A Lie Algebraic Approach to Order Parameters

Richard Bausch, H. K. Janssen, Y. Yamazaki

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

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.)

Mojtaba Jafarpour, Davood Afshar

We present a new Lie algebraic method to study the eigenvalues and eigenfunctions of quantum anharmonic oscillators. We consider the Hamiltonians for the simple harmonic and anharmonic oscillator as the two generators of a Lie algebra, whose other generators may be found exactly or up to any order of the parameter involved. Speciflcally, the closed commutator algebra for the quartic anharmonic oscillator is established in a perturbation sense. An element of this Lie group, turning out to be the four-photon operator, transforms the quartic anharmonic oscillator Hamiltonian to the harmonic one in a perturbation sense; thus, facilitating the calculation of the eigenvalues and eigenfunctions of the former.

M. Gaudin

A.K Mishra, G. Rajasekaran

Song Tong-Qiang, Feng Jian, Gao Yun-Feng

We discuss the unitary operator corresponding to the general two-mode coordinate-momentum mixed transformation , where and are arbitrary real numbers. Suitably selecting the parameters and , we obtain a new two-mode bosonic realization of the Lie algebra. We also study the squeezing effects of the squeezed vacuum associated with the new two-mode bosonic realization of the Lie algebra. The results show that the new squeezed vacuum does not possess second-order squeezing, but exhibits higher-order squeezing.