Kekule versus hidden superconducting order in graphene-like systems: Competition and coexistence

We theoretically study the competition between two possible exotic superconducting orders that may occur in graphene-like systems, assuming dominant nearest-neighbor attraction: the gapless hidden superconducting order, which renormalizes the Fermi velocity, and the Kekule order, which opens a superconducting gap. We perform an analysis within the mean-field theory for Dirac electrons, at finite temperature and finite chemical potential, as well as at half filling and zero temperature, first excluding the possibility of the coexistence of the two orders. In that case, we find the dependence of the critical (more precisely, crossover) temperature and the critical interaction on the chemical potential. As a result of this analysis, we find that the Kekule order is preferred over the hidden order at both finite temperature and finite chemical potential. However, when the coexistence of the two superconducting orders is allowed, by solving the coupled mean-field gap equations, we find that above a critical value of the attractive interaction a mixed phase sets in, in which these orders coexist. We show that the critical value of the interaction for this transition is greater than the critical coupling for the hidden superconducting state in the absence of the Kekule order, implying that there is a region in the phase diagram where the Kekule order is favored as a result of the competition with the hidden superconducting order. The latter, however, eventually sets in and coexists with the Kekule state. According to our mean-field calculations, the transition from the Kekule to the mixed phase is of the second order, but it may become first order when fluctuations are considered. Finally, we investigate whether these phases could be possible in honeycomb superlattices of self-assembled semiconducting nanocrystals, which have been recently experimentally realized with CdSe and PbSe.

Figure 1

Dispersion relation in Eq. (28) is plotted for $\mu /t={\mathrm{\Delta }}^{\prime }/t=0,\tilde{V}/t=v_F/t=1$, and $m=1$ [panel (a)], showing that the Kekule order $m$ opens a gap. In panel (b), the renormalization of the Fermi velocity by the hidden order parameter, given by Eq. (29), is shown.

Figure 2

Evolution of the thermodynamical potential in Eq. (26) for the Kekule (a) and hidden order (b) with $v_F/t=\mathrm{\Lambda }=N=1,k_{B}T/t=0.1$, and $\mu /t=0$. (a) Kekule order: $\tilde{V}/t=2\pi$ (solid), $\tilde{V}/t=3\pi$ (black), $\tilde{V}/t=5\pi$ (large dashed), $\tilde{V}/t=10\pi$ (small dashed), and $\tilde{V}/t=25\pi$ (dotted). (b) Hidden order: $\tilde{V}/t=15\pi$ (solid), $\tilde{V}/t=18\pi$ (black), $\tilde{V}/t=21\pi$ (large dashed), $\tilde{V}/t=30\pi$ (small dashed), and $\tilde{V}/t=50\pi$ (dotted).

Figure 3

Behavior of the critical interaction for the Kekule (blue) and hidden order (red) with increasing temperature with $v_F/t=\mathrm{\Lambda }=1,\mu /t=0$, and $m/t={\mathrm{\Delta }}^{\prime }/t=0$.

Figure 4

Solutions of the finite-temperature gap equation for the Kekule (a) and hidden order (b) with $v_F/t=\mathrm{\Lambda }=1$. In panel (a) ${\mathrm{\Delta }}^{\prime }/t=0$ and $\tilde{V}/t=4\pi$, while in panel (b) $m/t=0$ and $\tilde{V}/t=20\pi$. In the plots we use the following values of the chemical potential: $\mu /t=0$ (solid), $\mu /t=0.1$ (large dashed), $\mu /t=0.25$ (small dashed), and $\mu /t=0.5$ (dotted).

Figure 5

Behavior of the critical temperature for the Kekule (blue) and hidden order (red) with increasing chemical potential with $v_F/t=\mathrm{\Lambda }=1$. For the Kekule order (blue), we chose $\tilde{V}/t=3\pi$, and for the hidden order (red) $\tilde{V}/t=18\pi$.

Figure 6

Solutions of the finite-temperature gap equation for the Kekule (a) and hidden order (b) with $v_F/t=\mathrm{\Lambda }=1$ for different values of the chemical potential: $\mu /t=0$ (solid), $\mu /t=0.1$ (large dashed), $\mu /t=0.25$ (small dashed), and $\mu /t=0.5$ (dotted). $m\left(T_c\right)=0$ and ${\mathrm{\Delta }}^{\prime }/t=0$ for (a), and ${\mathrm{\Delta }}^{\prime }\left(T_c^{\prime }\right)=0$ and $m/t=0$ for (b).

Figure 7

Behavior of the Kekule (blue) and hidden (red) order parameters with interaction strength $\tilde{V}$, given by Eqs. (45) and (44), respectively. We use the critical coupling for the Kekule and hidden order in the absence of the competition $\widetilde{V_c}=1$ and $\widetilde{V_c^{\prime }}=5$, respectively.

Figure 8

Phase diagram of the system as a function of the nearest-neighbor attraction $\tilde{V}$. As this coupling increases, at a critical value $\widetilde{V_c}$ the system first enters the Kekule superconducting state (blue region). In the region labeled by dashed blue lines, the Kekule order is favored over the hidden order. The latter would in the absence of the Kekule order set in for $\tilde{V}>\widetilde{V_c^{\prime }}$, but eventually coexists with the Kekule superconductor above the critical interaction ${\widetilde{V_c}}^{\mathrm{m}}>\widetilde{V_c^{\prime }}$ (red solid line).

Figure 9

Dependence of the critical coupling for the onset of the mixed phase ${\widetilde{V_c}}^{\mathrm{m}}$ on the critical coupling for the hidden order in absence of the competition, $\widetilde{V_c^{\prime }}$. The critical coupling for the Kekule order ${\tilde{V}}_c/\mathrm{\Lambda }=1$.