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Quantum Fluctuation of Conductivities in Quantum Hall Effect

Manabu Machida (U. Tokyo, Japan)

In collaboration with

• N. Hatano (U. Tokyo, Japan) and
• J. Goryo (Aoyama Gakuin Univ., Japan)

Dynamics and Relaxation in Complex Quantum and Classical Systems and Nanostructures (Dresden, 2006 summer)

Conductivity in a periodic potential

B Ey y x

Jx = xyEy, Jy = yyEy

[D. J. Thouless, M. Kohmoto, M.P. Nightingale, and M. den Nijs, PRL 9 (1982) 405. ]

H t

( ) = 1

2m p

• + eA
• t

( )

( )

2 +U x,y

( )

A

• t

( ) =

Bx Eyt

• U x, y

( ) = U1 cos 2x

a

• +U2 cos 2y

b

My calculation

Linear response for finite time

xy = e2 h NCh + [Oscillation in time] yy = 0 + [Oscillation in time] Ey t

Switch on abruptly at

Ey t = 0

Energy levels

N = 0 N = 3 N = 2 N = 1

U(x,y) p

EF

m0 +1 m0

• =

abB 2 / e = p q

Wavefunction in the weak potential

umk

x,y

( ) =

1 N dm

• n
• n =1

p

• e

eB 2 x+ ky eB qa n qa p

• 2

=

• e

ikx xqa n qa/ p

( )e

2iy p+ n b

dm

• n 1 +

n dm

• n + *dm
• n +1 = mk
• 1

( ) dm

• n

n = U1e qb 2 pa cos qbky / p + 2

n q / p

( )

= U2 2 e

qa 2 pbeiqakx / p

[D. J. Thouless, M. Kohmoto, M.P. Nightingale, and M. den Nijs, PRL 9 (1982) 405. ]

Greenwood’s linear response theory

[D. A. Greenwood, Proc. Phys. Soc. 71 (1958) 585.]

d dt t

( ) = 1

i H t

( ), t ( )

• ( ) + Ey

1

( )

(0) = d 2k

• (2)2

MBZ

• mk
• f Emk
• (

) mk

• m=1

p

• Jx Ey

( ) = Tr t

( ) H t ( )

Ax

Jx = xyEy Jy = yyEy

Density Matrix

mnk

= umk e-ik

• x
• eik
• x
• unk

,

mnk

• ( ) = fmmn

d dt mnk

• 1

( ) = 1

i mk

nk

• (

)mnk

• 1

( ) + e

fm fn

( )

umk

• ky

unk

• m n

( ),

d dt nnk

• 1

( ) = 0

mnk

• 1

( ) = ie umk

• ky

unk

• fm fn

mk

nk 1 e i m k

n k

• (

)t /

Conductivity

xy = e2 2 NCh + e2

• d2k
• 2

( )

2

gmn

xy

• k

( )

mn

• MBZ
• sin

mk

nk

• (

)t / mn k

• ( )

( )

yy = e2

• d2k
• 2

( )

2

gmn

yy

• k

( )

mn

• MBZ
• sin

mk

nk

• (

)t /

( )

nk

• F

mk

Details

NCh = fm

m

• d2k
• MBZ
• Im umk
• ky

umk

• kx

gmn

xy k

• ( ) = fm fn

( )r

mn k

• ( )

gmn

yy k

• ( ) = fm fn

( )

umk

• ky

unk

• 2

r

mn k

• ( )e

imn k

• ( ) =

umk

• ky

unk

• unk

umk

• kx

Parameters

B 10T U1,U2 0.1meV

100nm

B 10T, a,b 100nm, U1,U2 0.1meV U 1K 10mK

( )

[C.T. Liu, et al., Appl. Phys. Lett. 58 (1991) 2945] [M.C. Geisler, et al., PRL 92 (2004) 256801 and PRB 72 (2005) 045320]

Example 1

• = p

q = 65 2

p = 65

Example 2

p

Long-time average

yy = e2

• d2k
• 2

( )

2

gmn

yy

• k

( )

mn

• MBZ
• sin

mk

nk

• (

)t /

( )

xy = Ch + xy, Ch = e2 2 NCh xy = e2

• d2k
• 2

( )

2

gmn

xy

• k

( )

mn

• MBZ
• sin

mk

nk

• (

)t / mn k

• ( )

( )

yy lim

T

1 T dt

T

• yy = 0,

xy = Ch

xy yy

1 LxLy

jy =(M y 1)/2q M y /2q

• jx =(M x 1)/2q

M x /2q

• and

Long-time variance

var yy

( ) yy ( )

2 yy

( )

2

= lim

T

1 T dt

T

• 1

LxLy

• 2

e2 gmn

yy

• k

( )

• 2

sin2 mk

nk

• (

)t /

( )

mn

• jx jy
• 1

LxLy

• 2

max e2 gmn

yy

• k

( )

• 2

mn

• jx jy
• =

1 qabLxLy max e2 gmn

yy

• k

( )

• 2

mn

• L
• var yy

( )

L T

• 0,

var xy

( )

L T

Long-time behavior

If bands are flat, and

• nly a pair of m0th and (m0+1)th bands contributes,

For T , L , xy = Ch, yy = 0, var xy

( ) = var yy ( ) 0

Decays fast Decays slow

Summary

■Quantum Hall effect of 2D Bloch electrons in a periodic potential

Conductivity for finite time

Conductivities oscillate in short time. Eventually the time dependence will cease. Behavior depends on the Fermi energy.

Manabu Machida, Naomichi Hatano, and Jun Goryo,

• J. Phys. Soc. Jpn. 75 (2006) 063704.