In this talk, we shall introduce a CFT factory – a novel algorithm of methodically generating 2D lattice models that would flow to conformal field theories (CFTs) in the infrared. We realise these models by engineering the boundary conditions of 3D topological orders (SymTOs) described by string-net models. The critical points are induced by a commensurate condensation of non-commuting anyons. Our structured method generates an infinite family of critical lattice models, including the A-series minimal models, and uncovers previously unknown critical points. Notably, we discover at least three novel CFTs (with central charge c around 1.3, 1.8, and 2.5) that preserves the Haagerup symmetries, in addition to recovering previously reported examples. The non-invertible symmetries preserved at these points are dictated by a novel “refined condensation tree”. The condensation trees predict large swathes of phase boundaries and sieves out second order phase transitions. This predictive power is illustrated not only in well-studied examples, such as the 8-vertex model associated with the A5 category, but also in new cases involving Haagerup symmetries, validated by an improved symmetry-preserving tensor-network renormalization group method. The critical couplings are precisely encoded in algebraic data (the Frobenius algebras and quantum dimensions of unitary fusion categories), thereby establishing a powerful and systematic route to the discovery and potential classification of new CFTs.

The classical wave system has demonstrated itself as an excellent platform to realize and investigate novel phenomena and physics. The bedrock principle is to utilize the macroscopic quantities obtained from the homogenization or mean-field treatment. However, it usually deals with Hermitian problems and averages out fluctuations. Therefore, the presentation will cover two topics: non-Hermitian physics and Casimir effect. The first part focuses on the impact of non Hermitian ingredients on soliton formation and dynamics. By constructing a soliton phase diagram, two distinct soliton phases and their transitions are identified. A Wannier-function-based nonlinear Hamiltonian shows that soliton formation critically depends on how skin-mode localization and band nonreciprocity suppress or enhance wave dispersion. Both soliton phases have been demonstrated to be dynamically accessible from bulk and edge excitations. The second part discusses the influence of the metal’s surface electrons on Casimir forces. A three-dimensional frame transformation method has been established by embedding mesoscopic boundary conditions of electromagnetic fields. We find that mesoscopic Casimir forces are sensitive to the surface electron behavior, including spill-in and spill-out, as verified by the multiple scattering method and proximity force approximation. The mechanism has finally been revealed as Casimir softening distances rooted in quantum surface responses of electrons.