Symmetry-adapted Wannier functions lecture and tutorials

Yusuke Nomura and Valerio Vitale

26 March, 2020 Wannier90 School

Collaborators (SAWF part): Yoshiro Nohara, Takashi Koretsune, and Ryotaro Arita

Symmetry-adapted Wannier functions lecture and tutorials

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If you have questions, please put them in chat ! Mute your audio and unmute it when necessary

Now available in Wannier90 ※ Please use version 3.1.0

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Wannier State Bloch state

Maximally localized Wannier functionsN. Marzari and D. Vanderbilt, Phys. Rev. B. 56 12847 (1997) I. Souza et al., ibid. 65, 035109 (2001)

The unitary matrix U is obtained by minimizing the spread functional Ω

where,

Symmetry breaking of “maxloc" Wannier functions : CopperI. Souza et al., Phys. Rev. B 65 035109 (2001)

initial projections with atom-centered s and d orbitals, i.e., 6 orbital model → results in interstitial-centered s-like “maxloc”

(“maxloc” procedure does not care about symmetry at all)

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Symmetry-adapted Wannier functions = “maxloc” procedure + symmetry constraint → obtain irreducible representations of a subgroup of full symmetry group

(irreducible representations of site-symmetry group)

Now available in Wannier90 ※ Please use version 3.1.0 !!!

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Site-symmetry group

Example: two-dimensional square latticeFull symmetry group : 8 symmetry operations (sym. ops.)

(0,0) (1/2,0)

A subgroup of full-symmetry group, whose elements leave the site-position unchanged

Wyckoff position (0,0) : Site-symmetry group : 8 sym. ops. Multiplicity : 1

Wyckoff position (1/2,0), (0,1/2) : Site-symmetry group : 4 sym. ops. (without C4) Multiplicity : 2

(0,1/2)

Multiplicity = (# full sym. ops.)/(# sym. ops. in site-symmetry group)

Concept of Symmetry-adapted Wannier functions (SAWF)

SAWF = “maxloc" procedure + symmetry constraint → obtain irreducible representations of site-symmetry group

“maxloc” Wannier (global minimum)

SAWF (local minimum)

All unitary transformation Umn(k)

symmetry-adapted Umn(k)

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Symmetry-adapted Wannier functionsRei Sakuma, Phys. Rev. B 87 235109 (2013)

Symmetry-adapted Wannier functions = “maxloc” procedure + symmetry constraint → obtain irreducible representations of a subgroup of full symmetry group

(irreducible representations of site-symmetry group)

U(Rk) = d̃(g,k)U(k)D†(g,k)<latexit sha1_base64="54K67hgXmjqmCJGTvIuKLHGYeHU=">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</latexit><latexit sha1_base64="54K67hgXmjqmCJGTvIuKLHGYeHU=">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</latexit><latexit sha1_base64="54K67hgXmjqmCJGTvIuKLHGYeHU=">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</latexit><latexit sha1_base64="54K67hgXmjqmCJGTvIuKLHGYeHU=">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</latexit>

U(k) = d̃(gk,k)U(k)D†(gk,k)<latexit sha1_base64="+mOCY/nxjbE7FYJh1nB9vy9mWuY=">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</latexit><latexit sha1_base64="+mOCY/nxjbE7FYJh1nB9vy9mWuY=">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</latexit><latexit sha1_base64="+mOCY/nxjbE7FYJh1nB9vy9mWuY=">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</latexit><latexit sha1_base64="+mOCY/nxjbE7FYJh1nB9vy9mWuY=">AAACz3ichVFNaxNBGH6yftW0mmgvhV6WhpYKEt4VQSkUSu3BY9M2SaGpYXcz2UyzX+xOUtol4lX/gAdPFkSkP6MXj1489CdIjxV7KeibzeJX0b7DzLzzzPO888yMFboyVkTHOe3K1WvXb4zdzI9P3LpdKN65W4uDXmSLqh24QbRpmbFwpS+qSipXbIaRMD3LFXWr+2S4X++LKJaBv6H2QrHtmY4v29I2FUPN4k51PmlYbb07uKfri3pDSbclktZAZ9gzVSfydGfQzCj3f1J/U608Sxot03FEdImoWSxRmdLQLyZGlpSQxWpQfI8GWghgowcPAj4U5y5MxNy2YIAQMraNhLGIM5nuCwyQZ22PWYIZJqNdHh1ebWWoz+thzThV23yKyz1ipY5Z+kwf6JQ+0iF9ofN/1krSGkMvezxbI60Im4VXU+tnl6o8nhU6v1T/9azQxuPUq2TvYYoMb2GP9P3916frC2uzyRwd0An7f0vHdMQ38Ptf7XcVsfYGef4A4+/nvpjUHpQNKhuVh6Wl5ewrxjCNGczzez/CEp5iFVU+9xO+4XsOWkXb1Z5rL0ZULZdpJvFHaC9/AD7hsMo=</latexit>

Constraint on U(k) for k in irreducible BZ

Unitary matrix for other k points

matrices related to symmetry

gkk = k<latexit sha1_base64="Fi8TiozTDHaidk4kBO1rq4pb/Lk=">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</latexit><latexit sha1_base64="Fi8TiozTDHaidk4kBO1rq4pb/Lk=">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</latexit><latexit sha1_base64="Fi8TiozTDHaidk4kBO1rq4pb/Lk=">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</latexit><latexit sha1_base64="Fi8TiozTDHaidk4kBO1rq4pb/Lk=">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</latexit>

Important quantities: D and d matrices ~

num_wann×num_wann matrix

num_bands×num_bands matrix

Transformation of symmetry-adapted (Wannier-gauge) Bloch functions by sym. ops.

Transformation of Kohn-Sham Bloch functions by sym. ops.

written in seedname.dmn file8

Flow of calculation (after nscf calculations and creating .nnkp file)

pw2wannier90.x

seedname.win site_symmetry = .true.

pw2wan.in write_dmn = .true.

input executable output/input

seedname.wout …

seedname.dmn (on top of amn, mmn)

wannier90.x

seedname.sym (not used in wannier90.x)

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Example of Dmn : CaCuO2 d-p model

Famous family of high-Tc cuprates

Quasi-2D (layered) material

16 symmetry operations (P4/mmm)

CuO2 layer

Ca layer

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x2-y2 px

py

Example of Dmn : CaCuO2 d-p modelEnergy (eV)

Γ X M Γ Z R A Z

D matrix → how does the SAWF transform by sym. ops.

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(We do not discuss d matrix because the global phase of KS wave function can be random)~

1 dim. irrep. site multiplicity: 2

1 dim. irrep. site multiplicity: 1

1 dim. irrep. site multiplicity: 2

Example of Dmn (90° rotation, Γ): CaCuO2 d-p modelafter 90° rotation

Wannier-gauge Bloch at Γ ΨΓ,x2-y2,(0,0,0)　

→ (-1)ΨΓ,x2-y2,(0,0,0)　　

ΨΓ,px,(1/2,0,0)　 ΨΓ,py,(0,1/2,0)　

→ ΨΓ,py,(0,1/2,0)　　 → (-1)ΨΓ,px,(1/2,0,0)　　

R̂⇡/2 �n(r) =X

m

Dmn �m(r)

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Example of Dmn (90° rotation, Γ): CaCuO2 d-p modelafter 90° rotation

Wannier-gauge Bloch at Γ ΨΓ,x2-y2,(0,0,0)　

→ (-1)ΨΓ,x2-y2,(0,0,0)　　

ΨΓ,px,(1/2,0,0)　 ΨΓ,py,(0,1/2,0)　

→ ΨΓ,py,(0,1/2,0)　　 → (-1)ΨΓ,px,(1/2,0,0)　　

R̂⇡/2 �n(r) =X

m

Dmn �m(r)

D =

0

@�1 0 00 0 �10 1 0

1

A

px pyx2-y2

Block diagonal for each irrep. (block size: dimension of irrep × site multiplicity)

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Energy (eV)

Γ X M Γ Z R A Z

x2-y2 px

py

In this case, SAWF = “maxloc” Wannier function

SAWFs : CaCuO2 d-p model

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SAWFs : CopperRei Sakuma, Phys. Rev. B 87 235109 (2013)

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Applications to H3S ①

bcc, 48 sym. ops.

Γ H N Γ P N

0

20

40

Energy (eV)

Band structure of Im-3m H3S

A. P. Drozdov et al., Nature 2015 M. Einaga et al., Nat. Phys. 2016 R. Akashi et al., PRL 2016

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Im3-m H3SMaximally localized Wannier functions

Applications to H3S ②

S s

S p

S d

H s

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Im3-m H3SSymmetry adapted Wannier functions

Applications to H3S ③

S s

S p

S d

H s

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Examples in Wannier90

example21: Gallium Arsenide - Symmetry-adapted Wannier functions

example22: Copper - Symmetry-adapted Wannier functions

—— atom_centered_As_sp —— atom_centered_Ga_p —— atom_centered_Ga_s —— atom_centered_Ga_sp —— bond_centered

—— s_at_0.00 —— s_at_0.25 —— s_at_0.50

You can try SAWF ※ Please use version 3.1.0 !!!

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Be careful …

One cannot always get atom-centered Wannier orbitals with symmetry-adapted mode

Example: MoS2 monolayer

sulphur p orbitals cannot be exactly atom-centered 20

Summary and future perspective

SAWF = “maxloc" procedure + symmetry constraint → obtain irreducible representations of site-symmetry group

“maxloc” Wannier (global minimum)

SAWF (local minimum)

All unitary transformation Umn(k)

symmetry-adapted Umn(k)

Summary

Future perspective Frozen window (need to be symmetry-adapted) Extension to noncolliner case

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Tutorials

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Instructions

Tutorials can be found in “Tutorials_SAWF”

Basic tutorial: “GaAs”

(A little bit) advanced tutorials: “H3S” and “Cu”

Work flow “GaAs” → “H3S” and “Cu”

Follow the instruction pdf file or README in each directory

For “GaAs” and “Cu”, refer also to Wannier90 Tutorials and Tutorial Solutions

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Flow of calculation (after nscf calculations and creating .nnkp file)

pw2wannier90.x

seedname.win site_symmetry = .true.

pw2wan.in write_dmn = .true.

input executable output/input

seedname.wout …

seedname.dmn (on top of amn, mmn)

wannier90.x

seedname.sym (not used in wannier90.x)

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Reference: H3S

Γ H N Γ P N

0

20

40

Energy (eV)

12 band model

Γ H N Γ P N

0

20

40

Energy (eV)

7 band model

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Flow of calculation in Cu tutorial : advanced usage (after nscf calculations and creating .nnkp file)

pw2wannier90.x

seedname.win site_symmetry = .true.

pw2wan.in write_dmn = .true. read_sym = .true.

input executable output/input

seedname.wout …

seedname.dmn (on top of amn, mmn)

wannier90.x

seedname.sym (prepare it by ourselves)

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