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We outline a holographic framework that attempts to unify Landau andbeyond-Landau paradigms of quantum phases and phase transitions. Leveraging amodern understanding of symmetries as topological defects/operators, theframework uses a topological order to organize the space of quantum systemswith a global symmetry in one lower dimension. The global symmetry naturallyserves as an input for the topological order. In particular, we holographicallyconstruct a String Operator Algebra (SOA) which is the building block ofsymmetric quantum systems with a given symmetry $G$ in one lower dimension.This exposes a vast web of dualities which act on the space of $G$-symmetricquantum systems. The SOA facilitates the classification of gapped phases aswell as their corresponding order parameters and fundamental excitations, whiledualities help to navigate and predict various corners of phase diagrams andanalytically compute universality classes of phase transitions. A novelty ofthe approach is that it treats conventional Landau and unconventionaltopological phase transitions on an equal footing, thereby providing aholographic unification of these seemingly-disparate domains of understanding.We uncover a new feature of gapped phases and their multi-critical points,which we dub fusion structure, that encodes information about which phases andtransitions can be dual to each other. Furthermore, we discover that self-dualsystems typically posses emergent non-invertible, i.e., beyond group-likesymmetries. We apply these ideas to $1+1d$ quantum spin chains with finiteAbelian group symmetry, using topologically-ordered systems in $2+1d$. Wepredict the phase diagrams of various concrete spin models, and analyticallycompute the full conformal spectra of non-trivial quantum phase transitions,which we then verify numerically.

Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics