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 $\kappa \propto \zeta (3)/L^3$, 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 ${\eta }_{r-s}$, 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 ${\eta }_{r-s}$ yields logarithmic, parity-dependent transverse-state coupling, while large ${\eta }_{r-s}$ 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.

A revised replacing the previously withdrawn manuscript. Title refined, the current version is substantially expanded with Brillouin-zone analysis and regularization of the one-dimension problem. 16 pages, 2 figures and Glossary included