Neumann's principle based eigenvector approach for deriving non-vanishing tensor elements for nonlinear optics

Area of Science:

• Solid State Physics
• Materials Science
• Quantum Information Technologies

Background:

• Physical properties are often described by tensors, like optical susceptibilities.
• Conventional methods for deriving tensor elements in symmetric systems are complex and error-prone, relying on intuitive sign-flipping after symmetry operations.
• Neumann's principle, stating that physical property symmetries mirror geometric symmetries, provides a foundation for a more systematic approach.

Purpose of the Study:

• To develop a novel matrix-based approach for deriving non-vanishing tensor elements of physical properties.
• To provide a physically intuitive and mathematically rigorous method for analyzing tensor properties based on Neumann's principle.
• To extend the method for higher-rank tensors and applications involving magnetization for spin polarization measurements.

Main Methods:

• Mathematical application of Neumann's principle to tensor expressions.
• Development of a procedure based on eigensystems to derive non-vanishing tensor elements.
• Implementation of a generalized Mathematica code for arbitrary symmetries and nonlinear processes.

Main Results:

• Demonstrated the approach's validity for second and third-order nonlinear susceptibilities in chiral/achiral surfaces with complex symmetries (D6, Oh).
• Successfully applied the method to higher-rank tensors relevant for 2D and high-order spectroscopy.
• Extended the approach to derive nonlinear tensor elements involving magnetization, crucial for surface spin polarization analysis.

Conclusions:

• The matrix-based approach offers a simplified, intuitive, and accurate method for determining tensor elements in physical properties.
• This generalized method is applicable to a wide range of symmetries, tensor ranks, and nonlinear optical/magnetic phenomena.
• The provided Mathematica code facilitates the analysis of complex material properties for fundamental research and technological applications, including quantum information.