Adventure in Topological Phase Transitions in 3+1-D: Non-Abelian Deconfined Quantum Criticalities and a Possible Duality

Continuous quantum phase transitions beyond the conventional paradigm of fluctuations of a symmetry-breaking order parameter are challenging for theory. These phase transitions often involve emergent deconfined gauge fields at the critical points as demonstrated in phase transitions between different broken-symmetry states of 2 + 1-dimensional quantum magnets, as well as those between symmetry-protected topological (SPT) phases. In this paper, we present several examples of deconfined quantum critical points between SPT phases in 3 + 1-D for both bosonic and fermionic systems. These critical theories can be formulated as non-Abelian gauge theories either in the infrared-free regime or in the conformal window when they flow to the Banks-Zaks fixed points. We explicitly demonstrate several interesting quantum critical phenomena. We describe situations in which the same phase transition allows for multiple universality classes controlled by distinct fixed points. We exhibit the possibility-which we dub unnecessary quantum critical points-of stable generic continuous phase transitions within the same phase. We present examples of interaction-driven, band-theory-forbidden, continuous phase transitions between two distinct band insulators. The understanding we develop leads us to suggest an interesting possible 3 + l-D field theory duality between SU(2) gauge theory coupled to one massless adjoint Dirac fermion and the theory of a single massless Dirac fermion augmented by a decoupled topological field theory.