The Gutzwiller Density Functional Theory · 1. Gutzwiller wave functions 2. Ferromagnetism in...
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The Gutzwiller Density Functional Theory
IV) Further developments
I) Introduction
II) Gutzwiller variational theory
Jörg Bünemann, BTU Cottbus
III) The Gutzwiller density functional theory
1. Model for an H2-molecule
2. Transition metals and their compounds
1. Gutzwiller wave functions
2. Ferromagnetism in two-band models
3. Magnetic order in LaFeAsO
1. Remainder: The Density-Functional Theory
2. The Gutzwiller Kohn-Sham equations
3. Example: lattice parameters of iron pnictides
[2. The time-dependent Gutzwiller theory]
1. Superconductivity: beyond the Gutzwiller approximation
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I) Introduction
1. Model for an H2-molecule:
energies:
: Coulomb interaction (intra-atomic)
(spin: )
t
t
t t
basis:
matrix element for transitions:
t
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1.1. Perspective of elementary quantum mechanics:
ground state:
matrix
(ground-state energy)
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1.2. Perspective of solid-state theory
ground state:
('molecular orbitals': 'bonding' & 'anti-bonding')
first: solve the 'single-particle problem' ( ) :
ground state is 'single-particle product state'
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Effective single-particle ('Hartree-Fock') theory:
Idea: find the single-particle product state with the lowest
energy
Problems:
('correct' spin symmetry: eigenstate of with eigenvalue )
energy
Hartree-Fock ground state breaks
i) for , is never a single-particle product state
ii) the HF ground state with the 'correct' spin symmetry is
iii)
spin-symmetry for :
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1.3. Perspective of many-particle theory
(operator for double occupancies)
with increasing , double occupancies are more
and more suppressed in the ground stateGutzwiller's idea:
Gutzwiller wave function: M. C. Gutzwiller PRL 10, 159 (1963) ( )
compare:
proper choice of the variational parameter
reproduces the exact ground state
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2. Transition metals and their compounds
partially filled d-shell: with and
2.1. Transition-metal atoms
in cubic environment:orbitals
orbitals
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with the local 'atomic' Hamiltonian:
('multi-band Hubbard models”)
combined spin-orbital index
2.2. Lattice models
Hamiltonian for transition metals with partially filled d-shells:
local Coulomb interaction
orbital energies
('single-band Hubbard model')
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Local Hamiltonian for d-orbitals in cubic environment:
with 10 independent parameters
in spherical approximation: 3 Racah parameters
use mean valuesalternatively:
and
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3d wave functions are rather localised
in solids, the local Coulomb interaction and the band-width are of the same order of magnitude
- magnetism
- metal-insulator transitions
- magneto-resistance ('giant', 'colossal')
- orbital order
- high-temperature superconductivity
Experiment:
Theory: effective single-particle theories often fail
simplest example: fcc nickel
2.3. Interaction effects in transition-metal compounds
(Hartree-Fock or local density approximation in density-functional theory)
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1. Gutzwiller wave functions
for the single-band Hubbard model
one defines: (Gutzwiller wave function)
mit i) arbitrary single-particle product wave function
ii)
alternative formulation:
with 'atomic' eigenstates
and variational parameters :
1.1 Definitions
und
II) Gutzwiller variational theory
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multi-band Hubbard models
: atomic eigenstates with energies
e.g.: 3d-shell: dimensional local Hilbert space
10 spin-orbitals
with
generalised Gutzwiller wave function:
: matrix of variational parameters (in this lecture )
problem: is still a complicated many-particle wave function
cannot be evaluated in general
mit
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1.2. Evaluation: Diagrammatic expansion
We need to calculate
is a single-particle product wave function
Wick theorem applies, e.g.,
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Diagrammatic representation of all terms, e.g.,
with lines
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1.3. Simplifications in infinite dimensions
A) Diagrams with three or more lines
Kinetic energy per site on a D-dimensional lattice:
for
(only n.n. hopping)
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B) Diagrams with two lines
Idea: make sure that all these diagrams cancel each other
This is achieved by the (local) constraints
C) Disconnected diagrams are cancelled by the norm
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Conclusion: All diagrams with internal vertices vanish
The remaining evaluation is rather straightforward:
='renormalisation matrix'
with
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1.4. Summary: energy functional in infinite spatial dimensions
(effective hopping parameters)
in the limit of infinite dimensions leads to
and are analytic functions of
and
Evaluation of
Remaining numerical problem:
Minimisation of with respect to and
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1.5. Minimisation of the energy functional
We consider as a function of and of the non-interacting
density matrix with the elements
'non-interacting'
Minimisation with respect to leads to the effective single
particle equation
bands are i) mixed ( ) and shifted ( )
ii) renormalised ( )
with
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2. Ferromagnetism in two-band models
16 local states:
local Hamiltonian:
-orbitals:
and
1 2
orbital
2-particle states energy Sym.
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-orbitals on a simple cubic 3-dimensional lattice
density of states:Gutzwiller wave function:
i) no multiplet coupling
ii) spin-polarisedFermi sea
:
Known: ferromagnetism is hardly found in single-band models
requires pathological densities of states and/or
very large Coulomb parameters
orbital degeneracy is an essential ingredient
therefore, we consider:
with hopping to nearest and next-nearest neighbours
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Results:
phase diagramMagnetisation ( )
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condensation energy
i) Orbital degeneracy and exchangeinteraction are essential forferromagnetic order
ii) Single-particle approaches areinsufficient
size of the local spin
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3. Magnetic order in LaFeAsO
3.1 Electronic structure of LaFeAsO
i) Metal with conductivity mainly in FeAs layers
ii) AF ground state with a magnetic moment of
Y. Kamihara et al., J. Am. Chem. Soc 130,3296 (2008)
(K. Ishida et al. J. Phys. Soc. Jpn. 78, 062001 (2009)) ( )
(DFT: )
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3.2 LDA band-structure and effective tight-binding models
Eight-band model Five-band model:
Three-band model:
O. K Andersen and L. Boeri, Ann. Physik 523, 8 (2011)( )
(S. Zhou et al., PRL 105, 096401 (2010))
(S. Graser et al., New J. Phys. 11, 025016 (2009))
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3.3 Magnetic order in three-band models
Without spin-flip terms:
(S. Zhou et al., PRL 105, 096401 (2010))
With spin-flip terms:
Hartree-Fock: Conclusion (?):
Gutzwiller theory yields a reasonable magnetic moment without fine-tuningof model parameters
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3.4 Five-band model
A) Hartree-Fock:
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Conclusions:
- small magnetic moments appear only in a small range of correlation parameters
- no orbital order (in contrast to Hartree-Fock)
still no satisfactory explanation for the magnetic orderobserved in LaFeAsO
T. Schickling et al., PRL 106, 156402 (2011) ( ) 3.4 Five-band model
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3.5 Electronic properties of LaAsFeO (eight-band model)
Phase diagram: Magnetic moment:
Reasonable values of the magnetic moment over alarge range of Coulomb parameters
T. Schickling et al., PRL 108, 036406 (2012) ( )
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III) The Gutzwiller Density Functional Theory
1. Remainder: The Densitiy Functional Theory
Electronic Hamiltonian in solid-state physics
Hohenberg-Kohn theorem:
Existence of a universal functional of the density such that
has its minimum at the exact ground-state density of
('universal' = independent of )
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One usually writes
with : 'kinetic energy functional'
: 'exchange correlation functional'
common approximations:free electron gas
free electron gas
HF approximation for
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Kohn-Sham scheme:
Instead of , consider the effective single-particle Hamiltonian
with the 'Kohn-Sham potential' Kinetic energy of non-interacting
particles
and have the same ground-state density
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Kohn-Sham equations:
We introduce a basis of local orbitals
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2. The Gutzwiller Kohn-Sham equations
We distinguish 'localised' ( ) and 'delocalised' orbitals ( )
where and
are now functionals of the density
Problem: Coulomb interaction is counted twice in the localised
orbitals 'double-counting problem'
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Density in the Gutzwiller ground state
depends on and
Gutzwiller-DFT energy functional:
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Minimisation
leads to 'Gutzwiller Kohn-Sham equations' with
and
for( )
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Correlation-induced changes of :
i)
differs from the DFT expression
ii) Correlated bands are shifted (via )
change of change of
iii) Correlated bands are renormalised in
change of and in
change of
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Problems:
1. The local Coulomb interactions are usually
considered as adjustable parameters
'ab-initio' character is partially lost
2. Double-counting problem:
Coulomb interaction appears in and in
One possible solution: subtract
from
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3. Example: Lattice parameters of iron pnictides
Interlayer distance and (average) band renormalisation
from G. Wang et. al, Phys. Rev. Lett. 104, 047002 (2010)
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Elastic constants:
softening of the corresponding phonon mode
in agreement with experiment
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1. Superconductivity: beyond the Gutzwiller approximation
1.1 Diagrammatic expansion
In the single-band case we need to calculate
Procedure:
A) Proper choice of the expansion parameter
B) Use of Wick's theorem and the linked-cluster theorem
Diagrammatic representation
C) Numerical evaluation in real space
i)
ii)
iii)
J. Bünemann et al., EPL 98, 27006 (2012) ( )
IV) Further developments
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This fixes three parameters and it remains only one ( or )
A) Proper choice of the expansion parameter
Main idea: Choose parameters such that
with!
Main advantage of the HF-operators: no 'Hartree bubbles'
e.g.:
with
In contrast:
(Wick's theorem)
,
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The missing of Hartree bubbles has two consequences:
i) Number of diagrams is significantly reduced, e.g.
Diagrams are fairly localised and the power series in converges rapidly (can be tested for )
ii) Each line is ( number of spatial dimensions)
This suggests the following strategy:
i) Calculate
in momentum space (i.e., with negligible numerical error)
ii) Calculate all diagrams (power series in ) in real space up
iii) Minimise the energy with respect to
to a certain order in
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1.2 The one-dimensional Hubbard model
In one dimension one can evaluate Gutzwiller wave functions exactly [W. Metzner and D. Vollhardt, PRL 59, 121 (1987)]
convergence of our approach can be tested
With the exact results we may calculate analytically each order of double occupancy and the kinetic energy :
i) Double occupancy ii) Kinetic energy
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1.3 Fermi-surface deformations in two dimensions
The Gutzwiller wave function
,
contains as variational objects the parameters (i.e., ) and the wave function . Without breaking translational or spin-Symmetry the only remaining degree of freedom in a one-band model is the shape of the Fermi surface:
Only constraint:
(particle number conservation)
Minimisation with respect to :
with
,
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Pomeranchuk instabilities
According to fRG calculations it may happen that the Fermi surface spontaneously breaks the rotational symmetry of the system at finite ('Pomeranchuk instability').
But:
i) fRG is a perturbative approach
ii) Little is know quantitatively, in particular for larger values of .
Obvious question:
Do we find a Pomeranchuk instabilityin our approach?
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1.4 Hamiltonian with only nearest-neighbor hopping
'Normal' Fermi surface:
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Pomeranchuk phase:
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1.5 Superconductivity in two-dimensional Hubbard models
Pairing beyond is relevant
collaboration with J. Kaczmarczyk, Krakow ( )
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Consequence 1: enhancement of stability region
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Consequence 2: gap-structure
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Phase diagram ( )
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with
“Kubo formula”:
(Fourier transformation)
2.1 Two-particle excitations: linear response theory
Aim: calculate the density matrix with the elements
with
equation of motion
2. The time-dependent Gutzwiller theory
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2.2 Time-dependent Hartree-Fock Theory (RPA):
Equation of motion
Approximation:
decouples (Wick's theorem)
is a closed set of differential equations for
In linear order with respect to this leads to the RPA result
with and
with
is assumed to be a single-particle wave function
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2.3 The time-dependent Gutzwiller theory G. Seibold und J. Lorenzana PRL 86, 2605 (2001) ( )
Comparison:
RPA Time-dep. Gutzwiller theory
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“Stoner continuum”
Spin-flip-excitation in the “exchange field”
Stoner criterion: ferromagnetism
finite exchange interaction
with
2.4 RPA for the spin susceptibility of a single-band model
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2.5 Spin excitations in single-band models
A) Hypercubic lattices in infinite dimensions
i) simple cubic (sc) ii) half cubic (hc)
densities of states
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Two-particle response functions in large dimensions only depend on the real parameter
i) Phase diagram of the hc-lattice:
instability of the paramagnet: spin-wave stability
(F. Günther et al., PRL 98, 176404 (2007))
(DMFT: G. Uhrig, PRL 77, 3629 (1996))
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instability of the paramagnet:
stability of the ferromagnet:
ii) Phase diagram of an sc lattice:
stablespin waves
(DMFT: Obermeier et al., PRB 56, 8479 (1997))
2.6 Spin excitations in a two-band model
Hartree-Fock phase diagrams:
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Gutzwiller phase diagrams:
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V) Conclusions
1. The Gutzwiller variational approach provides with us a numerically
'Cheap' way to study multi-band Hubbard models for transition
metals and their compounds.
2. The modest numerical efforts of this method make it particularly
suitable for a self-consistent merger with the DFT.
3. Apart from ground-state properties the method allows us to
calculate quasi-particle excitations and two-particle response
functions.
4. Systematic improvements of the infinite D limit are possible and
sometimes necessary to study, e.g., correlation-induced forms of
superconductivity.