# Examples of Simple Hamiltonians Giving Different Phases?

I am trying to study about simple condensed matter models that I can simulate numerically and use to calculate some topological invariant that defines a (topological) phase. My interest is in Hamiltonians that I can write in terms of momentum space coordinates and Pauli spin matrices, and that have parameters that can be tweaked to achieve different phases.

• Example 1: From 1, I have the so-called Qi-Wu-Zhang (QWZ) model: $$H(k) = \sin{k_x}\sigma_x+\sin{k_y}\sigma_y+[u+\cos{k_x}+\cos{k_y}]\sigma_z,$$ where $$u$$ tunes the energy gap of the spectrum. For $$|u|>2$$, the Chern number $$Q = 0$$, while for $$-2, $$Q = -1$$ and for $$0, $$Q = +1$$. The gap closes when $$u \in (-2,0,2)$$. This model fits my requirements because I can use the given Hamiltonian easily given its dependence on momenta $$(k_x,k_y)$$, and the fact that I can use $$u$$ to achieve different phases by calculating the Chern number that describes them.

• Example 2: Problem 2.3 of the same resource 1 gives a massive Dirac Hamiltonian: $$H(k_x,k_y)=m\sigma_z+k_x\sigma_x+k_y\sigma_y,$$ but I could not find more literature to identify the different phases this Hamiltonian gives, nor anything else to compare my results against.

• Example 3: A third example I found, from 2, was the 2D Haldane model: \begin{align} h_0 &= 2 t_2 \cos \phi \sum_{i=1}^3 \cos(k \cdot b_i) \\ h_x &= t \left[ 1 + \cos(k \cdot b_1) + \cos(k \cdot b_2) \right] \\ h_y &= t \left[ \sin(k \cdot b_1) - \sin(k \cdot b_2) \right] \\ h_z &= M - 2 t_2 \sin \phi \sum_{i=1}^3 \sin(k \cdot b_i) \end{align} Here, $$M$$ is the symmetry-breaking term, and $$\phi$$ is the phase that reflects the chosen flux inside the Haldane cell. When $$|M|> 3\sqrt{3} t_2 \sin{\phi}$$, $$Q = 0$$, but when $$|M|<3\sqrt{3} t_2 \sin{\phi}$$, $$Q = \pm 1$$. Again, this example fits my requirements.

Both examples 1 and 3 are for the quantum anomalous Hall effect. However, I now want to keep studying cases that fit my criteria above (in bold), but I cannot find examples that fit them. The reason I have these criteria in the first place is because I want to numerically discretize the parameter space and identify locations of Berry curvature singularities. The parameter space does not have to be momentum space. I then want to compare my Chern number calculations with what's known in the literature to verify my work, and this is by plotting a phase diagram such as Figure 7 in 2.

• Example 4: A semi-different example I have in mind is the 2D quantum XY chain (from 4). Here, the different phases are paramagnetic and ferromagnetic. However, I am not sure about how to rewrite the Hamiltonian in terms of Pauli spin matrices here, nor identify which parameters to vary for a good phase diagram.

Given all this, do you have any resources/models to refer me to? Preferably these will be 2D two-band models that fit my criteria, and have existing phase diagrams for me to compare my results against. I ask this question on here because I scoured the internet for more examples, but could not find any that were simple enough to implement. Or, the Hamiltonians given would be in unfamiliar notation (involving tensor products and real-space coordinates). I did not want to risk rewriting these on my own.

• Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files. – Qmechanic Apr 27 at 22:01