Electron-hole pairing of Fermi-arc surface states in a Weyl semimetal bilayer

The topological nature of Weyl semimetals (WSMs) is corroborated by the presence of chiral surface states, which connect the projections of the bulk Weyl points by Fermi arcs (FAs). We study a bilayer structure realized by introducing a thin insulating spacer into a bulk WSM. Employing a self-consistent mean-field description of the interlayer Coulomb interaction, we propose that this system can develop an interlayer electron-hole pair condensate. The formation of this excitonic condensate leads to partial gapping of the FA dispersion. We obtain the dependence of the energy gap and the critical temperature on the model parameters, finding, in particular, a linear scaling of these quantities with the separation between the Weyl points in momentum space. A detrimental role is played by the curvature of the FAs, although the pairing persists for moderately small curvature. A signature of the condensate is the modification of the quantum oscillations involving the surface FAs.

Figure 1

Illustration of a WSM bilayer, where two macroscopic slabs featuring an identical WSM phase are stacked in the $z$ direction, separated by an insulating spacer of thickness $t$.

Figure 2

Coulomb matrix element $V_{\mathbf{k},{\mathbf{k}}^{\prime }}^{\mathbf{Q}=0}$ for $\mathbf{k}$ and ${\mathbf{k}}^{\prime }$ on the $x$ axis, keeping $k_x^{\prime }$ fixed and varying $k_x\in (-K_0,K_0)$. The interaction becomes asymmetric and decays faster as a function of $|k_x-k_x^{\prime }|$ for $k_x^{\prime }$ approaching $K_0$ due to the increasing spatial extent of the FA states, which merge with the bulk states at $\pm K_0$.

Figure 3

Gap parameter (a) ${\mathrm{\Delta }}_{\mathbf{k}=k_x\widehat{\mathbf{x}}}$ and (b) ${\mathrm{\Delta }}_{\mathbf{k}=k_y\widehat{\mathbf{y}}}$, calculated for various values of $K_0$ and $T=0$. ${\mathrm{\Delta }}_{\mathbf{k}=k_x\widehat{\mathbf{x}}}$ scales roughly linearly with $K_0$. ${\mathrm{\Delta }}_{\mathbf{k}=k_y\widehat{\mathbf{y}}}$ features a peak at $k_y=0$ with a width on the order of $K_0$ and approaches zero for $|k_y|\gg K_0$.

Figure 4

Energy dispersion of surface states of a WSM bilayer in the regime of nonzero particle-hole coherence. Results have been obtained for $T=0$ in the limit $K_0\ll k_{\text{cut}}$ and $K_{0}t\ll 1$, with ${\epsilon }_r=10$, $K_0=0.01{\mathrm{nm}}^{-1}$, and $v_F=100\mathrm{meV}\mathrm{nm}$.

Figure 5

Maximum gap parameter $\mathrm{\Delta }$ at $T=0$ as a function of ${\epsilon }_r{v}_F$, calculated for various values of $K_0$. In the inset, $\mathrm{\Delta }$ is plotted as a function of $K_0$ for fixed ${\epsilon }_r{v}_F=1000\mathrm{meV}\mathrm{nm}$.

Figure 6

Maximum gap parameter $\mathrm{\Delta }$ as a function of temperature, calculated for various values of $K_0$ and ${\epsilon }_r$.

Figure 7

Maximum gap parameter $\mathrm{\Delta }$ calculated at $T=0$ as a function of $K_{0}t$ for three values of $K_0$. Inset: $\mathrm{\Delta }$ at $T=0$ as a function of $K_0/k_{\text{cut}}$.

Figure 8

Dispersion curves of the FA states for (a) nonzero chemical potential $\mu$ and (b) nonzero interlayer potential bias $V$. (c) Model of curved FAs. In all panels, dashed and solid lines refer to surface states belonging to opposite surfaces.

Figure 9

Maximum gap parameter $\mathrm{\Delta }$ at $T=0$ for a model with curved FAs as a function of the curvature parameter $\gamma$ for various values of $K_0$, tuning $\mathbf{Q}$ to the optimal nesting conditions. Inset: maximal value of the curvature parameter compatible with electron-hole coherence as a function of the Fermi velocity $v_F$.

Figure 10

Sketch of a magnetic orbit extending over the two WSM layers connected by the electron-hole condensate. The orbit includes FAs at the outermost surfaces and vertical bulk portions; see the text and Ref. [61].