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Symmetr: a Python package for determining symmetry properties of crystals

Jakub Železný

TL;DR

A Python package for determining the symmetry-restricted forms of tensors describing physical properties of crystals, focusing particularly on magnetic materials.

Abstract

Condensed matter compounds typically form crystals, which break the rotational and translational invariance of space but remain invariant under a discrete set of symmetry operations. Understanding the effects allowed by this symmetry breaking, as well as the constraints imposed by the crystal structure, is a crucial problem in condensed matter physics. Here, we present a Python package for determining the symmetry-restricted forms of tensors describing physical properties of crystals, focusing particularly on magnetic materials. The primary focus is on response tensors; however, the program can also describe equilibrium properties and other physical properties, such as magnetic interactions. The program can describe the symmetry using the conventional magnetic space groups, as well as using the spin groups that describe the non-relativistic limit. Additional functionality includes the treatment of quantities projected onto a particular site and expansions in the magnetic order parameter. The code can be used either from the command line or via a Python API.

Symmetr: a Python package for determining symmetry properties of crystals

TL;DR

A Python package for determining the symmetry-restricted forms of tensors describing physical properties of crystals, focusing particularly on magnetic materials.

Abstract

Condensed matter compounds typically form crystals, which break the rotational and translational invariance of space but remain invariant under a discrete set of symmetry operations. Understanding the effects allowed by this symmetry breaking, as well as the constraints imposed by the crystal structure, is a crucial problem in condensed matter physics. Here, we present a Python package for determining the symmetry-restricted forms of tensors describing physical properties of crystals, focusing particularly on magnetic materials. The primary focus is on response tensors; however, the program can also describe equilibrium properties and other physical properties, such as magnetic interactions. The program can describe the symmetry using the conventional magnetic space groups, as well as using the spin groups that describe the non-relativistic limit. Additional functionality includes the treatment of quantities projected onto a particular site and expansions in the magnetic order parameter. The code can be used either from the command line or via a Python API.
Paper Structure (24 sections, 20 equations, 2 figures)

This paper contains 24 sections, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Methodology
  3. Transformation under rotation
  4. Transformation under inversion
  5. Transformation under translations & atomic permutations
  6. Time-reversal symmetry
  7. Spin rotations
  8. Magnetic interactions
  9. Expansion in magnetization
  10. Non-relativistic symmetry
  11. Symmetrizing tensors
  12. Using the code
  13. Response mode
  14. Atomic projections
  15. Expansion in the order parameter
  16. ...and 9 more sections

Figures (2)

  • Figure 1: Simplified flow of the program for response mode without magnetization expansion.
  • Figure 2: (a) The crystal structure of CuMnAs. Blue atoms are Cu, red and purple are Mn, and brown are As. (b) Crystal structure of Mn$_3$Sn. Purple atoms are Mn, grey denotes Sn. (c) Scaling of the running time for the expansion of the order-parameter conductivity of Fe. The blue and orange lines correspond to the case without and with the flag, respectively.