Quantifying the fragility of unprotected quadratic band crossing points

We examine a basic lattice model of interacting fermions that exhibits quadratic band crossing points (QBCPs) in the noninteracting limit. In particular, we consider spinless fermions on the honeycomb lattice with nearest-neighbor hopping $t$ and third-nearest-neighbor hopping $t^{\prime \prime }$, which exhibits fine-tuned QBCPs at the corners of the Brillouin zone for $t^{\prime \prime }=t/2$. In this situation, the density of states remains finite at the Fermi level of the half-filled band and repulsive nearest-neighbor interactions $V$ lead to a charge-density-wave (CDW) instability at infinitesimally small $V$ in the random-phase approximation or mean-field theory. We examine the fragility of the QBCPs against dispersion renormalizations in the $t-t^{\prime \prime }-V$ model using perturbation theory and find that the $t^{\prime \prime }$ value needed for the QBCPs increases with $V$ due to the hopping renormalization. However, the instability toward CDW formation always requires a nonzero threshold interaction strength; i.e., one cannot fine-tune $t^{\prime \prime }$ to recover the QBCPs in the interacting system. These perturbative arguments are supported by quantum Monte Carlo simulations for which we carefully compare the corresponding threshold scales at and beyond the QBCP fine-tuning point. From this analysis, we thus gain a quantitative microscopic understanding of the fragility of the QBCPs in this basic interacting fermion system.

Figure 1

Contour plots of lines of constant positive band energy (top panels) near the Dirac point at $K\approx (2.42,0)$ and the corresponding DOS (bottom panels) for the noninteracting model, at $t^{\prime \prime }=0.4t$ (left panels) and $t^{\prime \prime }=0.5t$ (right panels). In the left lower panel, the DOS drops to zero linearly below $\epsilon \approx 0.025t$. It does not reach zero because of the finite numerical resolution. The right lower plot shows the DOS for the QBCP condition.

Figure 2

(a) Static noninteracting CDW susceptibility as a function of $t^{\prime \prime }$ for different temperatures. (b) The corresponding results for the threshold value $V_c$, according to the Stoner criterion.

Figure 3

(a) Real and imaginary part of the (non-self-consistent) momentum-dependent Fock self-energy along the edge of the irreducible Brillouin zone for $V=t$. The underlying lines show the real and imaginary parts of the nearest-neighbor hopping form factor $h_1\left(\mathbf{k}\right)$ in Eq. (3). (b) Hopping renormalization $\delta t$ from the first-order self-energy contribution for $V=t$, calculated using different values of $T$.

Figure 4

(a) Comparison of the bare ($V_{c,0}$) and hopping-renormalized (using self-consistent first-order perturbation theory, $V_{c,1\mathrm{st}}$) critical interaction strengths as functions of $t^{\prime \prime }$, for $T={10}^{-5}t$. Also indicated by the dashed line is the position of the QBCP condition ($V_{\mathrm{QBCP}}$) within self-consistent first-order perturbation theory. The panel (b) focuses on the region near $t^{\prime \prime }=0.5t$.

Figure 5

(a) The static CDW susceptibility as a function of the detuning $\mathrm{\Delta }=t^{\prime \prime }/t-1/2$. Also indicated is a fit of the low-$T$ data to a logarithmic dependence. (b) The critical interaction values $V_c$ according to the generalized Stoner criterium for different, fixed values of $b$.

Figure 6

QMC results for the correlation ratio $R_{\mathrm{CDW}}$ as a function of $V$ for different systems sizes at $t^{\prime \prime }/t=1/2$. Error bars are smaller than the symbol size.

Figure 7

QMC results for the crossing points in the correlation ratio $R_{\mathrm{CDW}}$ of consecutive system sizes $L$ and $L+3$ for various values of $t^{\prime \prime }/t$. Dashed lines indicate linear extrapolations in the low-$1/L$ regime to extract the thermodynamic limit value of the CDW threshold interaction $V_c$.

Figure 8

Result for the phase diagram of the $t-t^{\prime \prime }-V$ model from QMC simulations in terms of the extrapolated CDW threshold interaction $V_c$ as a function of $t^{\prime \prime }$. The lower tick of the uncertainty region (error bar) results from the linear extrapolation, while the upper tick indicates the estimate for $V_c$ from the crossing point in $R_{\mathrm{CDW}}$ for the largest simulated systems.

Figure 9

The CDW order parameter $m_{\mathrm{CDW}}$ as a function of in the interaction $V$ within finite-size Hartree-Fock mean-field theory for (a) different values of detuning $t^{\prime \prime }/t$ and (b) different linear system sizes $L$ in the vicinity of the dip of $V_c$ in Fig. 8 at $t^{\prime \prime }/t=0.34$, which exhibits significant nonmonotonous finite-size behavior.

Figure 10

The evolution of the noninteracting dispersion as a function of $t^{\prime \prime }/t$ shows that the QBCPs are split into four Dirac cones. Arrows indicate the shift of the satellite cones upon decreasing $t^{\prime \prime }/t$. While one Dirac cone remains at the $K$ ($K^{\prime }$) point, the satellite cones move toward the $M$ point where they eventually annihilate with their counterpart carrying the opposite Berry flux. The size and the color of the circles encode the Berry phase, as indicated in the second panel (from the top) and the bottom panel.