Instanton confinement and the topological phases of interlinked quantum loops

We argue that quantum liquids of interlinked flux loops, characterized by a Hopf fibration, can be stable thermodynamic phases distinguished from the conventional disorder by a generalized Wilson loop operator. Topological order of loop-linking is naively possible only at T=0 in d=4 spatial dimensions, but correlated “pseudogap” phases without topological order can exist at finite temperatures even in d=3. Flux loops can be made of vortices or skyrmions; the latter leads to new kinds of spin liquids. Loop fluctuations which change the topological Hopf index are closely related to the chiral quantum anomaly. Furthermore, matter coupled to the flux gauge field promotes two separate phases, one which protects only the Gauss’ linking number and another which may protect additional knot invariants. We establish the stability of these phases with an exact renormalization group of the instanton gas, and characterize the basic universal features of the unlinking phase transitions. Instanton confinement below a critical temperature shapes many correlated phases, e.g. monopole topological orders at T>0 in d≥3, quantum Hall liquids, and possibly also a “vortex liquid” pseudogap state in cuprates.