Braiding and exchange statistics of liquid crystalline Majorana quasiparticles

Abstract

Liquid crystalline defects in 3D can be viewed as geometric spinors, whose emergent properties are reminiscent of those of topological excitations in quantum condensed matter, such as Majorana quasiparticles. However, it is unclear how deep this analogy is, and whether this is a purely mathematical mapping, or it extends to key physical features, such as the exchange statistics or braiding behaviour. To address this question, here we consider a simple pattern made up of four nematic Majorana-like defect profiles, and ask how the defect profiles change as we braid them repeatedly around each other. Surprisingly, we find that in a large range of parameter space the defect profiles behave as classical analogues of non-Abelian anyons, which can be described in our case by defect bivectors moving on a Bloch-like hemisphere. Elastic interactions and dynamical effects enhance the complexity of the gates which can be performed by braiding these quasiparticles, making these liquid crystalline spinors promising candidates as components of topological computers.

Figures (10)

• Figure 1: Sketch of the $4$-defect Majorana square, with alternating $-1/2$ (orange) and $+1/2$ (cyan) defects. The overall pattern behaves as a "nematic bit", which can be described by a geometric spinor on the Bloch sphere (see text; in this starting configuration the spinor is aligned along $\pm \hat{\bf z}$). We also sketch the trajectories simulated when exchanging defects $1$ with $2$ (i), with opposite winding number (i), and $2$ with $4$ (ii), with equal winding number. Black spots indicate regions with $\chi = 0$, where the simulated laser is applied.
• Figure 2: (a,b) Liquid crystal patterns associated with one cycle of the repeated exchange between defects $1$ and $2$ in the Majorana square. Snapshots in the first row (a) correspond to the first exchange, and those in the second row (b) to the second exchange, which returns the system to a configuration which is related to the starting one by a $90^{\circ}$ in-plane overall rotation. Defects of winding $-1/2$ (orange) and $+1/2$ (cyan) have been tracked using the local charge density in Eq. (\ref{['defDij']}), hence the small variation in the color of the background nematic pattern when the director rotates out of the plane. (c) Free energy versus time, showing that the states obtained after each exchange have the same free energy. (d) Value of $Q_{xy}$ in the middle of the square as a function of time. Light- and dark-gray shading marks the time windows of the first (a) and second (b) exchange events, respectively. (e) Plot of the defect bivector $\Omega$ (corresponding to the axis of the Majorana square bivector) for configurations corresponding to the three successive square configurations in (a,b). (f) Superposition of spherical harmonics (up to $99$th order) in a spherical harmonics expansion of the orbits on the Bloch sphere, corresponding to $64$ exchanges of the first two defects.
• Figure 3: (a-d) Liquid crystal patterns following repeated exchange between defects $2$ and $4$ in the Majorana square. (e) Free energy versus time, showing that after each exchange the states have the same, or nearly the same, free energy. (f) Value of $Q_{xy}$ in the middle of the square as a function of time.
• Figure 4: (a) Angle between square bivectors after an exchange, $\theta$, as a function of $l_n/d$. Shadings denote the regimes discussed in the text (Inv = Invariant). (b) Superposition of spherical harmonics (up to $99$th order) in a spherical harmonics expansion of the orbit on the Bloch sphere, of a configuration starting from the state in Fig. \ref{['fig1']}, following repeated exchange between defects $(1,2)$ (see Fig. \ref{['fig1']}), for $l_n\simeq 0.14$. (c,d) Corresponding representation for an orbit of a configuration which is braided according to the pattern (braid word) BDACBDACBDACBDAC (c) and ABACABACACACACAC (d), where A, B, C, D denote exchanges between particles $(1,2)$, $(1,4)$, $(3,4)$, $(2,3)$ respectively.
• Figure 5: (a) Spherical harmonics representation of the Bloch sphere orbit (i) and free energy versus time [(ii); part of the time series only shown for clarity] for random braiding for $K=0.04$. It can be seen that the free energy following each exchange (every $500000$ time steps) is degenerate. (b) Corresponding representation of the Bloch sphere orbit (i) and free energy versus time [(ii); part of the time series only shown for clarity] for $K=0.1$. For both (a) and (b), other parameters are as listed in the text.