Bloch states of an infinite one-dimensional lattice under transverse electric field

Abstract:We consider an effectively one-dimensional periodic atomic chain subjected to a weak static electric field transverse to the chain axis. The intrinsic lattice is represented in reciprocal space by a harmonically weighted periodic pseudopotential, while the transverse field produces a weak elastic and polarizable response whose minimum-energy distribution is approximated by a periodically repeated catenary profile. Bloch reduction places both contributions in a common reciprocal-space Hamiltonian. The intrinsic lattice produces a \ (|r-s|^ {-1} \) hierarchy, whereas the catenary contribution is a symmetric Toeplitz correction with coefficients proportional to \((1+\pi^2(r-2) ^ {-1} \) and therefore asymptotically to \ (|r-s|^ {-2} \). The complete nonzero reciprocal weight of the catenary contribution is absolutely convergent and sums to \ (|V_ {\rm c} |/e \). At the unit-cell level, the lowest intrinsic harmonic together with the catenary term gives a Mathieu equation containing both cosine and hyperbolic-cosine modulations. A first-Brillouin-zone reduction then yields the field-dependent band energies and the corresponding hierarchy of Bragg gaps. The model isolates a simple mechanism by which a transverse electric field can reorganize the Bloch spectrum of a low-dimensional periodic system. Keywords: Bloch theory; transverse electric field; catenary potential; Mathieu equation; Brillouin zone; band structure.

Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)

arXiv:2608.22199 [quant-ph]

(or arXiv:2608.22199v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.22199 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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