transverse oscillator length. A regularized one dimensional extension is also introduced through Coulomb regularization, yielding a finite diagonal offset and an exponentially decaying even-transfer coupling in reciprocal space. Small ηr−s yields logarithmic, parity-dependent transverse-state coupling, while large ηr−s approaches the one-dimensional Coulomb limit. In the finite-order spectrum, the latter approaches the Kronig Penney reference toward the band edge while retaining distinct Brillouin zone curvature. The formulation therefore identifies two characteristic quantities of the alternating lattice: the transverse restoring stiffness and the one-dimensional Madelung form, while retaining the two dimensional structure required to describe the response to a transverse electric field. Keywords: alternating charge lattice; Bloch states; Coulomb interaction; transverse electric field; regularization; reciprocal space coupling; Brillouin zone spectrum.
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BibTeX
@misc{kumar2026bloch,
title = {{Bloch states of an infinite alternating-charge lattice under a transverse electric field}},
author = {Anand Aruna Kumar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.22199},
eprint = {2608.22199},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22199},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Bloch states of an infinite alternating-charge lattice under a transverse electric field
AU - Anand Aruna Kumar
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2608.22199
UR - https://arxiv.org/abs/2608.22199
AB - We consider an infinite one dimensional array of alternating charges embedded in a two-dimensional configuration space and subjected to a weak static electric field transverse to the lattice axis. Expansion of the Coulomb interaction about a lattice site gives a transverse restoring stiffness κ∝ ζ(3)/L^3, defining the oscillator length a_y, while Bloch periodicity is retained along the lattice direction and the transverse motion is represented in a harmonic basis. The Coulomb matrix elements are governed by the dimensionless scale η_r−s, which couples the reciprocal-lattice transfer to the transverse oscillator length. A regularized one dimensional extension is also introduced through Coulomb regularization, yielding a finite diagonal offset and an exponentially decaying even-transfer coupling in reciprocal space. Small η_r−s yields logarithmic, parity-dependent transverse-state coupling, while large η_r−s approaches the one-dimensional Coulomb limit. In the finite-order spectrum, the latter approaches the Kronig Penney reference toward the band edge while retaining distinct Brillouin zone curvature. The formulation therefore identifies two characteristic quantities of the alternating lattice: the transverse restoring stiffness and the one-dimensional Madelung form, while retaining the two dimensional structure required to describe the response to a transverse electric field. Keywords: alternating charge lattice; Bloch states; Coulomb interaction; transverse electric field; regularization; reciprocal space coupling; Brillouin zone spectrum.
ER -