Classical criticality establishes quantum topological order

We establish an important duality correspondence between topological order in quantum many-body systems and criticality in ferromagnetic classical spin systems. We show how such a correspondence leads to a classical and simple procedure for characterization of topological order in an important set of quantum entangled states, namely the Calderbank-Shor-Steane (CSS) states. To this end, we introduce a particular quantum Hamiltonian which allows us to consider the existence of a topological phase transition from quantum CSS states to a magnetized state. We study the ground state fidelity in order to find nonanalyticity in the wave function as a signature of a topological phase transition. Since hypergraphs can be used to map any arbitrary CSS state to a classical spin model, we show that fidelity of the quantum model defined on a hypergraph H is mapped to the heat capacity of the classical spin model defined on dual hypergraph (H) over tilde. Consequently, we show that a ferromagnetic-paramagnetic phase transition in a classical model is mapped to a topological phase transition in the corresponding quantum model. We also show that magnetization does not behave as a local order parameter at the transition point while the classical order parameter is mapped to a nonlocal measure on the quantum side, further indicating the nonlocal nature of the transition. Our procedure not only opens the door for identification of topological phases via the existence of a local and classical quantity, i.e., critical point, but also offers the potential to classify various topological phases through the concept of universality in phase transitions.