At temperatures close to absolute zero, where quantum mechanics governs macroscopic behaviour, matter reveals unexpected phases not classifiable by classical models such as solids, liquids or gases; among these, topological phases stand out, characterised not by local arrangements of atoms but by global properties—topological invariants—that remain stable under smooth deformations, making them ideal candidates for robust quantum memory systems and hosts to exotic quasi-particles beyond the boson-fermion dichotomy; yet, unlike conventional phase transitions explained through symmetry breaking and associated order parameters (à la Noether), topological phases elude this framework, hiding their symmetries at a deeper structural level that requires a redefinition of state representation—precisely where tensor networks enter as a mathematical language for quantum many-body systems; these networks—composed of tensors generalising vectors and matrices—map the correlations among particles by contracting multi-dimensional arrays along edges that represent entanglement, providing a highly efficient and physically faithful description of low-temperature quantum states, especially in two dimensions; the symmetries of the tensors themselves, often describable via weak Hopf algebras rather than classical groups, encode the topology of the phase and enable a classification of phases beyond Landau’s symmetry paradigm; crucially, this tensorial approach has made it possible to derive actual interactions for Resonating Valence Bond (RVB) states, theorised by Nobel laureate P.W. Anderson as spin liquids potentially underpinning high-temperature superconductivity, but long elusive until tensor symmetries gave them physical realisability; as such, what Shivaji Sondhi called a “quantum state in search of interactions” has found its dynamical ground through parity-symmetric tensor networks; while these advances have unified the known two-dimensional topological phases, the challenge of extending this language to three-dimensional quantum matter remains open, promising not only mathematical breakthroughs but novel platforms for quantum computation and exotic condensed matter phenomena; in this sense, tensor networks are not mere calculational tools—they are the geometrical grammar of quantum topology, encoding a new ontology where matter, symmetry and information intertwine.
