From bosonic topological transition to symmetric fermion mass generation

A bosonic topological transition (BTT) is a quantum critical point between the bosonic symmetry-protected topological phase and the trivial phase. In this work, we investigate such a transition in a (2+1)-dimensional lattice model with the maximal microscopic symmetry: an internal $\mathrm{SO}\left(4\right)$ symmetry. We derive a description for this transition in terms of compact quantum electrodynamics (QED) with four fermion flavors ($N_f=4$). Within a systematic renormalization group analysis, we identify the critical point with the desired $\mathrm{O}\left(4\right)$ emergent symmetry and all expected deformations. By lowering the microscopic symmetry, we recover the previous $N_f=2$ noncompact QED description of the BTT. Finally, by merging two BTTs we recover a previously discussed theory of symmetric mass generation, as an $\mathrm{SU}\left(2\right)$ quantum chromodynamics-Higgs theory with $N_f=4$ flavors of $\mathrm{SU}\left(2\right)$ fundamental fermions and one $\mathrm{SU}\left(2\right)$ fundamental Higgs boson. This provides a consistency check on both theories.

Figure 1

RG flow diagram for $N_f=4$ ($N=1$) in the ${\mathbb{Z}}_2$-symmetric plane.

Figure 2

Phase diagram of the model Eq. (24) and $K$ matrices in different parameter regimes. The $K$ matrices are given in the basis of $\mathcal{A}=(a,A_{\uparrow }^3,A_{\downarrow }^3)$. We assume $m_{\uparrow }>0$, and choose it as the mass scale. The solid lines are physical phase transitions, while the dashed lines are not.

Figure 3

Honeycomb lattice and space group symmetries. The lattice can be partitioned into $A$ (red) and $B$ (blue) sublattices. The sublattice sign ${(-)}^i$ is $+1$ on $A$ and $-1$ on $B$. The black arrows mark the $T_{1,2}$ translation vectors. The background arrow (light gray) shows Haldane's second neighbor hopping direction ${\nu }_{ij}$.

Figure 4

Schematic phase diagram of both the bilayer honeycomb model Eq. (33) tuned by $\lambda ,J$ and the field theory Eq. (55) tuned by $m_{\text{QSH}},r$. There is a featureless Mott phase and two bosonic symmetry protected topological (BSPT) phases, labeled by the topological index $\nu$ or equivalently the quantum spin Hall conductance ${\sigma }_{\text{sH}}$. Different phases are separated by quantum phase transitions: a fermionic transition (in green) corresponding to the Dirac semimetal (SM) and two bosonic topological transitions (BTTs) (in blue). The three transition lines meet at the symmetric mass generation (SMG) tricritical point (in red). The dashed line is a “faked transition” in the field theory that does not correspond to any physical transition.

Figure 5

The physical fermion $c$ carries three $\mathrm{SU}\left(2\right)$ symmetry charges. Two of them, the $\mathrm{SU}{\left(2\right)}_{\uparrow }\times \mathrm{SU}{\left(2\right)}_{\downarrow }$ charges (in blue), are carried by the fermionic parton $\psi$; and the remaining $\mathrm{SU}{\left(2\right)}_{\ell }$ charge (in green) is carried by the bosonic parton $\varphi$. Both partons carry the $\mathrm{SU}\left(2\right)$ gauge charge (in red). The real fields $\mathrm{c}$, $\varphi$, and $\psi$ will inherit the charge assignments.

Figure 6

Illustration of excitation gaps through the semimetal to featureless Mott transition via (a) an SMG point and (b) an intermediate $\mathrm{SO}{\left(3\right)}_{\ell }$ symmetry breaking phase. $J_c$ is an SMG critical point, $J_{c1}$ is a Gross-Neveu critical point, and $J_{c2}$ is an $\mathrm{O}\left(3\right)$ Wilson-Fisher critical point. In the strong interaction $J\to \infty$ limit, we expect ${\mathrm{\Delta }}_c\sim 9J/8$, ${\mathrm{\Delta }}_N\sim 3J/2$, and ${\mathrm{\Delta }}_M\sim J$ to match the onsite interaction spectrum listed in Table 3.