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arXiv:2603.18278

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

Cheng-hao He, Yi-Zhuang You, Xiao Yan Xu

Published Mar 18, 2026• 0 citations
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Abstract

A central question beyond the Landau paradigm is the non-perturbative critical theory of the symmetric mass generation (SMG) transition, where strong interactions gap Dirac fermions in (2+1) dimensions without triggering spontaneous symmetry breaking or topological order. While previous studies have already provided evidence for direct SMG transitions in (2+1) dimensions, the fermion scaling dimension -- the key observable for distinguishing candidate critical theories -- has not been determined in a controlled unbiased way. In this Letter, using 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 establish a direct continuous transition by observing 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 further obtain the first controlled unbiased estimate of the fermion anomalous dimension, $η_ψ= 0.071(1)$, which deviates significantly from the large-$N$ prediction ($η_ψ\approx 0.595$) and variational Monte Carlo estimates ($η_ψ\approx 0.62$), thereby placing direct quantitative constraints on SMG criticality. By contrasting with a related $\mathrm{Spin}(5)\times\mathrm{U}(1)/\mathbb{Z}_2$ model that develops an intermediate excitonic phase, we show that pure non-Abelian symmetry plays a decisive role in stabilizing the direct SMG transition.

DOI
10.48550/arxiv.2603.18278

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Symmetric Mass Generation in a Bilayer Honeycomb Lattice with SU(2)×SU(2)×SU(2)/Z2\mathrm{SU}(2)\times\mathrm{SU}(2)\times\mathrm{SU}(2)/\mathbb{Z}_2SU(2)×SU(2)×SU(2)/Z2​ Symmetry
A central question beyond the Landau paradigm is the non-perturbative critical theory of the symmetric mass generation (SMG) transition, where strong interactions gap Dirac fermions in (2+1) dimensions without triggering spontaneous symmetry breaking or topological order. While previous studies have already provided evidence for direct SMG transitions in (2+1) dimensions, the fermion scaling dimension -- the key observable for distinguishing candidate critical theories -- has not been determined in a controlled unbiased way. In this Letter, using large-scale determinant quantum Monte Carlo (DQMC) simulations of a bilayer honeycomb lattice model with SU(2)×SU(2)×SU(2)/Z2\mathrm{SU}(2)\times\mathrm{SU}(2)\times\mathrm{SU}(2)/\mathbb{Z}_2SU(2)×SU(2)×SU(2)/Z2​ symmetry, we establish a direct continuous transition by observing the simultaneous opening of single-particle and bosonic gaps at a critical coupling Jc≈2.6J_c \approx 2.6Jc​≈2.6 with correlation length exponent ν=1.14(2)ν= 1.14(2)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 further obtain the first controlled unbiased estimate of the fermion anomalous dimension, ηψ=0.071(1)η_ψ= 0.071(1)ηψ​=0.071(1), which deviates significantly from the large-NNN prediction (ηψ≈0.595η_ψ\approx 0.595ηψ​≈0.595) and variational Monte Carlo estimates (ηψ≈0.62η_ψ\approx 0.62ηψ​≈0.62), thereby placing direct quantitative constraints on SMG criticality. By contrasting with a related Spin(5)×U(1)/Z2\mathrm{Spin}(5)\times\mathrm{U}(1)/\mathbb{Z}_2Spin(5)×U(1)/Z2​ model that develops an intermediate excitonic phase, we show that pure non-Abelian symmetry plays a decisive role in stabilizing the direct SMG transition.
Abstract Symmetric mass generation (SMG) transitions defy the conventional Landau–Ginzburg–Wilson paradigm by opening a many-body gap without spontaneous symmetry breaking or topological order, attracting intense interest across particle physics and condensed matter physics. Here, we utilize unbiased quantum Monte Carlo simulations to investigate the equilibrium and nonequilibrium critical dynamics of the SMG transition in a bilayer honeycomb lattice model. We unambiguously confirm the existence of an SMG transition at J c = 2.584(4) that separates the Dirac semimetal phase from a symmetry-preserving SMG phase. High-precision extraction of the critical exponents reveals a novel universality class that profoundly departs from mean-field theory. We then extend our study to the nonequilibrium regime, exploring the driven dynamics of the SMG transition. Notably, despite the breakdown of the prerequisites for the celebrated Kibble–Zurek mechanism, the nonequilibrium SMG transition still follows the generalized finite-time scaling. By bridging equilibrium criticality and nonequilibrium dynamics, our work uncovers the universal critical properties of SMG transitions, providing a solid theoretical basis for future experimental studies of SMG physics.
The symmetric mass generation (SMG) phase is an insulator in which a single-particle gap is intrinsically opened by the interaction, without involving symmetry spontaneously breaking or topological order. Here, we perform unbiased quantum Monte-Carlo simulation and systematically investigate a bilayer fermionic model hosting Fermi surface SMG in the strongly interacting regime. With increasing interaction strength, the model undergoes a quantum phase transition from an exciton insulator to an SMG phase, belonging to the (2+1)-dimensional O(4) universality class. We access the spectral properties of the SMG phase, resembling a Mott insulating phase with relatively flat dispersion and pronounced spectral broadening. The dispersion of Green's function zeros is extracted from spectral function, featuring a surface at zero frequency precisely located at the original non-interacting Fermi surface, which constitutes a hallmark of the Fermi surface SMG phase. The bilayer model we study is potentially relevant to the newly discovered high-TcT_cTc​ superconductor La3Ni2O7\rm{La}_3 \rm{Ni}_2 \rm{O}_7La3​Ni2​O7​. Our results in SMG phase qualitatively capture the salient features of spectral function unveiled in recent ARPES experiments, shedding new insight on the underlying physics of La3Ni2O7\rm{La}_3 \rm{Ni}_2 \rm{O}_7La3​Ni2​O7​.
This paper investigates the symmetric mass generation transition in a bilayer honeycomb lattice model using variational Monte Carlo simulations. By measuring the fermion scaling dimension at the critical point, the authors provide evidence for fermion fractionalization into deconfined emergent particles, supporting the hypothesis that the transition is a fermionic deconfined quantum critical point. The numerical results, including the critical exponent and fermion dimension, agree with theoretical predictions.
We study the phase diagram of interacting spinless fermions on the honeycomb bilayer at charge neutrality using large-scale quantum Monte Carlo simulations. In the noninteracting limit, the low-energy spectrum features quadratically dispersing bands that touch at the corners of the hexagonal Brillouin zone. Weak to intermediate interactions induce a splitting of each of the quadratic band touching points into four Dirac points, located along high-symmetry directions of the reciprocal lattice. Strong interactions lead to the formation of a layer-polarized charge density wave, which spontaneously breaks the Z2\mathbb Z_2Z2​ layer inversion symmetry and opens an insulating gap in the spectrum. We show that the semimetal-to-insulator quantum phase transition as a function of interaction is continuous and characterized by emergent relativistic symmetry. Our results for the values of the correlation-length exponent ννν, the order-parameter anomalous dimension ηϕη_ϕηϕ​, and the fermion anomalous dimension ηψη_ψηψ​ agree with those of the theoretically predicted 2+1D Gross-Neveu-Ising universality class with eight two-component Dirac fermions within less than 5\%\ deviation. We also determine the crossover scale as a function of interaction strength between the nonrelativistic semimetal state at high temperatures, characterized by dynamical critical exponent z=2z = 2z=2, and the Dirac semimetal state at intermediates temperatures, characterized by z=1z=1z=1. Further reducing the temperature below the crossover scale at a fixed value of the interaction strength above the quantum critical point results in a classical ordering transition in the 2D Ising universality class.
Motivated by the physics of spin-orbital liquids, we study a model of interacting Dirac fermions on a bilayer honeycomb lattice at half filling, featuring an explicit global SO(3)×U(1) symmetry. Using large-scale auxiliary-field quantum Monte Carlo (QMC) simulations, we locate two zero-temperature phase transitions as function of increasing interaction strength. First, we observe a continuous transition from the weakly interacting semimetal to a different semimetallic phase in which the SO(3) symmetry is spontaneously broken and where two out of three Dirac cones acquire a mass gap. The associated quantum critical point can be understood in terms of a Gross-Neveu-SO(3) theory. Second, we subsequently observe a transition toward an insulating phase in which the SO(3) symmetry is restored and the U(1) symmetry is spontaneously broken. While strongly first order at the mean-field level, the QMC data are consistent with a direct and continuous transition. It is thus a candidate for a new type of deconfined quantum critical point that features gapless fermionic degrees of freedom.
Symmetric mass generation (SMG) is a mechanism to give gapless fermions a mass gap by nonperturbative interactions without generating any fermion bilinear condensation. The previous studies of SMG have been limited to Dirac/Weyl/Majorana fermions with zero Fermi volume in the free fermion limit. In this paper, we generalize the concept of SMG to Fermi liquid (FL) with a finite Fermi volume and discuss how to gap out the Fermi surfaces (FSs) by interactions without breaking the U(1) loop group symmetry or developing topological orders. We provide examples of FS SMG in both (1+1)(1+1)(1+1)-dimensional [(1+1)D][(1+1)\mathrm{D}][(1+1)D] and (2+1)(2+1)(2+1)-dimensional FL systems when several FSs together cancel the FS anomaly. However, the U(1) loop group symmetry in these cases is still restrictive enough to rule out all possible fermion bilinear gapping terms, such that a nonperturbative interaction mechanism is the only way to gap out the FSs. This symmetric FS reconstruction is in contrast to the conventional symmetry-breaking gapping mechanism in the FL. As a side product, our model provides a pristine one-dimensional lattice regularization for the (1+1)D(1+1)\mathrm{D}(1+1)D U(1) symmetric chiral fermion model (e.g., the 3-4-5-0 model) by utilizing a lattice translation symmetry as an emergent U(1) symmetry at low energy. This opens up the opportunity for efficient numerical simulations of chiral fermions in their own dimensions without introducing mirror fermions under the domain wall fermion construction.