fiber bundles and topology for condensed matter …fiber bundles and topology for condensed matter...

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

Krakow, 24 April 2015 2/31

1. Tangent bundles and curvature


The tangent bundle of the sphere

p

TpS2

S2

v e1

e2


Parallel transport of vectors on S2

1 1 2

2

• Comparison of different tangent spaces

by parallel transport along a path

• Levi Civita connection


Covariant derivative

t t0 s(t)

• Covariant derivative:

0wD :transport Parallel

tscoefficien Connection :

wwe:wD

tdt

dsu

twtwttPtt

1limwDu

0

00

0tt 0


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


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


2. Fiber bundles


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


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


3. Topological quantum numbers


Euler characteristic

χE=0 χE=0

χE=-4 χE=-4

χE=2

χE=-2


Euler characteristic

edges # vertices # faces #E


4. Applications of fiber bundles


Fiber bundle in cosmology

spacetime M , 3,1SO,M or TM 444


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:


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


5. Topological quantum states


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


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


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


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


6. Basics for general topological classification


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


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


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


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


7. Summary


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


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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fiber bundles and topology for condensed matter …fiber bundles and topology for condensed matter...

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