Quantum Fluctuation of Conductivities in Quantum Hall Effect - PowerPoint PPT Presentation

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

Conductivity in a periodic potential � � � � 2 + U x , y ( ) ( ) = 1 ( ) ( ) + eA H t 2 m p t ) = U 1 cos 2 � x � + U 2 cos 2 � y � � � � ( B U x , y � � � � � � � a b � � � � y E y 0 ( ) = A t � � Bx � E y t � � J x = � xy E y , J y = � yy E y x [D. J. Thouless, M. Kohmoto, M.P. Nightingale, and M. den Nijs, PRL 9 (1982) 405. ]

My calculation t = 0 Switch on abruptly at E y E y t 0 Linear response for finite time � xy = e 2 h N Ch + [Oscillation in time] � yy = 0 + [Oscillation in time]

� 2 � � / e = p abB Energy levels = � 0 q N = 3 p U ( x , y ) N = 2 N = 1 m 0 + 1 E F N = 0 m 0

Wavefunction in the weak potential 2 � � k y � eB � � qa � � � eB n qa � p 2 � x + 1 � � � � ( ) = � � � p � x , y n u mk d m e N n = 1 � � = �� � 2 � i y � p + � n ( ) e � i k x x � � qa � � � e n qa / p b n � 1 + � � n + � * d m n + 1 = � mk ( ) d m � � � � � d m 1 n n d m � � � qb ( ) 2 pa cos � qbk y / p + 2 � � � � n = U 1 e n q / p � � qa � = U 2 2 pb e � i qak x / p 2 e [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.] ( ) + E y � ( ) � � � 0 1 � � � d 2 k p ( ) mk � (0) = � � mk f E mk � (2 � ) 2 MBZ m = 1 d ( ) = 1 ( ) , � t ( ) d t � t � � i � H t � � J x = � xy E y ( ) ( ) � H t ( ) = Tr � t J y = � yy E y J x E y � A x

Density Matrix � � � � ( ) = f m � mn � = u mk � x � x � mnk � e i k � mnk � e -i k � , 0 u nk � � u mk ( ) � mnk ( ) = 1 ( ) + e d � ( ) � � � nk d t � mnk i � � mk � f m � f n 1 1 u nk � � � � � k y ( ) , m � n ( ) = 0 d d t � nnk 1 � ( ) = i e � u mk f m � f n ( ) t / � � � � � � n k � � i � m k � � mnk � 1 � e 1 u nk � � � � � � � nk � k y � mk

Conductivity � nk � xy = e 2 � 2 � � N Ch � F � � mk � � ( ) + e 2 d 2 k � � xy g mn k ( ) 2 � 2 � MBZ m � n � ( ) ( ) ( ) t / � � � mn k � � � nk � sin � mk � � � ( ) ( ) � yy = e 2 d 2 k ( ) t / � � � � � � nk � mk yy g mn k sin � ( ) 2 � 2 � MBZ m � n

Details � Im � u mk � u mk d 2 k � � � � N Ch = f m � � k y � k x MBZ m � � ( ) = f m � f n ( ) ( ) r xy k g mn mn k 2 � � u mk ( ) = f m � f n � ( ) yy k g mn u nk � � k y � � u mk � � u mk ( ) e � ( ) = � � i � mn k r mn k u nk u nk � � k y � k x

Parameters B � 10T B � 10T, a , b � 100nm, U 1 , U 2 � 0.1meV ( ) �� � U � 1K � 10mK � 100nm U 1 , U 2 � 0.1meV [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 � 0 2 p = 65

Example 2 p

Long-time average � xy � yy and � Ch = e 2 � xy = � Ch + � � xy , 2 � � N Ch � � � ( ) ( ) ( ) � � xy = e 2 d 2 k ( ) t / � � � mn k � � � � � nk � mk xy g mn k sin � ( ) 2 � 2 � MBZ m � n � � M y /2 q ( ) M x /2 q ( ) � yy = e 2 d 2 k ( ) t / � 1 � � � � � � � nk � mk yy g mn k sin � ( ) 2 � 2 � L x L y MBZ j x = � ( M x � 1)/2 q j y = � ( M y � 1)/2 q m � n 1 T � � yy � lim � yy = 0, � xy = � Ch dt T �� T 0

Long-time variance 2 � � yy ( ) ( ) � � yy ( ) 2 var � yy 2 � � 2 � � � ( ) ( ) e 2 ( ) t / � 1 1 � � T � � � � nk = lim � mk yy sin 2 dt � � � g mn k � � � � � � � T �� T L x L y 0 m � n j x j y 2 � � 2 � � � ( ) max e 2 1 � � � yy � � � g mn k � � � � � � L x L y m � n j x j y 2 � � � ( ) max e 2 1 � L �� = � �� 0 � yy � g mn k � � � � qabL x L y m � n L �� L �� ( ) ( ) T �� T �� var � yy � �� 0, � var � xy � �� 0 �

Long-time behavior If bands are flat, and only a pair of m 0 th and (m 0 +1)th bands contributes, For T � � , L � � , ( ) = var � yy ( ) � 0 � xy = � Ch , � yy = 0, var � xy 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.