Accessing topological order in fractionalized liquids with gapped edges

We consider manifestations of topological order in time-reversal-symmetric fractional topological liquids (TRS-FTLs), defined on planar surfaces with holes. We derive a formula for the topological ground-state degeneracy of such a TRS-FTL, which applies to cases where the edge modes on each boundary are fully gapped by appropriate backscattering terms. The degeneracy is exact in the limit of infinite system size, and is given by $q^{N_{\mathrm{h}}^{}}$, where $N_{\mathrm{h}}^{}$ is the number of holes and $q$ is an integer that is determined by the topological field theory. When the degeneracy is lifted by finite-size effects, the holes realize a system of $N_{\mathrm{h}}^{}$ coupled spinlike $q$-state degrees of freedom. In particular, we provide examples where ${\mathbb{Z}}_q^{}$ quantum clock models are realized on the low-energy manifold of states. We also investigate the possibility of measuring the topological ground-state degeneracy with calorimetry, and briefly revisit the notion of topological order in $s$-wave BCS superconductors.

Figure 1

Gluing argument for the special case $\Delta =0$. In this case, the CS theory consists of two independent copies, with equal and opposite $K$ matrices. The tunneling processes (dotted lines) that gap out each pair of counterpropagating edge modes couple the two annuli, and the conditions (2.9) ensure that the two copies of the theory can be consistently “glued ” together.

Figure 2

Coordinate system on the annulus $A=[0,\pi ]\times S^1$. The inner boundary is at $x_1^{}=0$, while the outer boundary is at $x_1^{}=\pi$. The coordinate $x_2^{}$ is defined on the circle $S^1$.

Figure 3

A punctured TRS-FTL with gapped edges. (a) Schematic representation of an “artificial” spinlike system. In the limit $D\gg d,R$, each hole (white square) carries with it a $q$-fold topological degeneracy that is split exponentially by tunneling processes that encircle (red lines) or connect (green lines) the holes. (b) Wilson loops defined in Eq. (3.1). The dashed line represents the product of the two Wilson loops above it, which connects the two holes.

Figure 4

Total heat capacity for a monolayer TRS-FTL with $N_{\mathrm{a}}^{}={10}^{14}$. The topological contribution is shown (above background) for $q=2,4,$ and 6. The parameters used for the topological contribution were $\nu =5\times {10}^{-6}$ ($\sim {22000}^2$ holes) and $h/k_{\mathrm{B}}^{}\approx 0.321$ K, which leads to a maximum excess (for $q=6$) of $\sim 30\%$ over the background (blue curve) near $T=0.1$ K.

Figure 5

Trapping a flux quantum inside a superconducting ring. Confining the flux inside the ring costs no energy for the electrons inside the superconductor, but there is an electromagnetic energy cost obtained by integrating the enclosed magnetic field intensity over the interior of the dashed cylinder, which we denote $\mathcal{V}$.