千葉大学理学部集中講義

2015年7月9日-10日

バルクエッジ対応の物理の多様性と普遍性
L Iy RVx

量子ホール効果・Laughlinの議論
Mother of topological phases
TKNN# & Chern #筑波大学数理物質系 物理学域
初貝 安弘

’80, K.v.Klitzing et al.

Topol. char. by edges

VR. Willett et al.

topologically cylinder

Conventional QHE
Landau Level and Integer QHE

⇥k

D(E)

µF

D(E) =(E⇥n )nµF
1/23/2

5/2

E[]

3

⇤xy (µF ) =2
1/2

3/2

5/22mk2

1n = ⇥(n + )2e2n, n = 1, 2, 3, · · ·n 1 < µF < nµF

12[e/h]xyE(B = 0) =E2E[]

QHE of Semiconductors
Landau Level of Valence band (Holes)⇥kE(B = 0) = ±Eg⇥n )1⇥(n + 2 )Eg /2 +⌅⌅⇤n⇥0n =1)/2+⇥(n+E⌅g2⌅⇥
n<0n(EµF⇤xy (µF ) =--1--2µF-+1 +2--(Eg+ 5)/2 -(Eg+-3)/2 --(Eg+1)/2--3

xy [e /h] D(E) =k22mEn : zero point energy 1/2µF22E[]e2n(n > 0)
n + 1 (n ⇥ 0)n 2
1 < µF < nexy :hinteger

QHE of Semiconductors
Landau Level of Conduction band (Electrons)⇥kE(B = 0) = ±Eg⇥n )µFk22mEn : zero point energy 1/21⇥(n + 2 )n(EµFEg /2 +⌅⌅⇤n⇥0n =1)
/2+
⇥(n+E
⌅
g2
⌅
⇥
n<0

⇤xy (µF ) =--1--2µF-+1 +2(Eg+1)/2 (Eg+3)/2 (Eg+5)/2--3

xy [e /h] D(E) =22E[]e2n(n > 0)
n + 1 (n ⇥ 0)n 2
1 < µF < nexy :hinteger

Edge states are topologically stableCyclotron motion by Lorentz force

F =Topol. char. by edges

ev ⇥ B

Currents are canceled in the bulk but induces a boundary current

Edge states

Edge states are chiral
One way going !!
Cannot stop !
No back scattering

Stable for impurities !!

X

Topological stability
ofChiral edge states

Edge states in quantum Hall effects

Levinson’s theorem to the Friedel’s sum ruleSurface states of Semiconductors & polarization
Solitons in polyacetylene

Everything started from here
テキスト
Local moments in integer
spin chainslaternear the impurities
discuss
Zero bias conductance peaks of the d-wave superconductors
Zero energy localized states of graphene
Quantum Spin Hall Edge states
Edge states in 2D cold atoms in optical lattice
One-way edge modes in gyromagnetic photonic crystals
Spin Ladder with ring exchanges

Topol. char. by edges

Stability of the quantized Hall Conductance
Edge states and Hall conductanceWithout boundaries E Landau gauge in yxy Halperin ‘82

H2D =LH1D (ky )

RkyEF
ky =

With boundaries
Edge potential

E2(ny +Ly=0

H1D (ky ): harmonic osc. centered atky0) ny = 0, ±1, ±2, · · ·⇥x⇤

2 states are carried from L to R

2B ky=0

Gapless excitations

EFLeft edge

Right edge Left edge
Laughlin’s undetermined n : # of Landau Levels below EF

Edge states are essential in the QHE !

Right edge

Topol. char. by edges

Stability of the quantized Hall Conductance
Gauge invariance and quantization ofadiabatic process to increaseyx

L Iy RVx
Gauge transformation

Byers-Yang formulaA⇥A =A+⇤ ,⇥Laughlin ‘81

xy=(A⇥A)⇥ ⇥ ⇥ ⇥ = ei2⇥ / 0 ⇥ one particle stateh0!
= 0==eflux quantumyx
2⇥neVxeE==nVx =Iy =⇥h/ehh
E = n · eVx⇥ = ⇥0 = ,eAll states are invariant up to phase after the process:Some n states are carried from L to the Rn : generic integer ( but undetermined )yx Vx

Emerging Dirac Fermions in condensed matter
Bloch Electrons in a magnetic field ( Hofstadter problem )Graphene & 2D gapless superconductors

Non Abelian gauge structures in condensed matter
Quantum Hall effects, especially of Graphene
Lattice gauge fields in a parameter space

Dirac Fermion & Zero modes
Universality for the Zero modes of Dirac fermionsBulk--Edge correspondence of topological ordered states

Random Dirac ferimion & “real” Gauge field
Ripples as Random gauge field in graphene
Realization of exact fix point of random Dirac fermionsEnergy spectrum of Bloch Electrons in a magnetic field
Hofstadter ’76† i ijci e cj + h.c.

H=ij⇥

ci : electron annihilation operator

Electrons in a 2D periodic potential+ magnetic field

It might be realized by cold atoms?

The spectrum is fractal as a function of magnetic flux per plaquetteHofstadter’s Butterfly
Self-similar

Fractal in condensed matter

C. Albrecht, J.H. Smet, K. von Klitzing, D. Weiss, V. Umansky, H. Schweizer:
Evidence of Hofstadter's Fractal Energy Spectrum in the Quantized Hall Conductance.
Phys. Rev. Lett. 86(1), 147-150 (2001).