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Short version: Is it possible to arrange the fluxes for the Kagomé lattice with triangle flux $\phi_\triangle=\frac{\pi}2$ and hexagon flux $\phi_{hex}=0$ using a single unit cell?

Longer version: I am looking at fermionic mean field theories on the Kagomé lattice that describe a chiral spin liquid state for spin-1/2. Skipping the derivation (in 1 and 2), the mean field Hamiltonian is

$H = -\sum\limits_{\langle i, j\rangle,\sigma}~\rho~e^{iA_{ij}}~f^\dagger_{i\sigma}~f_{j\sigma} + H.c.$

This basically says that $f$-fermions hop along nearest-neighbor links, picking up a phase $A_{ij}$ one way along the bond, and $-A_{ij}$ hopping in the reverse direction. Phases (aka gauge field) are assigned to each lattice link in a unit cell, often depicted with arrows. The flux $\phi$ is defined as the sum of phases on a closed lattice plaquette along, say, the counter-clockwise direction.

In Marston's paper (Ref. 1), and Ran's paper (Ref. 2), they introduce a state with triangle flux $\frac{\pi}2$ and hexagon flux 0 (equivalent to Chua's (Ref. 5) (d)), and SL-[$\frac{\pi}2$,0] in Kim's diagram (Ref. 6).

Flux assignments From Ref. 5

Phase and Flux assignments From Ref. 6, referencing Ref. 2

My problem is with that with Kim's (Ref. 6) phase assignment for the triangles, I don't see how it is possible to achieve zero (or 2$\pi$) hexagon flux, unless one uses a doubled magnetic unit cell, as is claimed in Chua's work (Ref. 5). Ran (Ref. 2) doesn't explicitly state whether the unit cell has to double or not. Am I missing something here?

J. B. Marston and C. Zeng, J. Appl. Phys. 69, 8 (1991), Spin‐Peierls and spin‐liquid phases of Kagomé quantum antiferromagnets

Y. Ran, M. Hermele, P. A. Lee, and X.-G. Wen, PRL 98, 117205 (2007), Projected-Wave-Function Study of the Spin-1/2 Heisenberg Model on the Kagome ́Lattice

V. Chua, H. Yao and G. A. Fiete, PRB 83, 180412(R) (2011), Exact chiral spin liquid with stable spin Fermi surface on the kagome lattice

Jung Hoon Kim & Jung Hoon Han talk, http://manybody.skku.edu/manybody/ppt/Chiral_Spin_States_Pyrochlore-A3-Shanghai.pdf

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