1. Topological Insulators and Superconductors
Akira Furusaki
2012/2/8
YIPQS Symposium

2. 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)

3. Outline Topological insulators: introduction Examples:
Integer quantum Hall effect
Quantum spin Hall effect
3D Z2 topological insulator
Topological superconductor
Classification
Summary and outlook

4. 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)

5. Recent developments
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
L. Fu, C.L. Kane & E.J. Mele 2007; J. Moore & L. Balents 2007; R. Roy 2007
Z. Hasan’s group (Princeton) Bi1-xSbx
Bi2Se3 , Bi2Te3 , Bi2Tl2Se, …..

6. Topological insulators
in broader sense
　band insulators
　characterized by a topological number (Z or Z2)
Chern #, winding #, …
　gapless excitations at boundaries
free fermions (ignore e-e int.)
Topological
insulator
non-topological
(vacuum)
stable
Examples: integer quantum Hall effect, polyacetylen,
quantum spin Hall effect, 3D topological insulator, ….

7. Band insulators An electron in a periodic potential (crystal)
Bloch’s theorem Brillouin zone
empty
occupied
Band insulator
band gap

8. Energy band structure:
a mapping
Topological equivalence (adiabatic continuity)
Band structures are equivalent if they can be continuously deformed
into one another without closing the energy gap

9. Topological distinction of ground states
deformed “Hamiltonian”
m filled
bands
n empty
map from BZ to Grassmannian
IQHE (2 dim.)
homotopy class

10. Berry phase of Bloch wave function
Berry connection
Berry curvature
Berry phase
Example: 2-level Hamiltonian (spin ½ in magnetic field)

11. Integer QHE

12. Integer quantum Hall effect (von Klitzing 1980)
Quantization of Hall conductance
exact, robust against disorder etc.

13. Integer Quantum Hall Effect
(TKNN: Thouless, Kohmoto, Nightingale & den Nijs 1982)
Chern number
integer valued
= number of edge modes crossing EF
bulk-edge correspondence
filled band
Berry connection

14. Lattice model for IQHE (Haldane 1988)
Graphene: a single layer of graphite
Relativistic electrons in a pencil
Geim & Novoselov: Nobel prize 2010
B A py px E K K’
K K’ A B
Matrix element for hopping
between nearest-neighbor sites: t

15. Dirac masses Staggered site energy (G. Semenoff 1984)
Complex 2nd-nearest-neighbor hopping (Haldane 1988)
No net magnetic flux through a unit cell
Breaks inversion symmetry
Breaks time-reversal symmetry
Hall conductivity
Chern insulator

16. Massive Dirac fermion: a minimal model for IQHE
parity anomaly
(2+1)d Chern-Simons theory for EM
Domain wall fermion

17. Quantum spin Hall effect (2D Z2 topological insulator)

18. 2D Quantum spin Hall effect
Kane & Mele (2005, 2006); Bernevig & Zhang (2006)
time-reversal invariant band insulator
spin-orbit interaction
gapless helical edge mode (Kramers’ pair)
up-spin electrons
down-spin electrons
conduction band
valence band
Sz is not conserved in general.
Topological index: Z Z2

19. Quantum spin Hall insulator
Bulk energy gap & gapless edge states
Helical edge states:
Half an ordinary 1D electron gas
Protected by time reversal symmetry

20. 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
up-spin electrons
down-spin electrons

21. Effective Hamiltonian
complex 2nd nearest-neighbor hopping (Haldane)
spin-flip hopping
staggered site potential (Semenoff)
Time-reversal symmetry
Chern # = 0
complex conjugation

22. Z2 index Kane & Mele (2005); Fu & Kane (2006)
Quantum spin Hall insulator
Trivial insulator
valence band
conduction band
valence band
conduction band
an odd number
of crossing
an even number
of crossing
Bloch wave of occupied bands
Time-reversal invariant momenta:
antisymmetric
Z2 index

23. Time reversal symmetry
Time reversal operator
Kramers’ theorem
time-reversal pair
All states are doubly degenerate.

24. Z2: stability of gapless edge states
(1) A single Kramers doublet E k E k
Kramers’ theorem
stable
(2) Two Kramers doublets E k E k
Two pairs of edge states are unstable against perturbations that respect TRS.

25. Experiment HgTe/(Hg,Cd)Te quantum wells QSHI CdTe HgCdTe CdTe
Konig et al. [Science 318, 766 (2007)]
Trivial Ins.
QSHI

26. Z2 topological insulator in 3 spatial dimensions

27. 3 dimensional Topological insulator
Band insulator
Metallic surface: massless Dirac fermions
(Weyl fermions)
Z2 topologically nontrivial
Theoretical Predictions made by:
Fu, Kane, & Mele (2007)
Moore & Balents (2007)
Roy (2007)
an odd number of Dirac cones/surface

28. Surface Dirac fermions
topological
insulator
“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’

29. Experimental confirmation
Bi1-xSbx 0.09<x< 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
Other topological insulators:
Bi2Te3, Bi2Te2Se, …

30. Response to external EM field
Qi, Hughes & Zhang, 2008
Essin, Moore & Vanderbilt 2009
Integrate out electron fields to obtain effective action for the external EM field
axion electrodynamics (Wilczek, …)
time reversal
trivial insulators
topological insulators
(2+1)d Chern-Simons theory
topological insulator

31. Topological magnetoelectric effect
Magnetization induced by electric field
Polarization induced by magnetic field

32. Topological superconductors

33. 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 , 3He

34. Majorana fermion Particle that is its own anti-particle Neutrino ?
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
This happens at E=0.

35. 2D p+ip superconductor (similar to IQHE)
(px+ipy)-wave Cooper pairing
Hamiltonian for Nambu spinor (spinless case)
Majorana Weyl fermion along the edge
angular momentum =
wrapping # = 1
px+ipy
px-ipy

36. (p+ip) superconductor
Majorana zeromode in a quantum vortex
Zero-energy Majorana bound state
(p+ip) superconductor
vortex
zero mode
energy spectrum
near a vortex
Majorana fermion
If there are 2N vortices, then the ground-state degeneracy = 2N.

37. (p+ip) superconductor
interchanging vortices braid groups, non-Abelian statistics
(p+ip) superconductor i
i+1
D.A. Ivanov, PRL (2001)
topological quantum computing ?
Majorana zeromode is insensitive to external disturbance (long coherence time).

38. 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, ….)
InAs, InSb wire B
s-SC

39. Classification of topological insulators and superconductors

40. A: There are 5 classes of TPIs or TPSCs in each spatial dimension.
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

41. 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)
anti-unitary
spin 0
spin 1/2
triplet
singlet
anti-unitary

42. 10 random matrix ensembles (symmetric spaces) Altland & Zirnbauer (1997)
TRS PHS Ch
time evolution operator
IQHE
Wigner-
Dyson
Z2 TPI
chiral
px+ipy
super-
conductor
Wigner-Dyson ( ): “three-fold way” complex nuclei
Verbaarschot & others ( ) chiral phase transition in QCD
Altland-Zirnbauer (1997): “ten-fold way” mesoscopic SC systems

43. 10 random matrix ensembles (symmetric spaces) Altland & Zirnbauer (1997)
TRS PHS Ch
time evolution operator
IQHE
Wigner-
Dyson
Z2 TPI
chiral
px+ipy
super-
conductor
“Complex” cases: A & AIII
“Real” cases: the remaining 8 classes

44. 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
(with no adjustable parameter)
Z2 top. term
WZW term
bulk: d dimensions
boundary: d -1 dimensions
-term

45. NLSM topological terms
Z2: Z2 topological term can exist in d dimensions
d+1 dim. TI/TSC
Z: WZW term can exist in d-1 dimensions
d dim. TI/TSC

46. Classification of topological insulators/superconductors
Standard
(Wigner-Dyson)
A (unitary)
AI (orthogonal)
AII (symplectic)
TRS PHS CS d= d= d=3 Z
Z Z2
AIII (chiral unitary)
BDI (chiral orthogonal)
CII (chiral symplectic)
Chiral
Z Z Z
Z Z2
D (p-wave SC)
C (d-wave SC)
DIII (p-wave TRS SC)
CI (d-wave TRS SC)
Z Z Z
Z Z Z Z
BdG
IQHE
QSHE
Z2TPI
polyacetylene (SSH)
p+ip SC
p SC
d+id SC
3He-B
(p+ip)x(p-ip) SC
Schnyder, Ryu, AF, and Ludwig, PRB (2008)

47. Periodic table of topological insulators/superconductors
A. Kitaev, AIP Conf. Proc. 1134, 22 (2009); arXiv: K-theory, Bott periodicity
Ryu, Schnyder, AF, Ludwig, NJP 12, (2010) massive Dirac Hamiltonian
Ryu, Takayanagi, PRD 82, (2010) Dp-brane & Dq-brane system

48. 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?

49. 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)

50. 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