First-Principles Calculation of Topological Invariants (Wannier Functions Approach) Alexey A. Soluyanov

Transcription

1 First-Principles Calculation of Topological Invariants (Wannier Functions Approach) Alexey A. Soluyanov ES'12, WFU, June 8, 212

2 The present work was done in collaboration with David Vanderbilt

3 Outline: Part I: Overview Wannier functions (WFs) Topological insulators Chern insulators Z 2 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

4 PART I OVERVIEW

5 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 '1) computing topological invariants (Soluyanov, Vanderbilt '11) Figures from Marzari et. al. ArXiv '12

6 Gauge freedom: Consider a band insulator with N occupied bands: A particular choice of representative Bloch states does not matter gauge freedom: U(1) freedom: U(N) freedom:

7 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: = P x Is gauge invariant mod a, but not each of them separately! Resta et. al. PRB'1 Marzari, Vanderbilt PRB'97 Let us take maximally localized in x

8 Wannier charge centers as a function of k j : Usually smooth lines that come back to the original value in the end x 1 - k y but not always not in topological insulators.

9 Wannier charge centers as a function of k j : Usually smooth lines that come back to the original value in the end x 1 - k y but not always not in topological insulators.

10 Wannier charge centers as a function of k j : Usually smooth lines that come back to the original value in the end... x 1 x - k y but not always not in topological insulators.

11 Topological insulators: The first example: Haldane model for IQHE without external magnetic field. (Haldane PRL'88) - two band tight binding model - two inequivalent sites - complex hoppings - exhibits chiral edge states Broken time reversal symmetry (TRS) 1 x - k y

12 Topological insulators: The first example: Haldane model for IQHE without external magnetic field. (Haldane PRL'88) - two band tight binding model - two inequivalent sites - complex hoppings - exhibits chiral edge states Broken time reversal symmetry (TRS) (Thonhauser, Vanderbilt PRB'6) x - k y

13 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'6 and Kane et. al. PRL'5

14 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

15 Chern insulator (multiband case): Berry connection Berry curvature Integrated over the 2D Brillouin zone gives an integer first Chern number

16 Z 2 topological insulators: Add spin to Haldane model and restore TRS (Kane, Mele PRL'5) H KM = H Chern (k) [H Chern (-k)]* Spin-up block Spin-down block A Kramers pair of counter-propagating edge states - Quantized spin Hall effect. S z non-conserving terms are usually present.

17 Z 2 topological insulators: Spin-orbit interaction brings in spin-mixing terms (Kane, Mele PRL'5) H KM = H H (k) SO SO [H H (-k)]* Quantum spin Hall effect (QSH) not quantized. Is there any topological protection of this phase? YES, this phase is topological.

18 Z 2 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. Z 2 invariant distinguishes these two phases: it is the number of Kramers pairs at the edge mod 2. From Kane, Mele PRL'5

19 Z 2 invariant: How to compute? There are many ways. But let us concentrate on the original expression: Fu, Kane PRB'6 k2 k1 where BZ and This formula works only when the gauge is smooth Soluyanov, Vanderbilt PRB'12

20 Synopsis for PART II: Hybrid Wannier functions reveal information about the underlying topology: Zig-zag or not zig-zag?! This determines the Z 2 invariant.

21 PART II Computing topological invariants by means of Wannier functions

22 With inversion symmetry: - product of parities of occupied Kramers pairs Fu, Kane PRB'7 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. Fukui, Hatsugai JPSJ'6 Yu et. al. PRB'11 Zhang et. al. Nature'9

23 Wannier charge center interchange: Z 2 -even Z 2 -odd (Soluyanov, Vanderbilt PRB'11)

24 Wannier charge center zig-zag: Z 2 -even Z 2 -odd Do hybrid Wannier centers zig-zag when going from to or not? How would you track that on a not so dense mesh of k- points when connectivity of lines is not obvious?

25 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

26 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

27 Parallel transport (multiband case): SVD is used to make the overlap matrices Hermitian at each This leads to Wilczek, Zee PRL'84 Mead RMP'92 Marzari, Vanderbilt PRB'97

28 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

29 1x 1x Tracking the largest gap between WCC: k y k y k y

30 Tracking the largest gap between WCC: 1 x k y 1 k y x k y

31 Tracking the largest gap between WCC: 1 x k y 1 k y x k y

32 Tracking the largest gap between WCC: x k y k y

33 Tracking the largest gap between WCC: x 2 k y 1 k y

34 Tracking the largest gap between WCC: x 2 3 k y 2 k y

35 Tracking the largest gap between WCC: x k y k y 3

36 1x Tracking the largest gap between WCC: k y 4 5 k y

37 1x 1x Tracking the largest gap between WCC: k y k y

38 1x 1x Tracking the largest gap between WCC: k y k y

39 1x 1x Tracking the largest gap between WCC: k y k y

40 Automated procedure: Directed area of a triangle: DAT + - And for large enough mesh. Completely automated procedure: no plotting needed.

41 3D Z 2 insulators: Insulators with metallic surfaces. Metallic surface states are topologically protected. k2 T-symmetric planes are equivalent to BZ of a 2D T-symmetric insulator. T k1 k3 BZ (Fu et. al. PRL'6; Moore, Balents PRB'6; Roy PRB'6)

42 3D Z 2 insulators: Insulators with metallic surfaces. Metallic surface states are topologically protected. k2 T-symmetric planes are equivalent to BZ of a 2D T-symmetric insulator. T k1 k3 BZ (Fu et. al. PRL'6; Moore, Balents PRB'6; Roy PRB'6)

43 3D Z 2 insulators: Insulators with metallic surfaces. Metallic surface states are topologically protected. k2 T-symmetric planes are equivalent to BZ of a 2D T-symmetric insulator. T k1 k3 BZ (Fu et. al. PRL'6; Moore, Balents PRB'6; Roy PRB'6)

44 3D Z 2 insulators: Insulators with metallic surfaces. Metallic surface states are topologically protected. k2 T-symmetric planes are equivalent to BZ of a 2D T-symmetric insulator. T k1 k3 BZ (Fu et. al. PRL'6; Moore, Balents PRB'6; Roy PRB'6)

45 DFT + LDA First-principles calculations: ABINIT package Spin-orbit included in calculations via HGH pseudopotentials 1x1x1 k-mesh

46 Testing with centrosymmetric materials: Semimetallic Bi (lower bands are separated by a gap at each k) Space group #166

47 x x Testing with centrosymmetric materials: Bi 1 1 k 1 = : 4 jumps k 1 =/a : 1 jumps Topologically trivial manifold

48 Testing with centrosymmetric materials: Insulating Bi 2 Se 3 Space group #166

49 Testing with centrosymmetric materials: Bi 2 Se x x k 1 = : 1 jump k 1 =/a : jumps Topologically non-trivial manifold

50 Application to noncentrosymmetric materials: Insulating GeTe Space group #16

51 Application to noncentrosymmetric materials: GeTe 1 1 x x k 1 = : jumps k 1 =/a : jumps Topologically trivial manifold

52 Application to noncentrosymmetric materials: [111] epitaxially strained HgTe +2% +5% No gap closure in between

53 x x Application to noncentrosymmetric materials: +2% strain 1 k 1 = : 1 jump 1 k 1 =/a : jumps Topologically non-trivial manifold

54 Other candidate binary compounds: Ordinary insulators: FeSi OsSi OsSi 2 Topological insulators: [1] epitaxially strained AlBi but not BBi hypothetical compound FeSi 2 WSe 2 PbTe InSb... Al (B) Figure from Bi

55 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. Wannier9 add-on is under construction.