Observable order parameter for topological order and spin liquids from conformal geometry

Long-range entangled phases of matter, such as quantum spin liquids and fractional quantum Hall phases, are characterized through the notion of topological order. However, unambiguous detection of such order has proven hard due to the lack of a general observable order parameter, i.e an order parameter based on expectation values of Hermitian operators. As a consequence, most experimental effort is focused either on interferometry of low-lying excitations or simulating the wavefunction on very small lattices and obtaining information theoretic measures from multiple wavefunction snapshots. I will present a different approach which allows one to extract an order parameter from ground state expectation values in toy models of two dimensional topological order with boundaries, based on their quantum loop gas description. Our construction uses a combination of the unique structure of the superposition in the quantum loop gas description and the corresponding connection to the critical classical loop models to extract a universal quantity from the topologically ordered ground state. I shall discuss the extent to which the universal quantity obtained describes the anyon content of the underlying topological order, and its robustness as compared to other information theoretic quantities. Further, I shall comment on how such an order parameter could be used to unambiguously detect quantum spin liquids, possibly in two dimensional materials or large quantum simulation platforms.