Constrained RPA: Calculating the Hubbard U from FirstPrinciples

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Constrained RPACalculating the Hubbard U from
First-Principles
Bonn 2008.01.10-12
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)
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Motivations and 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-principl
es (parameter-free) method. Adjustable
parameters may give nice results but not
necessarily for good reasons.
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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
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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
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Polarization function
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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)
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(No Transcript)
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Static Hubbard U for the 3d transition metal
series
?For a more accurate model it may be necessary to
include nearest-neighbour U
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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
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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 U6 eV
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U as a function of eliminated bands
Ni
Band 2 to 6 are eliminated
Ce
?compare with cLDA5.4 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
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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
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Roles of screening channels Vanadium vs
Nickel (early vs late transition metals)
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• 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.
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Connection between constrained LDA and
constrained RPA
Janaks 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)
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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).
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SrVO3
t_2g
Only O2p screening
e_g
self- screening
?compare with cLDA9 eV
U as a function of eliminated transitions
(c.f. similar result, Solovyev, cond-mat/0506632)
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SrVO3 1 d system
Eliminating all transitions from the 3d bands
(red curve) has almost no influence on U(0)
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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.
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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.
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Energy dependence of U
Gd
Ce
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Influence of energy dependence of U
Spectral function of Ni from the Hubbard model
with a static U, compared with the true one
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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
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Self-energy of Ni from the Hubbard model with an
energy-dependent Hubbard U
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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
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On-site interaction at w0
LMTO-ASA (the head partial wave)
Maxloc Wannier
Hubbard U
Ni
Fully screened W
Full-Potential LMTO-GW (Takashi Miyake)
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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.
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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)
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We have defined an anti-Hermitian matrix F
Steepest ascent
which ensures that
Construct
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The Hubbard U calculated in maxU Wannier
orbitals are surprisingly close to the values
calculated in the maximally localised Wannier
orbitals.
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Check the procedure
Have not found the global maximum of U?
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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.