UV perspective on mixed anomalies at critical points between bosonic symmetry-protected phases

Symmetry-protected phases are gapped phases of matter which are distinguished only in the presence of a global symmetry $G$. These quantum phases lack any symmetry-breaking or topological order and have short-range entangled ground states. Based on this short-range entanglement property, we give a general argument for the existence of an emergent antiunitary (and sometimes also a unitary) ${\mathbb{Z}}_2$ symmetry at a critical point separating two different bosonic symmetry-protected phases in any dimension. Often, the emergent symmetry group at criticality has a mixed global anomaly. For those phases classified by group cohomology, we identify a criterion for when such a mixed global anomaly is present, and write down representative cocycles for the corresponding anomaly class. We illustrate our results with a series of examples and make connections to recent results on $(2+1)$-dimensional beyond-Landau critical points.

Figure 1

A path of local $G$-symmetric, short-range entangled bosonic Hamiltonians $H\left(\lambda \right)$, with a critical point at $\lambda ={\lambda }^{*}$, depicted by the red star. For $\lambda <{\lambda }^{*}$ ($\lambda >{\lambda }^{*}$), the Hamiltonian realizes a trivial (nontrivial) $G$-SPT phase. For two Hamiltonians $H\left({\lambda }_0\right)$ and $H\left({\lambda }_1\right)$ at the same distance ${\xi }_0^{-1}={\xi }_1^{-1}$ from the critical point, there exists a locality-preserving unitary $\mathcal{U}$ mapping the ground state of $H\left({\lambda }_0\right)$ to the ground state of $H\left({\lambda }_1\right)$.

Figure 2

The choice of branching structure to define a finite group cohomology fixed-point wave function. As explained in the main text, the arrows on the links of the triangular lattice induce a local ordering on the sites around each plaquette. To obtain the nontrivial $G$-SPT ground-state wave function one assigns a phase factor ${\omega }_3$ or ${\omega }_3^{-1}$ to each plaquette, depending on the choice of branching structure. With our choice, upward-pointing triangles get assigned a factor ${\omega }_3$ and downward-pointing triangles a factor ${\omega }_3^{-1}$.

Figure 3

Coupled wire construction to study the trivial to BIQH transition. Each wire labeled by $y$ is a nonchiral Luttinger liquid where one chiral boson field is charged under the $U\left(1\right)$ symmetry (denoted by $c$), while the field with opposite chirality is neutral ($n$). Each unit cell consists of two wires labeled by $(2y,2y+1)$ and the intra- and interunit cell coupling strength between the wires is given by respectively $t$ and $t^{\prime }$.

Figure 4

An illustration of the symmetric mass generation (SMG) transition. The multicritical point depicted by a red circle can be interpreted as a SMG transition because the trivial-${\left[\omega \right]}^2$ critical line is anomaly free. By stacking the phase diagram with the $G$-SPT ${\left[\omega \right]}^{-1}$, the anomaly-free critical line separates the $G$-SPT phases ${\left[\omega \right]}^{-1}$ and $\left[\omega \right]$. This is the SMG scenario discussed in Ref. [20].