Fiber bundles and topology for condensed matter systems
Hans-Rainer Trebin
Institut für Theoretische und Angewandte Physik der Universität Stuttgart, Germany
Krakow, 24 April 2015
1/31
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1. Tangent bundles and curvature
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The tangent bundle of the sphere
p
TpS2
S2
v e1
e2
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Parallel transport of vectors on S2
1 1 2
2
• Comparison of different tangent spaces
by parallel transport along a path
• Levi Civita connection
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Covariant derivative
t t0 s(t)
• Covariant derivative:
0wD :transport Parallel
tscoefficien Connection :
wwe:wD
tdt
dsu
twtwttPtt
1limwDu
0
00
0tt 0
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Curvature of S2
• Curvature: transport along closed path rotates vector
(holonomy)
• Rotation angle:
• Curvature tensor two-dimensional manifold:
21
21
areaencircled
R
1
R
1KK
curvature Gaussian
, d
1212
1122
2211
2121 RRRR
R
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Principal bundle
• For description of parallel transport and curvature:
attach rotation group SO(2) at each point, yields
principal bundle (S2,SO(2))=(S2,S1)
• Change of origin (unit operation): gauge
transformation
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2. Fiber bundles
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Fiber bundles
• Consist of a basic differential manifold M
• At each point attached: fiber Fp which is either copy of a vector
space (“vector bundle”) or of a (gauge) group (“principle
bundle”)
• Prescriptions for glueing the fibers together e.g. Moebius strip
(S1,R)
M=S1
Fp=R
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Fiber bundles
• Prescription for parallel transport : covariant derivative and
curvature
• Topological quantum numbers, e.g. for 2d tangent bundles:
Euler characteristic:
,R
R
:tensor Curvature
matrix tcoefficien Connection :
w
ww ,wwD
wwe:wD
l
j
i
l
l
j
i
l
i
j
i
j
i
j
2
1
ij
i
j
j
M=S1
Fp=R
g22, d2
1
2M
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3. Topological quantum numbers
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Euler characteristic
χE=0 χE=0
χE=-4 χE=-4
χE=2
χE=-2
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Euler characteristic
edges # vertices # faces #E
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4. Applications of fiber bundles
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Fiber bundle in cosmology
spacetime M , 3,1SO,M or TM 444
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Fiber bundle in electrodynamics
Classical relativistic physics:
AA ,qAp
m
1v :fieldnetic electromag In
mpv :velocity four ,p
cEp :momentum Four
dsubmanifol closed d2
12
211
xyz
xzy
yzx
zyx
Fdxdx2
1ch
number Chern First :number quantum lTopologica
0BBE
B0BE
BB0E
EEE0
F
,R cp. AAF
tensor fieldnetic electromag the equals Curvature
Quantum mechanics:
bundle principle 1U,RR
Adxq
iexp0D
Dim
Aq
iim
v ,i
p
m?Parallelis
bundle-C,RR a through section i.e. ,C
3
1U change phase
0
3
Topology:
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Magnetic monopole of strength γ
2F dd2
1ch :number chern First
sin 01
10R̂ ,cos
01
10ˆ ,0ˆ
:R on acting basis lorthonorma in sphere the on connection Civita-Levi Cp.
sin iiF ,cos iiA ,0iA
:C on acting tensor fieldnetic electromag and potential Vector
2
0 0
1
ii
2
ninteractio Strong :3SU,RR
ninteractiok Electrowea :2SU1U,RR
netismElectromag :1U,RR
3
3
3
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5. Topological quantum states
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The Berry phase
1U M, bundle principle is system ,e factor phase a to up determined is m
, ,SM :norientatioarbitrary
but strength, fixed of fieldmagnetic in particle21-Spin :Example
manifold parameter M
m : parameter on dependent states ground mechanical Quantum
i
2
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The Berry phase
1ch ,sin2
1iiF ,cos
2
1iiA- ,0iA-
:particle-21-spin Example
Fdxdx2
1Mch
space parameter 2d for number chern First
mmmm iF curvatureBerry
mm iA connectionBerry
withiAD derivative Covariant
holonomy change, phase :motion Circular
m state ousinstantane in remain particles
:gapenergy and motionAdiabatic
1
M
12
211
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The Quantum Hall Effect (von Klitzing 1980)
nh
e
e
h
n
1
Ej jE
fieldmagnetic in crystal 2d
2
xy2yx
yxyxxyxy
zB
xy
zB
xy
zB
Ey
jx
Bz Bz
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The Quantum Hall Effect
n
2
bandsoccupied
12
xy
F curvatureBerry
1221
bands occupied Torus
22
xy
1U,Tchh
e
uuuui kd2
1
h
e
:tyconductivi Hall for formula transport Kubo
12
Hasan MZ, Kane CL 2010 RMP 82, 3045
1K
2K
2Tk
1U,T bundle Principle
TK,KmodRk :periodic also is space Wavevector
ku:k,kk wavevectorby labelled functions Wave
2
2
21
2
21
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6. Basics for general topological classification
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Search for topological insulators
,T,T
that such space standard knownally mathematic
equivalent anby Replace :Procedure
mappings? differentlly topologica of ,T Set
gap. with nsHamiltonia-Bloch of Set ?
kH k
T
mapping aby described :Insulator
dd
d
d
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Topological classification of insulators 1
UkUkH
mnUkUU
k
k
k
k
UkHUjsskHrrijkHik
kH of seigenstate j ,i skHrkH
:matrices Hamilton-Bloch ediagonaliz :step First
n
1
1
m
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Topological classification of insulators 2
2,3d for Z CG,T
:isolators for numbers quantum lTopologica
nUmU/nmU CG
space coset an"Grassmanni" the to isomorphic is
UD US U D S UnUmU S
nUmU :D of group Fixpoint
nmU group the of action the under D of Orbit"" is
UD UU
1
1
1
1
UkK
1k ,1k
gap closing without structure band deform :step Second
mn,m
d
mn,m
m
n
m
n
m
n
m
n
m
n
ij
+1
-1
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The spin quantum Hall effect
currents surface 3d, in TeBi ,SBi ,SbBi
currents edge polarized Spin
2d in structures well quantum HgTe/CdTe
2,3d ,1,0Z CG, /T
by tionclassificaNew
/T manifold baseNew
k-k :symmetry reversal timeby manifold base the on nRestrictio
3232xx-1
2m,nm
d
d
1K
2K
2Tkk
Hasan MZ, Kane CL 2010 RMP 82, 3045
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7. Summary
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Summary
• Up to 1980: Quantum numbers based on symmetry
• Easy to break, lift of degeneracies
• Since 1980: Topological quantum numbers
• New states of quantum matter
• Robust
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Literature
• Hasan MZ, Kane CL 2010 Rev.Mod.Phys. 82, 3045
• Qi X-L, Chang S-Ch 2011 Rev.Mod.Phys. 83, 1057
• Budich JC, Trauzettel B 2013 phys.stat.sol.(RRL) 7, 109
• v. Klitzing K, Dorda G, Pepper M 1980 PRL 45, 494
• Thouless DJ, Kohmoto M, Nightingale MP, den Nijs M 1982 PRL 49, 405
• Kane CL, Mele EJ 2005 PRL 95, 146802
• Bernevig BA, Zhang S-Ch 2006 PRL 96, 106802
• Bernevig BA, Hughes DL, Zhang S-Ch 2006 Science 314, 1757
• König M, Wiedmann S, Brüne Ch, Roth A, Buhmann H, Molenkamp W, Qi X-L, Zhang S-Ch 2007 Science 318, 766
• Hsieh D, Qian D, Wray L, Xia Y, Hor YS, Cava RJ, Hasan MZ 2008 Nature 452, 970
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The End