Topology-driven quantum phase transitions in time-reversal-invariant anyonic quantum liquids

Abstract

The degrees of freedom in our microscopic models are so-called Fibonacci anyons, one of the simplest types of non-abelian anyons [1,2]. The Fibonacci theory has two distinct particles, the trivial state 1 and the Fibonacci anyon τ , which can be thought of as a generalization (or more precisely a 'truncated version') of an 'angular momentum' when viewing the Fibonacci theory as a certain deformation [3] of SU(2). We will now make this notion more precise and illustrate it in detail. In analogy to the ordinary angular momentum coupling rules, we can write down a set of 'fusion rules' for the anyonic degrees of freedom which are analogs of the Clebsch-Gordon rules for coupling of ordinary angular momenta,where the last fusion rule reveals what is known as the non-abelian character of the Fibonacci anyon: Two Fibonacci anyons τ can fuse to either the trivial particle or to another Fibonacci anyon. In more mathematical terms, these fusion rules can also be expressed by means of so-called fusion matrices N j whose entries (N j ) j1 j2 equal to one if and only if the fusion of anyons of types j 1 and j 2 into j is possible. The fusion rules are related to the so-called 'quantum dimensions' d j of the anyonic particles bywhere |d j � is the ('Perron Frobenius') eigenvector corresponding to the largest positive eigenvalue of the 2 × 2-matrix N j . [The sense in which these numbers are 'dimensions' will become apparent in section II A 1 below.] For the particles in the Fibonacci theory the quantum dimensions are d 1 = 1 and d τ = ϕ ≡ (1 + √ 5)/2 and the total quantum dimension of the theory is then given by D = (To define our Hamiltonian, some additional indegredients of the theory of anyons are required. In analogy to the 6j-symbols for ordinary SU(2) spins, there exists a basis transformation F that relates the two differents ways three anyons can fuse to a fourth anyon, depicted asThe left hand side (l.h.s.) represents the quantum state that arises when anyon a first fuses with anyon b into an anyon of type e, which, subsequently, fuses with anyon c into an anyon of type d. Similarly, the right hand side (r.h.s.) denotes the quantum states that arises when anyon b first fuses with anyon c into anyon type f which, in turn, fuses with anyon a into anyon type d. Whilst keeping all external labels, the types of the three anyons (a, b, c) as well as the resulting anyon type d fixed, the states on the l.h.s and r.h.s. are fully specified by the labels e and f , respectively. Eq. (3) says the so-specified states are linearly related to each other by the so-called F -matrix [4] with matrix elements (Ff e . In general, the F -matrix is uniquely defined (up to 'gauge transformations') by the fusion rules through a consistency relation called the pentagon equation [6]. Similarly, the braiding properties of anyons are given by the so-called R-matrix (which however is not needed here) that is uniquely determined by the hexagon equation [6].For the Fibonacci theory, it is straightforward to verify that in most cases there is only o...

Microscopic models of interacting Yang–Lee anyons

“…The first term favors the trivial label 1 on each rung of the ladder, while the second term favors the no-flux state for all plaquettes. As shown in [28], the projector onto the flux through a square plaquette can be expressed in terms of the unitary/non-unitary F-matrices in equations (6) and (7). This term is equivalent to the plaquette term in the Levin-Wen models [51], which are defined on a different lattice, the honeycomb lattice.…”

“…The analytic solution of the Fibonacci ladder model at the two exactly solvable points has been described in detail in (the supplementary material of) [28]. Thus, we will be rather brief here, and point out some of the crucial steps of the mapping of the ladder model onto an exactly solved (RSOS) model.…”

“…In this way, the model can be mapped on the D 6 version of the RSOS model and, precisely when J r = J p = ±1, the model is critical, and can be solved exactly. We will not go into the details here (which can be found in [28]), but we will quickly state the result for this critical behavior in the original,…”

“…Fibonacci anyons in [28], where we introduce the model in section 3.1. We then discuss the phase diagram in section 3.2, present the analytical solution at special critical points in section 3.3 and finish with exact numerical results in section 3.4.…”

Interacting Fibonacci anyons in a Rydberg gas

“…The Hilbert space maps then directly on the fusion space of Fibonacci anyons and a proper tuning of the laser parameters renders the system into an interacting topological liquid of non-Abelian anyons. We discuss the low-energy properties of this system and show how to experimentally measure anyonic observables.There is great interest in studying many-body systems of interacting anyons as they often exhibit exotic quantum phases [5,[7][8][9]. A further motivation is that the exchange of anyons -the braiding -permits the implementation of robust protocols for quantum information processing [10][11][12].…”