Unconventional pairing symmetry of interacting Dirac fermions on a $\pi$-flux lattice

The pairing symmetry of interacting Dirac fermions on the $\pi$-flux lattice is studied with the determinant quantum Monte Carlo and numerical linked-cluster expansion methods. The $s^{*}$- (i.e., extended $s$-) and $d$-wave pairing symmetries, which are distinct in the conventional square lattice, are degenerate under the Landau gauge. We demonstrate that the dominant pairing channel at strong interactions is an unconventional $d{s}^{*}$-wave phase consisting of alternating stripes of $s^{*}$- and $d$-wave phases. A complementary mean-field analysis shows that while the $s^{*}$- and $d$-wave symmetries individually have nodes in the energy spectrum, the $d{s}^{*}$ channel is fully gapped. The results represent a new realization of pairing in Dirac systems, connected to the problem of chiral $d$-wave pairing on the honeycomb lattice, which might be more readily accessed by cold-atom experiments.

Figure 1

(a) The $\pi$-flux lattice in the Landau gauge. The solid (dashed) lines represent positive (negative) hoppings. The $d{s}^{*}$-wave pairing symmetry is schematically shown. A gauge transformation on sites marked by the white bars shows that $s^{*}$ and $d$ waves are equivalent. (b) The noninteracting energy spectrum, which shows that the system is a semimetal with two inequivalent Dirac points. The corresponding density of state is linear for low energies and has a Van Hove singularity at $E/t=2$.

Figure 2

DQMC results for the $\mathbf{q}=0$ (uniform) $s_{xy}$-wave, $d_{xy}$-wave, $p_x$-wave, $p_y$-wave, and $p_{xy}$-wave pairing structure factors as a function of temperature. Here $U=8t$ and the densities are: (a) $n=1.00$; (b) $n=0.95$; (c) $n=0.90$; (d) $n=0.85$. All channels are repulsive except for weakly attractive $p_{xy}$.

Figure 3

The $d{s}^{*}$-wave, uniform $d$-wave, and $s^{*}$-wave pairing structure factors vs temperature for $U=8t$ at densities $n=1.00,0.95,0.90,0.85$. $s^{*}$-wave and $d$-wave are identical to the accuracy of our calculations. Symbols are from the DQMC. Thin dashed and dotted lines are bare NLCE results for the seventh and eighth orders, respectively. Thick solid lines are results after the Euler resummation (see Appendix pp7).

Figure 4

(a) Temperature dependence of $d{s}^{*}$-wave pairing structure factor at density $n=0.9$ for different values of the interaction. The inset shows the structure factor vs $U$ at a fixed temperature $T=0.4$. A maximum is present at intermediate coupling. Symbols and lines in the main panels are the same as in Fig. 3. (b) The $d{s}^{*}$-wave pairing susceptibility as a function of the temperature at $n=0.9$ for different values of $U$.

Figure 5

The lower-energy dispersion within the mean-field theory near the Fermi energy for $s^{*}$ wave (or $d$ wave) (a) and $d{s}^{*}$ wave (b). Here the parameters are $\mu =0.8,\mathrm{\Delta }=0.2$.

Figure 6

Six of the nine the pairing symmetries available when the down spin fermion is created on a $3\times 3$ lattice about the location of the up spin fermion at the center. On-site $s$ wave, where the down spin fermion is created at the same point as the up spin fermion, is not shown, nor are $p_y$ and $p_{yx}$, which are just ${90}^{\circ }$ rotations of the $p_x$ and $p_{xy}$ symmetries illustrated in the two right-hand panels.

Figure 7

The $\pi$-flux lattice under other gauges. The corresponding $d{s}^{*}$-wave pairing symmetry is schematically shown. The lattice and the pairing symmetry is transformed from the one under Landau gauge [see Fig. 1 in the main text] by a gauge transformation $c_{i,\sigma }\left(c_{i,\sigma }^{†}\right)\to -c_{i,\sigma }(-c_{i,\sigma }^{†})$ on the sites marked by blue crosses.

Figure 8

The correlated pairing structure factors for two different lattice sizes, $L=10$ (black circles) and $L=12$ (blue triangles). The absolute difference for the densities $n=1.00,0.95,0.90,0.85$ at $U=8$ is of order ${10}^{-3}$, which is comparable to the statistical error bars (the corresponding axis is marked by the red arrow).

Figure 9

The DQMC and ED results on $4\times 4$ lattice for $n=1$ and $U=4$. The finite-temperature DQMC values tend to those of ED at zero temperature.

Figure 10

The comparison of the local moment (a) and NN spin correlation (b) for fermions with linear and quadratic dispersions. The results are extrapolated to the continuous imaginary time limit using two separate simulations with $\mathrm{\Delta }\tau =\frac{1}{16}$ and $\mathrm{\Delta }\tau =\frac{1}{12}$.

Figure 11

(a) The inverse of the $d{s}^{*}$-wave pairing susceptibility as a function of the temperature at $n=0.9$ for different values of $U$. (b) The inverse of the $d{s}^{*}$-wave pairing susceptibility divided by the local uncorrelated susceptibility at $r=0$.