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Weyl Semimetals Journal Club, 09/08/2013 Semimetals: basic questions ● ● Is it possible for a gapless system to have a defining topological feature? How can one distinguish phases of a system with gapless degrees of freedom? ● ● 09/08/2013 Weyl semimetal, as an example JC: Semimetals 2 Brief review: Dirac equation ● Dirac equation for a spin-1/2 particle with mass Dirac algebra ● Weyl representation: ● 09/08/2013 is diagonal (Dirac representation: JC: Semimetals is diagonal) 3 Brief review: Dirac equation ● In momentum space, with ● Using the Weyl representation are eigenstates of ● : helicity is a good quantum number 09/08/2013 JC: Semimetals 4 Weyl semimetals ● Valence and conduction bands touch ● Non-degenerate band: time-reversal symmetry is broken ● One possible Hamiltonian ● Expanding around ● where Weyl equation 09/08/2013 JC: Semimetals 5 How robust are the Weyl nodes? ● The system is 3D! Expanding around a node ● Solve for 3 equations; 3 unknowns 09/08/2013 JC: Semimetals 6 Magnetic Monopoles (review) ● What is the magnetic charge of a Dirac node? ● Consider the eingestates of ● The Berry flux is 09/08/2013 JC: Semimetals 7 Topological Response: Chiral Anomaly ● ● ● ● Breaks down the conservation of particles at a given Weyl point Independent equations for R/L ? QFT: Cut-off violates conservation (Adler-Bell-Jackiw anomaly) Another derivation: Starting with B in the z-direction ● ● 09/08/2013 Landau Levels JC: Semimetals 8 Topological Response: Chiral Anomaly ● Solving the Dirac equation with an applied magnetic field, 09/08/2013 JC: Semimetals 9 Topological Response: Chiral Anomaly ● Effect of the electric field ● Rate of increase/decrease of electrons force density of states density of modes ● Change of momentum per unit volume 09/08/2013 JC: Semimetals 10 Topological Response: Anomalous Hall Effect Using the Lorentz force and the basic equations for the QHE and 09/08/2013 JC: Semimetals 11 Is Graphene a Topological Semimetal? ● ● ● If just nearest-neighbor hopping is taken into account, Sublattice symmetry: changing the sign of amplitudes on one sublattice reverses the sign 1D: Class AIII, edge states with zero-energy zigzag model 09/08/2013 JC: Semimetals 12 Experimental Proposal ● Goal: distinguish anomalyrelated physics from conventional metallic behavior 09/08/2013 JC: Semimetals 13