Quantum Phase diagrams and transitions for Chern topological insulators

TL;DR

This work develops a comprehensive geometry-driven framework for Chern topological insulators, showing that any phase diagram with integer Chern-number transitions can be realized by pulling back a basic spin-1/2 Hamiltonian along higher-degree maps, with wall crossings encoded by rose curves $r=\cos(k\theta)$ and Dirac points counting as the local charges. The authors connect abstract bundle-theoretic invariants to concrete lattice constructions, using momentum-space Hamiltonians and commensurate sublattices to implement arbitrary Chern numbers $C$ via degree-$d$ maps, including $C=N^2$ through distant-neighbor couplings. They provide explicit lattice realizations on Gaussian and Eisenstein lattices (square, triangular) and extend the approach to honeycomb and Kagome lattices, deriving criteria for admissible distant neighbors to preserve lattice symmetry and result in higher Chern numbers. The results offer a principled method to design phase diagrams and engineer lattice models with desired topological transitions, with potential implications for topological quantum computation and materials science where high-Chern-number phases are sought.

Abstract

Topological invariants such as Chern classes are by now a standard way to classify topological phases. Introducing and varying parameters in such systems leads to phase diagrams, where the Chern classes may jump when crossing a critical locus. These systems appear naturally when considering slicing of higher dimensional systems or when considering systems with parameters. As the Chern classes are topological invariants, they can only change if the "topology breaks down". We give a precise mathematical formulation of this phenomenon and show that synthetically any phase diagram of Chern topological phases can be designed and realized by a physical system, using covering, aka. winding maps. Here we provide explicit families realizing arbitrary Chern jumps. The critical locus of these maps is described by the classical rose curves. These realize the lower bound on the number of Dirac points necessary obtained from viewing them as local charges. We treat several concrete models and show that they have the predicted generic behavior. In particular, we focus on different types of lattices and tight-binding models, and show that effective winding maps, and thus higher Chern numbers, can be achieved using k-th nearest neighbors. We give explicit formulas for a family of 2D lattices using imaginary quadratic field extensions and their norms. Our study includes the square, triangular, honeycomb and Kagome lattices.

Figures (20)

• Figure 1: Rose curves for different values of $t,d$ and $d'$. The origin is in the image for $t=\frac{1}{2}$. The winding numbers are constant for $0\leq t<\frac{1}{2}$ and $\frac{1}{2}<t\leq 1$. These can be readily be read off by using a ray.
• Figure 2: Surface plot of the function given in Eq. (\ref{['eq:MBphase']}) for $M=B=1$.
• Figure 3: Phase diagram corresponding to the phases listed in Table \ref{['tab:phases']}.
• Figure 4: Haldane model on a honeycomb lattice with two different atom species (blue and red). The interactions are $t_1$ between the first nearest neighbors and $t_2 e^{\pm i\phi}$ between the second nearest neighbors using the convention shown.
• Figure 5: A slice of the 3-parameter family of Hamiltonians with $t_2 = 1$ in the Haldane Hamiltonian Eq. \ref{['equation:HaldaneH']}. The Chern number takes values 0,-1,1 for different values of the Hamiltonian parameters $\phi$ and $m$.

Theorems & Definitions (42)

• Theorem 2.1
• proof
• Corollary 2.1
• Corollary 2.2
• Lemma 2.1
• proof
• Theorem 2.2
• Definition 2.1
• Proposition 2.1
• Definition 3.1