Berry phase Masatsugu Sei Suzuki

Department of Physics, SUNY at Binghamton (Date: May 16, 2015)

Sir Michael Victor Berry, FRS (born 14 March 1941), is a mathematical physicist at the University of Bristol, England. He was elected a fellow of the Royal Society of London in 1982 and knighted in 1996. From 2006 he has been editor of the journal, Proceedings of the Royal Society. He is famous for the Berry phase, a phenomenon observed e.g. in quantum mechanics and optics. He specializes in semi-classical physics (asymptotic physics, quantum chaos), applied to wave phenomena in quantum mechanics and other areas such as optics. He is also currently affiliated with the Institute for Quantum Studies at Chapman University in California.

Prof. Micheal Berry (Melville Wills Professor of Physics (Emeritus), University of Bristol) https://michaelberryphysics.wordpress.com/ 1. What is the Berry phase?

In classical and quantum mechanics, the geometric phase, Pancharatnam–Berry phase (named after S. Pancharatnam and Sir Michael Berry), Pancharatnam phase or most commonly Berry phase, is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian. The phenomenon was first discovered in 1956, and rediscovered in 1984. It can be seen in the Aharonov–Bohm effect and in the conical intersection of potential energy surfaces. In the case of the Aharonov–Bohm effect, the adiabatic parameter is the magnetic field enclosed by two interference paths, and it is cyclic in the sense that these two paths form a loop. In the case of the conical intersection, the adiabatic parameters are the molecular coordinates. Apart from quantum mechanics, it arises in a variety of other wave

systems, such as classical optics. As a rule of thumb, it can occur whenever there are at least two parameters characterizing a wave in the vicinity of some sort of singularity or hole in the topology; two parameters are required because either the set of nonsingular states will not be simply connected, or there will be nonzero holonomy. http://en.wikipedia.org/wiki/Geometric_phase 2. General formula for phase factor M.V. Berry, Quantum Phase Factors Accompanying Adiabatic Changes, Proc. R. London A392,

45-57 (1984).

Let the Hamiltonian H be changed by varying parameter R [R = (x, y, z)] on which it depends. Then the excursion of the system between times t = 0 and t = T can be pictured as

transport round a closed path R(t) in parameter space, with Hamiltonian ))((ˆ tH R and such that

)0()( RR T . The path is called a circuit and denoted by C. For the adiabatic approximation to

apply, T must be large.

The state vector ))(t of the system evolves according to Schrödinger equation given by

))())((ˆ))())( ttHtitt

i R

.

At any instant, the natural basis consists of the eigenstates )(Rn (assumed discrete) of )(ˆ RH

for )(tRR , that satisfy

))(())(())(())((ˆ tntEtntH n RRRR ,

with energies ))(( tEn R . The eigenvalue equation implies no relation between the phases of the

eigenstates )(Rn at different R.

Adiabatically, a system prepared in one of these states ))0((Rn will evolve with

H and so be in the state ))(( tRn at t

))(()](exp[)](exp[

))(()](exp[]'))'((exp[)(0

tntiti

tntidttEi

t

nn

n

t

n

R

RR

where )(tn is a geometric phase, and the dynamical phase factor )(tn is defined by

t

nn dttEt0

'))'((1

)( R

. )(1

)( tEt nn

Plugging the solution form into the this Schrödinger equation, we get

))(())((]))(()())(())(())(([ tntEtntitntEi

tnt

i nnn RRRRRR

or

0))(()())(( tntitn n RR .

Taking the inner product with ))(( tn R we get

0))(())(()())(())(( tntntitntn n RRRR ,

Since 1))(())(( tntn RR , we have

))(())(()( tntnitn RR

))(( tn R depends on t because there is some parameter )(tR in the Hamiltonian that changes

with time.

)())(())(( ttntn RRR R

so that

)())(())(()( ttntnitn RRR R

and thus

f

i

dtntnitn

R

R

R RRR ))(())(()(

3. Expression of )(Cn

We calculate the geometric phase )(Cn as follows.

For 1nn (normalization), we have

0

x

nnn

x

n,

0

y

nnn

y

n,

0

z

nnn

z

n.

For 0mn ( mn ) (orthogonality)

0

x

mnm

x

n,

0

y

mnm

y

n,

0

z

mnm

z

n.

The rotation of the vector nn 0A is given by

z

yx

zyx

x

nn

yy

nn

x

z

nn

xx

nn

zy

nn

zz

nn

y

z

nn

y

nn

x

nn

zyx

nn

e

ee

eee

A

][

][][

0

nm

nm

m

z

x

nm

y

mn

y

nm

x

mn

x

nmm

y

n

y

nmm

x

n

x

nmm

y

n

y

nmm

x

n

x

n

y

n

y

n

x

n

x

n

yn

x

n

y

n

y

n

xn

y

n

x

n

x

nn

yy

nn

x

][

][

][

)( 0A

where we use the closure relation and the relations

x

mnm

x

n

, y

mnm

y

n

.

Then we obtain

nm nm

nm mnnmmnnm

nmz

EE

nHx

mmHy

nnHy

mmHx

n

EE

nHx

m

EE

mHy

n

EE

nHy

m

EE

mHx

n

x

nm

y

mn

y

nm

x

mn

]

ˆˆˆˆ

[

]

ˆˆˆˆ

[

][)(

2

0A

where we use the relations

mn

ii EE

nHmnm

ˆ

, nm

ii EE

mHnmn

ˆ

with i = x, y, and z. We note that

nHz

mnHy

mnHx

m

mHz

nmHy

nmHx

n

zyx

mnnm

ˆˆˆ

ˆˆˆ

eee

PP

where

mHnnmˆP , nHmmn

ˆP .

Then we get

nm nm

nm nm

mnnm

EE

nHmmHn

EE

2

20

ˆˆ

PPA

So we have

nnd

nninndi

nndi

diCn

R

R

R

AR

Im

)]Im())[Re(

)( 0

since 0])Re[ nn . Using the Stokes’ theorem, we get

n

n

d

nndC

Va

a ][Im)(

where da denotes are elements in R space

nm nm

nEE

nHmmHn2)()(

)()(ˆ)()()(ˆ)(Im)(

RR

RRRRRRRV RR

The notation for nV is the same used by Berry in the original paper (1984).

4. Spin in Magnetic Field (the adiabatic approximation)

A particle with the angular momentum J interacts with a magnetic field B via the Hamiltonian:

BJB ˆ)(ˆ

BJgH

,

where Jg is the Lande-g factor. Note that

)()()(ˆ BBB mmmJz ,

where )(Bm is the eigenstate of zJ with the eigenvalue )(Bm .

For any fixed value of B, we have

)()()()(ˆ BBBB mEmH m

Schrodinger equation:

)())(()()]([ˆ)( ttEttHtt

i m BB

with

))0(()0( tmt B

where ))0((Bm is the eigenstate of ))0((ˆ tH B .

)](exp[)](exp[))((

)](exp[]'))'((exp[))(()(0

tititm

tidttEi

tmt

mm

m

t

m

B

BB

where

t

mm dttEt0

'))'((1

)( B

Plugging the solution form into the this Schrodinger equation, we get

))(())((])(

))(())(())(())(([ tmtEt

titmtmtE

itm

ti m

mm BBBBBB

or

t

ttmtmi m

)(

))(())((

BB

Taking the inner product with ))(( tm B we get

))(())(()(

))(())(( tmtmt

ttmtmi m BBBB

,

Since 1))(())(( tmtm BB , we have

))(())(()(

tmtmit

tm BB

or

t

m dttmtmit0

'))'(())'(()( BB

Note that )(tm is real, since

0]))(())((Re[))(())(())(())(( tmtmtmtmtmtm BBBBBB

The geometrical character of the Berry phase emerges when the variation of the instantaneous energy eigenstates with time is restated as their variation with field;

)()()(

))(())(( BBBB

BBB Bmm

dt

tdtm

ttm

This expresses the phase as an integral over field values;

)()(Im

)])()(Im())()([Re(

)()()(

BBB

BBBBB

BBB

B

BB

B

mmd

mmimmdi

mmdiCm

(1)

Note that

0])()(Re[)()()()( BBBBBB BBB mmmmmm

Stokes’ theorem applied to Eq.(1) gives, in an abbreviated notation.

)(Im

)()()()(Im

)()(Im)(

Ba

BBBBa

BBa

BB

BB

m

mn

m

Vd

mnnmd

mmdC

where

)()()()()( BBBBB BB mnnmVmn

m

da denotes area element in B space and exclusion in the summation is justified by

)()( BB Bnn being imaginary. The off-diagonal elements )()( BB Bmn are obtained as

follows. Since

)()()()(ˆ BBBB mEmH m , (Eigenvalue problem)

we get

)()()()()()(ˆ)()(ˆ BBBBBBBB mEmEmHmH BmmBBB

But the )(Bn is an orthogonal set, so for mn , we have

)()()()()()()()(ˆ)()()(ˆ)( BBBBBBBBBBBB mEnmEnmHnmHn BmmBBB

or

)()()()()()()()()()()(ˆ)( BBBBBBBBBBBB mnEmEnmnEmHn BmmBBnB .

Since

0)()()()()()( BBBBBB mnEmEn mBmB

we get

)()(

)()(ˆ)()()(

BB

BBBBB B

Bnm EE

mHnmn

Hence

nm nmm EE

mHnnHm2)]()((

)()(ˆ)()()(ˆ)([)(

BB

BBBBBBBV BB

where

BBJBJ Bgg

H eJBJB ˆˆ)(ˆ

, JBB

ˆ)(ˆ

BJgH

Then we get

nmm nm

mnnm

B 222 )]()([

)(ˆ)()(ˆ)([

1)(

BB

BJBBJBBV

)(ˆ)()(ˆ)()(ˆ)(

)(ˆ)()(ˆ)()(ˆ)()(ˆ)()(ˆ)(

321

321

321

BBBBBB

BBBBBB

eee

BJBBJB

mJnmJnmJn

nJmnJmnJmmnnm

)(ˆ)(2

1)(ˆ)(

2

1

)(ˆˆ)(2

1)(ˆ)( 1

BBBB

BBBB

nJmnJm

nJJmnJm

)(ˆˆ)(2

1)(ˆ)( 2 BBBB nJJm

inJm

nmnnmnnJm ,3 )(ˆ)( BB

1,

1,

))(1(

)1)((

1)1)(()(ˆ)(

nm

nm

mjmj

njnj

nmnjnjnJm

BB

1,

1,

))(1(

)1)((

1)1)(()(ˆ)(

nm

nm

mjmj

njnj

nmnjnjnJm

BB

So we get

[

nmm nm

mJnnJmmJnnJm

B 22332

221 )]()([

)(ˆ)()(ˆ)()(ˆ)()(ˆ)([Im

1)](Im[

BB

BBBBBBBBBV

or

0)(Im 1 BVm .

nmm nm

mJnnJmmJnnJm

B 23113

222 )]()([

)(ˆ)()(ˆ)()(ˆ)()(ˆ)([Im

1)(Im

BB

BBBBBBBBBV

or

0)(Im 2 BVm .

nmm nm

mJnnJmmJnnJm

B 21221

223 )]()([

)(ˆ)()(ˆ)()(ˆ)()(ˆ)([Im

1)(Im

BB

BBBBBBBBBV

Since 1)]()([ 2 BB nm , we have

2

2

322

122122

12

121223

)(

)(

)(ˆ)(Im1

)(ˆˆˆˆ)(Im1

])(ˆ)()(ˆ)(

)(ˆ)()(ˆ)([Im1

)](Im[

B

mB

m

mJimB

mJJJJmB

mJnnJm

mJnnJmB mn

m

B

B

BB

BB

BBBB

BBBBBV

Here we use the commutation relation

31221ˆˆˆˆˆ JiJJJJ .

We can put this in a form that does not depend on choice of the 3-axis to lie along B:

BB

BV3

)()](Im[

B

mm

We note that is

)(41

BB B

where

3

1

BB

BB

This singularity is spherically symmetric. The Berry phase is given by

)()0()()(

)(Im)(3

CmmCB

mdVdC mm BB

BaBa

where )(C is the solid angle subtended by C as seen from the origin in field space .

)()1

(1

22

3Cd

BdB

Bd BB eeBa

((Note)) Vector analysis: Gauss’ law

41

)]1

([)]1

[ 3

2 B

Bd

Bd

Bd

Bd aaBB BBBB

where

3

1

BB

BB .

We have

)(41

)1

( 2 BBB BB B

where )(B is the Dirac delta function.

)cos1(2sin2

00

dd

5. Example: spin ½ under the magnetic field which undergoes a precession

adiabatically

The magnetic field is given by

cos

sinsin

cossin

0BB

with

t .

Spin magnetic moment: σSμ ˆˆ2ˆ B

Bs

.

where

mc

eB 2

. (Bohr magneton)

The spin Hamiltonian is given by

nσBσBμ ˆ)()(ˆ)(ˆ)(ˆ tBtttH BBs

where σ is the Pauli spin operator. The eigenstates are given by

)()()(ˆ ttt nnnσ , )()()(ˆ ttt nnnσ

where

01

10ˆ x ,

0

0ˆ

i

iy ,

10

01ˆ z

2sin

2cos

)(

ietn ,

2cos

2sin

)(

ietn

The energy eigenstate:

)()()()(ˆ ttBttH B nn , )()()()(ˆ ttBttH B nn

)(tn is the eigenstate of )(ˆ tH with the energy eigenvalue )(tBE B . )(tn is the eigenstate

of )(ˆ tH with the energy eigenvalue )(tBE B .

We note that we change the parameters: and in

2sin

2cos

)(

ietn ,

Then we get

2cos

2sin

)(

ietn

)(

2cos

2sin

2

)(sin

2

)(cos

)(t

ee iin

except for the minus sign, when and (parity operation).

In the spherical co-ordinate,

2sin

0

sin

1

2cos

2

12

sin2

11

2sin

2cos

sin

1

2sin

2cos1

)(sin

1)(

1)()(

ii

ii

r

ierer

erer

tr

tr

tr

t

ee

ee

nenenen

2cos

0

sin

1

2sin

2

12

cos2

11

2cos

2sin

sin

1

2cos

2sin1

)(sin

1)(

1)()(

ii

ii

r

ierer

erer

tr

tr

tr

t

ee

ee

nenenen

Then we get

sin2

sin

2sin

0

2sin

2cos

sin

1

2cos

2

12

sin2

1

2sin

2cos

1)()(

2

r

i

ieer

ee

rtt

ii

i

i

e

e

enn

where

2sin

2cos)(

ietn .

sin2

cos

2cos

0

2cos

2sin

sin

1

2sin

2

12

cos2

1

2cos

2sin

1)()(

2

r

i

ieer

ee

rtt

ii

i

i

e

e

enn

where

2cos

2sin)(

ietn .

)cos1(2

sin2

sinsin

2sin

)()()(

2

2

0

2

drr

ii

dttiC

ee

rnn

)cos1(2

cos2

sinsin

2cos

)()()(

2

2

0

2

drr

ii

dttiC

ee

rnn

The solid angle

)cos1(2sin2)(0

dC

TBdttE BT

0

0

')'(1

The final state after one rotation where 0)( BTB is then given by

)0()exp(]cos1(exp[

)()](exp[)](exp[)(

0 nTBii

TnTiTiT

B

nn

We see that the dynamical phase factor depends on the period T of the rotation, but the geometrical phase depends only on the special geometry of the problem. In this case it depends on the opening angle q of the cone that the magnetic field traces out.

______________________________________________________________________________ REFERENCES

M.V. Berry, Quantum Phase Factors Accompanying Adiabatic Changes, Proc. R. London A392, 45-57 (1984).

A. Shapere and F. Wilczek, Geometric Phases in Physics (World Scientific, 1989). J.J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 2nd edition (Addison-Wesley,

2011). R. Shankar, Principles of Quantum Mechanics, second edition (Springer, 1994). D. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995). L.E. Ballentine, Quantum mechanics: A Modern Development (World Scientific, 2000). A. Böhm, A. Mostafazadeh, H. Koizumi, Q. Niu, and J. Zwanziger, The Geometric Phase in

Quantum Systems (Springer, 2003). D. Chruściński and A. Jamiołkowski, Geometric phase in classical mechanics and quantum

mechanics (Springer, 2004). S. Weinberg, Lectures on Quantum Mechanics (Cambridge, 2013).

__________________________________________________________________________ APPENDIX Derivation of Green’s function

)(412 rr

,

where

),,( zyxr , 222 zyxr .

We consider a sphere with radius ( )0

)1

(111

rda

rd

rd

rd narr

where

222 zyxr , ),,(r

z

r

y

r

x

r r er

n , dad na

and

3

1

rr

r ,

23

1)(ˆ

1

rrr

r

rn

We now

surface of

connected

Since

Using the

V

or

A

0)1

( r

e

consider the

f sphere A' (v

d to an approp

0)1

( r

ov

Gauss's law,

1

'

V r

dr

)1

(A rda n

except at the

volume integ

volume V', rad

priate cylinde

ver the whole

we get

(

'

2

A

VV

da

rd

n

r

'

'('A

da n

origin.

gral over the

dius )0r.

volume V - V

'(')1

1

'

A

dar

r

n

'

(')1

A

dar

n

e whole volum

. We note tha

V' we have

0)1

r

)1

rn

me (V - V') b

at the outer su

between the

urface and th

surface A an

he inner surfac

nd the

ce are

where r' nn and dr is over the volume integral. Then we have

)(441

4)1

()1

(2

22

rrn

dr

dar

daA

Using the Gauss's law, we have

VVA

dr

dr

da )(4)1

()1

( rrrn

or

)(41

rr

.

or

)()4

1( r

r

.

((Mathematica))

Clear"Gobal`";

Needs"VectorAnalysis`"SetCoordinatesCartesianx, y, zCartesianx, y, z

r1 x, y, z; r r1.r1

x2 y2 z2

Grad1

r Simplify

x

x2 y2 z232,

y

x2 y2 z232,

z

x2 y2 z232

Laplacian1

r Simplify

0