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Duncan Haldane constructed a two-dimensional lattice model with quantized Hall conductivity without a net applied magnetic field. The Haldane model separates the topological mechanism of the quantum Hall effect from the usual formation of Landau levels in a uniform field. Changing model parameters closes and reopens an energy gap, separating phases with different Hall responses. This provides a compact theoretical example in which the global structure of occupied electronic states determines a measurable transport coefficient.^[ref]

A physicist researching condensed matter physics.^[ref]

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 $a$, $t_1>0$, $t_2\ge 0$ and ordered sublattice basis $(A,B)$, chooseThe following Bloch Hamiltonian defines $\varphi$; reversing directed second-neighbour hopping sends $\varphi \mapsto -\varphi$:The Pauli matrices act on sublattice amplitudes, as in sublattice pseudospin in graphene. The identity matrix term $d_{0}I$ 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 ${\min }_{\mathbf{k}}{E}_{+}>{\max }_{\mathbf{k}}{E}_{-}$. We use $t_2/t_1=0.08$ in the figures, within the original paper's sufficient nonoverlap restriction $|t_2/t_1|<1/3$.^[ref]