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

Topological insulators supporting non-Abelian anyonic excitations are in the center of attention as candidates for topological quantum computation. In this paper, we analyze the ground-state properties of disordered non-Abelian anyonic chains. The resemblance of fusion rules of non-Abelian anyons and real-space decimation strongly suggests that disordered chains of such anyons generically exhibit infinite-randomness phases. Concentrating on the disordered golden chain model with nearest-neighbor coupling, we show that Fibonacci anyons with the fusion rule $\tau \otimes \tau =1\oplus \tau$ exhibit two infinite-randomness phases: a random-singlet phase when all bonds prefer the trivial fusion channel and a mixed phase which occurs whenever a finite density of bonds prefers the $\tau$ fusion channel. Real-space renormalization-group (RG) analysis shows that the random-singlet fixed point is unstable to the mixed fixed point. By analyzing the entanglement entropy of the mixed phase, we find its effective central charge and find that it increases along the RG flow from the random-singlet point, thus ruling out a $c$ theorem for the effective central charge.

Figure 1

Sample random-singlet ground state of a spin-1/2 antiferromagnetic chain. Each line represents a singlet. Singlets form in a random fashion, mostly between nearest neighbors but occasionally between distant sites giving rise to an average correlation that decays algebraically.

Figure 2

(a) A state in the Hilbert space: the labels 1 and $\tau$ specify the total topological charge of all the sites to the left (or equivalently right) of the bond, with the fusion rules obeyed at each trivalent node. (b) The same state in graph notation—we only draw the $\tau$’s.

Figure 3

The $F$-matrix relations. (a) Graphs that can be disconnected by cutting one edge are equal to 0 (the no tadpole condition) and disconnected loops are worth $\tau$. (b) Local reconnection rules are given by the $F$ matrix defined in the text. Here $m$ and $n$ are binary variables equal to either $\tau$ or 1, i.e., the link is either there or not.

Figure 4

The trivalent graph on the left-hand side expressed in terms of the link basis.

Figure 5

(a) Projection operator $P_i^A$ on a pair of sites. (b) Projection operator $P_i^F$ on a different pair of sites.

Figure 6

A rainbow diagram. The partition bond is in the middle.

Figure 7

Particles 2 and 3 are decimated ferromagnetically. The effect on an AFM projection acting on 1 and 2 is simply to turn it into an FM projection. For an AFM projector (FM projector) there is a numerical factor associated to the graphical representation above of ${\tau }^{-1}$ $\left({\tau }^{-1/2}\right)$. Combining these, we get ${\tau }^{-1}{\left({\tau }^{-1/2}\right)}^2={\tau }^{-2}$, of which ${\tau }^{-1/2}$ gets absorbed in the normalization for the new FM projector and another ${\tau }^{-1/2}$ goes into the normalization of the wave function, leaving ${\tau }^{-1}$. Thus $P^A$ on sites 1 and 2 turns into $\frac{1}{\tau }{P}^F$. A similar graphical argument shows that $P^F$ turns into $\frac{1}{\tau }{P}^A$.

Figure 8

(Color online) Flow diagram for the mixed Fibonacci chain. Two fixed points exist. The AFM fixed point is unstable with respect to the introduction of FM bonds. A stable fixed point exists at the symmetric FM-AFM point.

Figure 9

Decomposition of the total Hilbert space into superselection sectors.

Figure 10

RG history near the partition bond (represented by the dashed vertical line). The partition bond is at some point decimated with equal probabilities of having an AFM and FM decimation. In the case of an AFM decimation, we get another partition bond, which will at some point be decimated by an AFM or FM decimation, again with equal probability. In the case of an FM decimation, however, we produce a partition site, which can decimate, either via AFM or FM decimation, with either the site to its left or the site to its right. All four of these possibilities are equally likely. Carried out further, this process will generate complicated treelike structures. Note that we have ignored decimations that do not involve the partition bond or partition site.

Figure 11

(a) Trivalent graph representing a ground state generated by the real-space RG. It can be decomposed into tree components, which are separated by AFM decimations of the partition bond, denoted in the figure by dotted lines. There could also be tree diagrams that do not straddle the partition bond and hence do not contribute to the entanglement entropy; we omit them from the illustration for clarity. (b) After applying $F$-matrix relations we can reduce the trivalent graph in (a) to a superposition of graphs of this form. Here the dashed lines denote either a 1 (trivial) or $\tau$ (nontrivial) line.

Figure 12

Application of one $F$-matrix move decomposes the graph on the left into the superposition on the right.

Figure 13

A trivalent graph whose labelings give a basis for the Hilbert space left of the partition bond. Here we have a tree for each region $A_i$, with the line leading out of $A_i$ fixed to be $\tau$ (for $i=1,3,4$) or 1 (for $i=2$). The rest of the lines, including the lines connecting different $A_i$, can be labeled at will, consistent with the fusion rules. The label of the dashed line gives the total topological charge of the left side of the system.

Figure 14

(a) A rainbow diagram. (b) Typical state at the FM-AFM point. The dashed lines denote the location of AFM decimations, which separate the thickened stripes.

Figure 15

(Color online) Flow diagram of the pure and disordered golden chain. In the pure chain, assuming no intervening fixed points exist, the FM fixed point is unstable to flow to the AFM fixed point, as inferred from the Zamolodchikov $c$ theorem. In the disordered chain, however, the flow is in the opposite direction, with the mixed FM/AFM phase, which is most likely the terminus of the flow from the pure FM phase, being stable relative to the random-singlet phase, which is the result of disordering the pure AFM phase. The fixed-point (effective) central charges are also quoted.