A data-centered approach to understanding quantum behaviors in - - PowerPoint PPT Presentation

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A data-centered approach to understanding quantum behaviors in - - PowerPoint PPT Presentation

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A data-centered approach to understanding quantum behaviors in materials Lucas K. Wagner Department of Physics Institute for Condensed Matter Theory National Center for Supercomputing Applications University of Illinois at Urbana-Champaign

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A data-centered approach to understanding quantum behaviors in materials

Lucas K. Wagner Department of Physics Institute for Condensed Matter Theory National Center for Supercomputing Applications University of Illinois at Urbana-Champaign June 5, 2018

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Effective models in materials physics

Interacting balls with potentials Band structure Interacting spins

Leblanc, Whitehead, Plumer. J. Phys. Cond. Mat. 25 296004

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The challenge

Can we systematically build interacting quantum models based

  • n fine-grained simulations?
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The challenge

Can we systematically build interacting quantum models based

  • n fine-grained simulations?

Brian Busemeyer Hitesh Changlani Huihuo Zheng Joao Nunes Rodrigues Kiel Williams

Thanks to: Blue Waters, Simons foundation, Department of Energy EFRC Center for Emergent Superconductivity (DEAC0298CH1088) , DOE FG02-12ER46875 , NSF Grant No. DMR 1206242

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Outline

Quantum mechanics of electrons in materials Effective models as a data analysis problem Applications to real materials

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Quantum mechanics of electrons in materials

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Basics of quantum mechanics

The state of a system is described by a wave function. For a collection of particles at a given time t, Ψ(r1, r2, r3, . . . , t). |Ψ(r1, r2, r3, . . . , t)|2 gives the probability of each particle being at the given position at the given time. Expectation values: Q =

  • Ψ∗(r1, r2, . . .) ˆ

QΨ(r1, r2, . . .)

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The Hamiltonian

Hamiltonian ˆ H is an operator. For nuclei and electrons, it is −1 2

  • i

∇2

i +

  • i<j

1 rij −

  • α,i

Zα riα +

  • α<β

ZαZβ rαβ Special wave functions are eigenfunctions: ˆ HΦk = EkΦk If you know the eigenfunctions and their eigenenergies, then you know the dynamics of the system.

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An example: H2

An H2 molecule: two atoms and two electrons. Electrons exist in a continuum. Ψ(r1, r2) = Φ(r1)Φ(r2) Color is wave function of electron 1 given that electron 2 is at the green dot. No correlation here! Not an eigenfunction.

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An example: H2

Better approximation to the lowest energy eigenstate Ψ(r1, r2) = Φ(r1)Φ(r2) Electron 1 avoids the atom where electron 2 is nearby. Competes with quantum “spreading out” effect.

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Quantum Monte Carlo

Expectation values: E =

  • Ψ∗(r1, r2, . . .) ˆ

HΨ(r1, r2, . . .) If Ψ is not factorizable..use Monte Carlo!

−2 2

x1

−2 2

x2

β=0

−2 2

x1

β=0.5

We technically use projection methods for higher accuracy but it doesn’t affect the point here.

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Quantum Monte Carlo

1 2 3 4 5

Experimental gap (eV)

1 2 3 4 5

Theoretical gap (eV)

MnO FeO ZnO VO2 (rutile) VO2 (monoclinic) La2CuO4 NiO ZnSe FN-DMC DFT(PBE)

Can create wave functions that are very close (but not quite exactly equal) to the ground state and some excited states. Open-source code QWalk: http://qwalk.org

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Effective models as a data analysis problem

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Effective quantum models

Detailed Coarse-grained Wave function Ψ(r1, r2, r3, . . .) [0.1,-0.1,. . . ] Hamiltonian − 1

2

  • i ∇2

i + i<j 1 rij + . . .

Matrix Expectation values integral ΨTMΨ

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Model for H2

Reduced wave function representation      (1, 1) (1, 2) (2, 1) (2, 2)      (1,1) means that both electrons are near atom 1.

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Model Hamiltonian

Want to find matrix that operates on the state vector, such that the eigenstates are the same as the original one. Ψ =      (1, 1) (1, 2) (2, 1) (2, 2)      , ˆ Heff =      U t t t t t t t t U     

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How to map (Hitesh Changlani)

E =

  • a

b c d

    U t t t t t t t t U           a b c d      = U (a2 + d2)

  • double occupancy

+t (ab + ac + . . .)

  • hopping
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How to map (Hitesh Changlani)

E =

  • a

b c d

    U t t t t t t t t U           a b c d      = U (a2 + d2)

  • double occupancy

+t (ab + ac + . . .)

  • hopping

E = U(double occupancy) + t(hopping) All quantities in blue can be evaluated using detailed (real space) simulations.

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The algorithm

H = U(double occupancy) + t(hopping)

  • 1. Generate wave functions in the low-energy space of interest
  • 2. Accumulate expectation value of energy and “descriptors”

(e.g. double occupancy and hopping)

  • 3. Ek ≃

i cidk[Ψk]; find ci to minimize deviation

Proofs that this is the right thing to do: Frontiers in Physics. DOI:10.3389/fphy.2018.00043

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A chain of H atoms (Kiel Williams)

Good model when atoms are well-separated, poor when the atoms are too close (need more long-range terms)

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Applications to real materials

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More complex materials (Brian Busemeyer)

Fe=Se molecule 21 symmetry-allowed parameters

1 2 3 4 5 6 7

MP step

−1 1 2 3 4

Parameter (eV)

J Ud ǫδ,Fe ǫpz ǫs tσ,d tσ,s

1 2 3 4 5 6 7

MP step

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

RMS Error (eV)

Use matching pursuit to select best parameters

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Vanadium dioxide (Huihuo Zheng)

Monoclinic Rutile Charge density [110] Spin density [110] Spin density 3D

V O

0.125 0.015

a b c am ar dm bm cm

br

− − − − Å

(c) (b)

0.16

Metal-insulator transition at 340 K. Insulating state not well

  • understood. Spin excitation?

Measurement: 460 meVa Calculation: 440(24) meVb

aHe et al. PRB 94, 161119 (2016). bZheng, Wagner PRL 120, 059901(E) (2018)

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MgTi2O4 (Brian Busemeyer)

Another spin excitation: used Blue Waters to make a prediction of an excitation at 350(50) meV. Hoping experiments will test this!

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Predicting superconductivity classes (Joao Nunes Rodrigues)

BaCo2As2 CrGeTe3 Sr2MnO4 K2CoF4 NiPSe3 K2CuF4 BaMn2As2 Sr2CrO4 La2NiO4 BaCr2As2

  • ­FeSe

t­FeSe FeTe Sr2VO4 FeS BaFe2As2 La2CoO4

Material

0.0 0.2 0.4 0.6 0.8 1.0

P(SC| , Mcalc)

U = 0

BaCo2As2 CrGeTe3 K2CuF4 Sr2MnO4 NiPSe3 La2NiO4 BaMn2As2 BaCr2As2 Sr2CrO4 NbSe2 TaS2 K2CoF4 TaSe2 La2CoO4 SrCuO2 CaCuO2 BaFe2As2 Sr2VO4 T­La2CuO4 FeS

  • ­FeSe

FeTe t­FeSe T′­La2CuO4

Material

U = 5

Superconducting? no yes

BaCo2As2 K2CuF4 CrGeTe3 NiPSe3 La2NiO4 BaMn2As2 Sr2MnO4 BaCr2As2 Sr2CrO4 T­La2CuO4 NbSe2 MoS2 WSe2 BaFe2As2 La2CoO4 FeTe Sr2VO4 TiSe2

  • ­FeSe

TaSe2 CaCuO2 FeS K2CoF4 TaS2 SrCuO2 t­FeSe T′­La2CuO4

Material

U = 10

Predictor derived through the model fitting technique. Separates superconductors from non-superconductors with high fidelity.

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Relevance of Blue Waters

Need to compute expectation values of many many-particle wave functions via Monte Carlo: O =

  • Ψ∗(r1, r2, . . .) ˆ

OΨ(r1, r2, . . .) Massively parallel and high throughput. Need to generate a moderate amount of high-cost data.

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Open questions

  • Can we automatically sample good quality wave functions?

Usually we rely a lot on physical understanding.

  • Can we make QMC faster? (probably algorithms)
  • Best representation of coarse-grained model: finding basis.
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The end

→      U t t t t t t t t U     