Abstract
We find a new relation between the spectral problem for Bloch electrons on a two-dimensional honeycomb lattice in a uniform magnetic field and that for quantum geometry of a toric Calabi-Yau threefold. We show that a difference equation for the Bloch electron is identical to a quantum mirror curve of the Calabi-Yau threefold. As an application, we show that bandwidths of the electron spectra in the weak magnetic flux regime are systematically calculated by the topological string free energies at conifold singular points in the Nekrasov-Shatashvili limit.
Highlights
That the honeycomb lattice system corresponds to an unconventional moduli identification while the triangular lattice to a more natural one
We find a new relation between the spectral problem for Bloch electrons on a two-dimensional honeycomb lattice in a uniform magnetic field and that for quantum geometry of a toric Calabi-Yau threefold
We show that a difference equation for the Bloch electron is identical to a quantum mirror curve of the Calabi-Yau threefold
Summary
We start with a short review of an electron system in a two-dimensional honeycomb lattice. We denote the two sub-lattices as A and B, as shown in figure 1. We turn on a magnetic field perpendicular to the lattice plane. As seen in [5], the eigenvalue equations of the electron are given by the following two-dimensional difference equations:. The magnetic flux φ is normalized as φ = 2πΦ/Φ0 where Φ is the flux per unit cell and Φ0 = hc/e. These eigenvalue equations are our starting point. If the magnetic field is turned off (φ = 0), the difference equation leads to λψB(x) = 2 cos. We show the spectra of λ and of E as functions of rational φ in figure 2
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