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

TL;DR

The paper extends the concept of deconfined quantum criticality to 3+1 dimensions, constructing bosonic and fermionic DQCPs via non-Abelian SU(2) gauge theories that separate trivial and SPT phases. It reveals multiple universality classes for the same phase boundary, introduces stable yet unnecessary continuous transitions within a single phase, and identifies band-theory-forbidden transitions between band-allowed insulators. A key result is the proposal of a 3+1D duality between SU(2) gauge theory with a massless adjoint fermion and a theory of a free Dirac fermion plus a decoupled topological order, with anomaly matching guiding the consistency checks. The work highlights the role of spectator fields and higher-form symmetry anomalies in shaping the infrared physics and topological character of the massive phases, opening new directions for numerical tests and higher-dimensional dualities.

Abstract

Continuous quantum phase transitions that are 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 2+1-dimensions. Examples include phase transitions in quantum magnetism as well as those between Symmetry Protected Topological phases. In this paper, we present several examples of Deconfined Quantum Critical Points (DQCP) between Symmetry Protected Topological phases in 3+1-D for both bosonic and fermionic systems. Some of the critical theories can be formulated as non-abelian gauge theories either in their Infra-Red 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+1-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.

Figures (10)

• Figure 1: (A) Deconfined quantum criticality at the trivial to SPT phase boundary of systems of either bosons or of fermions. (B) Multiple universality classes for the same phase transition. (C) "Unnecessary quantum critical points" that live within a single phase of matter (D) Band-theory-forbidden QCP between two band insulators.
• Figure 2: (a) demonstrates the renormalization group flow of the gauge coupling in three different regimes: 1. IR free (green curve); 2. Banks-Zaks fixed point (red curve), conformal; 3. IR confined (blue curve). (b) shows the conformal window for $SU(N_c)$ gauge theories with $N_f$ flavors of fundamental or adjoint fermion fields. The upper edge of the conformal window is sharply defined by the condition $\beta_0(N_c,N_f,R)=0$. The lower edge of the conformal window can only be determined through numerical simulations. Therefore, we should not take the numbers on the dotted line very seriously.
• Figure 3: On the left is a schematic demonstration of renormalization flow in $g^2$-$m$ plane for large $N_f$ in the IR free case. The gauge coupling $g^2$ is a dangerous irrelevant operator for the $m=0$ critical point. On the right is the finite temperature phase diagram for the deconfined quantum phase transition. It features two interesting crossover scales. At temperature $T>m$ (or length scale $l<\xi\sim 1/m$), the physics is controlled by the critical point and the system has deconfined massless fermions with weakly interacting gluons. For temperature $m<T<m^y$ (or length scale $1/m<L<1/m^y$) with $y>1$ is a universal exponent, the system has deconfined but massive fermions and weakly interacting gluons. For temperature lower than $\sim m^y$ (or $L >1/m^y$), the gauge theory flows to strong coupling and the system is in a confined phase.
• Figure 4: A schematic renormalization flow diagram for degenerate quantum critical points.
• Figure 5: A sketch for the conformal window of $Sp(N_c)$ gauge theories (numbers on the $N_f$ axis are not precies). The red and green dots are different gauge theories. The red ones are free and the green one is strongly interacting. However, they all describe the topological phase transition from the trivial state to the same $PSp(N_f)\times Z_2^T$ bosonic symmetry protected topological phase.