Symmetric Mass Generation in a Bilayer Honeycomb Lattice with $\mathrm{SU}(2)\times\mathrm{SU}(2)\times\mathrm{SU}(2)/\mathbb{Z}_2$ Symmetry

Abstract

Symmetric mass generation (SMG) is a mechanism for generating mass gaps in fermionic systems without breaking any symmetries or developing topological order, challenging the conventional Landau paradigm. In this Letter, we provide numerically exact evidence for SMG in (2+1) dimensions through large-scale determinant quantum Monte Carlo (DQMC) simulations of a bilayer honeycomb lattice model with $\mathrm{SU}(2)\times\mathrm{SU}(2)\times\mathrm{SU}(2)/\mathbb{Z}_2$ symmetry. We observe the simultaneous opening of single-particle and bosonic gaps at a critical coupling $J_c \approx 2.6$ with correlation length exponent $ν= 1.14(2)$, while an exhaustive search over all 19 symmetry-inequivalent fermion bilinear order parameters confirms the absence of any symmetry breaking. We estimate the fermion anomalous dimension to be $η_ψ= 0.071(1)$, which deviates significantly from the large-$N$ prediction ($η_ψ\approx 0.595$) and variational Monte Carlo estimates ($η_ψ\approx 0.62$), pointing to a distinct universality class. By contrasting with a related $\mathrm{Spin}(5)\times\mathrm{U}(1)/\mathbb{Z}_2$ model that develops an intermediate excitonic phase, we demonstrate the crucial role of pure non-Abelian symmetry in prohibiting bilinear condensates and enforcing the direct SMG transition.

Figures (14)

• Figure 1: (a) Bilayer honeycomb lattice. (b) Phase diagram of the bilayer honeycomb lattice model. A direct SMG transition occurs between the gapless Dirac semimetal phase and the gapped SMG phase without any symmetry breaking. (c) Real-space and (d) momentum-space coordinates of the honeycomb lattice.
• Figure 2: (a) Single-particle gap $\Delta_{sp}$ and (b) bosonic gap $\Delta_{b}$ as a function of $J$, extrapolated using the power-law form $\Delta=a\left(\frac{1}{L}\right)^{b}+c$ to the thermodynamic limit $L\to \infty$. $\Delta_{sp}$ and $\Delta_{b}$ open around $J=2.60$ and $2.55$, indicating a direct transition from the gapless phase to the gapped SMG phase.
• Figure 3: Structure factors of ferromagnetic (FM), charge density wave (CDW), spin density wave (SDW), and EC/SC order parameters as a function of $1/L$ for different $J$: (a) Dirac semimetal phase ($J=2.4$) and (b) SMG phase ($J=2.8$). All order parameters extrapolate to zero in the thermodynamic limit using power-law fits, indicating the absence of spontaneous symmetry breaking in the gapped phase.
• Figure 4: Finite-size scaling analysis for $\Delta_{sp}$ and $\Delta_b$ with a fitting window of four sizes. $L_{min}$ represents the minimum system size included in the fitting. (a) and (c) show representative data collapses for $\Delta_{sp}$ ($L_{min}=12$) and $\Delta_b$ ($L_{min}=9$), respectively. (b) and (d) show the evolution of the critical point $J_c$ and correlation length exponent $\nu$ as a function of $1/L_{min}$. The $\Delta_{sp}$ analysis converges, yielding $J_c=2.597(1)$ and $\nu=1.14(2)$. The $\Delta_b$ analysis does not fully converge; in particular, $\nu$ shows non-monotonic behavior due to large statistical noise at large sizes and imaginary times. Linear extrapolation gives $J_c=2.57(3)$ and $\nu=1.15(15)$.
• Figure 5: Scaling analysis of the fermion anomalous dimension $\eta_\psi$ from the off-diagonal Green's function in (a) momentum space and (c) real space. (b) shows the evolution of $\eta_\psi$ as a function of $1/L_{min}$ in momentum space; linear extrapolation gives $\eta_\psi = 0.071(1)$. (d) shows how $\eta_\psi$ obtained in real space varies with $J$ and gives $\eta_\psi=0.064(3)$ at $J=2.55$ and $\eta_\psi=0.088(3)$ at $J=2.60$, which are close to the result in momentum space.