Issaka Dialga
email: issakadialga@univ-nantes.fr

Issaka Dialga Lemna

Changing the Africa's impoverishing economic model: Towards a rewarding sustainable specialization model with a new factor of production

Keywords: impoverishing specialization, African economies, neo-factorial model, natural resources, values' chain

This paper highlights the impoverishing character of Africa's economic model in its world merchandise trade. With a view to reduce the ousting of the wealth of the continent by its trading partners, we develop a neo-factorial specialization model in which we introduce technology and raw materials as endogenous factors of production of manufactured goods. In addition, we distinguish between skilled labor and unskilled. Considering raw commodities as a factor production (natural capital) and making technology and skilled labor factors endogenous allows us to understand why Africa is historically specialized in raw material exports. We show how Africa can, thanks to its advantage in natural resources, accumulate technology and human capital necessary to its industrialization in the second phase of the model, allowing it to eliminate the impoverishing effects of trade. The model predictions are quite optimistic in the second phase of Africa's opening process to the world. Calibrating the model on real data, results are consistent with some goals of sustainable development particularly in its economic and social dimensions. The environmental dimension is however difficult to reconcile with both others.

), we analyze the overall level of employment in Africa at T1 and T2. At the point 𝐸𝐸 0 , the equilibrium is established between imports 𝐼𝐼𝑀𝑀 and exports 𝑋𝑋. 𝐿𝐿 0 * and 𝐻𝐻 0 * are determined on the labor market. Assuming that labor supply is always satisfied, that is the case in African economies because imports exceed exports.

At T2, Africa realizes a trade surplus of a unit amount equivalent to 𝑝𝑝 𝑚𝑚 -𝑟𝑟 and in terms of volume this trade surplus is 𝛥𝛥 𝑋𝑋=𝑄𝑄 𝑚𝑚 -𝑄𝑄 𝑀𝑀 . This induces an additional demand for labor especially in skilled labor 𝐻𝐻. The level of employment increases from 𝐿𝐿 1 * to 𝐿𝐿 1 * +𝐻𝐻 1 * with 𝐿𝐿 1 * < 𝐿𝐿 0 * +𝐻𝐻 0 * < 𝐿𝐿 2 * +𝐻𝐻 2 * ≤ 𝐿𝐿 + 𝐻𝐻; 𝐿𝐿 + 𝐻𝐻 being the total workforce. The 𝐻𝐻 2 * = 𝐿𝐿 2 * +𝐻𝐻 2 * -𝐿𝐿 1 *

Introduction

The last decade was marked by a sharp acceleration in world merchandise trade.

The volume of world merchandise exports increased by 3.5% per year between 2005 and 2012, while the world GDP grew by only 2% a year over the same period2 . Africa has not remained on the sidelines of globalization. The volume of trade on the continent with the rest of the world rose by 11% in exports and 13% in imports between 2005 and 2012 against an average annual growth rate of 2.01% in exports and -0.57% imports between 1953 and 2003. However, African products to global trade remain poorly diversified [START_REF] Asiedu | Foreign Direct Investment in Africa: The Role of Natural Resources, Market Size, Government Policy, Institutions and Political Instability[END_REF][START_REF] Harding | Exports, imports and foreign exchange windfalls[END_REF]Cadot et al. 2013;[START_REF] Cadot | Trade Diversification, Income, and Growth: What Do We Know?[END_REF][START_REF] Kaplinsky | Developing industrial clusters and supply chains to support diversification and sustainable development of exports in Africa[END_REF][START_REF] Anyanwu | Factors Affecting Economic Growth in Africa: Are There Any Lessons from China?[END_REF]) and are mainly composed of primary products, particularly agricultural commodities and natural resources [START_REF] Jeanneney | Régimes de change et commerce Chine-Afrique[END_REF]OCDE-BAD, 2013;Hugon, 2013). In 2010, the share of industrial products in total African exports is about 20% according to [START_REF] Gelb | Diversification de léconomie des pays riches en ressources Naturelles[END_REF], between 10 and 11% according to Hugon (2013), [START_REF] Jeanneney | Régimes de change et commerce Chine-Afrique[END_REF] or according to the United Nations Economic Commission for Africa (2013).

In other words, 80% of African exports are commodities with over 59% of non-renewable natural resources including oil and minerals. The predominance of raw materials in Africa's export products seem consistent with traditional theories of international trade according to which each country specializes in the production and marketing of goods in which it has an absolute advantage (Smith) relative (Ricardo) or factorial (Heckscher-Ohlin-Samuelson (HOS)) 3 . The impoverishing specialization refers to an economic model in where one exports lot of goods and earns less in return in terms of money.

This paper aims to show, with stylized facts and mathematical proofs, that the current specialization of Africa in world merchandise trade presents facts that contrast with some predictions of international trade patterns. We develop a neo-factorial 4 specialization model which predicts that Africa can move from an impoverishing specialization in the first period (current specialization) to an enriching global exchange through an optimization of the values chain by industrialization.

The article is organized as follows: Section 2 presents stylized facts of African international trade and defines a specialization rule. Section 3 develops the neo-factorial specialization model. We present the theoretical foundations of the model and its assumptions before analyzing the predictions of the model in terms of new specialization gains and sustainability of African economies. Section 4 concludes by returning to the important results of the model and the future challenges of research on the issue.

Stylized facts and specialization rule

The abundance in Natural Resources (NR) and agricultural products in Africa seems to justify its exclusive exports of raw materials. In consequence, agricultural products account for 25% of Africa's GDP. Africa has on average 52 % of global reserves in NR, shared as follows:

12% of global oil reserves; 40% of world gold reserves; 85 to 95% of metal reserves of chromium and platinum group; 85% of phosphate reserves; more than 50 % of cobalt reserves; and a third of bauxite reserves 5 . According to the HOS model of factor endowments, each country gains from international trade by specializing in the production and export of goods that use more factors for which the country is relatively well endowed. The specialization index 6 can help us to verify the compliance of international exchange facts from Africa with the predictions of specialization models. Let 𝑀𝑀, raw materials, 𝑚𝑚, finished product, 𝐼𝐼𝑀𝑀 𝑀𝑀 , 𝐼𝐼𝑀𝑀 𝑚𝑚 , 𝑋𝑋 𝑀𝑀 and 𝑋𝑋 𝑚𝑚 , the import and export quantities of both types of goods respectively.

We calculate the Africa 'specialization index. According to the predictions of the HOS model, the 𝑀𝑀 and 𝑚𝑚 goods export ratio (

𝑋𝑋 𝑀𝑀 𝑋𝑋 𝑚𝑚

) is greater than the 𝐼𝐼𝑀𝑀 𝑀𝑀 𝐼𝐼𝑀𝑀 𝑚𝑚 by volume, due to the 𝑀𝑀model". The model is consider as new, hence the prefix "neo-", because we made the following changes: introduction of natural capita, abstraction of capital factor, which is decisive in the traditional patterns of international trade, and making technology factor endogenous. . The SI is based on the logic of calculation of Leontief specialization coefficient and is interpreted in the same way i.e. if a country exports products which are more intensive in factor for which the country is well endowed, its 𝑆𝑆𝐼𝐼 < 0. This is obviously in agreement with the predictions of HOS models, also emphasized by [START_REF] Prebisch | Five stages in my thinking on development[END_REF] in the so-called dependence theory or in unequal exchange model (see [START_REF] Arghiri | L'échange inégal essai sur les antagonismes dans les rapports économiques internationaux[END_REF]. However, do specialization models really justify the exclusive choice exports of raw commodities in Africa with regard to its current specialization's gains?

More obvious that may seem, it is difficult to validate the export choice of raw materials based on the foundations of traditional international trade models. First, these models base their analysis on tradable manufactured goods, not raw materials. There are relative gains when it is exchange of finished products, which generate added value. Then, we can raise a contradiction between the stylized facts of Africa and predictions of specialization models.

Indeed, these specialization patterns in international trade predict that all countries, participating in the international merchandise trade, gain. So, there is no loser. However, Africa loses more than it gains. For proof, it suffices to calculate a Trade Gain Index (TGI)7 defined as follows: In other words, a 𝑇𝑇𝑇𝑇𝐼𝐼 > 1 means that exports of raw materials without processing enable Africa to import from its trading partners manufactured goods and capital goods that it does not produce. Its exchanges with the rest of the world are in this case enriching. Using WTO data, we find an index of 0.709. Fosu (2011) highlighted similar results. This result suggests that around 30% of expected gains of Africa in the international exchange of tradable goods are ousted by its trading partners. Gross exports of raw materials of Africa are, in view of this result, an impoverishing specialization's way insofar as it is obliged to export more to satisfy its demand for manufactured goods and equipment.

The [START_REF] Prebisch | Five stages in my thinking on development[END_REF] center-periphery relationship and the extraversion of African economies of [START_REF] Bayart | L'Afrique dans le monde : une histoire d'extraversion[END_REF][START_REF] Hugon | L'économie de l'Afrique. 6e édition[END_REF][START_REF] Obeng-Odoom | Africa's Failed Economic Development Trajectory: A Critique[END_REF] can serve to explain the stylized facts about Africa's international trade. In [START_REF] Prebisch | Five stages in my thinking on development[END_REF], global economic relations can be summarized as a system with two sets. There is the center of the system focused on transformation means and creation of value added products, and the periphery of the system, in charge of providing necessary materials to the center for the production of goods manufactured (see [START_REF] Koulibaly | The increasing returns revolution in trade and geography[END_REF]. In global trade, Africa would incarnate the role of provider of raw materials. For [START_REF] Bayart | L'Afrique dans le monde : une histoire d'extraversion[END_REF] and Hugon (2013), the current specialization of Africa in world trade is the result of the colonial pact 8 that continues to exist due to the lack of profound changes in its economic structures after the independences. Assuming that Africans recognize the current trade patterns are impoverishing, the neo-factorial specialization model provides the conditions for a rewarding specialization for Africa and analyzes the possible effects of this new specialization in terms of sustainable development goals.

The neo-factorial specialization model

The theoretical foundations of the model

The model is based on the postulate that there are technology endowment and natural resources abundance differences between countries. It further postulates that the distinction between skilled and unskilled labor in the production of tradable goods internationally is relevant. Unlike traditional specialization models, advanced or technological endowment gives country preferential gains in international exchange. Finally, the model postulates that the industrialization of Africa is a necessary condition to reverse the situation of [START_REF] Bhagwati | Immiserizing Growth: A Geometrical Note[END_REF] impoverishing specialization and pointed out in Obeng-Odoom (2013).

Technological differences and specialization of countries

Taking into account the technological differences between countries in the specialization models is not new. It is even the source of theories of specialization of Smith and Ricardo. [START_REF] Vernon | International investment and international trade in the product cycle[END_REF] already highlighted the technological differences in the production of tradable goods through its product cycle model 9 .

Based on spatial analysis, Mérenne-Schoumaker (2011) supports that large urban areas in developed countries often gather the conditions conducive to launching new products while 8 The colonial pact is a device conceived at the time of colonization by settlers in which economies under colonial rule had to resolve to produce enough exotic products, raw materials and resources and then, should export them to the mainland to feed and support as long as possible the economic structures of the invader. 9 For specific literature about technology in tradable goods production, see [START_REF] Aliouat | Les Stratégies de coopération industrielle[END_REF]; [START_REF] Ciroth | Geographical and Technological Differences in Life Cycle Inventories Shown by the Use of Process Models for Waste Incinerators Part I. Technological and Geographical Differences[END_REF]; [START_REF] Perrain | la R&D dans les PME, les ETI et les grandes entreprises[END_REF]; [START_REF] Huet | Les technologies comme facteur de développement des pays[END_REF]; [START_REF] Rousseau | Technological differences in job characteristics, employee satisfaction, and motivation: A synthesis of job design research and sociotechnical systems theory[END_REF]; [START_REF] Bessen | Technology Adoption Costs and Productivity Growth: The Transition to Information Technology[END_REF]; [START_REF] Madsen | The Indian growth miracle and endogenous growth[END_REF]; [START_REF] Taylor | Trade, Tragedy, and the Commons[END_REF].

peripheral areas especially the "Third World'' meet the requirements to accommodate the manufacture of products in maturity. This is the design "center-periphery" of international trade [START_REF] Prebisch | Five stages in my thinking on development[END_REF] in which innovations and new technologies start from developed countries (center) and then spread to the rest of the world while the periphery supplies raw materials for the development application of these discoveries. [START_REF] Feenstra | Putting Things in Order: Trade Dynamics and Product Cycles[END_REF] empirically validated the model of Vernon by US data. However, the rapid mutations in technology reduce the relevance of the analysis. Indeed, technological advances in R & D, particularly the boom in ICT, make that the time required for a discovery to become obsolete is relatively short. So, other countries benefit from the invention at the same time as its inventor. The recent illustration is Asian countries including China [START_REF] Wolf | China and Latecomer-Industrialisation Processes in Sub-Saharan Africa : Situating the Role of (Industrial) Policy[END_REF][START_REF] Kaplinsky | China and the terms of trade: The challenge to development strategy in Sub-Saharan Africa[END_REF][START_REF] Busse | China's Impact on Africa -The Role of Trade, FDI and Aid[END_REF][START_REF] Anyanwu | Factors Affecting Economic Growth in Africa: Are There Any Lessons from China?[END_REF][START_REF] Sindzingre | Fostering Structural Change? China's Divergence and Convergence with Africa's Other Trade and Investment Partners[END_REF]WTO 2011).

Technology is the foundation of dynamic models [START_REF] Solow | A Contribution to the Theory of Economic Growth[END_REF]). Indeed, studying the sources of growth in the US over a long period , Solow realizes that much of the economic growth is not explained by the sum of the relative contributions of rates traditional growth factors namely capital and labor. The growth of capital per capita for 40 years in the United States has explained a seventh of the growth of total production.

Solow concludes that other unseen factors influence total factor productivity. Those include such factors as the technical advances in the broad sense in that subsequent studies have highlighted the Solow residual differences following the country's level of development (see [START_REF] Mankiw | A Contribution to the Empirics of Economic Growth[END_REF]. Empirical evidences indicate that this residue is only 14% of the Total Factor Productivity (TFP) in a sample of 145 countries, while TFP reaches 34% for developed countries [START_REF] Mankiw | A Contribution to the Empirics of Economic Growth[END_REF].

More recent analyzes have also attempted to capture the technological differences between countries, showing that there is a technological gap between the North and the South [START_REF] Hellier | North-South Globalization and Inequality[END_REF] or between Asian emerging countries and African countries [START_REF] Kaplinsky | China and the terms of trade: The challenge to development strategy in Sub-Saharan Africa[END_REF]WTO 2011;[START_REF] Busse | China's Impact on Africa -The Role of Trade, FDI and Aid[END_REF][START_REF] Anyanwu | Factors Affecting Economic Growth in Africa: Are There Any Lessons from China?[END_REF][START_REF] Sindzingre | Fostering Structural Change? China's Divergence and Convergence with Africa's Other Trade and Investment Partners[END_REF][START_REF] Wolf | China and Latecomer-Industrialisation Processes in Sub-Saharan Africa : Situating the Role of (Industrial) Policy[END_REF]).

These differences, according to the authors, lead to differences in terms of productivity gains in a first time and then a catch of seconds by the firsts in the case of North-South analysis, through a learning process and adaptation (learning-by-doing). In this latter case, technology is an exogenous factor in the production and is analyzed in terms of contribution to overall productivity factors.

If the models that consider technology help explaining differences in productivity, neofactorial model makes this factor endogenous as in [START_REF] Stadler | Business Cycle Models with Endogenous Technology[END_REF]; [START_REF] Xu | Endogenous Technology Bias, International Trade, and Relative Wages[END_REF]; Gerlagh and van der Zwaan (2003); Gil [START_REF] Moltó | Cooperative R&D with Endogenous Technology Differentiation[END_REF]; [START_REF] Taylor | Trade, Tragedy, and the Commons[END_REF]; [START_REF] Madsen | The Indian growth miracle and endogenous growth[END_REF] or in [START_REF] Anzoategui | Endogenous Technology Adoption and R&D as Sources of Business Cycle Persistence[END_REF]. Make technology endogenous helps us to explain why most of African countries do not produce high valueadded goods. This consideration helps explaining how the availability or not of technology influences the specialization rule.

Natural resource endowment and specialization in the production of manufactured goods

Excepted the [START_REF] Stiglitz | Growth with Exhaustible Natural Resources: Efficient and Optimal Growth Paths[END_REF] model in where NR are partially considered as factors of production, or the [START_REF] Russo | The Emergence of Sustainable Industries: Building on Natural Capital[END_REF], [START_REF] Giraud | Economie des matières premières[END_REF] framework analyses integrating natural capital, it exists, to our knowledge, very few models which put NR as factor of production. The criticism on African economies' extraversion [START_REF] Bayart | L'Afrique dans le monde : une histoire d'extraversion[END_REF] finds that Africa resolves to export only its natural resources without developing an economic modeling of this type of exchange. In this model we make distinction between natural resources (renewable and non-renewable) and land as factors of production. The latter served as the analytical basis of Thünen, (1826) and HOS models.

To test empirically relevance of NR as factor of production, [START_REF] Ayres | Exergy, power and work in the US economy, 1900-1998[END_REF] showed that the growth factors in an economy are diverse and cannot be limited to only those factors identified in traditional growth models. They showed that, unlike the conventional designs according to which the consumption of raw materials is a consequence of economic growth, these raw materials, in particular energy, are the cause of the economic growth. The availability in NR allows producing at reduced costs and quality of inputs allows for economies of scale. In a competitive situation, the drop in production costs and economies of scale induce a fall in prices of goods that result. Overall demand increases inducing new investments, which in turn stimulate overall production. From the spatial point of view, as "the territories appear more and more as resources that companies can activate" (Mérenne-Schoumaker 2011 p.239-240), the greater or lesser availability of NR, including those used in finished products, becomes an important advantage to be taken into account in the models of production and international trade.

First, it should be noted that NR are not ubiquitous: some areas are better endowed in resources than others. This characteristic is a first point of difference in factor endowment.

Any industrial company which comes closer to raw materials' place of supply can be more competitive than its potential competitors since transport costs are negligible (see [START_REF] Bouvard | La localisation des activités économiques au sein de l'Aire Urbaine de Lyon[END_REF]. The proximity to raw materials allows companies that settled to expand their potential market: the countries possessing these resources are also a significant potential10 market flow of tradable goods. The China-Africa economic relationships illustrate this new configuration of the international exchange11 . Indeed, the volume of manufactured exports towards some developing countries in particular towards sub-Saharan Africa countries has dramatically increased. According to [START_REF] Jeanneney | Régimes de change et commerce Chine-Afrique[END_REF] China's exports of manufactured goods to Africa represents 95% of African imports in 2010 while Chinese exports to the rest of the world account for only 23% of total Chinese exports the same date.

In return, China imports 90% of African primary products, mainly raw materials for Chinese industries of which 59% oil and 26% minerals [START_REF] Park | China's Impacts on Africa's Development[END_REF]. China diverts Africa from its old economic partner namely Europe. However, the increase of this southern market -from the perspective of growth -is not accompanied by an improvement in the purchasing power of the population that is relatively very low12 .

Then, the issues related to the depletion of raw materials oblige to rethink economic development models. Studies indicate that firms are relocating to escape purely environmental regulations considered very restrictive in the North: "Measures relating to the protection of the environment are becoming an obstacle for the most polluting companies. But regulations concerning pollution vary across countries, cities and regions. The most polluting companies move of the most regulated areas towards the more tolerant." (Bouvard and Million, 2008, p.28). However, note that environmental costs are very marginal in total costs, the environment constraints are not the main justifications of the factories' relocation.

Environmental constraints and the depletion of natural resources make that they will become less and less available following the rule of [START_REF] Hotelling | The Economics of Exhaustible Resources[END_REF]. But experiences show that exogenous prices determined by raw materials market are inefficient to regulate NR depletion (see [START_REF] Ruta | International Trade in Natural Resources: Practice and Policy[END_REF][START_REF] Chamaret | Une démarche top-down / bottom-up pour l'évaluation en termes multicritères et multi-acteurs des projets miniers dans l'optique du développement durable. Application sur les mines d'Uranium d'Arlit (Niger)[END_REF]. The challenge in this economic renewal is to ensure that consumption of raw materials is the cause of economic growth and not the consequence. In that vision, the raw materials should be considered as factors in the economic model of production as well as physical capital or labor. This amounts to more explicitly assign a price to endogenous factor.

Model hypotheses

The model is based on three assumptions: the sovereignty of Africa in the extraction and marketing of its natural resources, a difference in endowment of production factors, a difference in goods in international exchange and finally the existence of a supranational regulator.

H1: the model assumes that Africa is sovereign of the rational use of its natural resources and agricultural raw materials 𝑀𝑀. It has the opportunity to decide how much quantity to produce.

This implies that Africa can decide to transform its NR noted 𝑀𝑀 locally into manufactured goods 𝑚𝑚 before export. This assumption is fundamental to the validation of the model results. capital is assumed to be 1. Indeed, Mérenne-Schoumaker (2011) supports that if the cost and availability of capital have played a major role in the start of industrialization, this role has currently decreased at regional and local level due to the very high fluidity of this input.

Taking into account the considerations above allows us to understand why Africa is historically specialized in raw material exports and how to improve the trade gains under sustainable development goals considerations.

The production of goods 𝑀𝑀 and 𝑚𝑚 generates different levels of pollution, proportionally to the quantities 𝑄𝑄 𝑀𝑀 and 𝑄𝑄 𝑚𝑚 of raw materials and manufactured goods produced. Technology is assumed to be constant returns to scale. Production functions are given by: Africa specializes in the exclusive export of 𝑀𝑀 and 𝑚𝑚 in period 1 and 2 respectively. Raw materials 𝑀𝑀 are considered as goods when they are exported in the state, and are considered as factor when they enter the production of manufactured goods 𝑚𝑚.

𝐴𝐴 𝑀𝑀 (𝐴𝐴 𝑀𝑀 , 𝐿𝐿) = 𝐴𝐴 α 𝑀𝑀 𝐿𝐿 β 𝑀𝑀 =(1 + 𝑛𝑛 𝑀𝑀 )𝑄𝑄 𝑀𝑀 (1) 𝐴𝐴 𝑚𝑚 (𝐴𝐴 𝑚𝑚 , 𝐻𝐻, 𝑀𝑀) = 𝐴𝐴 α 𝑚𝑚 𝐿𝐿 β 𝑚𝑚 𝑀𝑀 η 𝑚𝑚 = (1 + 𝑛𝑛 𝑚𝑚 )𝑄𝑄 𝑚𝑚 (2) with 0 < α ≤ β ≤ η < 1,
H3: there is a public and supranational regulator (African Union) who plans the exploitation of NR and spreads them over three periods (T1 to T3). It also regulates the level of tolerable pollution Ē in Africa. Although studies indicate that regulation by carbon markets is economically more efficient than emission standard, the lack of this type of market in Africa [START_REF] Thiombiano | [END_REF] leads us to retain the standard. The environmental constraint is given by

∑ 𝑛𝑛 𝑖𝑖 𝑛𝑛 =1
𝑄𝑄 𝑖𝑖 ≤ Ē. In the event of a pollution proportional to economic activity, environmental constraint indicates that the pollution emitted by all economic activities should not exceeded the tolerable standard set by the regulator.

One assumes that in period T1, Africa has neither the technology 𝐴𝐴 nor enough skilled labor 𝐻𝐻to produce goods 𝑚𝑚. It specializes in the production and export of 𝑀𝑀. The regulator sets exploitation contracts of share of NR planned for the period 1. The regulator receives a share 𝑎𝑎 of NR extracted as an annuity (0 <𝑎𝑎 <1). In addition, contracts are indexed to changes in global commodity prices 𝑟𝑟. The negotiated unit rent is 𝑎𝑎𝑟𝑟. Let 𝑁𝑁𝑁𝑁 1 the share of NR extracted at T1. The total rent got by the regulator is 𝑎𝑎𝑟𝑟𝑁𝑁𝑁𝑁 1 . The total rent is assigned to cover both technology 𝐴𝐴 and labor 𝐻𝐻 training costs at T2. At period 2, Africa has all the factors essential to produce both goods 𝑀𝑀 and 𝑚𝑚. However, goods M is used as factor, so Africa exports only manufactured goods.

Reserves in NR in the period T3 imposed by the regulator are assumed to be used in the same conditions as described at T2. The optimization program over the two firsts periods and the optimal equilibria are showed in Appendix 1, 2, 3. Analyzing the equilibria, Equation ( 8) shows that with a view to optimizing the exploitation of raw materials, the technology appears as a very crucial factor. However, under social sustainable consideration, there is a trade-off between the level of technology and the volume of human employment in endogenous growth models that is not exactly the message send by Equation ( 7). The future challenge is to find specification which reflects this trade-off. As illustrated in Figure 1, a hyper sophistication of the economy by an advanced or an abundance of technology (robotics) evicts some potential jobs that would result from an economic boom.

The employment level ousted by robotics is estimated at 30% of total employment in 2030.

The research's challenge is to determine the optimal level of technology 𝐴𝐴 opt required to make effective the production of both goods and then, to be also consistent with the optimal volume of employment 𝐸𝐸 opt .

Equation ( 15 As the equilibria are determined sequentially, they are unstable, making some of them under optimal. Because, this can be interpreted as a weakness of the findings of the model, the challenge is to find mathematical technique which allows determining these equilibria jointly.

The impacts of the model equilibria on SD goals

The model distinguishes two phases of specialization (T1 and T2) in Africa's international trade.

At T1, Africa, due to its disadvantage in technology and skilled labor shortage, specializes in the production and export of raw materials 𝑀𝑀 . Levels of employment and pollution at T1 are low due to low diversification in production in the continent (Figure 2). Stylized facts show that given an optimal level of exported raw materials 𝑄𝑄 𝑀𝑀 * , trade revenues 13 do not cover the needs of the continent imports because Africa's TGI< 1 (Table 1). The depletion of NR does not follow by other types of wealth accumulation as [START_REF] Hartwick | Intergenerational Equity and the Investing of Rents from Exhaustible Resources[END_REF] suggests At T2, Africa exports only good 𝑚𝑚. This new configuration changes the trade equilibria of Africa as shown in Figure 2. These equilibria can be analyzed in terms of SD goals.

13 According to the WTO definition, exports are valued on the basis of the transaction value, including the cost of transportation and insurance for the delivery of goods to the border of the exporting country or territory (value "FOB "). Imports are valued on the basis of the transaction value plus the cost of transportation and insurance to the frontier of the importing country or territory (value "cost, insurance and freight"), WTO Report, 2013, p.198.

Figure 2-Trade equilibrium and Employment in Africa before and after neo-factorial specialization

On the 𝑀𝑀 and 𝑚𝑚 goods market

𝐸𝐸 0
On the 𝑳𝑳 and 𝑯𝑯 Labor market

The model predicts that for equal quantities of raw materials 𝑄𝑄 𝑀𝑀 * produced in periods 1 and 2, Africa realizes preferential benefits to the international exchange by exporting manufactured goods. This advantageous situation leads in improved TGI (see Figure 3).

Proof of this TGI's evolution is showed in Appendix 4. In Figure 2, the trade surplus is represented by 𝑝𝑝 𝑚𝑚 -𝑟𝑟, where 𝑝𝑝 𝑚𝑚 is the international prices of manufactured goods and 𝑟𝑟, the price of commodity. In volume terms, the trade surplus is

𝛥𝛥 𝑋𝑋=𝑄𝑄 𝑚𝑚 -𝑄𝑄 𝑀𝑀 .
However, how this advantage could be maintained when one integrates the production costs of manufactured goods, including costs of technology and engineering education? To answer that, we calculate the overall profit of the production chain of goods 𝑚𝑚 at T2. show that the overall profit is positive when the green unit price covers the average cost of producing the good 𝑚𝑚 i.e. the remuneration of workers 𝐿𝐿 and 𝐻𝐻 taking into account the cost remediation of the Environment caused by the exploitation of raw materials and pollutants discharged.

How do the model results affect the social dimension of SD? Social variable is labor. As showed by Figure 2, the specialization of Africa in period 2 is not only enriching economically; it has positive effects on the social dimension.

𝑇𝑇

0 1 𝑇𝑇1 𝑇𝑇2 𝑋𝑋 𝑀𝑀 < 𝐼𝐼𝑀𝑀 𝑚𝑚 𝐼𝐼𝑀𝑀 𝑚𝑚 -α𝑋𝑋 𝑚𝑚 < (1 -α)𝑋𝑋 𝑚𝑚
gap is called structural unemployment. This level of underemployment in period 1 is endogenous to the choice of specialization that Africa suffers at T1. Figure 2 highlighted the wage differential between skilled and unskilled workers as pointed out in [START_REF] Hellier | North-South Globalization and Inequality[END_REF]. The skill premium is much higher in Africa than that expected in the international equilibrium, i.e.

at the point 𝐸𝐸 0 .

𝑄𝑄 𝑚𝑚 ⇒ (-) Social Dynamism Employment 𝐿𝐿 1 * < 𝐿𝐿 0 * +𝐻𝐻 0 * ⇒ (+) 𝐿𝐿 1 * < 𝐿𝐿 0 * +𝐻𝐻 0 * < 𝐿𝐿 2 * +𝐻𝐻 2 * ≤ 𝐿𝐿 + 𝐻𝐻 ⇒ (++) Environment Equilibrium Pollution 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 ≤ Ē ⇒ (-) 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 + 𝑛𝑛 𝑚𝑚 𝑄𝑄 𝑚𝑚 ≤ Ē ⇒ (--) Pollution treatment 𝜏𝜏 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 ⇒ (+) 𝜏𝜏 𝑛𝑛 𝑚𝑚 𝑄𝑄 𝑚𝑚 +𝜏𝜏 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 ⇒ (+) NR Depletion 𝑆𝑆 0 -𝑁𝑁𝑁𝑁 1 < 𝑆𝑆 0 (1 + 𝒂𝒂) ⇒ (-) 𝑆𝑆 0 -( 𝑄𝑄 𝑀𝑀 + 𝑁𝑁𝑁𝑁 1 ) < 𝑆𝑆 0 (1 + 𝑎𝑎) ⇒ (--)
Source : Author (+) ; (-) indicate a positive effect and a negative effect of the model's results on the SD goals; (+ +) ; (--) indicate a positive impact and greater negative effects respectively; 𝑆𝑆 0 is the stock of NR. 𝑎𝑎 denotes the renewal rate of the raw material.

At T3, we will attend a coexistence of two economic models of production. Countries that have built up reserves in NR, will continue producing manufactured goods 𝑚𝑚, as at T2. However, they use more efficient production technologies due to the gradual depletion of the stock of certain resources. Countries that have exhausted their stocks factors 𝑀𝑀 during period 2 or those that have none, adopt the circular economy model. They therefore invest more in sustainable production patterns such as renewable sources. In the longer term, the scarcity of factor 𝑀𝑀 will lead to a more stable equilibrium of factors of production than its level at T2.

Model simulation

In this section, we simulate the model using data on NR endowment in Africa. We 2 and Table 3 are an illustration of a representative firm in both 𝑀𝑀 and 𝑚𝑚 goods production.

85.872>34.218

Hypotheses confirmed

𝑖𝑖𝑖𝑖 𝑝𝑝 𝑚𝑚 -𝜏𝜏 𝑛𝑛 𝑚𝑚 < [ω 𝐿𝐿 𝐿𝐿+ 𝜏𝜏 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 +ω 𝐻𝐻 𝐻𝐻] 𝑄𝑄 𝑚𝑚 (-)

85.872>34.218

Hypotheses rejected equivalent/year against a tolerable threshold of 1299984t / year in Africa.

𝐿𝐿 1 * < 𝐿𝐿 0 * +𝐻𝐻 0 * < 𝐿𝐿 2 * +𝐻𝐻 2 * ≤ 𝐿𝐿 + 𝐻𝐻 ⇒ (++) (

Conclusion

This paper aimed to show, through the stylized facts, that the current specialization of Africa in world merchandise trade is an impoverishing one. We then developed a model of neofactorial specialization that highlights the crippling factors industrialization of the continent. (5) , one obtains

𝐴𝐴 ′ 𝑀𝑀 (𝐴𝐴 𝑀𝑀 ) 𝐴𝐴′ 𝑀𝑀 (𝐿𝐿) � = 𝑐𝑐 ′�𝐴𝐴 𝑀𝑀 � ω 𝐿𝐿 (6)
Equation ( 6) indicates that at the optimum, production factors 𝐴𝐴 and 𝐿𝐿 are paid up to their respective marginal productivities.

From equation ( 6), optimal function of each input is determined.

We have:

𝐴𝐴 ′ 𝑀𝑀 (𝐴𝐴 𝑀𝑀 ) = α 𝑀𝑀 𝐴𝐴 α 𝑀𝑀-1 𝐿𝐿 β 𝑀𝑀 and 𝐴𝐴′ 𝑀𝑀 (𝐿𝐿) = β 𝑀𝑀 𝐴𝐴 α 𝑀𝑀 𝐿𝐿 β 𝑀𝑀-1 Hence, 𝐴𝐴 ′ 𝑀𝑀 (𝐴𝐴 𝑀𝑀 ) 𝐴𝐴′ 𝑀𝑀 (𝐿𝐿) � = α 𝑀𝑀 𝐿𝐿 β 𝑀𝑀 𝐴𝐴 � = 𝑐𝑐 ′�𝐴𝐴 𝑀𝑀 � ω 𝐿𝐿 ⇒ 𝐿𝐿 * = β 𝑀𝑀 𝑐𝑐 ′�𝐴𝐴 𝑀𝑀 � α 𝑀𝑀 ω 𝐿𝐿 𝐴𝐴 (7)
Equation ( 7) shows that the number of low-skilled labor depends on the technology employed, respective yields of the two factors of production and of course the wage rate of unskilled labor. The optimal output 𝑄𝑄 𝑀𝑀 * is given by:

𝐴𝐴 𝑀𝑀 (𝐴𝐴 𝑀𝑀 , 𝐿𝐿) = 𝐴𝐴 α 𝑀𝑀 ( β 𝑀𝑀 𝑐𝑐 ′(𝐴𝐴 𝑀𝑀 ) α 𝑀𝑀 ω 𝐿𝐿 𝐴𝐴) β 𝑀𝑀 = 𝐴𝐴 α 𝑀𝑀+β 𝑀𝑀 ( β 𝑀𝑀 𝑐𝑐 ′(𝐴𝐴 𝑀𝑀 ) α 𝑀𝑀 ω 𝐿𝐿 ) β 𝑀𝑀
As returns to scale are assumed constant, then α 𝑀𝑀+β 𝑀𝑀 = 1 ; hence

𝐴𝐴 𝑀𝑀 (𝐴𝐴 𝑀𝑀 , 𝐿𝐿) = 𝑄𝑄 𝑀𝑀 * = 𝐴𝐴( β 𝑀𝑀 𝑐𝑐 ′�𝐴𝐴 𝑀𝑀 � α 𝑀𝑀 ω 𝐿𝐿 ) β 𝑀𝑀 (8)
Appendix 3-Determination of factors 𝐀𝐀 𝐦𝐦 and 𝐇𝐇 Similarly, we determine the optimal amounts of factors 𝐴𝐴 𝑚𝑚 and 𝐻𝐻 used in the production of manufactured goods at T2.

At each period (see equation ( 3)), the Lagrangian is written:

𝐿𝐿 𝑚𝑚 (𝐴𝐴 𝑚𝑚 , 𝐻𝐻, 𝑀𝑀, 𝜆𝜆) = 𝐴𝐴 𝑚𝑚 (𝐴𝐴 𝑚𝑚 , 𝐻𝐻, 𝑀𝑀) + 𝜆𝜆(𝐶𝐶( 𝑄𝑄 𝑚𝑚 , 𝐸𝐸 𝑚𝑚 ) -𝑐𝑐(𝐴𝐴 𝑚𝑚 ) -ω 𝐻𝐻 𝐻𝐻 -𝑟𝑟𝑀𝑀 -𝜏𝜏 𝑛𝑛 𝑚𝑚 𝑄𝑄 𝑚𝑚 )
The FOC imply that:

𝐿𝐿 𝑚𝑚 (𝐴𝐴 𝑚𝑚 ,𝐻𝐻,𝑀𝑀,𝜆𝜆) 𝜕𝜕𝐴𝐴 𝑚𝑚 =0⟺𝐴𝐴 ′ 𝑚𝑚 (𝐴𝐴 𝑚𝑚 ) -𝜆𝜆𝑐𝑐 ′(𝐴𝐴 𝑚𝑚 ) = 0 ⇒ 𝐴𝐴 ′ 𝑚𝑚 (𝐴𝐴 𝑚𝑚 ) = 𝜆𝜆𝑐𝑐 ′(𝐴𝐴 𝑚𝑚 ) (9) 𝐿𝐿 𝑚𝑚 (𝐴𝐴 𝑚𝑚 ,𝐻𝐻,𝑀𝑀,𝜆𝜆) 𝜕𝜕𝐻𝐻 =0⟺𝐴𝐴′ 𝑚𝑚 (𝐻𝐻) -𝜆𝜆ω 𝐻𝐻 = 0 ⇒ 𝐴𝐴′ 𝑚𝑚 (𝐻𝐻) = 𝜆𝜆ω 𝐻𝐻 (10) 𝐿𝐿 𝑚𝑚 (𝐴𝐴 𝑚𝑚 ,𝐻𝐻,𝑀𝑀,𝜆𝜆) 𝜕𝜕𝑀𝑀 =0⟺𝐴𝐴′ 𝑚𝑚 (𝑀𝑀) -𝜆𝜆𝑟𝑟 = 0 ⇒ 𝐴𝐴′ 𝑚𝑚 (𝑀𝑀) = 𝜆𝜆𝑟𝑟 (
Equation ( 13) indicates that at the optimum, production 𝐴𝐴 𝑚𝑚 and 𝐻𝐻 are paid up to their respective marginal productivities.

From equation ( 13), optimal function of each input is determined.

𝐴𝐴 ′ 𝑚𝑚 (𝐴𝐴 𝑚𝑚 ) = α 𝑚𝑚 𝐴𝐴 α 𝑚𝑚-1 𝐻𝐻 β 𝑚𝑚 𝑀𝑀 η 𝑚𝑚 et 𝐴𝐴′ 𝑚𝑚 (𝐻𝐻) = β 𝑚𝑚 𝐴𝐴 α 𝑚𝑚 𝐻𝐻 β 𝑚𝑚-1 𝑀𝑀 η 𝑚𝑚 Hence, 𝐴𝐴 ′ 𝑚𝑚 (𝐴𝐴 𝑚𝑚 ) 𝐴𝐴′ 𝑚𝑚 (𝐻𝐻) � = α 𝑚𝑚 𝐻𝐻 β 𝑚𝑚 𝐴𝐴 𝑚𝑚 � = 𝑐𝑐 ′(𝐴𝐴 𝑚𝑚 ) ω 𝐻𝐻 ⇒ 𝐻𝐻 * = β 𝑚𝑚 𝑐𝑐 ′(𝐴𝐴 𝑚𝑚 ) α 𝑚𝑚 ω 𝐻𝐻 𝐴𝐴 𝑚𝑚 (14)
Equation ( 14) indicates that the number of skilled workers to be employed also depends on the technology used, respective yields of the two factors of production and the wage rate of skilled workers. The optimal output 𝑄𝑄 𝑚𝑚 * is given by 𝐴𝐴 𝑚𝑚 (𝐴𝐴 𝑚𝑚 , 𝐻𝐻, 𝑀𝑀) =

𝐴𝐴 𝑚𝑚 α 𝑚𝑚 ( β 𝑚𝑚 𝑐𝑐 ′(𝐴𝐴 𝑚𝑚 ) α 𝑚𝑚 ω 𝐻𝐻 𝐴𝐴 𝑚𝑚 ) β 𝑚𝑚 𝑀𝑀 η 𝑚𝑚 = 𝐴𝐴 α 𝑚𝑚+β 𝑚𝑚 ( β 𝑚𝑚 𝑐𝑐 ′(𝐴𝐴 𝑚𝑚 ) α 𝑚𝑚 ω 𝐻𝐻 ) β 𝑚𝑚 𝑀𝑀 η 𝑚𝑚
By replacing factor 𝑀𝑀 by its optimum value previously determined in equation ( 8), the maximum output obtained is manufactured goods given by:

𝐴𝐴 𝑚𝑚 (𝐴𝐴 𝑚𝑚 , 𝐻𝐻, 𝑀𝑀) = 𝑄𝑄 𝑚𝑚 * = 𝐴𝐴 α 𝑚𝑚+β 𝑚𝑚 ( β 𝑚𝑚 𝑐𝑐 ′(𝐴𝐴 𝑚𝑚 ) α 𝑚𝑚 ω 𝐻𝐻 ) β 𝑚𝑚 �𝐴𝐴( β 𝑀𝑀 𝑐𝑐 ′(𝐴𝐴 𝑀𝑀 ) α 𝑀𝑀 ω 𝐿𝐿 ) β 𝑀𝑀 � η 𝑚𝑚 𝑄𝑄 𝑚𝑚 * = 𝐴𝐴 (α 𝑚𝑚 +β 𝑚𝑚 +η 𝑚𝑚 ) ( β 𝑚𝑚 𝑐𝑐 ′(𝐴𝐴 𝑚𝑚 ) α 𝑚𝑚 ω 𝐻𝐻 ) β 𝑚𝑚 �( β 𝑀𝑀 𝑐𝑐 ′(𝐴𝐴 𝑀𝑀 ) α 𝑀𝑀 ω 𝐿𝐿 ) β 𝑀𝑀 � η 𝑚𝑚 (15)
Appendix 4-The Africa's TGA improvement demonstration

We know that at T1, the Africa's TGI is < 1, because of its disadvantage in technology and skilled labor. At T2, technology and training of skilled labor make possible the production of manufactured goods 𝑚𝑚. A fraction of this production replaces one part 𝐼𝐼𝑀𝑀 𝑚𝑚 of imports manufactured goods at T1. Let 𝑋𝑋 𝑚𝑚 , the amount of exportable manufactured goods produced by Africa. Imports 𝐼𝐼𝑀𝑀 𝑚𝑚 in period 2 are reduced by an amount α𝑋𝑋 𝑚𝑚 (0<α<1 One knows that α𝑋𝑋 𝑚𝑚 𝐼𝐼𝑀𝑀 𝑚𝑚 ≤ 1, because α𝑋𝑋 𝑚𝑚 is a faction of Africa's total imports at T2. So, 0< α𝑋𝑋 𝑚𝑚 𝐼𝐼𝑀𝑀 𝑚𝑚 ≤1. Moreover, 0<α<1, ⇒ 0 ≤ 1 -α < 1. Thus, Equation ( 18) indicates that the export of manufactured goods is more rewarding for Africa than its commodity exports.

As we know that at T1 TGI= 𝑋𝑋 𝑀𝑀 𝐼𝐼𝑀𝑀 𝑚𝑚 < 1 and that of period T2 equal 𝐼𝐼𝑀𝑀 𝑚𝑚 , we deduce that the situation in Africa in period 2 is much better than its specialization in period 1.

Appendix 5-Africa's manufactured goods trade advantage under Technology and High Skilled costs considerations

In the values chain, Africa is making two profits 𝜋𝜋 𝑀𝑀 and 𝜋𝜋 𝑚𝑚 linked to the production of good 𝑀𝑀 sold locally as a factor of production and export of goods 𝑚𝑚. The overall profit is

𝜋𝜋 𝐺𝐺 = 𝜋𝜋 𝑀𝑀 + 𝜋𝜋 𝑚𝑚 .
Let We know that technology costs 𝑐𝑐(𝐴𝐴 𝑀𝑀 ) + 𝑐𝑐(𝐴𝐴 𝑚𝑚 ) are financed by the rent 𝑎𝑎𝑟𝑟𝑁𝑁𝑁𝑁 1 constituted in period 1. In period 2, 𝑐𝑐(𝐴𝐴 𝑀𝑀 ) + 𝑐𝑐(𝐴𝐴 𝑚𝑚 ) becomes a depreciable fixed cost on all periods T1 and T2. So, we can assume that the annual fixed cost is approximately zero.

The overall benefit at T2 is then: The conditions for a positive unit profit are given in Table 1.

𝜋𝜋 𝐺𝐺 =

5

  Source: http://www.afdb.org/en/annual-meetings-2013/programme/africa%E2%80%99s-natural-resourceswhat-is-the-agenda/ last accessed at 09/07/2015.6 The specialization index (SI) in raw materials is defined by the reporting imports (exports) raw materials 𝑀𝑀 and finished products 𝑚𝑚. S𝐼𝐼= 𝐼𝐼𝑀𝑀 𝑀𝑀 /𝐼𝐼𝑀𝑀 𝑚𝑚 𝑋𝑋 𝑀𝑀 /𝑋𝑋 𝑚𝑚

  H2: there are three factors of production namely endogenous technology 𝐴𝐴, skilled labor 𝐻𝐻, unskilled labor 𝐿𝐿, raw materials 𝑀𝑀. Factors 𝐴𝐴 𝑀𝑀 and 𝐿𝐿 are necessary for the extraction of raw materials 𝑀𝑀. Analogically to the message send by the KLEM models (see van der Werf 2008; Löschel et Otto 2009), we consider technology(𝐴𝐴) endogenous all along this model. As in the Malthusian classical production, factors 𝐴𝐴 𝑚𝑚 , 𝐻𝐻 and 𝑀𝑀 are required to produce manufactured good 𝑚𝑚. Only factors 𝐴𝐴 and 𝑀𝑀 are internationally mobile. The stock of human capital (𝐿𝐿 + 𝐻𝐻) is very little mobile. Following the idea of fluid capital in nowadays (Taylor 2000), factor

  factors production yields ; 0 < 𝑛𝑛 𝑀𝑀 < 𝑛𝑛 𝑚𝑚 , pollution factors in the production of goods 𝑀𝑀 and 𝑚𝑚. Thus, we assume that the production of manufactured goods is less virtuous from environmental standpoint than raw materials extraction. The production functions of the two goods are Cobb-Douglas type(0 < α ≤ β ≤ η < 1). The endogenous technology respects the properties of Cobb-Douglas functions.

Figure 1 -

 1 Figure 1-relationship between employment and technology in endogenous growth models

  ) in Appendix 3 shows the cumulative positive effects reached by an endogenous technology. The factors of production yields are amplified when the values chain is extended in Africa. These are the cumulative effects of the efficiency in the production of manufactured 𝐴𝐴 𝑜𝑜𝑝𝑝𝑛𝑛 𝐸𝐸 𝑜𝑜𝑝𝑝𝑛𝑛 𝐴𝐴 goods that are at the origin of the creation of value added and explain differences in levels of economic development between industrialized countries and exclusive exporters of raw materials.

Figure 3 -

 3 Figure 3-Evolution of Africa's Trade Gain Index

  approximate technology costs to R&D expenses. Based on OECD data covering the period 2002-2012, the total annual cost of the technology is 𝑐𝑐(𝐴𝐴) =920, 955 million dollars (PPP 2005) and the average cost of a discovery 𝐶𝐶𝑀𝑀(𝐴𝐴) is estimated at 3% of the total cost. When fixed costs are zero, the average cost is equal to marginal cost 𝑐𝑐 ′ (𝐴𝐴). We define tolerable thresholds of harmful particulate emissions in tons of CO2 equivalent. From Health Standards of 20µg/m 3 per year of harmful particles tolerable recommended by WHO 14 , we calculate the share of particulate matter standard Ē to 1,299,984 tons / year of CO2 in Africa. The price 𝜏𝜏 of a ton of CO2 set by the France's Parliament is € 32 / ton of carbon. The commodity prices and average prices of manufactured goods are approximated to the harmonized index of prices and provided by the National Institute of Statistics and Economic Studies. The rates of skilled workers wages are based on the experiences of developed countries. However, the wage rate of unskilled workers is approximated to decent minimum income. Knowing the annual cost of technology and the average expenditure on R&D per year, we determine the volume of technology needed by year. The pollution coefficients are calculated from the relative contributions of the activities' sectors to overall pollution. Raw materials production and treatment of waste inherent contribute up to 49.5% against 50.4 in manufacturing and services sector. Finally the model parameters are calibrated according to the experiences learnt. The results summarized in Table

  This low industrialization of Africa explains its weak gains in world merchandise trade relative to its trading partners. From the theoretical point of view, the model brings added value to the traditional models of international trade by introducing a new factor (commodities 𝑀𝑀) of specialization in the function of production of manufactured goods. The model also makes the technology factor endogenous and internalized environmental constraints. The integration of the two factors helps explaining why Africa does not produce manufactured goods in Period 1. By leveraging its advantage in NR, the model shows that Africa, by staggering the exploitation of its NR, can constitute an annuity in the second period that allows importing technology from outside and trains the skilled workforce (engineers) essential to the production of manufactured goods. In Period 2, by extending its value chain by processing raw materials, Africa moves from a situation of impoverishing specialization (period 1) to a new specialization which is rewarding. The model also shows that the neofactorial specialization of Africa is consistent with the main SD goals that a supranational regulator (African Union) regulates the exploitation of raw materials and the inherent pollution. In the longer term, we expected the coexistence of two economic models namely the traditional and the dominant model and the circular economy. The future research's challenges are: Mathematical formulations which respect the trade-off between endogenous technology level and employment, the determination of equilibria jointly and to test the empirical validity of the model by robust econometric tools and more disaggregated data of African economies. 𝜕𝜕𝐿𝐿 𝑀𝑀 (𝐴𝐴 𝑀𝑀 ,𝐿𝐿,𝜆𝜆) 𝜕𝜕𝜆𝜆 =0⟺𝑟𝑟 𝑄𝑄 𝑀𝑀 = 𝑐𝑐(𝐴𝐴 𝑀𝑀 ) + ω 𝐿𝐿 𝐿𝐿 + 𝜏𝜏 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 By (4)

⇒

  𝑋𝑋 𝑚𝑚 > 𝑋𝑋𝑀𝑀 (18)

  In other words, Africa exports more raw materials than it imports. Using the WTO data from 1980 to 2012, the Africa's 𝑆𝑆𝐼𝐼 is 0.699.

		𝐼𝐼𝑀𝑀 𝑀𝑀
	abundance in Africa. It follows that	𝐼𝐼𝑀𝑀 𝑚𝑚 � ≤ 1. 𝑋𝑋 𝑀𝑀
		𝑋𝑋 𝑚𝑚

Table 1

 1

gives the conditions for which Africa wins in manufactured goods exports. Proof of Table

1

is given in Appendix 5.

Most interesting result is that Africa is competitive even integrating environmental costs. That differs from traditional business maximization input-output model. 𝑝𝑝 𝑚𝑚 -𝜏𝜏 𝑛𝑛 𝑚𝑚 is called "green unit price" of 𝑚𝑚 i.e. the price of the manufactured good after deducting the cost of pollution that its production induces. 𝑝𝑝 𝑚𝑚 -𝜏𝜏 𝑛𝑛 𝑚𝑚 is necessarily positive, if not, it is not rational to produce the good 𝑚𝑚. As the price 𝑝𝑝 𝑚𝑚 is set on the world market, equation (20) in Appendix 5 indicates the arbitration must do African regulator to encourage individual firms to produce the good 𝑚𝑚 in Africa while preserving the Environment. Equation (

21

) and Table

1

Table 1 -Results of neo-factorial specialization model and their impacts on the SD objectives

 1

	Pillars	Goals	Indicators					Effects of the model
				Period T1				Period T2
			Terms of trade	X M IM m < 1			⇒ (-)	(1-α)X m IM m -αX m >1	⇒ (+)
	Economy	Performance		𝑖𝑖𝑖𝑖 𝑟𝑟 >	ω 𝐿𝐿 L 𝑄𝑄 𝑀𝑀	+	𝑐𝑐(𝐴𝐴 𝑀𝑀 ) 𝑄𝑄 𝑀𝑀 𝐴𝐴 𝑀𝑀	⇒ (+) 𝑖𝑖𝑖𝑖 𝑝𝑝 𝑚𝑚 -𝜏𝜏 𝑛𝑛 𝑚𝑚 >	[ω 𝐿𝐿 𝐿𝐿+ 𝜏𝜏 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 +ω 𝐻𝐻 𝐻𝐻] 𝑄𝑄 𝑚𝑚 ⇒ (+)
			Profit	𝑖𝑖𝑖𝑖 𝑟𝑟 <	ω 𝐿𝐿 L 𝑄𝑄 𝑀𝑀	+	𝑐𝑐(𝐴𝐴 𝑀𝑀 ) 𝑄𝑄 𝑀𝑀 𝐴𝐴 𝑀𝑀	⇒ (-) 𝑖𝑖𝑖𝑖 𝑝𝑝 𝑚𝑚 -𝜏𝜏 𝑛𝑛 𝑚𝑚 <

[ω 𝐿𝐿 𝐿𝐿+ 𝜏𝜏 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 +ω 𝐻𝐻 𝐻𝐻]

Table 2 -Model parameters and optimal values

 2

	Variables and model parameters	Values	Units of measure
	α 𝑀𝑀 α 𝑚𝑚 β 𝑀𝑀 β 𝑚𝑚 η 𝑚𝑚 𝑐𝑐 ′(𝐴𝐴 𝑀𝑀 )	0.67 0.34 0.33 0.33 0.33 511,641	€/year
	𝑐𝑐 ′(𝐴𝐴 𝑚𝑚 )	511,641	€/year
	𝑛𝑛 𝑀𝑀	0.496

Table 3 -Effects of model results on SD goals during T2: Predictions and validations

 3

	Equations		Predictions (effects in ())	values	Conclusions
	(1-α)X m IM m -αX m >1		(+)	1.1887	Hypotheses confirmed
	𝑖𝑖𝑖𝑖 𝑝𝑝 𝑚𝑚 -𝜏𝜏 𝑛𝑛 𝑚𝑚 >	𝑄𝑄 𝑚𝑚 [ω 𝐿𝐿 𝐿𝐿+ 𝜏𝜏 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 +ω 𝐻𝐻 𝐻𝐻]	(+)

Table 2 and

 2 Table 3 results show rather optimistic results as a whole. Indeed, the

	industrialization of Africa in phase 2 of the model allows expecting on average 462
	employees, including 224 in manufacturing per firm per year. The TGI is improving with a
	net gain of + 18.87% of wealth from trade against a net loss of 30% at T1. The adoption of
	technology will cost to Africa of € 9,209.55 billion over a period of ten years. This requires a
	depletion of its NR of 3,026,332 tons sold during period 1 in the global raw materials' prices.
	Moreover, without clean technology, industrialization of the continent generates higher
	pollution to tolerable standards recommended by WHO; either 3,962,695t CO2

  ). Africa thus exports the rest of its manufactured production for a volume (1 -α)𝑋𝑋 𝑚𝑚 .As (1 -α)>0 and 𝐼𝐼𝑀𝑀 𝑚𝑚 -α𝑋𝑋 𝑚𝑚 ≥ 0, thus(1 -α)𝑋𝑋 𝑚𝑚 𝐼𝐼𝑀𝑀 𝑚𝑚 -α𝑋𝑋 𝑚𝑚 > 𝑋𝑋 𝑀𝑀 𝐼𝐼𝑀𝑀 𝑚𝑚 ⟺ (1 -α)𝑋𝑋 𝑚𝑚 (𝐼𝐼𝑀𝑀 𝑚𝑚 -α𝑋𝑋 𝑚𝑚 )𝑋𝑋 𝑀𝑀 > -α)𝑋𝑋 𝑚𝑚 (𝐼𝐼𝑀𝑀 𝑚𝑚 -α𝑋𝑋 𝑚𝑚 )𝑋𝑋 𝑀𝑀 > -α)𝑋𝑋 𝑚𝑚 (1 -α)(𝐼𝐼𝑀𝑀 𝑚𝑚 -α𝑋𝑋 𝑚𝑚 )𝑋𝑋 𝑀𝑀 >

	The Africa's TGI at T2 is thus written: 𝑇𝑇𝑇𝑇𝐼𝐼= 𝐼𝐼𝑀𝑀 𝑚𝑚 -α𝑋𝑋 𝑚𝑚 . One has to demonstrate that (1-α)𝑋𝑋 𝑚𝑚
	(1-α)𝑋𝑋 𝑚𝑚 𝐼𝐼𝑀𝑀 𝑚𝑚 -α𝑋𝑋 𝑚𝑚 >	𝑋𝑋 𝑀𝑀 𝐼𝐼𝑀𝑀 𝑚𝑚 .
	Let's suppose that	(1-α)𝑋𝑋 𝑚𝑚 𝐼𝐼𝑀𝑀 𝑚𝑚 -α𝑋𝑋 𝑚𝑚 >	𝑋𝑋 𝑀𝑀 𝐼𝐼𝑀𝑀 𝑚𝑚 .	(16)
					𝑋𝑋 𝑀𝑀
					𝑋𝑋 𝑀𝑀 𝐼𝐼𝑀𝑀 𝑚𝑚
					⟺	(1 1 𝐼𝐼𝑀𝑀 𝑚𝑚
					⟺	(1 1 (1 -α)𝐼𝐼𝑀𝑀 𝑚𝑚
					⟺	𝑋𝑋 𝑚𝑚
					⟺	𝑋𝑋 𝑚𝑚 𝑋𝑋 𝑀𝑀 >	(𝐼𝐼𝑀𝑀 𝑚𝑚 -α𝑋𝑋 𝑚𝑚 ) (1 -α)𝐼𝐼𝑀𝑀 𝑚𝑚
					⟺	𝑋𝑋 𝑚𝑚 𝑋𝑋 𝑀𝑀 >	(1 -(1 -α)𝐼𝐼𝑀𝑀 𝑚𝑚 α𝑋𝑋 𝑚𝑚 𝐼𝐼𝑀𝑀 𝑚𝑚 )𝐼𝐼𝑀𝑀 𝑚𝑚
	⟺	𝑋𝑋 𝑚𝑚 𝑋𝑋 𝑀𝑀 >	(1-(1-α) α𝑋𝑋 𝑚𝑚 𝐼𝐼𝑀𝑀 𝑚𝑚 )	(17)

(𝐼𝐼𝑀𝑀 𝑚𝑚 -α𝑋𝑋 𝑚𝑚 )𝑋𝑋 𝑀𝑀 > 1 (1 -α)𝐼𝐼𝑀𝑀 𝑚𝑚 ⟺ 𝑋𝑋 𝑚𝑚 (𝐼𝐼𝑀𝑀 𝑚𝑚 -α𝑋𝑋 𝑚𝑚 ) (𝐼𝐼𝑀𝑀 𝑚𝑚 -α𝑋𝑋 𝑚𝑚 )𝑋𝑋 𝑀𝑀 > (𝐼𝐼𝑀𝑀 𝑚𝑚 -α𝑋𝑋 𝑚𝑚 ) (1 -α)𝐼𝐼𝑀𝑀 𝑚𝑚

  𝑁𝑁𝑇𝑇 𝑀𝑀 = 𝑟𝑟𝑄𝑄 𝑀𝑀 , 𝑁𝑁𝑇𝑇 𝑚𝑚 = 𝑝𝑝 𝑚𝑚 𝑄𝑄 𝑚𝑚 , 𝐶𝐶� 𝑄𝑄 𝑀𝑀 , 𝐸𝐸 𝑀𝑀 � = 𝑐𝑐(𝐴𝐴 𝑀𝑀 ) + ω 𝐿𝐿 𝐿𝐿 + 𝜏𝜏 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 , 𝑛𝑛𝑛𝑛 𝐶𝐶� 𝑄𝑄 𝑚𝑚 , 𝐸𝐸 𝑚𝑚 � = 𝑐𝑐(𝐴𝐴 𝑚𝑚 ) + ω 𝐻𝐻 𝐻𝐻 + 𝑟𝑟𝑄𝑄 𝑀𝑀 + 𝜏𝜏 𝑛𝑛 𝑚𝑚 𝑄𝑄 𝑚𝑚 respective total revenues and total costs of goods 𝑀𝑀 and 𝑚𝑚, it follows that 𝜋𝜋 𝐺𝐺 = (𝑁𝑁𝑇𝑇 𝑀𝑀 + 𝑁𝑁𝑇𝑇 𝑚𝑚 ) -[𝐶𝐶( 𝑄𝑄 𝑀𝑀 , 𝐸𝐸 𝑀𝑀 ) + 𝐶𝐶( 𝑄𝑄 𝑚𝑚 , 𝐸𝐸 𝑚𝑚 )] (19). ⇒ 𝜋𝜋 𝐺𝐺 = [𝑟𝑟𝑄𝑄 𝑀𝑀 + 𝑝𝑝 𝑚𝑚 𝑄𝑄 𝑚𝑚 ] -[𝑐𝑐(𝐴𝐴 𝑀𝑀 ) + ω 𝐿𝐿 𝐿𝐿 + 𝜏𝜏 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 + 𝑐𝑐(𝐴𝐴 𝑚𝑚 ) + ω 𝐻𝐻 𝐻𝐻 + 𝑟𝑟𝑄𝑄 𝑀𝑀 + 𝜏𝜏 𝑛𝑛 𝑚𝑚 𝑄𝑄 𝑚𝑚 ] ⇒ 𝜋𝜋 𝐺𝐺 = 𝑄𝑄 𝑚𝑚 [𝑝𝑝 𝑚𝑚 -𝜏𝜏 𝑛𝑛 𝑚𝑚 ] -[𝑐𝑐(𝐴𝐴 𝑀𝑀 ) + 𝑐𝑐(𝐴𝐴 𝑚𝑚 ) + ω 𝐿𝐿 𝐿𝐿 + 𝜏𝜏 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 + ω 𝐻𝐻 𝐻𝐻]

  𝑄𝑄 𝑚𝑚 [𝑝𝑝 𝑚𝑚 -𝜏𝜏 𝑛𝑛 𝑚𝑚 ] -[ω 𝐿𝐿 𝐿𝐿 + 𝜏𝜏 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 + ω 𝐻𝐻 𝐻𝐻] 𝑝𝑝 𝑚𝑚 -𝜏𝜏 𝑛𝑛 𝑚𝑚 ] -[ω 𝐿𝐿 𝐿𝐿+ 𝜏𝜏 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 +ω 𝐻𝐻 𝐻𝐻] [𝑝𝑝 𝑚𝑚 -𝜏𝜏 𝑛𝑛 𝑚𝑚 ] ≥ [ω 𝐿𝐿 𝐿𝐿 + 𝜏𝜏 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 + ω 𝐻𝐻 𝐻𝐻]

				(20)
	⟺	𝜋𝜋 𝐺𝐺 𝑄𝑄 𝑚𝑚	=[𝑄𝑄 𝑚𝑚
	𝜋𝜋 𝐺𝐺 𝑄𝑄 𝑚𝑚	≥ 0 ⇒ 𝑄𝑄 𝑚𝑚	(21)
	The unit maximized profit is:
	𝜋𝜋 𝐺𝐺 *

𝑄𝑄 𝑚𝑚 * = [𝑝𝑝 𝑚𝑚 -𝜏𝜏 𝑛𝑛 𝑚𝑚 ] -[ω 𝐿𝐿 L 2 * + 𝜏𝜏 𝑛𝑛 𝑀𝑀 𝑄𝑄 𝑀𝑀 * + ω 𝐻𝐻 𝐻𝐻 2 * ] 𝑄𝑄 𝑚𝑚 *

(

22)

This paper was presented at the African Economics and Finance Conference, held at the Kwame Nkurmah University of Science and Technology, Kumasi, Ghana from August 11 th to 12 th

The author thanks all participants who made comments on the previous manuscript.

Data from the World Trade Organization, available at https://www.wto.org/french/res_f/statis_f/statis_f.htm, last accessed at 09/07/2015.

See[START_REF] Koulibaly | The increasing returns revolution in trade and geography[END_REF];[START_REF] Levchenko | The evolution of comparative advantage: Measurement and welfare implications[END_REF];[START_REF] Cadot | Trade Diversification, Income, and Growth: What Do We Know?[END_REF] for updated discussions.

Because of introducing Natural capital including Natural Resources and Agricultural Commodities, as a new factor in the Cobb-Douglas' production function, the model takes the name "neo-factorial specialization

We prefer the Trade Gains Index (TGI) to the terms of trade (TOT) because the terms of trade are determined by exogenous variables, notably international prices of tradable goods, while the TGI accounts for the structure of the concerned economies.

The African population represents 15.7% of the world population in 2014 and will be 25% in 2050, see http://esa.un.org/unpd/wpp/Excel-Data/population.htm

see[START_REF] Kaplinsky | China and the terms of trade: The challenge to development strategy in Sub-Saharan Africa[END_REF];WTO (2011);[START_REF] Anyanwu | Factors Affecting Economic Growth in Africa: Are There Any Lessons from China?[END_REF];[START_REF] Sindzingre | Fostering Structural Change? China's Divergence and Convergence with Africa's Other Trade and Investment Partners[END_REF];[START_REF] Busse | China's Impact on Africa -The Role of Trade, FDI and Aid[END_REF];[START_REF] Wolf | China and Latecomer-Industrialisation Processes in Sub-Saharan Africa : Situating the Role of (Industrial) Policy[END_REF];[START_REF] Park | China's Impacts on Africa's Development[END_REF].

The average income per capita in purchasing power parity in sub-Saharan Africa was $ 2,480 (PPP2005) against $ 30,007 in the euro area and $ 33,000 in the OECD higher income countries . In 2014, revenues are respectively $ 3,613 in sub-Saharan Africa, $ 38 694 in the euro area and $ 42,707 in the OECD. Data from http://data.worldbank.org/indicator/NY.GDP.PCAP.PP.CD last accessed at 09/07/2015.

http://www.who.int/mediacentre/factsheets/fs313/en/

Appendix

Appendix 1-Optimization program over T1-T3

Following the hypotheses made in section 3.2, the optimization program of production of both tradable goods on all three periods T1, T2 and T3 can be formalized as follows:


Alberto D Verga

Edge states in a two-dimensional quantum walk with disorder

Keywords: XX, XX, XX No code given

Edge states in a two-dimensional quantum walk with disorder

Introduction

Thirty years ago Feynman devised quantum algorithms using unitary transformations of an initial quantum state [1]. In 1993, Aharonov et al. [2] introduced the notion of quantum walk through a coin and a shift operator in analogy with the classical random walk, and found that information propagates at a ballistic rate instead of a diffusive one. Also in the nineties, Meyer [3] defined quantum cellular automata in one particle sector, and demonstrated that its continuous limit leads to a Dirac equation in two dimensions. The genealogy of quantum walks shows their rich physical content, ranging for quantum information to condensed matter [4,5,6,7].

Quantum walks are especially interesting because they provide an original point of view of quantum systems based on the properties of quantum states instead of the more usual approach based on the energy levels and eigenstates of a Hamiltonian. This point of view shed a new light on the dynamics of quantum systems and the mechanisms by which they explore the available Hilbert space. At variance to the standard definition of a quantum system by its Hamiltonian, a quantum walk is defined by an evolution unitary operator. In particular, the relationship between the quantum state and information allows to relate concepts from quantum information theory to material systems through the introduction of a quantum walk effective Hamiltonian. We may also invert the reasoning and ask whether it is possible to get some insight on the behavior of quantum walks using concepts from condensed matter. Some classes of quantum walks possess remarkable topological properties whose origin can be traced back to the structure of their evolution operator which links the particle's spin (coin operator) with its momentum (shift operator), akin to the spin-orbit coupling in solid state, leading a E-mail: Alberto.Verga@univ-amu.fr to nontrivial Berry phases, edge states and non vanishing Chern numbers [8,9,10]. Experiments [11,12,13] show that in the presence of disorder, the quantum walk losses its coherent interference pattern and, depending on the nature of the noise, can localize or transit to a classical diffusion regime. Noise also affects the topological phases and the localization-delocalization properties of one dimensional [14] and two dimensional quantum walks [15]. In the absence of disorder, a one dimensional quantum walk can still localize at the interface between two distinct topological regions [16]. These phenomena are also characteristic of topological phases in condensed matter, like quantum Hall [START_REF] Prange | The Quantum Hall Effect[END_REF] or spin Hall insulators [18] where edge states appear. However, it is worth noting that such bound states are protected against disorder. As a consequence, the transport properties (quantization of the conductance, for instance) are preserved as long as the topology is not changed.

Our aim is to investigate the effect of spatial noise on the edge states localized at the interface between two regions differing in their topology.

Quantum walk topology and effective Hamiltonian

We consider a discrete time quantum walk in a square lattice, with nonzero Chern number [8]. The walker's Hilbert space is the set of spinors |ψ(t) depending on time t. They are given by the Kronecker product ⊗ of the walker's position and spin state |x ⊗ |s , where x = (x, y) ∈ Z 2 is a lattice node (the lattice constant a = 1 is the length unit), and s =↑, ↓ takes "up" and "down" values. The state evolution is determined by a "coin" R operator acting in spin space, and a "shift" T operator. At a given time step, the motion direction depends on the spin orientation. At the end of the walk, a measure of the position density probability distribution P (x, y, t), is performed. The initial state,

|ψ(0) = |0 ⊗ 1 √ 2 | ↑ + i| ↓ , (1)
is a superposition of spin up ψ ↑ and down ψ ↑ amplitudes:

ψ ↑ = 1/ √ 2 , ψ ↓ = i/ √ 2 ,
given equal probabilities of the spin orientation in the z direction. The coin operator rotates the spin around the y axis by an angle θ,

R(θ) = 1 x ⊗ e -iθσy/2 = 1 x ⊗ cos θ/2 -sin θ/2 sin θ/2 cos θ/2 , (2)
where 1 x is the unit matrix in position space. The shift operator moves the particle to a neighboring node according to its spin projection:

T (p) = x |x + e p , ↑ x, ↑ |+ |x -e p , ↓ x, ↓ | , p = x, y (3)
where e p is a unit vector in the p direction. One time step is executed by the unitary operator,

U (θ, α) = T (x)R(θ)T (y)R(α)T (y)T (x)R(θ) , (4)
where the product T (y)T (x) = T (x)T (y) shift the position by ±1 on both directions (x, y) → (x ± 1, y ± 1).

According to the values of the rotation angles the family of effective Hamiltonians,

H(θ, α) = i log U (θ, α) ,
is characterized by a Chern number taken the values C = {-1, 0, 1}. The effective Hamiltonian is not uniquely defined, its eigenvalues are determined modulo 2π (quasienergies). The topological properties of the walk related to the symmetries of the effective Hamiltonian [8], depend on the values of the two parameters (θ, α), as schematically represented in Fig. 1. It is worth noting that the topological classification of the quantum walks is richer than the one inferred from the sole effective Hamiltonian symmetries: taking into account the properties of the (one period) evolution operator implies the existence of a pair of topological invariants [19]. As a consequence, topological protected edge states can appear at the interface of two trivial effective Hamiltonian phases [9,15].

The explicit form of H is easily obtained using the momentum representation k = (k x , k y ) of the unitary operator U , in the base of the Pauli matrices σ:

U (k) = T (k x )R(θ)T (k y )R(α)T (k x + k y )R(θ), (5)
where

R(θ) = cos θ σ 0 -i sin θ σ y , T (k) = cos k σ 0 + i sin k σ z from which one readily obtains, U (k) = d 0 σ 0 -id • σ, d = (d x , d y , d z ), (6)
or equivalently,

H(k) = E(k)n(k) • σ , (7)
where the energy spectrum is given by,

cos E(k) = cos α 2 cos θ cos k x cos(k x + 2k y )- sin k x sin(k x + 2k y ) -sin α 2 sin θ cos 2 k x , (8)
and the unit vector n(k) = d(k)/d(k), with

d x = sin k x cos α 2 sin θ cos(k x + 2k y )- 2 sin α 2 sin 2 θ 2 cos k x , d y = sin α 2 cos θ cos 2 k x + sin 2 k x + cos α 2 sin θ cos k x cos(k x + 2k y ) , d z = sin α 2 sin θ sin k x cos k x - cos α 2 cos θ sin k x cos(k x + 2k y )+ cos k x sin(k x + 2k y ) . (9)
A quantum walk with effective Hamiltonian (7) possesses a nontrivial topology if the n vector (or equivalently d), surrounds the origin when the k vector scans the Brillouin zone [-π, π] 2 . In Fig. 2 we represented two cases differing in their Chern number. We observe in the parametric plot of the vector d over the Brillouin zone that depending on the choice of the pair (θ, α), the origin is surrounded, signaling a nontrivial topology, or it is not surrounded.

The random walk thus defined possesses particle-hole symmetry but not time-reversal symmetry. Indeed, from the explicit form (7), we verify that under conjugation (i → -i, k → -k) the effective Hamiltonian changes sign: the one step operator ( 4) is real implying that the walk has particle-hole symmetry. The walk defined by (4) with particle-hole symmetry and broken time reversal symmetry, is thus reminiscent to class D in the usual classification [20].

To investigate the consequences of the nontrivial topology, in particular the edge states, we perform a series of numerical computations for different disorder types. We use Eq. ( 4) to calculate the walker state evolution for one time step, in a square lattice with periodic boundary conditions. We split the lattice into two regions separated by an interface at x = 0, differing in their Chern number: for x ≤ 0 we take (θ, α) = (π/2, 3π/7)), and for

x > 0, (θ, α) = (-(1 + √ 5)/2, √ 2/2), with C = 1, -1
, respectively (Fig. 1). In the one dimensional case the presence of an edge state at the origin leads to a localization of the quantum walk [8,16]. In two dimensions one may expect propagation along the edge state localized at the interface. This is precisely what we observe, as demonstrated in Fig. 3 where a realization of the quantum walk ( 4) is shown. The propagation along the interface is ballistic, as in the case of a one dimensional quantum walk. This phenomenon is observed for initial states having an overlap with the edge bound state, otherwise the quantum walk do not localize and explore the whole lattice with ballistic speed.

Spatial disorder

In order to investigate the effect of disorder we consider a set of randomly distributed sites I, we call impurities, where the coin operator change. The spatial concentration of impurities is given by the probability p for a site to be occupied by an impurity. In Fig. 4 we show two of such distributions with p = 1% and p = 10%. In such sites the simplest modification of the coin rule (2) is to make the angles θ(x) position dependent,

θ → θ(x) = θ + Jδθ(x) , δθ ∼ U(0, 2π)
with x ∈ I the set of impurity sites, and U the uniform probability distribution in the given interval (i.e. rotation disorder). The parameter J measures the strength of the disorder. For J small enough to let the system's topology unchanged, one may expect the edge state protected; in the opposite case one expects localization (or diffusive spreading in same special cases [15]) of the quantum walk (Fig. 1).

It is important to emphasize that the introduction of spatial disorder do not change the unitary evolution of the walker, nor the symmetries of the coin operator; however it modifies in a nontrivial manner the physics of the system by breaking the translational invariance, only preserved in a statistical sense, and by introducing new correlations between different sites that should change the walk interference patterns and hence its spreading properties. The effects of spatial noise in quantum walks are well studied, especially in one dimension; they can localize the walker or change its spreading rate [21,22,11]. A more subtle behavior appears in topologically nontrivial walks, where localized zero energy modes may coexist with delocalized nonzero energy modes [23]. We investigate in the case of two dimensions effects alike to this one.

We show a numerical computation of the random rotation quantum walk in Fig. 5, to compare the weak and strong noise cases with the clean system of Fig. 3. The presence of the interface introduces an anisotropy; actually, the evolution of the initial condition leads to an asymptotically inhomogeneous distribution of the position probability of the walker with different spreading properties on the interface and in the bulk. We find that the known general picture is confirmed, expectedly a localization transition is observed at a finite value of the disorder strength, as occurs in the one dimensional split-step quantum walk [14]. The point is that in two dimensions, even if the interference pattern of the walk is lost away form the interface and the bulk distribution is localized or spread slowly, the ballistic transport along the edge is not affected for weak enough disorder.

To quantify the effect of noise on the walker spreading, we measure the width of the probability density:

P (x, t) = |ψ ↑ (x, t)| 2 + |ψ ↓ (x, t)| 2 (10)
using the definition

w(t) = x |x| 2 P (x, t) - x xP (x, t) 2 (11)
which gives std r(t) = w(t), and an analogous definition for std y 0 (t) for the width on the interface x = 0: std y 0 (t) = y y 2 P (0, y, t) y P (0, y, t)

1/2 . ( 12
)
(Note the time dependent normalization and the absence of mean term in order to catch running away distributions.) Figure 6 shows the width for different values of the disorder strength J. Moreover we plot the probability to stay in a neighborhood of the origin P (x = 0, y ∈ [-100, 100], t) for weak and strong disorder. The measure of the isotropic width shows that the presence of disorder destroys the ballistic regime. However, the edge state transport remains ballistic for weak enough disorder. The width of the position distribution at the interface (x = 0), increases linearly with time only for weak disorder. Concomitantly, the probability to find the walker near the origin tends to zero, in contrast to the persistence observed for stronger disorder. The deviation of stdr(t) to a straight line together with the linear stdy 0 (t) is a manifestation of the underlying anisotropy of the walk in the weak noise regime, which tends to localize in the bulk and to propagate at the interface. // Fig. 5. Position probability distribution at t = 1000 for p = 0.03 and J = 0.05 (a-b) and J = 0. 2 (c-d). The coupling J = 0.2 is strong enough to change the topology (c.f. 1); as a consequence the edge state is no longer protected and the walk localizes. Comparison with Fig. 3 shows that in the low noise case J = 0.05, the edge state is protected, but it is wider than in the clean case, and moreover, diffusion destroys the walk ballistic spreading outside the interface.

Nonlinear spatial disorder

It is interesting to investigate the effect of disorder under a more complex setting, in particular generalizing the walk to take into account nonlinear (many-body) interactions [24,?,25, 26]. Pursuing our analogy with matter systems, we may think the impurities as being fixed spins whose interaction with the itinerant spin of the walker is of the exchange type. This "magnetic disorder" would introduce into the coin operator a phase factor φ S J (φ) = e -iJφ(x,t)σz , where x ∈ I and J the coupling constant, which leads to the redefinition

R(θ) → R J (θ, φ) = S J (φ)R(θ) . ( 13
)
The simplest choice is to take φ as a random angle, uniformly distributed on the circle [15] (i.e. phase disorder). However, if we assume that the orientation of the fixed spin is determined by the orientation of the itinerant one (its z spin component), φ will be related to the walker's state, φ = πs(x, t):

s(x, t) = |ψ ↑ (x, t)| 2 -|ψ ↓ (x, t)| 2 , x ∈ I . ( 14
)
leading to a nonlinear coupling. The coin operator R J (θ, φ) replaces R in (4); it is now a composition of a rotation around the y axis (with a uniform angle in each region) followed by a rotation around the z axis at the occupied sites of the lattice. The spatial distribution of the defects remains random and we refer to the choice ( 13)-( 14) as "nonlinear disorder". However, the values of φ at different sites are correlated by (14) with the walker's state, making Note the similar distribution in (a) with the one of the clean state (Fig. 3).

the nonlinear disorder essentially different to the rotation or magnetic disorder types. One important consequence of the introduction of the coin operator (13) with respect to the random rotation one of the previous section, is that the form R J (θ, φ) breaks the so-called particle-hole symmetry. The unitary operator of the walker is no longer real: the rotation spin axis has now a z component. As a matter of fact, the coin operator can be put in a form of a rotation around an axis depending on φ of angle also depending on φ. This is true even if the angle φ is taken randomly instead of being related to the walker's state, although the implications on the existence of edge states in both cases, nonlinear disorder and random phases, can in principle strongly differ. The breaking of the particle-hole symmetry change the topological properties of the quantum walk putting it in the same class as the integer quantum Hall system (see Ref. [8], Appendix B).

To highlight the specific properties of the nonlinear walk, we chose the coupling parameter and the concentration in a range about J ∼ 1 -100, and p ∼ 0.1, respectively. These large values of J are justified because the spin density is of the order of p and decreases at a given site at least as 1/t because of the probability spreading over the lattice (this is the scaling corresponding to the ballistic transport over the interface, supposing no spreading in the transverse direction). Significant results are obtained for values with J > 1/p; the probability density on the interface scales as P ∼ 1/t in the ballistic regime. It is a remarkable fact that smaller values do not give a big difference with the clean case, for times up to t = 1000. The most striking observation is that the ballistic transport is fully preserved, even in the direction perpendicular to the interface. As shown in Fig. 7, where we plot the density and walk width at the interface in the case of nonlinear disorder, we observe ballistic spreading proportional to t; though not shown in the figure, we find in particular that the isotropic width, after an initial transient, is also linear in time.

Merely for extreme values of the coupling constant J 1/p, the well structured interference pattern of the clean walk tends to loss its organization. This start to be visible on the small scale amplitudes as shown on the three panels of Fig. 8, where we compare the spin density for the clean case with the nonlinear walk for two values of J. For smaller values of J the nonlinear walk behavior remains close to the clean one, for the same initial condition starting at the interface. In sharp contrast with the ballistic transport of the nonlinear walk, if we replace the self-consistent phases φ = φ(ψ) in ( 13) by random anφ ∼ U(0, 2π), we numerically observe that the edge channel is broken and the walk cannot spread at a ballistic rate (Fig. 7c). This can be understood as a consequence of the random phase shifts in the z direction that for strong enough disorder lead to spin flips hence changing locally the walker's direction. By the way of comparison we remark the special behavior of the system governed by the nonlinear coin in which the edge states appear to be robust. Therefore, the nonlinear disorder restores the ballistic behavior lost for random phases, even for much larger values of the coupling constant.

In fact, the edge state not only provides a channel to transport the information, it is also an attractor to the nonlinear walk. Indeed, if in the absence of disorder one starts with an initial state that does not overlap with the edge state, for instance at x = -10, the quantum walk propagates freely on the whole lattice. However, adding the nonlinear interaction, the walk is partially trapped at the interface where it propagates at a ballistic rate, in spite of the spatial disorder (Figs. 9 and 10). In Fig. 9 we compare at t = 200 the density distribution at the interface, and in Fig. 10 the distribution over the lattice. The initial particle position is chosen in the left region (x = -10 at t = 0), and the spin set up, to favor the propagation towards the right, allowing the walker encounter the interface. Figure 10 clearly shows the difference between the clean case (a), the random rotation angles (b) and phases (c) disordered cases, and finally the nonlinear walk (d). The anisotropy is a consequence of the choice of the initial state (spin up). Only in the nonlinear case an accumulation of probability density on the interface is observed. This behavior, for which the interface edge states act as an attractor, is reminiscent to a generalization to two dimensions, of the observed phenomenon of localization at a topological defect in one dimension [26]. This is related to the fact that at low energy and for small values of the rotation angle, the one dimensional walk limits to the nonlinear Dirac equation that has soliton solutions [25].

Indeed, using the explicit expression of the evolution operator in momentum space, it is easy to obtain its "hydrodynamic" limit k → 0, U ≈ u 0 + u x ik x + u y ik y :

(averaged spin distribution) to the Dirac like equation in 2 + 1 dimensions:

∂ ∂t -2σ z ∂ ∂x + ∂ ∂y + θσ x ∂ ∂x ψ(x, t)+ i θ + α 2 σ y + gs(x, t)σ z ψ(x, t) = 0 , ( 16
)
where ψ is the position representation of the spinor, and g = 3πpJ the effective coupling constant (cf. [25,?] for a more formal calculation in the one dimensional case).

In addition to the "mass" term which gives the boundary separating two different topologies (the mass vanishes for α + 2θ = 0, in the present approximation for small angles). The effective coupling constant is of the order of pJ, which justifies the choice of J ∼ 1/p as the order of magnitude of the nonlinear noise in the numerical computations. The term in θ∂ x is the lowest order term showing the anisotropy of the walk, the "velocity" θ being smaller than the "light velocity" c = 2 in the diagonal direction. We note that the nonlinearity appears as a self-consistent gauge field. The nonlinear term, being proportional to iσ z , adds naturally to the first order spatial derivatives also proportional to σ z , in a way similar to a "vector" potential [27]. This is somewhat different to the more usual nonlinear Gross-Neveu model in which the nonlinear term adds to the mass term [25], or the state dependent rotation of Ref. [26] which preserves the particle-hole symmetry.

Conclusion

In conclusion, we investigated the effect of spatial disorder in a two dimensional discrete quantum walk. We demonstrated the ballistic propagation along the edge state localized at the interface between two distinct topological phases. This edge state is robust against quenched disorder, provided it cannot change the system's topology. In particular, we found an anisotropic state with a bulk localized density distribution coexisting with ballistic propagation at the interface.

In the case of a coin operator depending on nonlinear phases randomly distributed in space, the ballistic propagation of the information, mostly as in the absence of disorder, is preserved even for very strong couplings. This is also in contrast with the behavior of a walk perturbed by random phases breaking the particle-hole symmetry, in which case the ballistic propagation is no longer possible. In addition a new phenomenon arises, the walk can be trapped at the interface. The nonlinear walk can be mapped to a Dirac equation in the continuous limit with a mass term depending on the rotation angles and a nonlinear term proportional to the smoothed spin density, known to possess propagating localized solutions along the interface (the transverse direction being localized).

We focused here on the rich and striking phenomenology of quantum walks and their topological edge states; the analysis of the mechanisms behind the observed effects of disorder, especially in the nonlinear case, deserves further consideration.

We benefited from useful discussions with Laurent Raymond and Thomas Krajewski. We thank Giuseppe di Molfetta for his interest and thorough comments. This work was partially supported by CNRS UMR 7332, and Université de Toulon.

1 Fig. 1 .

 11 Fig.1. Phase diagram (θ, α), the triangular sectors corrspond to different Chern numbers C (see Ref.[8]). Rotation angles for x < 0 (red point, C = 1) and x > 0 (blue point, C = -1). The circles represent the distribution of random angles for J = 0.1 (inner circle) and J = 0.2 (outer circle). In the second case the noise may change the topology of the quantum walk.

Fig. 2 .

 2 Fig. 2. The topology of the effective Hamiltonian H = E(k)d(k) • σ change when the vector d(k) vanishes. In the top panel θ = π/2, α = 3π/7, the d vector surrounds the origin (black point), which corresponds to a nontrivial topology; in the bottom panel θ = √ 2/2, α = (1 + √ 5)/2, the topology is trivial. The two colors correspond to the inner and outer surfaces.

Fig. 3 .

 3 Fig. 3. Position probability of a quantum walk with an interface between two regions having distinct topology: P = |ψ ↓ | 2 + |ψ ↑ | 2 . The top panel (a) shows that the walker distribution at time t = 1000, is essentially concentrated along the interface. The bottom panel (b) shows the same data in a logarithmic scale to reveal the structure of the small amplitudes (the wave folding due to the periodic boundary is not visible in a linear scale).

1 Fig. 4 .

 14 Fig. 4. Random distribution of impurities for p = 0.01 (left) and p = 0.1 (right) over a square lattice of size 201 × 201.

Fig. 6 .

 6 Fig.6. Quantum walk in the presence of rotation disorder. Width of the density distribution along the interface (a) as a function of time, for J = 0.05 (blue lines) and J = 0.2 (red lines), and p = 0.03 (solid lines) and p = 0.01 (dotted lines); the upper black line corresponds to the clean case. (b) Radial (black line) and at the interface spreading in logarithmic scale; J = 0.05 (blue, scaled by a factor 0.4 to fit inside the frame), J = 0.2 (red). The dashed lines correspond to the fitting exponents 1 and 1/2. For weak disorder the propagation is ballistic (exponent 1), and approaches diffusion for strong disorder (exponent 1/2). (c) Probability to be on the interface in a neighborhood of the origin, x = 0 and y ∈ (-100, 100), J = 0.05 (lower blue line), and J = 0.2 (upper red line).

Fig. 7 .

 7 Fig.7. Nonlinear and phase disorder quantum walk. Probability distribution of the nonlinear walk, (a) for J = 1 and (b) J = 10, (p = 0.1, t = 1000). In (a) superposed to the nonlinear walk (in gray), we show the random phase walk using the same parameters, localized near the origin (in red). (c) Width in the y direction for x = 0, at the interface, showing ballistic propagation for J = 0 (dotted line), J = 1 (blue dashed line), and J = 10 (blue solid line), in the nonlinear case; one can compare with the random phase disorder (red dashed line) J = 0.1 and (red solid line) J = 1, showing diffusive spreading. Note the similar distribution in (a) with the one of the clean state (Fig.3).

Fig. 8 .

 8 Fig. 8. Zoom on the small scales of the spin density distribution s × 10 6 , s = |ψ ↑ | -|ψ ↓ | 2 , for J = 0 (a), J = 10 (b), and J = 100 (c) (p = 0.1, t = 750). In the nonlinear case (b-c), spin up and down waves are present on both sides of the interface.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Short time (t = 200) position distribution of a quantum walk for an initial condition at x = -10, in the spin up state. (a) Clean case, the walk freely propagates on the lattice; (b) with rotation disorder it is localized (J = 0.2, p = 0.03), and (c) with nonlinear disorder it is partially trapped at the interface restoring the ballistic propagation (J = 100, p = 0.1).Only the nonlinear quantum walk is trapped at the interface.


Alberto Verga

Spin-transfer torque and topological changes of magnetic textures

Keywords: Magnetism, magnetic vortices, Landau-Lifshitz equation, spin-transfer torque

The electric manipulation of magnetic textures in nanostructures, important for applications in spintronics, can be realized through the spin-transfer torque mechanism: a spin-polarized current can modify the magnetization of skyrmions and magnetic vortices, and eventually change the topology of the magnetization. The spin-transfer torque and the intrinsic space and time scales of the topological changes are essentially quantum mechanical. We model the interaction between itinerant and fixed spins with a simple tight-binding hamiltonian in a square lattice. The dynamics is described by the Schrödinger equation for the electrons and the Landau-Lifshitz equation for the evolution of the magnetic texture. We investigate the phenomenology of the topological change of a Belavin-Polyakov skyrmion under the action of a spin-polarized current and show that adding an exchange dissipation term, regularizes the transition towards a ferromagnetic state.

INTRODUCTION

The equilibrium magnetization field in ferromagnetic nanodots and in helical metals, often possesses a nontrivial topology. Magnetic vortices in permalloy [START_REF] Van Waeyenberge | Magnetic vortex core reversal by excitation with short bursts of an alternating field[END_REF]] and skyrmion lattices in transition metal compounds [START_REF] Muhlbauer | Skyrmion Lattice in a Chiral Magnet[END_REF]] were experimentally observed. These configurations are interesting because a change between states having different topologies (switching of the vortex core, motions and annihilation of skyrmions), can be used as basic states in nonvolatile memories and spintronic devices [START_REF] Romming | Writing and deleting single magnetic skyrmions[END_REF]]. The basic physical mechanism that can trigger these topological changes, without magnetic fields, is the spin-transfer torque [START_REF] Ralph | Spin transfer torques[END_REF]]. It describes the interaction between itinerant spins s, produced by a spin-polarized current j s , and the magnetic moments (fixed spins S) of the magnetic material, when the underlying magnetization is non-uniform. Within the framework of micromagnetism, to account for the spin-transfer torque one has to extend the usual Landau-Lifshitz equation [START_REF] Landau | On the theory of the dispersion of magnetic permeability in ferromagnetic bodies[END_REF]] with terms proportional to the current and the magnetization gradients j s • ∇S [START_REF] Zhang | Roles of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets[END_REF]]. However, this approach neglects strong non-adiabatic effects, such as the generation of current inhomogeneities due the scattering of electrons on the magnetization gradients. In order to investigate these effects, we recently proposed a self-consistent model, where electrons obey to quantum dynamics [START_REF] Elías | Topological changes of two-dimensional magnetic textures[END_REF]]. Here we develop further this model, and investigate the influence of dissipation.

In this paper we are interested in the skyrmion-ferromagnetic transition induced by a polarized current. The basic magnetic texture is taken to be a stabilized lattice version of the [START_REF] Belavin | Metastable states of two-dimensional isotropic ferromagnets[END_REF]] skyrmion in a periodic system. After a presentation of the basic equations coupling the Schrödinger equation for the itinerant spins (electrons) to the Landau-Lifshitz equation for the ferromagnetic spins (fixed classical magnetic moments), defined on a square lattice, we present a qualitative model to show the basic mechanism of the transition. The transition from the skyrmion state to the ferromagnetic state implies a change in topology, and thus a violation of the topological charge conservation. This is possible because of the large fluctuations of the electron spin field (it acts a stochastic perturbation to the magnetization field) and because of the discreteness of the system (the lattice constant introduces a microscopic length cutoff). We performed a series of numerical computations to study the phenomenology of the transition and to identify the mechanism of the topological change.

MODEL

The motion of an electron in a lattice of step a and size L 2 is given by the Heisenberg equation for the two components (for the spin up and down) annihilation operator

c i at site i, i ċi (t) = H e (t, S i )c i (t) , (1)
where H e is the time dependent hamiltonian,

H e = - i,j e iφi,j (t) c † i c j -J s i S i • (c † i σc i ) -B p • i c † i σc i (2)
where the fixed S i and itinerant spins c † i σc i (σ is the vector of Pauli matrices) are coupled by the exchange constant J s ;

is the energy to jump from site i = (x i , y i )/a = x i /a to its neighbor j. The system is subject to a constant electric field E in the x-direction, responsible of the phase factor appearing in the kinetic energy term, 1 z Λ Figure 1. Skyrmions of topological charge Q = 1 and size λ defined by the stereographic projection on the complex plane w = w(z), for (left) w = z/λ, (center) w = (1 + i)z/λ, and (right) w = (1 -i)z/λ. Arrows give the magnetization in the plane, and color, from blue to red, gives its z-component: at the center S = (0, 0, 1). A constant z-polarized current induces the precession of the skyrmion magnetization and tends to reduce it core size.

φ i,j (t) = (i -j) • x eaEt
(with -e the electron charge). The last term in (2) contains the current polarization effective magnetic field B p in energy units.

The magnetic texture follows the dynamics given by the Landau-Lifshitz equation,

∂ ∂t S i = S i × (f i -αS i × f i + J s s i ) -β∇ 2 f i , (3)
where the effective field,

f i = - δH S δS i , (4)
is derived from the coarse-grained free energy,

H S = J 2 i (∇S i ) 2 + K 2 i S 2 zi - D 2 i S i • (∇ × S i ) (5)
J is the Heisenberg exchange constant, K is the anisotropy (easy-plane if positive, and easy-axis if negative), and D is the Dzyaloshinski-Moriya spin-orbit coupling energy. The term in β represents a dissipation of the magnetization related to the exchange energy (proportional to the gradients ∇S i ) [START_REF] Baryakhtar | Soliton relaxation in magnets[END_REF]]. This equation is coupled to the equation for the electrons through the torque in J s , where the electron spin is computed by the formula

s i (t) = c † i (t)σc i (t)
where the braket is for the quantum mean value. We use periodic boundary conditions in order to have a well defined topology. Note that ∇ is a difference operator acting on the lattice sites. In practice, the difference operators are computed in Fourier space and transformed to the lattice space. In units such that = a = = e = 1, typical parameters are:

J s = 1, J = 0.4, K = D = 0 (for the Belavin-Polyakov skyrmion), B p = n e = 0.1, L = 128.
Without the dissipation term, the system (1-5) conserves the magnetization modulus |S| = 1, and the topological charge

Q = Q(t), Q = dx 4π q(x, t), Q + = dx 4π |q(x, t)|, q = S • ∂ x S × ∂ y S , (6)
where the integration is over the lattice; Q + (t) is a rough measure of the number of vortices.

SKYRMION

To study the skyrmion-ferromagnetic transition it is convenient to work in the continuous limit, and to transform the magnetization field (which has only two independent components) by the stereographic projection:

S x = w + w 1 + |w| 2 , S y = 1 i w - w 1 + |w| 2 , S z = 1 -|w| 2 1 + |w| 2 , (7) with w = S x + iS y 1 + S z . ( 8
)
When S z = 1, w goes to zero, and in the opposite pole, S z = -1, w goes to infinity. This projection maps the vector field over the unit sphere S = S(x, t) to the field over the complex plane w = w(z, z, t) (where z = x + iy).

The Landau-Lifshitz equation, in absence of dissipation, becomes,

i∂ t w = -J∂ ∂w + 2J w 1 + |w| 2 ∂w ∂w -1 2 s + + s z w + 1 2 s -w 2 , ( 9
) (only the exchange field is considered here) where we defined the complex derivative ∂ = ∂/∂x -i∂/∂y an its complex conjugate ∂. The second line in (9) corresponds to the spin-transfer torque term, where s ± = s x ± is y . Equilibrium solutions of (9) are arbitrary analytic functions w = w(z) and s = 0. The simplest one is the simple zero, w = z/λ, the Belavin-Polyakov skyrmion of charge Q = 1, centered at the origin and of characteristic size λ ∈ R (spin up at the origin and spin down at infinity).

In order to investigate how the spin torque perturbs the skyrmion state w = z/λ, we focus on two simple limiting cases: first, small deviations from the skyrmion state by a uniform polarized current s = (0, 0, s z ); and second, a small circular region around the skyrmion core relevant to track the transition towards the ferromagnetic state |w| → ∞. 0.0 0.5 1.0 1.5 2.0 In the first case, we linearize (9) around the skyrmion state, w = z/λ + f (z, z, t):

t/t 0 ×10 -3 -1 0 1 Q(t) 0.0 0.5 1.0 1.5 2.0 t/t 0 ×10 -3 -1 0 1 2 Q(t) 0 1 2 3 4 5 6 t/t 0 ×10 -3 -1 0 1 2 Q(t)
i∂ t f = -J∂ ∂f + 2J z λ 2 + |z| 2 ∂f + s z λ z , (10)
the spin torque appears as a source term in this approximation. An interesting particular solution is readily found:

f = f 0 (z, t) = -i s z t λ z .
The pure imaginary factor has the effect of changing the orientation of the magnetization field around the center of the skyrmion, passing successively in time from left to right chirality (as shown in Fig. 1). A sequence observed in the numerical simulations, in addition to the fact that the effective size of the core reduces: λ → λ/ 1 + (s z t) 2 (even if, for long times, the perturbation analysis ceases its validity).

We turn now to the second case, for which we assume that the collapse of the skyrmion core is radially symmetric w = w(r, t), with r the polar radius. We are interested in the asymptotic limit of very large |w|, and propose a self-similar form [START_REF] Zakharov | Collapse of langmuir waves[END_REF]]:

w(r, t) = 1 (t * -t) α f r (t * -t) β (11)
to describe the approach of w to infinity (when t → t * , t * is the collapse time). Equation ( 9) becomes,

i∂ t w(r, t) = - J r ∂ r (r∂ r w) + 2J w (∂ r w) 2 + 1 2 s -w 2 (12)
Inserting ansatz (11) into (12) one obtains, two conditions that determine the unknown exponents:

α = 1 , β = 1/2 . ( 13
)
It is worth noting that the exchange interaction, which is scale invariant, do not permit to select the α exponent; its value is determined by the coupling term with the spin polarized current. A crude estimation of the finite time singularity is t * ∼ λ/s 0 a, where s 0 ∼ n e B p is the typical itinerant spin strength per site.

RESULTS

We investigate now, the role of dissipation in the transition towards the ferromagnetic state, using the numerical computation of equations ( 1) and ( 3). The first effect of the dissipation is to break the invariance of the topological charge. We observe in particular that the exchange dissipation term favorises the transition from the skyrmion state to the ferromagnetic state. Indeed, we plot in Fig. 2 the evolution of the topological charge as a function of the dissipation strength. In addition to the increasing stability of the skyrmion state, it is worth noting that the characteristic transition time sharply increases with dissipation. This is, as expected, a regularization effect. In the limit of vanishing dissipation the transition produces, ultimately, by the change of a single spin.

A quantitative explanation of the mechanism behind the transition is based on the behavior of the electron spins.

As we can observe in Fig. 3, although the electrons dynamics is almost stochastic (due to multiple scattering and interference effects), their spins organize near the skyrmion core. This can be verified by measuring the topological b-field, defined in a similar way as the density q of topological charge, but substituting s to S:

b = n • ∂ x n × ∂ y n, n = s/|s| ; (14)
represented in figure 3 by the color density. We verify that the transition is associated with the nucleation of a well localized electron vortex having a topological charge density opposite to the one of the background skyrmion (the white spots that appear near the skyrmion core at times t = 1100, 1748, 5936 for the three values of the dissipation, respectively). We remark, that in spite of the dissipation, the characteristic length scale of these structures is comparable with a few lattice steps.

CONCLUSION

We investigated the transition between a skyrmion state and a ferromagnetic state driven by a spin polarized electron current. We demonstrated that the torque exerted by the itinerant spins modifies the distribution of the magnetization around the skyrmion core, and tends initially to reduce its size. When the system is dominated by the exchange interaction, a self-similar collapse of the core in a finite time produces, changing the topology of the system. In addition, this topological change strongly depends on the dissipation mechanism. In absence of dissipation the collapse time explicitly depends on the lattice cutoff. At variance, in the case where dissipation is effective, the singularity tends to regularize, and the life time of the skyrmion state reduces with increasing dissipation. However, the microscopic mechanism of topological change is in the dissipative case, similar to the nondissipative one (even if the time and length scales may differ). It is related to the appearance of a peculiar electronic structure possessing a net charge opposite to the one in the skyrmion. The synchronization of the fixed spins with this electron vortical structure, leads to the annihilation of the topological charge, which pass from its initial value Q = -1 (skyrmion state) to zero (homogeneous ferromagnetic state).

Figure 2 .Figure 3 .

 23 Figure 2. Topological charge Q as a function of time (in adimensional units) for different values of the exchange dissipation parameter: (left) β = 0.1, (center) β = 0.01, and (right) β = 0.001. The red line is for Q + . The initial skyrmion charge is Q = -1, and the current is polarized in the -z direction. The change from Q = -1 to Q = 0 corresponds to a transition from the skyrmion to the ferromagnetic sate. Decreasing the dissipation strength results in an increase of the time necessary to reach the transition: t =1236, 1748 and 5936, for β =0.1, 0.01 and 0.001, respectively. The electric field is E = 10 -3 .(1080) (1100)

ACKNOWLEDGEMENTS

We thank R. G. Elías for useful discussions.