Transcript pptx
Topological Insulators and
Superconductors
Akira Furusaki
2012/2/8
YIPQS Symposium
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Condensed matter physics
• Diversity of materials
– Understand their properties
– Find new states of matter
“More is different” (P.W. Anderson)
• Emergent behavior of electron systems at low energy
– Spontaneous symmetry breaking
crystal, magnetism, superconductivity, ….
– Fermi liquids (non-Fermi liquids)
(high-Tc) superconductivity, quantum criticality, …
– Insulators
Mott insulators, quantum Hall effect, topological insulators, …
Outline
• Topological insulators: introduction
• Examples:
–
–
–
–
–
Integer quantum Hall effect
Quantum spin Hall effect
3D Z2 topological insulator
Topological superconductor
Classification
• Summary and outlook
Introduction
• Topological insulator
– an insulator with nontrivial topological structure
– massless excitations live at boundaries
bulk: insulating, surface: metallic
• Many ideas from field theory are realized
in condensed matter systems
– anomaly
– domain wall fermions
–…
• Recent reviews:
– Z. Hasan & C.L. Kane, RMP 82, 3045 (2010)
– X.L. Qi & S.C. Zhang, RMP 83, 1057 (2011)
Recent developments 2005• Insulators which are invariant under time reversal
can have topologically nontrivial electronic structure
• 2D: Quantum Spin Hall Effect
– theory
C.L. Kane & E.J. Mele 2005; A. Bernevig, T. Hughes & S.C. Zhang 2006
– experiment
L. Molenkamp’s group (Wurzburg) 2007 HgTe
• 3D: Topological Insulators
in the narrow sense
– theory
L. Fu, C.L. Kane & E.J. Mele 2007; J. Moore & L. Balents 2007; R. Roy 2007
– experiment
Z. Hasan’s group (Princeton) 2008 Bi1-xSbx
Bi2Se3 , Bi2Te3 , Bi2Tl2Se, …..
Topological insulators
• band insulators
in broader sense
free fermions (ignore e-e int.)
• characterized by a topological number (Z or Z2)
Chern #, winding #, …
• gapless excitations at boundaries
stable
Topological non-topological
(vacuum)
insulator
Examples: integer quantum Hall effect, polyacetylen,
quantum spin Hall effect, 3D topological insulator, ….
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Band insulators
• An electron in a periodic potential (crystal)
2
d2
x V x x E x
2
2m dx
V x a V x
• Bloch’s theorem
k x eikxuk x
Brillouin zone
uk x a uk x
a
k
a
E
empty
band gap
Band insulator
occupied
a
0
a
k
Energy band structure:
a mapping k H k , En k , or un k
Topological equivalence (adiabatic continuity)
Band structures are equivalent if they can be continuously deformed
into one another without closing the energy gap
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Topological distinction of ground states
deformed “Hamiltonian”
n empty
bands
0 †
1n
Q k U
U U m n
U
0 1m
Q2 1, Q† Q, trQ m n
Q : Brillouin zone (k -space)
m filled
bands
U m n U m U n
ky
kx
2 U m n U mU n
map from BZ to Grassmannian
IQHE (2 dim.)
homotopy class
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Berry phase of Bloch wave function
Berry connection
A k i u k k u k
Berry curvature
F k k A k
Berry phase
C
C
A k dk Fd 2 k
S
Example: 2-level Hamiltonian (spin ½ in magnetic field)
dz
H k d k
d x id y
d x id y
d z
H k u k d k u k
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Integer QHE
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Integer quantum Hall effect
(von Klitzing 1980)
xy H
h
25812 .807
2
e
xx
Quantization of Hall conductance
e2
xy i
h
i : integer
exact, robust against disorder etc.
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Integer Quantum Hall Effect
xy H
B
xx
y
x
xy
e2
C
h
(TKNN: Thouless, Kohmoto, Nightingale & den Nijs 1982)
Chern number
integer valued
1
2
C
d
k k A k x , k y = number of edge modes crossing EF
2 i
bulk-edge correspondence
filled band
A kx , k y k k k
k kx , k y
Berry connection
13 eB y
kx
Lattice model for IQHE
(Haldane 1988)
• Graphene: a single layer of graphite
– Relativistic electrons in a pencil
Geim & Novoselov: Nobel prize 2010
B
A
K
E
K’
K’
K
K’
K
K
Matrix element for hopping
between nearest-neighbor sites: t
3ta
c
vF
2
300
px
H vF
K’
px ip y
0
ip y
0
0
0
0
0
0
0
px ip y
0
py
px
A
0
B
px ip y A
0
B
0
Dirac masses
• Staggered site energy
M , on A sites
(G. Semenoff 1984)
Breaks inversion symmetry
M , on B sites
• Complex 2nd-nearest-neighbor hopping (Haldane 1988)
t2ei
No net magnetic flux through a unit cell
Breaks time-reversal symmetry
K point: H K vF px x p y y m z
m M 3 3t2 sin
K ' point: H K ' vF px x p y y m ' z
m ' M 3 3t2 sin
Hall conductivity xy
e2
C
h
1
C sgn m sgn m '
2
C 0
C 1
C 0
C 1
Chern insulator
C0
Massive Dirac fermion: a minimal model for IQHE
H iv x x y y m z
parity anomaly
xy sgn m
1
2
(2+1)d Chern-Simons theory for EM
mx
Domain wall fermion
H iv x x y y mx z
x
x
1
1
x, y expiky y mx'dx'
0
v
i
E vk
m0
m0
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Quantum spin Hall effect
(2D Z2 topological insulator)
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2D Quantum spin Hall effect
Kane & Mele (2005, 2006); Bernevig & Zhang (2006)
• time-reversal invariant band insulator
• spin-orbit interaction
L S
• gapless helical edge mode (Kramers’ pair)
B
E
up-spin electrons
conduction band
B
down-spin electrons
valence band
kx
Sz is not conserved in general.
Topological index: Z
Z2
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Quantum spin Hall insulator
Bulk energy gap & gapless edge states
Helical edge states:
(i) Half an ordinary 1D electron gas
(ii) Protected by time reversal symmetry
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Kane-Mele model
• Two copies of Haldane’s model (spin up & down)
+ spin-flip term
down-spin electrons
up-spin electrons
HKane-Mele HHaldane 2 HHaldane 2 Hspin-flip
Hspin-flip iR ci† (s rij ) z c j
s : electron spin
n. n .
• Invariant under time-reversal transformation
i i s s
isy K
2 1
• Spin-flip term breaks U (1) U (1) symmetry
– R 0 two copies of Chern insulators C 1 C
– R 0 C 0 a new topological number: Z2 index 0, 1
Effective Hamiltonian
•
•
•
: z 1 A sublattice, z 1 B sublattice
: z 1 K point, z 1 K' point
s : s z 1 up spin, s z 1 down spin
H 0 i vF x z x y y
HSO SO z z sz
complex 2nd nearest-neighbor hopping (Haldane)
H R R x z s y y sx
H M M z
spin-flip hopping
staggered site potential (Semenoff)
Time-reversal symmetry
1H total H total
i x sy K
2 1
complex conjugation
Chern # = 0
Z2 index
Kane & Mele (2005); Fu & Kane (2006)
1
0
Quantum spin Hall insulator
Trivial insulator
E
E
conduction
band
conduction
band
EF
0
k
Time-reversal invariant momenta: a
4
1
a 1
Pf w a
1
det w a
a
a G
0
an even number
of crossing
k
wmn k um k un k
un k
Bloch wave of occupied bands
Z2 index
EF
valence band
valence band
an odd number
of crossing
w a w a
antisymmetric
Time reversal symmetry
Time reversal operator
k * k
k * k
Kramers’ theorem
2 1
time-reversal pair
All states are doubly degenerate.
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Z2: stability of gapless edge states
(1) A single Kramers doublet
E
E
Kramers’ theorem
stable
k
k
(2) Two Kramers doublets
E
E
k
k
Two pairs of edge states are unstable against perturbations that respect TRS.
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Experiment
HgTe/(Hg,Cd)Te quantum wells
CdTe
HgCdTe
CdTe
Konig et al. [Science 318, 766 (2007)]
Trivial Ins.
QSHI
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Z2 topological insulator
in 3 spatial dimensions
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3 dimensional Topological insulator
• Band insulator
Z2 topologically nontrivial
• Metallic surface: massless Dirac fermions
(Weyl fermions)
E
y
ky
x
Theoretical Predictions made by:
Fu, Kane, & Mele (2007)
Moore & Balents (2007)
Roy (2007)
kx
an odd number of Dirac cones/surface
Surface Dirac fermions
topological
insulator
• “1/4” of graphene
K
E
K’
K’
K
K
K’
ky
kx
Hsurface i y x i x y
• An odd number of Dirac fermions in 2 dimensions
cf. Nielsen-Ninomiya’s no-go theorem
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Experimental confirmation
• Bi1-xSbx
0.09<x<0.18
theory: Fu & Kane (PRL 2007)
exp: Angle Resolved Photo Emission Spectroscopy
Princeton group (Hsieh et al., Nature 2008)
5 surface bands cross Fermi energy
• Bi2Se3
ARPES exp.: Xia et al., Nature Phys. 2009
photon
p, E
a single Dirac cone
Other topological insulators:
Bi2Te3, Bi2Te2Se, …
Response to external EM field
Qi, Hughes & Zhang, 2008
Essin, Moore & Vanderbilt 2009
ieA
Integrate out electron fields to obtain effective action for the external EM field
Seff
e2
e2
3
3
dtdx
F
F
dtdx
EB
2
2
32 c
4 c
2
0
axion electrodynamics (Wilczek, …)
time reversal
trivial insulators
topological insulators
vacuum 0
topological insulator
FF d AdA
(2+1)d Chern-Simons theory
surface
xy
dtdx 2 A A
e2
2h
Topological magnetoelectric effect
Seff
2
e2
e
3
3
dtdx
F
F
dtdx
EB
2
2
32 c
4 c
Magnetization induced by electric field
S e2
M
E
B 2hc
Polarization induced by magnetic field
S e2
P
B
E 2hc
Topological superconductors
Topological superconductors
•
•
•
•
BCS superconductors
Quasiparticles are massive inside the superconductor
Topological numbers
Majorana (Weyl) fermions at the boundaries
stable
topological
superconductor
vacuum
(topologically trivial)
Examples: p+ip superconductor, fractional QHE at
5 3
, He
2
Majorana fermion
Ettore Majorana
mysteriously disappeared in 1938
• Particle that is its own anti-particle
• Neutrino ?
• In superconductors:
condensation of Cooper pairs
nothing (vacuum)
particle
hole
Quasiparticle operator
uc vc†
† if u v
This happens at E=0.
2D p+ip superconductor
• (px+ipy)-wave Cooper pairing
(similar to IQHE)
angular momentum =
• Hamiltonian for Nambu spinor c p (spinless case)
p
2m
Hp
px ip y
pF
2
†
c p
px ip y
S2
pF
d p
ˆd d
d
px , p y
2
p
2m
wrapping # = 1
x H* p x H p
• Majorana Weyl fermion along the edge
px+ipy
E
2
px-ipy
k †k
k
x
e
ikx
k 0
† x
k eikx †k dk
S2
Majorana zeromode in a quantum vortex
Zero-energy Majorana bound state
vortex
hc
e
E
(p+ip) superconductor
0
zero mode
0 0
0 0
Majorana fermion
energy spectrum
near a vortex
If there are 2N vortices, then the ground-state degeneracy = 2N.
interchanging vortices
i
i+1
braid groups, non-Abelian statistics
i i 1
i 1 i
(p+ip) superconductor
D.A. Ivanov, PRL (2001)
topological quantum computing ?
Majorana zeromode is insensitive to external disturbance (long coherence time).
Engineering topological superconductors
• 3D topological insulator + s-wave superconductor (Fu & Kane, 2008)
s-SC
S-wave SC
Dirac mass for the (2+1)d surface Dirac fermion
Similar to a spinless p+ip superconductor
Z2 TPI
Majorana zeromode in a vortex core (cf. Jakiw & Rossi 1981)
• Quantum wire with strong spin-orbit coupling + B field + s-SC
(Das Sarma et al, Alicea, von Oppen, Oreg, … Sato-Fujimoto-Takahashi, ….)
InAs, InSb wire
B
px2
H0
gpx s y Bsx
2m
s-SC
• Race is on for the search of elusive Majorana!
Classification of topological
insulators and superconductors
Q: How many classes of topological
insulators/superconductors exist in nature?
A: There are 5 classes of TPIs or TPSCs
in each spatial dimension.
Generic Symmetries:
time reversal
charge conjugation (particle hole) SC
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Classification of free-fermion Hamiltonian fi Hij f j
in terms of generic discrete symmetries
• Time-reversal symmetry (TRS) 0
anti-unitary
no TRS
TRS 1 TRS with 2 1 spin 0
1 TRS with 2 1 spin 1/2
T
TH *T 1 H
• Particle-hole symmetry (PHS)
BdG Hamiltonian
P
1
PH P H
*
anti-unitary
0 no PHS
PHS 1 PHS with 2 1
1 PHS with 2 1
triplet
singlet
• TRS PHS = Chiral symmetry (CS)
TPHTP H
1
T RS PHS 0, Ch 1
3 3 1 10
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10 random matrix ensembles
(symmetric spaces) Altland & Zirnbauer (1997)
TRS
PHS Ch
time evolution operator exp iHt
IQHE
WignerDyson
Z2 TPI
chiral
px+ipy
superconductor
• Wigner-Dyson (1951-1963): “three-fold way” complex nuclei
• Verbaarschot & others (1992-1993)
chiral phase transition in QCD
42
• Altland-Zirnbauer (1997): “ten-fold way”
mesoscopic SC systems
10 random matrix ensembles
(symmetric spaces) Altland & Zirnbauer (1997)
TRS
PHS Ch
time evolution operator exp iHt
IQHE
WignerDyson
Z2 TPI
chiral
px+ipy
superconductor
“Complex” cases: A & AIII
“Real” cases: the remaining 8 classes
43
TH *T 1 H or PH * P 1 H
How to classify topological insulators and SCs
• Gapless boundary modes are topologically protected.
• They are stable against any local perturbation.
(respecting discrete symmetries)
• They should never be Anderson localized by disorder.
Nonlinear sigma models for Anderson localization
of gapless boundary modes
S d d 1r tr Q + topological term (with no adjustable parameter)
2
QM
bulk: d dimensions
boundary: d -1 dimensions
Z2 top. term
d 1 M Z 2
WZW term
d M Z
-term
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NLSM topological terms
Z2: Z2 topological term can exist in d dimensions
Z: WZW term can exist in d-1 dimensions
d G H
d+1 dim. TI/TSC
d dim. TI/TSC
Classification of topological insulators/superconductors
Standard
(Wigner-Dyson)
Chiral
BdG
TRS
PHS
CS
d=1
d=2
A (unitary)
0
0
0
--
Z
AI (orthogonal)
+1
0
0
--
-QSHE --
AII (symplectic)
1
0
0
--
Z2
Z2 Z2TPI
--
Z
AIII (chiral unitary)
0
0
1
Z
BDI (chiral orthogonal)
+1
+1
1
Z
CII (chiral symplectic)
1
1
1
Z
D (p-wave SC)
0
+1
C (d-wave SC)
0
1
DIII (p-wave TRS SC)
1
+1
CI (d-wave TRS SC)
+1
1
-Z
0
Z
1
Z2
IQHE
--
polyacetylene
(SSH)
---
0 p SC Z2
--
d=3
Z2
p+ip SC
--
d+id SC
Z2
1 (p+ip)x(p-ip)
---SC
-Z
3He-B
Z
Schnyder, Ryu, AF, and Ludwig, PRB (2008)
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Periodic table of topological insulators/superconductors
period
d=2
period
d=8
A. Kitaev, AIP Conf. Proc. 1134, 22 (2009); arXiv:0901.2686
K-theory, Bott periodicity
Ryu, Schnyder, AF, Ludwig, NJP 12, 065010 (2010) massive Dirac Hamiltonian
Ryu, Takayanagi, PRD 82, 086914 (2010) Dp-brane & Dq-brane system
47
Summary and outlook
• Topological insulators/superconductors are new
states of matter!
• There are many such states to be discovered.
• Junctions: TI + SC, TI + Ferromagnets, ….
• Search for Majorana fermions
• So far, free fermions. What about interactions?
Outlook
• Effects of interactions among electrons
– Topological insulators of strongly correlated electrons??
– Fractional topological insulators ??
• Topological order
–
–
–
–
(no symmetry breaking)
Fractional QH states
Chern-Simons theory
Low-energy physics described by topological field theory
Fractionalization
Symmetry protected topological states
(e.g., Haldane spin chain in 1+1d)
X.-G. Wen
• Strongly correlated many-body systems
– have been (will remain to be) central problems
• High-Tc SC, heavy fermion SC, spin liquids, …
– but, very difficult to solve
• Theoretical approaches
– Analytical
• Application of new field theory techniques?
• ….
AdS/CMT?
– Numerical
• Quantum Monte Carlo (fermion sign problem)
• Density Matrix RG (only in 1+1 d)
• New algorithms: tensor-network RG, ….
Quantum information theory