Optical Activities as Computing Resources for Space-time Symmetries

Computing Floquet Hamiltonians with Symmetries

Unitary matrices arise in many ways in physics, in particular as a time evolution operator. For a periodically driven system one frequently wishes to compute a Floquet Hamilonian that should be a Hermitian operator $H$ such that $e^{-iTH}=U(T)$ where $U(T)$ is the time evolution operator at time corresponding the period of the system. That is, we want $H$ to be equal to $-i$ times a matrix logarithm of $U(T)$. If the system has a symmetry, such as time reversal symmetry, one can expect $H$ to have a symmetry beyond being Hermitian. We discuss here practical numerical algorithms on computing ma…

阿贝尔对称性下局域性诱导的非普适性

Locality Induced Non-Universality for Abelian Symmetries

本文针对量子计算中2-局域幺正生成复合系统任意幺正变换的普适性结论，研究了所有阿贝尔对称性下该普适性不成立的对应情况，拓展了对称性与局域性关联的相关研究。

Instantons to the people: the power of one-form symmetries

We show that the non-perturbative dynamics of $\mathcal{N}=2$ super Yang-Mills theories in a self-dual $Ω$-background and with an arbitrary simple gauge group is fully determined by studying renormalization group equations of vevs of surface operators generating one-form symmetries. The corresponding system of equations is a {\it non-autonomous} Toda chain, the time being the RG scale. We obtain new recurrence relations which provide a systematic algorithm computing multi-instanton corrections from the tree-level one-loop prepotential as the asymptotic boundary condition of the RGE. We exempli…

非局部对称在计算有理二阶常微分方程的李维利安首积分中的新应用

A new use of nonlocal symmetries for computing Liouvillian first integrals of rational second order ordinary differential equations

本文提出一种基于非局部对称性的高效方法，用于计算有理二阶常微分方程的李维利安首积分。

Lie-Poisson神经网络（LPNets）：基于数据的具有对称性的哈密顿系统计算

Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetries

提出Lie-Poisson神经网络（LPNets），通过精确保持泊松括号和Casimir不变量的变换，实现具有对称性的哈密顿系统长期演化的数据驱动预测。

Symmetries of Differential Equations via Cartan's Method of Equivalence

We formulate a method of computing invariant 1-forms and structure equations of symmetry pseudo-groups of differential equations based on Cartan's method of equivalence and the moving coframe method introduced by Fels and Olver. Our apparoach does not require a preliminary computation of infinitesimal defining systems, their analysis and integration, and uses differentiation and linear algebra operations only. Examples of its applications are given.