Fibonacci topological phase in arrays of anyonic chains

Fibonacci anyon is an exotic quasi-particle excitation, which would signal a major step forward in realization of a quantum computer. In this paper, we consider $SU(2)_4$ topological phase and theoretically demonstrate a way to construct a Fibonacci topological phase, a topological phase which has only one non-trivial excitation described by the Fibonacci anyon. We will show that arrays of anyonic chains created by excitations of the $SU(2)_4$ phase leads to the Fibonacci phase. The logic presented in this paper can be applied to other topological phases.


Introduction
One of the fascinating property of topologically ordered phases, such as fractional quantum Hall states [1,2] and spin liquid phases [3][4][5], is that they admit an exotic fractionalized excitation, namely anyon [6,7]. Non-Abelian anyon [8][9][10], whose braiding representation is described by a matrix in a degenerate ground state, is of particular importance from quantum information perspective. For practical purposes, the Fibonacci anyon [11,12], τ which is subject to the fusion rule τ × τ = I + τ with vacuum being denoted by I, attracts a lot of interests as braiding these anyons yields a complete set of quantum operations in ground state manifold, which could be useful for quantum computations. Therefore, it would be desirable to obtain a topological phase which admits the Fibonacci anyon as an excitation.
There are several studies which construct the Fibonacci phase, a.k.a the (G 2 ) 1 topological phase which habors only two types of excitations, vacuum I and the Fibonacci anyon τ . Mong et al have shown that couplings of trenches of ν = 2/3 fractional quantum Hall edge modes generates the Fibonacci phase [13]. Hu and Kane [14] have demonstrated that the Fibonacci phase arises in interacting p-wave superconductors. The goal of this paper is to present an alternative construction of the Fibonacci phase by introducing the SU(2) 4 topological phase and an arrays of nucleated anyonic excitations in the bulk. It turns out that couplings between these arrays drives the system to the desired Fibonacci phase.
Another motivation of this work is related to nucleation of a topological liquid. When we populate a topologically order phase with a set of non-Abelian anyons, it yields degeneracy in a ground state manifold. Interactions between the anyons lift the degeneracy, giving rise to a new collective ground states. It is known that one-dimensional (1D) alignment of interacting non-Abelian anyons, that we call anyonic chain in this paper, realizes a non-trivial gapless collective mode [15,16]. It is interesting to investigate whether the interplay of interacting non-Abelian anyons forming in two-dimensional (2D) configuration allows us to obtain new kind of phases. Previous works [16,17] made an attempt to study this problem by considering two anyonic chains forming a ladder. One can naturally ask, what would happen if, instead of just two, he/she introduces arrays of anyonic chains to form a 2D ladder? (Fig. 1) In this paper, we address this question by demonstrating that an arrays of anyonic chains would yield a new kind of topological phase.
The organization of this paper is as follows. In the Sec. 2, we present a way to construct the Fibonacci phase in the SU(2) 4 topological phase, consisting of three steps. In Sec. 3, we briefly comment on how the logic developed in the preceding section to other cases of SU(2) k . Finally, Sec. 4 is devoted to conclusion and outlook. Technical details are relegated to Appendices.

Formulation
In this section, we demonstrate how the Fibonacci phase arises in the SU(2) 4 topological phase. There are three steps to realize it. Figure 1: A schematic picture of the SU(2) 4 topological phase (blue rectangle). An anyonic chain consisting of Z anyons depicted by dotted circles. The "anti-ferromagnetic" state of this anyonic chain yields the M(6, 5) minimal conformal field theory (CFT) with central charge c = 4/5 depicted by a red line. The coupling between adjacent anyonic chains is mediated by X anyon (black arrow). Consider a 1D alignment of anyonic excitations described by Z anyons in the bulk such that fusion of each adjacent Z anyon may occur, giving The chain of this Z anyons is reminiscent of a 1/2-spin chain with nearest neighbors coupling. Similarly to the fact that there are two orders in the spin chain, namely, antiferromagnetic/ferromagnetic state which favors spin singlet/triplet state in each bond depending on the sign of the coupling, the anyonic chain admits two different orders as studied by Refs. [15,16]. It is known that if the chain of the Z-anyons is "anti-ferromagnetic", meaning the Hilbert space is projected to the sector with fusion channel I, the anyon chain yields a silver of a gapped nucleated liquid carrying a gapless edge mode at the interface with the SU(2) 4 topological phase. The gapless theory is decried by the minimal CFT, M(6, 5) with central charge c = 4/5, the same universality class as the tetracritical Ising model. We envisage array of the critical Z-anyon chains (with the index j to denote the jth chain) in the bulk of the SU(2) 4 topological phase, each of which is described by 1 + 1-dimensional c = 4/5 CFT. We would like to couple these arrays by introducing a tunneling term between adjacent chains. To this end, we assume that the silver of a gapped liquid nucleated by the anyonic chain has finite width [16], thus we focus on the coupling between the closer counter-propagating edge modes of the adjacent chains, as indicated by black arrows in Fig. 1. More concretely, in this section, we assume that the closer counter-propagating edge modes are described by the holomorphic and anti-holomorphic sectors at jth and j + 1th chains, respectively. Further, we claim such a coupling is mediated via a bulk excitation of X anyon. Focusing on a pair of adjacent chains, e.g., the jth and j + 1th anyonic chains, the leading order (in the sense of being the most relevant) term of such a tunneling has the form of where φ j r,s represents a primary field with index (r, s) in holomorphic sector in the M(6, 5) minimal model at jth chain with conformal weight (see also Likewise, φ j+1 r,s denotes the primary field in anti-holomorphic sector at j + 1th chain with the conformal weight h r,s . To see how the form of the tunneling given in Eq. (2) arises, we recall that X-anyon is characterized by topological spin θ X = e 2πih X with h X = 1/8. Therefore, the tunneling in question is regarded as transfer of an excitation which has conformal spin 1 1/8 (mod 1), i.e., the tunneling involves an excitation at one chain with conformal spin 1/8 (mod 1) and the one with conformal spin −1/8 (mod 1) at the adjacent chain. 2 In the language of the primary fields in the M(6, 5) minimal CFT, the excitation η j carrying conformal spin 1/8 (mod 1) at jth chain is given by Similarly, the excitation η j+1 at j + 1th chain which carries conformal spin Since we focus on the coupling between the closer counter-propagating edge modes at the adjacent chains, the coupling is then given by 3 where we have omitted writing the terms φ j (1,1) and φ j+1 (1,1) as they are identity with conformal weight 0 [Eq.
(1,2) is intuitively understood as a pair of creation of the excitation characterized by the primary field with index (1, 2) at jth chain and annihilation of the one by the same primary field at j + 1th chain.

Perturbed minimal model: Φ (1,2) deformation
The holomorphic sector at jth chain and the anti-holomorphic sector at j + 1th chain constitute the 1 + 1-dimensional minimal CFT. With the tunneling that we have considered in the preceding subsection, the theory is described by where z and z are complex coordinate which originates from the one in 1 + 1-dimension, i.e., (t, x), a non-chiral primary field is defined by (1,2) (other primary fields are similarly defined), and λ is dimensional constant which scales as λ ∼ aM [2−2h (1,2) ] = aM 3/2 with a and M being dimensionless constant and mass scale, respectively.
The theory (6) is known as the Φ (1,2) deformation of the minimal model, which is studied in view of integrability and quantum group [18,19]. Based on these analysis, it turns out 1 Difference between conformal weight of holomorphic sector and the one of the anti-holomorphic sector. 2 To understand this argument better, let us give a simple example. Consider two p-wave superconductors in 2D where they sit next to each other with a small distance. Around the border of the two p-wave superconductors, the chiral Majorana edge modes are propagating in the opposite direction, defined by ψ 1 and ψ 2 . Between the p-wave superconductors, only the electron tunneling is allowed. Since the electron is characterized by topological spin e 2πih ψ with h ψ = 1/2, such a tunneling involves an excitation of the one edge with conformal spin 1/2 (namely, ψ 1 ) and the one at the other edge with conformal spin −1/2 (that is, ψ 2 ). Thus, the electron tunneling has the form ψ 1 ψ 2 .
that the theory (6) describes a gapped fluid with two fold ground state degeneracy admitting one type of kink excitation, which is associated with the Fibonacci anyon. Indeed, due to the fusion rule the Φ (1,2) perturbation mixes the states generated by Φ (r,s) and those by Φ (r,s±1) . It follows that the Hilbert space in this theory splits into two sectors: Here, V (r,s) [V (r,s) ] is the irreducible representation of the Virasoro algebra in the holomorphic [anti-holomorphic] sector with highest weight h r,s . Irrespective of the sign of λ in Eq. (6), the theory becomes gapped with two fold ground state degeneracy, associated to the field Φ (r,r) in the sector H r (r = 1, 2). Furthermore, we can associate these two ground states to spin-0 and 1 representations of the U q [sl (2)] quantum group symmetry which are denoted by |0 and |1 , and an elementary kink excitation which interpolates between the two vacua is also represented by spin-1 representation of the same quantum group symmetry. The fusion rule of this spin-1 representation is given by (Appendix. B) which coincides with the one of the Fibonacci anyon.

Hilbert space of the anyonic chain
In the previous subsection, we have seen that each pair of holomorphic/anti-holomorphic sectors of the adjacent anionic chains constitutes a 1D gapped fluid which admits the Fibonacci kink. We will see that array of such a gapped 1D fluid gives rise to the desired Fibonacci topological phase. To do so, we resort to the technique developed in Ref. [13] and investigate how the form of the Hilbert space of each anyonic chain put a constraint on the number of ground states in the bulk. As we have discussed, the adjacent anyonic chains are coupled via tunneling of the excitation corresponding to a primary field with index (r, s) = (1, 2). We envisage that the bulk topological phase is placed on a torus geometry by imposing periodic boundary condition along the direction perpendicular to the 1D anyonic chains. In the present case, this is done by introducing the tunneling in the form of Eq. (2) with the superscript j and j + 1 being replaced with N and 1 (N is the number of the anyonic chains). Focusing on one anyonic chain, the tunneling process is effectively regarded as transfer of the excitation with index (r, s) = (1, 2) within the same chain; at the jth chain, the excitation with (r, s) = (1, 2) comes in from the holomorphic sector of the j − 1th chain and the same type of the excitation gets out from the anti-holomorphic sector of the jth chain to the j + 1th chain. Therefore, the Hilbert space of the jth chain is given by where the index (r, s) and (t, u) are related by fusion with primary field with index (1, 2). For instance, (r, s) = (1, 1) and (t, u) = (1, 2) is allowed combination due to the fusion rule [Φ (1,2) ] × [Φ (1,1) ] = [Φ (1,2) ]. We derive the same form of the Hilbert space from another perspective, see Appendix. C. Consider three anyonic chains, the j − 1th, jth, and j + 1th chains. From the discussion in the previous subsection, between the j − 1th and jth chains as well as the jth and j + 1th chains, we have gapped theories each of which has two fold ground state degeneracy. Naively, one expects that such gapped theories contribute to four fold degeneracy, which, however, turns out to be incorrect; two states are eliminated, giving two fold ground state degeneracy. To see why, we denote multiplication of the ground states in areas between the j − 1th and jth chains and the ones between jth and j + 1th chains by |p j−1/2 |q j+1/2 (p, q = 0, 1). Suppose |0 j−1/2 |1 j+1/2 is allowed ground state. From Eq. (8), the state |0 j−1/2 belongs to the Hilbert space where we have introduced the superscript to denote the anyonic chain. Similarly, the state |1 j+1/2 belongs to the Hilbert space Eqs. (11) (12) implies that the state in question, |0 j−1/2 |1 j+1/2 forces the Hilbert space of the jth anyonic chain to have the form ∼ 5 s,s ′ V j (1,s) ⊗ V j (2,s ′ ) . However, this from is not allowed due to Eq. (10). Indeed, V j (1,s) and V j (2,s ′ ) is not related by fusion with the primary field of the index (1, 2). Therefore, the state |0 j−1/2 |1 j+1/2 is not allowed ground state. The similar argument shows that |1 j−1/2 |0 j+1/2 is also excluded, concluding that there are two ground states, |p j−1/2 |p j+1/2 (p = 0, 1).
The arrays of anyonic chains with tunneling between adjacent chains yields a 2D gapped phase in the same spirit as the coupled wire construction [20,21]. The iterative use of the argument on the form of the Hilbert space given above shows that the arrays of the anyonic chains produces a gapped phase which has two ground state degeneracy in the torus geometry, admitting the Fibonacci anyon as an excitation. Due to the well known fact that the number of the ground state degeneracy of a topological phase in a torus geometry is equivalent to the number of types of anyons (i.e., superselection sectors, in the present case, vacuum I and the Fibonacci anyon τ ), the phase we have obtained is the desired Fibonacci phase, as advertised.

Brief comments on other cases
The logic presented in the previous section, consisting of three steps, can be easily applied to other cases of the SU(2) k topological order phase. We summarize these three steps as follows: 1. Make an alignment of non-Abelian anyons corresponding to the φ k−1 2 primary field with fusion rule φ k−1 2 × φ k−1 2 = φ 0 + φ 1 in the SU(2) k topological phase. Assuming that the anyonic chain is in the "anti-ferromagnetic" state, i.e., Hilbert space is projected to the sector with fusion channel φ 0 , then the anyonic chain leads to a gapless collective mode. This gapless mode is characterized by M(k + 2, k + 1) minimal CFT. Find a relevant tunneling operator between adjacent anyonic chains. The tunneling term consists of a pair of primary fields in the holomorphic anti-holomorphic sector in the adjacent chains. We require that the primary field carry the same conformal spin (mod 1) as the one of an anyon in the bulk phase.
2. If you can successfully find an appropriate tunneling term, then the CFT consisting of a pair of holomorphic/anti-holomorphic sectors in the adjacent chains is perturbed by this tunneling term. The theory leads to 1D gapped fluid with non-trivial ground state degeneracy harboring kink excitations subject to the fusion rule determined by the quantum group symmetry.
3. Due to the analysis of the form of the Hilbert space in each anyonic chains, one can show that arrays of the anyonic chains with the tunneling between the adjacent chains considered in the previous steps manifests the 2D topological phase with the same ground state degeneracy as the one of the 1D gapped fluid in step 2 in a torus geometry. Furthermore, the phase admits anyons with the same fusion rule as the one of the spin representation of kink excitations in the gapped 1D fluid.
As an example, we can apply these three steps to the SU(2) 3 topological phase. The gapless collective mode of "anti-ferromagnetic" anyonic chain is characterized by the M(5, 4) minimal CFT with central charge c = 7/10 (a.k.a. tricritical Ising CFT). Through a bulk anyon excitation corresponding to the primary field φ 1 , adjacent anyonic chains are coupled, which corresponds to the Φ 1,3 perturbation of the c = 7/10 CFT; a 1D theory consisting of a pair of holomorphic/anti-holomorphic sectors of adjacent anyonic chains is described by When λ < 0 it is known that the theory is gapped with three fold degeneracy. The fundamental kink interpolating between the three vacua is described by spin-1/2 representation with fusion rule 1 2 reminiscent of the fusion rule of the Ising anyon (fusion rule of the spin-1 representation on the right hand side is given by 1 × 1 = 0). Therefore, we obtain the 2D Ising topological phase by coupling of the anyonic chains. Another example is the SU(2) 5 topological phase. The anti-ferromagnetic state of the anyonic chains is characterized by M(7, 6) minimal CFT with central charge c = 6/7. The adjacent anyonic chains are coupled via the excitation corresponding to the primary field φ 5/2 in the bulk of the SU(2) 5 topological phase, giving the Φ 1,2 perturbation in the c = 6/7 CFT. Similarly to the previous example, a 1D theory consisting of a pair of holomorphic/antiholomorphic sectors of adjacent anyonic chains is described by S = S CFT M(7,6) + λ dzdzΦ (1,2) (z,z). (15) It is known that the perturbation with λ < 0 leads to a gapped theory with three fold ground state degeneracy. A fundamental kink excitation interpolating between the vacua is given by spin-1 representation, subject to the fusion rule with 1 × 2 = 1, 2 × 2 = 0. These fusion rules coincide with the ones of the SO(3) 5 topological phase. Therefore, coupling of the anyonic chains leads to the SO(3) 5 topological phase. It is worth mentioning that braiding representations of the excitations of this phase constitute a complete set of quantum operation [22], similarly to the Fibonacci phase.
As an aside, the adjacent anti-ferromagnetic chains in the SU(2) 4 topological phase can be coupled via bulk excitation of the Y -anyon, giving the Φ 1,3 deformation of the M(6, 5) minimal model. When the sign of the couplings is negative, one obtains the SU(2) 3 topological phase.

Conclusions and outlook
In this work, we present a construction of the Fibonacci phase in array of anyonic chains in the SU(2) 4 topological phase. Combination of the analysis of the perturbed CFT and the form of the Hilbert space of each anyonic chain allows us to obtain the Fibonacci phase, which harbors the Fibonacci anyon as the only non-trivial excitation. The way we present in this work is applicable to other cases. Before closing this section, we give several comments on future directions.
For realization purposes, it is important to find Hamiltonian formalism to describe our phases. However, this task is challenging up to now. Especially, much is not known about how to write explicitly a concrete model to realize the anti-ferromagnetic anyonic chains, i.e., a model Hamiltonian which projects to sector I when two adjacent anyons are fused as demonstrated in Eq. (1). One useful direction to address this issue is to introduce a SU(2) current-current interaction in a fermionic system. As illustrated by Ref. [23], one can rewrite the interaction by parafermionic mass term where the mass is accompanied with a bosonic field. By adding a term which couples with the bosonic field, the spatial modulation of the mass of the parafermionic field may bind zero modes which are regarded as non-Abelian anyons, leading to an interacting anyonic chain. It turns out that one obtains a collective gapless state corresponding to the ferromagnetic anyonic chain [23,24]. It would be an interesting direction to investigate whether our construction is described by Hamiltonian in the similar fashion.
It would also be interesting to find a possibility to get a gapless theory by different means from critical anyonic chains. For instance, we prepare many copies of 2D topological phases with finite size carrying a chiral edge mode characterized by a CFT corresponding to the bulk topological phase. We place them consecutively. Focusing on a pair of adjacent topological phases, these two phases are separated by a pair of counter-propagating edge modes. Interactions or couplings between these modes may lead to a different gapless mode from the original edge mode. Systematic way to investigate the existence of such a gapless mode would be resorting to a recent work [25] which provides a way to find a partition function of the possible gapless mode of a given topological phase, based on bootstrap analysis. These new gapless modes may be coupled via an excitation of the bulk of the topological phase, similarly to our construction. Since the gapless CFT is different from the original edge modes, one expects that coupling of the CFTs would yield a new phase which is different from the original 2D topological phase.

Acknowledgment
The author thanks Biswarup Ash, Rohit R. Kalloor, Yuval Oreg for discussion. This work is in part supported by the Koshland postdoc fellowship.

B Quick review on quantum group
The quantum group SL q (2) [27], which is the deformation of the algebra of functions over SL (2). The universal enveloping algebra U q [sl (2)] is generated by {H, J + , J − } satisfying the relation The irreducible representations of U q [sl (2)] are labelled by j = 0, 1 2 , 1, · · · , acting on a Hilbert space V j with basis vector |j, m (−j ≤ m ≤ j) as follows: with [n] q = q n −q −n q−q −1 . When q → 1, the commutation relation (18) and the eigenvalue in Eq. (19) reduces to the ones of the standard SL(2) group, therefore the quantum group SL q (2) becomes the SL(2) group.
When q is not a root of unity, the irreducible representations have dimension 2j + 1. On the contrary, when q is a root of unity, the Hilbert space is truncated so that the allowed spins are {0, 1/2, · · · , j max } with j max = N/2 − 1 such that q N = ±1. Accordingly, introducing a, b, c which are spins of this representation, tensor product of these spins is given by When minimal CFT M(m + 1, m) is perturbed by λΦ (1,3) , (λ < 0), the theory is gapped and the kink interpolating between adjacent vacua is described by spin-1/2 representation of the quantum group SL q (2) with q = e iπ m+1 m . In the case of the minimal CFT M(m + 1, m) being perturbed by λΦ (1,2) , the fundamental kink excitation which interpolates between vacua is described by spin-1 representation of the same quantum group.

C Twisted partition function
We present an alternative way to derive the form of the Hilbert space at jth anyonic chain given in Eq. (10).
In Sec. 2, we consider gapless anyonic chains characterized by minimal CFT, M(6, 5) with central charge c = 4/5, coupled by Φ 1,2 , namely, primary field with index (1, 2). Focusing on one anyonic chain, we would like to find the Hilbert space of the chain. To do so, we take into account the presence of the tunneling of the primary field Φ 1,2 properly. Suppose we put the theory on a geometry of a cylinder, S 1 × R, where spatial direction is along the circle. The tunneling term at a local point of x is visualized by a line along the time R direction, tempting us to interpret the tunneling term as the topological defect line which modifies the quantization by a twisted periodic boundary condition. We claim that the Hilbert space in question can be read from a twisted partition function with topological defect line corresponding to the primary field Φ 1,2 being inserted along the time direction.
Let us briefly recall the defining property of the topological defect line [28,29]. All the physical observable are invariant under continuous deformation of the topological defect lines. They are denoted by L g where g is global symmetry operation. When a state |φ k crosses the L g , the symmetry g is acted on the state. More rigorously, the topological defect line is described by the Verlinde line [30,31] which acts on a state |φ k by L g |φ k = S gk S 0k |φ k (21) with S gk is modular S-matrix of the CFT. In the present case, for the (1, 2) primary field, we can defined the Verlinde line L (1,2) via L (1,2) φ (r,s) = S (1,2)(r,s) S (1,1)(r,s) φ (r,s) , where φ (r,s) is a state corresponding to the (r, s) primary field in the c = 4/5 CFT. The twisted partition function is calculated as follows. We start with the diagonal partition function with the topological defect line inserted along the spacial direction, which reads where χ (r,s) [χ (r,s) ] is the character corresponding to the (r, s) primary field in the holomorphic [anti-holomorphic] sector of the c = 4/5 CFT. Implementing S-transformation and using the Verlinde formula [30], one finds the desired twisted partition function: (1,2),(r,s) χ (r,s) χ (t,u) (24) with N (u,t) (1,2),(r,s) being the fusion coefficient of the c = 4/5 CFT. From the form of the twisted partition function (24), we obtain the Hilbert space given in Eq. (10).