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)