Generalized Wannier functions for an ab initio description of the electronic structure of chiral magnets

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Generalized Wannier functions for an ab initio description of the electronic

structure of chiral magnets

Jan-Philipp Hanke, Frank Freimuth, Stefan Blügel and Yuriy Mokrousov Peter Grünberg Institut and Institute for Advanced Simulation,

Forschungszentrum Jülich

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Outline

1 Motivation

2 1D toy model

3 Maximally localized Wannier functions

4 Generalized Wannier functions Construction of GWFs

Interpolation and Heisenberg model

5 Conclusions

March 31, 2014 Jan-Philipp Hanke, Frank Freimuth, Stefan Blügel and Yuriy Mokrousov Slide 2 12

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Motivation

occurence ofnontrivial magnetic structuresin nature

spin spirals (1D), skyrmions (2D) and others with fascinating properties

M. Bode et al., Nature 447, 7141 (2007)

ultimate goal: topological characterizationof complex magnetic structures inrealandmomentumspace Berry curvature inλ-space

Ωⁿ_ij(λ) =−2 Im P

m6=n hλn| ^∂

∂λiH(λ)|λmihλm| ^∂

∂λjH(λ)|λni (λn−_λm)²

⇒anomalous & topological Hall effect

March 31, 2014 Jan-Philipp Hanke, Frank Freimuth, Stefan Blügel and Yuriy Mokrousov Slide 3 12

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Motivation

spin spiral vector qas additional, tunable parameter topological characterization usingmixed Berry curvature

mixed Berry curvature in(k,q)-space Ωⁿ_kq =−2 Im P

m6=n

hkqn|_∂k^∂H(k,q)|kqmihkqm|_∂q^∂H(k,q)|kqni (kqn−kqm)²

contributions to pumping∆Pand anomalous velocity

challenge: poor convergence in ab initio calculation

needinterpolation of H(k,q)

M. Menzel et al., PRL 108, 197204 (2012)

March 31, 2014 Jan-Philipp Hanke, Frank Freimuth, Stefan Blügel and Yuriy Mokrousov Slide 4 12

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1D toy model

chain of atoms + helical spin spiral Hamiltonian

H =−^∆₂ +P

jΘj(z) [V₀+B₀ ˆn·~σ]

ˆ

n= (cos(q·ja), sin(q·ja), 0)^T

a b

B0

B0

V0

V0=−12 eV,B0=2 eV,a=3 Å,b=2.9 Å

antiferromagnetic groundstate Generalized Bloch theorem

Ψ_kqn(z) =e^ikz e^−iq2zu_kqn^↑ (z) e^iq2zu_kqn^↓ (z)

!

−14

−12

−10

−8

−6

−4

Γ X Γ

energyE(k,q)ineV

wave vectork AFMFM

−80

−60

−40

−20 0

Γ X Γ

E(q)−E(0)inmeV

spin spiral vectorq March 31, 2014 Jan-Philipp Hanke, Frank Freimuth, Stefan Blügel and Yuriy Mokrousov Slide 5 12

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Maximally localized Wannier functions

exact tight-binding basis

Maximally localized Wannier functions (MLWF)

|W_Ri= _N¹

k

P

k

e^−ikRU(k)|Ψ_ki=F_k(U(k)|Ψ_ki)

N. Marzari & D. Vanderbilt, PRB 56, 12847 (1997)

U(k)determined byspread minimization(e.g. wannier90)

A. A. Mostofi et al., Comput. Phys. Commun. 178, 685 (2008)

interpolatingH(k): main ingredienthW₀|H|W_Ri

information on fine grid by inverse FT &diagonalization

N. Marzari et al., Rev. Mod. Phys. 84, 1419 (2012) March 31, 2014 Jan-Philipp Hanke, Frank Freimuth, Stefan Blügel and Yuriy Mokrousov Slide 6 12

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Generalized Wannier functions

Construction of GWFs

includeadditional parameter q

aim: obtainuseful|W_RQnivia Fourier transformations

|W_RQi= _N¹

kNq

P

k,q

e^−ikRe^−iqQU(k,q)|Ψ_kqi

challenge: unitaryU(k,q)⇔spread functional inq-space gauge choiceU(k,q) =U(k)V(q)and assumeV(q) =I

Generalized Wannier functions

1 |W_R^qi=F_k(U(k)|Ψ_kqi)

2 |W_RQi= _N¹

q

P

qe^−iqQ|W_R^qi

March 31, 2014 Jan-Philipp Hanke, Frank Freimuth, Stefan Blügel and Yuriy Mokrousov Slide 7 12

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Generalized Wannier functions

Construction of GWFs 1 W_R^q(z)∼ F_k

U(k)e^ikze^∓iq2zu_kq

2 W_RQ(z)∼P

qe^−iqQW_R^q(z)

⇒changes implied byq&Q?

−0.8

−0.4 0 0.4 0.8

−Nk 0 N_k

ReWq↑ R(z)

zalong the chain in units ofa q=^2π_a

q=0

L=Nk

L=lcm(Nk,Nq)

−0.2

−0 0.2 0.4 0.6 0.8

−Nk 0 Nk

ReW↑ RQ(z)

zalong the chain in units ofa Q=^a₂

Q=0

Q=0→^a₂

⇒q∼modulation

⇒Q ∼shift

March 31, 2014 Jan-Philipp Hanke, Frank Freimuth, Stefan Blügel and Yuriy Mokrousov Slide 8 12

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Generalized Wannier functions

Interpretation ofQvariable

similar behaviour forW_RQ^↓ (z)but...

... tuningQ introduces discreterelative shiftbetween up- &

down-component of GWFs(at least for certain choices ofN_k,Nq)

center-of-massRandrelativeQ(unlike standard WFs)

Schematic plot

R−2Nk R−Nk R R+Nk R+2Nk

|Wσ RQ(z)|2

zalong the chain in units ofa

↑ ↓

↑ ↓

↑ ↓

Q=0 Q=^a₂

Q=^a₂ Q=a

Q=a

March 31, 2014 Jan-Philipp Hanke, Frank Freimuth, Stefan Blügel and Yuriy Mokrousov Slide 9 12

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Generalized Wannier functions

Interpolation and Heisenberg model

Interpolation scheme

challenge: overlaphΨ_k⁰_q⁰_n|Ψ_kqmi 6=δ_kk⁰δ_qq⁰δ_nm scheme H_α= (1−α)H₁+αH₂ gauge invariant?

proceed withhW_RQ|H|W_R⁰_Q⁰ias main ingredient

⇒generalized eigenvalue problem

Heisenberg HamiltonianH =−J_ijS_iS_j

allows for interpretation ofQ variable of GWFs

Exchange couplings

t_QQ⁰ =−¹₂M²sin(θ) J(Q⁰−Q)

March 31, 2014 Jan-Philipp Hanke, Frank Freimuth, Stefan Blügel and Yuriy Mokrousov Slide 10 12

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Conclusions

deal with additional parameters in Hamiltonian establishinterpolationscheme forH(k,q) generalize formalism of Wannier functions

Challenges

physical interpretation of shift by tuningQ cutoffQ_C for interpolation scheme implement withinab initioframework improve results byV(q)6=I

evaluation ofexchangeJ_ij andBerry curvature

March 31, 2014 Jan-Philipp Hanke, Frank Freimuth, Stefan Blügel and Yuriy Mokrousov Slide 11 12

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Thank you for your attention

March 31, 2014 Jan-Philipp Hanke, Frank Freimuth, Stefan Blügel and Yuriy Mokrousov Slide 12 12