pyqula

pyqula is a Python library for tight-binding models of electrons on lattices. It computes band structures, densities of states and spectral functions, self-consistent mean-field Hamiltonians for interacting electrons, superconductivity, topological invariants, response functions and quantum transport, and also classical spin and lattice-gas models.

This guide explains how to compute physical quantities. For each quantity it gives the Hamiltonian or the formula, its physical meaning, the code, and the keywords that matter. The code is on GitHub, and the whole guide is also available as one PDF.

Installation

pip install pyqula

pyqula is written in Python, with numba for the heavy loops, so it needs no compiler. To use the code of a clone of the repository, run pip install -e . from its root.

Note that import pyqula alone gives access to nothing. Always import the module you need, for example from pyqula import geometry.

A first calculation

Almost every calculation has four steps: we build a lattice, we generate its tight-binding Hamiltonian, we add terms to it, and we compute an observable.

from pyqula import geometry
g = geometry.honeycomb_lattice()   # 1. a lattice
h = g.get_hamiltonian()            # 2. its tight-binding Hamiltonian
h.add_zeeman([0.,0.,0.3])          # 3. add terms (in place)
(k,e) = h.get_bands()              # 4. compute an observable

Here the lattice is the honeycomb lattice of graphene, the Hamiltonian has first-neighbor hopping \(t=1\) and is spinful, and the Zeeman field \(B_z=0.3\) splits the two Dirac cones of the two spins by \(0.6\). h.get_bands() returns the k-points along a path in the Brillouin zone and the energies at each of them. Energies are in units of the hopping \(t\), and lengths in units of the first-neighbor distance.

Each code block in this guide has its own imports and runs alone, except when it continues the block just before it on the same page. The blocks stop at the computed arrays, and the plotting is left to you.

Where to find more

• examples/ in the repository contains several hundred runnable scripts, ordered by dimension (0d/ 1d/ 2d/ 3d/) and by topic (transport/, embedding/, wannier/, classicalspin/, latticegas/ and others). Most of them end with a figure.
• jupyter-notebooks/functionalities/ contains one executed notebook for each functionality listed in the README, with the physics and the plots. Look there to see what a result should look like.
• The Reference tab lists the methods of Geometry and Hamiltonian and their arguments.