Anyon Condensation and Continuous Topological Phase Transitions in Non-Abelian Fractional Quantum Hall States

Abstract

We find a series of possible continuous quantum phase transitions between fractional quantum Hall (FQH) states at the same filling fraction in two-component quantum Hall systems. These can be driven by tuning the interlayer tunneling and/or interlayer repulsion. One side of the transition is the Halperin (p, p, p − 3) Abelian two-component state while the other side is the non-Abelian Z4 parafermion (Read-Rezayi) state. We predict that the transition is a continuous transition in the 3D Ising class. The critical point is described by a Z2 gauged Ginzburg-Landau theory. These results have implications for experiments on two-component systems at ν = 2/3 and single-component systems at ν = 8/3.One of the most challenging problems in the study of quantum many-body systems is to understand transitions between topologically ordered states [1]. Since topological states cannot be characterized by broken symmetry and local order parameters, we cannot use the conventional Ginzburg-Landau theory. When nonAbelian topological states are involved, the transitions that are currently understood are essentially all equivalent to the transition between weak and strong-paired BCS states [2,3]. Over the last ten years, while there has been much work on the subject, there has not been another quantum phase transition in a physically realizable system, involving a non-Abelian phase, for which we can answer the most basic questions of whether the transition can be continuous and what the critical theory is. Here we present an additional example in the context of fractional quantum Hall (FQH) systems.The quasiparticle excitations in FQH states carry fractional statistics and fractional charge [1]. In particular, in a (ppq) bilayer FQH state [1,4], there is a type of excitation, called a fractional exciton (f-exciton), which is a bound state of a quasiparticle in one layer and an oppositely charged quasihole in the other layer. It carries fractional statistics. As we increase the repulsion between the electrons in the two layers, the energy gap of the f-exciton will be reduced; when it is reduced to zero, the f-exciton will condense and drive a phase transition. When the anyon number has only a mod n conservation, this can even lead to a non-Abelian FQH state [2], yet little is known about "anyon condensation" [5][6][7][8]. A better understanding of these phase transitions may aid the quest for experimental detection of non-Abelian FQH states, because one side of the transition -in our case the (330) state at ν = 2/3 -can be accessed experimentally [9,10]. The results of this paper suggest a new way of experimentally tuning to a non-Abelian state in bilayer FQH states, similar to the transition from the (331) state to the Moore-Read Pfaffian at ν = 1/2 [2, 3, 11].In the (ppq) state, when the energy gap of the f-exciton at k = 0 is reduced to zero, the f-exciton will condense [2]. The transition can be described by the φ = 0 → φ = 0 transition in a Ginzburg-Landau theory with a Chern-where θ is the statistical angle of the f-exciton. Suc...

Synthetic non-Abelian statistics by Abelian anyon condensation

“…As the external field can be turned on and off, the synthetic dislocations can be produced and moved around physically. On the other hand, the external field (or coupling) also drives intrinsic anyon condensation 10,16 along the branch-cut line, which demonstrates our designing principle of synthetic dislocations by anyon condensation. This makes synthetic dislocations more general than the lattice dislocations, since we can choose to condense different types of anyons, which in turn generates different kinds of topological degeneracies, associated to different synthetic dislocations with different projective non-Abelian statistics.…”

Topological Nematic States and Non-Abelian Lattice Dislocations

“…This is because the interlayer repulsion V inter is smaller than the intralayer repulsion V intra . When V inter % V intra , a quantum phase transition to a different many-body state is generally expected to occur [45][46][47]. In a flat band with Chern number C 1 ¼ 2, the different layers are separated by a lattice translation.…”

Phase transitions inZNgauge theory and twistedZNtopological phases

“…In our study of bilayer quantum Hall phase transitions in Ref. 21, it was suggested that this transition, in the presence of the δL term, may be dual to a 3D Ising transition. In those cases, one starts from an Abelian bilayer FQH phase and obtains the non-Abelian orbifold FQH states by tuning the interlayer tunneling and/or interlayer repulsion.…”

“…This formulation of the effective field theory allowed us to show that there is a continuous phase transition, in the 3D Ising universality class, between the Z 4 parafermion states and the Abelian (k, k, k − 3) states in bilayer quantum Hall systems. 21 Subsequently, it was found that for more general values of the coupling constants, k, l = 0, the U (1) × U (1) ⋊ Z 2 CS theory describes a series of non-Abelian FQH states -the orbifold FQH states. 22 The Z 4 parafermion FQH states are then a special case of these more general orbifold FQH states, which are separated from the (k, k, k−l) states by a continuous 3D Ising phase transition.…”