classification of topological insulators
TRANSCRIPT
Classification of
Topological Insulators
Victor Gurarie
Windsor, August 2012
1
Sunday, August 19, 12
Classes of topologically distinct Hamiltonians 2
matrix (Hamiltonian) eigenvaluesEn
En
0
En
0
smoothly deform
Two matrices are topologically equivalentif one can be deformed into another without any of its energy levels ever becoming equal to 0.
Hij
H(1)ij H(2)
ij
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Application of topological classes 3
H =X
ij
Hij a†i aj
En
0
En
0
smoothly deformH(1)ij H(2)
ij
filled states
empty states
gapless points
fill negative energy levels with fermions
negative = below the chemical potential
Sunday, August 19, 12
Topological invariants 4
Mathematical expressions which take integer values and change only if acquires a zero eigenvalue (acquires zero energy)
H
For example: N0 = # of levels below zero
very simple topological invariant
But there are many more less trivial invariants(more on that later)
Sunday, August 19, 12
Example: particle in 2D in a magnetic field 5
nx
ny Square lattice in 2D The spectrum consists of 1/q-bands (or “Landau levels”)
It is not possible to change the number of bands below 0 by smoothly changing the Hamiltonian (including by changing μ) without tuning through a point with zero energy single-particle states
(Thouless et al, 1982)
Bloch waves
topological invariant (Chern number)
En
(kx
, ky
)
H = tX
nx
,ny
ha†n
x
+1,ny
anx
,ny
+ e2⇡iqnx a†nx
,ny
+1anx
,ny
+ h.c.i�µ
X
nx
,ny
a†nx
,ny
anx
,ny
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Chern number in terms of Green’s functions 6
Gnx
,ny
= [i! �H]�1
Gab2 the basis(kx, ky) =
X
n
x
,n
y
2Bravais lattice
e�i(nx
k
x
+n
y
k
y
) G(nx
, ny
)
Chern number (an alternative form)N2 =
1
24⇡2
X
↵��
✏↵��
Zd!dk
x
dky
tr⇥G�1@
↵
GG�1@�
GG�1@�
G⇤
G ! G+ �G �N2 = 0
The only way to change N2 is by making G singular.That requires to have zero energy eigenvalues. H
α, β, γ take values ω, kx, ky
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Edge states 7
Particle hopping on a lattice with2π/3 magnetic flux through each plaquette
Periodic boundary conditions in the y-direction
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Edge states 7
Particle hopping on a lattice with2π/3 magnetic flux through each plaquette
Hard wall boundary conditions in the y-direction
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Edge states as a result of topology 8
NL2 NR
2
Zero energy states must live in the boundary
(x, y) ⇠ e
� |x|`
e
iky
y
E(ky) ⇠ ky � k0
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Other topological classes? 9
For a long time 2D particle in a magnetic field was considered to be the only example of topological classes of single-particle Hamiltonians
Generalizations to 4D, 6D, generally even d, was known, however
Nd = ��d2
�!
(2⇡i)d2+1(d+ 1)!
X
↵0↵1...↵d
✏↵0↵1...↵d
Zd!ddk tr
⇥G�1@↵0GG�1@↵1G . . .G�1@↵dG
⇤
d must be even all α take values ω, k1, k2, ..., kd
Topological classes in high dimensions - perhaps not very physical
⇡d+1 (GL(N ,C) = Zexistence of this topological invariant reflects the homotopy classif d is even
Fortunately, it turns out these are not the only topological insulatorsSunday, August 19, 12
Chiral systems 10
What if we have Hamiltonians with a special symmetry
⌃⌃† = 1
⌃H⌃† = �H⌃2 = 1It follows
t1 t2 t1
⌃n,n0 = (�1)n�nn0
H =X
n
ht1 a
†2n+1a2n + t2 a
†2n+2a2n+1 + h.c.
i
Example: systems with sublattice symmetry
=X
n1,n2
Hn1n2 a†n1an2
Ek
H =
0
BBBBBBBB@
0 t1 0 . . . 0 0t1 0 t2 . . . 0 00 t2 0 . . . 0 00 0 t1 . . . 0 0. . . . . . . . . . . . . . . . . .0 0 0 . . . 0 t10 0 0 . . . t1 0
1
CCCCCCCCA
E(k)
k
E(k) = ±q
t21 + t22 + 2t1t2 cos(k)
�⇡ ⇡Sunday, August 19, 12
Topological invariant for chiral systems 11
H =
✓0 VV † 0
◆ ✓aodd
aeven
◆basis
Nd = ��d�12
�!
(2⇡i)d+12 d!
X
↵1...↵d
✏↵1...↵d
Zddk tr
⇥V �1@↵1V . . . V �1@↵dV
⇤
General topological invariant But here d is odd
⌃ =
✓1 00 �1
◆
Sunday, August 19, 12
Topological invariant for chiral systems 11
H =
✓0 VV † 0
◆ ✓aodd
aeven
◆basis
Nd = ��d�12
�!
(2⇡i)d+12 d!
X
↵1...↵d
✏↵1...↵d
Zddk tr
⇥V �1@↵1V . . . V �1@↵dV
⇤
General topological invariant
H =
✓0 t1 + t2eik
t1 + t2e�ik 0
◆
N1 =
Z ⇡
�⇡
dk
2⇡i@k ln
�t1 + t2e
ik�
With some algebra, one can show
But here d is odd
⌃ =
✓1 00 �1
◆
Example: d=1
Sunday, August 19, 12
Topological invariant for chiral systems 11
H =
✓0 VV † 0
◆ ✓aodd
aeven
◆basis
Nd = ��d�12
�!
(2⇡i)d+12 d!
X
↵1...↵d
✏↵1...↵d
Zddk tr
⇥V �1@↵1V . . . V �1@↵dV
⇤
General topological invariant
t1 < t2N1 = 1
t1 + t2eik
H =
✓0 t1 + t2eik
t1 + t2e�ik 0
◆
N1 =
Z ⇡
�⇡
dk
2⇡i@k ln
�t1 + t2e
ik�
With some algebra, one can show
But here d is odd
⌃ =
✓1 00 �1
◆
Example: d=1
Sunday, August 19, 12
Topological invariant for chiral systems 11
H =
✓0 VV † 0
◆ ✓aodd
aeven
◆basis
Nd = ��d�12
�!
(2⇡i)d+12 d!
X
↵1...↵d
✏↵1...↵d
Zddk tr
⇥V �1@↵1V . . . V �1@↵dV
⇤
General topological invariant
t1 > t2N1 = 0
t1 + t2eik
H =
✓0 t1 + t2eik
t1 + t2e�ik 0
◆
N1 =
Z ⇡
�⇡
dk
2⇡i@k ln
�t1 + t2e
ik�
With some algebra, one can show
But here d is odd
⌃ =
✓1 00 �1
◆
Example: d=1
Sunday, August 19, 12
Edge states for 1D chiral systems 12
t1 t2 t1
H = 0
This works (decays for n>0) only if t1<t2
t1 t2
That is, if N1=1 (not when it is zero)
H =
0
BBBBBBBB@
0 t1 0 . . . 0 0t1 0 t2 . . . 0 00 t2 0 . . . 0 00 0 t1 . . . 0 0. . . . . . . . . . . . . . . . . .0 0 0 . . . 0 t10 0 0 . . . t1 0
1
CCCCCCCCA
|ψ|
1
t1* t1*t2
t2* t1 t2*
t2
tt1 2n�1 + t2 2n+1 = 0
2n+1 ⇠✓� t1t2
◆n
Sunday, August 19, 12
Symmetry classes 13
Class A (no symmetry)
Class AIII (chiral symmetry) Z
Z
Z
Z
Z
Z
1 2 3 4 5 6space dimension
Altland-Zirnbauer nomenclature
Sunday, August 19, 12
Other relevant symmetries 14
Time reversal
U †TH⇤UT = H
U †TH⇤UT = H
U †TH⇤UT = H U⇤
TUT =
⇢either +1
or �1
Particle-hole conjugation U †
CH⇤UC = �H U⇤CUC =
⇢either +1
or �1
If both symmetries are present, chiral symmetryis automatically present, with ⌃ = U⇤
TUC
7
Table 1. Listed are the ten generic symmetry classes of single-particleHamiltonians H, classified according to their behavior under time-reversalsymmetry (T ), charge-conjugation (or particle–hole) symmetry (C), as wellas ‘sublattice’ (or ‘chiral’) symmetry (S). The labels T, C and S representthe presence/absence of time-reversal, particle–hole and chiral symmetries,respectively, as well as the types of these symmetries. The column entitled‘Hamiltonian’ lists, for each of the ten symmetry classes, the symmetric space ofwhich the quantum mechanical time-evolution operator exp(itH) is an element.The column ‘Cartan label’ is the name given to the corresponding symmetricspace listed in the column ‘Hamiltonian’ in Élie Cartan’s classification scheme(dating back to the year 1926). The last column entitled ‘G/H (ferm. NL�M)’lists the (compact sectors of the) target space of the NL�M describing Andersonlocalization physics at long wavelength in this given symmetry class.Cartan label T C S Hamiltonian G/H (ferm. NL�M)
A (unitary) 0 0 0 U(N ) U(2n)/U(n) ⇥ U(n)
AI (orthogonal) +1 0 0 U(N )/O(N ) Sp(2n)/Sp(n) ⇥ Sp(n)
AII (symplectic) �1 0 0 U(2N )/Sp(2N ) O(2n)/O(n) ⇥ O(n)
AIII (ch. unit.) 0 0 1 U(N + M)/U(N ) ⇥ U(M) U(n)
BDI (ch. orth.) +1 +1 1 O(N + M)/O(N ) ⇥ O(M) U(2n)/Sp(2n)
CII (ch. sympl.) �1 �1 1 Sp(N + M)/Sp(N ) ⇥ Sp(M) U(2n)/O(2n)
D (BdG) 0 +1 0 SO(2N ) O(2n)/U(n)
C (BdG) 0 �1 0 Sp(2N ) Sp(2n)/U(n)
DIII (BdG) �1 +1 1 SO(2N )/U(N ) O(2n)
CI (BdG) +1 �1 1 Sp(2N )/U(N ) Sp(2n)
The only case when the behavior under the combined transformation S = T · C is not determinedby the behavior under T and C is the case where T = 0 and C = 0. In this case, either S = 0or S = 1 is possible. This then yields (3 ⇥ 3 � 1) + 2 = 10 possible types of behavior of theHamiltonian.
The list of ten possible types of behavior of the first quantized Hamiltonian under T , Cand S is given in table 1. These are the ten generic symmetry classes (the ‘tenfold way’),which are the framework within which the classification scheme of topological insulators(superconductors) is formulated.
Let us first point out a very general structure seen in table 1. This is listed in the columnentitled ‘Hamiltonian’. When the first quantized Hamiltonian H is ‘regularized’ (or ‘put on’)a finite lattice, it becomes an N ⇥ N matrix (as discussed above). The entries in the column‘Hamiltonian’ specify the type of N ⇥ N matrix that the quantum mechanical time-evolutionoperator exp(itH) is. For example, for systems that have no time-reversal or charge-conjugationsymmetry properties at all, i.e. for which T = 0, C = 0, S = 0, which are listed in the first rowof the table, there are no constraints on the Hamiltonian except for Hermiticity. Thus, H is ageneric Hermitian matrix and the time-evolution operator is a generic unitary matrix, so thatexp(itH) is an element of the unitary group U(N ) of unitary N ⇥ N matrices. By imposingtime-reversal symmetry (for a system that has, e.g., no other degree of freedom such as, e.g.,spin), there exists a basis in which H is represented by a real symmetric N ⇥ N matrix. This,in turn, can be expressed as saying that the time-evolution operator is an element of the coset
New Journal of Physics 12 (2010) 065010 (http://www.njp.org/)
From: Ryu, Schnyder,
Furusaki, Ludwig, 2010
This leads to 10 “symmetry classes”,
introduced by Altland and Zirnbauer
Sunday, August 19, 12
Classes with time reversal invariance only 15
Time reversal U⇤TUT =
⇢either +1
or �1
Class AI: time reversal for spinless particles or spin rotation invariant Hamiltonians
U †TH⇤(�k)UT = H(k)
Class AII: time reversal for spin-dependent spin-1/2 Hamiltonians (usually implies spin-orbit coupling)
Example: H↵�(k) =k2
2m�↵� ↵,� =", #
H↵�(k) =k2
2m�↵� + gSO
X
µ
kµ�µ↵,�
�yH⇤↵�(�k)�y = H(k) UT = �y UTU
⇤T = �1
Sunday, August 19, 12
Only time-reversal is present 16
These are classes AI, AIIG = [i! �H]�1
U †TG
TUT = G
Nd = ��d2
�!
(2⇡i)d2+1(d+ 1)!
X
↵0↵1...↵d
✏↵0↵1...↵d
Zd!ddk tr
⇥G�1@↵0GG�1@↵1G . . .G�1@↵dG
⇤Applying the symmetry to G, we can show that the invariant is identically zero if d = 2 + 4n
Consequence: no topological band structure for time reversal invariant systems in 2D.This is not quite true, however - there is a different topological invariant we haven’t yet looked at.
Class A (no symmetry)
Class AI (time reversal) Z
Z
Z
Z ZZ
1 2 3 4 5 6 7 8 space dimension
Class AII (time reversal withspin-1/2) Z Z
Green’s function transposed
GTab(k) = Gba(�k)
Sunday, August 19, 12
Only particle-hole is present 17
These are classes D, CU †CH⇤UC = �HG = [i! �H]�1
Nd = ��d2
�!
(2⇡i)d2+1(d+ 1)!
X
↵0↵1...↵d
✏↵0↵1...↵d
Zd!ddk tr
⇥G�1@↵0GG�1@↵1G . . .G�1@↵dG
⇤Applying the symmetry to G, we can show that the invariant is identically zero if
U †CG
T (!)UC = �G(�!)
d = 4n
Class A (no symmetry)
Class D (p-h, ) Z
Z
Z
Z ZZ
1 2 3 4 5 6 7 8 space dimension
Class C (p-h, ) Z Z
UCU⇤C = 1
UCU⇤C = �1
Sunday, August 19, 12
The origin of p-h symmetry 18
H =X
ij
h2hij a
†i aj +�ij a
†i a
†j +�†
ij aiaj
i=
X
ij
�a†i ai
�✓hij �ij
�†ij �hT
ij
◆✓aia†j
◆
H =
✓hij �ij
�†ij �hT
ij
◆UC
⌘ �x
This is class D
BCS superconductor
This describes Bogoliubov quasiparticles
Famous example: px+i py spin-polarized superconductor (important: it breaks time reversal)
H =
p
2
2m � µ �(px
+ ipy
)
�(px
� ipy
) � p
2
2m + µ
!
N2 = 1
This superconductor has edge states, just like a particle in a magnetic field
�x
H⇤�x
= �H
� = ��T h = h†
Ep = ±
s✓p2
2m� µ
◆2
+�2p2
H =X
µ
nµ�µ
N2 =1
8⇡
X
↵��
X
µ⌫
✏µ⌫�✏↵�
Zd2p
nµ@↵n⌫@�n�
n3
Sunday, August 19, 12
The origin of p-h symmetry 18
H =X
ij
h2hij a
†i aj +�ij a
†i a
†j +�†
ij aiaj
i=
X
ij
�a†i ai
�✓hij �ij
�†ij �hT
ij
◆✓aia†j
◆
H =
✓hij �ij
�†ij �hT
ij
◆UC
⌘ �x
This is class D
BCS superconductor
This describes Bogoliubov quasiparticles
Famous example: px+i py spin-polarized superconductor (important: it breaks time reversal)
H =
p
2
2m � µ �(px
+ ipy
)
�(px
� ipy
) � p
2
2m + µ
!
N2 = 1
This superconductor has edge states, just like a particle in a magnetic field
�x
H⇤�x
= �H
� = ��T h = h†
Ep = ±
s✓p2
2m� µ
◆2
+�2p2
H =X
µ
nµ�µ
N2 =1
8⇡
X
↵��
X
µ⌫
✏µ⌫�✏↵�
Zd2p
nµ@↵n⌫@�n�
n3
Sunday, August 19, 12
Class C 19
BCS spin-singlet superconductor
H =X
ij
2
4X
�=",#2hij a
†i�aj� +�ij(a
†i"a
†j,# � a†i,#a
†j") +�†
ij (ai,#aj," � aj,#ai,")
3
5
� = �T h = h†
= 2X
ij
⇣a†i," aj,#
⌘✓hij �ij
�†ij �hT
ij
◆✓aj,"a†j,#
◆
H =
✓hij �ij
�†ij �hT
ij
◆�yH⇤�y = �H UC = �y
UCU⇤C = �1
Example: d-wave superconductor with the order parameter dx
2�y
2 + idxy
Sunday, August 19, 12
Classes with both TR and PH 20
Automatically have chiral symmetry. Topological invariant in odd dimensional space.
Nd = ��d�12
�!
(2⇡i)d+12 d!
X
↵1...↵d
✏↵1...↵d
Zddk tr
⇥V �1@↵1V . . . V �1@↵dV
⇤
Nd = �1
2
�d�12
�!
(2⇡i)d+12 d!
X
↵1...↵d
Zddk tr
⇥⌃H�1@↵1H . . .H�1@↵dH
⇤
UCU⇤C = ✏C UTU
⇤T = ✏T
Nd = 0 if✏C✏T = 1 d = 3 + 4n✏C✏T = �1 d = 1 + 4n
Sunday, August 19, 12
Example: class DIII 21
7
Table 1. Listed are the ten generic symmetry classes of single-particleHamiltonians H, classified according to their behavior under time-reversalsymmetry (T ), charge-conjugation (or particle–hole) symmetry (C), as wellas ‘sublattice’ (or ‘chiral’) symmetry (S). The labels T, C and S representthe presence/absence of time-reversal, particle–hole and chiral symmetries,respectively, as well as the types of these symmetries. The column entitled‘Hamiltonian’ lists, for each of the ten symmetry classes, the symmetric space ofwhich the quantum mechanical time-evolution operator exp(itH) is an element.The column ‘Cartan label’ is the name given to the corresponding symmetricspace listed in the column ‘Hamiltonian’ in Élie Cartan’s classification scheme(dating back to the year 1926). The last column entitled ‘G/H (ferm. NL�M)’lists the (compact sectors of the) target space of the NL�M describing Andersonlocalization physics at long wavelength in this given symmetry class.Cartan label T C S Hamiltonian G/H (ferm. NL�M)
A (unitary) 0 0 0 U(N ) U(2n)/U(n) ⇥ U(n)
AI (orthogonal) +1 0 0 U(N )/O(N ) Sp(2n)/Sp(n) ⇥ Sp(n)
AII (symplectic) �1 0 0 U(2N )/Sp(2N ) O(2n)/O(n) ⇥ O(n)
AIII (ch. unit.) 0 0 1 U(N + M)/U(N ) ⇥ U(M) U(n)
BDI (ch. orth.) +1 +1 1 O(N + M)/O(N ) ⇥ O(M) U(2n)/Sp(2n)
CII (ch. sympl.) �1 �1 1 Sp(N + M)/Sp(N ) ⇥ Sp(M) U(2n)/O(2n)
D (BdG) 0 +1 0 SO(2N ) O(2n)/U(n)
C (BdG) 0 �1 0 Sp(2N ) Sp(2n)/U(n)
DIII (BdG) �1 +1 1 SO(2N )/U(N ) O(2n)
CI (BdG) +1 �1 1 Sp(2N )/U(N ) Sp(2n)
The only case when the behavior under the combined transformation S = T · C is not determinedby the behavior under T and C is the case where T = 0 and C = 0. In this case, either S = 0or S = 1 is possible. This then yields (3 ⇥ 3 � 1) + 2 = 10 possible types of behavior of theHamiltonian.
The list of ten possible types of behavior of the first quantized Hamiltonian under T , Cand S is given in table 1. These are the ten generic symmetry classes (the ‘tenfold way’),which are the framework within which the classification scheme of topological insulators(superconductors) is formulated.
Let us first point out a very general structure seen in table 1. This is listed in the columnentitled ‘Hamiltonian’. When the first quantized Hamiltonian H is ‘regularized’ (or ‘put on’)a finite lattice, it becomes an N ⇥ N matrix (as discussed above). The entries in the column‘Hamiltonian’ specify the type of N ⇥ N matrix that the quantum mechanical time-evolutionoperator exp(itH) is. For example, for systems that have no time-reversal or charge-conjugationsymmetry properties at all, i.e. for which T = 0, C = 0, S = 0, which are listed in the first rowof the table, there are no constraints on the Hamiltonian except for Hermiticity. Thus, H is ageneric Hermitian matrix and the time-evolution operator is a generic unitary matrix, so thatexp(itH) is an element of the unitary group U(N ) of unitary N ⇥ N matrices. By imposingtime-reversal symmetry (for a system that has, e.g., no other degree of freedom such as, e.g.,spin), there exists a basis in which H is represented by a real symmetric N ⇥ N matrix. This,in turn, can be expressed as saying that the time-evolution operator is an element of the coset
New Journal of Physics 12 (2010) 065010 (http://www.njp.org/)
UTU⇤T = �1
UCU⇤C = 1
Sunday, August 19, 12
Example: class DIII 21
7
Table 1. Listed are the ten generic symmetry classes of single-particleHamiltonians H, classified according to their behavior under time-reversalsymmetry (T ), charge-conjugation (or particle–hole) symmetry (C), as wellas ‘sublattice’ (or ‘chiral’) symmetry (S). The labels T, C and S representthe presence/absence of time-reversal, particle–hole and chiral symmetries,respectively, as well as the types of these symmetries. The column entitled‘Hamiltonian’ lists, for each of the ten symmetry classes, the symmetric space ofwhich the quantum mechanical time-evolution operator exp(itH) is an element.The column ‘Cartan label’ is the name given to the corresponding symmetricspace listed in the column ‘Hamiltonian’ in Élie Cartan’s classification scheme(dating back to the year 1926). The last column entitled ‘G/H (ferm. NL�M)’lists the (compact sectors of the) target space of the NL�M describing Andersonlocalization physics at long wavelength in this given symmetry class.Cartan label T C S Hamiltonian G/H (ferm. NL�M)
A (unitary) 0 0 0 U(N ) U(2n)/U(n) ⇥ U(n)
AI (orthogonal) +1 0 0 U(N )/O(N ) Sp(2n)/Sp(n) ⇥ Sp(n)
AII (symplectic) �1 0 0 U(2N )/Sp(2N ) O(2n)/O(n) ⇥ O(n)
AIII (ch. unit.) 0 0 1 U(N + M)/U(N ) ⇥ U(M) U(n)
BDI (ch. orth.) +1 +1 1 O(N + M)/O(N ) ⇥ O(M) U(2n)/Sp(2n)
CII (ch. sympl.) �1 �1 1 Sp(N + M)/Sp(N ) ⇥ Sp(M) U(2n)/O(2n)
D (BdG) 0 +1 0 SO(2N ) O(2n)/U(n)
C (BdG) 0 �1 0 Sp(2N ) Sp(2n)/U(n)
DIII (BdG) �1 +1 1 SO(2N )/U(N ) O(2n)
CI (BdG) +1 �1 1 Sp(2N )/U(N ) Sp(2n)
The only case when the behavior under the combined transformation S = T · C is not determinedby the behavior under T and C is the case where T = 0 and C = 0. In this case, either S = 0or S = 1 is possible. This then yields (3 ⇥ 3 � 1) + 2 = 10 possible types of behavior of theHamiltonian.
The list of ten possible types of behavior of the first quantized Hamiltonian under T , Cand S is given in table 1. These are the ten generic symmetry classes (the ‘tenfold way’),which are the framework within which the classification scheme of topological insulators(superconductors) is formulated.
Let us first point out a very general structure seen in table 1. This is listed in the columnentitled ‘Hamiltonian’. When the first quantized Hamiltonian H is ‘regularized’ (or ‘put on’)a finite lattice, it becomes an N ⇥ N matrix (as discussed above). The entries in the column‘Hamiltonian’ specify the type of N ⇥ N matrix that the quantum mechanical time-evolutionoperator exp(itH) is. For example, for systems that have no time-reversal or charge-conjugationsymmetry properties at all, i.e. for which T = 0, C = 0, S = 0, which are listed in the first rowof the table, there are no constraints on the Hamiltonian except for Hermiticity. Thus, H is ageneric Hermitian matrix and the time-evolution operator is a generic unitary matrix, so thatexp(itH) is an element of the unitary group U(N ) of unitary N ⇥ N matrices. By imposingtime-reversal symmetry (for a system that has, e.g., no other degree of freedom such as, e.g.,spin), there exists a basis in which H is represented by a real symmetric N ⇥ N matrix. This,in turn, can be expressed as saying that the time-evolution operator is an element of the coset
New Journal of Physics 12 (2010) 065010 (http://www.njp.org/)
UTU⇤T = �1
1. can be a superconductorUCU
⇤C = 1
Sunday, August 19, 12
Example: class DIII 21
7
Table 1. Listed are the ten generic symmetry classes of single-particleHamiltonians H, classified according to their behavior under time-reversalsymmetry (T ), charge-conjugation (or particle–hole) symmetry (C), as wellas ‘sublattice’ (or ‘chiral’) symmetry (S). The labels T, C and S representthe presence/absence of time-reversal, particle–hole and chiral symmetries,respectively, as well as the types of these symmetries. The column entitled‘Hamiltonian’ lists, for each of the ten symmetry classes, the symmetric space ofwhich the quantum mechanical time-evolution operator exp(itH) is an element.The column ‘Cartan label’ is the name given to the corresponding symmetricspace listed in the column ‘Hamiltonian’ in Élie Cartan’s classification scheme(dating back to the year 1926). The last column entitled ‘G/H (ferm. NL�M)’lists the (compact sectors of the) target space of the NL�M describing Andersonlocalization physics at long wavelength in this given symmetry class.Cartan label T C S Hamiltonian G/H (ferm. NL�M)
A (unitary) 0 0 0 U(N ) U(2n)/U(n) ⇥ U(n)
AI (orthogonal) +1 0 0 U(N )/O(N ) Sp(2n)/Sp(n) ⇥ Sp(n)
AII (symplectic) �1 0 0 U(2N )/Sp(2N ) O(2n)/O(n) ⇥ O(n)
AIII (ch. unit.) 0 0 1 U(N + M)/U(N ) ⇥ U(M) U(n)
BDI (ch. orth.) +1 +1 1 O(N + M)/O(N ) ⇥ O(M) U(2n)/Sp(2n)
CII (ch. sympl.) �1 �1 1 Sp(N + M)/Sp(N ) ⇥ Sp(M) U(2n)/O(2n)
D (BdG) 0 +1 0 SO(2N ) O(2n)/U(n)
C (BdG) 0 �1 0 Sp(2N ) Sp(2n)/U(n)
DIII (BdG) �1 +1 1 SO(2N )/U(N ) O(2n)
CI (BdG) +1 �1 1 Sp(2N )/U(N ) Sp(2n)
The only case when the behavior under the combined transformation S = T · C is not determinedby the behavior under T and C is the case where T = 0 and C = 0. In this case, either S = 0or S = 1 is possible. This then yields (3 ⇥ 3 � 1) + 2 = 10 possible types of behavior of theHamiltonian.
The list of ten possible types of behavior of the first quantized Hamiltonian under T , Cand S is given in table 1. These are the ten generic symmetry classes (the ‘tenfold way’),which are the framework within which the classification scheme of topological insulators(superconductors) is formulated.
Let us first point out a very general structure seen in table 1. This is listed in the columnentitled ‘Hamiltonian’. When the first quantized Hamiltonian H is ‘regularized’ (or ‘put on’)a finite lattice, it becomes an N ⇥ N matrix (as discussed above). The entries in the column‘Hamiltonian’ specify the type of N ⇥ N matrix that the quantum mechanical time-evolutionoperator exp(itH) is. For example, for systems that have no time-reversal or charge-conjugationsymmetry properties at all, i.e. for which T = 0, C = 0, S = 0, which are listed in the first rowof the table, there are no constraints on the Hamiltonian except for Hermiticity. Thus, H is ageneric Hermitian matrix and the time-evolution operator is a generic unitary matrix, so thatexp(itH) is an element of the unitary group U(N ) of unitary N ⇥ N matrices. By imposingtime-reversal symmetry (for a system that has, e.g., no other degree of freedom such as, e.g.,spin), there exists a basis in which H is represented by a real symmetric N ⇥ N matrix. This,in turn, can be expressed as saying that the time-evolution operator is an element of the coset
New Journal of Physics 12 (2010) 065010 (http://www.njp.org/)
UTU⇤T = �1
1. can be a superconductor2. has to be a spin-triplet (p-wave)superconductor
UCU⇤C = 1
Sunday, August 19, 12
Example: class DIII 21
7
Table 1. Listed are the ten generic symmetry classes of single-particleHamiltonians H, classified according to their behavior under time-reversalsymmetry (T ), charge-conjugation (or particle–hole) symmetry (C), as wellas ‘sublattice’ (or ‘chiral’) symmetry (S). The labels T, C and S representthe presence/absence of time-reversal, particle–hole and chiral symmetries,respectively, as well as the types of these symmetries. The column entitled‘Hamiltonian’ lists, for each of the ten symmetry classes, the symmetric space ofwhich the quantum mechanical time-evolution operator exp(itH) is an element.The column ‘Cartan label’ is the name given to the corresponding symmetricspace listed in the column ‘Hamiltonian’ in Élie Cartan’s classification scheme(dating back to the year 1926). The last column entitled ‘G/H (ferm. NL�M)’lists the (compact sectors of the) target space of the NL�M describing Andersonlocalization physics at long wavelength in this given symmetry class.Cartan label T C S Hamiltonian G/H (ferm. NL�M)
A (unitary) 0 0 0 U(N ) U(2n)/U(n) ⇥ U(n)
AI (orthogonal) +1 0 0 U(N )/O(N ) Sp(2n)/Sp(n) ⇥ Sp(n)
AII (symplectic) �1 0 0 U(2N )/Sp(2N ) O(2n)/O(n) ⇥ O(n)
AIII (ch. unit.) 0 0 1 U(N + M)/U(N ) ⇥ U(M) U(n)
BDI (ch. orth.) +1 +1 1 O(N + M)/O(N ) ⇥ O(M) U(2n)/Sp(2n)
CII (ch. sympl.) �1 �1 1 Sp(N + M)/Sp(N ) ⇥ Sp(M) U(2n)/O(2n)
D (BdG) 0 +1 0 SO(2N ) O(2n)/U(n)
C (BdG) 0 �1 0 Sp(2N ) Sp(2n)/U(n)
DIII (BdG) �1 +1 1 SO(2N )/U(N ) O(2n)
CI (BdG) +1 �1 1 Sp(2N )/U(N ) Sp(2n)
The only case when the behavior under the combined transformation S = T · C is not determinedby the behavior under T and C is the case where T = 0 and C = 0. In this case, either S = 0or S = 1 is possible. This then yields (3 ⇥ 3 � 1) + 2 = 10 possible types of behavior of theHamiltonian.
The list of ten possible types of behavior of the first quantized Hamiltonian under T , Cand S is given in table 1. These are the ten generic symmetry classes (the ‘tenfold way’),which are the framework within which the classification scheme of topological insulators(superconductors) is formulated.
Let us first point out a very general structure seen in table 1. This is listed in the columnentitled ‘Hamiltonian’. When the first quantized Hamiltonian H is ‘regularized’ (or ‘put on’)a finite lattice, it becomes an N ⇥ N matrix (as discussed above). The entries in the column‘Hamiltonian’ specify the type of N ⇥ N matrix that the quantum mechanical time-evolutionoperator exp(itH) is. For example, for systems that have no time-reversal or charge-conjugationsymmetry properties at all, i.e. for which T = 0, C = 0, S = 0, which are listed in the first rowof the table, there are no constraints on the Hamiltonian except for Hermiticity. Thus, H is ageneric Hermitian matrix and the time-evolution operator is a generic unitary matrix, so thatexp(itH) is an element of the unitary group U(N ) of unitary N ⇥ N matrices. By imposingtime-reversal symmetry (for a system that has, e.g., no other degree of freedom such as, e.g.,spin), there exists a basis in which H is represented by a real symmetric N ⇥ N matrix. This,in turn, can be expressed as saying that the time-evolution operator is an element of the coset
New Journal of Physics 12 (2010) 065010 (http://www.njp.org/)
UTU⇤T = �1
1. can be a superconductor2. has to be a spin-triplet (p-wave)superconductor3. has to have spin-orbit coupling
UCU⇤C = 1
Sunday, August 19, 12
Example: class DIII 21
7
Table 1. Listed are the ten generic symmetry classes of single-particleHamiltonians H, classified according to their behavior under time-reversalsymmetry (T ), charge-conjugation (or particle–hole) symmetry (C), as wellas ‘sublattice’ (or ‘chiral’) symmetry (S). The labels T, C and S representthe presence/absence of time-reversal, particle–hole and chiral symmetries,respectively, as well as the types of these symmetries. The column entitled‘Hamiltonian’ lists, for each of the ten symmetry classes, the symmetric space ofwhich the quantum mechanical time-evolution operator exp(itH) is an element.The column ‘Cartan label’ is the name given to the corresponding symmetricspace listed in the column ‘Hamiltonian’ in Élie Cartan’s classification scheme(dating back to the year 1926). The last column entitled ‘G/H (ferm. NL�M)’lists the (compact sectors of the) target space of the NL�M describing Andersonlocalization physics at long wavelength in this given symmetry class.Cartan label T C S Hamiltonian G/H (ferm. NL�M)
A (unitary) 0 0 0 U(N ) U(2n)/U(n) ⇥ U(n)
AI (orthogonal) +1 0 0 U(N )/O(N ) Sp(2n)/Sp(n) ⇥ Sp(n)
AII (symplectic) �1 0 0 U(2N )/Sp(2N ) O(2n)/O(n) ⇥ O(n)
AIII (ch. unit.) 0 0 1 U(N + M)/U(N ) ⇥ U(M) U(n)
BDI (ch. orth.) +1 +1 1 O(N + M)/O(N ) ⇥ O(M) U(2n)/Sp(2n)
CII (ch. sympl.) �1 �1 1 Sp(N + M)/Sp(N ) ⇥ Sp(M) U(2n)/O(2n)
D (BdG) 0 +1 0 SO(2N ) O(2n)/U(n)
C (BdG) 0 �1 0 Sp(2N ) Sp(2n)/U(n)
DIII (BdG) �1 +1 1 SO(2N )/U(N ) O(2n)
CI (BdG) +1 �1 1 Sp(2N )/U(N ) Sp(2n)
The only case when the behavior under the combined transformation S = T · C is not determinedby the behavior under T and C is the case where T = 0 and C = 0. In this case, either S = 0or S = 1 is possible. This then yields (3 ⇥ 3 � 1) + 2 = 10 possible types of behavior of theHamiltonian.
The list of ten possible types of behavior of the first quantized Hamiltonian under T , Cand S is given in table 1. These are the ten generic symmetry classes (the ‘tenfold way’),which are the framework within which the classification scheme of topological insulators(superconductors) is formulated.
Let us first point out a very general structure seen in table 1. This is listed in the columnentitled ‘Hamiltonian’. When the first quantized Hamiltonian H is ‘regularized’ (or ‘put on’)a finite lattice, it becomes an N ⇥ N matrix (as discussed above). The entries in the column‘Hamiltonian’ specify the type of N ⇥ N matrix that the quantum mechanical time-evolutionoperator exp(itH) is. For example, for systems that have no time-reversal or charge-conjugationsymmetry properties at all, i.e. for which T = 0, C = 0, S = 0, which are listed in the first rowof the table, there are no constraints on the Hamiltonian except for Hermiticity. Thus, H is ageneric Hermitian matrix and the time-evolution operator is a generic unitary matrix, so thatexp(itH) is an element of the unitary group U(N ) of unitary N ⇥ N matrices. By imposingtime-reversal symmetry (for a system that has, e.g., no other degree of freedom such as, e.g.,spin), there exists a basis in which H is represented by a real symmetric N ⇥ N matrix. This,in turn, can be expressed as saying that the time-evolution operator is an element of the coset
New Journal of Physics 12 (2010) 065010 (http://www.njp.org/)
UTU⇤T = �1
1. can be a superconductor2. has to be a spin-triplet (p-wave)superconductor3. has to have spin-orbit coupling
UCU⇤C = 1
This is 3He phase B.H =
X
p,↵=",#
✓p2
2m� µ
◆a†p↵ap↵+�
X
p,↵,�,�
pµ �y↵��
µ�� ap↵a�p�+�
X
p,↵,�,�
pµ �µ↵��
y�� a
†�p↵a
†p�
H =
0
@12
⇣p2
2m � µ⌘�↵� �
P�µ pµ�
µ↵��
y��
�P
�µ pµ�y↵��
µ�� � 1
2
⇣p2
2m � µ⌘�↵�
1
Athis must be chirally symmetric.Its invariant is N3=1.
3He is topological and has edge states(discovered only in ~2008 by Ludwig et al)
Sunday, August 19, 12
Another example of a 3D DIII insulator 22
t+ tB
�x
t� tB
�x
t+ tB
�x
t�tB�
y
t�tB�
y t+ tB�
y
t+tB�
y
t+t B
�z
t�t B
�zt+t B
�z
t+t B
�z
sublattice Asublattice B
N3 = 1
Sunday, August 19, 12
Example: Class CI 23
UTU⇤T = 1
UCU⇤C = �1
Spin-singlet time-reversal invariant superconductor
This is a conventional s-wave spin-singlet superconductor.
Can be topological in 3D
Conventional superconductors are not topological,but an example of a 3D CI topological superconductor is known (Ludwig et al)
Sunday, August 19, 12
Full classification table 24
11
Table 3. Classification of topological insulators and superconductors as afunction of spatial dimension d and symmetry class, indicated by the ‘Cartanlabel’ (first column). The definition of the ten generic symmetry classes of singleparticle Hamiltonians (due to Altland and Zirnbauer [29, 30]) is given in table 1.The symmetry classes are grouped into two separate lists, the complex andreal cases, depending on whether the Hamiltonian is complex or whether one(or more) reality conditions (arising from time-reversal or charge-conjugationsymmetries) are imposed on it; the symmetry classes are ordered in such a waythat a periodic pattern in dimensionality becomes visible [27]. (See also thediscussion in subsection 1.1 and table 2.) The symbols Z and Z2 indicate thatthe topologically distinct phases within a given symmetry class of topologicalinsulators (superconductors) are characterized by an integer invariant (Z) or aZ2 quantity, respectively. The symbol ‘0’ denotes the case when there exists notopological insulator (superconductor), i.e. when all quantum ground states aretopologically equivalent to the trivial state.
dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .
Complex case:A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0 . . .AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z . . .
Real case:AI Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 . . .BDI Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 . . .D Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 . . .DIII 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z . . .AII 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 . . .CII 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 . . .C 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 . . .CI 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z . . .
dimensions if and only if the target space of the NL�M on the d-dimensional boundary allowsfor either (i) a Z2 topological term, which is the case when ⇡d(G/H) = ⇡d�1(G/H) = Z2, or(ii) a WZW term, which is the case when ⇡d(G/H) = ⇡d+1(G/H) = Z. By using this rule inconjunction with table 2 of homotopy groups, we arrive with the help of table 1 at table 3 oftopological insulators and superconductors24.
A look at table 3 reveals that in each spatial dimension there exist five distinct classes oftopological insulators (superconductors), three of which are characterized by an integral (Z)topological number, while the remaining two possess a binary (Z2) topological quantity25.
24 To be explicit, one has to move all the entries Z in table 2 into the locations indicated by the arrows, and replacethe column label d by d = d + 1. The result is table 3.25 Note that while d = 0, 1, 2, 3-dimensional systems are of direct physical relevance, higher-dimensionaltopological states might be of interest indirectly, because, for example, some of the additional components ofmomentum in a higher-dimensional space may be interpreted as adiabatic parameters (external parameters onwhich the Hamiltonian depends and which can be changed adiabatically, traversing closed paths in parameterspace—sometimes referred to as adiabatic ‘pumping processes’).
New Journal of Physics 12 (2010) 065010 (http://www.njp.org/)
space dimensionality
symmetryclasses Kitaev, 2009;
Ludwig, Ryu, Schnyder, Furusaki, 2009.
Table from Ryu, Schnyder, Furusaki, Ludwig, 2010
White - nonchiralGrey - chiral
Sunday, August 19, 12
Full classification table 24
11
Table 3. Classification of topological insulators and superconductors as afunction of spatial dimension d and symmetry class, indicated by the ‘Cartanlabel’ (first column). The definition of the ten generic symmetry classes of singleparticle Hamiltonians (due to Altland and Zirnbauer [29, 30]) is given in table 1.The symmetry classes are grouped into two separate lists, the complex andreal cases, depending on whether the Hamiltonian is complex or whether one(or more) reality conditions (arising from time-reversal or charge-conjugationsymmetries) are imposed on it; the symmetry classes are ordered in such a waythat a periodic pattern in dimensionality becomes visible [27]. (See also thediscussion in subsection 1.1 and table 2.) The symbols Z and Z2 indicate thatthe topologically distinct phases within a given symmetry class of topologicalinsulators (superconductors) are characterized by an integer invariant (Z) or aZ2 quantity, respectively. The symbol ‘0’ denotes the case when there exists notopological insulator (superconductor), i.e. when all quantum ground states aretopologically equivalent to the trivial state.
dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .
Complex case:A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0 . . .AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z . . .
Real case:AI Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 . . .BDI Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 . . .D Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 . . .DIII 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z . . .AII 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 . . .CII 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 . . .C 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 . . .CI 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z . . .
dimensions if and only if the target space of the NL�M on the d-dimensional boundary allowsfor either (i) a Z2 topological term, which is the case when ⇡d(G/H) = ⇡d�1(G/H) = Z2, or(ii) a WZW term, which is the case when ⇡d(G/H) = ⇡d+1(G/H) = Z. By using this rule inconjunction with table 2 of homotopy groups, we arrive with the help of table 1 at table 3 oftopological insulators and superconductors24.
A look at table 3 reveals that in each spatial dimension there exist five distinct classes oftopological insulators (superconductors), three of which are characterized by an integral (Z)topological number, while the remaining two possess a binary (Z2) topological quantity25.
24 To be explicit, one has to move all the entries Z in table 2 into the locations indicated by the arrows, and replacethe column label d by d = d + 1. The result is table 3.25 Note that while d = 0, 1, 2, 3-dimensional systems are of direct physical relevance, higher-dimensionaltopological states might be of interest indirectly, because, for example, some of the additional components ofmomentum in a higher-dimensional space may be interpreted as adiabatic parameters (external parameters onwhich the Hamiltonian depends and which can be changed adiabatically, traversing closed paths in parameterspace—sometimes referred to as adiabatic ‘pumping processes’).
New Journal of Physics 12 (2010) 065010 (http://www.njp.org/)
IQHE
space dimensionality
symmetryclasses Kitaev, 2009;
Ludwig, Ryu, Schnyder, Furusaki, 2009.
Table from Ryu, Schnyder, Furusaki, Ludwig, 2010
White - nonchiralGrey - chiral
Sunday, August 19, 12
Full classification table 24
11
Table 3. Classification of topological insulators and superconductors as afunction of spatial dimension d and symmetry class, indicated by the ‘Cartanlabel’ (first column). The definition of the ten generic symmetry classes of singleparticle Hamiltonians (due to Altland and Zirnbauer [29, 30]) is given in table 1.The symmetry classes are grouped into two separate lists, the complex andreal cases, depending on whether the Hamiltonian is complex or whether one(or more) reality conditions (arising from time-reversal or charge-conjugationsymmetries) are imposed on it; the symmetry classes are ordered in such a waythat a periodic pattern in dimensionality becomes visible [27]. (See also thediscussion in subsection 1.1 and table 2.) The symbols Z and Z2 indicate thatthe topologically distinct phases within a given symmetry class of topologicalinsulators (superconductors) are characterized by an integer invariant (Z) or aZ2 quantity, respectively. The symbol ‘0’ denotes the case when there exists notopological insulator (superconductor), i.e. when all quantum ground states aretopologically equivalent to the trivial state.
dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .
Complex case:A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0 . . .AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z . . .
Real case:AI Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 . . .BDI Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 . . .D Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 . . .DIII 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z . . .AII 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 . . .CII 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 . . .C 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 . . .CI 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z . . .
dimensions if and only if the target space of the NL�M on the d-dimensional boundary allowsfor either (i) a Z2 topological term, which is the case when ⇡d(G/H) = ⇡d�1(G/H) = Z2, or(ii) a WZW term, which is the case when ⇡d(G/H) = ⇡d+1(G/H) = Z. By using this rule inconjunction with table 2 of homotopy groups, we arrive with the help of table 1 at table 3 oftopological insulators and superconductors24.
A look at table 3 reveals that in each spatial dimension there exist five distinct classes oftopological insulators (superconductors), three of which are characterized by an integral (Z)topological number, while the remaining two possess a binary (Z2) topological quantity25.
24 To be explicit, one has to move all the entries Z in table 2 into the locations indicated by the arrows, and replacethe column label d by d = d + 1. The result is table 3.25 Note that while d = 0, 1, 2, 3-dimensional systems are of direct physical relevance, higher-dimensionaltopological states might be of interest indirectly, because, for example, some of the additional components ofmomentum in a higher-dimensional space may be interpreted as adiabatic parameters (external parameters onwhich the Hamiltonian depends and which can be changed adiabatically, traversing closed paths in parameterspace—sometimes referred to as adiabatic ‘pumping processes’).
New Journal of Physics 12 (2010) 065010 (http://www.njp.org/)
IQHESu,
Schrieffer,Heeger
space dimensionality
symmetryclasses Kitaev, 2009;
Ludwig, Ryu, Schnyder, Furusaki, 2009.
Table from Ryu, Schnyder, Furusaki, Ludwig, 2010
White - nonchiralGrey - chiral
Sunday, August 19, 12
Full classification table 24
11
Table 3. Classification of topological insulators and superconductors as afunction of spatial dimension d and symmetry class, indicated by the ‘Cartanlabel’ (first column). The definition of the ten generic symmetry classes of singleparticle Hamiltonians (due to Altland and Zirnbauer [29, 30]) is given in table 1.The symmetry classes are grouped into two separate lists, the complex andreal cases, depending on whether the Hamiltonian is complex or whether one(or more) reality conditions (arising from time-reversal or charge-conjugationsymmetries) are imposed on it; the symmetry classes are ordered in such a waythat a periodic pattern in dimensionality becomes visible [27]. (See also thediscussion in subsection 1.1 and table 2.) The symbols Z and Z2 indicate thatthe topologically distinct phases within a given symmetry class of topologicalinsulators (superconductors) are characterized by an integer invariant (Z) or aZ2 quantity, respectively. The symbol ‘0’ denotes the case when there exists notopological insulator (superconductor), i.e. when all quantum ground states aretopologically equivalent to the trivial state.
dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .
Complex case:A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0 . . .AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z . . .
Real case:AI Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 . . .BDI Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 . . .D Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 . . .DIII 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z . . .AII 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 . . .CII 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 . . .C 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 . . .CI 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z . . .
dimensions if and only if the target space of the NL�M on the d-dimensional boundary allowsfor either (i) a Z2 topological term, which is the case when ⇡d(G/H) = ⇡d�1(G/H) = Z2, or(ii) a WZW term, which is the case when ⇡d(G/H) = ⇡d+1(G/H) = Z. By using this rule inconjunction with table 2 of homotopy groups, we arrive with the help of table 1 at table 3 oftopological insulators and superconductors24.
A look at table 3 reveals that in each spatial dimension there exist five distinct classes oftopological insulators (superconductors), three of which are characterized by an integral (Z)topological number, while the remaining two possess a binary (Z2) topological quantity25.
24 To be explicit, one has to move all the entries Z in table 2 into the locations indicated by the arrows, and replacethe column label d by d = d + 1. The result is table 3.25 Note that while d = 0, 1, 2, 3-dimensional systems are of direct physical relevance, higher-dimensionaltopological states might be of interest indirectly, because, for example, some of the additional components ofmomentum in a higher-dimensional space may be interpreted as adiabatic parameters (external parameters onwhich the Hamiltonian depends and which can be changed adiabatically, traversing closed paths in parameterspace—sometimes referred to as adiabatic ‘pumping processes’).
New Journal of Physics 12 (2010) 065010 (http://www.njp.org/)
IQHESu,
Schrieffer,Heeger
2D p-wave supercond
uctor
space dimensionality
symmetryclasses Kitaev, 2009;
Ludwig, Ryu, Schnyder, Furusaki, 2009.
Table from Ryu, Schnyder, Furusaki, Ludwig, 2010
White - nonchiralGrey - chiral
Sunday, August 19, 12
Full classification table 24
11
Table 3. Classification of topological insulators and superconductors as afunction of spatial dimension d and symmetry class, indicated by the ‘Cartanlabel’ (first column). The definition of the ten generic symmetry classes of singleparticle Hamiltonians (due to Altland and Zirnbauer [29, 30]) is given in table 1.The symmetry classes are grouped into two separate lists, the complex andreal cases, depending on whether the Hamiltonian is complex or whether one(or more) reality conditions (arising from time-reversal or charge-conjugationsymmetries) are imposed on it; the symmetry classes are ordered in such a waythat a periodic pattern in dimensionality becomes visible [27]. (See also thediscussion in subsection 1.1 and table 2.) The symbols Z and Z2 indicate thatthe topologically distinct phases within a given symmetry class of topologicalinsulators (superconductors) are characterized by an integer invariant (Z) or aZ2 quantity, respectively. The symbol ‘0’ denotes the case when there exists notopological insulator (superconductor), i.e. when all quantum ground states aretopologically equivalent to the trivial state.
dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .
Complex case:A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0 . . .AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z . . .
Real case:AI Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 . . .BDI Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 . . .D Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 . . .DIII 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z . . .AII 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 . . .CII 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 . . .C 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 . . .CI 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z . . .
dimensions if and only if the target space of the NL�M on the d-dimensional boundary allowsfor either (i) a Z2 topological term, which is the case when ⇡d(G/H) = ⇡d�1(G/H) = Z2, or(ii) a WZW term, which is the case when ⇡d(G/H) = ⇡d+1(G/H) = Z. By using this rule inconjunction with table 2 of homotopy groups, we arrive with the help of table 1 at table 3 oftopological insulators and superconductors24.
A look at table 3 reveals that in each spatial dimension there exist five distinct classes oftopological insulators (superconductors), three of which are characterized by an integral (Z)topological number, while the remaining two possess a binary (Z2) topological quantity25.
24 To be explicit, one has to move all the entries Z in table 2 into the locations indicated by the arrows, and replacethe column label d by d = d + 1. The result is table 3.25 Note that while d = 0, 1, 2, 3-dimensional systems are of direct physical relevance, higher-dimensionaltopological states might be of interest indirectly, because, for example, some of the additional components ofmomentum in a higher-dimensional space may be interpreted as adiabatic parameters (external parameters onwhich the Hamiltonian depends and which can be changed adiabatically, traversing closed paths in parameterspace—sometimes referred to as adiabatic ‘pumping processes’).
New Journal of Physics 12 (2010) 065010 (http://www.njp.org/)
IQHESu,
Schrieffer,Heeger
2D p-wave supercond
uctor
3He, phase B
space dimensionality
symmetryclasses Kitaev, 2009;
Ludwig, Ryu, Schnyder, Furusaki, 2009.
Table from Ryu, Schnyder, Furusaki, Ludwig, 2010
White - nonchiralGrey - chiral
Sunday, August 19, 12
Full classification table 24
11
Table 3. Classification of topological insulators and superconductors as afunction of spatial dimension d and symmetry class, indicated by the ‘Cartanlabel’ (first column). The definition of the ten generic symmetry classes of singleparticle Hamiltonians (due to Altland and Zirnbauer [29, 30]) is given in table 1.The symmetry classes are grouped into two separate lists, the complex andreal cases, depending on whether the Hamiltonian is complex or whether one(or more) reality conditions (arising from time-reversal or charge-conjugationsymmetries) are imposed on it; the symmetry classes are ordered in such a waythat a periodic pattern in dimensionality becomes visible [27]. (See also thediscussion in subsection 1.1 and table 2.) The symbols Z and Z2 indicate thatthe topologically distinct phases within a given symmetry class of topologicalinsulators (superconductors) are characterized by an integer invariant (Z) or aZ2 quantity, respectively. The symbol ‘0’ denotes the case when there exists notopological insulator (superconductor), i.e. when all quantum ground states aretopologically equivalent to the trivial state.
dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .
Complex case:A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0 . . .AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z . . .
Real case:AI Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 . . .BDI Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 . . .D Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 . . .DIII 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z . . .AII 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 . . .CII 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 . . .C 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 . . .CI 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z . . .
dimensions if and only if the target space of the NL�M on the d-dimensional boundary allowsfor either (i) a Z2 topological term, which is the case when ⇡d(G/H) = ⇡d�1(G/H) = Z2, or(ii) a WZW term, which is the case when ⇡d(G/H) = ⇡d+1(G/H) = Z. By using this rule inconjunction with table 2 of homotopy groups, we arrive with the help of table 1 at table 3 oftopological insulators and superconductors24.
A look at table 3 reveals that in each spatial dimension there exist five distinct classes oftopological insulators (superconductors), three of which are characterized by an integral (Z)topological number, while the remaining two possess a binary (Z2) topological quantity25.
24 To be explicit, one has to move all the entries Z in table 2 into the locations indicated by the arrows, and replacethe column label d by d = d + 1. The result is table 3.25 Note that while d = 0, 1, 2, 3-dimensional systems are of direct physical relevance, higher-dimensionaltopological states might be of interest indirectly, because, for example, some of the additional components ofmomentum in a higher-dimensional space may be interpreted as adiabatic parameters (external parameters onwhich the Hamiltonian depends and which can be changed adiabatically, traversing closed paths in parameterspace—sometimes referred to as adiabatic ‘pumping processes’).
New Journal of Physics 12 (2010) 065010 (http://www.njp.org/)
IQHESu,
Schrieffer,Heeger
2D p-wave supercond
uctor
3He, phase B
space dimensionality
symmetryclasses Kitaev, 2009;
Ludwig, Ryu, Schnyder, Furusaki, 2009.
Table from Ryu, Schnyder, Furusaki, Ludwig, 2010
New Kane-Mele topological insulators
White - nonchiralGrey - chiral
Sunday, August 19, 12
New crucial feature - Z2 invariant 25
Take class AII: time reversal invariance with spin-1/2 (with spin-orbit coupling)
In 4D it can be topological
Its 3D boundary has gapless excitations. These generally form a Fermi spheres.
U †TG
T (!,�k)UT = G(!,k)
px
pyq
3D boundary of a 4D insulator
TR invariancerelates p,q and -p,-q
Fermi surfaces at the edge
Declare q “unphysical” and reduce dimensions to 3D insulator with
2D boundary
Gphys(!, px, py) = G(!, px
, py
, q)|q=0
If the number of Fermi spheres wasodd, the physical 3D insulator has
gapless excitations.Otherwise, it does not.
Z2Sunday, August 19, 12
Full classification table 26
11
Table 3. Classification of topological insulators and superconductors as afunction of spatial dimension d and symmetry class, indicated by the ‘Cartanlabel’ (first column). The definition of the ten generic symmetry classes of singleparticle Hamiltonians (due to Altland and Zirnbauer [29, 30]) is given in table 1.The symmetry classes are grouped into two separate lists, the complex andreal cases, depending on whether the Hamiltonian is complex or whether one(or more) reality conditions (arising from time-reversal or charge-conjugationsymmetries) are imposed on it; the symmetry classes are ordered in such a waythat a periodic pattern in dimensionality becomes visible [27]. (See also thediscussion in subsection 1.1 and table 2.) The symbols Z and Z2 indicate thatthe topologically distinct phases within a given symmetry class of topologicalinsulators (superconductors) are characterized by an integer invariant (Z) or aZ2 quantity, respectively. The symbol ‘0’ denotes the case when there exists notopological insulator (superconductor), i.e. when all quantum ground states aretopologically equivalent to the trivial state.
dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .
Complex case:A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0 . . .AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z . . .
Real case:AI Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 . . .BDI Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 . . .D Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 . . .DIII 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z . . .AII 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 . . .CII 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 . . .C 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 . . .CI 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z . . .
dimensions if and only if the target space of the NL�M on the d-dimensional boundary allowsfor either (i) a Z2 topological term, which is the case when ⇡d(G/H) = ⇡d�1(G/H) = Z2, or(ii) a WZW term, which is the case when ⇡d(G/H) = ⇡d+1(G/H) = Z. By using this rule inconjunction with table 2 of homotopy groups, we arrive with the help of table 1 at table 3 oftopological insulators and superconductors24.
A look at table 3 reveals that in each spatial dimension there exist five distinct classes oftopological insulators (superconductors), three of which are characterized by an integral (Z)topological number, while the remaining two possess a binary (Z2) topological quantity25.
24 To be explicit, one has to move all the entries Z in table 2 into the locations indicated by the arrows, and replacethe column label d by d = d + 1. The result is table 3.25 Note that while d = 0, 1, 2, 3-dimensional systems are of direct physical relevance, higher-dimensionaltopological states might be of interest indirectly, because, for example, some of the additional components ofmomentum in a higher-dimensional space may be interpreted as adiabatic parameters (external parameters onwhich the Hamiltonian depends and which can be changed adiabatically, traversing closed paths in parameterspace—sometimes referred to as adiabatic ‘pumping processes’).
New Journal of Physics 12 (2010) 065010 (http://www.njp.org/)
space dimensionality
symmetryclasses
New Kane-Mele topological insulators
White - nonchiralGrey - chiral
Sunday, August 19, 12
Edge excitations 27
Take nonchiral insulator in d-dimensions (d even)It has a d-1 dimensional edge with gapless excitations
Here α go over !, k1, . . . , kd�1
n↵0 = ��d2 � 1
�!
(2⇡i)d2 (d� 1)!
X
↵1...↵d�1
✏↵0↵1...↵d�1
Zd!dd�2k tr
⇥G�1@↵1GG�1@↵2G . . .G�1@↵d�1G
⇤
X
↵
@↵n↵ = 0 except where G is singular or where there are zero-energy excitations
Nd =
Ids↵n↵
this integral must be equal to the bulk topological invariant of the insulator
similarly for d odd and chiral insulators
Sunday, August 19, 12
Example: quantum hall edge 28
Bn↵0 =
1
2⇡i
X
↵1
✏↵0↵1 tr⇥G�1@↵1G
⇤
kk
!
Ids↵n↵ = N0(⇤)�N0(�⇤)
N0(k) = tr
Zd!
2⇡iG�1@!G =
X
n
Zd!
2⇡i@! ln
✓1
i! � ✏n(k)
◆=
1
2
X
n
sign ✏n(k)
⇤�⇤
�⇤⇤ k
chiral edge state
✏n
Sunday, August 19, 12
Example: an edge of a 3D DIII insulator 29
This is 3He
H = �xkx
+ �yky
�x
H⇤(�k)�x
= �H(k) p.h.�yH⇤(�k)�y = H(k) t.r.
Z X
↵=x,y
ds↵ n↵
= 1.
�zH(k)�z = H(k) chiral
Sunday, August 19, 12
Example: an edge of a 3D DIII insulator 29
This is 3He
H = �xkx
+ �yky
�x
H⇤(�k)�x
= �H(k) p.h.�yH⇤(�k)�y = H(k) t.r.
Z X
↵=x,y
ds↵ n↵
= 1.
�zH(k)�z = H(k) chiral
H =
✓0 k
x
� iky
kx
+ iky
0
◆ Zds↵n
↵
=1
2⇡i
Idkµ@
kµ ln (kx
� iky
)
V = kx
� iky
Sunday, August 19, 12
Edge theory of AII 3D topological insulators 30
Literature states that its 2D edge is a Dirac fermion theory
because it is1. linear in momenta2. time-reversal invariantH(p) = �yH
⇤(�p)�y
But does it have the right edge invariant?
H = v (�x
px
+ �y
py
)� µ
Sunday, August 19, 12
Edge theory of AII 3D topological insulators 30
Literature states that its 2D edge is a Dirac fermion theory
because it is1. linear in momenta2. time-reversal invariantH(p) = �yH
⇤(�p)�y
But does it have the right edge invariant?
H = v (�x
px
+ �y
py
)� µ
H = v (�x
px
+ �y
py
+ �z
q)� µEnlarge dimensions to 4D (3D edge)
Sunday, August 19, 12
Edge theory of AII 3D topological insulators 30
Literature states that its 2D edge is a Dirac fermion theory
because it is1. linear in momenta2. time-reversal invariantH(p) = �yH
⇤(�p)�y
But does it have the right edge invariant?
H = v�x
px
+ v�y
py
± v��z
� µ
Fix q=+Λ or q=-Λ
Effectively 2D.
H = v (�x
px
+ �y
py
)� µ
q
H = v (�x
px
+ �y
py
+ �z
q)� µEnlarge dimensions to 4D (3D edge)
Sunday, August 19, 12
Edge theory of AII 3D topological insulators 30
Literature states that its 2D edge is a Dirac fermion theory
because it is1. linear in momenta2. time-reversal invariantH(p) = �yH
⇤(�p)�y
But does it have the right edge invariant?
H = v�x
px
+ v�y
py
± v��z
� µ
Fix q=+Λ or q=-Λ
Effectively 2D.N2(�)�N2(��) = 1 Well known relation
in this context.LFSG, 1994
H = v (�x
px
+ �y
py
)� µ
q
H = v (�x
px
+ �y
py
+ �z
q)� µEnlarge dimensions to 4D (3D edge)
Sunday, August 19, 12
Edge theory of AII 3D topological insulators 30
Literature states that its 2D edge is a Dirac fermion theory
because it is1. linear in momenta2. time-reversal invariantH(p) = �yH
⇤(�p)�y
But does it have the right edge invariant?
H = v�x
px
+ v�y
py
± v��z
� µ
Fix q=+Λ or q=-Λ
Effectively 2D.N2(�)�N2(��) = 1 Well known relation
in this context.LFSG, 1994
H = v (�x
px
+ �y
py
)� µ
q
H = v (�x
px
+ �y
py
+ �z
q)� µEnlarge dimensions to 4D (3D edge)
Yes, it does have the right edge invariant.
Sunday, August 19, 12
Disorder 31
Old idea of Thouless, Wu, Niu: impose phases across the system
✓x
✓y
Gij
(!, ✓x
, ✓y
. . . )
Summation over each ↵ = !, ✓1, . . . , ✓d is implied
Nd = Cd ✏↵0...↵d tr
Zd!dd✓G�1@↵0G . . .G�1@↵dG
It follows that an edge of a topological insulator does not localize in the presence of disorder
Sunday, August 19, 12
Disorder 31
Old idea of Thouless, Wu, Niu: impose phases across the system
✓x
✓y
Gij
(!, ✓x
, ✓y
. . . )
Summation over each ↵ = !, ✓1, . . . , ✓d is implied
Nd = Cd ✏↵0...↵d tr
Zd!dd✓G�1@↵0G . . .G�1@↵dG
N0(⇤)�N0(�⇤) = 1
✏
�⇤⇤ ✓
d = 2 : This edge level must be delocalized
It follows that an edge of a topological insulator does not localize in the presence of disorder
Sunday, August 19, 12
Sigma models with “topological” terms 32
Describe lack of localization at the boundary of an insulator
S ⇠ �
Zd
dx (@µQ)
2+ topological term
Topological term = either WZW term or “Z2” term.
Can be added if either ⇡d(T ) = Z2 ⇡d+1(T ) = Zor
It is believed that these sigma models with topological terms result in the absence of localization
T - target space
10 classes of sigma modelsfirst derived byAltland & Zirnbauerto describe localization properties
d = d� 1
Justifies 10 symmetryclasses of topological insulators
From: Ryu, Schnyder,
Furusaki, Ludwig, 2010
Sunday, August 19, 12