Technical foundations of the package¶

Pybandstructure is suited to calculate physical quantities of periodic models. Any periodic model is characterized by a basis of primitive translation vectors 1 $\bm{t}_{1}$ and $\bm{t}_{2}$ , whose integer linear combinations

(1)¶ $\bm{t}_{n} = n_{1}\bm{t}_{1} + n_{2}\bm{t}_{2} \quad \text{with } n = (n_{1},n_{2}) \in \mathbb{Z}\times\mathbb{Z}$

span a Bravais lattice. Accordingly, it is important to consider also the reciprocal lattice in the dual (or reciprocal) space. Indeed, the propagation wavevector $\bm{k}$ of a general plane wave $\exp (i \bm{k}\cdot\bm{r})$ has reciprocal length dimension, and can be conveniently represented in the reciprocal space. The reciprocal lattice is spanned by integer linear combinations $\bm{G}_{n}$ of the vectors $\bm{G}_{1}$ and $\bm{G}_{2}$ , which are obtained from the primitive translation vectors through the relation $\bm{t}_{i}\cdot\bm{G}_{j} = 2\pi\delta_{ij}$ .

The Brillouin Zone (BZ) is the region of the reciprocal space with the property that any point included in the BZ is closer to a chosen lattice point (say $\bm{G}_{n} = \bm{0}$ ) than to any other. The energy eigenvalues of a translational invariant model are labeled by the crystal wave vector $\bm{k}$ , which is inside the Brillouin zone.

To numerically compute the electronic states of a periodic model, one needs the matrix elements of its Hamiltonian for an orthonormal basis which is explicitly dependent on the crystal wave vector $\bm{k}$ . Two basis explicitly depending on $\bm{k}$ are commonly considered: Bloch sums and plane waves. Concrete examples of the usage of Bloch sums and plane waves are going to be presented when discussing the numerical calculation of the energy spectrum of TBG for, respectively, the tight-binding and continuum model. For the moment let just assume to have an orthonormal basis

(2)¶ $\{|\phi_{1}(\bm{k})\rangle, \cdots, |\phi_{N}(\bm{k})\rangle\ \} \text{ with } \bm{k}\in{\rm BZ}, \text{ and } \langle \phi_{i}(\bm{k})|\phi_{j}(\bm{k}^{\prime})\rangle = \delta_{ij}\delta_{\bm{k}\bm{k}^{\prime}}$

and to know the system’s Hamiltonian $\hat{H}$ matrix elements on this basis

$H_{ij}(\bm{k}) \equiv \langle \phi_{i}(\bm{k})|\hat{H}|\phi_{j}(\bm{k})\rangle.$

The matrix whose elements are $H_{ij}(\bm{k})$ is Hermitian by construction, and the spectral theorem assure it has $N$ real eigenvalues. Let us now call these eigenvalues $\epsilon_{\bm{k}\nu}$ and $(c_{1}(\bm{k},\nu), \ldots, c_{N}(\bm{k},\nu)) \in \mathbb{C}^{N}$ the corresponding eigenvectors, with $\nu = 1, \cdots , N$ indexing the different eigenvalues. The quantities $\epsilon_{\bm{k}\nu}$ and $(c_{1}(\bm{k},\nu), \ldots, c_{N}(\bm{k},\nu))$ are routinely obtained numerically thanks to standard diagonalization algorithms based on the “divide and conquer” approach, with a computational cost of the order of $\sim N^{3}$ floating-point operations. The Bloch eigenstate at crystal momentum $\bm{k}$ and (band) index $\nu$ is given simply by

$|\bm{k}\nu\rangle = \sum_{i = 1}^{N} c_{i}(\bm{k}, \nu)|\phi_{i}(\bm{k})\rangle.$

By knowing the matrix elements of a generic operator $\hat{O}$ on the basis Eq. [eq:NUM_basis],

$O_{ij}(\bm{k},\bm{k}^{\prime}) = \langle \phi_{i}(\bm{k})|\hat{O}|\phi_{j}(\bm{k}^{\prime})\rangle,$

the matrix elements of the same operator on the Bloch eigenstates are readily obtained

$\langle\bm{k}\nu|\hat{O}|\bm{k}\nu^{\prime}\rangle = \sum_{i = 1}^{N} \sum_{j=1}^{N} c_{i}(\bm{k}, \nu)^{*}O_{ij}(\bm{k},\bm{k}^{\prime})c_{j}(\bm{k}, \nu),$

which is just a vector-matrix-vector product. These remarks, albeit being rather trivial linear algebra facts, allow to actually compute the matrix elements appearing in the electrical conductivity Eq. [eq:local_cond_crystal], and density-density response function Eqs. [eqs:density_density_matrix_elements].

1: For the sake of simplicity we explicitly work with two-dimensional systems in this Section. Pybandstructure, however, can handle models of any spatial dimension.