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Bonn 2008.01.10-12. Constrained RPA: Calculating the Hubbard U from First-Principles. Ferdi Aryasetiawan Research Institute for Computational Sciences Tsukuba, Ibaraki 305-8568 – Japan. Collaborators: Takashi Miyake (Tsukuba) Masa Imada (ISSP, Tokyo)

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constrained rpa calculating the hubbard u from first principles

Bonn 2008.01.10-12

Constrained RPA:Calculating the Hubbard U from First-Principles

Ferdi Aryasetiawan

Research Institute for Computational Sciences

Tsukuba, Ibaraki 305-8568 – Japan

Collaborators:

Takashi Miyake (Tsukuba)

Masa Imada (ISSP, Tokyo)

Antoine Georges, Silke Biermann (Ecole Polytechnique, Palaiseau)

Krister Karlsson (Skoevde)

slide2

Motivationsand aims:

Many-electron Hamiltonian is too complicated to be solved directly.

Isolate correlated bands and downfold weakly correlated bands

Systematic way of performing the downfolding

First-principles (parameter-free) method.

Adjustable parameters may give nice results but not necessarily for good reasons.

slide3

Related works on the Hubbard U

Seminal work on U (constrained LDA):

O Gunnarsson, OK Andersen, O Jepsen, J Zaanen, PRB 39, 1708 (1989)

VI Anisimov and O Gunnarsson, PRB 43, 7570 (1991)

Improvement on constrained LDA

M Cococcioni and S de Gironcoli, PRB 71, 035105 (2005)

Nakamura et al (PRB 2005)

Random-Phase Approximation (RPA):

M Springer and FA, PRB 57, 4364 (1998)

T Kotani, J. Phys.: Condens. Matter 12, 2413 (2000)

FA, M Imada, A Georges, G Kotliar, S Biermann, AI Lichtenstein,

PRB 70, 195104 (2004)

IV Solovyev and M Imada, PRB 71, 045103 (2005)

IV Solovyev, cond-mat/05066632

slide4

Instead of downfolding the Hamiltonian directly we downfold the response function.

Divide the total polarization into:

The rest of the polarization

Typical electronic structure of strongly correlated materials

t_2g

e_g

Aim: To find the effective interaction among electrons living in the t_2g band

slide6

Effective interaction among electrons in a narrow band:

Constrained RPA (cRPA)

Identity:

can be viewed

as an energy-dependent effective

interaction between the 3d electrons

  • Advantages:
  • Energy-dependent U
  • Full matrix U
  • U(r,r’) is basis-independent

Asymptotically

Long range !

FA, M Imada, A Georges, G Kotliar, S Biermann, AI Lichtenstein, PRB 70, 195104 (2004)

slide8

Static Hubbard U for the 3d transition metal series

For a more accurate model it may be necessary to include nearest-neighbour U

slide9

Sensitivity to the choice of energy window

Green 3d

Red 4s

3d

3d-4s hybridised

E window (eV) U

(-2.0, 4.0) 3.7

(-3.0, 4.0) 6.3

(-4.0, 4.0) 7.0

4s

slide10

Energy window (eV) U

(-2.0, 1.5) 7.9

(-1.5, 1.5) 7.6

(-1.0, 1.5) 5.7

(-0.5, 1.5) 3.3

Energy window (eV) U

(-2.0, 1.7) 6.6

(-1.5, 1.7) 5.4

(-1.0, 1.7) 4.3

(-0.7, 1.7) 3.2

cLDA U=6 eV

slide11

U as a function of eliminated bands

Ni

Band 2 to 6 are eliminated

Ce

compare with

cLDA=5.4 eV

V

Band 2 to 8 are eliminated

The 4f bands of Ce correspond to band 2 to 8.

The 3d bands of Ni and V correspond to band 2 to 6

slide12

For the 3d transition metals and Cerium the following

hybrid criterion has been used:

Lower bound: eliminate the lowest band (4s)

Upper bound: use energy cut off

slide13

Roles of screening channels: Vanadium vs Nickel

(early vs late transition metals)

slide14

Vanadium:

  • Eliminating all transitions from the 3d bands has little effect on U(0)(green)
  •  In early transition elements the screening for U (0) is provided by the 4s electrons.
  • Nickel:
  • In contrast to vanadium, eliminating all transitons from the 3d bands
  • has a large effect on U(0) (green)
  •  In late transition elements screening from 4s electrons alone are not sufficient
  • to obtain U(0). The 3d electrons contribute significantly to screening.

Eliminating transitions from the 4s band has no effects on W(0) for both V and Ni(red)

3d screening is metallic, very efficient in screening a point charge without help from

the 4s electrons. W(0) is rather constant across the 3d series.

slide15

Connection between constrained LDA and constrained RPA

Janak’s theorem

From the Kohn-Sham equation:

Constrained RPA:

Constrain transitions in

Constrained LDA:

Constrain hoppings

dielectric function

U von Barth, The Electronic Structure of Complex Systems,Vol 113 NATO series B: Physics p67.

M Springer and FA, PRB 57, 4364 (1998)

slide16

Constrained LDA

Super Cell

Transition metal or

rare earth atom

Hopping from and to

3d orbitals is cut off

“impurity”

Change the 3d charge on the impurity, keeping the system neutral,

do a self-consistent calculation

and calculate the change in the 3d energy level  U(3d).

slide17

SrVO3

t_2g

Only O2p screening

e_g

“self-

screening”

compare with cLDA=9 eV

U as a function of eliminated transitions

(c.f. similar result, Solovyev, cond-mat/0506632)

slide18

SrVO3: 1 d system

Eliminating all transitions from the 3d bands (red curve) has almost

no influence on U(0)

slide19

Comparison between cRPA and cLDA

for 3d transition metals

U (cLDA)

U (cRPA)

W (RPA)

The comparison is not clear cut because the 3d band is not completely isolated.

slide20

Breathing or Orbital relaxation

In constrained LDA calculations, the 3d/4f orbitals are allowed to relax.

Relaxing the 3d orbitals are equivalent to polarising them.

In the language of RPA: 3d3d, 3d4d, 3d5d, etc. transitions

Not allowed

Allowed

In constrained LDA, the 3d/4f orbitals should be fixed.

In constrained LDA calculations

orbital relaxation compensates for the lack of self-screening.

slide22

Influence of energy dependence of U

Spectral function of Ni from the Hubbard model with a static U,

compared with the “true” one

slide23

The real and imaginary part of the self-energy from the Hubbard model

with a static U compared with the “true” self-energy.

“true”

“true”

Hubbard

model

Hubbard

model

The Hubbard model should work if the high energy part of

is well separated from the low energy part

slide24

Self-energy of Ni from the Hubbard model

with an energy-dependent Hubbard U

maximally localized generalized wannier function
Maximally localized generalized Wannier function

Marzari and Vanderbilt, PRB56, 12847 (1997)

Souza, Marzari and Vanderbilt, PRB65, 035109 (2001)

Wannier function

Spread of Wannier function

Use Wannier functions as basis for a model Hamiltonian

slide26

On-site interaction at w=0

LMTO-ASA

(the head – partial wave)

Maxloc Wannier

Hubbard U

Ni

Fully screened W

Full-Potential LMTO-GW

(Takashi Miyake)

slide27

The screened exchange interaction J of some 3d metals

Fe

Ni

Filled black triangle:

fully screened J

Cu

Empty blue triangle:

J calculated according to cRPA

A non-negligible reduction of about 20 % from

the bare atomic value is found.

slide28

Wannier orbitals obtained by maximising U

Form a linear combination of maxloc Wannier orbitals in real space:

Max. loc. Wannier

Edmiston and Ruedenberg, Rev. Mod. Phys. 35, 457 (1963)

slide29

We have defined an anti-Hermitian matrix F

Steepest ascent:

which ensures that

Construct

slide30

The Hubbard U calculated in maxU Wannier orbitals

are surprisingly close to the values calculated in the

maximally localised Wannier orbitals.

slide31

Check the procedure

Have not found the global maximum of U?

summary
Summary
  • cRPA allows for a systematic calculation of U: Full U matrix, energy-dependent U
  • In early transition metals, the 4s electrons do most of the screening for U
  • but in late transition metals, the 3d-electron screening contributes significantly to U.
  • In transition metals, the 3d electrons are very efficient in screening
  • a point charge (metallic screening)  W is almost constant across the series.
  • One source of discrepancy between cLDA and cRPA may be attributed to self-screening
  • * 3dnon-3d transitions in transition metals,
  • * O2p3d in SrVO3,
  • Orbital relaxation?
  • Self-screening and orbital relaxation tend to cancel each other.
  • Energy dependence of U can be large, even at low energy.
  • How to find a static U that takes into account the variation in energy.
  • How to solve an impurity model with an energy-dependent U
  • Maximally localised Wannier orbitals together with constrained RPA
  • provide an unambiguous way of constructing low-energy model Hamiltonians.
  •  Applications to GW+DMFT