From Bosonic Topological Transition to Symmetric Fermion Mass Generation

TL;DR

This work presents an alternative, field-theoretic description of the bosonic topological transition (BTT) and connects it to the symmetric mass generation (SMG) transition within a unified parton framework. The BTT is captured by a compact $N_f=4$ QED with emergent $O(4)$ symmetry, while SMG is described by an $SU(2)$ QCD–Higgs theory with four fermionic flavors; the two are bridged on a bilayer honeycomb lattice, and their compatibility is reinforced by a mass-muning argument that merges the two theories into a single low-energy $N_f=4$ QED. The results illuminate how deconfined quantum criticality and symmetry-protected topological transitions can be related via parton fractionalization and gauge dynamics, offering a non-Landau route to gap formation without symmetry breaking. The findings also suggest intermediate phases and monopole physics consistent with lattice symmetries, emphasizing the role of emergent gauge structures in 2+1D quantum criticality.

Abstract

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

Figures (6)

• 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. \ref{['eq: BTT-DQCP']} 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 dash 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 2nd neighbor hopping direction $\nu_{ij}$.
• Figure 4: Schematic phase diagram of both the bilayer honeycomb model Eq. \ref{['eq: KMH']} tuned by $\lambda, J$ and the field theory Eq. \ref{['eq: BSPT0']} 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 $\mathrm{SU}(2)$ symmetry charges. Two of them, the $\mathrm{SU}(2)_\uparrow\times\mathrm{SU}(2)_\downarrow$ charges (in blue), are carried by the fermionic parton $\psi$; and the remaining $\mathrm{SU}(2)_\ell$ charge (in green) is carried by the bosonic parton $\phi$. Both partons carry the $\mathrm{SU}(2)$ gauge charge (in red). The real fields $\mathrm{c}$, $\upphi$ and $\uppsi$ will inherit the charge assignments.