  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 $\pi_3(S^2)$ homotopy
group, i.e. Hopf fibration. Their topologically non-trivial 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 with 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.

Strongly Correlated Electrons