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Z 2 invariant for time reversal two dimensional topological insulators Shkolnikov Vlad 26.01.2017
Z2 invariant for time reversal two dimensional
topological insulators
Shkolnikov Vlad 26.01.2017
Plan1)Topologically trivial and nontrivial phases of time reversal topological insulator-definition of Z2-invariant 2)How to calculate Z2-invariant 3) BHZ model as an example of different types of topological behaviour 4) Z2-invariant in real physical systems 5) Alternative ways to introduce topological invariant
Literature1) J.K.Asboth, L. Orozlany, A. Palyi Lecture_Notes_arXiv:1509.02295
2) Liang Fu, C.L. Kane Time reversal polarisation and a Z2 adiabatic spin pump Phys. Rev. B 74, 195312
3)Rui Yu, Xiao Liang Qi, Andrei Bernevig, Zhong Fang, Xi Dai Equivalent expression of Z2 topological invariant for band insulators using the non-Abelian Berry connection
Phys. Rev. B 84, 075119
Topologically trivial and nontrivial phases of time reversal topological insulator-definition of Z2-invariant
1) We start with a bulk Hamiltonian . . Reinterpret ky as time: is a bulk 1D Hamiltonian of an adiabatic pump.
2) We sweep the time and track the motion of the particles, that we associate with the Wannier centres flow.
3) If an adiabatic change to the Hamiltonian exists that would turn off the pump (the Wannier centres don’t move), we say that the topological insulator is in the trivial phase(Z2=0), if no such change exists-then it’s in the nontrivial phase(Z2=1)
H(kx
, ky
)H(k
x
, ky
! t)
How to calculate Z2-invariant
|kx
>= 1pN
Pm
eimk
x |m >
n 2 1, 2, ..., NF
m 2 1, 2, ..., N,
X =P
m e2⇡imN |m >< m|
P =P
n,k
x
| n
(kx
) >< n
(kx
)|
| n
(kx
) >= |ky
> ⌦|kx
> ⌦|un
(kx
, ky
)
Xp = P XP
Wannier functions W(j,n) are eigenfunctions of with eigenvalues . The position of the centre of the Wannier function can be associated with .
Xp
�(n, j)N2⇡ Im log(�(n, j))
How to calculate Z2-invariant
P =P
n,k
x
| n
(kx
) >< n
(kx
)|
Xp = P XP
Xp
= P XP =P
n,k
x
Pn,k
x
< n
(kx
)|X| n
(kx
) > | n
(kx
) >< n
(kx
)|
How to calculate Z2-invariant
P =P
n,k
x
| n
(kx
) >< n
(kx
)|
Xp = P XP
Xp
= P XP =P
n,k
x
Pn,k
x
< n
(kx
)|X| n
(kx
) > | n
(kx
) >< n
(kx
)|
|kx
>= 1pN
Pm
eimk
x |m >
| n
(kx
) >= |kx
> ⌦|un
(kx
) >
X =P
m e2⇡imN |m >< m| =
Pm ei�km|m >< m|
How to calculate Z2-invariant
P =P
n,k
x
| n
(kx
) >< n
(kx
)|
Xp = P XP
Xp
= P XP =P
n,k
x
Pn,k
x
< n
(kx
)|X| n
(kx
) > | n
(kx
) >< n
(kx
)|
|kx
>= 1pN
Pm
eimk
x |m >
X =P
m e2⇡imN |m >< m|
| n
(kx
) >= |kx
> ⌦|un
(kx
) >
=P
m ei�km|m >< m|
Xp
=P
n,n,k
x
< un
(kx
+ �k)|un
(kx
) > | n
(kx
+ �kx
) >< n
(kx
)|
How to calculate Z2-invariant
| n
(kx
) >= |kx
> ⌦|un
(kx
) >
Xp
=P
n,n,k
x
< un
(kx
+ �k)|un
(kx
) > | n
(kx
+ �kx
) >< n
(kx
)|
Wm,n
(kx
) =< um
(kx
)|ur
(kx
+ (N � 1)�k) > ⇤⇤ < u
r
(kx
+ (N � 1)�k)|uh
(kx
+ (N � 2)�k) > ...... < un
(kx
+ �k)|un
(kx
) >
(Xp
)N =P
m,n,k
x
Wmn
(kx
)| m
(kx
) >< n
(kx
)|
W (kx
) = ABCD
W (kx
+ �k) = DABC
ABCD ⇤ v = � ⇤ vDABC ⇤Dv = D ⇤ABCD ⇤ v = �Dv
Eigenvalues of the Wilson loop W(kx) don’t depend on kx
How to calculate Z2-invariant
Wm,n
(kx
) =< um
(kx
)|ur
(kx
+ (N � 1)�k) > ⇤⇤ < u
r
(kx
+ (N � 1)�k)|uh
(kx
+ (N � 2)�k) > ...... < un
(kx
+ �k)|un
(kx
) >
(Xp
)N =P
m,n,k
x
Wmn
(kx
)| m
(kx
) >< n
(kx
)|
Eigenvalues of the Wilson loop W(kx) don’t depend on kx
(Xp
)N (P
k
x
v(kx
)) = �i
(P
k
x
v(kx
))
W (kx
)v(kx
) = �i
v(kx
), i 2 1, ...NF
Then Xp has N ⇤Nf eigenvalues:
�i(j) = |�i|1/Ne(2⇡N ⇥i+2⇡ j
N )
N2⇡ Im log
˜�i(j) = ⇥i + j Positions of the Wannier centres
Example, BHZ model
Z2 invariant is 0Z2 invariant is 1
Z2 invariant in Graphen
Changing the parameter of the Hamiltonian it’s possible to switch
between different topological regimes
�v
Alternative ways to define Z2 invariant
1) We could as well calculate the number of edge state(Kramers pairs of edge states)
2) It turns out that adiabatic deformation of the Hamiltonian can only destroy pairs of Kramers
states, which means that the parity of the number of edge states is also a topological number
3)Theorem: Invariant defined in such a way is equivalent to the one we defined
Conclusion
1) We introduced the Z2 invariant for time reversal topological insulators, which defines whether the material will exhibit nontrivial topological behaviour or not. 2) A general procedure to calculate the invariant was shown 3) The method was tested on real system of graphene
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