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arXiv:2603.22259 · Physical review. B./Physical review. B

Emergent relativistic symmetry from interacting fermions on the honeycomb bilayer

Zi Hong Liu, Lukas Janssen

Published Sep 22, 2026• 0 citations
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Abstract

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 $\mathbb Z_2$ 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 $ν$, the order-parameter anomalous dimension $η_ϕ$, and the fermion anomalous dimension $η_ψ$ 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 $z = 2$, and the Dirac semimetal state at intermediates temperatures, characterized by $z=1$. 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.

DOI
10.1103/51zq-f75g

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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.
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We characterize, by means of large-scale fermion quantum Monte Carlo simulations, metallic and deconfined quantum phase transitions in a bilayer honeycomb model in terms of their quantum critical and finite-temperature properties. The model features three distinct phases at zero temperature as a function of interaction strength. At weak interaction, a fully symmetric Dirac semimetal state is realized. At intermediate and strong interaction, respectively, two long-range-ordered phases, each breaking different symmetries are stabilized. The ordered phases feature partial and full gap openings in the fermion spectrum, respectively. We clarify the symmetries of the different zero-temperature phases and the symmetry-breaking patterns across the two quantum phase transitions between them. The first transition between the disordered and long-range-ordered semimetallic phases has previously been argued to be described by the (2+1)(2+1)(2+1)-dimensional Gross-Neveu-SO(3) field theory. By performing simulations with an improved symmetric Trotter decomposition, we further substantiate this claim by computing the critical exponents $1/\ensuremath{\nu}$, ${\ensuremath{\eta}}_{\ensuremath{\phi}}$, and ${\ensuremath{\eta}}_{\ensuremath{\psi}}$, which turn out to be consistent with the field-theoretical expectation within numerical and analytical uncertainties. The second transition between the two long-range-ordered phases has previously been proposed as a possible instance of a metallic deconfined quantum critical point. We further develop this scenario by analyzing the spectral functions in the single-particle, particle-hole, and particle-particle channels. Our results indicate gapless excitations with a unique velocity, supporting the emergence of Lorentz symmetry at criticality. We also compute the finite-temperature phase boundaries of the ordered states above the fully gapped state at large interaction. The phase boundary vanishes smoothly in the vicinity of the putative metallic deconfined quantum critical point, in agreement with the expectation for a continuous or weakly first-order transition.
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This Thesis explores the formation of novel phases in Dirac systems that are driven by strong many-body interactions and collective quantum fluctuations. The central focus is the half-filled honeycomb lattice, which is the prototypical Dirac semimetal with low-energy excitations described by Dirac fermions. Of particular interest are exotic phases of matter that are not favoured at mean-field but are stabilised by fluctuations. Self-consistent path integral techniques are used to demonstrate that extended repulsive interactions favour charge order that breaks translational and other lattice symmetries. This charge order outcompetes interaction induced topological insulating phases, which are dominant at the mean-field level. The interacting quantum critical point was found to be near the critical point of a topological Lifshitz transition. Interacting Dirac fermions exhibit perhaps the simplest example of fermionic quantum criticality. Gross-Neveu-Yukawa field theories describe the universality class of semimetal-insulator transitions, which are driven by on-site and nearest neighbour repulsive interactions on the honeycomb lattice. The Dirac fermions couple to the dynamical order parameter fields and play a crucial role in determining the universal behaviour. Naturally, this variety of criticality is outside of the Ginzburg-Landau-Wilson paradigm. This idea is extended to include the translational symmetry breaking charge order. Here it is found that the breaking of crystal symmetries corresponds to Lorentz violating effective field theories. The criticality is analysed with the renormalisation group. The low energy excitations at the topological transition are gapless semi-Dirac fermionic quasiparticles that disperse linearly in one direction and quadratically in the other. The instabilities induced by generic short-range repulsions are investigated. The criticality of this anisotropic system is analysed with the renormalisation group in two spatial dimensions, and it is found to exhibit anisotropic order parameter correlations.
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.