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Emergent Gauge Fields and Topological Effects in Dirac Matter

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Emergent Gauge Fields and Topological Effects in Dirac Matter UNIVERSIDAD AUTONOMA´ DE MADRID Departamento de F´ısicade la Materia Condensada EMERGENT GAUGE FIELDS AND TOPOLOGICAL EFFECTS IN DIRAC MATTER Tesis doctoral presentada por Yago Ferreir´osBas Programa de Doctorado de F´ısicade la Materia Condensada y Nanotecnolog´ıa Directores: Alberto Cortijo Mar´ıa Angeles´ Hern´andezVozmediano Tutor: Nicol´asAgra¨ıt Madrid, Junio 2016 Agradecimientos Esta tesis no hubiera sido posible sin el incondicional apoyo de muchas personas. El principal motivo de que yo est´een este momento escribiendo estas l´ıneas,es el apoyo de Geli y Alberto, mis tutores de tesis. Siempre estar´eagradecido por la manera en la que me han apoyado, y por todo el tiempo que han dedicado para ayudarme en mi formaci´oncomo cient´ıfico,as´ıcomo en otros aspectos de la vida. Trabajar con ellos ha sido todo un placer, a la vez que un bonito reto, y cualquier logro que est´epor venir en mi carrera se lo deber´ea ambos. Quiero dar las gracias tambi´ena Tony, uno de los mejores f´ısicosque he conocido y una pieza fundamental en mi formaci´on,y a Nicol´aspor haber accedido a tutelar esta tesis y haberse mostrado tan accesible. Durante el transcurso de la tesis, he tenido el honor de colaborar con grand´ısimos cient´ıficos. La sabidur´ıay claridad de ideas de Karl ha sido un ejemplo para m´ı,y los consejos de Fito, Fernando y H´ector,tanto de f´ısicacomo de otros aspectos de la vida investigadora, han supuesto una ayuda inestimable. He podido aprender mucho de Paco, uno de los grandes, en especial de su inigualable intuici´onpara la f´ısica,as´ı como de Bel´en,Rafa y Pablo. My stay in Nijmegen was a fantastic experience, and I want to really thank Misha for his hospitality, his time to discuss physics with me, and specially for making a fruitful scientific collaboration possible. I really enjoyed the discussions with Frank, and the warm atmosphere at the department: Zhenya, Guus, Hylke... all very nice people. La agradable vida en el ICMM ha contribu´ıdoa que todo fuera m´asf´acil.Agradezco a todos los compa~neros del grupo, o que alguna vez lo fueron, por el magn´ıficotiempo que hemos pasado juntos, especialmente durante las comidas: Angel,´ Jose, Franc- cesca, Luca, Bruno, Nacho, Laura, Mauricio, Vincenzo, Vicente, Raquel... He de dar las gracias tambi´ena mis compa~nerosde despacho, M´onica,Fernando y Jose Carlos, por contribuir a crear un ambiente agradable, en el cual me he sentido realmente a gusto. No podr´ıadejar de agradecer el apoyo y el tiempo pasado con los compa~nerosde las mesas. Gracias a Peibol, Edu, Paula, Miguel, Manuela, Paloma, Clara, Luis, Drino, Emilio y todos aquellos que estuvieron presentes durante los a~nosde universidad. Quiero hacer tambi´enuna menci´onespecial a los amigos de la infancia en el colegio, 2 cuya amistad he mantenido hasta hoy, y que han contribu´ıdoa moldear la persona en la que me he convertido: Pelu, Galla, Frodo, ´I~nigo,Marcos y Rom´an.En especial a uno de ellos, Pelu, le estar´esiempre agradecido por saber que siempre va a estar ah´ı,tanto en los buenos como en los malos momentos. Para ir terminando, quiero agradecer a mis padres su apoyo infinito y su dedicaci´on absoluta. Es sobre todo gracias a ellos que he encontrado la fuerza necesaria para acometer este desafiante reto. Es realmente tranquilizador saber que siempre estar´an ah´ı,apoyando mi carrera. Por ´ultimo,quiero dedicar unas palabras a Tamara. Nunca he tenido m´astran- quilidad y convicci´onpara llevar a buen puerto mi carrera investigadora como desde que estamos juntos. Es el motor que me impulsa, nada de esto ser´ıaposible sin ella. 3 Publication list The research performed during the development of this thesis has led to the following publications: 1. Domain wall motion in junctions of thin-film magnets and topological insulators. Yago Ferreiros, Alberto Cortijo, Phys. Rev. B 89, 024413 (2014). 2. Large conduction band and Fermi velocity spin splitting due to Coulomb inter- actions in single-layer MoS2. Yago Ferreiros, Alberto Cortijo, Phys. Rev. B 90, 195426 (2014). 3. Dirac electrons and domain walls: a realization in junctions of ferromagnets and topological insulators. Yago Ferreiros, Frank Buijnsters, M. I. Katsnelson, Phys. Rev. B 92, 085416 (2015). 4. Elastic gauge fields in Weyl semimetals. Alberto Cortijo, Yago Ferreiros, Karl Landsteiner, Maria A. H. Vozmediano, Phys. Rev. Lett. 115, 177202 (2015). 5. Visco elasticity in 2D materials. Alberto Cortijo, Yago Ferreiros, Karl Landsteiner, Maria A. H. Vozmediano, 2D Mater. 3 011002 (2016). 6. Unconventional electromagnetic mode in neutral Weyl semimetals. Yago Ferreiros, Alberto Cortijo, Phys. Rev. B 93, 195154 (2015). Other publications during this period: Quantum corrections to vortex masses and energies. • Yago Ferreiros, Antonio Gonzalez-Arroyo, Phys. Rev. D 90, 025004 (2014). 4 Chirality-dependent transmission of spin waves through domain walls. • Frank Buijnsters, Yago Ferreiros, A. Fasolino, M. I. Katsnelson, Phys. Rev. Lett. 116, 147204 (2016). 5 Abstract In this thesis we deal with systems hosting Dirac quasiparticles. The main focus is put on the physics arising from the interplay of non trivial topology and the emer- gence of gauge fields, stemming from the interaction of electrons with either magnetic and elastic degrees of freedom. The investigation relies on tools from quantum field theory, in particular the theory of quantum anomalies and topological field theory. The physical systems involved are two dimensional Dirac materials, like graphene and single layer transition metal dichalcogenide semiconductors; three dimensional time reversal invariant topological insulators, from which we focused on the two dimen- sional physics occurring at the edge; and Weyl semimetals. Under certain conditions, all these materials can be expected to be in a topological phase of matter, denoted quantum Hall state. Most of the results derived in this thesis are consequences of the special properties of this state of matter. 6 Resumen En esta tesis tratamos con sistemas que albergan quasipart´ıculas de Dirac. La atenci´onse centra en la f´ısicaque surge de la interrelaci´onentre la topolog´ıay los campos gauge emergentes en ciertos materiales, derivados de la interacci´onde los electrones con los grados de libertad magn´eticosy el´asticos.La investigaci´onse so- porta en las herramientas de la teor´ıacu´antica de campos, en particular en la teor´ıa de anomal´ıascu´anticas y la teor´ıade campos topol´ogica.Los sistemas f´ısicostrata- dos son materiales de Dirac en dos dimensiones, como el grafeno y semiconductores dicalcogenuros de metales de transici´on;aislantes topol´ogicostridimensionales, en los cuales nos centramos en la f´ısicabidimensional de su superficie; y semimetales de Weyl. Bajo ciertas condiciones, se puede esperar que estos materiales se encuentren en una fase topol´ogicade la materia, llamada estado Hall cu´antico. La mayor´ıade los resultados obtenidos en esta tesis son consecuencia de las propiedades especiales de este estado de la materia. 7 Introduction Dirac matter refers to those materials whose low energy electronic quasiparticle exci- tations behave as Dirac fermions. The most popular example of a gapless 2D Dirac material is graphene[1,2], in which the spin of the Dirac spinor is not the real spin, but a pseudospin related to the sublattice degrees of freedom. Beyond graphene, other important materials hosting 2D Dirac pseudospinors are the transition metal dicalco- genide (TMD) semiconductors [3,4] which, unlike graphene, have a gaped spectrum which could be important for technological applications. With the emergence of 3D topological insulators (TI)[5,6,7], a new realization of 2D Dirac fermions arises as gapless excitations localized on the boundary of a 3D material, as a consequence of the topology of the bulk. The spinorial degree of freedom of these boundary fermions is the real spin of the electrons, and their gapless nature implies that their spin ori- entation is aligned with their propagation direction. This maximizes the effects of spin-orbit coupling which is very valuable for spintronic applications[8]. More recently, new materials hosting gapless Dirac fermions have been discovered in 3D, which are called Dirac semimetals[9]. In those Dirac semimetals having inver- sion or time reversal (T) symmetry broken, the four component spinor splits into two two component Weyl spinors, separated in four-momentum space. These are called Weyl semimetals (WSMs), which were first proposed to occur in a class of iridate materials[10], and their experimental realization during the last year[11, 12, 13, 14, 15] has given rise to an intense theoretical and experimental activity associated to new material types, and to the possibility of exploring the exotic physics associated to chiral matter. An important aspect of Weyl fermions in quantum field theory is the presence of quantum anomalies. They are produced when a symmetry transformation of a classical system, either external (space-time transformations like translations or rota- tions) or internal (like global phase or gauge transformations), is broken by quantum fluctuations. The most prominent example is the chiral anomaly of Weyl fermions in even space-time dimensions, which represents a non conservation of chiral charge at the quantum level, but there are others equally sound like the gravitational anomaly or the trace anomaly (see for example [16]). With the advent of Dirac matter, the 8 possible experimental consequences of the physics of anomalies has become a very active subject in condensed matter physics. When 2D Dirac fermions are gaped, T symmetry is broken and their quantum effective electromagnetic behavior is described by a topological field theory (TFT), characterized by a Chern-Simons (CS) term[17, 18]. A similar thing happens in 3D, where a new scale in the system can be introduced, inducing a separation of the two chiral projections of the Dirac fermion in momentum space.
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