Tight Binding Moiré Hamiltonian Construction

The tight-binding Hamiltonian 1 is a widely used model in solid-state physics and quantum chemistry to describe the electronic structure of solids — especially in crystals and layered materials. In this model, electrons are considered localized around atomic sites but can hop to neighbouring atoms. To describe such a Hamiltonian for a Moiré system, we use the second quantized 2 form:

\[ H = \sum_{\alpha, \beta;\, r,r' \in L} t^1_{rr', \alpha\beta}c^{\dagger}_{r,\beta}c_{r',\alpha} + \sum_{\alpha, \beta;\, r,r' \in U} t^2_{rr', \alpha\beta}d^{\dagger}_{r,\beta}d_{r',\alpha} + \sum_{\alpha, \beta;\, r,r'} t^{\perp}_{rr', \alpha\beta}c^{\dagger}_{r,\beta}d_{r',\alpha} + \text{h.c.} \]

Here, \(c^{\dagger}_{r,\beta}\) and \(c_{r',\alpha}\) denote the electron creation and annihilation operators at lattice sites \(r\) and \(r'\) in the lower layer (\(L\)), associated with orbitals \(\beta\) and \(\alpha\), respectively. Likewise, \(d^{\dagger}_{r,\beta}\) and \(d_{r',\alpha}\) are the corresponding operators in the upper layer (\(U\)).

The terms \(t^1_{rr', \alpha\beta}\) and \(t^2_{rr', \alpha\beta}\) represent the intralayer hopping amplitudes, describing electron tunneling from orbital \(\alpha\) at site \(r'\) to orbital \(\beta\) at site \(r\) within the lower and upper layers, respectively. In the special case where \(r = r'\) and \(\alpha = \beta\), these terms correspond to the on-site potential—the energy of an electron residing in a particular orbital. The interlayer coupling is described by \(t^{\perp}_{rr', \alpha\beta}\), which governs the hopping of an electron from orbital \(\alpha\) at site \(r'\) in the upper layer to orbital \(\beta\) at site \(r\) in the lower layer.

For simplicity, consider only nearest-neighbour hopping with a single orbital per site (MoirePy can handle any arbitrary number of orbital systems). In such cases, the orbital indices \( \alpha \) and \( \beta \) can be omitted to simplify the notation. We can define the basis as:

\[ \Psi^{\dagger} = (c^{\dagger}_{1}, c^{\dagger}_{2}, \dots, c^{\dagger}_{n}, d^{\dagger}_{1}, d^{\dagger}_{2}, \dots, d^{\dagger}_{n}) \]

Here, \( c^{\dagger}_{i} \) (\( d^{\dagger}_{i} \)) is the creation operator at site \( i \) in the lower (upper) layer.

Then, the Hamiltonian takes the block matrix form:

\[ H = \Psi^{\dagger} \begin{pmatrix} h_{LL} & h_{LU} \\ h_{UL} & h_{UU} \end{pmatrix} \Psi \]

Here, \( h_{LL} \) and \( h_{UU} \) are the first-quantized \( n \times n \) Hamiltonians of the lower and upper layers, respectively. The blocks \( h_{LU} \) and \( h_{UL} \) represent interlayer couplings.

  1. Neil W. Ashcroft and N. David Mermin. Solid State Physics. Saunders College Publishing, 1976. ↩

  2. Henrik Bruus and Karsten Flensberg. Many-Body Quantum Theory in Condensed Matter Physics: An Introduction. Oxford University Press, Oxford, 2004. ↩