Fractional Wannier Orbitals and Tight-Binding Gauge Fields for Kitaev Honeycomb Superlattices with Flat Majorana Bands

TL;DR

This paper introduces a systematic framework for deriving low-energy lattice models of fractional orbitals coupled to gauge fields, demonstrated on Majorana states in the Kitaev honeycomb model, revealing topological phase transitions and fractional Chern states.

Contribution

It presents a novel method to construct gauge-invariant tight-binding models for fractional orbitals, including a mean-field theory for interacting Majorana states, applicable to various gauge theories.

Findings

Constructed Wannier orbitals for Majorana states in Kitaev model.

Identified a phase transition between topological phases with flat bands.

Developed a gauge-invariant mean-field theory leading to fractional Chern states.

Abstract

Fractional excitations hold immense promise for both fundamental physics and quantum technologies. However, constructing lattice models for their dynamics under gauge fields remains a formidable challenge due to inherent obstructions. Here, we introduce a novel and systematic framework for deriving low-energy lattice models of fractional orbitals coupled to tight-binding gauge fields. Departing from conventional geometric approaches, our method systematically eliminates the high-energy states via virtual hopping, thereby deriving the gauge potential and quantum metric through a superexchange-like mechanism. We demonstrate the framework by constructing Wannier orbitals for Majorana states and a tight-binding $Z_2$ gauge field across various flux crystalline phases in the Kitaev spin model on a honeycomb lattice. Our study reveals a striking phase transition between two non-trivial topological phases characterized by gapless flat-band with extensive degeneracy. Furthermore, we develop a gauge-invariant mean-field theory for interacting Majorana orbitals, leading to a correlation-induced fractional Chern state. Our work establishes a general framework for gauge-mediated tight-binding models and a gauge-invariant mean-field theory for interacting fractional orbitals that can be readily extended to $U(1)$, $SU(N)$ lattice gauge theories.