Emergence of massless Dirac quasiparticles in correlated hydrogenated graphene with broken sublattice symmetry

Using the variational cluster approximation (VCA) and the cluster perturbation theory, we study the finite-temperature phase diagram of a half-depleted periodic Anderson model on the honeycomb lattice at half-filling for a model of graphone, i.e., single-side hydrogenated graphene. The ground state of this model is found to be ferromagnetic (FM) semimetal. The origin of this FM state is attributed to the instability of a flat band located at the Fermi energy in the noninteracting limit and is smoothly connected to the Lieb-Mattis–type ferromagnetism. The spin-wave dispersion in the FM state is linear in momentum at zero temperature but becomes quadratic at finite temperatures, implying that the FM state is fragile against thermal fluctuations. Indeed, our VCA calculations find that the paramagnetic (PM) state dominates the finite-temperature phase diagram. More surprisingly, we find that massless Dirac quasiparticles with the linear energy dispersion emerge at the Fermi energy upon introducing the electron correlation $U$ at the impurity sites in the PM phase. The Dirac Fermi velocity is found to be highly correlated to the quasiparticle weight of the emergent massless Dirac quasiparticles at the Fermi energy and monotonically increases with $U$. These unexpected massless Dirac quasiparticles are also examined with the Hubbard-I approximation and the origin is discussed in terms of the spectral weight redistribution involving a large energy scale of $U$. Considering an effective quasiparticle Hamiltonian which reproduces the single-particle excitations obtained by the Hubbard-I approximation, we argue that the massless Dirac quasiparticles are protected by the electron correlation. Our finding therefore provides a unique example of the emergence of massless Dirac quasiparticles due to dynamical electron correlations without breaking any spatial symmetry. The experimental implications of our results for graphone as well as a graphene sheet on transition-metal substrates are also briefly discussed.

Figure 1

Schematic honeycomb lattice structure on which the half-depleted periodic Anderson model is defined for a model of graphone, i.e., single-side hydrogenated graphene. Green and red circles denote the carbon conduction sites on $A$ and $B$ sublattices of the honeycomb lattice, respectively, and yellow circles indicate the hydrogen impurity sites. The primitive translational vectors of the honeycomb lattice ${\mathbf{d}}_1=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)a$ and ${\mathbf{d}}_2=\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)a$ are denoted by blue arrows, where $a$ is the lattice constant between the next-nearest-neighboring carbon sites. Clusters considered in the VCA and the CPT (gray shaded regions) include a twelve-site cluster containing four unit cells (upper left), a nine-site cluster containing 3 unit cells (upper right), and a six-site cluster containing two unit cells (lower center). The orange arrows indicate primitive translational vectors for each cluster. The six- and nine-site clusters are used for finite-temperature calculations and the twelve-site cluster is used for zero-temperature calculations.

Figure 2

(a) The energy band structure $E_{\mathbf{k}}$ (in unit of $t$) and (b) $\langle {\varphi }_{\mathbf{k}\sigma }^{\mathrm{flat}}\left|c_{\mathbf{k}\sigma A}^{†}{c}_{\mathbf{k}\sigma A}\right|{\varphi }_{\mathbf{k}\sigma }^{\mathrm{flat}}\rangle$ for the noninteracting case with $t_{sp}/t=1$, where $|{\varphi }_{\mathbf{k}\sigma }^{\mathrm{flat}}\rangle$ is the eigenstate of the flat band with $E_{\mathbf{k}}=0$ at momentum $\mathbf{k}=(k_x,k_y)$. The first Brillouin zone is indicated by the hexagon with solid lines in (b), where high-symmetric momenta are also denoted by $\mathrm{\Gamma }$: $(0,0), K$: $\frac{4\pi }{3a}(1,0), K^{\prime }$: $\frac{4\pi }{3a}(\frac{1}{2},\frac{\sqrt{3}}{2})$, and $M$: $\frac{4\pi }{3a}(\frac{3}{4},\frac{\sqrt{3}}{4})$.

Figure 3

Finite-temperature phase diagram of the periodic Anderson model $\mathcal{H}$ obtained by the VCA for the six- and nine-site clusters with $t_{sp}/t=1$ at $n=1$. PM-L (FM-Q) stands for a PM (FM) phase with the linear (quadratic) quasiparticle dispersion around $E_{\mathrm{F}}$. In the noninteracting limit with $U=0$, the flat band appears at $E_{\mathrm{F}}$ (red solid line). The FM state is stable in a blue (orange) shaded region below a solid (dashed) line for the nine-site (six-site) cluster.

Figure 4

The CPT results of the single-particle excitation spectra for (a)–(d) up and (e)–(h) down electrons in the FM state at $T=0$. The calculations are for $U/t=4$ and $t_{sp}/t=1$ using the nine-site cluster (containing three unit cells). (a), (e) $A_{\sigma }(\mathbf{k},\omega )={\sum }_{\alpha }{A}_{\sigma }^{\alpha \alpha }(\mathbf{k},\omega )$; (b), (f) $A_{\sigma }^{AA}(\mathbf{k},\omega )$; (c), (g) $A_{\sigma }^{BB}(\mathbf{k},\omega )$; and (d), (h) $A_{\sigma }^{HH}(\mathbf{k},\omega )$. A Lorentzian broadening of $\eta /t=0.05$ is used. The spectral intensity is indicated by a color bar in each figure. Notice that the different intensity scales are used for different figures. The Fermi energy $E_{\mathrm{F}}$ is located at $\omega =0$.

Figure 5

Same as Fig. 4 but the enlarged scale at the vicinity of the $K$ point near the Fermi energy $E_{\mathrm{F}}$ (white lines). (a) $A_{\sigma }(\mathbf{k},\omega )={\sum }_{\alpha }{A}_{\sigma }^{\alpha \alpha }(\mathbf{k},\omega )$, (b) $A_{\sigma }^{AA}(\mathbf{k},\omega )$, (c) $A_{\sigma }^{BB}(\mathbf{k},\omega )$, and (d) $A_{\sigma }^{HH}(\mathbf{k},\omega )$. Notice that the spectra $A_{\uparrow }^{\alpha \alpha }(\mathbf{k},\omega )$ and $A_{\downarrow }^{\alpha \alpha }(\mathbf{k},\omega )$ for up and down spins, respectively, are plotted with different colors below and above $E_{\mathrm{F}}$ at $\omega =0$. The region of momenta taken in the horizontal axis is $0.2\pi$ in the $K-M$ ($K-\mathrm{\Gamma }$) direction from $K$.

Figure 6

The CPT results of the single-particle excitation spectra in the PM state for $U/t=4$ and $t_{sp}/t=1$ obtained using the nine-site cluster (containing three unit cells) at $T/t=0.025$ around the $K$ point near the Fermi energy $E_{\mathrm{F}}$ (white lines). (a) $A_{\sigma }(\mathbf{k},\omega )={\sum }_{\alpha }{A}_{\sigma }^{\alpha \alpha }(\mathbf{k},\omega )$, (b) $A_{\sigma }^{AA}(\mathbf{k},\omega )$, (c) $A_{\sigma }^{BB}(\mathbf{k},\omega )$, and (d) $A_{\sigma }^{HH}(\mathbf{k},\omega )$. Here, we only show the spectra $A_{\uparrow }^{\alpha \alpha }(\mathbf{k},\omega )$ for up electrons, which are exactly the same as $A_{\downarrow }^{\alpha \alpha }(\mathbf{k},\omega )$. A Lorentzian broadening of $\eta /t=0.05$ is used. The spectral intensity is indicated by a color bar in each figure. Notice that the different intensity scales are used for different figures. The region of momenta taken in the horizontal axis is $0.2\pi$ in the $K-M$ ($K-\mathrm{\Gamma }$) direction from $K$.

Figure 7

The CPT results of the single-particle excitation spectra for $U/t=0,1,\cdots ,7$ (from top to bottom panels) and $t_{sp}/t=1$ in the PM state at $T=0$ obtained using nine-site cluster (containing three unit cells). $A_{\sigma }(\mathbf{k},\omega )={\sum }_{\alpha }{A}_{\sigma }^{\alpha \alpha }(\mathbf{k},\omega ), A_{\sigma }^{AA}(\mathbf{k},\omega ), A_{\sigma }^{BB}(\mathbf{k},\omega )$, and $A_{\sigma }^{HH}(\mathbf{k},\omega )$ are shown from left to right panels. Here, we only show the spectra for up electrons, which are exactly the same as $A_{\downarrow }^{\alpha \alpha }(\mathbf{k},\omega )$. A Lorentzian broadening of $\eta /t=0.05$ is used. The spectral intensity is indicated by a color bar in each figure. Note that the different intensity scales are used for different figures. The Fermi energy $E_{\mathrm{F}}$ is located at $\omega =0$.

Figure 8

$U$ dependence of the Dirac Fermi velocity $v_{\mathrm{F}}$ calculated using the CPT and the Hubbard-I (H-I) approximation. The CPT calculations are done for the six-site cluster (containing two unit cells) at $T=0.025t$ in the PM phase, and for the nine-site cluster (containing three unit cells), and the twelve-site cluster (containing four unit cells) at $T=0$ where the PM state is assumed. Both calculations are for $t_{sp}/t=1$ at $n=1$. The size of dots is proportional to the spectral weight ${\rho }_K^{BB}$ for $B$ orbital at the Fermi energy and $v_0=\sqrt{3}t/2$ is the Dirac Fermi velocity of the pure graphene model.

Figure 9

The CPT results of the single-particle excitation spectra in the PM state for $U/t=4$ and $t_{sp}/t=1$ obtained using the six-site cluster (containing two unit cells) at $T=0.025t$, the nine-site cluster (containing three unit cells) at $T=0$, and the twelve-site cluster (containing four unit cells) at $T=0$ (from top to bottom panels). $A_{\sigma }(\mathbf{k},\omega )={\sum }_{\alpha }{A}_{\sigma }^{\alpha \alpha }(\mathbf{k},\omega ), A_{\sigma }^{AA}(\mathbf{k},\omega ), A_{\sigma }^{BB}(\mathbf{k},\omega )$, and $A_{\sigma }^{HH}(\mathbf{k},\omega )$ are shown from left to right panels. A Lorentzian broadening of $\eta /t=0.05$ is used. The spectral intensity is indicated by a color bar in each figure. Note that the different intensity scales are used for different figures. The Fermi energy $E_{\mathrm{F}}$ is located at $\omega =0$. Although the emergent massless Dirac quasiparticles for the twelve-site calculations (bottom panels) are not as clear as the other cases, it becomes apparent in the enlarged scale near $E_{\mathrm{F}}$, as shown in Fig. 10.

Figure 10

Same as Fig. 9 but the enlarged scale at the vicinity of the $K$ point near the Fermi energy $E_{\mathrm{F}}$ (white lines). A Lorentzian broadening of $\eta /t=0.01$ is used. The region of momenta taken in the horizontal axis is $0.2\pi$ in the $K-M$ ($K-\mathrm{\Gamma }$) direction from $K$.

Figure 11

The spectral weight ${\rho }_K^{BB}$ for $B$ orbital at the Fermi energy $E_{\mathrm{F}}$ and momentum $\mathbf{k}=K$ versus $v_{\mathrm{F}}^2$ calculated using the CPT and the Hubbard-I (H-I) approximation. The CPT calculations are done for the six-site cluster (containing two unit cells) at $T=0.025t$ in the PM phase, and for the nine-site cluster (containing three unit cells), and the twelve-site cluster (containing four unit cells) at $T=0$ where the PM state is assumed. Both calculations are for $t_{sp}/t=1$ at $n=1$ with various values of $U$ shown in Fig. 8. Here, $v_0=\sqrt{3}t/2$ is the Dirac Fermi velocity of the pure graphene model.

Figure 12

The mean-field results of the single-particle excitation spectra for (a)–(d) up and (e)–(h) down electrons in the FM state at $T=0$. The calculations are for $t_{sp}/t=1$ and $\mathrm{\Delta }/t=2$ at $n=1$. (a), (e) $A_{\sigma }(\mathbf{k},\omega )={\sum }_{\alpha }{A}_{\sigma }^{\alpha \alpha }(\mathbf{k},\omega )$; (b), (f) $A_{\sigma }^{AA}(\mathbf{k},\omega )$; (c), (g) $A_{\sigma }^{BB}(\mathbf{k},\omega )$; and (d), (h) $A_{\sigma }^{HH}(\mathbf{k},\omega )$. A Lorentzian broadening of $\eta /t=0.05$ is used. The spectral intensity is indicated by a color bar in each figure. Notice that the different intensity scales are used for different figures. The Fermi energy $E_{\mathrm{F}}$ is located at $\omega =0$.

Figure 13

Single-particle excitation dispersion ${\omega }_{\mathbf{k}}$ (in unit of $t$) for $U/t=1$ (a), 3 (b), and 5 (c) with $t_{sp}/t=1$ obtained by the Hubbard-I approximation. Note that the inner (outer) two bands correspond to ${\omega }_{\mathbf{k}}=\pm {\omega }_{-,\mathbf{k}}$ ($\pm {\omega }_{+,\mathbf{k}}$). The Fermi energy $E_{\mathrm{F}}$ is located at ${\omega }_{\mathbf{k}}=0$. The inner two bands display the massless Dirac dispersions at $K$ and $K^{\prime }$ points with the Dirac points exactly at $E_{\mathrm{F}}$, and the outer two bands correspond to the upper and lower Hubbard bands.

Figure 14

Orbital resolved density of states (a) $D_{A/B}\left(\omega \right)$ and (b) $D_H\left(\omega \right)$ for various values of $U$ (indicated in the figures) with $t_{sp}/t=1$ obtained by the Hubbard-I approximation. Notice that the results are shifted by $U$ for clarity. (c) The enlarged plot of $D_H\left(\omega \right)$ in (b) near the Fermi energy. The diverging density of states due to the flat band for $U=0$ is represented by the vertical line at $\omega =0$. The Fermi energy $E_{\mathrm{F}}$ is located at $\omega =0$.

Figure 15

Density of states $D_B\left(\omega \right)$ for $B$ orbital near the Fermi energy for various values of $U$ (indicated in the figures) with $t_{sp}/t=1$ obtained by (a) the Hubbard-I approximation and (b) the CPT. The CPT calculations are for the PM state at $T/t=0.025$ using the six-site cluster (containing two unit cells) with a Lorentzian broadening of $\eta /t=0.005$. For comparison, density of states $D_0\left(\omega \right)$ per orbital for the pure graphene model is plotted in (c) and also indicated by a shaded region in (a) and (b). Red straight lines ($\frac{2\sqrt{3}}{3\pi }\frac{\left|\omega \right|}{t^2}$) in (c) represent the initial slope of $D_0\left(\omega \right)$ around the Fermi energy $E_{\mathrm{F}}$ at $\omega =0$.

Figure 16

Single-particle excitation spectra for $U/t=0,1,\cdots ,7$ (from top to bottom) and $t_{sp}/t=1$ obtained by the Hubbard-I approximation. $A_{\sigma }(\mathbf{k},\omega )={\sum }_{\alpha }{A}_{\sigma }^{\alpha \alpha }(\mathbf{k},\omega ), A_{\sigma }^{AA}(\mathbf{k},\omega ), A_{\sigma }^{BB}(\mathbf{k},\omega )$, and $A_{\sigma }^{HH}(\mathbf{k},\omega )$ are shown from left to right panels. Here, we only show the spectra $A_{\uparrow }^{\alpha \alpha }(\mathbf{k},\omega )$ for up electrons, which are exactly the same as $A_{\downarrow }^{\alpha \alpha }(\mathbf{k},\omega )$. A Lorentzian broadening of $\eta /t=0.05$ is used. The spectral intensity is indicated by a color bar in each figure. Note that the different intensity scales are used for different figures. The Fermi energy $E_{\mathrm{F}}$ is located at $\omega =0$.

Figure 17

$U$ dependence of the effective dynamical bonding strength $L_{AB}$ and $T_{AB}$ (see the text for definition) for $t_{sp}/t=1$ calculated using the Hubbard-I approximation.

Figure 18

Schematic representations for (a) the periodic Anderson model $\mathcal{H}$ in the noninteracting limit and (b) the effective Hamiltonian ${\mathcal{H}}_{\text{H-I}}$ where the electron correlation $U$ is represented as the hybridization between $H$ orbital and auxiliary $X$ orbital. Both models are defined on bipartite lattices with the hopping between different sublattices, and preserve the chiral symmetry. The $A-B$ sublattice symmetry (the Hamiltonian is invariant under the exchange of $A$ and $B$ orbitals) as in the pure graphene model is apparently broken in both models. Although the sublattice balance is also broken in model (a), i.e., $\left|a\right|\ne \left|b\right|$, it is preserved in model (b) due to the presence of auxiliary $X$ orbital.

Figure 19

The FM spin-wave dispersion for the effective spin Hamiltonian ${\mathcal{H}}_{\mathrm{RKKY}}$ obtained in the linear spin-wave approximation at zero temperature. Here, $J_{\mathrm{K}}$ is set to be one as the energy unit. The black dots in the ${\mathrm{\Omega }}_{\mathit{q}}=0$ plane indicate high-symmetric momenta such as $\mathrm{\Gamma }, M, K$, and $K^{\prime }$ [see Fig. 2].

Figure 20

(a) Temperature dependence of the FM spin-wave dispersion ${\mathrm{\Omega }}_{\mathbf{q}}$ for the effective spin Hamiltonian ${\mathcal{H}}_{\mathrm{RKKY}}$ obtained in the linear spin-wave approximation. Here, the temperature dependence of the RKKY interaction $J_{\mathbf{R}}$ in Eq. (A13) is explicitly considered. (b) Same as (a) but the enlarged scale at the vicinity of the $\mathrm{\Gamma }$ point near zero energy. The region of momenta taken in the horizontal axis is $0.5\pi$ in the $\mathrm{\Gamma }-M$ ($\mathrm{\Gamma }-K$) direction from $\mathrm{\Gamma }$. Temperatures are indicated in (a) and $J_{\mathrm{K}}$ is set to be one as the energy unit.

Figure 21

The ground-state phase diagrams of ${\mathcal{H}}_{\mathrm{HM}}$ for $t_{sp}=t$ at half-filling. The results are obtained by numerically exactly diagonalizing the twelve-site clusters, containing four unit cells (see Fig. 1), with (a) periodic boundary conditions (PBC) and (b) open boundary conditions (OBC). The total spin $S$ of the ground state is indicated in the figures. The ground state with $S=2$ is found all unique. ${\mathcal{H}}_{\mathrm{HM}}$ with $U_C=0$ (indicated by red dashed lines) corresponds to the periodic Anderson model $\mathcal{H}$ studied in the main text.