By B. Andrei Bernevig

This graduate-level textbook is the 1st pedagogical synthesis of the sector of topological insulators and superconductors, essentially the most fascinating parts of study in condensed topic physics. featuring the most recent advancements, whereas offering all of the calculations useful for a self-contained and entire description of the self-discipline, it really is excellent for graduate scholars and researchers getting ready to paintings during this quarter, and it'll be a vital reference either inside of and out of doors the classroom.

The e-book starts off with basic thoughts reminiscent of Berry levels, Dirac fermions, corridor conductance and its hyperlink to topology, and the Hofstadter challenge of lattice electrons in a magnetic box. It strikes directly to clarify topological stages of topic corresponding to Chern insulators, - and 3-dimensional topological insulators, and Majorana p-wave wires. also, the publication covers 0 modes on vortices in topological superconductors, time-reversal topological superconductors, and topological responses/field conception and topological indices. The ebook additionally analyzes contemporary issues in condensed topic conception and concludes by way of surveying energetic subfields of analysis equivalent to insulators with point-group symmetries and the soundness of topological semimetals. difficulties on the finish of every bankruptcy provide possibilities to check wisdom and have interaction with frontier learn matters. Topological Insulators and Topological Superconductors will offer graduate scholars and researchers with the actual knowing and mathematical instruments had to embark on learn during this quickly evolving field.

B. Andrei Bernevig is the Eugene and Mary Wigner Assistant Professor of Theoretical Physics at Princeton collage. Taylor L. Hughes is an assistant professor within the condensed topic concept team on the collage of Illinois, Urbana-Champaign.

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**Extra resources for Topological Insulators and Topological Superconductors**

**Sample text**

We introduce time reversal as a symmetry T of the Hamiltonian II of a system: [H, 71 =0. 2) which means that if lip) is an eigenstate of H, then so is Tl'lf). In many cases Tll/f) = II/I), but very interesting things happen if the two states are different. This is exactly the difference between integer and half-integer spin particles. 1 Time Reversal for Spinless Particles The TR operator changes only the arrow of time. As such, lt leaves the position operator x unchanged-in particular, time reversal commutes with any spatial symmetry.

We have T. 20) This result is of fundamental importance: acting with time reversal twice hence rotates the spin by 2n, which for particles with integer spin is equivalent to the identity operator, whereas for particles with half-integer spin, it gives a factor of -1. We remember this is identical to what was obtained in the Berry phase calculation. For spin-~ particles, the matrix exponential can be easily performed. We pick as spin-! the usual S = (h/2)(rr. "Y· er) and obtain -· e 17ruy/2 = L 00 -17r(I,.

If we can prove that they arc actually orthogonal, then the spectrum is doubly degenerate. , U is antisymmetric and unitary. 24) ...... Note that the T 2 = -1 was crucial to proving this theorem. 25) m~p So, we have proved that {Tl{llHllJI) = -(TIJllHllJI) = 0. If we had a set of n excitations and we were to calculate the probability that n left-movers are scattered back to n right-movers, we would get a factor of (-l)n. We can immediately show that for an even number of excitations, the scattering process between the Kramers' pairs does not vanish, but for an odd number of excitations it does.