pyqula

Official Resources

• Homepage: https://github.com/joselado/pyqula
• Repository: https://github.com/joselado/pyqula
• License: GPL-3.0

Overview

pyqula (Python Quantum Lattice) is a powerful standard-library for simulating quantum tight-binding models, with a distinctive focus on interacting systems and topological phases. Unlike many pure TB codes, pyqula includes self-consistent mean-field solvers for Hubbard and Heisenberg interactions, allowing it to simulate the interplay between correlation (magnetism, superconductivity) and topology. It also implements the Kernel Polynomial Method (KPM) for large-scale calculations.

Scientific domain: Correlated Topological Matter, Quantum Transport Target user community: Theorists studying interacting topological phases (e.g., twisted bilayers, magnetic TIs)

Theoretical Methods

• Tight-Binding: Slater-Koster or manual hopping.
• Mean-Field Theory:
  • Hubbard Model (Self-consistent $U$).
  • BdG Superconductivity (Self-consistent $\Delta$).
• Topological Invariants:
  • Chern Numbers (Real-space and k-space).
  • $\mathbb{Z}_2$ Invariants.
  • Local Chern Marker (for amorphous topology).
• KPM: Order-N expansion for DOS and Spectral Functions.

Capabilities

• Model Construction:
  • Built-in generators for Honeycomb, Kagome, Square, Lieb lattices.
  • Twisted Bilayer Graphene (TBG) constructors.
• Interactions:
  • Non-collinear magnetism.
  • Unconventional superconductivity (d-wave, p-wave).
• Observables:
  • Band structures.
  • Local Density of States (LDOS).
  • Quantum Transport (Conductance).
  • Berry Curvature maps.

Key Strengths

• Correlations + Topology: One of the few Python codes that seamlessly integrates mean-field order parameters into topological analysis. You can start with a TI and turn on Hubbard U to see if it becomes an antiferromagnet.
• Real-Space Topology: Implements the Local Chern Marker, enabling topological characterization of disordered or amorphous systems where $k$ is not a good quantum number.
• KPM Scalability: Can handle millions of atoms for spectral functions, overcoming the limits of exact diagonalization.

Inputs & Outputs

• Inputs: Python scripts defining geometry and interaction strengths.
• Outputs:
  • Matplotlib plots (bands, Fermi surfaces).
  • NumPy arrays of observables.

Interfaces & Ecosystem

• Core: Built on NumPy/SciPy.
• Visualization: Extensive internal plotting library using Matplotlib.

Performance Characteristics

• Speed: Mixed. Core arithmetic is NumPy (fast), but Python loops in self-consistent cycles can be slower than Fortran. KPM is highly efficient.
• Scalability: KPM scales to $O(N)$; Exact Diagonalization to $O(N^3)$.

Comparison with Other Codes

• vs. Pybinding: Pybinding is faster (C++) but non-interacting. Pyqula allows interactions.
• vs. Kwant: Kwant is better for transport in arbitrary geometries. Pyqula is better for bulk interacting phases and topological markers.

Application Areas

• Twisted Moiré Systems: Simulating correlated insulating states in TBG.
• Topological Superconductivity: Majorana modes in hybrid nanowires.
• Amorphous Topology: Chern numbers in random lattices.

Community and Support

• Development: Jose Lado (Aalto University).
• Source: GitHub.

Verification & Sources

• Repository: https://github.com/joselado/pyqula
• Verification status: ✅ VERIFIED
  • Widely cited research code.