Alexis Schotte

Jose Carrasco

Bram Vanhecke

Laurens Vanderstraeten

Jutho Haegeman

Frank Verstraete

Julien Vidal

Tensor-network approach to phase transitions in string-net models

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Since its discovery in the late 1980s, topological order aroused much interest in physics. The long-range entanglement structure as well as the exotic quasiparticle excitations associated with this order may prove essential in attempts to achieve scalable fault-tolerant quantum computers or quantum memories [START_REF] Nayak | Non-Abelian Anyons and Topological Quantum Computation[END_REF] . As such, it is of paramount importance to understand how perturbations generate dynamics and interactions between the anyonic excitations and induce a breakdown of topological phases.

One of the most famous models hosting topological order was proposed by Levin and Wen in 2005 [START_REF] Levin | String-net condensation: A physical mechanism for topological phases[END_REF] . The stringnet Hamiltonian allows to describe all doubled achiral topological phases. Thus, it has been the starting point for many studies concerning phase transitions [START_REF] Gils | Topology-driven quantum phase transitions in time-reversal-invariant anyonic quantum liquids[END_REF][START_REF] Gils | Ashkin-Teller universality in a quantum double model of Ising anyons[END_REF][START_REF] Ardonne | Microscopic models of interacting Yang-Lee anyons[END_REF][START_REF] Burnell | Condensation of achiral simple currents in topological lattice models: Hamiltonian study of topological symmetry breaking[END_REF][START_REF] Schulz | Topological Phase Transitions in the Golden String-Net Model[END_REF][START_REF] Morampudi | Numerical study of a transition between Z2 topologicallyordered phases[END_REF][START_REF] Schulz | Ising anyons with a string tension[END_REF][START_REF] Schulz | Russian doll spectrum in a non-Abelian string-net ladder[END_REF][START_REF] Schulz | Frustrated topological symmetry breaking: geometrical frustration and anyon condensation[END_REF][START_REF] Schulz | Bound states in string nets[END_REF][START_REF] Burnell | Anyon Condensation and Its Applications[END_REF][START_REF] Vidal | Ising versus SU (2)2 string-net ladder[END_REF] . Nevertheless, in the absence of local order parameter, the nature of these transitions remains an open question in many cases since one cannot use Landau's theory of symmetry breaking.

Among all alternative methods developed to study these topological phase transitions, a particularly versatile framework for constructing variational states is provided by tensor networks. In two dimensions, the projected entangled-pair states (PEPSs) [START_REF] Verstraete | Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions[END_REF] are known to describe the string-net ground states [START_REF] Gu | Tensorproduct representations for string-net condensed states[END_REF][START_REF] Chen | Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order[END_REF] and directly encode the topological properties in the virtual symmetries of the local PEPS tensor [START_REF] Schuch | PEPS as ground states: Degeneracy and topology[END_REF][START_REF] Williamson | Characterizing Topological Order with Matrix Product Operators[END_REF][START_REF] Bultinck | Anyons and matrix product operator algebras[END_REF] . This feature has been exploited to detect possible topological phase transitions and to identify the associated anyon-condensation mechanism [START_REF] Burnell | Anyon Condensation and Its Applications[END_REF] in Abelian [START_REF] Haegeman | Shadows of anyons and the entanglement structure of topological phases[END_REF][START_REF] Haegeman | Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems[END_REF][START_REF] Shukla | Boson condensation and instability in the tensor network representation of string-net states[END_REF][START_REF] Zhu | Gapless coulomb state emerging from a self-dual topological tensor-network state[END_REF] and non-Abelian [START_REF] Mariën | Condensation-driven phase transitions in perturbed string nets[END_REF][START_REF] Xu | Tricritical point with fractional supersymmetry from a Fibonacci topological state[END_REF] cases at the level of wave functions. However, variational PEPS calculations for concrete models have so far been restricted to Z N toric codes [START_REF] Gu | Tensorentanglement renormalization group approach as a unified method for symmetry breaking and topological phase transitions[END_REF][START_REF] Dusuel | Robustness of a Perturbed Topological Phase[END_REF][START_REF] Schulz | Breakdown of a perturbed ZN topological phase[END_REF] whose excitations are Abelian anyons. Recently, a family of PEPS based on perturbative expansions has been introduced to describe different ground states across a phase transition [START_REF] Vanderstraeten | Bridging Perturbative Expansions with Tensor Networks[END_REF] .

For a given Hamiltonian that exhibits a phase transi-tion, the procedure to build these "perturbative PEPSs" can be summarized as follows: (i) we start from a wave function that describes the phase transition at the meanfield level; (ii) we apply tensor-network operators which implement the perturbative expansions in an extensive way to wave functions on both sides of the transition; (iii) we promote the ad hoc coefficients of these expansions to variational parameters. In two dimensions, these perturbatively exact variational states are still PEPSs and the tensor-network machinery [START_REF] Fishman | Faster methods for contracting infinite two-dimensional tensor networks[END_REF][START_REF] Vanderstraeten | Gradient methods for variational optimization of projected entangled-pair states[END_REF] can be used to perform an efficient variational optimization. In Ref. 30, this method has been notably applied to the Z 2 toric code perturbed with a string tension [START_REF] Trebst | Breakdown of a Topological Phase: Quantum Phase Transition in a Loop Gas Model with Tension[END_REF][START_REF] Hamma | Adiabatic Preparation of Topological Order[END_REF] for which the virtual symmetry of the local PEPS tensor emerges as an order parameter.

In this work, we go one step beyond and implement a variational PEPS to study phase transitions in both Abelian (Z 2 ) and non-Abelian (Fibonacci) string-net models on the the honeycomb lattice. For the Z 2 case, we capture the second-order quantum phase transition known from the mapping onto the transverse-field Ising model on the triangular lattice [START_REF] Burnell | Condensation of achiral simple currents in topological lattice models: Hamiltonian study of topological symmetry breaking[END_REF] . In the Fibonacci case, we only find first-order phase transitions, in contrast to Ref. 7 but in agreement with the mean-field results [START_REF] Dusuel | Mean-field ansatz for topological phases with string tension[END_REF] .

II. STRING-NET MODELS

We consider the two-dimensional string-net model introduced by Levin and Wen 2 in the presence of a tension term. For simplicity, we focus on the simplest case where the microscopic degrees of freedom, defined on the links of a honeycomb lattice, can only be in two different states, 0 and 1 (when possible, we omit the ket notation to describe states). The Hilbert space H is defined as the set of configurations obeying the branching rules that stem from the fusion rules of the theory considered 2 .

In the present work, we discuss two different theories, Z 2 and Fibonacci, whose fusion rules are given by:

Z 2 : 0 × a = a × 0 = a, 1 × 1 = 0, (1) Fibonacci : 0 × a = a × 0 = a, 1 × 1 = 0 + 1, (2)
for a = 0, 1. As underlined in Ref. 2, there are actually two different theories obeying Z 2 fusion rules that give rise to either a doubled Z 2 (DZ 2 ) or a doubled semion (Dsem) topological phase. For the string tension considered thereafter, phase diagrams are the same for both theories.

At each vertex of the honeycomb lattice, the fusion rules must be satisfied 2 , i.e., if two links are in states a and b the third link must be in a state c ∈ a×b. Following Ref. 36, one can compute the dimension of the Hilbert space. For any trivalent graph with N v vertices, one then gets

Z 2 : dim H = 2 Nv 2 +1 , (3)
Fibonacci : dim H = (1 + ϕ 2 ) Nv 2 + (1 + ϕ -2 ) Nv 2 , ( 4
)
where

ϕ = 1+ √ 5 2
is the golden ratio. In order to analyze the breakdown of the topological phase originating from the string-net model, we consider the following Hamiltonian:

H = -J p p B p -J l l L l . (5)
The first term corresponds to the usual string-net Hamiltonian introduced by Levin for details]. The operator B p only modifies the six inner links of the plaquette p but its action depends (diagonally) on the six outer links 2 . For J p > 0 and J l = 0, all ground states are flux free and hence characterized by B p = 1 for all p, up to a topology-dependent degeneracy.

The second term is also a sum of mutually commuting projectors. Operators L l 's are diagonal in the canonical (link) basis and act as L l |a l = δ a,0 |a l , where |a l denotes the state of the link l. For J l > 0, this second term favors configurations with links in the state 0 and penalizes strings of links in the state 1, hence the name string tension.

For J l > 0 and J p = 0, the ground state is unique (trivial phase) and given by the product state |0 = ⊗ l |0 l for both Z 2 and Fibonacci fusion rules. For J l < 0 and J p = 0, the ground-state manifold depends on the fusion rules. Indeed, for Fibonacci fusion rules, the product state |1 = ⊗ l |1 l is allowed and is the unique ground state. By contrast, for Z 2 fusion rules, this state is not allowed (since 1×1 = 0) and the ground space is spanned by all allowed states with N v links in the state 1 and [START_REF] Nayak | Non-Abelian Anyons and Topological Quantum Computation[END_REF] 2 N v links in the state 0.

III. METHODOLOGY

The goal of this work is to analyze phase transitions from the topological phase existing for J p > 0 in the small |J l /J p | limit to the trivial phases found in the large |J l /J p | limit. To this aim, let us set J p = cos θ and J l = sin θ and consider first the region where θ ∈ [0, π/2]. Following the variational tensor-network approach introduced in Ref. 30, we consider the state

|α, β = N exp β l L l p (1 + αZ p )|0 , (6)
where Z p = 2B p -1, α and β are variational parameters, and N is a normalization factor. According to Ref. 30, a better description of the trivial phase would be obtained by adding an extra term exp(-γ p B p ). However, it considerably increases the complexity of the PEPS so that we do not consider it in the following. For β = 0, the state |α, 0 describes the phase transition at the mean-field level [START_REF] Dusuel | Mean-field ansatz for topological phases with string tension[END_REF] and the states |1, 0 and |0, 0 are the exact ground states for θ = 0 and θ = π/2, respectively. Furthermore, the first-order contribution in α to |α, β around (α, β) = (0, 0) corresponds to the firstorder perturbative correction to the exact ground state around θ = π/2. Likewise, the first-order contribution in β to |α, β around (α, β) = (1, 0) corresponds to the firstorder perturbative correction to the exact ground state around θ = 0 [START_REF] Vanderstraeten | Bridging Perturbative Expansions with Tensor Networks[END_REF] .

The state |α, β can be interpreted as a PEPS, whose bond dimension depends on the theory considered. The variational energy per plaquette

e 0 (α, β) = 1 N p α, β|H|α, β α, β|α, β , (7)
can be efficiently computed using the VUMPS algorithm [START_REF] Fishman | Faster methods for contracting infinite two-dimensional tensor networks[END_REF] for contracting two-dimensional tensor networks in the thermodynamic limit. Note that the previous approach reproduces the linear perturbative corrections up to second order both near θ = 0 and θ = π/2. As explained in Ref. 30, additional tensor-network operators can be added in order to reproduce higher-order perturbative corrections.

The PEPS framework allows for a natural characterization of the topological nature of the variational ground state. Indeed, the topological properties of a PEPS are related to the virtual symmetries of the local PEPS tensor [START_REF] Schuch | PEPS as ground states: Degeneracy and topology[END_REF] . In the Fibonacci theory, this virtual symmetry is described by a matrix product operator (MPO) [START_REF] Williamson | Characterizing Topological Order with Matrix Product Operators[END_REF][START_REF] Bultinck | Anyons and matrix product operator algebras[END_REF] . Since, as shown in the Appendix, the state |α, β exhibits such a virtual MPO symmetry only when α = 1, this parameter can be naturally interpreted as an order parameter [START_REF] Vanderstraeten | Bridging Perturbative Expansions with Tensor Networks[END_REF] to detect the transition between the topological phase (α = 1) and the trivial one (α < 1). Indeed, at α = 1, the expectation value of a Wilson loop operator changes from a trivial perimeter law for β = 0 to a non-trivial perimeter law for β > 0, still indicating deconfinement of the anyonic excitations, so that the state remains in the topological phase. For α < 1, the Wilson loop expectation value satisfies a non-trivial area law, indicating that anyons are confined and the state is in the trivial phase.

Yet, even in the presence of the virtual symmetry (α = 1), the parameter β can drive the state into a trivial phase by a spontaneous breaking of this symmetry, resulting in an area law for the Wilson loop. This process was shown to occur at β = [START_REF] Nayak | Non-Abelian Anyons and Topological Quantum Computation[END_REF] 2 ln(1 + √ 2) for the Z 2 case 37 ) and at β 0.168 for the Fibonacci case [START_REF] Mariën | Condensation-driven phase transitions in perturbed string nets[END_REF] . For the problem at hand, we checked that there is no spontaneous symmetry breaking so that α can be used as a bona-fide order parameter.

IV. RESULTS FOR THE Z2 THEORY

Let us first discuss the simplest theory and consider Z 2 (Abelian) fusion rules. As discussed in Ref. 6, the Hamiltonian (5) for any theory obeying Z N fusion rules can be exactly mapped onto the N -states Potts model in a magnetic field on the dual lattice. Thus, for N = 2, H is equivalent to the transverse-field Ising model on a triangular lattice where J p and J l are the strength of the magnetic field and of the spin-spin coupling, respectively. As a result, the sign of J p is irrelevant for this problem and we assume J p > 0 in the following.

In the antiferromagnetic case (J l < 0), the Ising model on a triangular lattice is highly frustrated. So, clearly, the ansatz |α, β is not adapted to that situation since the ground space has an extensive degeneracy for J p = 0. In this region θ ∈ [-π/2, 0], a critical point in the universality class of the three-dimensional classical XY model was found for θ arctan(-1/65) -0.545 [START_REF] Isakov | Interplay of quantum and thermal fluctuations in a frustrated magnet[END_REF] .

Here, we rather aim at benchmarking our ansatz with the phase transition in the region θ ∈ [0, π/2] corresponding to ferromagnetic interactions (J l > 0). The phase diagram in this region has been studied by high-order series expansion and a second-order transition occurs at θ c arctan(0.2097) 0.207 [START_REF] He | High-temperature series expansion for the (2+1)-dimensional Ising model[END_REF] , the critical point belonging to the universality class of the three-dimensional classical Ising model. Our results obtained from the variational ansatz (6) are summarized in Fig. 1 (top panel). As already discussed in Ref. 35, for β = 0 (mean-field approximation), one obtains a continuous transition at θ = arctan(1/6) 0.165 which is qualitatively correct but about 20% off from θ c . Remarkably, by adding β as a second variational parameter, the transition remains continuous (no jump of the order parameter α) and shifts to θ 0.198 which is only 4% off from θ c . Given that our variational ansatz has only short-ranged correlations [except for α = 1 and β = 1 2 ln(1 + √ 2), which is not a variational optimum for any value of θ], and that the exact correlations decay algebraically near θ c , these results can be considered as unexpectedly good. This can also be seen by comparing the variational ground-state energy with the numerical results obtained from exact diagonalization on a 25-plaquettes system with periodic boundary conditions, as shown in Fig. 1 (bottom panel). Note, finally, that for the Z 2 theory, our ansatz satisfies |α, β ∼ |α -1 , β , such that the expansion of the energy density e 0 (α, β) around α = 1 + δα can only contain even terms in δα. As a consequence, the way α deviates from 1 around the transition point is as |δα| = (θ -θ c ) 1/2 , both for the mean-field ansatz (with fixed β = 0) and for the ansatz where β is also optimized.

⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯

V. RESULTS FOR THE FIBONACCI THEORY

The phase diagram of the Hamiltonian (5) for the Fibonacci theory has already been computed by combining exact diagonalization results with high-order series expansions for the ground-state and gap energies [START_REF] Schulz | Topological Phase Transitions in the Golden String-Net Model[END_REF] . The doubled Fibonacci (DFib) topological phase has been found to extend from θ 2 -0.63 to θ 1 0.255 identifying θ 1 and θ 2 as critical points. Our variational results in this case are displayed in Figs. 2 and3 for θ ∈ [0, π/2] and θ ∈ [0, -π/2], respectively.

⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯
Order parameter α In the region θ ∈ [0, π/2], the mean-field approach 35 corresponding to β = 0 indicates a first-order transition for θ = arctan 1+ϕ 6+3ϕ 0.237. This result is qualitatively different from the one proposed in Ref. 7 although the position of the transition point is only 7% off from θ 1 . Since (i) there is no prior reason to believe that the meanfield result is exact 35 and (ii) higher-order series expansions need to be extrapolated to provide reliable informations about the nature of the transition, it is interesting to see what the ansatz (6) can bring to the understanding of the transition. As can be seen in Fig. 2, by adding β as variational parameter, one still obtains a first-order transition characterized by a jump of the order parameter, but the transition point is shifted to θ 0.254, which is less than 1% off from θ 1 . This leads us to conclude that in the region θ ∈ [0, π/2], there is a unique transition point located near θ 0.255 (in agreement with Ref. 7) corresponding to a first-order transition with a small gap at the transition point (weakly first-order). Note that the proximity of the transition points obtained by the two approaches could suggest that the flux-flux correlation length is finite, which is a favorable case for a PEPS description of the ground state and justifies the relevance of our ansatz.

In the region θ ∈ [-π/2, 0], we investigate the phase transition by slightly modifying the ansatz. Indeed, the state defined in Eq. ( 6) is designed for interpolating between |1, 0 and |0, 0 , that are the exact ground states at θ = 0 and θ = π/2, respectively. However, for θ = -π/2, the ground state of the Hamiltonian ( 5) is unique (topologically trivial phase) and given by |1 = ⊗ l |1 l . Consequently, to study the parameter range θ ∈ [-π/2, 0], we consider as variational ansatz

⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯
|α, β -= N exp β l L l p (1 + αZ p )|1 , (8)
where, for simplicity, we kept the same notations as in Eq. ( 6). The PEPS tensor encoding this state is described in the Appendix. It is important to note that the "reference" states |0 and |1 are very different. Indeed, for β = 0, a key property of the ansatz ( 6) is the factorization property that reads

α, 0| n p=1 B p |α, 0 = α, 0|B p |α, 0 n , (9)
for any set of n plaquettes. This identity does not hold for the state (8) with β = 0, which can no longer be interpreted as a mean-field ansatz. Yet, by construction, it is perturbatively exact near θ = 0 and it also matches the exact ground state at θ = -π/2. As such, this ansatz is a good candidate to capture the transition between the DFib phase and the trivial phase. As can be seen in Fig. 3, for β = 0, one obtains a continuous transition at θ -0.317 which is very far from the value θ 2 -0.63 obtained in Ref. 7. Interestingly, when including β, we obtain a discontinuous transition located at θ -0.630 which is in agreement with the extrapolated values of Ref. 7. Thus, we face a situation similar to the previous case, where the present variational study is in quantitative agreement with the series expansions studies. We emphasize that the series expansions in this region have to be resummed so that error bars on θ 2 are larger than for θ 1 . Regarding the nature of the transition, the same arguments as before favor the first-order scenario.

VI. CONCLUSIONS AND OUTLOOK

This work presents the first variational results based on tensor networks for a topological phase transition out of a non-Abelian topological phase. We have studied the transitions between the topological and trivial phases of the Levin-Wen Hamiltonian with string tension, both for Z 2 and Fibonacci fusion rules, by means of a simple twoparameters variational ansatz inspired by Ref. 30. For the Z 2 case, we recover the well-known second-order transition as predicted from the mapping onto the transversefield Ising model the triangular lattice. For the Fibonacci case, our results are in quantitative agreement with series expansions and exact diagonalizations 7 . However, we only find first-order transitions (as in the mean-field treatment [START_REF] Dusuel | Mean-field ansatz for topological phases with string tension[END_REF] ) whereas series expansions combined with exact diagonalizations rather plead in favor of secondorder transitions [START_REF] Schulz | Topological Phase Transitions in the Golden String-Net Model[END_REF] . This qualitative discrepancy is likely due to the extrapolation of the series expansion and finite-size effects in the exact diagonalizations, but we cannot exclude that the present variational approach is not sufficient to properly describe the transitions in this model. Going beyond would require more sophisticated ansätze that can be systematically constructed by using the ideas developed in Ref. 30. Using recently developed contraction methods for three-dimensional tensor networks [START_REF] Vanderstraeten | Residual entropies for three-dimensional frustrated spin systems with tensor networks[END_REF] , we stress that such an approach can also be applied in three-dimensional systems, as recently illustrated in Ref. 41.

where |0 = ⊗ l |0 l , and N is a normalization factor. The operator O p i corresponds to inserting a i-loop inside the plaquette [START_REF] Levin | String-net condensation: A physical mechanism for topological phases[END_REF] , and then resolving it into the lattice using F -moves .

(A8)

The gray lines above are initially in the |0 state. To find the state from this graphical notation, one has to resolve the loops appearing in Eq. (A8) into the lattice. This is done in two steps. First we fuse the neighboring loops along each edge, using an F-move:

= k F µµ 0 νν k k = k d k d µ d ν δ µνk k . ( A9
)
The k appearing in the sum, will be the physical degree of freedom in every edge, once we are done resolving everything into the lattice. The second step consists of using the following equality in every vertex:

i j k = d λ d µ d ν G ijk λµν j k , ( A10
)
where

G ijk λµν = 1 √ d i d λ F νjλ kµi = 1 √ d ν d k F ijk λµν * . (A11)
The PEPS tensor is obtained by splitting the factor d k /d µ d ν in Eq. (A9) evenly between the two adjacent vertices, while splitting the factor dµp in Eq. (A7) evenly between all 6 vertices appearing in plaquette p. The result is

i j k i j k = 6 dλ dµ dν 4 d i d j d k G ijk λµν δ ii δ jj δ kk δ λλ δ µµ δ νν , (A12)
where i , j and k represent the physical degrees of freedom. Each virtual index is associated to two physical indices (appearing in the tensors at the vertices connected by that edge). The first three δ-functions in Eq. (A12) guarantee that these two physical indices are always equal. The PEPS tensor for the vertices with the inverse orientation is obtained by rotating Eq. (A12).

Ansatz for

θ ∈ [-π/2, 0]
For θ ∈ [-π/2, 0], the ansatz is defined similarly: where the gray lines are now in the |1 state initially. As done for θ ∈ [0, π/2], we first fuse the loops along every edge: To finish resolving everything into the lattice, these objects must be reduced to trivalent vertices. This is done by applying Eq. (A10) multiple times:

|γ -= N p (1 + γd 1 O p 1 )|1 , (A13) = N p i di O p i |1 . ( A14
= x F 11 0 µµ x x , (A16) = x k F 11 0 µµ x F xx0 ννk x x k , = x k d k d 1 d µ v ν δ µ1x δ xνk x x k
i j k x y z = d 3 1 d λ d µ d ν d x d y d z G xy1 11ν G xjz λ1y G ijk zµx j k , (A17)
and analogously:

i j k x y z = d 3 1 d λ d µ d ν d x d y d z G zx1 11µ G kxy 1λz G ijk yxν j k . ( A18
)
By splitting the factors appearing in Eq. (A16) equally between the two adjacent vertices, we obtain the following PEPS tensors:

i j k x y z k j i = 6 dλ dµ dν 4 d i d j d k d x d y d z d 3/4 1 G xy1 11ν G xjz λ1y G ijk zµx δ ii δ jj δ kk δ λλ δ µµ δ νν , ( A19
) i j k k x y z j i = 6 dλ dµ dν 4 d i d j d k d x d y d z d 3/4 1 G zx1 11µ G kxy 1λz G ijk yxν δ ii δ jj δ kk δ λλ δ µµ δ νν . (A20)
Note that there is one more virtual leg per side compared to Eq. (A12). This is due to the extra sum appearing in Eq. (A16). Although, for γ = 1, the physical states given in Eqs. (A3) and (A13) are identical, the PEPS tensors representing these two states have very different properties. Indeed, the double-layer transfer matrix with PEPS tensors (A19) and (A20) appears to be critical with a central charge which is twice that of the three-state Potts model while the PEPS tensor given by Eq. (A12) does not share this property. This observation clearly deserves further investigations.

Reducing the bond dimensions

The PEPS tensor defined in Eq. (A12) has a bond dimension (both physical and virtual) of 2 [START_REF] Gils | Topology-driven quantum phase transitions in time-reversal-invariant anyonic quantum liquids[END_REF] . However, the G-symbol present in its definition, imposes certain rules which need to be met for the tensor to take a non-zero value. These rules can be exploited to rewrite this tensor as one with a lower bond dimensions: for the Z 2 theory (1), the bond dimension can be reduced to 4, while for the Fibonacci theory (2) it can be reduced to 5.

The structure of the PEPS tensor, can also be exploited to reduce the bond dimension of the double-layer transfer matrix MPO tensors. Using the more efficient encoding of the tensor we just mentioned, the double layer bond dimension already gets reduced from 64 to 16 for Z 2 and to 25 for Fibonacci. The Kronecker delta functions appearing in the right-hand side of Eq. (A12) allow us to further reduce these to 8 and 13 respectively.

The same tricks can be applied to the tensors (A19) and (A20) (note that we only use this ansatz for the Fibonacci theory). The virtual bond dimension can be reduced from 2 4 to 8, while the physical bond dimension can be reduced from 2 3 to 5. The bond dimension of double-layer MPO tensors can be reduced from 2 8 , to 34.

As mentioned in the main body of this paper, the variational energy per plaquette (7) is calculated using the VUMPS algorithm. Due to memory constrains, the bond dimension of the boundary MPS has to be limited to 100 for the Z 2 model, and for the Fibonacci model with θ ∈ [0, π/2]. Due to the higher bond dimension of the double-layer MPO obtained from ansatz (A19) and (A20), the bond dimension of the boundary MPS has to be limited to 60 for the Fibonacci model with θ ∈ [-π/2, 0].

4 FIG. 1 .

 41 FIG.1. Variational results for the Z2 theory obtained for β = 0 (red) and β = 0 (green). Top: order parameter α indicating a continuous transition from a topological phase (α = 1) to a trivial phase (α < 1). Bottom: ground-state energy per plaquette e0 compared with exact diagonalization data (blue crosses) (see inset for a broader range). Dashed lines give the position of the transition point obtained from series expansions[START_REF] He | High-temperature series expansion for the (2+1)-dimensional Ising model[END_REF] (blue) and from the order parameter behavior [red[START_REF] Dusuel | Mean-field ansatz for topological phases with string tension[END_REF] and green (this work)].

4 FIG. 2 .

 42 FIG.2. Variational results for the Fibonacci theory (same conventions as in Fig.1). Blue dashed lines indicate the position of the transition point computed from series expansions 7 .

FIG. 3 .

 3 FIG.3. Variational results for the Fibonacci theory (same conventions as in Fig.1). Blue dashed lines indicate the position of the transition point computed from series expansions[START_REF] Schulz | Topological Phase Transitions in the Golden String-Net Model[END_REF] which coincides with the one obtained with the ansatz (8), i.e., θ -0.630.

  loop can not happen across a plaquette, we treat plaquettes as if they have a puncture in their center. Applying these rules gives the full form of O p i : 1 and d1 = γ d 1 one can write |γ = N representation of the operators O p i , |γ can be represented as |γ = N µ1,µ2,... dµ1 dµ2 . . .

)

  Note that we are now acting on the |1 = ⊗ l |1 l product state, as opposed to the |0 state that we used for θ ∈ [0, π/2]. We can use the same graphical representation of this state: |γ -= N µ1,µ2,... dµ1 dµ2 . . .

.

ACKNOWLEDGMENTS

JC would like to thank the Laboratoire de Physique Théorique de la Matière Condensée of the Sorbonne Université and the Department of Physics and Astronomy of the Ghent University for their warm hospitality. This research was supported by the Research Foundation Flanders, and ERC Grants No. QUTE (647905), No. ERQUAF (715861) and QTFLAG.

Appendix A: Construction of the PEPS tensor

In the following section, we elaborate on the construction of the PEPS tensors representing the ansatz |α, β + = |α, β and |α, β -. It is sufficient to construct the PEPS for the one-parameter ansätze |α, 0 ± . The PEPS tensor for |α, β ± ∝ l exp (βL l ) |α, 0 ± is then readily found by applying the right operator on the physical level.

For simplicity, we will assume that quantum dimensions d 0 and d 1 are non-negative real numbers, and that the F -symbols are all real. Both models studied in this work satisfy these assumptions (see Ref. 2 for details about Z 2 and Fibonacci theories). In the general case, where one relaxes these assumptions, the construction of the PEPS can be done in a similar fashion.

In order to simplify the calculations in the following section, we will work with the states

To go back to the ansatz used in the main body of the paper, one can simply use

where D = d 2 0 + d 2 1 is total quantum dimension. The one-parameter ansatz is given by


Mohd Yusoff

Bin Omar

Mohd Fairuz Shiratuddin
email: f.shiratuddin@murdoch.edu.au

Dayang Rohaya
email: dayangrohaya.ar@utp.edu.my

Awang Rambli

Frédéric Merienne
email: frederic.merienne@ensam.eu

Suziah Sulaiman
email: suziahs@utp.edu.my

Delphine Vanhalle
email: delphine.vanhalle@yahoo.fr

✁✂✄✂☎✆✝ ✞✂✟✠✆✄✡☛☞✌ ✍✆☎✎✏✟ ✑✂✝✆✌ ✂✒ ✆✁☎☛☞☎✎✓✆ ✔☞✕✆ ✒✂✁ ✖✄✄✆✁ ✌✎✕✗ ✒✂✁ ✘✡✁✂✙✆ ✚✆✛☞✗✎✌✎✡☞✡✎✂✟

Keywords: ✬ Stroke Rehabilitation Game, Kinect Game, Conceptual Model, Game architecture

Abstract✜

The gamification of stroke rehabilitation may ✢ ✣✤✥✦✧★✦ ✩✧✪✢ ✦✣✪✫★ motivation and engagement towards their rehabilitation activity and hence contributes to faster recovery. Apart from the interactive experience that games offer, it also may be used as a persuasive tool. In this view, games may be used to persuade stroke patients in shaping a good behavior towards their rehabilitation activity. This paper reports our proposed conceptual design model of building persuasive game for stroke patients that follows the Persuasive System Design Model. We incorporate the persuasive software features that includes reduction, tailoring, self-monitoring, and rewards. The proposed game architecture is also discussed in this paper.

I. INTRODUCTION

Game based stroke rehabilitation has gained a lot of attention lately [START_REF] Laver | Virtual reality for stroke rehabilitation[END_REF][START_REF] Hung | What Do Stroke Patients Look for in Game-Based Rehabilitation: A Survey Study[END_REF][START_REF] Hondori | A Review on Technical and Clinical Impact of Microsoft Kinect on Physical Therapy and Rehabilitation[END_REF]. This is because the gamification of stroke rehabilitation may in✭✮✯✰✱✯ ✲✰✳✴✯✵✳✶✱ ✯✵✷✰✷✯✸✯✵✳✱ ✰✵✹ affect their result positively [START_REF] Laver | Virtual reality for stroke rehabilitation[END_REF]. In addition, the wide availability of low cost game technology employing motion sensors such as Kinect camera (or commonly known as exergame) and their adequate accuracy contribute to the development and proliferation of such study [START_REF] Hondori | A Review on Technical and Clinical Impact of Microsoft Kinect on Physical Therapy and Rehabilitation[END_REF].

Apart from entertaining the players, game also can be used as persuasive technology through its rich interaction that includes multimedia elements (eg: video, audio, graphics) and interactive experiences offered in the gameplay [START_REF] Fogg | Persuasive technology: using computers to change what we think and do[END_REF]. With the provision of fun experience, game can also can persuade stroke patients in shaping a good behavior and attitude towards their rehabilitation therapy. This is necessary because many stroke patients were reported as not motivated to continue with their rehabilitation at home and being highly dependent on rehabilitation services from therapists and hospitals [START_REF] Hung | What Do Stroke Patients Look for in Game-Based Rehabilitation: A Survey Study[END_REF]. Through the gamification of rehabilitation activity, patients may become more motivated and engaged in their rehabilitation therapy and may recover faster consequently. This may bring future benefits because the global burden of stroke is increasing due to the aging populations and the worrying trend of stroke that start attacking young adults [START_REF] Hankey | Stroke[END_REF].

To the best of our knowledge, there is limited literature about designing persuasive game for stroke patients. Therefore, we develop the conceptual persuasive game design for stroke rehabilitation following the Persuasive System Design (PSD) Model from our previous work [START_REF] Omar | Designing Persuasive Stroke Rehabilitation Game: An Analysis of Persuasion Context[END_REF]. This paper describes the persuasive features implementation, the game architecture and its modules.

II. RELATED WORK

There have been many studies about incorporating game technology in stroke rehabilitation. The principle of this study are to provide patients with interactive user interface and to implement different task scenarios. Such studies can be divided under customized development category and adoptions of commercial games category [START_REF] Hung | What Do Stroke Patients Look for in Game-Based Rehabilitation: A Survey Study[END_REF].

Motion Rehab AVE 3D is an example of customized game that requires patients to follow an avatar performing ball juggling with their hands and legs by the beach [START_REF] Trombetta | Motion Rehab AVE 3D: A VR-based exergame for post-stroke rehabilitation[END_REF]. An assessment of a cooking game, Kinelabs, concluded that the game may contribute to the improvement chronic stroke patients [START_REF] Askin | Effects of Kinectbased virtual reality game training on upper extremity motor recovery in chronic stroke[END_REF]. The development of a sword fighting game which requires patients to wear VR headset and robotic exoskeleton arm is able to accurately map ✳✺✯ ✲✰✳✴✯✵✳ ✶✱ limb in 3 dimensional spaces and may support arm rehabilitation [START_REF] Tageldeen | A virtual reality based serious games for rehabilitation of arm[END_REF]. Some adoptions of commercial exergame for stroke rehabilitation can be found such as, Kinect Sports package [START_REF] Turkbey | Clinical feasibility of Xbox Kinect training for stroke rehabilitation: A single-blind randomized controlled pilot study[END_REF], Kinect Adventure [START_REF] Sin | Additional Virtual Reality Training Using Xbox Kinect in Stroke Survivors with Hemiplegia[END_REF] and Nintendo Wii [START_REF] Paquin | Survivors of chronic stroke✂participant evaluations of commercial gaming for rehabilitation[END_REF]. These studies showed that these commercial games adoptions were ✱✺✻✼✯✹✱✴✷✵✴✽✴✭✰✵✳✲✰✳ ✴✯✵✳✱✴✸✲✮✻✾✯✸✯✵✳ ✶✱✰✵✹appeared to be safe and feasible for rehabilitation. However, not many commercial games adoption are suitable for rehabilitation purposes due to the interaction that requires full range of motion and fast movements of a normal and healthy player and therefore may be too difficult for stroke patients [START_REF] Hung | What Do Stroke Patients Look for in Game-Based Rehabilitation: A Survey Study[END_REF][START_REF] Pickrell | Exploring inhospital rehabilitation exercises for stroke patients: informing interaction design[END_REF]. Some games also are reported as costly and difficult for therapists to setup [START_REF] Pickrell | Exploring inhospital rehabilitation exercises for stroke patients: informing interaction design[END_REF].

Many of game based stroke rehabilitation need further strategic and longitudinal study and also with more variety of games to gain more understandings. Aminov et al. [START_REF] Aminov | What do randomized controlled trials say about virtual rehabilitation in stroke? A systematic literature review and meta-analysis of upper-limb and cognitive outcomes[END_REF] has reviewed the randomized controlled trials and concluded that game applications can give significant improvement on ✲✰✳✴✯✵✳✶✱ ✻✿✳✭✻✸✯✱ ✰✵✹ ✯✵✭✻✿✮✰✷✯✹ ✸✻✮✯ games implementation research. Similarly, Bonnechère et. al [START_REF] Bonnechère | The use of commercial video games in rehabilitation: a systematic review[END_REF] suggested that games applications may prevent monotonous and boring rehabilitation and suggested that specific game for rehabilitation may produce better therapeutic outcome. In addition, Laver et al. [START_REF] Laver | Virtual reality for stroke rehabilitation[END_REF] suggested that games applications showed more beneficial when games are used for additional rehabilitation activity but not to replace the conventional rehabilitation activity.

However, the use of exergame for rehabilitation is seemed as underutilized. The ability to store vast amount of data from gameplay sessions such as which games have been played and how much the player moves can provide information for rehabilitation adherence study. In addition, a personalized and ✳✰✴ ✻✮✯✹ ✱✁✱✳✯✸✶✱ ✭✻✵✳✯✵✳ ✰✵✹ ✹✯ ✴✾✯✮✁ for rehabilitation ✲✿✮✲✻✱✯ ✰ ✱✻ ✱✿✷✷✯✱✳✯✹ ✂✯✭✰✿✱✯ ✴✵ ✳✺✴✱ ✭✻✵✳✯✄✳☎ ✳✺✯✮✯ ✴✱ ✵✻ ✆✻✵✯ ✱✴✝✯ ✽✴✳✱ ✰ ✶ ✰✲✲✮✻✰✭✺ ✹✿✯ ✳✻ ✳✺✯ ✾✰✮✴✯✳✁ ✻✽ ✳✺✯ ✲✰✳✴✯✵✳✱✶ ✵✯✯✹✱ and conditions [START_REF] Skjaeret | Exercise and rehabilitation delivered through exergames in older adults: An integrative review of technologies, safety and efficacy[END_REF]. To use game as a persuasive technology, the persuasion context analysis from Persuasive System Design (PSD) model should be ✴✹✯✵✳✴✽✴✯✹ ✳✻ ✲✮✯✾✯✵✳ ✆✰✹ ✺✻✭✶ ✰✵✹ ✆✂ ✰✭✞ ✂✻✄✶ ✱✁✱✳✯✸ ✹✯✾✯ ✻✲✸✯✵✳ approach [START_REF] Kelders | Health Behavior Change Support Systems as a research discipline; A viewpoint[END_REF]. An example of PSD Model application can be found in iLift [START_REF] Kuipers | iLift: A health behavior change support system for lifting and transfer techniques to prevent lower-back injuries in healthcare[END_REF]. ✟✺✴✱ ✠✴✵✯✭✳ ✂✰✱✯✹ ✷✰✸✯✶✱ ✱✳✿✹✁ ✲✿✮✲✻✱✯ is to train nursing personnel to practice a proper lifting and transfer technique in performing their daily ward tasks. The results showed that the participants exhibited a correct lifting technique as they progressed further in playing the game.

III. PERSUASIVE GAME DEVELOPMENT

As an exploratory study, our focus is developing a persuasive game by following the PSD model [START_REF] Oinas-Kukkonen | Persuasive Systems Design: Key Issues, Process Model, and System Features[END_REF]. The purpose of our game is to persuade stroke patients to comply to the rehabilitation exercise as advised by therapist. This can be done by providing game environment that offer rich interactions to the patients. From the second step of PSD model, we provide the persuasion context analysis which involve describing the persuasion intent, persuasion event and the persuasion strategy (see Table 1).

In following the third step of the PSD Model, we select and discuss selected four persuasive software features which are reduction, tailoring, self-monitoring and reward to be implemented in our proposed game.

From the fishing game scenario [START_REF] Omar | Designing Persuasive Stroke Rehabilitation Game: An Analysis of Persuasion Context[END_REF], we emulated the upper limb shoulder with straight arm movement that ranges from 90 to 120 degrees (see Fig. 1). This slow and careful with high repetition of shoulder and straight arm movement will ✯✄✯✮✭✴✱✯ ✳✺✯ ✱✺✻✿ ✹✯✮ ✸✿✱✭ ✯ ✰✵✹ ✭✰✵ ✴✸✲✮✻✾✯ ✳✺✯ ✲✰✳✴✯✵✳✶✱ quality and range of motion and subsequently can prevent shoulder dislocation and muscle stiffness [START_REF] Winstein | Guidelines for Adult Stroke Rehabilitation and Recovery[END_REF]. To do this, the game should have reduction feature as to simplified the gameplay activity from a complex arm movement (that involving various parts of joints such as shoulder, elbow and fingers) to a task specific movement (that involving only the shoulder with a straight arm). This would enable the patient to just concentrate on performing the rehabilitation therapy. In addition, the game should allow the patient to play in sitting position instead of standing up in order to maintain balance and ensure safety [START_REF] Skjaeret | Exercise and rehabilitation delivered through exergames in older adults: An integrative review of technologies, safety and efficacy[END_REF]. The game should also have tailoring feature because depending on the severity of the stroke incident, a patient may have a very limited range of motion. For this reason, the target ✲ ✰✭✯✸✯✵✳ ✴✵ ✳✺✯ ✷✰✸✯ ✱✺✻✿ ✹ ✂✯ ✰✹✰✲✳✰✂ ✯ ✳✻ ✳✺✯ ✲✰✳✴✯✵✳✶✱ current ability. As suggested in [START_REF] Pickrell | Exploring inhospital rehabilitation exercises for stroke patients: informing interaction design[END_REF] [START_REF] Skjaeret | Exercise and rehabilitation delivered through exergames in older adults: An integrative review of technologies, safety and efficacy[END_REF], there are variety of needs and conditions of rehabilitation purposes that demand different treatment and therefore the task and target in a game activity should be different form one patient to another. In some cases for example, patient may need to use his other hand to lift up his totally paralyzed arm to exercise the shoulder muscles.

Self-monitoring feature should be implemented in the game as the patient may want to see the progress that they have achieved [START_REF] Pickrell | Exploring inhospital rehabilitation exercises for stroke patients: informing interaction design[END_REF]. This can be done by allowing the game to save ✳✺✯ ✲✰✳✴✯✵✳✶✱ ✷✰✸✯✲ ✰✁ ✹✰✳✰ ✱✿✭✺ ✰✱ ✲✻✴✵✳✱☎ ✂✰✹✷✯✱☎ ✯✾✯ ✱☎ and/or video recording of game session. This data can be used by the therapy to discuss current progress, achievement and further treatment plan with the patient [START_REF] Skjaeret | Exercise and rehabilitation delivered through exergames in older adults: An integrative review of technologies, safety and efficacy[END_REF]. In addition, the exergame feature that shows an avatar representing the ✲ ✰✁✯✮✶✱ ✂✻✹✁ movement can provide a direct feedback for the patient in performing the arm movement [START_REF] Pinto | Adaptive gameplay and difficulty adjustment in a gamified upper-limb rehabilitation[END_REF].

As for the highly repetitive motion that is required, the game should provide reward for a completed repetition that is targeted in a rehabilitation session. This commonly applied game element should be implemented as well in the proposed game to indicate a successful task completion [START_REF] Hannah | Gamification: Applications for Health Promotion and Health Information Technology Engagement[END_REF]. An example of reward in game are coins or badges collection that can unlock further levels of gameplay. This feature would ✯✵✺✰✵✭✯ ✲✰✳✴✯✵✳✶✱ ✸✻✳✴✾✰✳✴✻✵ ✰✵✹ ✯✵✷✰✷✯✸✯✵✳ ✴✵ ✳✺✯✴✮ rehabilitation therapy while playing the game [START_REF] Pinto | Adaptive gameplay and difficulty adjustment in a gamified upper-limb rehabilitation[END_REF]. From the sketch of the graph (see Fig. 6), an assessment can be done by looking at the line representation. For example, a good control of arm movement should be reflected by a smooth curve and a good range of motion should be reflected by a long curve. This graph can also be shared to the patient as to provide a feedback from his comparisons or achievements between rehabilitation session. We have proposed a conceptual game design model that follows Persuasive Design Model and incorporated the persuasive features into the game. To the best of our knowledge, this work may contribute as the first conceptual persuasive game design model for stroke rehabilitation. With this method, a descriptive model of proposed system level can be studied and analyzed. Our next step is to develop a prototype from the proposed game design model and conduct a demonstration with therapists to get feedback before testing it with stroke patients.

Fig. 1 .

 1 Fig.1. Gross Motor Upper Limb Movement[START_REF] Almousa | Requirements Elicitation and Prototyping of a Fully Immersive Virtual Reality Gaming System for Upper Limb Stroke Rehabilitation in Saudi Arabia[END_REF]

Fig. 5 .

 5 Fig. 5. Goal setting menu for therapist C. Therapist Module For the therapist module, we propose that the game should ✳✮✰✭✞ ✰✵✹ ✮✯✭✻✮✹ ✳✺✯ ✲✰✳✴✯✵✳✶✱ ✰✮✸ ✸✻✾✯✸✯✵✳ ✟✺✯ ✷✰✸✯ would translate the recordings into a graph so that the therapist can assess the ✲✰✳✴✯✵✳✶✱ ✁✿✰ ✴✳✁ ✻✽ ✳✺✯ ✸✻✾✯✸✯✵✳.From the sketch of the graph (see Fig.6), an assessment can be done by looking at the line representation. For example, a good control of arm movement should be reflected by a smooth curve and a good range of motion should be reflected by a long curve. This graph can also be shared to the patient as to provide a feedback from his comparisons or achievements between rehabilitation session.

Fig. 6 .

 6 Fig. 6. An example of graph for quality of movement assessment V. CONCLUSION

TABLE I

 I

	.	PERSUASION CONTEXT ANALYSIS [6]
	Intent	Description
	Persuader	✡ The rehabilitation center and therapist are the
		persuader
		✡ The game is designed with autogenous type of
		intent whereby the patient will
		become influenced as s/he play the game
		continuously
	Intended	✡ The aim of this game is to form a compliance
	Outcome/Change	of rehabilitation activity as suggested by the
		therapist
	Event	Description
	Use Context	✡ The objective of the game is to provide game
		environment for the patient to perform
		rehabilitation exercise
		✡ The game is made for gross motor upper limb
		rehabilitation.
		✡ The game allows goal settings by the therapist
		☛☞✌ ✍✎✏✑✒ ✎✓✒ ✔☛✎✕✒☞✎✖✍ ✔✑✏✗✕✘✒ ☛☞✌ ✙☛✚✒ ✑✒✛✏✑✌
		✡ The game will be played in rehabilitation
		✛✒☞✎✒✑ ✜☞✌✒✑ ✎✓✒✑☛✔✕✍✎✖✍ ✙✜✕✌☛☞✛✒ ☛☞✌
		assistance
	User context	✡ Advanced stroke patients who are able move
		the impaired arm with a limited range of
		motion
		✡ Patient that has no ability to move impaired
		arm may be assisted by using their normal arm
		✡ Therapists may adjust the targeted number of
		repetition in the game setting
	Technology	✡ The game will be developed using Unity and
	context	Kinect motion sensor to capture arm
		movement
	Strategy	Description
	Message and
	Route

✡ The message is to keep the patients in performing their rehabilitation activity by following the game✖✍ ✑✜✘✒ ☛☞✌ ✎✓✒ ✑✒✔✒✎✕✎✕✏☞ that is set by the therapist.