2014 kias-snu winter camp - lecture_yu
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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.
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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 -
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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
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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)
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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
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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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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
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Electric polarization in momentum space
22
instead of
!uk!
"=0 "=1
kx
ky
0 2#/a
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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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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
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