First-Principles Calculation of Topological Invariants ...es12.wfu.edu/pdfarchive/oral/ES12_A_Soluyanov.pdf · First-Principles Calculation of Topological Invariants (Wannier Functions

First-Principles Calculation of Topological Invariants

(Wannier Functions Approach)

Alexey A. Soluyanov

ES'12, WFU, June 8, 2012

The present work was done in collaboration with David Vanderbilt

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

PART I

OVERVIEW

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.


-π ky

0

1

x

π

2

3




-π ky

0

1

x

π



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.

PART II

Computing topological invariants by means of Wannier functions

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


x

ky

x0 k

y π

00

1

00

1 πk

y

π


x

ky

x0 k

y π

00

1

00

1 πk

y

π


00

1

1

x0 k

y π

πky

1


00

1

2

x0 1

πky

ky

π

1

2

22


00

13

3

x0 2 k

y π

πky

1 2

2

33


00

14

x0 3

4

ky

π

πky

3

3

2

2

144


00

1

x0

5

5

ky

π

πky

1 2

2

3

3

4

45


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


00

1

x

00

1

x πk

y

ky

π

1

1

2 3 4 5

2 3 45

2

233

44 5


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



k1

k2

k3



T

BZ



k1

k2

k3



BZ

T



k1

k2

k3



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.

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First-Principles Calculation of Topological Invariants ...es12.wfu.edu/pdfarchive/oral/ES12_A_Soluyanov.pdf · First-Principles Calculation of Topological Invariants (Wannier Functions