Berry's phase and the anomalous velocity of Bloch wavepackets

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by  Y. D. Chong

Berry’s phase and the anomalous velocity of Bloch wavepackets
∗
Y. D. Chong
Department of Applied Physics, Yale University, New Haven, Connecticut 06520
(Dated: January 25, 2010)
The semiclassical equations of motion for a Bloch electron include an anomalous velocity term
analogous to a k-space “Lorentz force”, with the Berry connection playing the role of a vector
potential. ByexaminingtheadiabaticevolutionofBlochstatesinamonotonically-increasingvector
potential,IshowthattheanomalousvelocitycanbeexplainedasthedifferenceintheBerry’sphase
acquired by adjacent Bloch states within a wavepacket.
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1 PACSnumbers: 72.10.Bg,72.15.Gd,72.20.My
0
2
When inversion or time-reversal symmetry is broken, connection (the “Berry curvature”). The idea is simple:
n
a the semiclassicalmotionforaBlochelectronis knownto a small DC electric field can be represented by a con-
J contain an additional non-vanishing term analogous to stant vector potential that increases monotonically with

5 the Lorentz force in momentum-space: time. Each Bloch state undergoes adiabatic evolution in
2 thisvectorpotential,andacquiresaBerry’sphase. Each
1
] r˙ = ~∇kEk+k˙ ×(∇k×Ak). (1) Bk-lsopcahcecotmrapjeocnteonrty,oafnadwacaqvueipraecskaedtiuffnedreenrtgoBeesrray’dsiffpehraesnet.
r
e Here,kisthereducedwave-vector,E isthebandenergy, The resulting Berry’sphase differences, characterizedby
h k
∇ denotes a k-space derivative, and A is defined by the Berry curvature, conspire to induce the anomalous
t k k
o term in the velocity of the wavepacketas a whole.
at. A~k = 1i ddru∗k(r)∇kuk(r), (2) netFiocrfiseimldp(liwcihtoyseofpprreesseenncteatdiooens, Inowtilallitgenrotrheethfienamlarge--
m ZΩ
sults). A Hamiltonian for a Bloch electron subjected to
withtheintegraltakenovertheunitcellΩandu (r) de-
- k an additional small electric field Exˆ is
d noting the Bloch function. The “anomalous velocity”—
on tinhaellsyecdoenrdivteedrmbyonKtahreprluigshtanhdanLdustitdinegoefr(11)in—wthaesiroreixg-- H′ =−2~m2 ∇2+V(r)−eEx, (3)
c
planation of the extraordinary Hall coefficients of fer-
[ where V(r) is the lattice potential, which can break in-
romagnetic materials, based on a careful examination
2 of wavepacket dynamics. Subsequently, Chang and Niu version symmetry. A gauge transformation φ → φ −
v re-derived the anomalous velocity using an effective- (1/c)∂tΛandA→A+∇Λ,whereΛ=−Ecxt, yieldsthe
6 equivalent periodic time-dependent Hamiltonian
Lagrangian technique, and pointed out that A~ is the
1 k
7 important quantity known as the Berry connection2. ~2 et 2
0 In its original context, the Berry connection describes H(t)= 2m −i∇+ ~E~ +V(r). (4)
. the gauge structure of a quantum state as it undergoes (cid:20) (cid:21)
2
1 adiabatic evolution, following a trajectory ~λ(t) in some The quantumstates of the new Hamiltonianhave an ex-
9 parameterspaceofthesystemHamiltonianH(λ(t)). The tra phase factor of exp(−ieExt/~), which is the same for
0 line integral of the Berry curvature, taken over ~λ(t), all states and therefore irrelevant.
v: yields “Berry’sphase”—anadditionalphase acquiredby As pointed out by Kittel6, H now has the form of a
i the quantum state during the adiabatic process3. As reduced Hamiltonian:
X
first appreciated by Simon4, the Berry connection also et
ar emergeswithinBlochsystems,inamannerthat appears H(t) = H(~q(t)), ~q(t)= ~ E~, (5)
to be quite different: the reduced wave-vector k serves ~2 2
as the “parameter” for the reduced Hamiltonian H(k), H(k) ≡ −i∇+~k +V(r). (6)
2m
and A~ describes the gauge structure of the Bloch func-
k h i
tions u (r) within the Brillouin zone. (In particular, the This allows us to describe the effects of the electric field
k
integral of A~ along the boundary of a two-dimensional in terms of adiabatic evolution. Similar considerations
k
Brillouin zone yields the TKNN number, which equals were used by Zak in a related one-dimensional model7.
the index of the integer quantum Hall effect5.) The Bloch states are eigenstates of H(~k) with band
I would like to present a derivation of the anoma- energies E(k) (suppressing the irrelevant band index):
lous velocity that clarifies its relationship with adiabatic
H(k)u (r)=E(k)u (r). (7)
quantum evolution, the context in which the Berry con- k k
nection first arose. One virtue of this derivation is that
It is easily seen that
it provides a simple geometrical explanation of why the
anomalousvelocityinvolvesthek-spacecurloftheBerry H(k′) uk′+k(r)eikr =E(k′+k) uk′+k(r)eikr . (8)
(cid:2) (cid:3) (cid:2) (cid:3)
2
and k. The k-space displacement is simply q(t). The
r-space displacement is determined by the phase factors
on the last line of (14), which should have the form
(overall phase factor)×exp[i∆k·r(t)],
FIG.1: k-spacelineintegralsgivingrisetotheBerry’sphases
where the overall phase factor comes from the phase of
in equation (16).
the central wavepacketk (t) and the second term comes
o
from the phase difference between k(t) and k (t).
o
From these considerations, we see that the phase dif-
Therefore, at each time t the Hamiltonian H(q(t)) pos-
sesses a set of instantaneous eigenstates u (r)eikr. ferences between k and k0 which arise from the band
q(t)+k
energies yield the usual group velocity:
(For t = 0, these are the usual Bloch states.) If the
electric field E is weak, the change is adiabatic.
d 1 t
Suppose we have the initial state ∆k·~v = dt′[E(k(t′))−E(k (t′))] (15)
g dt ~ 0
ψ(r,t=0)=uk(r)eikr. (9) (cid:26) 1Z0 (cid:27)
= ∆k· ∇ E(k (t)) . (16)
~ k 0
According to Berry’s theorem3, the state at time t is (cid:20) (cid:21)
i t I nowclaimthatthe Berry’sphasedifferences givethe
ψ(t)=u eikr exp − dt′E(k(t′)) eiγk(t), (10) anomalous velocity:
k(t) ~
(cid:20) Z0 (cid:21)
d k(t) k0(t)
where k(t)≡k+q(t). Berry’s phase, γk(t), is given by ∆k·~va = d~k′·A~k′ − d~k′′·A~k′′ . (17)
dt
(Zk(0) Zk0(0) )
γk(t) = i d~q′· ddr uq′+keikr ∗∇q′ uq′+keikr
To prove this, observe that the line integrals are taken
Z (cid:20)ZΩ (cid:21)
k(t) (cid:0) (cid:1) (cid:0) (cid:1) overthetopandbottomsegmentsoftheparallelogramin
= − d~k′·A~k′, (11) Fig.1. When∆kissufficientlysmall,integrals d~k′·A~k′
Zk(0) over the side segments become negligible; then the two
R
separateline integralscanbe replacedwith a single edge
with the integral taken over the trajectory of k(t).
integral,takenclockwisearoundtheboundaryΓ(t)ofthe
Now consider a wavepacketcomposed of Bloch states,
paralleogram:
initially centered on the Bloch state with k =k :
0
ψ(r,0) = f(|k−k0|)uk(r)eikr (12) ∆k·~va ≈ d d~k′·A~k′ (18)
dt
Xk (IΓ(t) )
= eik0r f(|∆k|)uk(r)ei∆k·r, (13) d
Xk = −dt(Za(t)da·(cid:16)∇k×A~k′(cid:17)). (19)
where ∆k = k − k and f is some envelope function.
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From (10) and (11), the wavepacketat time t is In time dt, the edge advances by k˙dt, and the additional
area (grey region in Fig. 1) is (k˙ ×∆k)dt. Thus,
ψ(r,t)=eik0r f(|∆k|)u (r)
k(t)
Xk ∆k·~va = ∇k×A~k0 · ∆k×k˙ (20)
×ei∆k·rexp −i tdt′E(k(t′)) eiγk(t). (14) = ∆(cid:16)k· k˙ ×∇(cid:17) ×(cid:16) A~ .(cid:17) (21)
~ k k0
(cid:20) Z0 (cid:21)
(cid:16) (cid:17)
Note that k(t)−k (t) = ∆k is time-independent. Com- This is precisely the anomalous velocity given in (1).
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paring (14) to (13), we observe displacements in both r I am grateful to P. A. Lee for helpful discussions.
∗ Electronic address: [email protected] 4 B. Simon, Phys.Rev. Lett.51, 2167 (1983).
1 R.Karplusand J. M. Luttinger, Phys.Rev.Lett. 95, 1154 5 D. J. Thouless, M. Kohmoto, M. Nightingale, and M. den
(1954). Nijs, Phys. Rev.Lett. 49, 405 (1982).
2 M.C.ChangandQ.Niu,Phys.Rev.Lett.75,1348(1995), 6 C. Kittel, Quantum Theory of Solids (Wiley, New York,
Phys.Rev.B 53, 7010 (1996). 1963), p. 190.
3 M. V. Berry,Proc. R. Soc. A392, 45 (1984). 7 J. Zak, Phys.Rev.Lett. 62, 2747 (1989).