Infinite-Randomness Fixed Points for Chains of Non-Abelian Quasiparticles

One-dimensional chains of non-Abelian quasiparticles described by $SU(2{)}_k$ Chern-Simons-Witten theory can enter random singlet phases analogous to that of a random chain of ordinary spin-$1/2$ particles (corresponding to $k\to \infty$). For $k=2$ this phase provides a random singlet description of the infinite-randomness fixed point of the critical transverse field Ising model. The entanglement entropy of a region of size $L$ in these phases scales as $S_L\simeq \frac{\ln d}{3}{log}_{2}L$ for large $L$, where $d$ is the quantum dimension of the particles.

Figure 1

(a) Two noncrossing singlet states for $SU(2{)}_k$ particles and their overlap. (b) Action of the singlet projection operators ${\Pi }_1^0$ (which acts on particles 1 and 2) and ${\Pi }_2^0$ (which acts on particles 2 and 3) on a particular noncrossing singlet state. The quantity $d$ appearing in (a) and (b) is the quantum dimension of the particles.

Figure 2

One step in the decimation procedure.

Figure 3

“Random singlet” view of a decimated random transverse field Ising model. A random singlet state (green) is overlaid with a dimer state (blue). In the dimer state bonds connect pairs of $SU(2{)}_2$ particles which are mapped onto the spin-$1/2$ degrees of freedom of the transverse field Ising model (bottom row of dots). Closed loops then correspond to decimated superspins, indicated by solid arrows.

Figure 4

Schmidt decomposition of a state of $N$ pairs of $SU(2{)}_k$ particles connected by singlet bonds. The states in the decomposition are expressed using a basis in which circles enclose particles in topological charge eigenstates. The sum is over all $s_2,s_3,\dots ,s_N$, consistent with the fusion rule (1).