Emergent relativistic symmetry from interacting fermions on the honeycomb bilayer

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.

Figures (10)

• Figure 1: (a) Illustration of the layer-polarized charge density wave (CDW) state. Blue and red spheres denote sites with an excess and a deficit, respectively, of charge carriers relative to the average half filling. Purple-shaded regions highlight interlayer dimers hosting a single delocalized fermion. (b) Phase diagram as a function of the nearest-neighbor repulsion $V$ and temperature $T$ from quantum Monte Carlo simulations. Blue dots denote finite-temperature Ising transition points separating the CDW insulator at lower temperatures from the disordered semimetal phase at higher temperatures. Red squares mark the crossover from the Dirac semimetal (DSM) regime at intermediate temperatures to the quadratic band touching (QBT) regime at high temperatures. Insets illustrate the low-energy spectra in the three distinct regimes. Lines are guides to the eye.
• Figure 2: (a) Correlation ratio $R_{\mathrm c}$ as a function of the interaction $V$ at zero temperature. The inset shows the finite-size scaling behavior of the quantum critical point $V(1/L)$ obtained from the crossing of $R_\mathrm c(L)$ and $R_\mathrm c(L+3)$. (b-c) Crossing-point analyses of the critical exponents $1/\nu$, $\eta_{\phi}+z$, and $\eta_{\psi}$ as functions of $1/L$. The dotted curves represent the power-law scaling functions $aL^{-p}+c$.
• Figure 3: (a) Rescaled charge susceptibility $L^{\eta-2}\chi$ as a function of $T$ for fixed $V = 1.15$. (b) Same as (a), but plotted as a function of $(T-T_\text{Ising})L^{1/\nu_\text{Ising}}$ with $\nu_\text{Ising} = 1$. (c,d) Same as (a,b), but for $V=1.25$.
• Figure 4: (a) Uniform charge susceptibility $\chi_{\text{uni}}$ as a function of $T$ for $V=1.00$ and $L=21$. The shaded yellow region indicates the confidence interval of the crossover temperature $T_{\text{cross}}$, determined by the equal-residual criterion. (b) Same as (a), but for $V=1.15$. (c) Fermionic single-particle spectral function $A(\mathbf k,\omega)$ for $V=1.00$, $L=24$ and $T=0.2$. (d) Same as (b), but for $T= 0.02$.
• Figure S1: (a) Correlation ratio $R_\mathrm c$ as a function of $V$ for different system sizes. (b) Same as (a), but for the order parameter $m^2_\text{CDW}$. (c) Same as (a), but for the quasiparticle weight $Z_\text{qp}$. (d) Correlation ratio $R_\mathrm c$ as a function of the rescaled tuning parameter $(V-V_\mathrm c)L^{1/\nu}$ for different system sizes, using $V_\mathrm{c} = 0.9$ and $1/\nu = 1.06$ as obtained from the crossing-point analysis presented in the main text. (e) Same as (d), but for the rescaled order parameter $L^{z+\eta_\phi} m^2_\text{CDW}$, using $z+\eta_\phi = 1.85$. (f) Same as (d), but for the quasiparticle weight $L^{\eta_{\psi}}Z_\text{qp}$, using $\eta_\psi = 0.02$.