Charge Order in the Holstein Model on a Honeycomb Lattice

The effect of electron-electron interactions on Dirac fermions, and the possibility of an intervening spin-liquid phase between the semimetal and antiferromagnetic (AF) regimes, has been a focus of intense quantum simulation effort over the last five years. We use determinant quantum Monte Carlo simulations to study the Holstein model on a honeycomb lattice and explore the role of electron- phonon interactions on Dirac fermions. We show that they give rise to charge-density-wave (CDW) order and present evidence that this occurs only above a finite critical interaction strength. We evaluate the temperature for the transition into the CDW which, unlike the AF transition, can occur at finite values owing to the discrete nature of the broken symmetry.

Figure 1

(a) A $4\times 4$ honeycomb lattice, with the trajectory (red dashed line) corresponding to the horizontal axis of (b), which shows charge correlations $c(\mathbf{r})$ at ${\lambda }_D=2/3$, ${\omega }_0=1$, and several temperatures. Here, and in all subsequent figures, when not shown, error bars are smaller than the symbol size.

Figure 2

(a) The charge structure factor as a function of $\beta$, for different lattice sizes ($L=4–8$), and its (b) best data collapse, with the 2D Ising critical exponents, which yields ${\beta }_c=5.8$. (c) The crossing plot for $S_{\mathrm{CDW}}/L^{\gamma /\nu }$, with vertical dashed lines indicating the uncertainty in the critical temperature. Here ${\lambda }_D=2/3$ and ${\omega }_0=1$.

Figure 3

CDW structure factor $S_{\mathrm{CDW}}$ as a function of dimensionless coupling ${\lambda }_D$. $S_{\mathrm{CDW}}$ becomes small for ${\lambda }_D\lesssim 0.25$. For the square lattice, $S_{\mathrm{CDW}}$ is large to much smaller values of ${\lambda }_D$. In addition, for the honeycomb (Hc.) lattice $S_{\mathrm{CDW}}$ does not change for the two lowest temperatures, whereas $S_{\mathrm{CDW}}$ continues to grow at weak coupling for the square (Sq.) lattice.

Figure 4

(a) The charge gap ${\mathrm{\Delta }}_c$ (see text) as a function of ${\lambda }_D$. (b) The electronic compressibility $\kappa$ as a function of ${\lambda }_D$ for square (open symbols) and honeycomb (filled symbols) lattices with linear sizes $L=8$ and 6, respectively.

Figure 5

Critical temperature for the CDW transition in the honeycomb Holstein model inferred from finite-size scaling analysis in Fig. 2. The inset shows the crossing of the invariant correlation ratio $R_c$ (see text), resulting in the indicated QCP, in good agreement with the value at which an extrapolated $T_c$ would vanish.