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Continuous symmetry analysis and systematic identification of candidate order parameters for interacting fermion models

Cheng-Hao He, Yi-Zhuang You, Xiao Yan Xu

Abstract

Symmetry plays a central role in modern physics, from classifying quantum states to characterizing phases of matter through spontaneous symmetry breaking. In interacting fermionic systems with multiple internal degrees of freedom, however, determining the full continuous symmetry group and classifying possible order parameters remain challenging. In this work, we present a systematic framework for analyzing continuous symmetries and identifying candidate order parameters in such systems. By mapping the Hamiltonian to a Majorana representation, we obtain the generators of continuous symmetries from the Lie algebra of operators that commute with the Hamiltonian. We then identify the structure of this Lie algebra using the theory of semisimple Lie algebras. Building on representation theory, we further develop a systematic method for exhaustively enumerating candidate order parameters. By decomposing the exterior-power representations induced by the symmetry algebra on the Majorana space and incorporating discrete lattice symmetries, we classify these order parameters according to the symmetries they break. (Abridged. Please see the PDF manuscript for the complete abstract and specific model applications.)

Continuous symmetry analysis and systematic identification of candidate order parameters for interacting fermion models

Abstract

Symmetry plays a central role in modern physics, from classifying quantum states to characterizing phases of matter through spontaneous symmetry breaking. In interacting fermionic systems with multiple internal degrees of freedom, however, determining the full continuous symmetry group and classifying possible order parameters remain challenging. In this work, we present a systematic framework for analyzing continuous symmetries and identifying candidate order parameters in such systems. By mapping the Hamiltonian to a Majorana representation, we obtain the generators of continuous symmetries from the Lie algebra of operators that commute with the Hamiltonian. We then identify the structure of this Lie algebra using the theory of semisimple Lie algebras. Building on representation theory, we further develop a systematic method for exhaustively enumerating candidate order parameters. By decomposing the exterior-power representations induced by the symmetry algebra on the Majorana space and incorporating discrete lattice symmetries, we classify these order parameters according to the symmetries they break. (Abridged. Please see the PDF manuscript for the complete abstract and specific model applications.)
Paper Structure (20 sections, 92 equations, 4 figures, 2 tables)

This paper contains 20 sections, 92 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Continuous Symmetry Analysis
  3. Majorana Representation
  4. Lie Algebra of Continuous Symmetries
  5. Application
  6. Hubbard Model
  7. Bilayer Spin-1/2 Model
  8. Identification of Candidate Order Parameters
  9. Irreducible Decomposition Algorithm
  10. Discrete Symmetries and Isomorphic Representations
  11. Application
  12. Hubbard Model on Honeycomb Lattice
  13. Bilayer Spin-1/2 Model on Honeycomb Lattice
  14. Conclusion
  15. Invariant Subspaces of the Lie Algebras
  16. ...and 5 more sections

Figures (4)

  • Figure 1: Flowchart summarizing the continuous symmetry analysis of a many-body Hamiltonian.
  • Figure 2: (a) Root system and (b) Dynkin diagram of $\mathfrak{g}_1$.
  • Figure 3: (a) Root system and (b) Dynkin diagram of $\mathfrak{g}_1^{\prime}$.
  • Figure 4: Flowchart illustrating the systematic decomposition of the induced exterior-power representation on the $m$-fermion operator space to identify candidate physical order parameters in interacting systems.