Jean-Noël Fuchs
email: fuchs@lptmc.jussieu.fr

Sourabh Patil
email: sourabhpatil.9@gmail.com

Julien Vidal
email: vidal@lptmc.jussieu.fr

Parity of Chern numbers in the Kitaev honeycomb model and the sixteen-fold way

In two dimensions, topological phases of free Majorana fermions coupled to a Z2 gauge field are known to be classified according to the Chern number ν ∈ Z. Its value modulo 16 specifies the type of anyonic excitations. In this work, we investigate triangular vortex configurations (and their dual) in the Kitaev honeycomb model and show that fourteen of these sixteen phases can be obtained by adding a time-reversal symmetry-breaking term. Missing phases are ν = ±7. More generally, we prove that any periodic vortex configuration with an odd number of vortices per geometric unit cell can only host even Chern numbers whereas odd Chern numbers can be found in other cases.

I. INTRODUCTION

Classifying gapped free-fermion systems in terms of topological invariants has been one of the major progresses in the understanding of quantum condensed matter [START_REF] Schnyder | Classification of topological insulators and superconductors in three spatial dimensions[END_REF][START_REF] Kitaev | Periodic table for topological insulators and superconductors[END_REF][START_REF] Ryu | Topological insulators and superconductors: tenfold way and dimensional hierarchy[END_REF][START_REF] Chiu | Classification of topological quantum matter with symmetries[END_REF]. At fixed space dimension, this classification scheme, known as the ten-fold way, is based on the analysis of time-reversal, particle-hole, and chiral symmetries which completely specify the topological invariant needed to characterize the corresponding quantum phase. For instance, in two dimensions, integer quantum Hall systems are class-A topological insulators characterized by a Chern number ν ∈ Z which gives the quantized Hall electric conductance [START_REF] Thouless | Quantized Hall Conductance in a Two-Dimensional Periodic Potential[END_REF][START_REF] Kohmoto | Topological invariant and the quantization of the Hall conductance[END_REF]. However, for a given class of topological insulators, one may still distinguish between several subclasses according to further criteria.

In his seminal paper [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF], Kitaev introduced a spin-1/2 model defined on the honeycomb lattice that can be mapped onto a free Majorana-fermion problem coupled to a static Z 2 gauge field. According to the tenfold way classification, this system is a class-D topological superconductor characterized by a Chern number ν ∈ Z. Importantly, Kitaev has shown that the anyonic properties of the excitations of this model solely depend on ν mod 16 giving rise to the celebrated sixteenfold way [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF][START_REF] Bernevig | Topological Superconductors and Category Theory[END_REF]. This classification relies on the topological spin θ = e iπν/8 of vortex excitations that are Abelian (non-Abelian) anyons if ν is even (odd). Experimental signatures of these sixteen different topological orders have been recently proposed for fractional quantum Hall states at half-integer filling factors [START_REF] Ma | The sixteenfold way and the quantum Hall effect at half-integer filling factors[END_REF].

The Chern number ν counts the number of chiral Majorana edge modes for a system with open boundary conditions. For real fermions, ν is associated to thermal transport whereas for complex fermions, it is related to electric transport. To obtain a nonvanishing Chern number, one must break the time-reversal symmetry. As early realized by Kitaev [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF], this symmetry can be bro-ken explicitly (e.g., by adding a magnetic field) or spontaneously (e.g., by adding odd cycles in the lattice [START_REF] Yao | Exact chiral spin liquid with non-abelian anyons[END_REF][START_REF] Dusuel | Perturbative study of the Kitaev model with spontaneous time-reversal symmetry breaking[END_REF][START_REF] Nasu | Thermodynamics of Chiral Spin Liquids with Abelian and Non-Abelian Anyons[END_REF]). However, an external magnetic field in the Kitaev model leads to interaction terms between Majorana fermions. To break time-reversal symmetry while preserving the integrability of the model, Kitaev suggested to introduce a three-spin term which amounts to add next-nearestneighbor hopping terms for the Majorana fermions on the honeycomb lattice. The corresponding model, which is closely related to the Haldane model for anomalous quantum Hall effect [START_REF] Haldane | Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly[END_REF], has been the subject of many studies (see, e.g. Refs. [START_REF] Lahtinen | Spectrum of the non-abelian phase in Kitaev's honeycomb lattice model[END_REF][START_REF] Kamfor | Kitaev model and dimer coverings on the honeycomb lattice[END_REF][START_REF] Lahtinen | Topological phase transitions driven by gauge fields in an exactly solvable model[END_REF][START_REF] Lahtinen | Interacting non-Abelian anyons as Majorana fermions in the honeycomb lattice model[END_REF][START_REF] Kells | Kaleidoscope of topological phases with multiple Majorana species[END_REF][START_REF] Lahtinen | Topological liquid nucleation induced by vortex-vortex interactions in Kitaev's honeycomb model[END_REF][START_REF] Lahtinen | Perturbed vortex lattices and the stability of nucleated topological phases[END_REF]), notably at finite temperature [START_REF] Self | Conformal energy currents on the edge of a topological superconductor[END_REF][START_REF] Self | Thermally induced metallic phase in a gapped quantum spin liquid: Monte Carlo study of the Kitaev model with parity projection[END_REF].

The goal of the present work is to investigate topological phases that can be found in different sectors of the Kitaev honeycomb model in the presence of this timereversal symmetry-breaking term. More precisely, we analyze the phase diagram of triangular vortex configurations (and their dual) and show that fourteen (among sixteen) different anyon theories can be generated when varying the strength of the corresponding coupling term. We also derive a general result about the parity of the Chern number: any periodic vortex configuration with an odd number of vortices per geometric unit cell can only host even Chern numbers whereas odd Chern numbers can be found in other cases. In addition, we elucidate the origin of gapless phases emerging for a family of vortex configurations lately observed [START_REF] Zhang | Toward Kitaev's sixteenfold way in a honeycomb lattice model[END_REF].

This paper is organized as follows: in Sec. II, we introduce the model directly in the Majorana fermion language. Its symmetries are discussed in Sec. III and the consequences for the parity of the Chern numbers are detailed in Sec. IV. The study of the triangular configurations is presented in Sec. V where a classification in terms of two indicators (vortex-number parity and inversion symmetry) is proposed to analyze the results. Finally, limiting cases and effective models are examined in Sec. VI.

II. MODEL AND DEFINITIONS

We consider the Kitaev honeycomb model in the presence of a three-spin term that breaks time-reversal symmetry [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF]. As explained by Kitaev, the original spins 1/2 defined on the vertices of the honeycomb lattice can be replaced by Majorana operators. This transformation leads to an effective quadratic fermionic Hamiltonian given by

H = i 4 j,k A jk c j c k , (1)
where the sum is performed over all sites of the honeycomb lattice. The matrix A is a real skew-symmetric matrix whose elements depend on Z 2 link variables u jk = -u kj = ±1 defined on each link of the lattice, and where c j is a Majorana operator acting on site j (see Ref. [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF] for a detailed derivation). Hence, this Hamiltonian describes noninteracting Majorana fermions, coupled to a Z 2 gauge field. For the problem at hand, one has:

A jk = 2 J u jk , if j and k are nearest neighbors, A jk = 2 κ u jl u lk , if j and k are next-nearest neighbors, A jk = 0, otherwise.
The term proportional to J involves only one gauge variable u jk and readily satisfies A jk = -A kj . Without loss of generality, we only consider the case κ 0 and set J = 1 in the following. By contrast, the term proportional to κ involves the product of two gauge variables u jl u lk (where the site l is connected to sites j and k) and requires a specific orientation choice to ensure the skew-symmetry of the matrix A. Here, we choose A jk = +2 κ u jl u lk if the triplet (k, l, j) is oriented clockwise (see Fig. 1 for illustration).

Following Kitaev, for each plaquette p, we define the Z 2 plaquette variable w p = (j,k)∈p u jk where j belongs to the white sublattice and k belongs to the black sublattice. If w p = -1 (w p = +1), we will say that there is a (no) vortex in the plaquette p. Two sets of link variables are said to be equivalent if they lead to the same map of w p 's, i.e., the same vortex configuration. The spectrum of the original spin model is obtained by studying all possible inequivalent configurations of the u jk which define a vortex sector. As a Majorana fermion problem, the spectrum in each sector is symmetric so that the ground-state energy is obtained by summing over all negative-energy levels (subtleties about the parity of physical states are discussed in Refs. [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF][START_REF] Pedrocchi | Physical solutions of the Kitaev honeycomb model[END_REF][START_REF] Zschocke | Physical states and finitesize effects in Kitaev's honeycomb model: Bond disorder, spin excitations, and NMR line shape[END_REF]). In the following, the gap of a given sector refers to the energy difference between the two states around zero energy.

As shown by Lieb [START_REF] Lieb | Flux Phase of the Half-Filled Band[END_REF] in a related problem, the (absolute) ground-state energy at κ = 0 is found in the vortex-free sector where w p = +1 for all plaquettes p. For κ = 0, the location of the ground state is still an open question which is not the focus of the present work. Let us simply mention that, among all configurations discussed below (triangular vortex lattice and their dual), FIG. 1. Pictorial representation of the Hamiltonian H for the standard gauge where u jk = +1 if j is a white site and k is a black site. This gauge leads to the vortex-free configuration (wp = +1 for all p). Black links correspond to nearest-neigbor hoppings J and dashed red links to next-nearest-neigbor hoppings κ.

we find that the ground state is in the vortex-free sector for κ < 1.301730 [START_REF] Schnyder | Classification of topological insulators and superconductors in three spatial dimensions[END_REF] and in the vortex-full sector otherwise.

III. SYMMETRIES

As a (free) Majorana fermion model, H is particlehole symmetric. In addition, for κ = 0, H breaks timereversal symmetry [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF] and therefore is a class-D topological superconductor according to the ten-fold classification [START_REF] Schnyder | Classification of topological insulators and superconductors in three spatial dimensions[END_REF][START_REF] Kitaev | Periodic table for topological insulators and superconductors[END_REF]. In this Section, we discuss the influence of symmetries on the spectrum of the Hamiltonian.

A. Translation symmetry

In this work, we shall focus on periodic vortex configurations defined by a geometric unit cell spanned by two vectors A 1 and A 2 (see Fig. 2 for illustration). Denoting by T 1 and T 2 , the magnetic translation operators associated to these vectors ([H, T 1 ] = [H, T 2 ] = 0), one has to distinguish between two different cases depending on the parity P w = p w p of the vortex number inside the geometric unit cell.

If P w = +1 (even number of vortices per geometric unit cell), then [T 1 , T 2 ] = 0 (see, e.g., Fig. 3). In this case, the energy levels ε can be labeled by a band index l which runs from 1 to N (number of sites in the geometric unit cell) and a momentum k that lies into the first Brillouin zone since one has

ε l (k + m A * 1 + n A * 2 ) = ε l (k) where A * 1 and A * 2 are the reciprocal lattice vectors and (m, n) ∈ Z 2 .
By contrast, if P w = -1 (odd number of vortices per geometric unit cell), one has {T 1 , T 2 } = 0 and hence [START_REF] Zhang | Toward Kitaev's sixteenfold way in a honeycomb lattice model[END_REF] (see, e.g., Figs. 2 and4). This indicates that one must double the unit cell to get commuting translation operators and use Bloch theorem. As a direct consequence, the band index l runs from 1 to

[T 2 1 , T 2 ] = [T 1 , T 2 2 ] = 0
⨯ A 1 A 2 n 1 n 2 FIG. 2.
Triangular vortex configuration with vortex density ρ = 1/3. Gray (white) plaquettes indicate plaquette with (without) a vortex. In this case, Pw = -1, so that the Hamiltonian unit cell (thin rhombus) is twice larger that the geometric unit cell spanned by the vectors A1 and A2. The blue cross indicates the inversion center. Black thin (green thick) links corresponds to u jk = +1 (u jk = -1) if j is a white site and k is a black site. The gauge transformation G is shown as ± (red symbols) for sites within the Hamiltonian unit cell. The dashed line indicates the string of flipped links between the two gray plaquettes inside the unit cell (see Appendix A 1 for a systematic contruction).

2N and one has:

ε l (k + m A * 1 /2 + n A * 2 /2) = ε l (k). (2)

B. Inversion symmetry

Some vortex configurations may also have an inversion symmetry with a center that depends on the configuration considered. The inversion operator P = G I is actually composed of a pure spatial inversion I and a Z 2 gauge transformation G. The pure spatial inversion I acts as

I : c j → c -j , (3)
where ±j stands for the site at position ±r j , so that I 2 = 1. The inversion center stands either in the middle of a link (as in Fig. 2) or at the center of a plaquette (as in Figs. 3 and4). As such, I exchanges black and white sites. The Z 2 gauge transformation acts as

G : c j → g j c j , (4)
where g j = ±1 are Z 2 site variables that depend on the links variable u's. Again, one has G 2 = 1. The resulting inversion operator P is traceless and its square acts as

P 2 : c j → g -j g j c j , (5)
so that P 2 = ±1. Indeed, if [H, P] = 0, one also has [H, P 2 ] = 0 which implies that the product g -j g j is independent of j although it can take two values ±1. If P 2 = +1, the gauge transformation is symmetric under spatial inversion ([I, G] = 0 and g -j = g j ). By contrast, if P 2 = -1, the gauge transformation is antisymmetric under spatial inversion ({I, G} = 0 and g -j = -g j ).

Finally, if the Hamiltonian has the inversion symmetry, i.e., if [H, P] = 0], one has:

ε l (-k) = ε l (k). (6)

C. Particle-hole symmetry

The operator C associated to the particle-hole symmetry is an antiunitary operator which, in the present case, is equivalent to complex conjugation (C 2 = 1) so that {H, C} = 0 for all vortex configurations. As a result, the spectrum has the following symmetry ε l (k) ↔ -ε l (-k).

Interestingly, one has [P, C] = 0. Hence, following Zhang et al. [START_REF] Zhang | Toward Kitaev's sixteenfold way in a honeycomb lattice model[END_REF], one can define the operator R = P C which is a momentum-conserving antiunitary operator such that {H, R} = 0. As a direct consequence, if H has the inversion symmetry, its spectrum has the following symmetry ε l (k) ↔ -ε l (k).

IV. PARITY OF CHERN NUMBERS

As a two-dimensional class-D topological superconductor [START_REF] Schnyder | Classification of topological insulators and superconductors in three spatial dimensions[END_REF][START_REF] Kitaev | Periodic table for topological insulators and superconductors[END_REF], when the system is gapped, it is characterized by a Chern number ν ∈ Z. In this Section, we establish a criteria to determine the parity of this Chern number for any periodic vortex configuration.

The general considerations on translation symmetries have an important consequence on the parity of the Chern numbers. Indeed, as discussed above, when the number of vortices per geometric unit cell is odd (P w = -1), the spectrum has an extra periodicity in the Brillouin zone stemming from the anticommutation relation of the magnetic translation operators. For clarity, let us assume that the Hamiltonian unit cell is defined by 2A 1 and A 2 (see Fig. 2 for illustration).

Equation [START_REF] Kitaev | Periodic table for topological insulators and superconductors[END_REF] indicates that if the gap closes at a given point k 0 in the first Brillouin zone, then it also closes at k 0 + A * 2 /2 which is not related to k 0 by a translation of the reciprocal lattice (A * 1 /2, A * 2 ). In other words, when P w = -1, one always has an even number of points in the Brillouin zone where the gap vanishes. Furthermore, as discussed in Ref. [START_REF] Bellissard | Change of the Chern number at band crossings[END_REF], one can associate to each point where the gap closes a Berry index (which is an integer). The sum of all Berry indices over all such points gives the variation of the Chern number. In the present context, this variation must be understood as the result of a process where a transition between two gapped phases is obtained by varying some parameters in the Hamiltonian. Thus, since gap-closure points always arise in pairs (k 0 , k 0 + A * 2 /2) carrying the same Berry index, we straightforwardly obtain that the variation of the Chern number ∆ν for configurations with P w = -1 is always even.

Up to now, we focused on isotropic couplings (J does not depend on the link orientation). However, if one rather considers anisotropic couplings and the perturbative limit where one of them is much larger than the two others (the isolated-dimer limit), the effective Hamiltonian at κ = 0 is unitarily equivalent to the toric code Hamiltonian [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF] (see also Refs. [START_REF] Schmidt | Emergent Fermions and Anyons in the Kitaev Model[END_REF][START_REF] Vidal | Perturbative approach to an exactly solved problem : the Kitaev honeycomb model[END_REF] for a more detailed discussion). Thus, in this strongly anisotropic limit and for κ = 0, the system is always gapped and time-reversal symmetric so that ν = 0 for all vortex configurations. Since ∆ν is even when one varies the couplings, we conclude that all gapped vortex configurations with P w = -1 only host even Chern numbers.

By contrast, for P w = +1, there are no constraint on ∆ν so that any Chern number parity can be found in such configurations.

V. TRIANGULAR CONFIGURATIONS

Although class-D topological superconductors are classified by a Chern number ν ∈ Z in two space dimensions, Kitaev has shown that the properties of the anyons in the corresponding topological phase are solely characterized by ν mod 16 [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF]. This celebrated sixteen-fold way also indicates that anyons are Abelian if ν is even, whereas they are non-Abelian for odd ν. The results derived in Sec. IV implies that this latter case can only be found in vortex configurations with P w = +1. In this Section, we investigate different vortex configurations in order to exhibit these sixteen different phases. Since changing the sign of κ changes the sign of ν [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF], it is actually sufficient to find all possible Chern number |ν| 8.

Any vortex configuration may be of interest in itself and display nontrivial features. Here, we restrict our investigation to triangular vortex configurations and their dual obtained by changing w p → -w p in each plaquette p. By construction, these triangular vortex configurations are spanned by two vectors A 1 and A 2 that are parametrized by two integers p and q such that:

A 1 = p n 1 + q n 2 , (7)
A 2 = -q n 1 + (p + q) n 2 , (8)
where n 1 and n 2 are the Bravais vectors of the honeycomb lattice (see Fig. 2). The corresponding vortex density is given by ρ = 1/n where n = p 2 + q 2 + p q. For dual configurations, the same vectors A 1 and A 2 lead to a vortex density ρ = (n -1)/n. Examples are shown in Figs. 2 and 3 for p = 1 and q = 1. All these configurations are invariant under inversion symmetry P.

A. Vortex lattices (Pw = -1, P 2 = -1)

By construction, all triangular vortex configurations have one vortex per geometric unit cell so that P w = -1. Furthermore, these configurations are invariant under inversion P = GI and it is always possible to find a configuration of the link variables and a symmetry center such that the gauge transformation G is trivial: g j = +1 if j is a black site and g j = -1 if j is a white site (see Fig. 2 for a concrete example and Appendix A 1 for a general construction). As can be seen in Eq. ( 5), such a transformation corresponds to P 2 = -1.

Since for these configurations, [H, P] = 0, one can write H = H + ⊕ H -, where the subscript ± refers to eigenvalues ±i of P. In this case, the particle-hole symmetry acts as:

CH ± C = -H ∓ so that: spec(H ± ) = -spec(H ∓ ). ( 9
)
In other words, the spectrum of H + and H -just differ by a sign. In the absence of any additional symmetry, level repulsion makes it unlikely that two eigenvalues of H coincide. We conclude that one cannot get extended gapless phases for vortex configurations with inversion symmetry if P 2 = -1. In this case, one expects gapped phases separated by gapless points.

For the triangular vortex configurations considered in this Section, we computed the Chern number in each gapped phase found in the range κ from 0 to 10. We performed a systematic study of all configurations with ρ = 1/n with n < 30. Results are summarized in Table I. As anticipated in Sec. III, since P w = -1 for these configurations, one only finds even Chern numbers. In addition, when n is a multiple of 3, the system is gapped at κ = 0 where the system is time-reversal symmetric [START_REF] Kamfor | Fate of Dirac points in a vortex superlattice[END_REF], and hence, ν = 0. Since changing the sign of κ changes the sign of ν, these triangular vortex configurations allow one to generate all possible even ν mod 16, and hence, all Abelian topological phases expected for free Majorana fermions coupled to a Z 2 gauge field in two dimensions [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF]. To generate odd Chern numbers, one must consider configurations with P w = +1. A simple way to generate P w = +1 configurations is to take triangular vortex configurations with ρ = 1/n (n odd) and to change w p into -w p in each plaquette. This gives rise to a dual vortex lattice with an even number of vortices per geometric unit cell and a vortex density ρ = (n -1)/n. As previously, we can systematically find a set of link variables and a symmetry center such that the gauge transformation G is trivial (see Fig. 3 for a concrete example and Appendix A 2 for a general construction) so that, one again finds P 2 = -1 for this inversion operator. Thus, again, one only gets gapped phases separated by gapless points. II where we used the same conventions as Table I. Since P w = +1, one can get Chern number with odd (ν = ±1, 3, -5) or even (ν = 0, ±2) parity. Contrary to triangular vortex configurations, these systems are always gapless at κ = 0. Thus, we do not have a simple explanation for these phases with ν = 0 emerging at infinitesimal κ > 0 since we cannot connect them to a gapped time-reversal symmetric point. Nevertheless, we observed (up to n = 39) that such phases are always found for n = 0 mod 3 (n > 3).

1 0 -1 1 2 2 2 2 2 2 -2 -2 1 -1 1 -1 1 -1 -5 -2 -2 -2 -2 -2 -2 1

Results are summarized in Table

C. Dual vortex lattices with n even

(Pw = -1, P 2 = +1)

Let us now consider dual vortex configurations obtained from triangular vortex configurations with ρ = 1/n (n odd) by changing w p into -w p in each plaquette. These configurations have a vortex density ρ = (n -1)/n with n even so that P w = -1 (odd number of vortices per geometric unit cell) and consequently, gapped phases can only host even Chern number. However, a salient feature of these configurations is that, contrary to previous cases, the gauge transformation G involved in the inversion symmetry P is nontrivial (see Fig. 4 for a concrete example and Appendix A 2 for a general construction). As a result, for these configurations one gets P 2 = +1.

In this case, one can write H = H + ⊕ H -, where the subscript ± refers to eigenvalues ±1 of P. The particlehole symmetry then acts as:

CH ± C = -H ± so that: spec(H ± ) = -spec(H ± ). ( 10
)
The spectra of H + and H -are not related and the corresponding energy levels therefore have no reason to repel, so that overlapping bands are possible. As a consequence, one cannot exclude extended gapless phases for vortex configurations with inversion symmetry if P 2 = +1. Results for this family of configurations are given in Table III. As expected, since P w = -1, one only gets gapped phases with even Chern number separated by gapless phases. Note also that when n = 0 mod 12, the perturbation due to the triangular lattice of white plaquettes leads to a nesting of the Dirac points in the vortex-full model and opens a gap at κ = 0 (see Ref. [START_REF] Kamfor | Fate of Dirac points in a vortex superlattice[END_REF] for a related discussion). Hence, in this case, one gets ν = 0 at κ = 0 due to time-reversal symmetry as can be seen for ρ = 11/12. Otherwise, the system remains gapless at small κ.

VI. THE LARGE-DILUTION LIMITS

A close inspection of Tables I-III unveils several features in the large-n limit. In this Section, we give some arguments to understand the Chern numbers found in this limit.

A. Around the vortex-free background

The limit of sparse vortex configurations has been discussed by Lahtinen et al. in Refs. [START_REF] Lahtinen | Topological liquid nucleation induced by vortex-vortex interactions in Kitaev's honeycomb model[END_REF][START_REF] Lahtinen | Perturbed vortex lattices and the stability of nucleated topological phases[END_REF] who derived an effective model to describe the spectrum and the Chern number of such configurations (see also Ref. [START_REF] Grosfeld | Electronic transport in an array of quasiparticles in the ν = 5/2 non-Abelian quantum Hall state[END_REF] for a related study). In the vortex-free configuration, plaquette excitations consists in pairs of plaquettes with w p = -1 (gray plaquettes, a.k.a., vortices). Since, in this sector, ν is odd, there is a single unpaired Majorana zero mode bound to each vortex inside the bulk gap [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF]. Thus, in the presence of a dilute vortex lattice in a vortexfree background, the spectrum can be split into highenergy bands (coming from the vortex-free background) and low-energy bands (coming from tunneling of Majorana fermions between zero modes attached to vortices). The total Chern number ν = ν h + ν l is then simply the sum of the high-energy bands Chern number ν h = 1 and the low-energy bands Chern number ν l . For triangular When gray (white) plaquettes are strongly dilute in a vortexfree (vortex-full) background, the bandwidth of low-energy bands W vanishes, ∆ is given by the gap of the vortex-free (vortex-full) sector, and δ vanishes (remains finite).

vortex configurations, including only the most relevant tunneling terms, this effective model predicts ν = 1 ± 1 or 1 ± 3 [START_REF] Lahtinen | Topological liquid nucleation induced by vortex-vortex interactions in Kitaev's honeycomb model[END_REF][START_REF] Lahtinen | Perturbed vortex lattices and the stability of nucleated topological phases[END_REF] which is in agreement with our findings for density ρ = 1/n at large n (see Table I).

Of course, this effective model is only valid if the total band width of the low-energy bands (which is essentially given by the tunneling terms) is much smaller than the gap between high-energy bands. Using notations of Fig. 5, this corresponds to the regime δ +2W ∆. However, the dependence of the tunneling terms on κ and n is nontrivial and each case must be considered carefully to determine the validity range of this approach. Finally, any infinitesimal tunneling term opens a gap at zeroenergy so that the strongly-dilute limit is nontrivial as can be seen in Table I.

B. Around the vortex-full background

Following the study of Lahtinen et al. [START_REF] Lahtinen | Topological liquid nucleation induced by vortex-vortex interactions in Kitaev's honeycomb model[END_REF][START_REF] Lahtinen | Perturbed vortex lattices and the stability of nucleated topological phases[END_REF] around the vortex-free lattice, let us now consider the vortex-full lattice where w p = -1 for all plaquettes. This vortex configuration (ρ = 1) has been studied in Refs. [START_REF] Lahtinen | Spectrum of the non-abelian phase in Kitaev's honeycomb lattice model[END_REF][START_REF] Lahtinen | Topological phase transitions driven by gauge fields in an exactly solvable model[END_REF] in the small-κ limit where a gapped phase with ν = 2 has been found. However, as shown in Fig. 6, there exists a phase transition separating a phase with ν = 2 for 0 < κ < 1/2 from a phase at ν = -2 for κ > 1/2. In the vortex-full configuration, elementary plaquette excitations consists in pairs of plaquettes with w p = +1 (white plaquettes). Since, in this sector, ν is even, a white plaquette does not bind an unpaired Majorana mode at zero energy [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF] but rather several modes at finite energy [START_REF] Volovik | Fermion zero modes on vortices in chiral superconductors[END_REF]. In the presence of a dilute white-plaquettes lattice in a vortex-full background, the spectrum splits into high-energy bands (coming from the vortex-full background) and low-energy bands (coming from the coupling between modes bound to white plaquettes) as depicted in Fig. 5. Thus, again, the total Chern number is the sum of high-and lowenergy bands Chern numbers, i.e., ν = ν h + ν l . Interestingly, when white plaquettes are sufficiently far from each other (large-dilution limit), the tunneling terms between modes bound to these plaquettes vanish so that the low-energy bandwidth W goes to zero. However, on each white plaquette, the two lowest-energy bound modes have an energy ±δ/2, where δ depends on κ (see Fig. 6).

The corresponding low-energy (flat) bands are topologically trivial (ν l = 0) since they correspond to an "atomic" limit. Hence, provided δ is smaller than the bulk gap ∆, we expect the total Chern number to be ν = ν h for all sparse white-plaquette configurations. Thus, in the largedilution limit, the phase diagram is the same as the one of the vortex-full.

As can be seen in Tables II andIII, one indeed gets ν = 2 for 0 κ 1/2 and ν = -2 for 1/2 κ (see Appendix B for phase boundaries). Nevertheless, for finite but large n, one observes the existence of intermediate gapped phases (with ν = 0, ±1) or gapless phases near the points κ = 0, 1/2 where both δ and ∆ go to zero (see Fig. 6). The extension of these intermediate phases shrinks to zero when n increases. In addition, one also observes in Table III that the phase at ν = -2 found for κ > 1/2 ends up at a value κ which increases with n (see Appendix B) giving rise to a gapless phase and possibly other phases for larger κ.

VII. OUTLOOK

We have shown that a wide variety of topological phases characterized by the Chern number ν mod 16 could be produced in the Kitaev honeycomb model in the presence of a simple time-reversal breaking term [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF].

In the triangular vortex configurations (and their dual) studied in this work, we found all possible values except ν = ±7. Although, only a finite number of examples were considered, we believe that ν = ±7 does not exist for these families of configurations. Indeed, we proved that odd Chern numbers can only be found when the geometric unit cell contains an even number of vortices (P w = 1). For the configurations considered here, this concerns dual triangular lattices with vortex density ρ = (n -1)/n and odd n. As can be seen in Table II, ν = ±7 is absent for n < 30. For larger n, we argued that one simply recovers the vortex-full phase diagram (ν = ±2) except in narrow regions near κ = 0 and κ = 1/2 where one only gets ν = 0, ±1. Nevertheless, the missing Chern numbers ν = ±7 are certainly present in other vortex sectors with P w = 1.

In this work, we investigated inversion-symmetric periodic vortex lattices and found that they are classified by two indicators: the parity of the number of vortices per geometric unit cell P w = ±1 and the square of the inversion operator P 2 = ±1. Here, we found only three classes: (P w , P 2 ) = (-1, -1) (see Sec. V A), (+1, -1) (see Sec. V B), and (-1, +1) (see Sec. V C). For other inversion-symmetric vortex lattices, one may create the fourth class (P w , P 2 ) = (+1, +1), which should exhibit even and odd Chern numbers as well as gapless phases.

Finally, it would be worth developing an effective model for dilute lattices of white plaquettes in a vortex-full background following the general strategy of Refs. [START_REF] Lahtinen | Topological liquid nucleation induced by vortex-vortex interactions in Kitaev's honeycomb model[END_REF][START_REF] Lahtinen | Perturbed vortex lattices and the stability of nucleated topological phases[END_REF]. A main difference is that low-energy bands are built from finite-energy mid-gap states bound to white plaquettes instead of unpaired Majorana zero modes bound to vortices. Such an effective model should help us to understand how the infinitely dilute whiteplaquette limit nucleates non-trivial phases as the density of white plaquettes is increased.

To conclude, we hope that the present work will motivate further studies to complete the quest of the sixteen topological orders in the Kitaev honeycomb model recently initiated by Zhang et al. [START_REF] Zhang | Toward Kitaev's sixteenfold way in a honeycomb lattice model[END_REF]. 7. Triangular vortex configurations (Pw = -1) with ρ = 1/7 (top) and ρ = 1/12 (bottom). The dashed line indicates the string of flipped links (green) between two vortices. The inversion center lies in the middle of a link for ρ = 1/7 and in the center of a white plaquette for ρ = 1/12. In both cases, the gauge transform G (± red symbols) is antisymmetric (g-j = -gj) so that P 2 = -1. involved in the inversion operator P = GI. In both cases, we start from the standard gauge of the vortex-free configuration depicted in Fig. 1. With respect to this standard gauge, flipping a link (u jk → -u jk ) is depicted graphically as a thick green link while a thin black link refers to an unflipped one.

⨯ A 1 A 2 ⨯ A 1 A 2 FIG.

Triangular vortex lattices

For the triangular vortex lattices with vortex density ρ = 1/n (P w = -1), there is a systematic way to build a Z 2 gauge field which is symmetric with respect to the inversion center. First, one has to select an inversion center which, for odd (even) n, can always be chosen as the middle of a link (white plaquette). Then, one creates a pair of vortices that one moves until they are in is antisymmetric (g-j = -gj) so that P 2 = -1. By contrast, for ρ = 11/12 (Pw = -1), the gauge transform G is symmetric (g-j = gj) so that P 2 = +1. a relative position given by A 1 . This is easily achieved by considering a string of flipped links chosen symmetrically with respect to the inversion center (see Figs. 2 and7 for examples). Finally, we repeat this procedure periodically with the translation vectors 2A 1 and A 2 .

By construction, the resulting Z 2 gauge field is symmetric under inversion. Hence, the gauge transform G is trivial: g j = +1 (-1) if j is a black (white) site and has the periodicity of the honeycomb. This transformation G is antisymmetric under inversion, (g -j = -g j ), so that P 2 = -1.

Dual lattices

To build the dual lattice with vortex density ρ = (n -1)/n, we start from the gauge used for the tri-angular lattice configuration with ρ = 1/n and we add the missing links to change w p into -w p for all plaquettes. This last step can always be achieved by respecting the Hamiltonian unit cell periodicity. However, one has to distinguish between two cases according to the parity of n.

Indeed, when n is odd (P w = +1), this completion step can be done symmetrically with respect to the inversion center which is the middle of a link [see Figs. 3 and 8 (top) for examples]. Thus, in this case, the resulting Z 2 gauge field is symmetric under inversion; the gauge transform G is trivial: g j = +1 (-1) if j is a black (white) site; G is antisymmetric under inversion, (g -j = -g j ), so that P 2 = -1.

By contrast, when n is even (P w = -1), one cannot add the missing links symmetrically because the inversion center is the center of a gray plaquette which, as such, must contain an odd number of flipped links [see Figs. 4 and 8 (bottom) for examples]. The corresponding gauge transform G is less trivial than previously and can be built according to the following rules: if inversion maps a flipped (unflipped) link (j, k) onto a flipped (unflipped) link then g j = -g k . By contrast, if a flipped (unflipped) link (j, k) is mapped onto an unflipped (flipped) link, then g j = g k . Such a gauge transform G is symmetric under inversion (g -j = g j ) and one has P 2 = +1.

⨯ A 1 A 2 FIG. 3 .

 123 FIG.3. Dual vortex configuration ρ = 2/3 (see Fig.2for notations). In this case, Pw = +1, so that the Hamiltonian unit cell coincides with the geometric unit cell (see also Appendix A 2).

B.

  Dual vortex lattices with n odd (Pw = +1, P 2 = -1)

2 FIG. 4 .

 24 FIG.[START_REF] Chiu | Classification of topological quantum matter with symmetries[END_REF]. Dual vortex configuration ρ = 3/4 (see Fig.2for notations). In this case, Pw = -1, so that the Hamiltonian unit cell is twice larger that the geometric unit cell. The dashed line indicates the string of flipped links connecting the two white plaquettes inside the unit cell (see also Appendix A 2).

FIG. 5 .

 5 FIG.5. Schematic energy spectrum of dilute configurations. When gray (white) plaquettes are strongly dilute in a vortexfree (vortex-full) background, the bandwidth of low-energy bands W vanishes, ∆ is given by the gap of the vortex-free (vortex-full) sector, and δ vanishes (remains finite).

1 FIG. 6 .

 16 FIG.6. Energy gaps ∆ (black) of the vortex-full configuration (see Appendix C for an analytical expression) and δ (green) of isolated white plaquettes in a vortex-full background as a function of κ. Both gaps vanish for κ = 0, 1/2 whereas ∆ ∼ 2 √ 3 κ and δ = 0.7865(1) in the large-κ limit.

⨯ A 1 A 2 ⨯ A 1 A 2 FIG. 8 .

 12128 FIG.8. Dual vortex configurations with ρ = 6/7 (top) and ρ = 11/12 (bottom). The dashed line indicates the string of flipped links (green) between two white plaquettes. For ρ = 6/7 (Pw = +1), the gauge transform G (± red symbols) is antisymmetric (g-j = -gj) so that P 2 = -1. By contrast, for ρ = 11/12 (Pw = -1), the gauge transform G is symmetric (g-j = gj) so that P 2 = +1.

TABLE II .

 II Chern numbers for dual configurations with vortex densities ρ = (n -1)/n and odd n. Conventions are the same as in TableI.

TABLE III .

 III Chern numbers for dual configurations with vortex densities ρ = (n -1)/n and even n. Conventions are the same as in TableI. Gapless phases are denoted by -.

	3/4 11/12 15/16 27/28
	-	0	-	-
	2	-	2	2
	-	2	-	-
	-2	-	-2	-2
	-	-2	-	-
	-4	-	0
		0

ACKNOWLEDGMENTS

We thank A. Auerbach, B. A. Bernevig, B. Douçot, S. Iblisdir, A. Mesaros, F. Piéchon, and G. J. Sreejith for useful discussions. S. Patil is supported by an Erasmus+ International Credit Mobility Grant.

Appendix A: Systematic construction of the Z2 gauge field and of the gauge transformation G

In this Appendix, we propose a systematic construction of the Z 2 gauge field (i.e., the u jk 's defining the vortex configuration) with the minimal periodicity for the triangular vortex lattices (and their duals) as well as the gauge transformation G [i.e., the g j 's defined in Eq. ( 4)] Appendix B: Phase boundaries of triangular vortex lattices and their dual

In this Appendix, we give the boundaries of each phase obtained for all triangular vortex configurations and their dual discussed in the main text. Changing the relative sign of κ/J simply changes the sign of the Chern number. Hence, Without loss of generality, we set J = 1 and only consider the case κ 0. Furthermore, for simplicity, we restrict our discussion to the range κ ∈ [0, 10] and give only a relative precision of 10 -6 on the phase boundaries. For each integers p and q considered, we show two figures representing a piece of infinite lattice for the triangular vortex configuration with vortex density ρ = 1/n (left) and its dual corresponding to ρ = (n-1)/n (right) where n = p 2 + q 2 + p q. We denote each Chern number by ν i ρ where the superscript i labels each phase.

Finally, in many cases, we also found other phases at larger κ (e.g., ν = -14 for ρ = 1/7 and κ > 16.4268) but none of them has a Chern number ±7 which is the missing one in our list to complete the sixteen-fold classification [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF].

1. (p, q) = (1, 0)

• ρ = 0:

• ρ = 1/3: • ρ = 2/3:

3. (p, q) = (2, 0)

• ρ = 1/4: • ρ = 3/4:

• ρ = 6/7:

6/7 = -2 for 0.518436 < κ < 0.816497, ν e 6/7 = -5 for 0.816497 < κ < 1.80057, ν e 6/7 = +1 for 1.80057 < κ 10.

5.

(p, q) = (3, 0)

• ρ = 1/9: • ρ = 1/16: • ρ = 24/25: ν a 24/25 = +1 for 0 < κ < 0.11127, ν b 24/25 = +2 for 0.11127 < κ < 0.496047, ν c 24/25 = +1 for 0.496047 < κ < 0.514012, ν d 24/25 = -2 for 0.514012 < κ 10.

12. (p, q) = (3, 3)

• ρ = 1/27: In this appendix, we give the exact expressions of the energy spectrum and of the gap ∆ for the vortex-free and vortex-full sectors that are both defined by A 1 = n 1 and A 2 = n 2 , i.e., p = 1 and q = 0.

• The spectrum of the Hamiltonian in the vortex-free sector (ρ = 0) consists in two bands [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF]: • The spectrum of the Hamiltonian in the vortex-full sector (ρ = 1) consists in four bands [START_REF] Lahtinen | Spectrum of the non-abelian phase in Kitaev's honeycomb lattice model[END_REF][START_REF] Lahtinen | Topological phase transitions driven by gauge fields in an exactly solvable model[END_REF]:

with

(C5)

Note several typos in Eq. ( 26) of Ref. [START_REF] Lahtinen | Spectrum of the non-abelian phase in Kitaev's honeycomb lattice model[END_REF] where this result was first derived.

The gap in this sector is given by: ∆ = 2 min k f (k) -2 g(k). (C6)


Youcef Kemari

Abdelouahab Mekhaldi

Gilbert Teyssèdre

Madjid Teguar

Correlations Between Structural Changes and Dielectric Behavior of Thermally Aged XLPE

Keywords: power cable insulation, cross-linked polyethylene insulation, aging, Fourier spectroscopy, calorimetry, dielectric measurements, dielectric materials

This study aims at elucidating the correlations between structural changes and the variation of dielectric properties of cross-linked polyethylene (XLPE) during thermal aging. For this purpose, thermal aging experiments were carried out on XLPE samples at 80, 100, 120 and 140 °C for different aging times for a maximum total duration of 5040 hours. Chemical changes were assessed by Fourier Transform Infrared Spectrum (FTIR). Besides, the crystallinity ratio and the melting temperature were evaluated using Differential Scanning Calorimetry (DSC). Frequency domain dielectric spectroscopy was used to investigate the evolution of the dielectric constant and losses of XLPE during aging. The results show that aging at 80 and 100 °C could help to improve the crystalline state of XLPE which leads to the decrease of the dielectric constant. However, for higher thermal stress, the main aging mechanism of XLPE is thermal oxidation which leads to deteriorate the morphology and increase both the polarization and losses, mainly at low frequencies. It has been found also that in some cases, the non-homogeneity of the color of XLPE seems to indicate the increase of interfaces in the material which could lead to increase the dielectric losses due to the interfacial polarization.

INTRODUCTION

POWER cables represent an important part of electrical transmission and distribution networks. Moreover, the use of polymeric insulation materials in underground cables offers a significant reduction of the electrical system volume [START_REF] Liu | Dielectric Spectroscopy of Very Low Loss Model Power Cables[END_REF] Therefore, special attention has been paid to the continuous development of insulating materials used in power cables. Among these materials, cross-linked polyethylene (XLPE) remains a material of choice for medium and high-voltage cables [START_REF] Teyssedre | Advances in High-Field Insulating Polymeric Materials Over the Past 50 Years[END_REF]. In fact, XLPE presents many advantages such as a low dissipation factor of about 0.03 % at 20 °C, low dielectric constant of 2.2-2.5, good thermo-mechanical properties and high operating temperature of about 90 °C [START_REF] Metwally | The Evolution of Medium Voltage Power Cables[END_REF].

However, irreversible changes may occur after the exposure of the cable to a variety of stresses. Among them, the thermal constraint is considered as one of the most severe factors leading to change the properties of the insulation and then to the deterioration of the cable [START_REF] Nedjar | Effect of thermal aging on the electrical properties of crosslinked polyethylene[END_REF]. To address these issues, many researchers have focused on physicochemical analyses to study the morphology changes in XLPE due to aging [START_REF] Andjelkovic | Influence of accelerated aging on mechanical and structural properties of cross-linked polyethylene (XLPE) insulation[END_REF][START_REF] Gulmine | Correlation between structure and accelerated artificial aging of XLPE[END_REF][START_REF] Li | The role of thermo-oxidative aging at different temperatures on the crystal structure of cross-linked polyethylene[END_REF]. These studies showed that the main degradation mechanisms in XLPE are thermal oxidation and scission of the molecular chains. In addition, some researchers have investigated the dielectric response of XLPE during thermal aging [START_REF] Nedjar | Effect of thermal aging on the electrical properties of crosslinked polyethylene[END_REF][START_REF] He | Research on mechanical and dielectric properties of XLPE cable under accelerated electrical-thermal aging[END_REF][START_REF] Kim | Investigation of dielectric behavior of thermally aged XLPE cable in the high-frequency range[END_REF] This approach has revealed the evolution of some dielectric parameters (e.g. conductivity, losses factor, dielectric constant...) with aging.

To our knowledge, the large number of works already done in this area were based on experimental databases collected from samples that have been aged in a very accelerated manner or for short periods [START_REF] Li | Accelerated inhomogeneous degradation of XLPE insulation caused by copper-rich impurities at elevated temperature[END_REF][START_REF] Zhang | Conductivity properties of XLPE insulation used for HVDC cable after accelerated thermal aging[END_REF] which could lead to skip some intermediate changing states of XLPE properties during the aging process. Besides, some aging mechanisms not in operation in normal service conditions can be triggered by overstressing. Especially the evolution of the material can substantially change when going through the melting temperature range.

The study of correlations between structural changes and dielectric properties represents an important task for the development of more resistant materials and additives in order to increase the insulation lifetime. Such a study could help to predict the dielectric properties according to physicochemical changes and vice versa and to contribute to the development of diagnosis tools for aging monitoring.

Through this paper, we focused in particular on the physicochemical and the dielectric behaviors of XLPE material versus aging. The selected material is a commercial XLPE used for the insulation of medium voltage cables. Samples were subjected to a controlled thermal aging at four different temperatures ranging from 80 to 140 °C and at different aging times for a maximum total duration of 5040 hours (210 days).

Moreover, our work consists of evaluating the physicochemical and dielectric properties of the samples before and after aging in order to characterize the degree of degradation of XLPE. Besides, we aim to study the correlations between these different properties, which leads to a better understanding of aging and degradation mechanisms.

EXPERIMENTAL

MATERIALS AND SAMPLES PREPARATION

The studied material is intended for the extrusion of power cables operated in phase to phase voltage of U = 30 kV (i.e. U 0 = 18 kV between the phase and the screen). The base materials are pellets of cross-linkable resin (LE4201R, manufactured by Borealis company). These pellets are loaded with di-cumyl peroxide (DCP) as cross-linking agent with 2 % in content. The material contains also di-tert-butyl peroxide and Irganox antioxidant. Samples were manufactured in thick plaque form and supplied by ENICAB Subsidiary company of General Cable of Biskra, Algeria.

For manufacturing the XLPE plaques, the pellets were preheated for 5 minutes at 125 °C then molded and crosslinked at 185 °C under a pressure of 300 MPa using a pressurized heat press. These conditions (300 MPa and 185 °C) are maintained for about 10 minutes to ensure complete cross-linking of the material. At the end, the press is cooled with water by maintaining always the same pressure.

The diameter of the obtained circular plates of XLPE is 12.9 cm, whereas their thickness is (2 ± 0.2) mm. In order to perform physicochemical and electrical experiments, the XLPE plates were cut into circular shaped samples with a diameter of about 46 mm and a thickness of (2 ± 0.2) mm. Samples were cut using a cutting press with a circular cutting die tool.

THERMAL AGING EXPERIMENTS

Accelerated thermal aging was performed according to IEC 60216 standard [START_REF]Guide for the determination of thermal endurance properties of electrical insulating materials. Part 1: General guidelines for aging procedures and evaluation of test result[END_REF]. The circular samples were vertically arranged inside air-circulating ovens at aging temperatures of 80, 100 and 120 °C. Samples were taken out every 1008 hours (42 days) for total aging time of 5040 hours (210 days).

Moreover, since the degradation process of XLPE is very fast at 140 °C, samples were taken out every 504 hours (21 days) for a total period of 1512 hours (63 days). All these samples are intended for FTIR and dielectric spectroscopy tests. On the other hand, DSC characterization was performed on unaged and aged samples that have been taken at the final stage of aging at each temperature (after 5040 hours at 80, 100 and 120 °C and after 1512 hours at 140 °C).

FOURIER TRANSFORM INFRARED SPECTRUM

CHARACTERIZATION (FTIR) FTIR characterizations were carried out using a Vertex 70 FTIR spectrometer from Bruker equipped with a platinum diamond accessory. Moreover, our samples present relatively large thickness of about 2 mm which imposed us to use FTIR analysis in Attenuated Total Reflection (ATR) mode.

All samples were analyzed and each experiment started with a background spectrum acquisition without the sample. Then, 24 scans were recorded in the wavenumber range of 430-5000 cm -1 with a resolution of 2 cm -1 for each sample. The obtained ATR-FTIR spectra represent the average of three spectra recorded from three different samples that have been aged in the same conditions (temperature and time).

DIFFERENTIAL SCANNING CALORIMETER (DSC)

The variations of phase transitions of XLPE due to thermal aging have been monitored using differential scanning calorimetry DSC 2010 TA Instruments with a computer data system. Samples of about 10 mg of weight were analyzed by two repeated heating and cooling cycles at a constant rate of 10 °C/min under nitrogen atmosphere in the temperature interval 20-150 °C. The DSC thermogram obtained in the first heating run is strongly influenced by the thermal history of the material. Therefore, we based the analysis on the second heating/cooling cycle to determine the DSC parameters such as the crystallinity rate and the melting temperature.

FREQUENCY DOMAIN DIELECTRIC SPECTROSCOPY

The dielectric spectroscopy measurements were performed using Novocontrol Alfa-A High Performance Analyzer dielectric spectrometer with a ZGS active sample cell. The system can control the temperature under nitrogen atmosphere by a Quatro cryosystem within ± 0.1 °C. In addition, the control and the evaluation of dielectric measurements were done using WinDETA® software. Prior to the dielectric measurements, gold electrodes were deposited onto both surfaces of the XLPE circular samples by sputtering. The diameter and the thickness of the sputtered electrodes are respectively 35 mm and 30 nm. For the sake of evaluating the dielectric response of XLPE samples with thermal aging, dielectric constant ε' r and dielectric losses tan() were measured in the frequency range 10 -2 to 10 5 Hz at 20 °C.

RESULTS AND DISCUSSIONS

COLOR CHANGES

Figure 1 shows the changes in aspect of XLPE samples at different aging stages. We observe that the color changed from gray before aging to light yellow after 5040 h at 80 °C and then to light orange after 5040 h at 100 °C. On the other hand, the color has become darker at 120 °C. Also, it can be noticed that after 5040 h at 120 °C and 504 h at 140 °C, XLPE color is a dark orange that is not uniformly distributed over the surface of the aged sample.

Moreover, the change in color at 140 °C is much faster and a dark-brown color has been obtained after 1512 h. These changes have been attributed to the formation of vinylidene and vinyl groups due to degradation by thermal oxidation [START_REF] Li | Accelerated inhomogeneous degradation of XLPE insulation caused by copper-rich impurities at elevated temperature[END_REF][START_REF] Kemari | Experimental investigation and signal processing techniques for degradation assessment of XLPE and PVC/B materials under thermal aging[END_REF]. It could correspond to the reaction of the antioxidant during the oxidation phase [START_REF] Teyssedre | Semi-quantitative analysis of photoluminescence in thermoelectrically aged cables: I-identification of optical signatures[END_REF].

ATR-FTIR SPECTROSCOPY ANALYSIS

The FTIR spectra of XLPE samples before and after thermal aging are presented in Figure 2. Before identifying the chemical functions related to the degradation of the material, the characteristic bands of the virgin XLPE have been found. The peak at wavelength of 719 cm -1 corresponds to the rocking methylene groups -CH 2 . The peak at 1466 cm -1 is attributed to the wag vibration of -CH2. The other peaks at 2849 cm -1 and 2914 cm -1 correspond respectively to symmetric and asymmetric stretching vibration of -CH 2 group [START_REF] Nobrega | Investigation and analysis of electrical aging of XLPE insulation for medium voltage covered conductors manufactured in Brazil[END_REF].

No significant change in the FTIR spectra of XLPE was observed after aging at 80 and 100 °C. However, due to thermo-oxidative reactions, the main obtained changes in the FTIR spectra after aging at 120 and 140 °C consist in the formation of C=O carbonyl groups. Three bands characteristic to the C=O group are observed between 1650 and 1800 cm -1 .

The first one at 1713 cm -1 shows an absorption of the carboxylic acid and ketones [RC(=O) OH, RC(=O) R'], where R and R' represent a chain of the aged polymer and its fragment, respectively. The second band at 1732 cm -1 characterizes the absorption of aldehyde or the ester [RC(=O) OR']. The third band corresponds at 1774 cm -1 and is attributed to the carboxylic anhydride [START_REF] Stancu | Space charge and electric field in thermally aged multilayer joints model[END_REF].

On the other hand, the decrease of the IR absorption in the XLPE characteristic bands (719, 1466, 2849 and 2914 cm -1 ) after aging, could be a result of molecular chains scission. In addition, an increase of the IR absorption band between 800 and 1128 cm -1 is observed with aging. This band could be associated with the C=C double bond of the unsaturated groups, vinylidene and trans-vinyl [START_REF] Nobrega | Investigation and analysis of electrical aging of XLPE insulation for medium voltage covered conductors manufactured in Brazil[END_REF][START_REF] Ouyang | The role of microstructure changes on space charge distribution of XLPE during thermooxidative aging[END_REF]. Also, the C=C bond of the vinylene group could be found between 1560 and 1650 cm -1 [START_REF] Liu | Effect of thermal stress on the space charge distribution of 160 kV HVDC cable insulation material[END_REF]. Moreover, it is suggested that the increase in absorption between 1169 and 1414 cm -1 is associated with the vibrations of the -C-O-C-bond [START_REF] Nobrega | Investigation and analysis of electrical aging of XLPE insulation for medium voltage covered conductors manufactured in Brazil[END_REF]. Finally, in the region between 3000 and 3500 cm -1 , a slight increase in the absorption of the -OH function of the hydroxyl groups could be noticed after aging [START_REF] Nobrega | Investigation and analysis of electrical aging of XLPE insulation for medium voltage covered conductors manufactured in Brazil[END_REF][START_REF] Ouyang | The role of microstructure changes on space charge distribution of XLPE during thermooxidative aging[END_REF].

Also, the IR absorption of the Irganox antioxidant, which is characterized by the presence of the C-O-C bond, was not detectable through ATR-FTIR analysis. This feature could be explained by the fact that the antioxidant absorption is superimposed on the IR absorption peaks of the oxidation products which occur at about 1078, 1159 and 3604 cm -1 [START_REF] Nobrega | Investigation and analysis of electrical aging of XLPE insulation for medium voltage covered conductors manufactured in Brazil[END_REF].

During thermo-oxidative reactions, the radicals generated by chains scission of the polymer could react in the presence of oxygen with another polymer chain in order to generate a peroxyl radical ROO°. Then, this last radical could capture hydrogen atoms from other polymer chains to generate hydroperoxide (ROOH) and alkyl (R°) radicals and keep the oxidative reactions auto-accelerated [START_REF] Ouyang | The role of microstructure changes on space charge distribution of XLPE during thermooxidative aging[END_REF]. The termination step involves all possible reactions between the two radicals ROO° and R°. Also, it has been shown in reference [START_REF] Cruz | Thermo-oxidative degradation of additive free polyethylene. Part I. Analysis of chemical modifications at molecular and macromolecular scales[END_REF] that peroxyl radicals could lead to the formation of a pair of alkoxy radicals (RO°) in a cage. In references [START_REF] Cruz | Thermo-oxidative degradation of additive free polyethylene. Part I. Analysis of chemical modifications at molecular and macromolecular scales[END_REF][START_REF] Fuse | Remaining issues of the degradation models of polymeric insulation used in nuclear power plant safety cables[END_REF], several possible reaction paths are explored leading to a termination and also to a non-terminating combination (diffusion out of the cage leading to the priming of a new oxidation chain). Moreover, the reactions of alkoxy radicals could result in the formation of more polar and carbonyl groups such as aldehydes and ketones [START_REF] Khelidj | Oxidation of polyethylene under irradiation at low temperature and low dose rate. Part I. The case of "pure" radiochemical initiation[END_REF].

For the sake to obtain qualitative information concerning the chemical changes caused by the thermal aging, carbonyl products were evaluated by the carbonyl index CI, characterized by the ratio of the relative intensities absorbed at 1732 cm -1 (absorption of the aldehyde) and 1466 cm -1 (methylene uptake). In addition, hydroxyl index HI was calculated as the ratio of the peak area of hydroxyl groups in the region between 3000 and 3500 cm -1 , CI and HI are shown in Figure 3.

The presence of carbonyl groups has been observed in all samples, before and after aging. The CI of the virgin sample is low and it is caused by impurities from the manufacturing process. At aging temperatures of 80 and 100 °C, the growth rate of hydroxyl groups started earlier than that of carbonyl where HI begins to increase significantly after about 1008 h at 100 °C (Figure 3b). On the other hand, thermal aging at 80 and 100 °C had a very weak effect on CI which exceeded slightly the values of the virgin sample.

For aging temperature of 120 °C, the thermo-oxidation process becomes more intense, which is reflected by a significant increase of both indexes. In addition, it has been noticed that CI increases rapidly at 120 °C after 4032 h of thermal aging. Moreover, the rise of CI is very fast at the beginning of aging at 140 °C. It has been found also that after 504 h at 140 °C, CI increases with aging time with a relatively slow rate (Figure 3a). Increasing the aging temperature obviously increases the rate of oxidation reaction and thus accelerates the generation of aging products.

Aging of XLPE at temperatures higher than the melting point (which is around 105 °C) could facilitate the oxygen diffusion into the space between the molecular chains and then leads to accelerate thermo-oxidative reactions. As a result, CI and HI at high aging temperature are significantly greater than those at lower temperatures.

DSC RESULTS

The DSC thermograms corresponding to the second heating and cooling cycles of the unaged and the aged XLPE samples are shown respectively in Figures 4 and5. To provide quantitative analysis, we considered the melting (T m) and the crystallization (T c ) temperatures, the enthalpies of fusion H m and of crystallization H c , the endothermic peak shape and the lamellar thickness.

The crystallinity rate is calculated using the Equation 1:

% 100 0   H H m c    (1)
where H0 is the theoretical melting enthalpy for the complete crystallinity PE, H0=287.3 J/g [START_REF] Nobrega | Investigation and analysis of electrical aging of XLPE insulation for medium voltage covered conductors manufactured in Brazil[END_REF]. Besides, the average lamellar thickness was calculated based on Thompson-Gibbs equation as follows:

m e m m m H T T T L   2 0 0   (2)
where Tm0 is the equilibrium melting temperature of an infinitely thick crystal T m0 = 414.6 K, T m is the melting point, ∆Hm is the enthalpy of fusion and e = 60.9×10 -3 J.m -2 is the surface free energy per unit area of basal face [START_REF] Contat-Rodrigo | Biodegradation studies of LDPE filled with biodegradable additives: Morphological changes. I[END_REF] The obtained results by DSC experiments are presented in Table 1.

The DSC thermograms in Figure 4 show an endothermic peak associated with the melting of the crystalline part in XLPE material. As the degradation goes on, the area and the intensity of the melting and crystallization peaks decrease while it moves to lower temperatures. Additionally, one main exothermic peak, characterizing the XLPE crystallization temperature, appeared on all the thermograms of Figure 5. This peak is shifted toward lower temperatures after aging at 120 and 140 °C. Therefore, based on DSC results, two steps of XLPE aging process could be distinguished.

The first step corresponds to aging temperatures below the melting point (80 and 100 °C), thermo-oxidation is inhibited by the antioxidant. Meanwhile, small parts of the lamellar crystals may melt and the oxidation mainly affects the amorphous regions. The lamellar thickness was practically not affected. In addition, low molecular weight segments could appear in the amorphous regions due to the scission of the ends of the macromolecular chains. These resulting segments crystallize and could act to enhance crystallization. This is known as a post-cross-linking phenomenon, which leads to higher homogeneity in the crystalline region. Hence, melting temperature, enthalpy of fusion, and crystallinity may increase slightly [START_REF] Ouyang | The role of microstructure changes on space charge distribution of XLPE during thermooxidative aging[END_REF].

The second phase corresponds to the higher aging temperatures (120 and 140 °C), the oxidation becomes autoaccelerated rapidly since the antioxidant has been consumed. Also, more small chain segments and dangling bonds would be released as a consequence of the main chain scissions in XLPE [START_REF] Liu | Effect of thermal stress on the space charge distribution of 160 kV HVDC cable insulation material[END_REF]. The crystalline parts, then tend to disappear, which makes the diffusion of oxygen faster [START_REF] He | Research on mechanical and dielectric properties of XLPE cable under accelerated electrical-thermal aging[END_REF][START_REF] Ouyang | The role of microstructure changes on space charge distribution of XLPE during thermooxidative aging[END_REF].

RESULTS OF THE FREQUENCY DIELECTRIC SPECTROSCOPY TEST

4.4.1 DIELECTRIC CONSTANT Figure 6 shows the evolution of the dielectric constant of XLPE samples at 20 °C for different aging conditions. The dielectric constant increases slightly as the frequency decreases. This increase is a little faster between about 10 and 10 5 Hz, which corresponds to the orientation polarization of dipoles [START_REF] Nedjar | Effect of thermal aging on the electrical properties of crosslinked polyethylene[END_REF].

Under 10 Hz, the interfacial polarization (Maxwell-Wagner-Sillars) associated with the accumulated charges on the surfaces of the electrodes occur and the dielectric constant tends to a constant value. Moreover, the dielectric constant remains practically unchanged as the aging time increases at 80 °C (Figure 6a). At this temperature, the material is in a state of scleroid solidity and thermal oxidation has a negligible effect on the dielectric constant of XLPE [START_REF] He | Research on mechanical and dielectric properties of XLPE cable under accelerated electrical-thermal aging[END_REF].

On the other hand, Figure 6b shows that the dielectric constant for the sample aged for 2016 h at 100 °C is greater than before aging. After that, it decreases with aging time and reaches a minimum after 5040 h with a value that is smaller than the unaged one. In fact, it can be assumed that when aging at temperatures below the melting point of the virgin sample (about 105 °C), the combined effects of thermal oxidation and post-cross-linking occur simultaneously during aging as shown by FTIR and DSC tests. Thermal oxidation reactions lead to the formation of polar groups, such as carbonyl, peroxide and hydroxyl groups, which contribute to increasing the dielectric constant. Nevertheless, the post-cross-linking could optimize the crystallinity of the samples leading to mitigate the accumulation of space charges and to make molecular orientation polarization more difficult [START_REF] Liu | Effect of thermal stress on the space charge distribution of 160 kV HVDC cable insulation material[END_REF]. Therefore, to our point of view the influence of thermal aging on XLPE samples would depend on which process is playing the dominating role, thermal oxidation or post-cross-linking and recrystallization. At the aging temperature of 100 °C, small parts of the crystalline regions are melted and certain aging time is necessary (longer than 2016 h) for the post-crosslinking to wield the dominant influence on the dielectric constant, which decreases with aging time [START_REF] He | Research on mechanical and dielectric properties of XLPE cable under accelerated electrical-thermal aging[END_REF][START_REF] Liu | Effect of thermal stress on the space charge distribution of 160 kV HVDC cable insulation material[END_REF].

In contrast, when samples were aged at 120 °C, the degradation process by thermo-oxidative reactions is quickly dominating. First, the dielectric constant increases slowly at the beginning of aging and remains almost unchanged until 3024 h but it then begins to increase over time, reaching a higher value after 5040 h (Figure 6c). At 140 °C, chain scission by thermal oxidation overwhelms the post-crosslinking which leads to increase the amorphous phase and make the macromolecule chains more flexible to interact under the polarization effect. At the same time, more polar groups are being generated resulting in increasing the dielectric constant drastically.

Moreover, it can be seen in Figure 6d that the dielectric constant curves at 140 °C are very close at the high frequency end (> 1 kHz) and they are spaced out as the frequency decreases. In other words, the variation of the dielectric constant as a function of aging time is more important at low frequencies due to the local accumulation of space charges in XLPE [START_REF] Li | The effect of accelerated water tree aging on the properties of XLPE cable insulation[END_REF].

DIELECTRIC LOSSES FACTOR

As far as the dielectric losses are concerned, we present in Figure 7 the variations of tan () of the XLPE for different thermal aging temperatures and durations. It can be observed that tan () is almost constant versus frequencies above 10 2 Hz. This is associated with the  relaxation due to the movement of molecular chains in the amorphous phase of XLPE [START_REF] Nedjar | Effect of thermal aging on the electrical properties of crosslinked polyethylene[END_REF][START_REF] Li | The effect of accelerated water tree aging on the properties of XLPE cable insulation[END_REF]. In this range, the dielectric losses are attributed to the orientation polarization of dipoles. Also, it is well known that dielectric losses are generally related to interfacial polarization and to conduction at low frequencies.

During aging at 80 and 100 °C, combined effects of thermal oxidation and recrystallization occur simultaneously, but remain practically without effect on the dielectric loss factor which shows a minimum peak around 0.1 Hz. On the other hand, when the degradation becomes more accentuated, at 140 °C and after 5040 h at 120 °C, the losses significantly rise. This behavior results from the generation of polar groups due to the decomposition of the material by thermal degradation.

In fact, it could be affirmed according to the DSC results that the clarity of the relaxation  at high frequencies in the most degraded samples is due to the decrease of XLPE crystallinity after aging (increase of the amorphous part). For instance, the sample aged for 5040 h at 120 ° C has a crystallinity degree of 24.07 % (against 39.75 % for the unaged sample as shown in Table 1) and  relaxation peak intensity higher than that of the unaged XLE.

Moreover, in the case of samples aged at 120 ° C for 5040 h and at 140 ° C for 504 h, the displacement of tan ( ) curves towards high frequencies (as indicated by arrows in Figure 7) could be attributed to the non-homogeneity of the physical structure of the material which could also be reflected by the non-homogeneity of the color of the samples (Figure 1d and1e). The non-homogeneity of the XLPE color seems to indicate the effect of increasing the interfacial zones and then promoting losses by Maxwell-Wagner-Sillars polarization at short times (high frequencies). This result remains very important indicating the advanced degradation state of XLPE material.

CONCLUSION

The physicochemical changes occurring in XLPE samples during thermal aging were investigated and put in relation to changes in the dynamic dielectric behavior of the insulating material. The results can be summarized as follows: 1) FTIR characterization results show that the chemical degradation of XLPE during thermal aging includes mainly the consumption of the antioxidant followed by thermo-oxidative reactions. Chemical degradation is accompanied by the generation of polar groups such as carbonyl, peroxide and hydroxyl groups.

2) The color change of XLPE to dark orange may reflect the beginning of the material deterioration. This change is attributed to the formation of vinylidene and vinyl groups.

3) DSC experiments show that the physical degradation consists in the deterioration of the crystalline structure of XLPE and decreasing of the melting temperature. 4) Post-cross-linking reactions could occur at the beginning of aging which leads to improve the thermal and dielectric properties due to the recrystallization of some parts in XLPE as shown by DSC results. 5) As the thermal stress increases, thermo-oxidative reactions cause molecular chain scission and generation of more polar groups. 6) Through the dielectric spectroscopy measurements, it was found that analysis at the low frequencies range seems to be more useful for evaluating the advanced degradation of XLPE samples. 7) The origin of dielectric losses at low frequencies is the interfacial polarization due to the space charge accumulated in the volume of the insulation, in particular, near the electrodes. 8) The non-homogeneity of XLPE color seems to indicate the increase of interfaces in the material which could lead to rise the dielectric losses due to the interfacial polarization. To sum up, the aging process can be divided into two phases: at the beginning of aging, the mechanisms of degradation are inhibited by the antioxidant introduced into the material. Also, post-cross-linking reactions could occur which leads to decrease the polarization due to the recrystallization of some parts in XLPE. During the second phase of aging, as the thermal stress increases, thermooxidative reactions take the leading role, which causes molecular chain scission and generation of more polar groups. This implies a significant increase in the dielectric constant and the dielectric losses, mainly at low frequencies. The nonhomogeneity color (structure) of aged XLPE could be very dangerous as it may indicate the appearance of more interface areas that enhances space charges accumulation.

Figure 2 .

 2 Figure 2. ATR-FTIR spectra of different XLPE samples aged at: (a) 80 °C, (b) 100 °C, (c) 120 °C and (d) 140 °C. The up and down arrows indicate respectively the increase and the decrease of the chemical group absorbance.

Figure 1 .

 1 Figure 1. Color change of XLPE samples: (a) before aging, (b) aged at 80 °C for 5040 h, (c) aged at 100 °C for 5040 h, (d) aged at 120 °C for 5040 h, (e) aged at 140 °C for 504 h, (e) aged at 140 °C for 1512 h

Figure 3 .

 3 Figure 3. Evolution of (a) Carbonyl index and (b) Hydroxyl index, versus aging time.

Figure 4 .

 4 Figure 4. DSC heating thermograms of XLPE samples before and after thermal aging at 80, 100, 120 and 140 °C.

Figure 5 .

 5 Figure 5. DSC cooling thermograms of XLPE samples before and after thermal aging at 80, 100, 120 and 140 °C.

Figure 6 .

 6 Figure 6. Variation of the dielectric constant of XLPE samples versus frequency at aging temperatures of: (a) 80 °C, (b) 100 °C, (c) 120 °C and (d) 140 °C, for different durations.

Figure 7 .

 7 Figure 7. Variation of the dielectric constant of XLPE samples versus frequency at aging temperatures of: (a) 80 °C, (b) 100 °C, (c) 120 °C and (d) 140 °C, for different durations.

Table 1 .

 1 RESULTS OBTAINED FROM DSC MEASUREMENTS (SECOND SCAN).

		Unaged	80 °C, 5040 h	100 °C, 5040 h	120 °C, 5040 h	140 °C, 1512 h
	T m (°C)	104.43	104.41	104.80	82.77	70.09
	T c (°C) Hm (J.g -1) H c (J.g -1) c (%)	91.17 -111.30 113.80 39.75	90.52 -112.70 110.80 40.25	91.30 -118.70 117.40 42.39	61.85 -67.40 66.80 24.08	51.05 -35.87 39.75 12.81
	L (nm)	4.74	4.73	4.78	2.99	2.45

60 70 80 90 100 110 120 130 140 150 -