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By Bertrand Duplantier, Vincent Rivasseau, Jean-Nöel Fuchs (eds.)
This 15th quantity of the Poincare Seminar sequence, Dirac topic, describes the amazing resurgence, as a low-energy powerful conception of engaging in electrons in lots of condensed topic structures, together with graphene and topological insulators, of the recognized equation initially invented via P.A.M. Dirac for relativistic quantum
mechanics. In 5 hugely pedagogical articles, as befits their starting place in lectures to a wide medical viewers, this booklet explains why Dirac issues. Highlights contain the designated "Graphene and Relativistic Quantum Physics", written by means of the experimental pioneer, Philip Kim, and dedicated to graphene, a form
of carbon crystallized in a two-dimensional hexagonal lattice, from its discovery in 2004-2005 by means of the longer term Nobel prize winners Kostya Novoselov and Andre Geim to the so-called relativistic quantum corridor impact; the evaluate entitled "Dirac Fermions in Condensed topic and Beyond", written by way of popular theoreticians, Mark Goerbig and Gilles Montambaux, who give some thought to many different fabrics than graphene, jointly often called "Dirac matter", and provide a radical description of the merging transition of Dirac cones that happens within the strength spectrum, in a variety of experiments concerning stretching of the microscopic hexagonal lattice; the 3rd contribution, entitled "Quantum delivery in Graphene: Impurity Scattering as a Probe of the Dirac
Spectrum", given by means of Hélène Bouchiat, a number one experimentalist in mesoscopic physics, with Sophie Guéron and Chuan Li, exhibits how measuring electric delivery, particularly magneto-transport in genuine graphene units - infected by means of impurities and consequently displaying a diffusive regime - permits one to deeply probe the Dirac nature of electrons. The final contributions concentrate on topological insulators; within the authoritative "Experimental Signatures of Topological Insulators", Laurent Lévy stories contemporary experimental growth within the physics of mercury-telluride samples below pressure, which demonstrates that the outside of a three-d topological insulator hosts a two-dimensional massless Dirac steel; the illuminating ultimate contribution by way of David Carpentier, entitled "Topology of Bands in Solids: From Insulators to Dirac Matter", presents a geometrical description of Bloch wave capabilities by way of Berry levels and parallel delivery, and in their topological category by way of invariants resembling Chern numbers, and ends with a point of view on three-d semi-metals as defined through the Weyl equation. This booklet can be of extensive basic curiosity to physicists, mathematicians, and historians of science.
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This situation arises when particular symmetries are imposed on the system that we discuss in the next paragraph. One of the symmetries that protect stable band-contact points in Hamiltonian (1) is time-reversal symmetry, and it imposes ∗ = Hk H− k → f y = −f y , x f− = fkx , k −k k z f− = fkz . k (3) Notice that in this argument, we have omitted the spin degree of freedom. 1 Another relevant symmetry is inversion symmetry. Consider that the diagonal elements of Hamiltonian (1) represent intra-sublattice (or intra-orbital) couplings.
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Y x 3 3 2 2 2 1 1 1 0 k ya 3 k ya k ya ±(a∗1 − a∗2 )/3, where a∗1 and a∗2 are elementary reciprocal lattice vectors, at these high-symmetry points is, however, a rather exceptional situation. Consider for example the “brickwall” lattice depicted on Figure 3-a. It has the same couplings between sites as in graphene, but due to the square symmetry, the Brillouin zone is a square. The Dirac points are now located inside the first Brillouin zone (BZ) (Figure 3-b). Although this brickwall crystal may appear impossible to realize in 0 0 1 1 1 2 2 2 3 3 3 3 2 1 0 kxa 1 2 3 3 2 1 0 kxa 1 2 3 3 2 1 0 kxa 1 Figure 3.