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Dirac and Weyl fermions in condensed matter systems: an introduction Fa Wang ( 王垡 ) ICQM, Peking University 第二届理论物理研讨会 2015/12/6@PKU Preamble: Dirac/Weyl fermions ● ● Dirac equation: reconciliation of special relativity and quantum mechanics – Positive and “negative” energy solutions (anti-particles). – Lorentz invariance. Weyl equation: w/ Dirac mass m=0, “chirality” is conserved, project Dirac equation (4x4) onto subspaces, it becomes two decoupled Weyl (2x2) equations 2015/12/6@PKU Outline ● Recap of solid state physics ● 2D/3D “Dirac/Weyl semimetals” in CMP ● – 2D materials: graphene, TI surface – 3D materials: Dirac/Weyl semimetals Phenomena related to Dirac/Weyl fermions in CMP – “monopole” of Berry curvature, surface “Fermi arc” – Landau levels and “chiral anomaly” 2015/12/6@PKU Recap: electrons in crystals ● ● Ref.: any solid state physics textbook Electrons in solid state systems should in principle be treated as non-relativistic fermions – ● Energy scale ~ 10eV, much smaller than electron rest mass Electron bands: as 0th approx., assume the nuclei form static periodic lattice, and electrons move in the periodic potential. – Bloch theorem: electron eigenstates are products of planewave and a periodic Bloch function u, for lattice translation vector R. – Electron energy eigenvalues are periodic in k-space, for reciprocal lattice vector G. 2015/12/6@PKU Recap: electrons in crystals ● Electron bands(cont'd) – Usually just give in the 1st Brillouin zone(BZ), the “unit cell” in k-space. – Depending on whether the Fermi energy is inside a band or in band gap, the system can be metal or insulator. – “effective mass” m* is different from free electron, if m*<0, it is hole-like carrier. – Effective Hamiltonian at band degeneracy points nodes may be Dirac/Weyl-like [Herring, PhysRev'37] 2015/12/6@PKU Recap: electrons in crystals ● Effective Hamiltonian: for bands n...(n+m-1), – Usually use k-indep. Bloch basis, w/ hermitian “Hamiltonian” – m bands are degenerate [H(k) prop. to identity matrix] : in general k needs to satisfy (m2-1) conditions – Symmetries may reduce the number of conditions – Without symmetry, only m=2 and k in 3D can have generic point solution of band degeneracy (usually 3D Weyl points). – Usually expand H(k) around the degenerate points (“nodes”) to linear order in k: 3D Dirac/Weyl are vi are (anisotropic) velocities. 2015/12/6@PKU “Trivial” realization of Dirac fermions in solid state systems: 1D metal ● For a 1D (spinless) metal with two “Fermi points” at ±kF, low energy electrons are described by “right/left mover” under the 1D Dirac equation – Mass term (back-scattering) is usually forbidden by lattice translation symmetry (conservation of momentum mod G) – Electron interactions are important. Free fermion description is generically not valid in 1D. 2015/12/6@PKU Outline ● Recap of solid state physics ● 2D/3D “Dirac/Weyl semimetals” in CMP ● – 2D materials: graphene, TI surface – 3D materials: Dirac/Weyl semimetals Phenomena related to Dirac/Weyl fermions in CMP – “monopole” of Berry curvature, surface “Fermi arc” – Landau levels and “chiral anomaly” 2015/12/6@PKU 2D Dirac fermions: graphene ● Ref.: Geim et al. RevModPhys'09. ● 2D honeycomb lattice of carbon atoms ● States close to Fermi level are pz orbitals Y.Yao et al Phys.Rev.B'07 ● Brillouin zone Two spin-degenerate bands (4 bands) at BZ corners (K&K') touch at Fermi level and disperse linearly 2015/12/6@PKU 2D Dirac fermions: graphene ● Under basis of pz orbitals effective Hamiltonians around K and K'=-K are – This is determined by the crystal & time-reversal symmetries – The Dirac mass is due to spin-orbit coupling and is tiny. – Breaking A/B sublatt. symmetry can induce mass term – Many phenomena related to Dirac fermions have been predicted/observed in graphene. ● e.g.: Klein “paradox”: perfect transmission for normal incidence on a potential barrier 2015/12/6@PKU 2D Dirac/Weyl fermions: surface states of 3D topological insulators(TI) ● Ref.: Qi&Zhang, RevModPhys'11 ● Many 3D TI materials: BiSb alloy, Bi2(Se,Te)3, ● Simplest model: massive Dirac fermions (4 bands) in 3D, – Trivial/non-trivial TI: M0&M2 are of same/different sign. – Surface: mass domain wall, M0 changes sign. – Surface state: Jackiw-Rebbi soliton on mass domain wall. e.g. TI(M0<0) in z<0 and vacuum(M0>0) in z>0, two surface states satisfy and and follow 2x2 gapless Dirac Hamiltonian (half of graphene) 2015/12/6@PKU 2D Dirac/Weyl fermions: surface states of 3D topological insulators(TI) ● Time-reversal(TR) symmetry protects the gaplessness of TI surface states. – Zeeman field can open gap on TI surface and lead to quantum anomalous Hall effect: each surface contribute e2/2h Hall conductance [observed by CZChang et al. Science'13] 2015/12/6@PKU YLChen et al. Science'10 2D Dirac/Weyl fermions: surface states of 3D topological insulators(TI) ● Topological field theory description of 3D TI – Couple bulk 3D TI to EM field, integrate out fermions, the action contains a “topological θ-term” – θ is analogous to the “axion field”. Due to TR symmetry, θ=0(trivial) or θ=π (TI) mod 2π. – Surface preserves TR, but θ cannot smoothly change from trivial to TI without breaking TR. To reconcile these, fermions on surface should be gapless. – On surface, θ jumps by ±π, it seems to produce a ChernSimons term for surface action (quantum anomalous Hall) 2015/12/6@PKU 3D Dirac fermions: Dirac semimetals ● 4-band degeneracy point in k-space with linear dispersions. – ● Special crystal symmetry is needed to forbid the Dirac mass. Example: Cd3As2. – Two Dirac points at ±kD on 4-fold rotation axis. – Dirac mass is “forbidden” by C4v symmetry: 4 states are two pairs of 2dim'l irreducible rep. of C4v; energy difference of two pairs change sign along Γ-Z. 2015/12/6@PKU Neupane et al. NatCommun'14 3D Weyl fermions: Weyl semimetals ● ● 2-band degeneracy point in k-space with linear dispersions. – NO symmetry needed to protect this degeneracy. – Such Weyl nodes must appear in pairs of opposite chirality. – Weyl nodes of opposite chirality can be mutually gapped out if they meet in k-space. Break inversion and/or TR symmetry to separate them. [Burkov&Balents, PhysRevLett'11] – Weyl nodes are difficult to locate (usually not on high symmetry lines) Example: TaAs. no inversion symmetry, 12 pairs of Weyl nodes. 2015/12/6@PKU Outline ● Recap of solid state physics ● 2D/3D “Dirac/Weyl semimetals” in CMP ● – 2D materials: graphene, TI surface – 3D materials: Dirac/Weyl semimetals Phenomena related to Dirac/Weyl fermions in CMP – “monopole” of Berry curvature, surface “Fermi arc” – Landau levels and “chiral anomaly” 2015/12/6@PKU Prerequisite: Berry curvature in k-space ● For a band with Bloch function its (Abelian) Berry connection Berry curvature – Analogue of vector potential & mag. field in k-space. Semiclassical equations of motion of electron on this band: red term: anomalous velocity[Sundaram&Niu, PhysRevB'99] – The Chern number over a closed surface in k-space, is an integer, if the band is non-degenerate on this surface. – Intrinsic anomalous Hall conductivity for 2D system Fully occupied band w/ Chern number shows quantum anomalous Hall effect. 2015/12/6@PKU 2D “Chern insulator” and chiral edge states ● ● Fully occupied bands with nonzero total Chern number: “Chern insulator(CI)” w/ quantum anomalous Hall effect. Simplest model: massive Dirac fermion (2 bands) in 2D – Occupied band, has nonzero Chern# if M0&M2 are of different sign (non-trivial). – Define unit vector skyrmion number of n, – Edge: CI(M0<0) in x<0 and vacuum(M0>0) in x>0. Jackiw-Rebbi soliton: edge states satisfy and and are 1D chiral fermion 2015/12/6@PKU Chern# = Weyl node as “monopole” of Berry curvature ● For a Weyl point at kW , the Berry curvature of occupied band, is – Total Berry flux thru. surface enclosing Weyl node(s) is – Weyl node~“monopole” in k-space, “topologically” stable. – due to periodicity in k-space, the total Berry flux thru. surface of entire 3D BZ must be zero, therefore sum of all Weyl points' chirality in 3D BZ must be zero: Weyl points appear in pairs of opposite chirality. Balents, Physics'11 2015/12/6@PKU Surface “Fermi arc” from Weyl nodes ● For two Weyl nodes of opposite chirality at kx=±kW ,ky=kz=0. – The Chern# of |kx|<kW and |kx|>kW k-planes differ by 1. Suppose |kx|<kW k-planes have nonzero Chern#. – With real space surface at z=0, kx,y are still conserved.It is a 2D system in yz-plane with edge at z=0. For |kx|<kW there will be chiral edge states – The Fermi surface on z=0 real space surface (2D kxky-system) is a non-closed “arc” connecting the projection of Weyl nodes on kxky-plane. 2015/12/6@PKU Turner&Vishwanath arXiv:1301.0330 Surface “Fermi arc” from Weyl nodes ● Surface “Fermi arc” is now used as the “smoking gun” signature of Weyl semimetals: – Claimed ARPES observations for TaAs [SYXu et al. Science'15, BQLv et al. PhysRevX'15] 2015/12/6@PKU SYXu et al. Science'15 Landau levels(LLs) from Dirac/Weyl fermions ● For uniform mag. field B=Bz, define Weyl Hamiltonian becomes – Energy eigenvalues are and w/ eigenstate – In 2D (kz=0): unevenly spaced Landau level energy – Single Weyl point produces a chiral 0th Landau level. for n>0, STM observation of Landau levels in graphene, Miller et al. Science'09 LLs from 3D Weyl points of opposite chirality 2015/12/6@PKU “Chiral anomaly”: 1D metal ● Consider 1D spinless metal, the low energy theory is – It seems that right/left-mover number are separately conserved – However under an “electric field” F, – Semiclassical picture: occupied states in k-space “move to the right”, particles are transferred from left-mover to right-mover 2015/12/6@PKU Chiral anomaly in 3D Weyl semimetals ● Chiral(Adler-Bell-Jackiw) anomaly: for simplest Weyl semimetal, the right/left-handed Weyl fermion number are not conserved under EM field, – Semiclassical picture: 1D chiral anomaly on n=0 LL, LL degeneracy – If there are back-scattering, this will produce a steady current and lead to negative magnetoresistance(MR) [DTSon&Spivak, PhysRevB'13] – This effect also applies to gapless Dirac semimetal. 2015/12/6@PKU 3D (massless) Dirac fermions in ZrTe5. ● ● “Chiral anomaly” observed in MR Optical transition between Landau levels observed [RYChen, ZGChen, XYSong, Schneeloch, GDGu, FWang, NLWang, PhysRevLett'15] Q.Li et al. arXiv:1412.6543 RYChen et al. PhysRevLett'15: magneto-optical reflectance, peaks indicate transitions between nth and (n+1)th Landau levels, 2015/12/6@PKU Summary ● Dirac/Weyl fermions emerge in 2D/3D solid state systems as special band degeneracy points – ● Many high-energy physics phenomena related to Dirac/Weyl fermions have analogy in CMP. – ● Topological surface chiral fermions ~ Jackiw-Rebbi solition. Chiral (Adler-Bell-Jackiw) anomaly: negative magnetoresistance Beyond these? – Interaction (with EM field)? – Emergent “supersymmetry”? [Grover et al. Science'14] 2015/12/6@PKU