Perturbative/nonperturbative aspects of Bloch electrons in a honeycomb lattice

Abstract

We revisit the spectral problem for Bloch electrons in a two-dimensional bipartite honeycomb lattice under a uniform magnetic field. It is well-known that such a honeycomb structure is realized in graphene. We present a systematic framework to compute the perturbative magnetic flux expansions near two distinct band edges. We then analyze the nonperturbative bandwidth of the spectrum. It turns out that there is a novel similarity between the spectrum near the Dirac point in the honeycomb lattice and the spectrum in the supersymmetric sine-Gordon quantum mechanics. We finally confirm a nontrivial vacuum-instanton-bion threesome relationship. Our analysis heavily relies on numerical experiments.

24 pages, v2: references added

References in corpus (7)

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• Deconstructing zero: resurgence, supersymmetry and complex saddles
• Hofstadter's Butterfly in Quantum Geometry
• The grin of Cheshire cat resurgence from supersymmetric localization
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Cited by in corpus (6)

• Bion non-perturbative contributions versus infrared renormalons in two-dimensional $\mathbb{C}{P}^{N-1}$ models
• Instantons in the Hofstadter butterfly: difference equation, resurgence and quantum mirror curves
• Topological states in the Hofstadter model on a honeycomb lattice
• Bloch electrons on honeycomb lattice and toric Calabi-Yau geometry
• The quantum Hall effect at a weak magnetic field
• Dirac fermion spectrum of the fractional quantum Hall states