Pseudogap phases, chiral anomaly, and topological order with quantum loop entanglement

A many-body quantum system whose topological defects are conserved, abundant, and mobile is a correlated quantum liquid. Since topological defects can be classified by homotopy groups, each homotopy identifies a class of quantum liquids. Here we explore the quantum liquids based on the ${\pi }_3\left(S^2\right)$ homotopy group, i.e., Hopf fibration. Their topologically nontrivial dynamics emerges from the interlinking between magnetic flux or skyrmion loops in the charge and spin sectors respectively. We lay down a field theory foundation for analyzing such states by naturally incorporating the well-known framing regularization into the theory, and constructing the appropriate topological Lagrangian terms. We show that at least two strongly correlated phases of interlinked loops can exist in $d=3$ spatial dimensions at zero and low-finite temperatures. These phases are closely related to the chiral quantum anomaly and do not have an obvious topological order, but they are distinguished from the trivial disordered phase by a generalization of the Wilson loop operator. In $d=4$ spatial dimensions, interlinked loops are able to produce topological order at zero temperature, featuring charge, angular momentum, and braiding fractionalization. We discuss some possible experimental signatures of loop entanglement in the quantum noise of charge currents.