Topological Color Codes and Two-Body Quantum Lattice Hamiltonians

Topological color codes are among the stabilizer codes with remarkable properties from quantum information perspective. In this paper we construct a four-valent lattice, the so called ruby lattice, governed by a 2-body Hamiltonian. In a particular regime of coupling constants, degenerate perturbation theory implies that the low energy spectrum of the model can be described by a many-body effective Hamiltonian, which encodes the color code as its ground state subspace. The gauge symmetry $\mathbf{Z}_{2}\times\mathbf{Z}_{2}$ of color code could already be realized by identifying three distinct plaquette operators on the lattice. Plaquettes are extended to closed strings or string-net structures. Non-contractible closed strings winding the space commute with Hamiltonian but not always with each other giving rise to exact topological degeneracy of the model. Connection to 2-colexes can be established at the non-perturbative level. The particular structure of the 2-body Hamiltonian provides a fruitful interpretation in terms of mapping to bosons coupled to effective spins. We show that high energy excitations of the model have fermionic statistics. They form three families of high energy excitations each of one color. Furthermore, we show that they belong to a particular family of topological charges. Also, we use Jordan-Wigner transformation in order to test the integrability of the model via introducing of Majorana fermions. The four-valent structure of the lattice prevents to reduce the fermionized Hamiltonian into a quadratic form due to interacting gauge fields. We also propose another construction for 2-body Hamiltonian based on the connection between color codes and cluster states. We discuss this latter approach along the construction based on the ruby lattice.


I. INTRODUCTION
Topological color codes (TCC) are a whole class of models that provide an instance of an interdisciplinary subject between Quantum Information and the physics of Quantum Many-Body Systems.
Topological color codes were introduced [1] as a class of topological quantum codes that allow a direct implementation of the Clifford group of quantum gates suitable for entangle-ment distillation, teleportation and fault-tolerant quantum computation. They are defined on certain types of 2D spatial lattices. They were extended to 3D lattices [2] in order to achieve universal quantum computation with TCCs. This proposal of topological quantum computation relies solely on the topological properties of the ground state sector of certain lattice Hamiltonians, without resorting to braiding of quasiparticle excitations. In addition to these applications in Quantum Information, topological color codes have also a natural application in strongly correlated systems of condensed matter with topological orders. In [3] was found that TCCs can be extended to arbitrary dimensions, giving rise to topological orders in any dimension, not just 2D. This is accomplished through the notion of D-colexes, which are a class of lattices with certain properties where quantum lattice Hamiltonians are defined. This corresponds to a new class of exact models in D=3 and higher dimensions that exhibit new mechanisms for topological order: i/ brane-net condensation; ii/ existence of branyons; iii/ higher ground-state degeneracy than other codes; iv/ different topological phases for D ≥ 4 etc. In what follows, we shall focus only on 2D lattices.
Physically, TCCs are exotic quantum states of matter with novel properties. They are useful for implementing topological quantum computation, but they have also an intrinsic interest by their own. Then, a natural question arises as to how to implement experimentally these new quantum systems by means of light, atoms or some other means. This is a challenge since TCCs are formulated in terms of Hamiltonians with many-body terms, the simplest having 6-body interactions in a hexagonal lattice. But the most common interactions in nature are typically 2-body interactions.
There are several approaches to trying to solve this challenge, depending on the type of scenario we envision to be in practice and the practical rules we are supposed to be allowed to have at our disposal.
Let us start first for what we may call a 'quantum control scenario'. By this we simply mean that we are able to perform very controllable quantum operations on our system that we have prepared artificially. In particular, we suppose that we can perform quantum measurements on the qubits and having ancilla qubits at will. Under these circumstances, we can resort to cluster states [4] and measurement-based quantum computation [5,6]. This is because TCCs can be described by a certain cluster state construction [7] within this scenario. Then, it is possible to use a technique to obtain graph states as ground states of 2-body qubit Hamiltonians [8,9]. We show this construction in A. However, this scenario is experimentally very demanding and it is left for the future when will it be achieved completely. Therefore, it is convenient to seek other alternatives.
Thus, let us move onto a 'condensed matter scenario'. The terminology is intended just to be illustrative, rather than precise. In fact, the scenario goes beyond condensed matter and may well be a quantum simulation of our system by means of engineering a set of photons, atoms or the like. The important difference now is that external measurements on the system, or ancilla qubits, are not allowed in order to obtain the desired Hamiltonian for the TCCs. We want to remain in a framework based on Hamiltonians with solely 2-body interactions [10].
This is so because, on one hand, the Hamiltonian system that we introduce is able to reproduce the quantum computational properties of the topological color codes (TCC) [1-3] at a non-pertubative level as explained in Sect.IV. This is an important step towards obtaining topological protection against decoherence in the quest for scalability. On the other hand, the fact that the interactions in the Hamiltonian appear as 2-body spin (or qubit) terms makes it more suitable for its realization by means of a quantum simulation based on available physical proposal with light and atoms.
In a framework of strongly correlated systems in Quantum Many-Body Systems, one of the several reasons for being interested in the experimental implementation of this Hamiltonian system is because it exhibits exotic quantum phases of matter known as topological orders, some of its distinctive features being the existence of anyons [24][25][26]. In our everyday 3D world, we only deal with fermions and bosons. Thus, exchanging twice a pair of particles is a topologically trivial operation. In 2D this is no longer true, and particles with other statistics are possible: anyons. When the difference is just a phase, the anyons are called abelian. Anyons are a signature of topological order (TO) [27,28], and there are others as well: • there is an energy gap between the ground state and the excitations; • topological degeneracy of the ground state subspace (GS); • this degeneracy cannot be lifted by local perturbations; • localized quasiparticles as excited states: anyons; • edge states; etc.
These features reflect the topological nature of the system. In addition, a signature of the TO is the dependence of that degeneracy on topological invariants of the lattice where the system is defined, like Betti numbers [3].
But where do we find topological orders? These quantum phases of matter are difficult to find. If we are lucky, we may find them on existing physical systems such as the quantum Hall effect. But we can also engineer suitable quantum Hamiltonian models, e.g., using polar molecules on optical lattices [14,15,20], or by some other means. There are methods for demonstrating topological order without resorting to interferometric techniques [29].
In this paper we present new results concerning the realization of 2-body Hamiltonians using cluster states techniques on one hand, and without measurement-based computations on the other. In this latter case, we present a detailed study of the set of integrals of motion (IOM) in a 2-body Hamiltoinan, fermionic mappings of the original spin Hamiltonian that give information about the physics of the system and which complements previous results using bosonic mapping techniques [10].
This paper is organized as follows: in Sect. II we present color codes as instances of topological stabilizer codes with Hamiltonians based on many-body interacting terms and then introduce the quantum Hamiltonian model based solely on 2-body interactions between spin-1 2 particles. The lattice is two-dimensional and has coordination number 4, instead of the usual 3 for the Kitaev model. It is shown in Fig. 3 and it is called ruby lattice. In Sect. III, we describe the structure of the set of exact integrals of motion (IOM) of the 2-Body model. We give a set of diagrammatic local rules that are the building blocks to construct arbitrary IOMs. These include colored strings and string-nets constant of motion, which is a distinctive feature with respect to the Kitaev's model. In Sect. IV, we establish a connection between the original topological color code and the new 2-Body color model. This is done firstly at a non-perturbative level using the colored string integrals of motion that are related with the corresponding strings in the TCC. Then, using degenerate perturbation theory in the Green function formalism, it is possible to describe a gapped phase of the 2-Body color model that corresponds precisely to the topological color code. In Sect. V, we introduce a mapping from the original spin-1 2 degrees of freedom onto bosonic degrees of freedom in the form of hard-core bosons which also carry a pseudospin. This provides an alternative way to perform perturbation theory and obtain the gapped phase corresponding to the TCC. It also provides a nice description of low energy properties of the 2-Body model and its quasiparticles. In Sect. VI, we introduce another mapping based on spinless fermions which is helpful to understand the structure of the 2-Body Hamiltonian and the presence of interacting terms which are related to the existence of stringnets constants of motion.
Sect. VII is devoted to conclusions and future prospects. A describes how to obtain 2-Body Hamiltonians for topological color codes based on cluster states and measurements using ancilla qubits.

A. Color Codes as Topological Stabilizers
Some of the simplest quantum Hamiltonian models with topological order can be obtained from a formalism based on the local stabilizer codes borrowed from quantum error correction [30] in quantum information [31,32]. These are spin-1 2 local models of the form where the stabilizer operators S i constitute an abelian subgroup of the Pauli group P n of n qubits, generated by the Pauli matrices not containing −1. The ground state is a stabilizer code since it satisfies the condition and the excited states of H are gapped, and correspond to error syndromes from the quantum information perspective The seminal example of topological stabilizer codes is the toric code [33]. There are basically two types of known topological stabilizer codes [13]. It is possible to study this type of homological error correcting codes in many different situations and perform comparative studies [34][35][36][37][38][39][40][41][42]. Topological color code (TCC) is the another relevant example of topological stabilizer codes, with enhanced computational capabilities [1][2][3]. In particular, they allow the transversal implementation of Clifford quantum operations. The simplest lattice to construct them is a honeycomb lattice Λ shown in Fig.1, where we place a spin-1 2 system at each vertex. There are two stabilizer operators per plaquette: where τ ν 's (ν = x, y) are usual Pauli operators. There exist six kinds of basic excitations. To label them, we first label the plaquettes with three colors: Notice that the lattice is 3-valent and has 3-colorable plaquettes. We call such lattices 2-colexes [3]. One can define color codes in any 2-colex embedded in an arbitrary surface. There exists a total of 15 nontrivial topological charges as follows. The excitation at a plaquette arises because of the violation of the stabilizer condition as in (3). Consider a rotation τ y applied to a certain qubit. Since τ y anticommutes with plaquette operators B x f of neighboring plaquettes, it will put an excitation at corresponding plaquette. Similarly, if we perform a τ x rotation on a qubit, the plaquette operators B y f are violated. These are the basic excitations, two types of excitations per each colored plaquette. Regarding the color and type of basic excitations, different emerging excitations can be combined. The whole spectrum of excitations is shown in Fig.2(a). Every single excitation is boson by itself as well as the combination of two basic excitations with the same color. They form nine bosons. However, excitations of different color and type have semionic mutual statistics as in Fig.2(b). The excitations of different color and type can also be combined. They form two families of fermions. Each family of fermions is closed under fusion, and fermions from different families have trivial mutual statistics. This latter property is very promising and will be the source of invisible charges as we will discuss in Sect.V. The anyonic charge sectors are in one to one correspondence with the irreducible representations (irreps) of the underlying gauge group, and the fusion corresponds to decomposition of the tensor product of irreps. We describe all above excitations in terms of representation of the gauge group of the TCC. Before that, let us make a convention for colors which will be useful for subsequent discussions. We refer to colors by a bar operationc that transform colors cyclically asr = g, ḡ = b andb = r. The elements of the gauge group Z 2 × Z 2 are {e, r, b, g}. Each excitation carries a topological charge. The corresponding topological charge can be labeled by the pair (q, χ), where q ∈ Z 2 ×Z 2 and χ an irrep of this group [10]. We label them as χ e (c) = χ c (c) = −χ c (c) = 1. Therefore, there are nine bosons labeled by (c, χ e ), (e, χ c ) and (c, χ c ) and six fermions (c, χc) and (c, χc). Taking into account the vacuum with trivial charge (e, χ e ), color code has sixteen topological charges or superselection sectors. Regarding the fusion process, fusion of two charges (q, χ c ) and (q ′ , χ c ′ ) give rises to (qq ′ , χ c χ c ′ ) charge. Additionally, the braiding of charge (q, χ c ) around charge (q ′ , χ c ′ ) produces the phase χ c (q ′ )χ c ′ (q). An excitation at a c-plaquette has (c, It is also possible to use both types of topological stabilizer codes, either toric codes or color codes, to go beyond homological operations. This corresponds to performing certain types of operations called code deformations, which may alter the topology of the surface allowing an extension of the computational capabilities of these 2D codes [43][44][45][46][47].
Active error correction procedures are particularly interesting in the case of topological stabilizer codes. They give rise to connections with random statistical mechanical models like the random bond Ising model for the toric code [47] and new random 3-body Ising models for color codes [48]. The whole phase diagram p − T has been mapped out using Monte Carlo, which in particular gives the value of the error threshold p c . This particular point can also be addressed using multicritical methods [49]. There is experimental realization of topological error correction [50]. Without external active error correction, the effect of thermal noise is the most challenging problem in toric codes [51][52][53][54][55][56]. Finite temperature effects of topological order in color codes has also been studied [57].
In all, the type of entanglement exhibited by topological color codes is very remarkable [58][59][60][61][62][63][64]. A very illustrative way to see this is using the connection of the ground state of topological codes with standard statistical models by means of projective measurements [7,[65][66][67][68][69][70][71]. For TCCs, this mapping yields the partition function of a 3-body classical Ising model on triangular lattices [7]. This 3-body model is the same found in active error correcting techniques [48], but without randomness since there is no noise produced by external errors. This type of statistical mapping allows us to test that different computational capabilities of color codes correspond to qualitatively different universality classes of their associated classical spin models. Furthermore, generalizing these statistical mechan-ical models for arbitrary inhomogeneous and complex couplings, it is possible to study a measurement-based quantum computation with a color code state and we find that their classical simulatability remains an open problem. This is in sharp contrast with toric codes which are classically simulable within this type of scheme [66].

B. The Model
In nature, we find that interactions are usually 2-body interactions. This is because interactions between particles are mediated by exchange bosons that carry the interactions (electromagnetic, phononic, etc.) between two particles.
The problem that arises is that for topological models, like the toric codes and color codes, their Hamiltonians have many-body terms (5). This could only achieved by finding some exotic quantum phase of nature, like FQHE, or by artificially enginering them somehow.
Here, we shall follow another route: try to find a 2-body Hamiltonian on a certain 2D lattice such that it exhibits the type of topological order found in toric codes and color codes. In this way, their physical implementation looks more accessible.
In fact, Kitaev [72] introduced a 2-body model in the honeycomb lattice that gives rise to an effective toric code model in one of its phases. It is a 2-body spin-1 2 model in a honeycomb lattice with one spin per vertex, and simulations based on optical lattices have been proposed [17].
It shows emerging free fermions in the honeycomb lattice. If a magnetic field is added, it contains a non-abelian topological phase (although not enough for universal quantum computation). Interestingly enough, another regime of the model gives rise to a 4-body model, which is precisely an effective toric code model. A natural question arises: Can we get something similar for color codes? We give a positive answer in what follows.
Motivated by these physical considerations related to a typical scenario in quantum manybody physics, either condensed matter, AMO physics or the like, we will seek a quantum spin Hamiltonian with the following properties: i/ One of its phases must be the TCC.
ii/ To have two sets of plaquette operators generating a Z 2 × Z 2 local, i.e. gauge, symmetry.
iii/ To have string-nets and colored strings IOM as in the TCC, but in all coupling regimes.
Thus, the reasons behind demanding these properties are to guarantee that the sought after model will host the TCC. For instance, property i/ means that we must be able to generate the 2D color code Hamiltonian consistently at some lowest order in perturbation theory (PT). This we shall see in Sect.IV B. Likewise, properties ii/ and iii/ are demanded in order to have the fundamental signatures regarding gauge symmetry and constants of motions associated with TCCs. Notice that we have not demanded that the model be exactly solvable. This is a mathematical requisite, rather than physical. We leave the door open for considering larger classes of models beyond exactly solvable models, which may be very interesting and contain new physics. For example, according to those properties, it would be possible to have models with a number of IOMs that scales linearly with N, the number of spins or qubits. Thus, the Kitaev model has a number of IOMs of 1 2 N. Our purpose is to present first the 2-body quantum Hamiltonian in 2D [10], and then to analyze diverse possible mappings in later sections, like using bosonic and fermionic degrees of freedom. The analysis of the set of IOMs will play also a crucial role in the understanding of our model as we shall see in Sect.III It is a 2-body spin-1/2 model in a 'ruby' lattice as shown in Fig.3. We place one spin per vertex. Links come in 3 colors, each color representing a different interaction.
x, red links y, green links z, blue links For a suitable coupling regime, this model gives rise to an effective color code model.
Furthermore, it exhibits new features, many of them not present in honeycomb-like models: • Exact topological degeneracy in all coupling regimes (4 g for genus g surfaces).
• String-net integrals of motion.
• Emergence of 3 families of strongly interacting fermions with semionic mutual statistics.
• Z 2 × Z 2 gauge symmetry. Each family of fermions sees a different Z 2 gauge subgroup.

III. STRING OPERATORS AND INTEGRALS OF MOTION
We can construct integrals of motion (IOM), I ∈ P n , [H cc , I] = 0, following a pattern of rules assigned to the vertices of the lattice, as shown in Fig.4. These rules are constructed to attach a Pauli operator of type σ x i , σ y i or σ z i to each of the vertices i. The lines around the vertices, either wavy lines or direct lines, are pictured in order to join them along paths of vertices in the lattice that will ultimately translate into products of Pauli operators, which will become IOMs. Clearly, σ z i operators are distinguished from the rest. The contribution of each qubit in the string operator is determined in terms of how it appears in the string.
Its contribution may be determined by the outgoing red and green links which have the qubit as their end point in the string. In this case the σ x or σ y Pauli operators contribute in the string IOM . If a typical qubit crossed only by a wavy line as shown in Fig.4(a), it contributes a σ z Pauli operator in the string. To have a clear picture of string operators, a typical example has also been shown in Fig.4 Fig.4. Let V f stand for a set of qubits on a plaquette. Note that each plaquette contains 18 qubits corresponding to six triangles around it. For first plaquette operator in Fig.5 we can write its explicit expression in terms of Pauli matrices as S A f = i∈V f σ ν i , where f denotes the plaquette and ν = x, y, depending on outgoing red or green links, respectively. Similarly the second plaquette operator has an expression as S B f = i∈V f σ ν i . The third string is only a closed wavy string which coincides to the inner hexagon of the plaquette. It's expression is S C f = i∈V h σ z i , where V h stands for six qubits on the inner hexagon. The three closed strings described above are not independent.
Using the Pauli algebra, it is immediate to check that they satisfy Thus, there exist 2 independent IOMs per plaquette: this is the Z 2 × Z 2 local symmetry of the model Hamiltonian (6). Plaquette operators commute with each other and with any other IOM. If a IOM corresponds to a nontrivial cycle c, it is possible to find another IOM that anticommutes with it, namely one that corresponds to a cycle that crosses once c ′ . Thus, IOMs obtained from nontrivial cycles are not products of plaquette operators.
Each string operator squares identity since we are working with qubits. Plaquette operators corresponding to different plaquettes commute with each other and also with terms in Hamiltonian in (6) since they share in zero or even number of qubits. Therefore, the closed strings with the underlying symmetry obtained above define a set of integrals of motion.
The number of integrals of motions is exponentially increasing. Let 3N be the total number of qubits, so the number of plaquettes will be N 2 . Regarding to the gauge symmetry of the model, the number of independent plaquette operators is N. This implies that there are 2 N integrals of motion and allow us to divide the Hilbert space into 2 N sectors being eigenspaces of plaquette operators. However, for closed manifold, for example a torus, all plaquette operators can not be independently set to +1 or −1 because they are subject to some constraint. All other closed string operators that are homologous to zero, i.e. they are homotopic to the boundary of a plaquette, are just the product of these elementary plaquette operators. It is natural that all of them are topologically equivalent up to a deformation and commute with the Hamiltonian of the model. The most general configuration that we may have is shown in Fig.6. We call them stringnets IOM since in the context of our model, they can be thought of as the string-nets introduced to characterize topological orders [79]. The key feature of these IOMs is the presence of branching points located at the blue triangles of the lattice. This is remarkable and it is absent in honeycomb 2-body models like the Kitaev model. When the stringnets IOM are defined on a simply connected piece of lattice they are products of plaquette operators. More generally, they can be topologically non-trivial and independent of plaquette operators.  Each qubit crossed by the string contributes a Pauli operator according to the rules in Fig.4.
Again, using Pauli algebra we can see that only two of them are independent, as with the plaquette IOMs.
where t is the number of triangles on the string. To distinguish properly the three types we have to color the lattice. We could already use the colors to label strings. Strings are then red, green or blue. This is closely related to the topological color code [1,10]. The latter relation shows that each string can be constructed of two other homologous ones, which is exactly the expression of the Z 2 × Z 2 gauge symmetry. Each non-contractible closed string operator of any homology commutes with all plaquette operators and with terms appearing in the Hamiltonian, so they are constant of motions. But, they don't always commute with each other. In fact, if the strings cross once then This latter anticommutation relation is a source of exact topological degeneracy [80] of the model independent of phase we are analyzing it.

A. Non-Perturbative Picture
In this subsection we discuss the ruby lattice is connected to the 2-colex even at the non-perturbative level. Then, in the subsequent sections we verify it using quantitative methods. From the previous discussion on IOMs, we have already seen a connection with the topological color codes. Now, we want to see how different strings introduced above are related to coloring of the lattice. To this end, consider the closed strings A, B, C in Fig.5.
The closed strings A and B can be visualized as a set of red and green links, respectively.
With such visualization, we put forward the next step to color the inner hexagons of the ruby lattice: a colored link, say red, connect the red inner hexagons. Accordingly, other inner hexagons and links can be colored, and eventually we are left with a colored lattice.
The emergence of the topological color code is beautifully pictured in Fig.8. Geometrically, it corresponds to shrinking the blue triangles of the original lattice into points, which will be referred as sites of a new emerging lattice, see Connection to the 2-colexes can be further explored by seeing how strings on the ruby lattice correspond to the colored strings on the effective honeycomb lattice. To this end, consider a typical string-net on the ruby lattice as shown in Fig.9(a). This corresponds to a non-perturbative picture of the IOMs of the model. The fat parts of the string-net connect two inner hexagons with the same color. In this way, the corresponding string-net on the effective lattice can be colored as in Fig.9(b). The color of each part of the string-net on the effective honeycomb lattice is determined by seeing which colored inner hexagons on the ruby lattice it connects. Three colored strings cross each other at a branching point, and its expression in terms of Pauli matrices of sites are given by product of Pauli operators written adjacent to the sites. How they are determined, will be clear soon.
It is possible to use colors to label the closed strings on the honeycomb lattice. Before that, let us use a notation for Pauli operators acting on effective spins of honeycomb lattice τ α , where α = x, y, z. We indicate the labels α as c|c := z,c|c := x,c|c := y, where we are using a bar operator. To each c-plaquette, we attach three operators each of one color. Let B c ′ f denotes such operators, where low and up indices stand for c-plaquette f and color of the closed string attached to the plaquette, respectively. With these notations, the plaquette operators read as follows: where the product runs over all vertices of the c-plaquette f in the honeycomb lattice in Fig.8(c). Thus, we can write the explicit expression of operators as follows: All these plaquette operators are constant of motions. Again, We can realize the gauge On a compact manifold, for example on the torus, all plaquettes are not independent. They are subject to the following constraint: where the product runs over all plaquettes f in the lattice Λ, and N is the total number of plaquettes.
We can also realize noncontractible strings on the effective lattice which are rooted in the topological degeneracy of the model. They are just the IOMs in Fig.7 when reduced on the effective honeycomb lattice. Once the inner hexagons of ruby lattice are colored, they correspond to colored strings as in Fig.9. Let S c µ stands for such string, where indices µ and c denote the homology and color of the string, respectively. This string operator is tensor product of Pauli operators of qubits lying on the string. Namely, the string operator is The contribution of each qubit is determined by the color of the hexagon that the string turns on it, see of Pauli operators of these two qubits in string S 1 is τ y 1 τ x 2 , while for string S 2 the contribution is τ Then, let S 2 be a green string. In this case the contribution of qubits will be τ y 1 τ z 2 , which explicitly shows that {S b 1 , S g 2 } = 0. The interplay in (7) can be translated into an interplay between color and homology as follows. body structure and physics. In particular, we show that this corresponds to the following set of couplings in the original 2-body Hamiltonian that is, a strong coupling limit in J z . The topological color code effectively emerges in this coupling regime. This can be seen using degenerate perturbation theory in the Green function formalism. Let H = H 0 + V be a Hamiltonian describing a physical system with two-body interaction, and we regard the V , the norm of V , be very small in comparison with the spectral gap of unperturbed H 0 . We also suppose that H 0 has a degenerate groundsubspace which is separated from the excited states by a gap ∆. The effect of V will be to break the ground state degeneracy in some order of perturbation. Now the interesting question is whether it is possible to construct an effective Hamiltonian, H eff , which describes the low energy properties of the perturbed Hamiltonian H. The effective Hamiltonian arises at orders of perturbation that break the ground state degeneracy. From the quantum information perspective the Hamiltonian H acts on the physical qubits while the effective Hamiltonian acts on the logical qubits projected down from the physical qubits.
We will clarify that how many-body Hamiltonian in (5) will present an effective description of low lying states of the 2-body Hamiltonian (6). We use the perturbation about the Hamiltonian in (6) considering the coupling regim (15). Here, the qubits on the triangles are physical qubits, and logical qubits are those living at the vertices of the 2-colex. We refer to triangles as sites, since they correspond to the vertices of the 2-colex. Thus a triangle will be shown by index v and its vertices by Latin indices i, j. In fact the low lying spectrum of 2-body Hamiltonian encodes the following projection from the physical qubits to the logical ones at each site: where | ⇑ and | ⇓ stands for the two states of the logical qubit at site v, and | ↑ (| ↓ ) is usual up (down) states of a single spin in computational bases.
To this end, we split the 2-body Hamiltonian into two parts. The unperturbed part is In the limit of strong Ising interaction the system is polarized. The interactions between neighboring qubits on different triangles are included in V . They are σ x i σ x j and σ y i σ y j corresponding to red and green links in Fig.3, respectively. So, the transverse part of the Hamiltonian is In the case when J z ≫ J x , J y the low lying excitations above the fully polarized state can be treated perturbatively. We analyze the effect of V on the ground state manifold by using the degenerate perturbation theory [81] in couplings J x and J y . We are interested in how ground state degeneracy is lifted by including the interaction between triangles perturbatively. Let L stand for the ground state manifold with energy E (0) 0 and let P be the projection onto the ground state manifold L. The projection is obtained from the degenerate ground states as follows: The self-energy can be represented in terms of Feynman diagrams and can be computed for any order of perturbation: The energy E can also be expanded at different orders 0 . Now, we are at the position to determine different orders of perturbation. Each term of V acts on two neighboring physical qubits of different triangles. At a given order of perturbation theory, there are terms which are product of σ x and σ y acting on the ground state subspace. Each term when acts on the ground state manifold brings the ground state into an excited state. However, there may be a specific product of the σ x and σ y which takes the ground state into itself, i.e. preserve the polarized configurations of triangles.
At zeroth-order the effective Hamiltonian will be trivial H 0 . The first-order correction is given by the operator The effect of V is to move the states out of the ground state manifold because each term either σ x σ x or σ y σ y flip two qubits giving rise to two triangles being excited, i.e V P = P 2 V P , where the operator P 2 is the projection to second excited state manifold. Therefore, P V P = 0, and there is no first-order correction to the ground state energy.
The second-order correction to the ground state will be the eigenvalues of the following operator.
where the operator G ′ (E) = 1/(E − H 0 ) is the unperturbed Green's function and the superscript prime stands for the fact that its value be zero when acts on the ground state. The second-order correction only shifts the ground state energy, and therefore, the second-order effective Hamiltonian acts trivially on ground state manifold, In fact the first V flips the qubits and the second V flips them back. As we go to higher order of perturbation theory the terms become more and more complicated. However, if the first-order is zero as in our case, the terms becomes simpler. Thus, the third-order of perturbation will be zero and will leave corrections to energy and ground state intact: The forth-order of perturbation theory contributes the following expression to the correction of ground state manifold: where E 0 is the second order correction to the ground state energy obtained in (23). The first term includes four V and must act in the ground state in which the last V returns the state to the ground state manifold. The second term is like the second-order. There are many terms which must be calculated. However, since the forth-order only gives a shift to the ground state energy, we don't need them explicitly. So, we can skip the forthorder. Fifth-order correction yields terms each containing odd number of V , so it gives zero contribution to the effective Hamiltonian.
The sixth-order of perturbation leads to the following long expression.
Apart from the first term, other terms contain two or four V and as we discussed in the preceding paragraphs they only contribute a shift in the ground state energy. However, the first term gives the first non-trivial term breaking in part the ground state degeneracy. In the sixth order correction, there are some terms which are the product of σ x σ x and σ y σ y associated to the red and green links of the ruby lattice. Some particular terms, as seen below, may map ground state subspace into itself. For instance, consider the following product of links around an inner hexagon <i,j> where the first product runs over three red and three green links making an inner hexagon, V h stands for the set of its vertices and the prefactor ± depends on the ordering of links in the product. The action of a σ z on one vertex (qubit) of a triangle encodes an logical τ z v operator acting on the associated vertex of lattice Λ. This can explicitly be seen from the following relation: where σ z acts on one of the vertices of a triangle and P v is the projection defined in (16).
Thus, the expression of (27) can be related to the plaquette operator B z f = − τ z v , where the index f denotes a plaquette of effective lattice Λ as in Fig.1 and product runs over six sites around it. Now we go on to pick up the sixth-order correction to the ground state manifold. There are many terms which must be summed. Sixth-order correction up to a numerical constant contributes the following expression to the effective Hamiltonian: where δ is a positive numerical constant arising from summing up 720 terms related to the order of product of six links around an inner hexagon of ruby lattice. Although, its exact numerical value is not important, but knowing its sign is essential for our subsequent discussions. As it is clear from the first term in (26) operators B z f with eigenvalue +1. Thus, highly degenerate ground state of the unperturbed Hamiltonian is broken in part. The same plaquette operators B z f also appear at higher order of perturbation. For example, at eighth-order. Instead of giving the rather lengthy expression of the eighth-order correction, we only keep terms resulting in the plaquette operators as follows: where β > 0. This term is added to the one in (29) to give the effective Hamiltonian up to eighth-order, but the ground state structure remains unchanged. Further splitting in the ground state manifold is achieved by taking into account the ninth-order of perturbation.
The expression of ninth-order is very lengthy. However, the first term of the expression containing nine V gives some terms being able of mapping the ground state manifold into itself in a nontrivial way. These terms map a polarized triangle, say up, to a down one.
Indeed, when one or two qubits of the polarized triangle gets flipped, its state is excited.
However, flipping three qubits of the triangle returns back the ground state onto itself. This process encodes τ x and τ y logical operators acting on logical qubits arising through the projection. Let σ x 1 , σ x 2 and σ y 3 act on three qubits of a triangle. The encoded τ y operator will be If σ x 1 , σ y 2 and σ y 3 act on three qubits of a triangle, the encoded τ x logical operator will be As we already pointed out a plaquette of the ruby lattice is made up of an inner hexagon, an outer hexagon and the six blue triangles. It is possible to act on the polarized space of the blue triangles by making two different combinations of 9 link interactions: i/ Applying 6 link interactions on the outer hexagon (three of them of XX type and another three of YY type), times 3 link interactions of red type on the inner hexagon. Notice that every vertex of the blue triangles in the plaquette gets acted upon these 9 link interactions. The resulting effective operator is of type τ y due to (31). ii/ Applying 6 link interactions on the outer hexagon, times 3 link interactions of green type. Then, the resulting effective operator is of type τ x due to (32).
The effective Hamiltonian at this order then reads Again, the sign of coefficient γ is important. Nine V 's, six τ x or τ y , and eight Green's function imply that the γ must have positive sign.
Putting together all above obtained corrections lead to an effective Hamiltonian encoding color code as its ground state [1,58]. Therefore, up to constant terms, the effective Hamiltonian reads as follows where the k z , k x and k y are positive coefficients arising at different orders.
Since B x f B y f = B z f , the above effective Hamiltonian is just the many-body Hamiltonian of the color code as in (1). The terms appearing in the Hamiltonian mutually commute, so the ground state will be the common eigenvector of plaquette operators. Since each plaquette operator squares identity, the ground state subspace, C, spanned by vectors which are common eigenvectors of all plaquette operators with eigenvalue +1, i.e The group of commuting boundary closed string operators can be used as an alternative way to find the terms appearing in the effective Hamiltonian [82]. As we pointed out in the preceding section, the non-zero contribution from various orders of perturbation theory results from the product of red and green links which preserve the configurations of the polarized triangles, i.e maps ground state manifold onto itself. For instance consider the elementary plaquette operator A corresponding to a closed string in Fig.5. Each triangle contributes σ y σ y σ x to the expression of operator which is projected to τ x as in (32). Thus, the effective representation of plaquette operator reads as follows: Plaquette operators S B f and S C f can also be recast into the effective forms as follows: These are lowest order contributions to the effective Hamiltonian as we obtained in (34).
Higher order of perturbation will be just the product of effective plaquette operators. The nontrivial strings winding around the torus will have also effective representations and appear at higher orders of perturbation. In general, every string-net IOM on the ruby lattice are projected on an effective one as in Fig.9.

V. BOSONIC MAPPING
As we stated in Sect.III, one of the defining properties of our model is the existence of non-trivial integrals of motion IOM, called string-nets. As a particular example, the Kitaev's model on the honeycomb lattice has strings IOMs, but not string-nets. We are interested in models in which the number of IOMs is proportional to the number of qubits in the lattice, i.e., where I is the number of IOMs, N q is the number of qubits (spins) in the lattice, and η is a fraction: η = 1 2 for the Kitaev model [72], η = 1 3 for our model in (6) [10]. The fact that η is a fraction η ≤ 1 implies that these models based on string-net IOMs will not necessarily be exactly solvable. In the Kitaev's model, it turns out to be exactly solvable using an additional mapping with Majorana fermions, but this need not be a generic case. Therefore, we need to resort to other techniques in order to study the physical properties of these models. We consider here and next section, approximate methods based on bosonic and fermionic mappings. The application of the bosonic method to our model is based on a mapping from the original spins on the blue triangles to hardcore bosons with spin [10]. With this mapping, it is possible to use the PCUTs approach (Perturbative Continuous Unitary Transformations) [83]. This is inspired by the RG method based on unitary transformation introduced by Wegner (the Similarity RG method) [84]. Originally, the PCUTs method was applied to the Kitaev model [85]. As we will see below, the method paves the way to go beyond the perturbation approach presented in the preceding section which fits into a sector without any hard-core boson. The physics at other sectors is very promising and we study it here. Emergence of three families of strongly interacting anyonic fermions and invisible charges are among them, which are not present in the Kitaev's model or its variants. To start with, let us set for simplicity J z = 1/4, and consider the extreme case of J x = J y = 0. In this case the system consists of isolated triangles. The ground states of an isolated triangle are polarized states | ↑↑↑ and | ↓↓↓ with energy −3/4. The excited states that appear by flipping spins are degenerate with energy 1/4. In this limit, the spectrum of whole system is made of equidistance levels being well-suited for perturbative analysis of the spectrum: Green's function formalism as discussed in the preceding section, which may capture only the lowest orders of perturbation or another alternative approach based on the PCUT. The change from the ground state to an exited state can be interpreted as a creation of particles with energy +1. This suggest an exact mapping from the original spin degrees of freedom to quasiparticles attached to effective spins. The mapping is exact, i.e. we don't miss any degrees of freedom. Such a particle is a hard-core boson. At each site, we attach such a boson and also an effective spin-1 2 . Let choose the following bases for the new degrees of freedom where a and d stand for states of the effective spin and quasiparticle attached to it. The Hilbert space H C representing the hard-core bosons is four dimensional spanned by bases {|0 , |r , |g , |b }. Now the following construction relates the original spin degrees of freedom and new ones in (39) Within such mapping, the effective spins and hard-core bosons live at the sites of the effective hexagonal lattice Λ in Fig. 8 Annihilation and creation operators anticommute on a single site, and commute at different sites, that is why they are hard-core bosons. We can also label the Pauli operators of original spins regarding to the their color in Fig. 8(b) as σ w c with c = r, g, b. The mapping in (40) can be expressed in operator form as follows where ν = x, y, s x := −s y := 1, the symbols τ denote the Pauli operators on the effective spin and we are using the color parity operators p c and the color switching operators r c defined as Now we can forget the original ruby lattice and work on the effective lattice in which the bosons are living at its sites. With the above identification for Pauli operators, the 2-body Hamiltonian in (6) can be written in this language. Before that, let fix a simplified notation.
All spin and bosonic operators act on the sites of the effective lattice. We refer to a site by considering its position relative to a reference site: the notation O ,c means O applied at the site that is connected to a site of reference by a c-link. The 2-body Hamiltonian then becomes with N the number of sites, Q := Λ n the total number of hardcore bosons, the first sum running over the N sites of the reduced lattice, the second sum running over the 6 combinations of different colors c, c ′ and a sum of several terms for an implicit reference site, according to the notation convention we are using. The meaning of the different terms in (45) where we are using the notation We can also describe the plaquette IOM operators in Fig. 5 in terms of spin-boson degrees of freedom by means of the mapping in (42). For each plaquette f and color c, the plaquette operator is expressed as where c ′ is the color of the plaquette f , the product runs through its sites and ⋆ is just a convenient symmetric color operator defined by c ⋆ c := c, c ⋆c :=c ⋆ c :=c.
The relation in (48) is just a generalization of plaquette operators in (11) to other sectors of the system. In fact taking the zero particle sector, the expressions in (11) are recovered.
In the same way the nontrivial string operators in Fig.7 can be described with the above mapping as where µ denotes the homology class of the string. On closed surfaces, not all plaquette operators are independent. They are subject to the following constraints where s is the number of sites of a given plaquette f . The first equation can be further divided into products over subsets of plaquettes giving rise to other constants of motion, the so called color charges, as f ∈Λc In these products the spin degrees of freedom are washed out, since they appear twice and consequently square identity. By use of equation (43), the product over parity operators can be written as where Q c = Λ n c is the total number of c-bosons. It is simple to check that the above equation commutes with Hamiltonian in (44). For each family of bosons we can attach a charge. We suppose that each c-boson carries a charge as χ c , that is, an irrep of the gauge group. In particular, the Hamiltonian preserves the following total charge The statistics of vortices depend on their color and type as in Fig.2. But what about the statistics of c-bosons? As studied in [86], the statistics of quasiparticles can be examined using the hopping terms. These hopping terms are combined so that two quasiparticles are exchanged. In addition to usual hopping terms, we also need for composite hopping , that is, a c-boson hops from a c-plaquette to another, which is carried out by terms like Let us consider a state with two c-boson excitations located at two different sites separated from a reference site byc andc links. An illustrative example for the case of, say, blue bosons is depicted in Fig.11(a). Consider a process with the net effect of resulting into the exchange of two bosons. Each step of process can be described by hopping terms. Upon the combination of hopping terms, we are left with the following phase which explicitly show that the quasiparticles made of hardcore-bosons and effective spins have fermion statistics [10]. Thus we have three families of fermions each of one color.
These are high energy fermions interact strongly with each other due to the fusion term in the Hamiltonian. Fermions from different families have mutual semionic statistics, that is, encircling one c-fermion around ac-fermion picks up a minus sign. This can also be checked by examining the hopping terms as in Fig.11(a). Thus we are not only dealing with fermions but also with anyons.
The elementary operators in (46) have a remarkable property, and that, they all commute will plaquettes and strings IOM. This naturally implies any fermionic process leaves the vorticity content of the model unaffected. A fermionic process may correspond to hopping, splitting, fusion and annihilation driven by the terms in the Hamiltonian (45) and (46). A typical fermionic process is shown in Fig.11 In the zero-charge sector, only the effective spin degrees of freedom do matter. The effective Hamiltonian is just a many-body Hamiltonian with terms that are product of plaquette operators as follows where the first and second sum run over an arbitrary collection of colors and plaquettes of effective honeycomb lattice. The coefficients O's are determined at a given order of perturbation. The product of plaquette operators is nothing but the string-net operators.
Let us focus at lowest order of perturbation, where the model represents non-interacting vortices. First, let us redefine plaquette operators as where j w := J w /|J w |. At ninth order of perturbation the effective Hamiltonian is with [10] This is exactly the many-body Hamiltonian of topological color code obtained in (34) using degenerate perturbation theory with the additional advantage of knowing the coefficients exactly. Its ground state is vortex free and can be written explicitly by choosing a reference state as Other degenerate ground states can be constructed by considering the nontrivial string operators winding around the torus. Excitations above ground state don't interact. Going to higher order of perturbation, as equation in (56) suggests, the ground states remain unchanged, however the excitation spectrum changes and vortices interact with each other.
The one-quasiparticle sector can also be treated by examining the expressions in (46).
The effective Hamiltonian can be written as What the second term describes is nothing but the annihilation of a c-fermion at a reference site and then its creation at a site connected to the reference by a string-net R, as shown in Fig.11(b). Again notice that this string-net is just the product of green and red links of original 2-body Hamiltonian, in its effective form is given in terms of spin-boson degrees of freedom. The coefficients O's are determined at any order of perturbation. Notice that these coefficients are different from those in (56). In the first order, only the hopping term does matter. Let us consider the sector containing a c-fermion. Up to this order, the fermion can only hop around a c-plaquette. This implies that at first order the fermion perform an orbital motion around a plaquette of its color. Notice that the fermion can not hop from a c-plaquette to other c-plaquette at the first order, since it needs for a composite process which appears at second order. This composite process is a combination of splitting and fusion processes. This is a virtual process in the sense that the splitting of a c-fermion into twoc-andc-fermion takes the model from 1-quasiparticle sector into the 2-quasiparticle sector, but the subsequent process fuses two particles into a single one turning back to 1quasiparticle sector. Thus, at second order the c-fermion can jump from one orbit to other one.
At first order, for J = J x = J y , we get a −2J contribution to the energy gap coming from orbital motion. Going to second order we get a non-flat dispersion relation. The gap, at this order, is given by 1 − 2J − J 2 /2 and thus it closes at J ≃ 0.45. This is just an approximate estimation, since we are omitting all fermion interactions and, perhaps more importantly, we are taking J ≃ J z . However, it is to be expected that as the couplings J x ∼ J y grow in magnitude the gap for high-energy fermions will reduce, producing a phase transition when the gap closes. Such a phase transition resembles the anyon condensations discussed in [87][88][89]. There are three topological charges invisible to the condensed anyons.
This means that in the new phase there exists a residual topological order related to these charges. They have semionic mutual statistics underlying the topological degeneracy in the new phase.

C. Fermions and gauge fields
The emerging high-energy c-fermions always appear with some nontrivial gauge fields [10,86], and carry different representation of the gauge symmetry Z 2 × Z 2 of the model. Before clarifying this, we can see that the plaquette degrees of freedom correspond to Z 2 ×Z 2 gauge fields. This correspondence is established via introducing the following plaquette operators The gauge element q f ∈ Z 2 × Z 2 that can be attached to the plaquette f is determined by following eigenvalue conditions which always has a solution due to The ground state of color code Hamiltonian (58) is vortex free and corresponds to χ c = 1.
The fact that for a 2-colex with hexagonal plaquettes, the gauge fields can be related to representation of the group is immediate. One way to see this is to check the phase picked where sum runs over all squares and octagons. Notice that at 12th order of perturbation multiplaquette terms that are product of square plaquette operators are also appeared. It is simple to check that the coefficients k's have positive sign. As we can relate the plaquettes to the representation of gauge group, the ground state corresponds to vortex free sector.
In fact, the ground state of all 2-colexes with plaquettes of any shape is vortex free and correspond to χ c = 1 of gauge group. What is able to differentiate between ground states of 2-colex plaquttes with 4n (n an integer) edges from others is related to the gauge fields attached to a fermion. In particular, there is a background of π-fluxes in the ground states of such lattices, and the emerging c-fermions can detect them. To make sense of the existence of such fluxes, let us consider a simple fermionic process as explained above. When a cfermion turns on a plaquette, the combination of hopping terms yield −Bc f , −Bc f and −B c f , which clearly imply that This result exhibit that the ground states of color code models defined on lattices with 4nplaquettes carry fluxes. Thus, in its representation in terms of Z 2 × Z 2 gauge group, the fluxes must be subtracted away.
Our derivation for ground states with 0-and π-flux can be compared with Lieb theorem [90], which states for a square lattice the energy is minimized by putting π flux in each square face of the lattice. The connection to our models makes sense when we consider how 2-colexes with hexagonal and 4-8-8 plaquettes can be constructed from a square lattice by removing some edges. Then the total π fluxes in a set of square faces corresponding to a 2-colex plaquette amounts to the flux that it carries. It is simple to see that each hexagonal plaquette composed of two (imaginary) square faces, and flux π in each square face then implies flux 0 in the hexagon. The same strategy holds for fluxes carried by the 4-8-8 plaquette(and in general for all 4n-plaquettes). Once again we see that each plaquettes of latter 2-colxes is composed of odd number of square faces, thus they carry flux π in their plaquettes.
Now we can give a general expression for the gauge fields seen by emerging high-energy fermions. To do so, let us consider a process in which a c-fermion is carried around a region λ, as in Fig.13. The hopping process yields a phase where q λ = λ q f , n λ c denotes the number of c-fermions inside λ and n λ 4 the number of 2colex plaquettes inside λ with a number of edges that is a multiple of four. Thus, we can see that each family of fermions carries a different representation of gauge group given by values of q f inside the region. Moreover, it emphasizes that fermions with different color charges have mutual semionic statistics. Clearly for hexagonal lattices n λ 4 = 0, and the ground state carries no fluxes.

VI. FERMIONIC MAPPING
In this section we will come back to the original Hamiltonian of (6) on the lattice in order to use another approximate method based on fermionic mappings. This Hamiltonian can be fermionized by Jordan-Wigner transformation [73]. To do so, firstly it is convenient to present a lattice which is topologically equivalent to the lattice of Fig.3. This is a new type of "brick-wall" lattice as shown in Fig.14. The black and white sites are chosen such that, at the effective level, the lattice be a bipartite lattice, since the effective spins are located at the vertices of hexagonal lattice representing a bipartite lattice. Note that neither original lattice in Fig.3 nor the brick-wall one in Fig.14 are bipartite in their own. Also as we will see, the fermionization of the model needs some ordering of sites in the "brick-wall" lattice.
The unit cell of the brick-wall lattice is comprised of two triangles as shown in Fig.14 in a yellow ellipse. The translation vectors n 1 and n 2 connect different unit cells of the lattice.
The deformation of the original lattice into a "brick-wall" lattice allows one to perform the one dimensional Jordan-Wigner transformation. The one dimensional structure of the lattice is considered as an array of sites on a contour as shown in Fig.15. The sites on the contour can be labeled by a single index and the ordering of the sites is identified by the direction of the arrows in Fig.15. The expression of the Pauli operators in terms of spinless fermions will be: where spinless fermions satisfy the usual anticommutation relations as follows Next, we introduce Majorana fermions as follows: for black sites and for white sites. Majorana operators are Hermitian and satisfy the following relations: where k = c, d. blue links : σ z j σ z k = iÛ jk d j d k , horizontal r − links : σ x j σ x k = id j d k , horizontal g − links : σ y j σ y k = −id j d k , vertical r − links : σ whereÛ jk = −ic j c k and the operatorF I is a non-local operator. Interestingly enough, this non-local term has the following expression: where we have used the ordering of the brick-wall lattice and R stands for a set of spins crossed by two brown ribbons shown in Fig.14. Observe that for each unit cell we can realize such ribbons. The non-local operator can be written as a product of some plaquette operators each corresponding to a rectangle(inner hexagon) wrapped by the ribbons. Notice that an inner hexagon of the ruby lattice in Fig.3 looks like a rectangle in the brick-wall lattice. Thus, two ribbons are nothing but a combination of rectangles as in Fig.14. This plaquette operator precisely corresponds to operator S C f in Fig.5. Let f 1 , f 2 , ... stand for those rectangles (inner hexagons). Thus, we have: After all these transformations, we arrive at an exact fermionized Hamiltonian given by It is simple to check the following commutation relations: SinceF 2 I = 1, the nonlocal operators can be replaced by its eigenvalues F I = ±1 . Thus the nonlocal terms appearing in the Hamiltonian which are related to the vertical links of the brick-wall lattice can be reduced to quadratic terms. However, the first sum in (76) which is related to the triangles cannot be reduced to the quadratic term. This is because local fieldsÛ jk corresponding to three links of a triangle anticommute with each other as well as with someF I 's. Due to these anticommutation, all fields cannot be fixed independently.
This fact is in sharp contrast with Kitaev's model and its variants. These latter models are defined on trivalent lattices. The different fields live at spatially separated links allowing a free fermion exact solution [72,73]. The obtained quadratic Hamiltonian describes free majorana fermions in a background of Z 2 charges. Instead, the lattice of our model is 4valent, a sharp difference that prevents complete solvability and gives rise to very interesting features not present in the mentioned models. Note that the fieldsÛ jk are highly interacting since on a triangle three fields go to vacuum in the sense thatÛ jkÛklÛlj = −1. This is resemblance of vertex interaction in high energy fermions that we have seen in Sect.V with the bosonic mapping. However, this latter relation doesn't coincide with the symmetry of the model as they do not commute with each other. This will be considered next.
Thus far, we have considered fields that are present in the Hamiltonian. In what follows, we introduce another set of fields which have the Z 2 × Z 2 symmetry commuting with each other and with the Hamiltonian. To this end, consider a plaquette f . As before, by a plaquette we mean an outer and an inner hexagon with six triangles between them. Let V f and V h stand for sets of vertices of plaquette and inner hexagon, respectively. It is natural that V h ⊂ V f . To each plaquette we attach the three following fields: where by f \h we simply mean V f − V h . Each φ f squares identity. They commute with each other and with Hamiltonian andF I : Also, the fields φ f are responsible of the Z 2 × Z 2 gauge symmetry since φ 1 The above fields are related to the plaquette operators.Using the transformations we have introduced in (73), we can fermionize the conserved plaquette operators obtained in Fig.5.
They are associated to the above constants of motion as follows Although the above gauge fields make it possible to divide the Hilbert space into sectors in which be eigenspaces of gauge fields (or eigenspaces of plaquette operators), they do not allow us to reduce the Hamiltonian in (76) into a quadratic form. TheF I 's can be fixed as they commute with the Hamiltonian. But, we are not able to reduce the four-body interaction terms in the Hamiltonian into quadratic form. In fact, the anticommutation of U jk 's on a blue triangle prevents them to be fixed consistently with gauge fixing.
We have shown that this new model exhibits enough novel interesting and relevant properties so as to justify further research. Some of these possible lines of study are as follows: We have only studied a particular phase of the system, although we are able to study non-perturbative effects as well. The fact that all phases show a topological degeneracy anticipates a rich phase diagram. In this regard, one may explicitly break the color symmetry that the model exhibits and still keep the features that we have discussed.
It would be particularly interesting to check whether any of the phases displays non-abelian anyons. The model has many integrals of motion, although not enough to make it exactly solvable. This becomes another appealing feature of the model since other methods of study, like numerical simulations and experimental realizations will help to give a complete understanding of all its phases. operators. Thus we have: Finally, a graph state can be related to a color code within a set of measurements in the Z basis on all qubits corresponding to the set U 2 .
We suppose there is a two dimensional lattice of physical qubits that is governed by a 2-body Hamiltonian. Physical qubits of the lattice are projected to logical qubits. The point is that this projection is achieved by going to some order in perturbation theory. We think of vertices of the graph in Fig.16(a) as logical qubits. The lattice with 2-body interaction is shown in Fig.16(b), where the number of physical qubits corresponding to the vertices of the graph equals the number of links crossing the vertex. The new resulting lattice consists of triangles and hexagons and physical qubits live on their vertices. Triangles and hexagons are in one to one correspondence with the white and black vertices (U 1 and U 2 ) of the graph in Fig.16(b), respectively. Note that each triangle is linked with three neighboring hexagons and each hexagon is linked with six neighboring triangles.
The low lying spectrum of a well constructed 2-body Hamiltonian defined on the lattice composed of hexagons and triangles may describe a cluster state. To this end, we need for the following projection from the physical qubits to the logical ones: where | ⇑ L and | ⇓ L with L = v, f stand for the two states of the logical qubits obtained within the above projections, or alternatively they are states of logical qubits of the graph in Fig.16(a). We set the following Hamiltonian: where H 0 is the unperturbed Hamiltonian which can be treated exactly and λ is a small quantity which allow us to treat the term λV perturbatively. We refer to each triangle (hexagon) and its vertices by a site index v (f ) and indices i, j, respectively. The unperturbed part of the Hamiltonian included in H 0 is as follows.
where the first sum runs over all triangles and hexagons (sites) and <i, j > stand for the nearest-neighbor qubits around the corresponding triangle or hexagon connected by the solid lines as in the Fig.16(b). The interaction between qubits of triangles and qubits of hexagons are included in V : where <vi, f j > denotes two neighboring qubits which are connected by the dashed line.
The unperturbed Hamiltonian H 0 can easily be diagonalized for triangles and hexagons.
Ground state vectors of a triangle or hexagon form a two dimensional space spanned by the following polarized eigenvectors: Also the action of a σ z on one qubit of a triangle or hexagon is equivalent to an Z operator acting on the related logical qubit, since Now we can go on in order to calculate the third order perturbation: where the product runs over three black vertices linked to the v and δ = 3 8 . The operator K v is a stabilizer for the logical qubits which are projected down from the triangles. Since K 2 v = 1, the ground states correspond to the values of k v = +1. We skip the forth and fifth order of perturbation because they have trivial effects.
Like in 3 rd order perturbation, we are faced with a nontrivial term in the 6 th order perturbation theory. We will see that by considering this order, the ground state degeneracy is lifted completely. This nontrivial effect arises from the product of terms in the perturbation λV corresponding to the links around a hexagon. Finally, for the 6 th order perturbation we have: and the product runs over six white vertices linked to the hexagon f . The coefficient γ has positive sign and its precise value is unimportant. We would like to emphasize that at six order in perturbation theory some other terms appear which are product of two distinct K v .
However, we skip them as they all commute. Equations (A13) and (A15) provides all we need to adopt the cluster state in (A3) as ground state of the low energy effective theory of Hamiltonian in (A5), which up to six order of perturbation can be written as follows We see that the above effective Hamiltonian is completely different from that of in (34). The latter equation gives rise directly to the topological color code as its ground state, but the ground state (cluster state) of former one needs further local measurements to encode the desired color code.