topological insulators and superconductors€¦ · topological insulators examples: integer quantum...
TRANSCRIPT
Topological Insulators and Superconductors
Akira Furusaki
2012/2/8 1 YIPQS Symposium
Condensed matter physics
• Diversity of materials
– Understand their properties
– Find new states of matter
• 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, …
“More is different” (P.W. Anderson)
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
Examples: integer quantum Hall effect, polyacetylen,
quantum spin Hall effect, 3D topological insulator, ….
• band insulators
• characterized by a topological number (Z or Z2) Chern #, winding #, …
• gapless excitations at boundaries
stable
free fermions (ignore e-e int.)
6
in broader sense
Topological insulator
non-topological (vacuum)
Band insulators • An electron in a periodic potential (crystal)
• Bloch’s theorem Brillouin zone
2 2
2
2
dx V x x E x V x a V x
m dx
ikx
k k k kx e u x u x a u x ka a
E
ka
a
empty
occupied
band gap Band insulator
0
8
Energy band structure:
kukEkHk nn
or , ,a mapping
Topological equivalence (adiabatic continuity)
Band structures are equivalent if they can be continuously deformed into one another without closing the energy gap
Topological distinction of ground states
m filled bands
n empty bands
xk
yk
map from BZ to Grassmannian
nUmUnmU2 IQHE (2 dim.)
homotopy class
deformed “Hamiltonian”
†1 0
0 1
n
m
Q k U U U U m n
9
2 †1, , trQ Q Q Q m n
: Brillouin zone ( -space) Q k U m n U m U n
Berry phase of Bloch wave function
10
Berry connection kukuikA
k
Berry curvature kAkF
k
Berry phase
C SC kFdkdkA 2
Example: 2-level Hamiltonian (spin ½ in magnetic field)
kukdkukH
zyx
yxz
didd
idddkdkH
Integer QHE
11
Integer quantum Hall effect (von Klitzing 1980)
Hxy
xx Quantization of Hall conductance
2
: integerxy
ei i
h
807.258122e
h
exact, robust against disorder etc.
12
Integer Quantum Hall Effect
Ch
exy
2
Chern number
21 ,
2k x yC d k A k k
i
filled band
,
, x y
x y k
k k k
A k k k k
integer valued
13
(TKNN: Thouless, Kohmoto, Nightingale & den Nijs 1982)
B
Hxy
xx
= number of edge modes crossing EF
Berry connection
bulk-edge correspondence
x
eBk y
x
y
Lattice model for IQHE (Haldane 1988)
• Graphene: a single layer of graphite
– Relativistic electrons in a pencil Geim & Novoselov: Nobel prize 2010
py
px
E
K
K’
K’
K’
K
K
0 0 0
0 0 0
0 0 0
0 0 0
x y
x y
F
x y
x y
p ip
p ipH v
p ip
p ip
A B A B
K K’
B A
Matrix element for hopping between nearest-neighbor sites: t
3
2 300F
ta cv
Dirac masses • Staggered site energy (G. Semenoff 1984)
• Complex 2nd-nearest-neighbor hopping (Haldane 1988)
No net magnetic flux through a unit cell
, on A sites
, on B sites
M
M
2
it e
point: K F x x y y zK H v p p m
'' point: 'K F x x y y zK H v p p m
23 3 sinm M t
2' 3 3 sinm M t
2
xy
eC
h Hall conductivity
1
sgn sgn '2
C m m 1C 1C
0C
0C
Chern insulator
0C
Breaks inversion symmetry
Breaks time-reversal symmetry
xm
x
Massive Dirac fermion: a minimal model for IQHE
zyyxx xmivH
zyyxx mivH
parity anomaly mxy sgn2
1
Domain wall fermion
i
dxxmv
ikyyxx
y
1''
1exp,
0
vkE 0m 0m
16
(2+1)d Chern-Simons theory for EM
Quantum spin Hall effect
(2D Z2 topological insulator)
17
2D Quantum spin Hall effect
• time-reversal invariant band insulator
• spin-orbit interaction
• gapless helical edge mode (Kramers’ pair)
Kane & Mele (2005, 2006); Bernevig & Zhang (2006)
B
B
up-spin electrons
down-spin electrons
L S
E
xk
Sz is not conserved in general. Topological index: Z Z2
18
valence band
conduction band
19
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
Kane-Mele model
• Two copies of Haldane’s model (spin up & down) + spin-flip term
• Invariant under time-reversal transformation
• Spin-flip term breaks symmetry
– two copies of Chern insulators
– a new topological number: Z2 index
Kane-Mele Haldane Haldane spin-flip2 2H H H H
up-spin electrons down-spin electrons
†
spin-flip
. .
( )R i ij z j
n n
H i c s r c
i i s s 2K 1yis
(1) (1)U U
0R 1C C
: electron spins
0R 0C 0, 1
Effective Hamiltonian •
•
•
: 1 A sublattice, 1 B sublatticez z
: 1 K point, 1 K' pointz z
: s 1 up spin, s 1 down spinz zs
0 F x z x y yH i v
SO SO z z zH s
R R x z y y xH s s
M zH M
complex 2nd nearest-neighbor hopping (Haldane)
spin-flip hopping
staggered site potential (Semenoff)
1
total totalH H
Time-reversal symmetry 2K 1x yi s Chern # = 0
complex conjugation
Z2 index Kane & Mele (2005); Fu & Kane (2006)
nu kBloch wave of occupied bands mn m nw k u k u k
Time-reversal invariant momenta: a a a G a aw w
antisymmetric
Z2 index
4
1
Pf1 1
det
a
aa
w
w
Quantum spin Hall insulator
E
kvalence band
conduction band
0
1 0
an odd number of crossing
Trivial insulator
E
kvalence band
conduction band
0
an even number of crossing
FEFE
Time reversal symmetry
23
Time reversal operator
*
*
k k
k k
2 1
Kramers’ theorem
All states are doubly degenerate.
time-reversal pair
Z2: stability of gapless edge states
(1) A single Kramers doublet
(2) Two Kramers doublets
24
E
k
E
k
Kramers’ theorem
E
k
E
k
stable
Two pairs of edge states are unstable against perturbations that respect TRS.
Experiment HgTe/(Hg,Cd)Te quantum wells
Konig et al. [Science 318, 766 (2007)]
CdTe HgCdTe CdTe
25
QSHI
Trivial Ins.
Z2 topological insulator
in 3 spatial dimensions
26
3 dimensional Topological insulator
• Band insulator
• Metallic surface: massless Dirac fermions
(Weyl fermions)
x
y
xk
yk
E
Theoretical Predictions made by: Fu, Kane, & Mele (2007) Moore & Balents (2007) Roy (2007)
Z2 topologically nontrivial
an odd number of Dirac cones/surface
Surface Dirac fermions
• “1/4” of graphene
• An odd number of Dirac fermions in 2 dimensions
cf. Nielsen-Ninomiya’s no-go theorem
ky
kx
E
K
K’
K’
K’
K
K
topological insulator
28
surface y x x yH i i
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
a single Dirac cone
p, E photon
Bi2Te3, Bi2Te2Se, …
Other topological insulators:
Response to external EM field
ieA
Integrate out electron fields to obtain effective action for the external EM field
2 23 3
eff 2 232 4
e eS dtdx F F dtdx E B
c c
axion electrodynamics (Wilczek, …)
Qi, Hughes & Zhang, 2008 Essin, Moore & Vanderbilt 2009
0
trivial insulators
topological insulators
2
time reversal
topological insulator
vacuum 0 FF d AdA 2
surfacedtdx A A
(2+1)d Chern-Simons theory
2
2xy
e
h
Topological magnetoelectric effect 2 2
3 3
eff 2 232 4
e eS dtdx F F dtdx E B
c c
2
2
S eM E
hcB
Magnetization induced by electric field
Polarization induced by magnetic field 2
2
S eP B
hcE
Topological superconductors
Topological superconductors
• BCS superconductors
• Quasiparticles are massive inside the superconductor
• Topological numbers
• Majorana (Weyl) fermions at the boundaries
Examples: p+ip superconductor, fractional QHE at , 3He
topological superconductor
vacuum (topologically trivial)
stable
2
5
Majorana fermion
• Particle that is its own anti-particle
• Neutrino ?
• In superconductors: condensation of Cooper pairs
particle
hole
nothing (vacuum)
†uc vc † if u v Quasiparticle operator
Ettore Majorana mysteriously disappeared in 1938
This happens at E=0.
2D p+ip superconductor (similar to IQHE)
• (px+ipy)-wave Cooper pairing
• Hamiltonian for Nambu spinor (spinless case)
• Majorana Weyl fermion along the edge
2
2
2
2
x y
F
p
x y
F
pp ip
m pH d p
pp ip
p m
angular momentum =
px-ipy px+ipy
d̂ d d 2,x yp p S
2S
wrapping # = 1
k
E
2
†
p
p
c
c
†
k k
†
0
†
ikx ikx
k k
k
x e e dk
x
*
x p x pH H
Majorana zeromode in a quantum vortex
Fn En / , 2
000
zero mode 00 00
Majorana fermion energy spectrum near a vortex
If there are 2N vortices, then the ground-state degeneracy = 2N.
(p+ip) superconductor
vortex
e
hc
Zero-energy Majorana bound state
E
0
interchanging vortices braid groups, non-Abelian statistics
1 ii
ii 1
D.A. Ivanov, PRL (2001)
(p+ip) superconductor
i i+1
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)
• Quantum wire with strong spin-orbit coupling + B field + s-SC
• Race is on for the search of elusive Majorana!
Z2 TPI
s-SC S-wave SC Dirac mass for the (2+1)d surface Dirac fermion
Similar to a spinless p+ip superconductor
Majorana zeromode in a vortex core (cf. Jakiw & Rossi 1981)
(Das Sarma et al, Alicea, von Oppen, Oreg, … Sato-Fujimoto-Takahashi, ….)
s-SC
B
2
02
xx y x
pH gp s Bs
m InAs, InSb wire
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
40
Classification of free-fermion Hamiltonian in terms of generic discrete symmetries
• Time-reversal symmetry (TRS)
• Particle-hole symmetry (PHS) BdG Hamiltonian
• TRS PHS = Chiral symmetry (CS)
HTTH 1*
HPPH 1*
2
2
0 no TRS
TRS 1 TRS with 1
1 TRS with 1
2
2
0 no PHS
PHS 1 PHS with 1
1 PHS with 1
spin 0
spin 1/2
HTPTPH 1
triplet
singlet
1Ch 0,PHSTRS
10133
jiji fHf
41
anti-unitary
anti-unitary
T
P
10 random matrix ensembles (symmetric spaces) Altland & Zirnbauer (1997)
42
Wigner- Dyson
chiral
super- conductor
IQHE
Z2 TPI
px+ipy
time evolution operator TRS PHS Ch
• Wigner-Dyson (1951-1963): “three-fold way” complex nuclei • Verbaarschot & others (1992-1993) chiral phase transition in QCD • Altland-Zirnbauer (1997): “ten-fold way” mesoscopic SC systems
exp iHt
10 random matrix ensembles (symmetric spaces) Altland & Zirnbauer (1997)
43
Wigner- Dyson
chiral
super- conductor
IQHE
Z2 TPI
px+ipy
time evolution operator TRS PHS Ch
“Complex” cases: A & AIII
“Real” cases: the remaining 8 classes HPPHHTTH 1*1* or
exp iHt
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
+ topological term 21 tr QrdS d
MQ
bulk: d dimensions boundary: d -1 dimensions
21 ZMd Z2 top. term
WZW term ZMd
-term
(with no adjustable parameter)
44
NLSM topological terms HGd
Z2: Z2 topological term can exist in d dimensions
Z: WZW term can exist in d-1 dimensions
d+1 dim. TI/TSC
d dim. TI/TSC
46
Standard (Wigner-Dyson)
A (unitary)
AI (orthogonal)
AII (symplectic)
TRS PHS CS d=1 d=2 d=3
0 0 0
+1 0 0
1 0 0
-- Z --
-- -- --
-- Z2 Z2
AIII (chiral unitary)
BDI (chiral orthogonal)
CII (chiral symplectic)
Chiral
0 0 1
+1 +1 1
1 1 1
Z -- Z
Z -- --
Z -- Z2
D (p-wave SC)
C (d-wave SC)
DIII (p-wave TRS SC)
CI (d-wave TRS SC)
0 +1 0
0 1 0
1 +1 1
+1 1 1
Z2 Z --
-- Z --
Z2 Z2 Z
-- -- Z
BdG
IQHE
QSHE Z2TPI
polyacetylene (SSH)
p+ip SC
d+id SC
3He-B
Classification of topological insulators/superconductors
Schnyder, Ryu, AF, and Ludwig, PRB (2008)
p SC
(p+ip)x(p-ip) SC
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
period d = 2
period d = 8
Periodic table of topological insulators/superconductors
47 Ryu, Takayanagi, PRD 82, 086914 (2010) Dp-brane & Dq-brane system
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 X.-G. Wen (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)
• 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