Disclination Classes, Fractional Excitations, and the Melting of Quantum Liquid Crystals

We consider how fractional excitations bound to a dislocation evolve as the dislocation is separated into a pair of disclinations. We show that some dislocation-bound excitations (such as Majorana modes and half-quantum vortices) are possible only if the elementary dislocation consists of two inequivalent disclinations, as is the case for stripes or square lattices but not for triangular lattices. The existence of multiple inequivalent disclination classes governs the two-dimensional melting of quantum liquid crystals (i.e., nematics and hexatics), determining whether superfluidity and orientational order can simultaneously vanish at a continuous transition.

Figure 1

(a), (b) $\pm \pi /2$-disclination dipoles that fuse into dislocations with odd, even Burgers’ vectors, respectively. (c) Two inequivalent classes of disclinations of a two-dimensional striped superfluid. Lines denote nodes of the order parameter; the sign of the condensate between these is indicated on the left. The crest-centered disclination (red circle) involves no sign frustration, whereas the trough-centered disclination (red square) does involve sign frustration.

Figure 2

(a) Phase diagram of a striped or square-lattice superfluid in two dimensions, for a fixed Frank constant, as a function of superfluid stiffness and temperature. Solid lines denote phase transitions, due to the proliferation of defects (indexed by their orientational and superfluid charges); the dashed line represents the temperature at which type-$A$ disclinations would (hypothetically) have proliferated if the system were still ordered. The four phases are nematic/hexatic (Ne/He), nematic/hexatic superfluid (Ne/He-SF), isotropic superfluid (SF), and isotropic normal (I). (b) Phase diagram of a triangular-lattice superfluid [case (a)]. The notation is the same as that in (a); note that a direct transition from the hexatic superfluid to the isotropic normal state, absent in (a), appears here.

Figure 3

Two possible varieties of hexagonal-lattice superfluid. (a) Hexagonal lattice of nodes of the order parameter, corresponding to a honeycomb lattice of the particle density, plusses, minuses represent the sign of Bose condensate $\psi$. (b) Hexagonal lattice in the particle density, corresponding to a honeycomb lattice of vortices, 0, $\pm 2\pi /3$ are phases of the condensate $\psi$.

Figure 4

(a–d) Inequivalent trivial $\pm 2\pi /3$ disclinations. None breaks the filled circle, open circle-sublattice order (no nearest vertices are of same type). Tricoloring ordering (no adjacent plaquettes have same color) is preserved for (a) and (b) and is broken for (c) and (d). (e) Topological $\pm \pi /3$-disclination dipole. Each breaks filled circle, open circle-sublattice and tricoloring order.