Time Reversal Symmetric Topological Exciton Condensate in Bilayer HgTe Quantum Wells

We investigate a bilayer system of critical HgTe quantum wells, each featuring a spin-degenerate pair of massless Dirac fermions. In the presence of an electrostatic interlayer Coulomb coupling, we determine the exciton condensate order parameter of the system self-consistently. Calculating the bulk topological ${\mathbb{Z}}_2$ invariant of the resulting mean-field Hamiltonian, we discover a novel time reversal symmetric topological exciton condensate state, coined the helical topological exciton condensate. We argue that this phase can exist for experimentally relevant parameters. Interestingly, due to its multiband nature, the present bilayer model exhibits a nontrivial interplay between spontaneous symmetry breaking and topology: Depending on which symmetry the condensate order parameter spontaneously picks in combined orbital and spin space, stable minima in the free energy corresponding to both trivial and nontrivial gapped states can be found.

Figure 1

Schematic of a bilayer HgTe system with spatially separated two-dimensional electron and hole gases. Electrostatic coupling of the layers (indicated by double arrows) leads to electron-hole pairing between the layers (indicated by orange strips). Under certain circumstances, specified in the text, a topologically nontrivial EC phase emerges. The bilayer system can be externally tuned through an interlayer potential bias $V$.

Figure 2

(a) Gap of the HTEC phase $xz0$ as a function of the interwell potential bias $V$ for $\epsilon =10$ at zero temperature for different values of $A$ and $D/B$. $M=0$ and $B=-1000\mathrm{meV}{\mathrm{nm}}^2$. In (b) and (c), we closely analyze the case in which $A=250\mathrm{meV}\mathrm{nm}$, $V=7.5\mathrm{meV}$, and $D/B=0.5$, with $B=-1000\mathrm{meV}{\mathrm{nm}}^2$. (b) Gap of the HTEC phase $xz0$ as a function of the temperature for $M=0$. (c) Gap of the HTEC phase $xz0$ as a function of the Dirac masses $M=M^{\prime }$ of electron and hole layers.

Figure 3

Ribbon spectra for a width of 20 sites in the $x$ direction and a lattice regularization with lattice constant $a=20\mathrm{nm}$. Energies are measured in meV, momenta in units of $a^{-1}$. Left: Topologically nontrivial order parameter ${\tau }_x{\sigma }_z{s}_0$. Spin-degenerate edge modes at opposite edges are marked in red (cross zero energy). Right: Topologically trivial order parameter ${\tau }_y{\sigma }_x{s}_y$ where no edge states appear. $A=275\mathrm{meV}\mathrm{nm}$, $B=-1300\mathrm{meV}{\mathrm{nm}}^2$, $D=-730\mathrm{meV}{\mathrm{nm}}^2$, and $M=0$ in both panels.