effective topological field theories in condensed matter physics theoretical prediction: bernevig,...
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Effective Topological Field Theories in Condensed Matter Physics
Theoretical prediction: Bernevig, Hughes and Zhang, Science 314, 1757 (2006) Experimental observation: Koenig et al, Science 318, 766 (2007)New Developments: Qi et al, Nature Physics 4, 273, 08’, Phy Rev B78, 195424, 08’, Science 323, 1184, 09’
The search for new states of matter
Crystal: Broken translational symmetry
Magnet: Broken rotational symmetry
Superconductor: Broken gauge symmetry
The search for new elements led to a golden age of chemistry.
The search for new particles led to the golden age of particle physics.
In condensed matter physics, we ask what are the fundamental states of matter?In the classical world we have solid, liquid and gas. The same H2O molecules can condense into ice, water or vapor.
In the quantum world we have metals, insulators, superconductors, magnets etc.Most of these states are differentiated by the broken symmetry.
The quantum Hall state, a topologically non-trivial state of matter
h
enxy
2
• TKNN integer=the first Chern number.
)()2( 2
2
kFkd
n
• Topological states of matter are defined and described by topological field theory:
AAxdtdS xy
eff 2
2
• Physically measurable topological properties are all contained in the topological field theory, e.g. QHE, fractional charge, fractional statistics etc…
Chiral (QHE) and helical (QSHE) liquids in D=1
k
kF-kF
The QHE state spatially separates the two chiral states of a spinless 1D liquid
No go theorems: chiral and helical states can never be constructed microscopicallyfrom a purely 1D model. (Wu, Bernevig, Zhang, 2006) Helical liquid=1/2 of 1D fermi liquid!
k
kF-kF
The QSHE state spatially separates the four chiral states of a spinful 1D liquid
2=1+1 4=2+2
x
x
Time reversal symmetry in quantum mechanics
• Wave function of a half-integer spin changes by -1 under 2 rotation.
• Kramers theorem, in a time reversal invariant system with half-integer spins, T2=-1, all states for degenerate doublets.• Application in condensed matter physics: Anderson’s theorem. BCS pair=(k,up)+(-k,down). General pairing between Kramers doublets.
Spin=1/2
• Wave function of a particle with integer spin changes by 1 under 2 rotation. Spin=1
-
The topological distinction between a conventional insulator and a QSH insulatorKane and Mele PRL, (2005); Wu, Bernevig and Zhang, PRL (2006); Xu and Moore, PRB (2006)
• Band diagram of a conventional insulator, a conventional insulator with accidental surface states (with animation), a QSH insulator (with animation). Blue and red color code for up and down spins.
k
Trivial TrivialNon-trivial
k=0 or
From topology to chemistry: the search for the QSH state
• Graphene – spin-orbit coupling only about 10-3meV. Not realizable in experiments. (Kane and Mele, 2005, Yao et al, 2006, MacDonald group 2006)
• Quantum spin Hall with Landau levels – spin-orbit coupling in GaAs too small. (Bernevig and Zhang, PRL, 2006)
5 .1 5 .2 5 .3 5 .4 5 .5 5 .6 5 .7 5 .8 5 .9 6 .0 6 .1 6 .2 6 .3 6 .4 6 .5 6 .76 .6
1 .0
1 .5
0 .5
0 .0
-0 .5
2 .0
2 .5
3 .0
3 .5
4 .0
4 .5
5 .0
6 .0
5 .5
B a n d g a p v s . la ttic e co n sta n t(a t ro o m te m p e ra tu re in z in c b le n d e s t ru c tu re )
Ban
dga
p e
nerg
y (e
V)
la ttice con sta n t a [Å ]0
• Type III quantum wells work. HgTe has a negative band gap! (Bernevig, Hughes and Zhang, Science 2006)• Tuning the thickness of the HgTe/CdTe quantum well leads to a topological quantum phase transition into the QSH state.
Band Structure of HgTe
S
P
S
P3/2
P1/2
S
P SP1/2
P3/2
CdTe
HgTe
CdTe
E1
H1
normal
HgTe
CdTe CdTe
H1
E1
inverted
Quantum Well Sub-bands
Let us focus on E1, H1 bands close to crossing point
Effective tight-binding model
),(,,(,,,, yxyx ippippss
Square lattice with 4-orbitals per site:
Nearest neighbor hopping integrals. Mixing matrix elements between the s and the p states must be odd in k.
)(0
0)(),(
kh
khkkH yxeff
mikkA
ikkAm
kdkmkikA
kikAkmkh
yx
yx
aa
yx
yx
)(
)(
)()()sin(sin
)sin(sin)()(
Relativistic Dirac equation in 2+1 dimensions, with a mass term tunable by the sample thickness d! m<0 for d>dc.
kx
E
Mass domain wall
Cutting the Hall bar along the y-direction we see a domain-wall structure in the band structure mass term. This leads to states localized on the domain wall which still disperse along the x-direction.
m>0
m<0 m
y
0
m>0
y
x
x
Bulk
Bulk
E
0
Experimental Predictions
k
k
x
x
Experimental evidence for the QSH state in HgTe
• Since the mass is proportional to the magnetization, a magnetization domain wall leads to a mass domain wall on the edge.
Fractional charge in the QSH state, E&M duality!
mxx
e/2
• The fractional charge e/2 can be measured by a Coulomb blockade experiment, one at the time! Jackiw+Rebbie, Qi, Hughes & Zhang
E G
Vg
G
Vg
E V=e/C
x
x
(b)
Se2
BiSe1
Quintuplelayer
B
B
CA
C
A
C
(a)
t1
t2 t3
z
x
y
(c)x
y
3D insulators with a single Dirac cone on the surface
(I) (II) (III)
(a)
Bi
Se
(eV)
E (
eV)
-0.2
0.2
0.6
0 0.2 0.4
(b)
c
Relevant orbitals of Bi2Se3 and the band inversion
(a) Sb2Se3 (b) Sb2Te3
(c) Bi2Se3 (d) Bi2Te3
Bulk and surface states from first principle calculations
Effective model for Bi2Se3, Zhang et al
Pz+, up, Pz-, up, Pz+, down, Pz-, down
Minimal Dirac model on the surface of Bi2Se3, Zhang et al
Surface of Bi2Se3 = ¼ Graphene !
Doping evolution of the FS and band structureDoping evolution of the FS and band structure
EF(undoped)BCB bottom
Dirac point position
Undoped Under-doped Optimally-doped Over-doped
BVB bottom
Arpes experiment on Be2Te3 surface states, Shen group
General definition of a topological insulator
• Z2 topological band invariant in momentum space based on single particle states.(Fu, Kane and Mele, Moore and Balents, Roy)• Topological field theory term in the effective action. Generally valid for interacting and disordered systems. Directly measurable physically. Relates to axion physics! (Qi, Hughes and Zhang)
• For a periodic system, the system is time reversal symmetric only when=0 => trivial insulator= => non-trivial insulator
• Arpes experiments (Hasan group)
term with open boundaries• = implies QHE on the boundary with
EM
j//
T breaking
• For a sample with boundary, it is only insulating when a small T-breaking field is applied to the boundary. The surface theory is a CS term, describing the half QH.• Each Dirac cone contributes xy=1/2e2/h to the QH. Therefore, = implies an odd number of Dirac cones on the surface!
h
exy
2
2
1
• Surface of a TI = ¼ graphene
The Topological Magneto-Electric (TME) effect• Equations of axion electrodynamics predict the robust TME effect.
4πP= /2B4πM= /2E
• P3=/2 is the electro-magnetic polarization, microscopically given by the CS term over the momentum space. Change of P3=2nd Chern number!
Wilzcek, axion electrodynamics
Low frequency Faraday/Kerr rotation(Qi, Hughes and Zhang, PRB78, 195424, 2008)
Topological contribution
topo» 3.6x 10-3 radnormal contribution
Adiabatic
Requirement:
(surface gap)
Eg
Seeing the magnetic monopole thru the mirror of a TME insulator, (Qi et al, Science 323, 1184, 2009)
TME insulator
q
(for =’, =’)similar to Witten’s dyon effect
higher order feed back Magnitude of B:
An electron-monopole dyon becomes an anyon!
322 P
New topological states of quantum matter
QH insulator (U(1) integer), QSH insulator (Z2 number), chiral (U(1) integer) and helical (Z2 number) superconductors.
Chiral Majorana fermions
Chiral fermions
massless Majorana fermions
massless Dirac fermions
Taking the square root in math and physics
i 1
DiracGordonKlein
trySupersymmeSymmetryTimeSpace
tySupergraviGravity
QHEofstateedgeChiralliquidspinlessD 1
QSHEofstateedgeHelicalliquidspinfulD 1
fermionChiralDirac
Summary: the search for new states of matter
Magnet s-wave superconductor
Crystal
Quantum Hall
Quantum Spin Hall
Recurrence of effective field theories
Dirac at MeV Schroedinger at eV Dirac at meV
Theta vacuum and axion of QCDTopological insulators in CM
Monopoles in cosmologytable top experiments in CM
To see the world in a grain of sand,To hold infinities in an hour!