Dirac and Weyl fermions in condensed matter systems: an introduction

Transcription

1 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

2 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

3 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

4 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 0 th 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.

5 Electron bands(cont'd) Recap: electrons in crystals Usually just give in the 1 st 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]

6 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 (m 2-1) 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 v i are (anisotropic) velocities.

7 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.

8 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

9 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 Two spin-degenerate bands (4 bands) at BZ corners (K&K') touch at Fermi level and disperse linearly Brillouin zone

10 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

11 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: M 0 &M 2 are of same/different sign. Surface: mass domain wall, M 0 changes sign. Surface state: Jackiw-Rebbi soliton on mass domain wall. e.g. TI(M 0 <0) in z<0 and vacuum(m 0 >0) in z>0, two surface states satisfy and and follow 2x2 gapless Dirac Hamiltonian (half of graphene)

12 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 e 2 /2h Hall conductance [observed by CZChang et al. Science'13] YLChen et al. Science'10

13 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 Chern- Simons term for surface action (quantum anomalous Hall)

14 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: Cd3 As 2. Two Dirac points at ±k D on 4-fold rotation axis. Dirac mass is forbidden by C 4v symmetry: 4 states are two pairs of 2dim'l irreducible rep. of C 4v ; energy difference of two pairs change sign along Γ-Z. Neupane et al. NatCommun'14

15 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.

16 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

17 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.

18 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 M 0 &M 2 are of different sign (non-trivial). Define unit vector Chern# = skyrmion number of n, Edge: CI(M 0 <0) in x<0 and vacuum(m 0 >0) in x>0. Jackiw-Rebbi soliton: edge states satisfy and and are 1D chiral fermion

19 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

20 Surface Fermi arc from Weyl nodes For two Weyl nodes of opposite chirality at kx =±k W,k y =k z =0. The Chern# of k x <k W and k x >k W k-planes differ by 1. Suppose k x <k W k-planes have nonzero Chern#. With real space surface at z=0, k x,y are still conserved.it is a 2D system in yz-plane with edge at z=0. For k x <k W there will be chiral edge states The Fermi surface on z=0 real space surface (2D k x k y -system) is a non-closed arc connecting the projection of Weyl nodes on k x k y -plane. Turner&Vishwanath arxiv:

21 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] SYXu et al. Science'15

22 Landau levels(lls) from Dirac/Weyl fermions For uniform mag. field B=Bz, define Weyl Hamiltonian becomes Energy eigenvalues are for n>0, and w/ eigenstate In 2D (k z =0): unevenly spaced Landau level energy Single Weyl point produces a chiral 0 th Landau level. STM observation of Landau levels in graphene, Miller et al. Science'09 LLs from 3D Weyl points of opposite chirality

23 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

24 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.

25 3D (massless) Dirac fermions in ZrTe 5. 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: RYChen et al. PhysRevLett'15: magneto-optical reflectance, peaks indicate transitions between nth and (n+1)th Landau levels,

26 Summary Dirac/Weyl fermions emerge in 2D/3D solid state systems as special band degeneracy points Topological surface chiral fermions ~ Jackiw-Rebbi solition. Many high-energy physics phenomena related to Dirac/Weyl fermions have analogy in CMP. Chiral (Adler-Bell-Jackiw) anomaly: negative magnetoresistance Beyond these? Interaction (with EM field)? Emergent supersymmetry? [Grover et al. Science'14]