Semiclassical equations of motion

Objective. We will obtain the semiclassical equation of mo-tion for Bloch electrons, accurate to the 1st order of externalelectromagnetic fields. The goal will be accomplished via tworoutes. In the first, we obtain a quantum Hamiltonian ac-curate to second order of perturbation, and subsequently ap-ply Hamilton dynamics. In the second approach a classicalLagrangian from the time-dependent variational principle ofquantum mechanics, from which the equations of motion arederived.

Key concepts: wavepacket; envelope function; Berry curva-ture and connection (first peep); effective mass, and effective-mass theory; group velocity; anomalous velocity; phase-spaceLagrangian; (generalized) Liouville theorem; time-dependentvariational principle;

1 Wavepacket

• Drude model: σ “ ne2τm ð kinetic theory of an ideal

classical gas of charged particles.

• Bloch waves are periodically modulated planewaves. Par-ticle mechanics should fail.

• A semiclassical theory can be formulated by wavepacket :

1

slowly modulated planewave

ψpx, 0q “ eikcpx´xcqe´px´xcq2

2σ2

“σ

?2π

ż 8

´8dk eikpx´xcqe´σ2pk´kcq2

2 (1)

• A planewave evolves as eipkx´ωtq. Expanding ωpkq “ωc ` ω1

cpk ´ kcq ` 12ω

2cpk ´ kcq2 ` ¨ ¨ ¨ ,

ψpx, tq „ exp rikc px ´ xcptqqs e´px´xcptqq2

2σptq2

xcptq “ ω1ct ” vgt, σptq2 “ σ2 `

ˆ

ω2c

σ

˙2

t2

• Equation of motion: 9xc “ ω1pkcq, 9kc “ ´V 1pxcq. Thatis, thinking in terms of xc and kc, a wavepacket under anexternal field moves like a classical particle.

2

2 Effective-mass theory

Synopsis. We will analyze the Schrodinger equation with anexternal potential slowly-varying in space:

pH0 ` V q|ψy “ E|ψy

We will expand the the wavefunction in terms of the Blochwavefunction of the perfect crystal. With H0 being the Hamil-tonian for a perfect crystal: H0|ψnky “ εnk|ψnky. As V isslowly varying in space the he expansion coefficients Fnk islarge only the neighborhood of k0. Then a main technical taskis to Taylor expand V around k0, from which we obtain aneffective Schrodinger equation in F , whose Hamiltonian is theeffective-mass Hamiltonian.

• The Bloch wavefunctions are orthonormal according to(Vc : unit cell volume)

ż

space

ddr ψ˚nkprqψn1k1prq “

p2πqd

Vcδnn1δpk ´ k1q,(2)

ô xunk|un1ky ”ż

uc

ddr u˚nkprqun1kprq “ δnn1 (3)

• The expansion: ψ “ř

n

ş

ddkFnkψnk, where Fnk is theFourier transform of envelope function Fnprq.

Fnk “ż

ddrFnprqe´ik¨r (4)

3

The Schrodinger equation becomes the secular equation

Fnkεnk `ÿ

n1

ż

ddk1 xψnk|V |ψn1k1yFn1k1 “ EFnk (5)

• Main assumptions of the effective-mass theory:(1) V is sufficiently slowly varying in space so that Fnk islocalized in real and reciprocal spaces (like a wavepacket);(2) Umklapp scattering is negligibly small:|V pk ´ k1 ` Gq| ! |V pk ´ k1q| if G ‰ 0.

• Eq. (5) then becomes

εnkFnk `ÿ

n1

ż

ddk1

p2πqdVk´k1xunk|un1k1yFn1k1 “ EFnk (6)

• Let’s rewrite in terms of matrices and vectors

εpkqF pkq `ÿ

n1

ż

ddk1

p2πqdV pk ´ k1qupkq:upk1qF pk1q

“ EF pkq (7)

where rupkq:upk1qsnn1 “ xunk|un1k1y.

• Suppose k,k1 are close to k0, and expand upkq, upk1qaround k0 to second order:

upkq:upk1q “ I ` iAαpk0 ` δkqδk2α

´1

2Gαβpk0qδk

2αδk

2β ` Opδk3q

4

where k2 “ δk ´ δk1. The (non-Abelian) Berry connec-tion is defined as

Apkq “ upkq:Bkupkq (8)

and the matrix-valued rank-2 tensor

Gαβ “ ´pu:BαBβuqk0 “ iBαAβ ` AαAβ (9)

• Putting the above back into Eq. (7) and with δk2 “δk ´ δk1 Ñ k1 and δk Ñ k, we obtain

εpkqF pkq `ż

ddk1

p2πqdV pk1q

“

I ` iAαpkqk1α

´1

2Gαβpk0qk

1αk

1β

ȷ

F pk ´ k1q “ EF pkq (10)

Using the identity fpkqgpkq “ş

ddre´ik¨rfp´i∇qgprq,we obtain a new Schrodinger equation in real space

rεp´i∇q ` ∇V prq ¨ Ap´i∇q

`1

2G : ∇∇V prq

ȷ

F prq “ EF prq (11)

Note: εp´i∇q “ εpk0´i∇q, Ap´i∇q “ Apk0´i∇q andG “ Gpk0q as implied by our replacement of variablesabove.

• Let’s single out the nth band, and ignore all other bandsand the interband matrix elements. And we will pick a

5

gauge An “ 12Ωn ˆ k, where the single-band (Abelian)

Berry curvature is defined as

Ωn “ ∇k ˆ An (12)

and the single-band connection as

An “ xunk|i∇k|unky (13)

Then the effective Hamiltonian becomes

H “ εpp!q`V prq´1

2!Ω ¨∇V prqˆp`

1

2g : ∇∇V prq

(14)where the band index n has been dropped, and Ω and gare evaluated for band n and k “ k0.

3 Digression: spin-orbit couplingfrom Dirac equation

Synopsis. We will take a digression to review the Dirac equa-tion for a relativistic spin-1/2 particle, and trace out the originof spin-orbit coupling. It will be clear at the end that the effec-tive Hamiltonian derived above has a remarkable resemblanceto the relativistic effect.

The Dirac equation describing a relativistic spin-1/2 particle,such as an electron, is actually four differential equations

ˆ

γµB

Bxµ`

mc

!

˙

ψ “ 0 (15)

6

where px0, x1, x2, x3q “ pict, x, y, zq. Here the Hermitian andtraceless gamma matrices, γµ, µ “ 0, 1, 2, 3, are given by

γk “ σ2 b σk, k “ 1, 2, 3

γ0 “ σz b σ0

and satisfy the following anticommutation relation

tγµ, γνu “ σ0 b tσµ, σνu “ 2δµν

The bispinor wavefunction ψ is commonly called the Diracspinor

ψ “

¨

˚

˚

˝

ψ1

ψ2

ψ3

ψ4

˛

‹

‹

‚

Multiplying both sides of Eq. (15) with γ0 we obtain a Schrodingerequation

i!Btψ “ pcα ¨ p ` mc2βqψ (16)

where αk “ σ1 b σk, k “ 1, 2, 3 and β “ γ0.

With electromagnetic potential Aµ “ piA0,Aq with A0 “φc in SI, the Dirac equation reads

i

ˆ

´i!B

Bxµ` eAµ

˙

γµψ ´ mcψ “ 0 (17)

7

which can be rewritten as

i rσ2 b ppk ` eAkqσk ` σ3 b pp0 ` ieA0qsψ

“

ˆ

´i!cBBt ´ eA0 p´i!∇ ` eAq ¨ σ

´p´i!∇ ` eAq ¨ σ i!cBBt ` eA0

˙

¨

˝

ψa

ψb

˛

‚

“ ´mc

¨

˝

ψa

ψb

˛

‚ (18)

where we have written the bispinor as a pair of spinors, ψ “pψa,ψbqT . Assuming Aµ to be time independent, we let ψ “ψprq expp´iEt!q. That is

rpp ` eAq ¨ σsψbprq “1

cpE ` eφ ´ mc2qψaprq

´rpp ` eAq ¨ σsψaprq “ ´1

cpE ` eφ ` mc2qψbprq

Thus, we obtain a second order differential equation for thesector a, which is exact

pp`eAq¨σc2

E ` eφ ` mc2pp`eAqψaprq “ pE`eφ´mc2qψaprq

(19)

Denote non-relativistic energy as ENR “ E ´ mc2. Approx-imations are now introduced, assuming E « mc2. Since thenonrelativistic energy is roughly the kinetic energy 1

2mv2, we

8

take an expansion in pvcq2

c2

E ` eφ ` mc2“

1

2m

2mc2

ENR ` eφ ` 2mc2

“1

2m

ˆ

1 ´ENR ` eφ

2m` ¨ ¨ ¨

˙

(20)

Using the operator identity pu ¨σqpv ¨σq “ u ¨v` iuˆv ¨σ,we have to the zeroth order in pvcq2,

„

pp ` eAq2

2m`

e!2m

B ¨ σ ´ eφ

ȷ

ψa “ ENRψa (21)

which is the Pauli equation. We have dropped the “minor”component ψb „ v

cψa, which is fine in a zeroth order theory.

To derive a 1st order theory, we put back the second termon the right-hand side of Eq. (20). Naıvely, we may write astationary Schrodinger equation like

„

pp ` eAq ¨ σ1

2m

ˆ

1 ´ENR ` eφ

2m

˙

pp ` eAq ¨ σ ´ eφ

ȷ

ψa

“ ENRψa

only to be bewildered by two problems. We will have some-thing like i!E ¨ pp ` eAq in the Hamiltonian, which is non-Hermitian. And the ψa is not normalized to the first order invc.

For simplicity, we will only deal with the case without a

9

magnetic field, so that A “ 0. We observe

1 “ż

d3r ψ:ψ “ż

d3r pψ:aψa ` ψ:

bψbq

«ż

d3r ψ:a

ˆ

1 `p2

4m2c2

˙

ψa

Thus, we can define a two-component wavefunction as

Ψ “ Pψa, P “ 1 `p2

8m2c2(22)

which is then normalized to the second order of vc. We thuswrite

P´1

„

p ¨ σ1

2m

ˆ

1 ´ENR ` eφ

2m

˙

p ¨ σ ´ eφ

ȷ

P´1Ψ “ ENRP´2Ψ

Then using P´1 « 1 ´ p28m2c2, V “ ´eφ and with a bit ofalgebra, we obtain

„

p2

2m` V `

!4m2c2

σ ¨ ∇V ˆ p `!2

8m2c2∇2V ´

p4

8m3c2

ȷ

ˆΨ “ ENRΨ (23)

This is called the nonrelativisitic approximation of the Diracequation. The third term of the Hamiltonian is called spin-orbit coupling interaction, because in a central field this termreduces to „ L ¨ S. And the fourth term is called the Darwinterm.

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4 Equation of motion

We now see that the effective-mass Hamiltonian in Eq. (14)formally resembles that of the Eq. (23). In the effective-masstheory is actually spinless, but the Berry curvature producesa spin-orbit like term. A Darwin term is also present, with∇2 Ñ ∇¨g¨∇, where the metric tensor g reflects the anisotropyof a crystal.

Without dwelling too much upon this connection to relativis-tic effects (which can be interesting in its own right), we nowobtain the equation of motion we set out for. We will drop theDarwin term and everything involving second derivative of V ,

HpR,pq “ εppp`eAq!q`V pRq`1

2Ω ¨∇V ˆppp`eAq!q

(24)We employ the Hamilton equations of motion

9p “ ´BHBR

, 9R “BHBp

(25)

There are a few caveats to heed, as explained below.

The position operator R is canonical, but does not representthe physical position. To see this

xRy “ż

ddRF ˚pRqF pRqR “ż

ddk

p2πqdF ˚pkqi∇kF pkq

But i∇k is not the correct representation of the position op-erator with respect to the envelope function. The key is to

11

observe that

xψnk|r|ψy “ i∇kFnk `ÿ

n1

Ann1pkqFn1k

Thus, the physical position in acting on the envelope functionsshould be

r “ i∇k ` Apkq. (26)

Since Fk is expected to be a sharply peaked function, it makessense to define its centroid

kc “ż

ddk

p2πqdF ˚pkqF pkqk. (27)

We expand Berry connection as A “ Ac`∇Ac ¨pk´kcq`¨ ¨ ¨ ,then lowest-order approximation says

rc “ R ` Apkcq (28)

We will just take the single band case, with the gauge choiceA “ 1

2Ω ˆ k. Therefore the physical position is given by

rc “ R `1

2Ω ˆ kc (29)

The other subtlety has to do with the momentum. The def-inition of kc means in coordinate representation kc looks tobe ´i∇ “ p!, which is not gauge-invariant under an electro-magnetic potential. The gauge-invariant momentum is p`eA,so we write

kc “p ` eApRq

!(30)

12

Consider 9kc, keeping only the 1st order terms in externalfields

9kc “dkc

dt

“1

!

ˆ

´∇V ´BεBR

` eBABRα

9Rα

˙

“1

!p´∇V q `

e

!

ˆ

BAβ

BRα

9Rα ´Bε

Bkα

BAα

BRβ

˙

eβ

Then, we can write

! 9kc “ ´eEpRq ´ e 9rc ˆ BpRq

Next we consider 9rc

9rc “Bε!Bk

´1

2!Ω ˆ ∇V `

BABkc,α

9kc,α

«Bε!Bk

` Ω ˆ 9kc

Finally, with the expansions fpRq “ fprcq ` f 1prcqpR´ rcq `¨ ¨ ¨ and fpkq “ fpkcq`f 1pkcqpk´kcq`¨ ¨ ¨ , and keeping onlythe lowest order terms, we have

9r “Bε!Bk

` Ω ˆ 9k (31)

! 9k “ ´eE ´ e 9r ˆ B (32)

where we have dropped all subscripts c to lighten up the no-tation.

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5 Equations of motion from vari-ational principle

In this section, we derive the equations motion by devisinga Lagrangian from the quantum time-dependent variationalprinciple.

The Hamilton dynamics given in Eq. (25) can be arrived atfrom a variational principle. Define a phase-space Lagrangian

Lpr, 9r,p, 9pq “ 9r ¨ p ´ Hpr,pq, (33)

which is a Legendre transform. The corresponding action is

S “ż

dtLpr, 9r,p, 9pq. (34)

The dynamics of a classical particle is fully characterized byits trajectory in phase space, rptq,pptq. Taking variation of Swe obtain

δS

δp“ 9r ´

BHBp

(35)

δS

δr“ ´ 9p ´

BHBr

(36)

hence the Hamilton dynamics.

The quantum time-dependent variational principle can beconstructed as follows. The Lagrangian is

L “ xψ|i!d

dt´ H|ψy (37)

14

with corresponding phase-space trajectory p|ψy, | 9ψy, xψ|, x 9ψ|qfollows the equations of motion

i!d

dt|ψy “ H|ψy, ´i!

d

dtxψ| “ xψ|H (38)

Consider a generalized quantum system embedded in a pa-rameter space, λ, the Lagrangian for describing its dynamicsin the parameter phase space is then

Lpλ, 9λq “ xψλ|i!d

dt´ H|ψλy “ !A ¨ 9λ ´ Hpλq, (39)

where we introduce the Berry connection of the parameterspace

Apλq “ xψλ|iBλ|ψλy (40)

The action is then

Srλs “ż

dt”

!A ¨ 9λ ´ Hpλqı

. (41)

Variation δλ yields

δS “ż

dt”

!BαAβ9λβ ´ BαHpλq ´ ! 9Aα

ı

δλα

“ż

dt”

!BαAβ9λβ ´ BαHpλq ´ !BβAα

9λβı

δλα

“ż

dt”

!Fαβ9λβ ´ BαHpλq

ı

δλα

where we have defined the Berry curvature F as a antisym-metric tensor

Fαβ “ BαAβ ´ pα Ø βq. (42)

15

Then equation of motion is then

!Fαβ9λβ “

BHBλα

. (43)

A few remarks regarding the equation of motion above iswarranted. First, because F is antisymmetric, the energy isconserved for the above equation of motion, d

dtHpλptqq “ 0.second, the action, and hence the equation of motion, are gaugeinvariant. What this means is that the action is invariant undera gauge transformation, |ψλy Ñ eiχpλq|ψλy, because

L Ñ !pAλ ´ Bλχq ¨ 9λ ´ Hpλq (44)

in which the second term becomes a total derivative of time,´! 9χ, and has no contribution up integration over time. Con-sequently, the equation of motion is also invariant.

How do we recover the Hamilton dynamics? Consider BαγFγβ “!´1δαβ . If now B is a constant tensor of the form

B “

„

0 ´11 0

ȷ

(45)

and we let λ “ pp, rq, we then find

9λα “ Bαγ BHBλγ

(46)

is precisely the Hamilton dynamics.

Let the system evolve over an infinitesimal period of time,δt,

t Ñ t1 “ t ` δt, λ Ñ λ1 “ λpt1q.

16

Then

λ1α « λα ` Bαγ BHBλγ

δt

The phase space volume element at t1 becomes

ddλ1 “ |Jpt1, tq|ddλ “

ˇ

ˇ

ˇ

ˇ

Bλ1

Bλ

ˇ

ˇ

ˇ

ˇ

ddλ . (47)

Here the Jacobian is to 1st order of δt

Jαβ “ δαβ `

BBαγ

Bλβ

BHBλγ

δt ` Bαγ B2H

BλβBλγδt (48)

If, as we have supposed, B is independent of its location onphase space, then detpJq “ 1. Here, we use the identity

log det J “ tr log J (49)

and the fact that the last term in Eq. (48) is traceless. There-fore, if B is independent of λ, we have

ddλ “ ddλ1 (50)

i.e., the phase space volume is conserved, which is nothing butthe classical Liouville theorem.

The more curious case arises when, in general, B is a non-constant function of phase space position λ. Suppose we achievethis by a change of variable, λ “ λpξq, with which F pξq is in-dependent of ξ.

?det F pξq ddξ “

?det F 1pξq ddξ1

17

Upon the change of variable, the Berry curvature becomes

Fpλqαβ “ F pξq

γη

Bξγ

Bλα

Bξη

Bλβ(51)

and we have

detF pλq “ detF pξq

ˇ

ˇ

ˇ

ˇ

BξBλ

ˇ

ˇ

ˇ

ˇ

2

(52)

Using the above relations, we obtain

PfpF pλqqddλ “ PfpF pλ1qqddλ1 (53)

where the Pfaffian is defined as the square root of the deter-minant of an antisymmetric (or anti-Hermitian) matrix. Thisresult is the generalized Liouville theorem.

We thus consider a wavepacket constructed from Bloch statesfrom one band (we will drop the band index), c.f. Eq. (1),

|W0y “ż

BZ

rdkswpkq|ψky (54)

where we use the notationş

BZ

rdks Ñş

BZ

ddkp2πqdVc

. The function

wpkq is localized around

kc “ż

[dks|wpkq|2k (55)

and is normalized asż

rdks|wpkq|2 “ 1. (56)

18

The center of mass for the wavepacket is

rc “ xW0|r|W0y. (57)

Under uniform and static external fields, E “ ´ 9apr, tq andB “ ∇ ˆ aprq, the system is described by the Hamiltonian

H “1

2mrp ` eapr, tqs2 ` Uprq. (58)

Introducing a gauge transformation to the wavepacket W0

|W qy “ e´ieaprc,tq!¨r|W0y (59)

for which the Hamiltonian transforms as

H “1

2mrp ` etapr, tq ´ aprc, tqus2 ` Uprq

« H0 `e

2mrapr, tq ´ aprc, tqs ¨ p ` H.c. (60)

with the unperturbed Hamiltonian

H0 “p2

2m` Uprq. (61)

Choosing the circular gauge, a “ ´Et ` 12B ˆ r, we have

H « H0 ` B ¨ M , (62)

whereM “

e

2mpr ´ rcq ˆ p. (63)

which describes the magnetization of the wavepacket arisingfrom revolution around its center of mass.

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The Lagrangian for the Hamiltonian given in Eq. (60) in theparameter space, λ “ pkc, rcq defining the center-of-mass ofthe packet in momentum and coordinate spaces, is

L “ xW |i!BBt

´ H|W y

« exW0| 9aprc, tq ¨ r|W0y `ż

rdksw˚pkqi!BBtwpkq

´εpkcq ´ B ¨ xW0|M |W0y.

Writing wpkq “ |wpkq|eiχpkq, we may write the Lagrangian as

L “ ´eaprc, tq ¨ 9rc ´ ! 9kc ¨ Bkcχpkcq ´ εpkcq ´ B ¨ Mpkcq

apart from an immaterial total time derivative. Using theidentity (how?)

rc “ż

rdks

„

w˚pkqiB

Bkwpkq ` |wpkq|2Apkq

ȷ

(64)

we have rc « ´Bkcχpkcq ` Apkcq. Here A is the one-bandBerry connection defined in Eq. (13). Then we have the La-grangian

Lpr,k, 9r, 9kq “ ´ea ¨ 9r ` !k ¨ 9r ` A ¨ ! 9k ´ ε (65)

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where ε is the energy of a Bloch state with the energy fromorbital magnetization in a magnetic field

εpkcq “ εpkcq ` B ¨ Mpkcq. (66)

Using the Lagrange equation, the equations of motion, Eqs.(31) and (32), follow straightforwardly.

One remaining piece of the problem is the correct measureof phase space volume element

d6λ “d3k d3r

p2πq3. (67)

We findPfpF q “ PfpΩ ` Bq (68)

where

Ω “

ˆ

´e!B 00 Ω

˙

. (69)

Here, B and Ω are rank-2 tensors, related to their pseudovectorcomponents according to

pαβ “ εαβγpγ. (70)

To the first order in B, the measure is

PfpF q « 1 `e

!B ¨ Ω. (71)

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