Abstract
We present an ab initio analysis of a continuous Hamiltonian that maps into the celebrated Haldane model. The tunnelling coefficients of the tight-binding model are computed by means of two independent methods - one based on the maximally localized Wannier functions, the other through analytic expressions in terms of gauge-invariant properties of the spectrum - that provide a remarkable agreement and allow to accurately reproduce the exact spectrum of the continuous Hamiltonian. By combining these results with the numerical calculation of the Chern number, we are able to draw the phase diagram in terms of the physical parameters of the microscopic model. Remarkably, we find that only a small fraction of the original phase diagram of the Haldane model can be accessed, and that the topological insulator phase is suppressed in the deep tight-binding regime.
Highlights
The Haldane model [1] is a celebrated lattice model describing a Chern insulator [2], characterized by the presence of quantum Hall effect [3] in the absence of a macroscopic magnetic field
This feature will be crucial for the analysis presented where we shall redraw the topological phase diagram in terms of the physical parameters—α, χA, and s—of the underlying continuous Hamiltonian
Owing to the above analysis, we suggest that a more appropriate way to draw the topological phase diagram is in terms of the physical parameters that characterize the underlying continuous Hamiltonian, namely, α, χA, and s
Summary
The Haldane model [1] is a celebrated lattice model describing a Chern insulator [2], characterized by the presence of quantum Hall effect [3] in the absence of a macroscopic magnetic field. Haldane showed that the properties of the system depend on the interplay between the phase acquired by t1 and the effect of parity breaking, affecting the topological phase diagram of the model [1]. We show that the two approaches considered provide remarkable agreement even in the presence of parity breaking, allowing for a precise determination of the tight-binding parameters of the model. By combining these results with the numerical calculation of the Chern number, we are able to redraw the topological phase diagram of the Haldane model in terms of the physical parameters of the microscopic model. In the appendices we present an analysis of the spread functional of the MLWFs (Appendix A) and additional remarks on the numerics (Appendix B)
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