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

Abstract

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 → ∞). 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 SL ≃ ln d 3 log 2 L for large L, where d is the quantum dimension of the particles.A particularly exotic form of quantum order is possible in two space dimensions -so-called non-Abelian order [1]. In states with non-Abelian order, when certain localized quasiparticle excitations are present there is a lowenergy Hilbert space whose dimensionality grows exponentially with the number of these quasiparticles. When these quasiparticles are well separated, this low-energy space becomes degenerate, and its states are characterized by purely topological quantum numbers, meaning they cannot be distinguished by local measurements. If these quasiparticles are then adiabatically moved around one another, unitary transformations corresponding to non-Abelian representations of the braid group are carried out on this degenerate space. Aside from their intrinsic scientific interest, recent attention has focused on the possibility of one day using non-Abelian states to perform fault-tolerant quantum computation [2,3].Recently Feiguin et al. [4] have studied models of interacting non-Abelian quasiparticles, specifically uniform chains in which neighboring quasiparticles are close enough together to lift the degeneracy of the topological Hilbert space. In this Letter we study a related class of random interacting chains of non-Abelian quasiparticles. We are motivated both by [4] and by recent work of Refael and Moore [5,6] showing that the entanglement entropy of certain random one-dimensional models scales logarithmically with a universal coefficient. We find the same is true here for an infinite class of models.Exact diagonalization studies [7,8,9] provide compelling evidence that the experimentally observed ν = 5/2 fractional quantum Hall (FQH) state is a nonAbelian state described by the Moore-Read "Pfaffian" state [1]. This state belongs to a wider class of nonAbelian FQH states introduced by Read and Rezayi [10], labeled by index k. In this class, the k = 1 state is an ordinary (Abelian) Laughlin state, the k = 2 state is the Moore-Read state, and all subsequent integer k values describe new non-Abelian states. There is some numerical evidence [10,11] that the k = 3 Read-Rezayi state describes the experimentally observed ν = 12/5 FQH state [12].The quasiparticle excitations of the Read-Rezayi states with index k can be viewed (up to Abelian phases) as par- ticle excitations in SU (2) k Chern-Simons-Witten theory [13]. These particles are characterized by their topological charge, a quantum number which can be viewed as a "q-deformed" spin [14]. At level k, topological charge can take the v...

c-theorem violation for effective central charge of infinite-randomness fixed points

“…This goal was partially achieved in Ref. 21 for a golden chain which contains only AFM bonds. As we shall see, including FM bonds in this analysis reveals a new fixed point, where the number of FM and AFM bonds is the same.…”

“…Carrying this out, Ref. 21 showed that for large number N of singlets, the entropy is asymptotically N log 2 . Thus the asymptotic contribution of each singlet is log 2 .…”

“…As an illustration of the complicated nature of the Hilbert space, we analyze the "rainbow" state that arises in the entanglement entropy calculation for the AFM chain 21 ͓see Fig. 6͑a͔͒.…”

“…The first analysis of the random Fibonacci chain 21 concentrated on the random antiferromagnetic Fibonacci chain. A random-singlet phase was found with an effective central charge reflecting the quantum dimension of the Fibonacci anyons; c eff =ln , with = 1+ ͱ 5 2 being the golden ratio.…”

Permutation-symmetric critical phases in disordered non-Abelian anyonic chains

“…[7][8][9][10] Because of the unique structure of their Hilbert spaces, disordered anyonic chains are particularly amenable to treatment via strong randomness renormalization-group ͑RG͒ methods and indeed have been shown to exhibit infiniterandomness fixed points. 11,12 They are thus an especially fertile ground for trying to discover and classify new universality classes of strongly random behavior. Indeed, even though no new universality classes were found, Ref.…”