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 \((1+π^2(r-2) ^ {-1} \) 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.