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 Incoming links: F. Duncan M. Haldane

F. Duncan M. Haldane's 1988 tight-binding model adds complex second-neighbour hopping and opposite sublattice offsets to a honeycomb lattice. Directed phases break time-reversal symmetry while each primitive cell has zero net magnetic flux. Smaller loops can enclose magnetic flux; ordinary Bloch states remain available.^[ref]

Extend the nearest-neighbour model of graphene, ignoring physical spin and interactions. With bond length

a

t_{1} > 0

t_{2} \geq 0

and ordered sublattice basis

(A, B)

, choose

δ_{1} b_{1} f (k) = a (0, 1), = δ_{3} - δ_{2}, = j = 1 \sum 3 e^{i k \cdot δ_{j}} . δ_{2} b_{2} = a (3 /2, - 1/2), = δ_{1} - δ_{3}, δ_{3} b_{3} = a (- 3 /2, - 1/2), = δ_{2} - δ_{1},

ϕ

; reversing directed second-neighbour hopping sends

ϕ \mapsto - ϕ

d_{0} d_{x} d_{z} H = 2 t_{2} cos ϕ j \sum cos (k \cdot b_{j}), = - t_{1} Re f, = M - 2 t_{2} sin ϕ j \sum sin (k \cdot b_{j}), = (d_{0} + d_{z} - t_{1} f^{*} - t_{1} f d_{0} - d_{z}), d_{y} E_{\pm} = t_{1} Im f, = d_{0} \pm t_{1}^{2} ∣ f ∣^{2} + d_{z}^{2} .

d_{0} I

shifts both energy eigenvalues equally at each momentum; it changes neither the eigenvectors nor the direct separation

E_{+} - E_{-}

. A genuine band insulator additionally needs

min_{k} E_{+} > max_{k} E_{-}

. We use

t_{2} / t_{1} = 0.08

in the figures, within the original paper's sufficient nonoverlap restriction

∣ t_{2} / t_{1} ∣ < 1/3