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

TL;DR

This paper introduces a systematic framework for analyzing continuous symmetries and identifying potential order parameters in interacting fermion models using Majorana representations and Lie algebra theory.

Contribution

It develops a comprehensive method combining Lie algebra and representation theory to classify order parameters and symmetries in complex fermionic systems.

Findings

Framework successfully classifies symmetries in fermionic models.

Method identifies candidate order parameters based on symmetry breaking.

Incorporates discrete lattice symmetries for comprehensive analysis.

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.)