2014 kias-snu winter camp - lecture_yu

8/13/2019 2014 KIAS-SNU Winter Camp - Lecture_YU

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2014 KIAS-SNU Physics Winter Camp, 2014.02.09-16

Topological Ideas

in Condensed Matter Systems

2014 KIAS-SNU Physics Winter Camp

2014.02.09-16


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Geometric phases in Physics

Jeeva Anandan, Joy Christian, and Kazimir Wanelik, American Journal of Physics 65, 180 (1997).


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Fi .1

Falling Cats

Have you ever wondered why a falling cat

always lands on her feet?Apparently, the problem of the cat is that

due to angular momentum conservation

there seems to be no way for the cat to

right herself.

However, by changing her shape she can

affect a rotation as a whole, a geometric

effect. In the quantum world, such

geometric effects give rise to additional

phase factors depend only on the way the

system evolves.

3R. Montgomery, Commun. Math. Phys. 128, 565 (1990).


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Foucaults Pendulum

4M. Berry, Scientific American, December 1988.

The start of the pendulums rotation hasshifted by a certain angle, called

Hannays angle which is equal to the

solid angle subtended by the pendulums

axis of rotation around the globe.


5/30


Aharonov-Bohm effect

5

B = 0

B6= 0

2 = ei22

1 = ei11

1 =

e

h

ZC1

Adl

2 =e

h

ZC2

Adl

Q=1+2= ei1

11

+(2/1)ei(e/h)B

B= (1 2)/(e/h) =IC12

Adl

Question:

geometric ortopological effect?


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Magnetic monopole and Dirac string

6Images from wikipedia http://en.wikipedia.org/wiki/Magnetic_monopole

B , 0

B = A

B = gC
http://en.wikipedia.org/wiki/Magnetic_monopole


7/30


Anholonomy and topology

7

Px =!- y

1-x =!+y

P

When transported along C,P moves fromytoy. When transported along C,P moves fromyto -y.

x =!- y

1-x =!+y

Anholonomy,e.g.,sgn(y),

comes from the topology of manifold.


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Fiber bundles, connection, curvature, and topology

Holonomy: Connection:

Curvature:

Topology

8

C = (s2)

(s1)(s1)

(s2)

A(r)

C = 2 1 +q

~

ZC12

A(r)dl

(r)

F

=

IS

~F ds =e

~

IS

A(r) ds

~F =e

~ A(r)


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Geometric phase in superconductors:magnetic vortex


10/30


GinzburgLandau theory

Superconducting order parameter: G-L free energy:

Superconducting current:

10

(r) = |(r) |ei(r)

(x)

x

0

1

j =2e~

m

| |2( 2e

~A)

(r)


11/30


What if

11

(r ,) = ?

supercurrent

along the edge?

j =2e~

m| |

2

( 2e

~ A)

The kinetic energy of the edge current can be suppressed if

A(r ) =~

2e

=

~

2er


12/30


Abrikosov vortex in type-II superconductors

12

(r,) = |(r) |ei

|(r = 0) | = 0

otherwise singular at r=0

STM image of Vortex lattice

H. F. Hess et al., Phys. Rev. Lett. 62, 214 (1989)

A(r > ro ,) =~

2er


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Geometric phase in quantum mechanics:Berry phase


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(C) =

ZS

1

i

dn

dx1

dn

dx2

dn

dx2

dn

dx1

!dx1 dx2

(C) =

iIC

n

n

dxdx

Parallel transport and anholonomy angle:Mathematical formulation

14Ming-Che Chang, Berry phase in solid state physics, http://phy.ntnu.edu.tw/~changmc/Paper/Berry_IFF_FF_4.pdf

12

34

vi

vf

r

e1

e2

r

e1

e2

~

r

e1

e2

r

u

v

fixed frame

u

v

e1e2

(x) = n(x)ei(x)

r = 0

e1 = ~ e1 e1 e2 = 0

e2 = ~ e2

e2

e1 = 0

=12

(e1 + ie2) Im

= 0

d = n dn i d
http://phy.ntnu.edu.tw/~changmc/Paper/Berry_IFF_FF_4.pdf


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Adiabatic theorem

15

i~t(x, t) = H(x, t)

(x, t) =X

n

cnn (x)eien t/~ Hn (x) = enn (x)

Time-dependent perturbation theory for tH(t) , 0

a general solution:(t) =

Xn

cn (t)n (t)ei(t)

wheren (t) =(1/~)

Z t

0

en (t0)dt0

cm (t) =X

n

cnhm |niei(nm )

=cmhm |niX

n,m

cn

hm |H |ni

en emei(nm )

adiabatic approximation

cm (t) = cm (0)exp

"

Z t

0

hm (t0) |m (t

0)idt0# = cm (0)e

im (t)

n (t) =n (t)ein (t)

ein (t) n (t) Berry phase


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Anholonomy in quantum mechanics:Berry phase

16

H(r,p;R,P)

slow variables

fast variables

|(t)i = ein(R)ei/~

R t

0 d t en(R(t)) |n;Ri

n (t) =

ihn|ni

n (C) = i

IC

*n|n

R

+ dR =

IC

A dR

A(R) = i *n|n

R+ Berry connection

n (C) = i

ZS

*n

R| |

n

R

+ d2R =

ZS

~F d2R

~F(R) =R A(R) Berry curvature


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A spin in a rotating solenoid

17Ming-Che Chang, Berry phase in solid state physics, http://phy.ntnu.edu.tw/~changmc/Paper/Berry_IFF_FF_4.pdf

S

x

y

z

H=

L2

2I+B

B

|+;B=

cos

2

ei sin 2

, |; B=

ei sin

2

cos 2

Electromagnetism quantum anholonomy

vector potential A(r) Berry connection A(R)

magnetic field B(r) Berry curvatureF(R)

magnetic monopole point degeneracy

magnetic flux (C) Berry phase(C)

A = 121 cos Bsin

F = 1

2

B

B2.

(C) = 12(C)
http://phy.ntnu.edu.tw/~changmc/Paper/Berry_IFF_FF_4.pdf


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Berry phase and Bloch state:Electric polarization

18

P

P

P

Unit cell

+

+

Electric polarization in a periodic solid is

an ill-defined quantity!

P() = q

V

i

i|r|i


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20/30


Berry phase in a discrete form

20

!u3!

!u2!

!un!=!u1!

!u4!

!un-1!

Check that a local gauge does not affect the Berry phase:

In the continuum limit:

!u!"

!=0!=1

"Nothing physical changes for a !-dependent gauge:

() ei()()


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Berry phase in momentum space

21

!uk!

kx

ky

0 2"/a

#

Bloch function


22/30


Electric polarization in momentum space

22

instead of

!uk!

"=0 "=1

kx

ky

0 2#/a


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24/30


Anomalous Hall effect:unquantized version of quantum Hall effect

24

xy = e2

h

Z dk(2)d

f(k)kxky

H= h2k2

2m + (k )ez z

= 2

2(2k2 +2)3/2


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Magnets:Spins and geometric phases


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Spin ice crystals

26D. J. P. Morris et al., Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7, Science 326, 411 (2009).

a b

| l |


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Monopoles in momentum space of SrRuO3

27

0

20

40

60

80

Res

xy

(W1

cm1)

Im

sxy

(W1

cm1)

100

E EF

k

m

120

2|m|

20

0

experiment

calculationT=10K

m=2meVm=1meVm=2meVm=3meV

20

40

60

0 2 4 6

energy (meV)

8 10 12

(a)

(b)

4

2

0

2

4

E(k)

(a)

21

01

2

10

1

kykx

H= kxsx+ kysy + msz

sxy=

e2

2 hsign(m)

bn(k) = k

2|k|3

N.Nagaosa, X.Z. Yu and Y. Tokura, Gauge fields in real and momentum spaces in magnets: monopoles and skyrmions,

Phil. Trans. R. Soc. A 370, 58065819 (2012)


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Anomalous Hall effect in Nd2Mo2O7

28

6

(a) (b)

umbrella structureM

Jfd

Nd 4fNd

Mo 4d

Mo

(c)6

4

2

0 50 100

2K

H||(100)

10K

20K30K40K

50K

60K

70K

80K

90K100K

T(K)

H=0.5 T

4

2rH

(106

Wc

m)

rH

(106

W

cm)

0 2 4magnetic field (T)

6 8 10

N.Nagaosa, X.Z. Yu and Y. Tokura, Gauge fields in real and momentum spaces in magnets: monopoles and skyrmions,

Phil. Trans. R. Soc. A 370, 58065819 (2012)


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Skyrmion

29N.Nagaosa, X.Z. Yu and Y. Tokura, Gauge fields in real and momentum spaces in magnets: monopoles and skyrmions,

Phil. Trans. R. Soc. A 370, 58065819 (2012)

ei= via0 1c

ai= h2e

(n vin n)

hi= [V a]i=hc

2ediz(n vxn vyn)

Effective electromagnetic fields


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2014 KIAS SNU Physics Winter Camp 2014 02 09 16

Liquid crystals

30

http://2012books.lardbucket.org/books/principles-of-general-chemistry-v1.0m/s15-08-liquid-crystals.html
http://2012books.lardbucket.org/books/principles-of-general-chemistry-v1.0m/s15-08-liquid-crystals.html

2014 kias-snu winter camp - lecture_yu

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