Post on 02-Nov-2018
First-Principles Calculation of Topological Invariants
(Wannier Functions Approach)
Alexey A. Soluyanov
ES'12, WFU, June 8, 2012
Outline:Outline:
• Part I: Overview– Wannier functions (WFs)– Topological insulators
• Chern insulators
• Z2 insulators as two copies of Chern insulators
• Part II: Computing topological invariants using WFs– Use of hybrid WFs to compute Z
2 invariant
– Application to real materials
Wannier functions for band insulators:Wannier functions for band insulators:• Wannier functions
• A real space representation by a set of well localized states that span the same Hilbert space as the occupied Bloch states.
• For an ordinary insulator with N occupied states there exists a set of N exponentially localized Wannier functions.
• Are used for– computing electronic polarization (King-Smith, Vanderbilt '93)– computing charge density (Marzari, Vanderbilt '97)– constructing model Hamiltonians (Souza, Marzari, Vanderbilt '01)– computing topological invariants (Soluyanov, Vanderbilt '11)
Figures from Marzari et. al. ArXiv '12
U(1) freedom:
U(N) freedom:
A particular choice of representative Bloch states does not matter – gauge freedom:
Consider a band insulator with N occupied bands:Gauge freedom:Gauge freedom:
Hybrid WFs and Wannier charge centers:Hybrid WFs and Wannier charge centers:
1D maximally localized Wannier functions (a=1):
1D:
2D:
Localized in the x-direction and extended in the y-direction.
Wannier charge centers in 1D:
Is gauge invariant mod a, but not each of them separately!
= Px
Resta et. al. PRB'01Marzari, Vanderbilt PRB'97
Let us take maximally localized in x
Wannier charge centers as a function of kWannier charge centers as a function of kj j ::
-π ky
0
1
x
π
2
3
Usually smooth lines that come back to the original value in the end...
… but not always – not in topological insulators.
Wannier charge centers as a function of kWannier charge centers as a function of kj j ::
-π ky
0
1
x
π
2
3
Usually smooth lines that come back to the original value in the end...
… but not always – not in topological insulators.
Wannier charge centers as a function of kWannier charge centers as a function of kj j ::
-π ky
0
1
x
π
Usually smooth lines that come back to the original value in the end...
… but not always – not in topological insulators.
x
Topological insulators:Topological insulators:• The first example: Haldane model for IQHE without external
magnetic field.
- two band tight binding model- two inequivalent sites - complex hoppings- exhibits chiral edge states
Broken time reversal symmetry (TRS)
-π ky
0
1
π
x
(Haldane PRL'88)
Topological insulators:Topological insulators:• The first example: Haldane model for IQHE without external
magnetic field.
- two band tight binding model- two inequivalent sites - complex hoppings- exhibits chiral edge states
Broken time reversal symmetry (TRS)
(Haldane PRL'88)
x
-π ky
π
(Thonhauser, Vanderbilt PRB'06)
Chern insulators:Chern insulators:
The edge states are topologically protected.
Phases with different values of Hall conductance are separated by a metallic phase and can not be adiabatically connected to each other.
Figures from Thonhauser et. al. PRB'06 and Kane et. al. PRL'05
Chern insulator (single band):Chern insulator (single band):• Berry connection
• Berry curvature
• Integrated over the 2D Brillouin zone gives an integer – first Chern number
C gives the value of Hall conductance
Chern insulator (multiband case):Chern insulator (multiband case):• Berry connection
• Berry curvature
• Integrated over the 2D Brillouin zone gives an integer – first Chern number
ZZ22 topological insulators: topological insulators:
• Add spin to Haldane model and restore TRS
• A Kramers pair of counter-propagating edge states - Quantized spin Hall effect.
• Sz non-conserving terms are usually present.
HKM
=
HChern
(k)
[HChern
(-k)]*
0
0
Spin-up block
Spin-down block
(Kane, Mele PRL'05)
ZZ22 topological insulators: topological insulators:
• Spin-orbit interaction brings in spin-mixing terms
• Quantum spin Hall effect (QSH) – not quantized.
• Is there any topological protection of this phase?
HKM
=
SO
SO
YES, this phase is topological.
HH
(k)
[HH
(-k)]*
(Kane, Mele PRL'05)
ZZ22 invariant: invariant:
• Phases with odd and even number of Kramers pairs of edge states are topologically distinct.
• If the number is even, such a Hamiltonian can be adiabatically connected to the ordinary insulating Hamiltonian that has no edge states.
• If the number is odd, then there is no adiabatic connection of this phase to the ordinary insulating phase.
• Z2 invariant distinguishes these two phases: it is the number of
Kramers pairs at the edge mod 2.
From Kane, Mele PRL'05
• How to compute? There are many ways.
But let us concentrate on the original expression: Fu, Kane PRB'06
ZZ22 invariant: invariant:
k1
k2
BZ
where
and
This formula works only when the gauge is smooth
Soluyanov, Vanderbilt PRB'12
Synopsis for PART II:
• Hybrid Wannier functions reveal information about the underlying topology:
Zig-zag or not zig-zag?!
This determines the Z2 invariant.
With inversion symmetry:With inversion symmetry:
Without inversion symmetry:Without inversion symmetry:
We suggest a new numerical method that gives topological invariants directly as a result of an
automated procedure and has a straightforward application in the majority of ab initio codes.
Fu, Kane PRB'07- product of parities of occupied Kramers pairs
Fukui, Hatsugai JPSJ'06Yu et. al. PRB'11Zhang et. al. Nature'09
Wannier charge center interchange:Wannier charge center interchange:
Z2-even Z
2-odd
(Soluyanov, Vanderbilt PRB'11)
Wannier charge center zig-zag:Wannier charge center zig-zag:
Z2-even Z
2-odd
Do hybrid Wannier centers zig-zag when going from 0 to π or not?
How would you track that on a not so dense mesh of k-points when connectivity of lines is not obvious?
Parallel transport (maximally localized hybrid WF):Parallel transport (maximally localized hybrid WF):
• Single band
Apply a U(1) transformation at in order to have
Berry PRS'84
Marzari, Vanderbilt PRB'97
MaxLoc hybrid Wannier (single band):MaxLoc hybrid Wannier (single band):
At a given value of produce parallel transport in
Hybrid Wannier charge center is given by:
The choice of branch cut corresponds to the choice of the unit cell in which the corresponding Wannier function resides
Parallel transport (multiband case):Parallel transport (multiband case):
SVD is used to make the overlap matrices Hermitian at each
This leads to
Wilczek, Zee PRL'84Mead RMP'92Marzari, Vanderbilt PRB'97
MaxLoc hybrid Wannier centers (multiband case):MaxLoc hybrid Wannier centers (multiband case):
A multiband generalization of Berry phase:
Two problems:– connectivity of individual Wannier charge centers– branch choice in the log for each Wannier charge center
Tracking the largest gap between WCCTracking the largest gap between WCC::
00
1
x
0 ky
0
1
x0
π
πky
ky
π
Tracking the largest gap between WCCTracking the largest gap between WCC::
x
ky
x0 k
y π
00
1
00
1 πk
y
π
Tracking the largest gap between WCCTracking the largest gap between WCC::
x
ky
x0 k
y π
00
1
00
1 πk
y
π
Tracking the largest gap between WCCTracking the largest gap between WCC::
00
1
2
x0 1
πky
ky
π
1
2
22
Tracking the largest gap between WCCTracking the largest gap between WCC::
00
13
3
x0 2 k
y π
πky
1 2
2
33
Tracking the largest gap between WCCTracking the largest gap between WCC::
00
14
x0 3
4
ky
π
πky
3
3
2
2
144
Tracking the largest gap between WCCTracking the largest gap between WCC::
00
1
x0
5
5
ky
π
πky
1 2
2
3
3
4
45
Tracking the largest gap between WCCTracking the largest gap between WCC::
00
1
x
00
1
x πk
y
ky
π
4
5
5
4
44 5
3
3
33
2
2
2
2
1
1
Tracking the largest gap between WCCTracking the largest gap between WCC::
00
1
x
00
1
x πk
y
ky
π
1
1
2 3 4 5
2 3 45
2
233
44 5
Tracking the largest gap between WCCTracking the largest gap between WCC::
00
1
x
00
1
x πk
y
ky
π
1
1
2 3 4 5
2 3 45
2
233
44 5
Automated procedureAutomated procedure::
Completely automated procedure: no plotting needed.
Directed area of a triangle:
And for large enough mesh.
+ -DAT
3D Z3D Z2 2 insulators:insulators:
• Insulators with metallic surfaces.• Metallic surface states are topologically protected.
k1
k2
k3
T-symmetric planes are equivalent to BZ of a 2D T-symmetric insulator.
(Fu et. al. PRL'06; Moore, Balents PRB'06; Roy PRB'06)
T
BZ
3D Z3D Z2 2 insulators:insulators:
• Insulators with metallic surfaces.• Metallic surface states are topologically protected.
k1
k2
k3
T-symmetric planes are equivalent to BZ of a 2D T-symmetric insulator.
(Fu et. al. PRL'06; Moore, Balents PRB'06; Roy PRB'06)
T
BZ
3D Z3D Z2 2 insulators:insulators:
• Insulators with metallic surfaces.• Metallic surface states are topologically protected.
k1
k2
k3
T-symmetric planes are equivalent to BZ of a 2D T-symmetric insulator.
(Fu et. al. PRL'06; Moore, Balents PRB'06; Roy PRB'06)
BZ
T
3D Z3D Z2 2 insulators:insulators:
• Insulators with metallic surfaces.• Metallic surface states are topologically protected.
k1
k2
k3
T-symmetric planes are equivalent to BZ of a 2D T-symmetric insulator.
(Fu et. al. PRL'06; Moore, Balents PRB'06; Roy PRB'06)
T
BZ
First-principles calculations:First-principles calculations:
• DFT + LDA
• ABINIT package
• Spin-orbit included in calculations via HGH pseudopotentials
• 10x10x10 k-mesh
Testing with centrosymmetric materials: Testing with centrosymmetric materials:
Semimetallic Bi(lower bands are separated by a gap at each k)
Space group #166
Testing with centrosymmetric materials: Bi Testing with centrosymmetric materials: Bi
k
1=0 : 4 jumps
k1=π/a : 10 jumps
Topologically trivial manifold
00
11
xx
Testing with centrosymmetric materials: Testing with centrosymmetric materials:
Insulating Bi2Se
3
Space group #166
Testing with centrosymmetric materials: BiTesting with centrosymmetric materials: Bi22SeSe
33
k
1=0 : 1 jump
k1=π/a : 0 jumps
Topologically non-trivial manifold
xx
00
11
Insulating GeTeSpace group
#160
Application to noncentrosymmetric materials:Application to noncentrosymmetric materials:
Application to noncentrosymmetric materials: GeTeApplication to noncentrosymmetric materials: GeTe
xx
00
11
k
1=0 : 0 jumps
k1=π/a : 0 jumps
Topologically trivial manifold
Application to noncentrosymmetric materials: Application to noncentrosymmetric materials: [111] epitaxially strained HgTe[111] epitaxially strained HgTe
+2% +5%
No gap closure in between
Application to noncentrosymmetric materials:Application to noncentrosymmetric materials:+2% strain+2% strain
xx
0
1
0
1
k
1=0 : 1 jump
Topologically non-trivial manifold
k1=π/a : 0 jumps
Other candidate binary compounds:Other candidate binary compounds:
• FeSi• OsSi
• OsSi2
• FeSi2
• WSe2
• PbTe• InSb• ...
• [001] epitaxially strained AlBibut not BBi
Ordinary insulators: Topological insulators:
hypothetical compound
Figure from http://ii-viworkshop.org
Al (B)
Bi
Conclusions:Conclusions:
• Hybrid Wannier functions can be used to determine topological invariants.
• A new numerical method for computing topological invariants is proposed.
•
• The method is easily applicable in most of the ab initio packages as well as in tight binding context.
• Tested in Abinit with Hartwigsen-Goedecker-Hutter pseudopotentials.
• Wannier90 add-on is under construction.