Dirac fermions in condensed matters

Transcription

1 Dirac fermions in condensed matters Bohm Jung Yang Department of Physics and Astronomy, Seoul National University

2 Outline 1. Dirac fermions in relativistic wave equations 2. How do Dirac fermions appear in condensed matters? 3. Band crossing, symmetry, and dimensionality 4. Dirac fermions and topology 5. Interacting Dirac fermions 6. New low energy excitations and their classification 7. Summary

3 Dirac equation Relativistic wave equation describing massive spin-1/2 fermions E E gap =2mc 2 2-fold degenerate p

4 Weyl fermions Massless limit of Dirac equation Weyl equations A massless Dirac fermion A massless Weyl fermion 4-fold degenerate 2-fold degenerate

5 Outline 1. Dirac fermions in relativistic wave equations 2. How do Dirac fermions appear in condensed matters? 3. Band crossing, symmetry, and dimensionality 4. Dirac fermions and topology 5. Interacting Dirac fermions 6. New low energy excitations and their classification 7. Summary

6 Dirac fermions in condensed matters Emergent low energy excitations More is different P.W.Anderson Fig. from C. Terakura Novel phenomena/effects/functions of solids which can never be expected for individual constituent elements

7 Electronic band structure in periodic solids Bloch s theorem: Greiner and Folling (2008) Metal Insulator

8 Low energy excitations in metals Electrons on a lattice form a band structure Low energy excitations are described by emergent particles Schrodinger particles e.g.) Ordinary metals Dirac particles e.g.) 2D graphene Holes E F Electrons

9 Dirac fermions: when and why? Effective two-band model H(k)= f 0 (k)+ f 1 (k) 1 + f 2 (k) 2 + f 3 (k) 3 E gap =2(f 1 2 +f 2 2 +f 3 2 ) 1/2 Bulk black phosphorus (J. Kim et al, Science, 2015)

10 Outline 1. Dirac fermions in relativistic wave equations 2. How do Dirac fermions appear in condensed matters? 3. Band crossing, symmetry, and dimensionality 4. Dirac fermions and topology 5. Interacting Dirac fermions 6. New low energy excitations and their classification 7. Summary

11 Energy Energy Accidental band crossing A Dirac particle can be created by band crossing! Band crossing Band anti-crossing C C v.s. V V m m (m: tuning parameter) Accidental band crossing is not easy to achieve!

12 Band crossing in generic systems E H= f 0 + f f f 3 3 k E =2(f 1 2 +f 2 2 +f 3 2 ) 1/2 f 1 (k x, k y, k z ) = 0 Gap closing : f 2 (k x, k y, k z ) = 0 f 3 (k x, k y, k z ) = 0 In 3D : crossing of three planes In 2D : crossing of three lines Weyl points can exist! No solution!

13 Symmetry and band degeneracy Time-reversal(T): E n, (k)=e n, (-k) Inversion(P): E n, (k)=e n, (-k) E(k) E n, (k)=e n, (k) 2-fold degenerate -k 0 k Four bands should cross to generate a Dirac point!

14 Band crossing and T, P symmetries The effective Hamiltonian should be a 4 by 4 matrix! E H= b 1 (k) 1 + b 2 (k) 2 + b 3 (k) 3 + b 4 (k) 4 + b 5 (k) 5 2-fold degenerate k where i is 4 by 4 matrix satisfying { i, j }=2 ij e.g.) 1 = z x, 2 = z y, 3 = z z, 4 = x, 5 = y H 2 = b 1 (k) 2 +b 2 (k) 2 +b 3 (k) 2 +b 4 (k) 2 +b 5 (k) 2 =E 2 E CB E VB = 2 b 1 2 +b 2 2 +b 3 2 +b 4 2 +b 5 2 Gap-closing condition : b 1 (k) = b 2 (k) = b 3 (k) = b 4 (k) = b 5 (k) = 0 In general, a band crossing is impossible unless additional symmetries other than T, P

15 SU(2) symmetry and graphene Graphene has both T and P symmetries H = b 1 (k) 1 + b 2 (k) 2 + b 3 (k) 3 + b 4 (k) 4 + b 5 (k) 5 = b 1 (k) z x + b 2 (k) z y + b 3 (k) z z + b 4 (k) x + b 5 (k) y x,y,z describe spin degrees of freedom Spin SU(2) symmetry requires b 1 (k) = b 2 (k) = b 3 (k) = 0 Gap closing condition: b 4 (k x, k y ) = 0 b 5 (k x, k y ) = 0 Dirac point! In the presence of spin-orbit coupling, graphene is a gapped quantum spin Hall insulator!

16 Massless fermion and crystalline symmetry Role of a rotation symmetry C n k=k k k =R n k C n : rotation by 2 /n about an axis [H(k), C n ]=0 with k on the rotation axis Bands carry quantized rotation eigenvalues J 1 Band inversion C n=2,3,4,6 J 2 Wang, Dai, Fang (2012, 2013); H = H CB V mix V mix H VB = H J1 0 0 H J2

17 Observation of 3D Dirac semimetals Cd 3 As 2, Na 3 Bi are confirmed as a 3D Dirac SM! Liu, Shen, Chen (Science,2014) Neupane, Hasan (Nat. Comm. 2014) C 3 C 4

18 Outline 1. Dirac fermions in relativistic wave equations 2. How do Dirac fermions appear in condensed matters? 3. Band crossing, symmetry, and dimensionality 4. Dirac fermions and topology 5. Interacting Dirac fermions 6. New low energy excitations and their classification 7. Summary

19 Let s break P or T symmetry PT symmetric E(k) -k 0 k P-breaking T-breaking E(k) E(k) -k 0 k -k 0 k 2 by 2 matrix description is possible!

20 Emergence of Weyl fermions Band inversion Weyl fermion +1-1 H(k)= (v 1 k) x + (v 2 k) y + (v 3 k) z Handedness (or chirality) Chirality = v 1 (v 2 v 3 ) v 1 (v 2 v 3 ) v 3 v 1 v 2 v 1 v 2 v 3 Right-handed (+1) Left-handed (-1)

21 Topological properties of 3D Weyl fermions Enhanced Berry curvature +B A Weyl point has a quantized topological (chiral) charge ( 1)! Quantum Hall effect? k z = Chern number of the torus = Total monopole charge in the torus C= 1

22 Surface Fermi arc of 3D Weyl SM A sample with finite length along z direction k Top surface z Surface states : Fermi Arc Topological invariant : chiral charge W.Witczak-Krempa, G.Chen, Y.B.Kim, L.Balents

23 Observation of Fermi arcs TaAs, NbP, NbAs, TaP Band calculation predicts 12 Weyl points Lv,Ding (PRX, 2015)

24 Outline 1. Dirac fermions in relativistic wave equations 2. How do Dirac fermions appear in condensed matters? 3. Band crossing, symmetry, and dimensionality 4. Dirac fermions and topology 5. Interacting Dirac fermions 6. New low energy excitations and their classification 7. Summary

25 Why is interaction important in Dirac systems H QC = v 1 k v 2 k v 3 k Scale invariance E(k) E F k k 1 =b k 1, k 2 =b k 2 H QC (k i )=b H QC (k i ), k 3 =b k 3 2. Screening of Coulomb potential V sc (r) = e2 1 ε 0 r e q TFr, q 2 TF D E F = 0 Long-range Coulomb potential! 3. Fermi points (similar to 1D system) Non-Fermi liquid (Luttinger liquid)

26 Coulomb interaction in Dirac systems Coupling constant There is a single dimensionless coupling constant v=c/300, ε=1~100, =0.1~1 Effective Lagrangian: Quantum electrodynamics Fermi velocity << Light velocity (No Lorentz invariance) Instantaneous Coulomb potential

27 Marginal interaction and log-corrections Coulomb interaction is marginally irrelevant 0 Logarithmic enhancement of various physical quantities V. N. Kotov et al., RMP (2012);D.T.Son,PRB (2007); Gonzalez et al., Nucl.Phys.B (1994)) Velocity renormalization and emergent Lorentz invariance v/c 1 Elias, Geim et al.(2011)

28 QCP between a Weyl SM to an insulator The chiral charge of a Weyl point guarantees its stability A pair-annihilation is required +1 k C =k C (m) -1 Weyl-SM m c Pair-annihilation (+1)+(-1)=0 Insulator m H = k k [(m-m c )+ k 2 3] 3

29 Anisotropic Weyl fermions at QCP H QCP = v k v k A k k 1 k 2 Anisotropic dispersion at QCP : direct consequence of zero chiral charge! k 2 3 H +Weyl = v k v k v k H -Weyl = v k v k v k m c Pair-annihilation m

30 Polarization Correlation effect at QCP Screened Coulomb interaction (A.A. Abrikosov) Anisotropic partial screening! In real space : Effective interaction between fermions became weaker! Electrons behaves like free particles!

31 Interacting anisotropic Dirac fermions Unconventional quantum criticality Unconventional screening, free quasi-particles Density of states Specific heat Compressibility Diamagnetic susceptibility (B.-J. Yang, N. Nagaosa et al., Nature Physics, 2014)

32 Outline 1. Dirac fermions in relativistic wave equations 2. How do Dirac fermions appear in condensed matters? 3. Band crossing, symmetry, and dimensionality 4. Dirac fermions and topology 5. Interacting Dirac fermions 6. New low energy excitations and their classification 7. Summary

33 New Weyl/Dirac fermions Double Weyl SM Chiral charge= 2 Type-II Weyl SM Tilted Dirac cone H=(k 2 x - k 2 y) 1 +2k x k y 2 +k z 3 Type I Type II C 4 or C 6 rotation symmetry is required HgCr 2 Se 4, SrSi 2 WTe 2 X. Gu, X. Dai, Z. Fang, PRL (2012) Soluyanov, Dai, Bernevig, Nature (2015)

34 Line node semimetals BaTaS 3 Ca 3 P 2 Surface flat band Liang, Weng, PRB (2016) Chan, Schnyder, PRB (2016)

35 Classification based on crystalline symmetry Variety of Dirac semimetals protected by symmetries! Class I Class II Class III Symmorphic rotation symmetry Type-I nonsymmorphic symmetry Type-II nonsymmorphic symmetry

36 Summary Variety of nodal semimetals protected by symmetries New emerging fermionic particles New topological responses New correlated phenomena

37 Applications Dirac fermions are a basis for non-dissipative electronics! Quantum Hall K. v. Klitzing, G. Dorda, and M. Pepper (1980); K. v. Klitzing (2005) 1D Chiral Dirac fermions on the sample boundary! 1. Quantized Hall Resistance (R H = h 1 e 2, N N C =integer ) c 2. Zero longitudinal resistance (No back scattering, dissipationless)

38 Emergent physics in the future Electron correlation Symmetry breaking Quantum criticality High-T C superconductivity Topology Quantum Hall physics Dirac/Weyl semimetals Berry phase effects Emergent physics in strongly correlated topological systems