Chiral lattice supersolid on edges of quantum spin Hall samples

We show that the edges of quantum spin Hall topological insulators represent a natural platform for the realization of an exotic magnetic phase which has all the properties of a lattice supersolid. On one hand, fermionic edge modes are helical due to the nontrivial topology of the bulk. On the other hand, a disorder at the edge or magnetic adatoms may produce a dense array of localized spins interacting with the helical electrons. The spin subsystem is magnetically frustrated since the indirect exchange favors the formation of helical spin order and the direct one favors (anti)ferromagnetic ordering of the spins. At a moderately strong direct exchange, the competition between these spin interactions results in the spontaneous breaking of parity and in the Ising-type order of the $z$ components at zero temperature. If the total spin is conserved, the spin order does not pin a collective massless helical mode which supports the ideal transport. In this case, the phase transition converts the helical spin order to the order of a chiral lattice supersolid. This represents a promising possibility for experimental studies of the elusive supersolidity.

Figure 1

Illustration of helical (upper panel) and supersolid (lower panel) spin orders. Red and green arrows show in-plane and $z$ components of the spins, respectively. The orange arrow exemplifies the total spin orientation in the supersolid phase. The dotted line stands for the helix; black circles are the lattice sites. We have assumed that the isotropy of the $xy$ plane is inherited from the bulk of QSH.

Figure 2

Phase diagram of the dense Kondo-Heisenberg array coupled to the interacting HEs at the QSH edge. Green and red lines show phases with helical and supersolid order at $T=0$, respectively. Light-green and light-red regions mark regimes where these orders are felt at finite $T$. The supersolid order disappears at $T\sim E_W$ [Eq. (14)]. The system becomes completely disordered at $T\sim \mathrm{\Delta }$ [Eq. (7)]. Dashed lines exemplify measurement protocols which could reveal different phases (see Conclusions).