Physics of massless Dirac electrons ― from 2D to 3D
NQS2014, Kyoto, Nov. 17, 2014
Mikito Koshino (Tohoku University)
Collaborations with Takahiro Morimoto (RIKEN) Masatoshi Sato (Nagoya) Yuya Ominato (Tohoku) Tsuneya Ando (Titech)
Organic metal α-(BEDT-TTF)2I3
Surface of 3D topological Insulator
Graphene
2D gapless electron
Dirac Hamiltonian (3D)
m = 0
3D gapless electron
Doubly-degenerate Dirac cones … Dirac semimetal
4-fold degenerate
4x4 massless Dirac Hamiltonian
kx ky, kz
E
Murakami, New J. Phys. 9, 356 (2007) Burkov and Balents, PRL 107, 127205 (2011)
Weyl semimetal Separate two Dirac points in k-space
Each single node is descibed by 2x2 Weyl Hamiltonian
3D gapless electron
1) Topological band touching points protected by spatial symmetry + chiral symmetry
This talk
2) Some characteristic physics in 2D and 3D Weyl electrons
--- Trasnport property (metallic or insulating at Weyl point?)
--- Orbital diamagnetism (Singularity at Weyl point)
--- New class of 2D and 3D Weyl electrons
Hamiltonian connects ○ - ●, but not ○ - ○, or ● - ●
0
0 Schroedinger eq.
Chiral symmetry
Energy spectrum in chiral symmetry
E=0
|N○ ー N●| zero energy modes
N○ = N●
Zero energy modes
… topological number (never changes without breaking chiral symmetry) Cf. Atiyah‒Singer
index theorem
N○ ≠ N●
Squared
Diagonalized a b
d c
a b d c
0 0 0 zero modes
Zero energy modes Schroedinger eq.
← R : mirror reflection symmetry on line 1-3
| N○ ー N●|
Chiral symmetry + Reflection symmetry
2 zero modes
Koshino, Morimoto, Sato, Phys. Rev. B 90, 115207 (2014)
|2-1| = 1
|0-1| = 1
← C3 (120 rotation) symmetric
| N○ ー N●|
|3-1| = 2
4 zero modes
Chiral symmetry + C3 symmetry
Eigenvalues of C3:1, ω, ω2 [ ∵(C3)3 = 1]
|0-1| = 1
|0-1| = 1
Koshino, Morimoto, Sato, Phys. Rev. B 90, 115207 (2014)
Application to Bloch eletrons
Chiral symmetric
Not chiral symmetric
Ex.) Honeycomb lattice (simplest model for graphene)
A B
--- C3 + chiral symmetry
k-space
K’ K
K
K’
K’
K
R3(k)
k
Bloch Hamiltonian
For generic k-points
For special k-points satisfying
… the previous argument applies
Reciprocal space
Koshino, Morimoto, Sato, Phys. Rev. B 90, 115207 (2014)
At K-point
none
| N○ ー N●|
1
1
0
2 zero modes
: rotation center
K’ K
K
K’
K’
K
Koshino, Morimoto, Sato, Phys. Rev. B 90, 115207 (2014)
K’ K
K
K’
K’
px
py
E
px
py
E
K
Gap closing 2 zero modes at each of K and K’
Weyl nodes (band touching point) at K and K’
Winding number (Berry phase)
Weyl nodes are protected also by the winding number
Are these arguments equivalent?? … No, there are different topological matters
Ex.) honeycomb lattice + superlattice distortion
real space
unit cell
k-space
Total winding number at Γ
Γ
Koshino, Morimoto, Sato, PRB 90, 115207 (2014)
Honeycomb lattice + superlattice distortion
1 4 5
6
2 3
3 2
2 3
By C3 rotation: 1 → 1 2 → 2 3 → 3 4 → 5 5 → 6 6 → 4
By C3 rotation: 1 → 1 2 → 2 3 → 3 4 → 5 5 → 6 6 → 4
1 → 1 2 → 2 3 → 3 4 → 5 5 → 6 6 → 4
1 4 5
6
2 3
3 2
2 3
Honeycomb lattice + superlattice distortion
Are these arguments equivalent??
1 4 5
6
2 3
3 2
2 3
| N○ ー N●| |3-1| = 2
4 zero modes = 2 Weyl nodes still touching |0-1| = 1
|0-1| = 1
Half-flux 2D square lattice (C2 symmetry)
| N○ ー N●|
1
1 2 zero modes (single Weyl node)
At K point:
Koshino, Morimoto, Sato, PRB 90, 115207 (2014)
i i
(even)
(odd)
Dirac semimetal (3D band touching point)
4 zero modes (double Weyl nodes)
At (K, π/2c): A1
B1 A2
B2
← C3Rz symmetric (120 rotation + reflection on xy-plane)
Eigenvalues of C3Rz = [ ∵(C3Rz)6 = 1]
1
-1 ω
-ω ω2
-ω2
none none
| N○ ー N●| 1
1 1
1
Koshino, Morimoto, Sato, PRB 90, 115207 (2014)
energy bands always 2-fold degenerate
Weyl nodes (4-fold degenerate)
… Dirac semimetal
σ : sublattice A, B ρ : layer 1, 2
Effective Hamiltonian (K-point)
Dirac semimetal (3D band touching point) Koshino, Morimoto, Sato, PRB 90, 115207 (2014)
4 zero modes (double Weyl nodes)
At π/(2a)(1,1,1):
A1, B2
← C2Rz (180 rotation + reflection on xy-plane) symmetric
Eigenvalues of C2Rz =
1
-1 B1, A2
| N○ ー N●|
2
2
Dirac semimetal (3D band touching point) Koshino, Morimoto, Sato, PRB 90, 115207 (2014)
How many independent topological numbers? Ex.) C3 symmetry in 2D
Topological numbers associated with a Weyl node
… Winding number
… | N○ ー N●| for eigenspaces of C3
constraint
Independent topological numbers:
Koshino, Morimoto, Sato, PRB 90, 115207 (2014)
How many independent topological numbers? Ex.) C2 symmetry in 2D
Topological numbers associated with a Weyl node
… Winding number
… | N○ ー N●| for eigenspaces of C2
constraint
Independent topological numbers:
Koshino, Morimoto, Sato, PRB 90, 115207 (2014)
Complete set?
Algebraic argument (Clifford algebra + K-theory)
… How to prove completeness? Independent topological numbers:
Ex.) C2 symmetry in 2D
Koshino, Morimoto, Sato, PRB 90, 115207 (2014)
1) Topological band touching points protected by spatial symmetry + chiral symmetry
This talk
2) Some characteristic physics in 2D and 3D Weyl electrons
--- Trasnport property (metallic or insulating at Weyl point?)
--- Orbital diamagnetism (Singularity at Weyl point)
--- New class of 2D and 3D Weyl electrons
Conductivity of Weyl electron
Graphene:
Conductivity of conventional metal
εF
εF : Fermi energy τ : scettering time
Current flows
εF = 0
-eE
εF → 0 … then τ → 0
conductivity at Weyl point?
τ τ
Theoretical calculation (short-range impurities, self-consitent Born approx.)
At Weyl point Off Weyl point
Conductivity of graphene (2D Weyl)
… Graphene is a “zero-gap metal”
εF
Shon and Ando, JPSJ, 67, 2421 (1998)
… independent of disorder strength
Conductivity of 3D Weyl electron?
εF = 0
At Weyl point:
2D
3D
… ?
Cf. Transport in 3D Weyl electron: Fradkin, PRB 33, 3263 (1986). Nandkishore, Huse, and Sondhi, arXiv:1307.3252 (2013). Kobayashi, Ohtsuki, Imura, Herbut, Phys. Rev. Lett. 112, 016402 (2014). Biswas and Ryu , arXiv: 1309.3278 (2013).
d0
Self-consistent Born approximation
Hamiltonian:
Self-consistent Born approximation (SCBA)
Disorder strength (dimensionless)
ni … density of scatterers
scattering potential
Ominato and Koshino, PRB 89, 054202 (2014)
Density of states (ε=0) 3D
See also, Fradkin, PRB 33, 3263 (1986). K. Kobayashi, et al, Phys. Rev. Lett. 112, 016402 (2014).
Critical point Wc ~ 1.8
Strong disorder Weak disorder
Ominato and Koshino, Phys. Rev. B 89, 054202 (2014)
2D
Disorder strength W
DOS
No critical disorder strength
What makes difference between 2D and 3D? 2D
Energy
DOS
Energy DO
S
Smaller density of states around the Weyl point
3D
1/Wc
Strong disorder … metallic
Weak disorder … insulating
Conductivity (ε=0) Ominato and Koshino, Phys. Rev. B 89, 054202 (2014)
2D 3D
“Universal conductivity”
∝ DOS
Difference between 2D and 3D?
Dimension of conductivity
2D
3D
… only the length scale
… selfenergy at E=0 (energy broadening)
SCBA calculation
Ominato and Koshino, Phys. Rev. B 89, 054202 (2014)
Diamagnetism of graphene
Susceptibility -χ
Graphene: Singular diamagnetism at Dirac point
McClure, Phys. Rev. 104, 666 (1956). Safran and DiSalvo, Phys. Rev. B 20, 4889 (1979). Fukuyama, J. Phys. Soc. Jpn. 76 043711 (2007) Koshino and Ando, Phys. Rev. B 75, 235333 (2007).
T → 0 :
low T
high T Singular diamagnetism at Dirac point
Dirac point
Fermi energy
Diamagnetism of 3D Weyl electron
2D
3D
Koshino and Ando, PRB 81, 195431 (2010)
McClure, Phys. Rev. 104, 666 (1956)
+Δ
-Δ
Inversion center
“Massive” graphene
Potential asymmetry between A and B opens an energy gap
Δ ≠ 0 effective mass
Massive Dirac equation (relativistic electron)
Analogy to conventional 2D electron
Energy
Landau dia: χD
Pauli para: χP = -3χD
-χ
Conventional electron: Dirac electron:
Energy
-χ Total
Koshino and Ando, PRB 81, 195431 (2010) Diamagnetism of massive graphene
Constant susceptibility shift Energy
-χ
Energy
-χ
Dirac electron: Massive graphene:
2Δ
∼1/Δ
Koshino and Ando, PRB 81, 195431 (2010)
Zero-gap limit
Diamagnetism of intrinsic graphene
Valley Zeeman energy Koshino and Ando, PRB 81, 195431 (2010)
K K’ K K’
Valley Zeeman energy
AB asymmetry
Landau level energies differ between K and K’ (analog of spin Zeeman splitting)
K-K’ splitting energy:
Δε
effective g-factor:
pseudo spin
pseudo spin
effective mass approximation
2Δ
Valley Zeeman energy
Hamiltonian in B-field
Koshino, PRB 84, 125427 (2011)
Magnetic moment caused by self-rotating orbital current
K K’
Koshino and Ando, PRB 81, 195431 (2010)
pz
3D = sum of 2D pz … “parameter” of 2D system
|vpz|
3D massive Dirac electron Hamiltonian (~ Bismuth)
Energy gap:
Composition of 2D Dirac bands
pz
Koshino and Ando, PRB 81, 195431 (2010)
Bismuth magnetism: Wolff J. Phys. Chem. Solids 25, 1057 (1964), Fukuyama and Kubo J. Phys. Soc. Jpn. 27, 604 (1969). Fuseya, Ogata, Fukuyama, PRL. 102, 066601 (2009).
1) Topological band touching points protected by spatial symmetry + chiral symmetry
Summary
2) Some characteristic physics in 2D and 3D Weyl electrons
--- Trasnport property (metallic or insulating at Weyl point?)
--- Orbital diamagnetism (Singularity at Weyl point)
--- New class of 2D and 3D Weyl electrons Koshino, Morimoto, Sato, PRB 90, 115207 (2014)
Koshino and Ando, PRB 81, 195431 (2010) Koshino, PRB 84, 125427 (2011)
Ominato and Koshino, PRB 89, 054202 (2014)