TOPOLOGICAL MATERIALS-BAND THEORY

Describtion of the course Seldom new states of matter are predicted and even less often are observed in the laboratory. This was the case with Topological Insulators like HgTe observed by Konig et al. Tillwe whis facts hints that this chaerge quantization are connected with the space time topology. The concepts which we will cover a) A central goal in Condensed Maatter is to charactezized phases of matter.A magnet or a Superconductor can be ounderstood in terms of the symetries that they spontaneusly break. It was become clear that a concept of entaglement must be used to describe the Quantum Hall effect and topological order. b) A central notion in the theory of eletrons is the formation of bands which are characterized by the concept of Topological Equivalence:The surface of a sphere (g = 0) and a doughnut (g = 1 are distinguishable topologically. The Mobius strip and the Mobius space are constuted by identifiying the left and the right edge of a sheet of paper giving it a twist c) The Bravais Lattice is obtained once the Hamiltonian obey H( r) = H( r + n1a1 + n2a2 + n3a3), where ni are integer and ai are unit vectors. In a periodic potential the Bloch theorem guaranties that in momentum space with the momentum K = k1b1 + k2b2 + k3b3 with the reciprocal unit vectors b3 = a1 × a2/|a3| obey ? ( r) = ei k· rU ( r) In momentum space we have the periodicity H( k) = H( k + G ) where G is ther eciprocal vector. d) The charge and conductance quantization is related to discreetete singulairties in the Brilouine Zone . Quantization follows from obstruction,in particular It is not possible to fix the phase for all the Billouine Zone. This leads to topological term caled Chern Invariants. In particular k and x = i?k do not commute in this way we can intoduce the Berry phase In a spin dependent space we have the wave functiuon Ua(K > which allows to introduce the spin connections. Aa,ß( k) =< Uß(K )|?a|Ua(K ) > which is the Berry phase. uch Aa,ß( k) allows to compute the curvature a a and connect the result tio a simple geometrical describtion.su e) The point symmetry souch as rotation and reflection and discreete symmetry as time eversal sym- metry.The evolution is symmetric in past and future.Such conceps are important for defining Topological Invariants. The prerequisite is. Any course which cover elements of Quantum Physics is apropriete. Number of Credits is 4 The Prerequisite is knowledge of Quantum Physics. Any course which cover this material is acceptable. The corequisite of the following courses is acceptable. Student can take in paralel the followings: a)551-Quantum I with the Pre:Math 391 and 346 ,Phys 351 b)552-Quantum II with the Pre:Phys 551 and ,Phys 361 c)554Solids State Physics with the Pre Phys 551 or Phys 321

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