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arXiv:0807.1123 · Physical Review B

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

Lukasz Fidkowski, Gil Refael, N. E. Bonesteel, Joel E. Moore

Published Dec 18, 2008• 38 citations
arXivHTMLPDF

Abstract

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 $\ensuremath{\tau}\ensuremath{\bigotimes}\ensuremath{\tau}=1\ensuremath{\bigoplus}\ensuremath{\tau}$ 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 $\ensuremath{\tau}$ 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.

DOI
10.1103/physrevb.78.224204

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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 \ensuremathτ\ensuremath⨂\ensuremathτ=1\ensuremath⨁\ensuremathτ\ensuremath{\tau}\ensuremath{\bigotimes}\ensuremath{\tau}=1\ensuremath{\bigoplus}\ensuremath{\tau}\ensuremathτ\ensuremath⨂\ensuremathτ=1\ensuremath⨁\ensuremathτ 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 \ensuremathτ\ensuremath{\tau}\ensuremathτ 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 ccc theorem for the effective central charge.
Non-Abelian anyon chains whose interactions favor a single fusion channel host quantum critical ground states described by minimal-model conformal field theories (CFTs) protected by a topological symmetry. We show that their measurement-only counterpart, where non-commuting fusion charge measurements act locally on the anyons of a chain of LLL sites, provides a fundamental example of a non-unitary CFT controlling a non-Abelian learning problem. Mapping the monitored dynamics to a random tensor network, we numerically find 1/L1/L1/L scaling Lyapunov gaps, logarithmic entanglement scaling with the chord length, and an effective central charge in the Shannon entropy of the measurement record. The conserved topological charge sharpens with dynamical exponent z=1z= 1z=1 and is learnable by a decoder on an O(L)O(L)O(L) time scale. As the topological symmetry is a non-invertible symmetry of the SU(2)kSU(2)_kSU(2)k​ anyon theories, we establish the monitored golden chain (k=3)(k=3)(k=3) as a non-Abelian generalization of the weakly (Kramers-Wannier) self-dual quantum critical point of the toric code (k=2)(k=2)(k=2), and the first of a family for k≥3k\geq 3k≥3.
We consider the effect of quenched spatial disorder on systems of interacting, pinned non-Abelian anyons as might arise in disordered Hall samples at filling fractions $\ensuremath{\nu}=5/2$ or $\ensuremath{\nu}=12/5$. In one spatial dimension, such disordered anyon models have previously been shown to exhibit a hierarchy of infinite randomness phases. Here, we address systems in two spatial dimensions and report on the behavior of Ising and Fibonacci anyons under the numerical strong-disorder renormalization group (SDRG). In order to manage the topology-dependent interactions generated during the flow, we introduce a planar approximation to the SDRG treatment. We characterize this planar approximation by studying the flow of disordered hard-core bosons and the transverse field Ising model, where it successfully reproduces the known infinite randomness critical point with exponent $\ensuremath{\psi}\ensuremath{\approx}0.49$. Our main conclusion for disordered anyon models in two spatial dimensions is that systems of Ising anyons as well as systems of Fibonacci anyons do not realize infinite randomness phases, but flow back to weaker disorder under the numerical SDRG treatment.
Topological phases supporting non-Abelian anyonic excitations have been proposed as candidates for topological quantum computation. In this paper, we study disordered non-Abelian anyonic chains based on the quantum groups SU(2)k\text{SU}{(2)}_{k}SU(2)k​, a hierarchy that includes the $\ensuremath{\nu}=5/2$ fractional quantum Hall state and the proposed $\ensuremath{\nu}=12/5$ Fibonacci state, among others. We find that for odd kkk these anyonic chains realize infinite-randomness critical phases in the same universality class as the Sk{S}_{k}Sk​ permutation symmetric multicritical points of Damle and Huse [Phys. Rev. Lett. 89, 277203 (2002)]. Indeed, we show that the pertinent subspace of these anyonic chains actually sits inside the ${\mathbb{Z}}_{k}\ensuremath{\subset}{S}_{k}$ symmetric sector of the Damle-Huse model, and this Zk{\mathbb{Z}}_{k}Zk​ symmetry stabilizes the phase.
Random spin chains at quantum critical points exhibit an entanglement entropy between a segment of length LLL and the rest of the chain that scales as log2L{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}Llog2​L with a universal coefficient. Since for pure quantum critical spin chains this coefficient is fixed by the central charge of the associated conformal field theory, the universal coefficient in the random case can be understood as an effective central charge. In this paper we calculate the entanglement entropy and effective central charge of the spin-1 random Heisenberg model in its random-singlet phase and also at the critical point at which the Haldane phase breaks down. The latter is the first entanglement calculation for an infinite-randomness fixed point that is not in the random-singlet universality class. Our results are consistent with a ccc-theorem for flow between infinite-randomness fixed points. The formalism we use can be generally applied to calculation of quantities that depend on the RG history in $s\ensuremath{\geqslant}1$ random Heisenberg chains.
In clean and weakly disordered systems, topological and trivial phases having a finite bulk energy gap can transit to each other via a quantum critical point. In presence of strong disorder, both the nature of the phases and the associated criticality can fundamentally change. Here we investigate topological properties of a strongly disordered fermionic chain where the bond couplings are drawn from normal probability distributions which are defined by characteristic standard deviations. Using numerical strong disorder renormalization group methods along with analytical techniques, we show that the competition between fluctuation scales renders both the trivial and topological phases gapless with Griffiths like rare regions. Moreover, the transition between these phases is solely governed by the fluctuation scales, rather than the means, rendering the critical behavior to be determined by an infinite randomness fixed point with an irrational central charge. Our work points to a host of novel topological phases and atypical topological phase transitions which can be realized in systems under strong disorder.
We construct models of interacting itinerant non-Abelian anyons moving along one-dimensional chains. We focus on itinerant Ising (Majorana) and Fibonacci anyons, which are, respectively, related to SU(2)2{(2)}_{2}(2)2​ and SU(2)3{(2)}_{3}(2)3​ anyons and also, respectively, describe quasiparticles of the Moore-Read and Z3{\mathbb{Z}}_{3}Z3​-Read-Rezayi fractional quantum Hall states. Following the derivation of the electronic large-U effective Hubbard model, we derive effective anyonic ttt-JJJ models for the low-energy sectors. Solving these models by exact diagonalization, we find a fractionalization of the anyons into charge and (neutral) anyonic degrees of freedom---a generalization of spin-charge separation of electrons which occurs in Luttinger liquids. A detailed description of the excitation spectrum can be performed by combining spectra for charge and anyonic sectors. The anyonic sector is that of a squeezed chain of localized interacting anyons and, hence, is described by the same conformal field theory (CFT), with central charge c=1/2c=1/2c=1/2 for Ising anyons and c=7/10c=7/10c=7/10 or c=4/5c=4/5c=4/5 for Fibonacci anyons with antiferromagnetic or ferromagnetic coupling, respectively. The charge sector is the spectrum of a chain of hard-core bosons subject to phase shifts which coincide with the momenta of the combined anyonic eigenstates, revealing a subtle coupling between charge and anyonic excitations at the microscopic level (which we also find to be present in Luttinger liquids), despite the macroscopic fractionalization. The combined central charge extracted from the entanglement entropy between segments of the chain is shown to be 1+c1+c1+c, where ccc is the central charge of the underlying CFT of the localized anyon (squeezed) chain.