Journal of the Physical Society of Japan
Vol. 75, No. 6, June, 2006, 063704#2006 The Physical Society of JapanLETTERS

Temporal Oscillation of Conductancesin Quantum Hall Eﬀect of Bloch Electrons
Manabu M ACHIDA, Naomichi HATANOy and Jun G ORYO1 z
Institute of Industrial Science, The University of Tokyo, Komaba, Meguro, Tokyo 153-8505
1
Department of Physics and Mathematics, Aoyama Gakuin University,
5-10-1 Fuchinobe, Sagamihara, Kanagawa 229-8558(Received December 31, 2005; accepted April 4, 2006; published June 12, 2006)

We study a nonadiabatic eﬀect on the conductances in the quantum Hall eﬀect of two-dimensional
electrons with a periodic potential. We found that the Hall and longitudinal conductances oscillate intime with very large frequencies due to quantum ﬂuctuation.
KEYWORDS: integer quantum Hall effect, Chern number, nonadiabatic effect, Greenwood linear-response theory
DOI: 10.1143/JPSJ.75.063704

Two years after the discovery of the quantum Halleﬀect,1,2) Thouless, Kohmoto, Nightingale and den Nijs
(TKNN)3,4) theoretically studied the quantum Hall eﬀect intwo-dimensional electrons with a periodic potential. They
expressed the Hall conductance as an integer multiplied bye2 =h. The integer is a topological invariant called the Chern
number. Such a system has been experimentally realized
as a superlattice structure in a semi-conductor heterojunction.5–7)
We can treat the electric ﬁeld as a time-dependent vector
potential. Then adiabatic approximation gives the same
expression of the Hall conductance as in the TKNN
theory.8,9) In this Letter, we consider the conductances ofthe same system taking a nonadiabatic eﬀect into account.
We determine the quantum ﬂuctuation of the Hall conductance xy and the longitudinal conductance yy ; i.e., xyand yy oscillate in time with very large frequencies.
We consider the nonadiabatic eﬀect on the conductanceswith the help of the Greenwood linear-response theory10) andstudy a correction to the adiabatic approximation. Wagner
studied the failure of the Kubo formula due to a nonadiabaticeﬀect on the dc current in a ﬁnite system.11,12) Grimaldi et al.
explained the high-Tc superconductivity of fullerene compounds by taking nonadiabatic eﬀects into account.13)
This Letter is organized as follows. We ﬁrst calculate
the conductances beyond the adiabatic approximation andobtain correction terms to xy and yy of the TKNN theory.
We then evaluate the oscillations in these correction terms
numerically.
We consider noninteracting electrons in a periodic potential in the x–y plane. A magnetic ﬁeld is applied in the z
direction, and an electric ﬁeld is applied in the y direction.
We treat the electric ﬁeld as a vector potential. Using
the Landau gauge, we express the Hamiltonian of the systemasHðtÞ ¼21 p þ eAðtÞ þUðxÞ;2mewhereAðtÞ ¼ Bx  Ey t ey ;2x2yþ Uy cos:UðxÞ ¼ Ux cosab

ð2Þ

We consider the ﬂux per unit cell of UðxÞ to be therational number p=q in the unit of the ﬂux quantum, which
produces p subbands. We label each subband by m (1 m  p) hereafter. Then we obtain the generalized crystal
momentum h k deﬁned in the magnetic Brillouin zone.14)
0  kx < 2=qa and 0  ky < 2=b.
We deﬁne H k ðtÞ asH k ðtÞ  eikx HðtÞeikx :ð3Þ

The instantaneous Hamiltonian H k ðtÞ is diagonalized as
H k ðtÞjumk ðx; tÞi ¼ mk ðtÞjumk ðx; tÞi:ð4Þ

Since H k ðtÞ ¼ H kx ;ky eEy t=h ð0Þ, we obtain for m 6¼ n
mk ðtÞ ¼ mk ð0Þ þ OðEy Þ;humk ðx; tÞju_nk ðx; tÞi
eEy @umk ðx; 0Þ 2¼
ð5Þ
unk ðx; 0Þ þ O Ey ;@kyhwhere the dot indicates the time derivative. We note that,by ﬁxing the arbitrary phase in each time, we can always
achieve humk ðx; tÞju_mk ðx; tÞi ¼ 0:
The time evolution of the density operator is given by thevon Neumann equationd1
ðtÞ ¼ ½HðtÞ; ðtÞ:
dtih

ð6Þ

Following the Greenwood linear-response theory,10) we
expand the density operator with respect to the electric ﬁeld
and take the zeroth- and ﬁrst-order terms into accountðtÞ ¼ ð0Þ þ Ey ð1Þ ðtÞ:ð7ÞThus, the matrix elements
ð1Þ

mnk  humk ðx; tÞjeikx ðtÞeikx junk ðx; tÞið8Þ

are approximated using
ð1Þmnk ¼ ð0Þmnk þ Ey mnk

¼ fm mn þ Ey ð1Þmnk ;E-mail: machida@iis.u-tokyo.ac.jp
E-mail: hatano@iis.u-tokyo.ac.jp
zE-mail: jungoryo@phys.aoyama.ac.jp
yð9Þ

where fm [¼ f ðmk Þ] is the Fermi distribution. Hereafter, welet mk denote mk ð0Þ and jumk i denote jumk ðx; 0Þi.
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J. Phys. Soc. Jpn., Vol. 75, No. 6

The time evolution of the oﬀ-diagonal elements (m 6¼ n)
is given byd ð1Þ
1¼ ðmk  nk Þð1Þmnk
dt mnk ih e@umk unk :þ ð fm  fn Þð10Þh@ky 
Therefore, we obtain the matrix elements of the density
operator as @umk fm  fn
unk
¼
ie
1  eiðmk nk Þt=h
ð1Þ
mnk

@ky
mk  nk
ðm 6¼ nÞ; ð11Þð1Þ
mmk ¼ const.

ð12Þ

Let us calculate the current J ( ¼ x; y) using
ð13ÞJ  Re hJ ðEy Þi  hJ ð0Þi ;@HðtÞhJ ðEy Þi ¼ Tr ðtÞ@A ðtÞe@H k ðtÞ¼ Tr k ðtÞ:
ð14Þh@k
By plugging eqs. (11) and (12) into eq. (14), we obtainZe2
d2 k XJ ¼ Ey Reið fm  fn Þ2hMBZ ð2Þ m6¼n   @umk@umk uu1  eiðmk nk Þt=h ; ð15Þ
nknk @ky @k
where MBZ denotes the magnetic Brillouin zone. Aftersome tedious but straightforward calculation, we obtain theconductances xy and yy in the formsJxxy EyZe2
e2d2 k X¼
ð fm  fn Þrmn ðkÞNCh þ2hh MBZ ð2Þ2 m6¼n
 sin ðmk  nk Þt=h  mn ðkÞ ;Jyyy Ey 2Z e2d2 k X @umk unk  fm  fn
¼
2h MBZ ð2Þ m6¼n @ky sinðmk  nk Þt=h ;ð16Þwhere rmn ðkÞ and mn ðkÞ are the real numbers that satisfy  
 @umk@umk urmn ðkÞeimn ðkÞ u:ð17Þ
nknk @k @kyx

Here, NCh is the Chern number with a ﬁnite-temperaturecorrectionX Z
d2 k@umk  @umkImfm:
ð18Þ
NCh @ky  @kx
MBZ m
We note that if we ignore the time dependence of ð1Þ
mnk ineq. (10), we obtain xy ¼ ðe2 =2h ÞNCh and yy ¼ 0, which
are consistent with the TKNN theory.
The above-mentioned equations imply that we obtainsinusoidally oscillating terms in addition to the Chernnumber term in the Hall conductance xy due to quantum
ﬂuctuation. The longitudinal conductance yy also oscillateswith zero mean. The oscillation period is determined by theM. MACHIDA et al.

energy-gap size.
Equation (16) is also a gauge invariant. We observe thisby treating the electric ﬁeld as the scalar potential A0 ðtÞ ¼
eEy y exp½i!t instead of the vector potential. The ﬁrstorder time-dependent perturbation theory gives eqs. (15) and(16) in the limit of ! ! 0.
To demonstrate the oscillation numerically, we assume
that the state lies in the lowest Landau level. Moreover, weassume that the ﬂux  (¼ abB) is given by the unit ﬂux 0(¼ 2h =e) multiplied by the rational number p=q as

abeB
p¼ ;
¼
02hq

ð19Þ

where p and q are coprime. Due to the periodic potential
Uðx; yÞ, each Landau level splits into p subbands with aq-fold degeneracy in each subband.
In the weak-potential limit jUðxÞjh eB=me , the perturbation theory gives the instantaneous eigenfunction jumk ðxÞias3)
p1
X
1 X
jumk ðxÞi ¼ pﬃﬃﬃﬃﬃ
dm ðkÞN ¼1‘¼1"   #eBh kyqa 2xþ
 exp  ‘qa 2hpeBqa exp ikx x  ‘qa p‘p þ;
ð20Þ
 exp 2iyb

where N is the normalization factor. Here, the coeﬃcients
fdm g and ﬁrst-order energy shift ð1Þ
mk are determined by thefollowing secular equation.
dm1 ðkÞ þ  dm ðkÞ þdmþ1 ðkÞ ¼ ð1Þ
mk dm ðkÞ:

ð21Þ

Here,
 ¼ Ux eqb=2 pa cosðqbky =p þ 2 q=pÞ;Uy qa=2 pb iqakx = pe:
ð22Þ
e
¼
2
Note that the unperturbed energy ð0Þ
mk is independent of
ð1Þ
m and k; i.e., mk  nk ¼ ð1Þ
mk  nk . After straightforward
calculation, we obtain


 X

p @umk  @umk
ddm  ddm
h;
ð23Þ
i
¼
eB
@kx @kydkx dky¼1
 
 Xp
 @umk  ddm
dn;
ð24Þ
unk 
¼
@kx
dkx
¼1
 
 X
p
 @umk  ddm
dn:
ð25Þ
unk 
¼@kydky¼1
Thus, we can calculate the conductances xy and yy ineq. (16) using the eigenvalues and eigenvectors in eq. (21).
As typical values, we use a ¼ 100 nm, b ¼ 130 nm, andUx ¼ Uy ¼ 0:1 meV. We assume p=q ¼ 65=2, implying
B ¼ 10:4 T. With this ﬂux ratio, the lowest Landau level
splits into 65 subbands. We note that the p subbands in thelowest Landau level range from 0.240 to 0.948 meV. The
center of the next Landau level is located at 1.78 meV. In
the numerical calculation, we obtain the derivatives of dm ðkÞby the ﬁnite-diﬀerence method. We consider the slices kx063704-2

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J. Phys. Soc. Jpn., Vol. 75, No. 6

0.60.58
0.56

Energy [meV]

0.62

0.62
0.60.58

π 2πqakx

00.2

π 2πbky0Fig. 1. The energy levels around the center of the p (¼ 65) subbands areshown as functions of (a) qakx and (b) bky . The ﬂux ratio p=q ¼ 65=2corresponds to B ¼ 10:4 T. We use Ux ¼ Uy ¼ 0:1 meV, a ¼ 100 nm,and b ¼ 130 nm. The Fermi energy (the dashed line) lies between the33rd and 34th levels.

0.3π 2πqakx

0.2πbky

02πFig. 3. The energy levels in the lowest part of the p (¼ 65) subbands areshown as functions of (a) qakx and (b) bky . The ﬂux ratio p=q ¼ 65=2corresponds to B ¼ 10:4 T. We use Ux ¼ Uy ¼ 0:1 meV, a ¼ 100 nm,and b ¼ 130 nm. The Fermi energy (the dashed line) lies between thesecond and third levels. In each panel, the lowest solid line below thedashed line expresses the degenerate ﬁrst and second levels.

50

σxy [μS]

4030012t [ns]340.50-0.5

100σyy [μS]σxy [μS]

0.30.56
0σyy [μS]0.4Energy [meV]0.40.64
Energy [meV]

Energy [meV]

0.64

M. MACHIDA et al.

-10012t [ns]34Fig. 2. Conductances corresponding to the situation in Fig. 1. In the upper
panel, the solid line shows xy as a function of time, and the dashed line
shows the Chern number term Ch ¼ ðe2 =2h ÞNCh . In this case, NCh ¼ 1.
The lower panel shows yy as a function of time.

and ky to be 2=100qa and 2=100b, respectively. The
Chern number term is obtained by Fukui, Hatsugai andSuzuki’s method.15)
First, we make the Fermi energy F lie between the 33rdand 34th levels, as shown in Fig. 1. The ﬂuctuations ofxy and yy in Fig. 2 consist of the oscillations corresponding
to the energy gaps between levels higher than F and levels
lower than F . Figure 2 shows that the time resolution
required to detect the quantum ﬂuctuation is about 10 ps.
Secondly, we make F lie between the second and thirdlevels, as shown in Fig. 3. Then we obtain almost periodic
oscillations of xy and yy (Fig. 4) in contrast to theseemingly random oscillation in the former case. This isbecause the energy gaps are almost constant in the lattercase.
To summarize, we studied a nonadiabatic eﬀect on theconductances in the TKNN theory. We found that both xyand yy oscillate due to quantum ﬂuctuation. The oscillation
period is of the order of 100 ps to 1 ns. It is a challenging but
interesting issue to detect these oscillations experimentally.
Although we studied the time dependence of currents at
a constant electric ﬁeld, by similar calculation, we observethat constant currents in reverse produce a time-dependent
electric ﬁeld. In this case, the oscillation period is also
determined by the energy-gap size.

012t [ns]34012t [ns]

340.5
0-0.5

Fig. 4. Conductances corresponding to the situation in Fig. 3. In the upperpanel, the solid line shows xy as a function of time, and the dashed line
shows the Chern number term Ch ¼ ðe2 =2h ÞNCh . In this case, NCh ¼ 0.
The lower panel shows yy as a function of time.

Acknowledgments
The authors thank Professor Tomoki Machida for valuablecomments from an experimental point of view. One of theauthors (M.M.) thanks Takahiro Fukui and Mikito Koshino
for indispensable discussions on the numerical calculationof the Chern numbers. M.M. also thanks Seiji Miyashita
for fruitful discussions on the Greenwood linear-response
theory. The study is partially supported by a Grant-in-Aidfor Scientiﬁc Research (No. 17340115) from the Ministry ofEducation, Culture, Sports, Science and Technology as well
as by Core Research for Evolutional Science and Technology of Japan Science and Technology Agency.
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LETTERS

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