Gutzwiller wave functions for itinerant ferromagnetism of transition metals Florian Gebhard fachbereich physik, philipps- universität marburg

Collaboration

Theory

J. Bünemann, Marburg; W. Weber, Dortmund

Experiment

R. Claessen, V. Strocov, Würzburg; A. Kakizaki, S. Shin, Tokyo; D. Ehm, G. Nicolay, Saarbrücken; A. Kimura, Hiroshima

Overview

Florian Gebhard : Itinerant Ferromagnetism – p. 2/34

Overview 1. Basic experimental results

Florian Gebhard : Itinerant Ferromagnetism – p. 2/34

Overview 1. Basic experimental results 2. Stoner–Slater theory: band magnetism

Florian Gebhard : Itinerant Ferromagnetism – p. 2/34

Overview 1. Basic experimental results 2. Stoner–Slater theory: band magnetism 3. From Hund to Heisenberg: magnetic insulators

Florian Gebhard : Itinerant Ferromagnetism – p. 2/34

Overview 1. Basic experimental results 2. Stoner–Slater theory: band magnetism 3. From Hund to Heisenberg: magnetic insulators 4. Unified description: Gutzwiller variational wave functions

Florian Gebhard : Itinerant Ferromagnetism – p. 2/34

Overview 1. Basic experimental results 2. Stoner–Slater theory: band magnetism 3. From Hund to Heisenberg: magnetic insulators 4. Unified description: Gutzwiller variational wave functions 5. Results for a generic two-band model

Florian Gebhard : Itinerant Ferromagnetism – p. 2/34

Overview 1. Basic experimental results 2. Stoner–Slater theory: band magnetism 3. From Hund to Heisenberg: magnetic insulators 4. Unified description: Gutzwiller variational wave functions 5. Results for a generic two-band model 6. Results for nickel

Florian Gebhard : Itinerant Ferromagnetism – p. 2/34

Overview 1. Basic experimental results 2. Stoner–Slater theory: band magnetism 3. From Hund to Heisenberg: magnetic insulators 4. Unified description: Gutzwiller variational wave functions 5. Results for a generic two-band model 6. Results for nickel 7. Summary

Florian Gebhard : Itinerant Ferromagnetism – p. 2/34

1 Basic experimental results

Florian Gebhard : Itinerant Ferromagnetism – p. 3/34

1 Basic experimental results Transition metals Fe, Co, Ni: itinerant ferromagnets

Florian Gebhard : Itinerant Ferromagnetism – p. 3/34

1 Basic experimental results Transition metals Fe, Co, Ni: itinerant ferromagnets 3d shell is not completely filled;

Florian Gebhard : Itinerant Ferromagnetism – p. 3/34

1 Basic experimental results Transition metals Fe, Co, Ni: itinerant ferromagnets 3d shell is not completely filled; 3d electrons contribute to the metallic conduction;

Florian Gebhard : Itinerant Ferromagnetism – p. 3/34

1 Basic experimental results Transition metals Fe, Co, Ni: itinerant ferromagnets 3d shell is not completely filled; 3d electrons contribute to the metallic conduction; 3d electrons carry almost all of the magnetic moment;

Florian Gebhard : Itinerant Ferromagnetism – p. 3/34

1 Basic experimental results Transition metals Fe, Co, Ni: itinerant ferromagnets 3d shell is not completely filled; 3d electrons contribute to the metallic conduction; 3d electrons carry almost all of the magnetic moment; g-factor is purely spin, g ≈ 2;

Florian Gebhard : Itinerant Ferromagnetism – p. 3/34

1 Basic experimental results Transition metals Fe, Co, Ni: itinerant ferromagnets 3d shell is not completely filled; 3d electrons contribute to the metallic conduction; 3d electrons carry almost all of the magnetic moment; g-factor is purely spin, g ≈ 2; magnetic order sets in at the Curie temperature TC with 

 TC = O 10 K 3

Florian Gebhard : Itinerant Ferromagnetism – p. 3/34

1 Basic experimental results Consequences:

Florian Gebhard : Itinerant Ferromagnetism – p. 4/34

1 Basic experimental results Consequences: 1. 3d electrons are delocalized over the specimen;

Florian Gebhard : Itinerant Ferromagnetism – p. 4/34

1 Basic experimental results Consequences: 1. 3d electrons are delocalized over the specimen; 2. itinerant 3d electrons are responsible for the magnetism (contribution from 4s and 4p is small);

Florian Gebhard : Itinerant Ferromagnetism – p. 4/34

1 Basic experimental results Consequences: 1. 3d electrons are delocalized over the specimen; 2. itinerant 3d electrons are responsible for the magnetism (contribution from 4s and 4p is small); 3. large values for the Curie temperature can only be understood from the competition between the electrons’ kinetic energy (bandwidth W = O(eV)) the electrons’ potential energy (mutual Coulomb repulsion U = O(eV)).

Florian Gebhard : Itinerant Ferromagnetism – p. 4/34

2 Stoner–Slater theory

Florian Gebhard : Itinerant Ferromagnetism – p. 5/34

2 Stoner–Slater theory 2.1

Starting point: non-interacting electrons ˆ0 = H

X ~ ~ l,m

X

(bσ ),(b′ σ ′ )

bσ ,b′ σ ′ + cˆ~ cˆ ~ ′ ′ t~ ~ l,bσ m,b σ l,m

=

XX ~ bσ k

~ b)dˆ+ ǫ(k, ~

k,bσ

dˆ~

k,bσ

Florian Gebhard : Itinerant Ferromagnetism – p. 5/34

2 Stoner–Slater theory 2.1

Starting point: non-interacting electrons ˆ0 = H

X ~ ~ l,m

X

(bσ ),(b′ σ ′ )

bσ ,b′ σ ′ + cˆ~ cˆ ~ ′ ′ t~ ~ l,bσ m,b σ l,m

=

XX ~ bσ k

~ b)dˆ+ ǫ(k, ~

k,bσ

dˆ~

k,bσ

Density of states per spin direction:  X  ~ b) − E δ ǫ(k, Dσ (E) = ~ k,b

In the ground state, all states are filled up to the Fermi energy EF :

Florian Gebhard : Itinerant Ferromagnetism – p. 5/34

2 Stoner–Slater theory 2.1

Starting point: non-interacting electrons ˆ0 = H

X ~ ~ l,m

X

(bσ ),(b′ σ ′ )

bσ ,b′ σ ′ + cˆ~ cˆ ~ ′ ′ t~ ~ l,bσ m,b σ l,m

=

XX ~ bσ k

~ b)dˆ+ ǫ(k, ~

k,bσ

dˆ~

k,bσ

Density of states per spin direction:  X  ~ b) − E δ ǫ(k, Dσ (E) = ~ k,b

In the ground state, all states are filled up to the Fermi energy EF : metal (OK); purely electronic g-factor, g ≈ 2 (OK); paramagnetic (not OK). Florian Gebhard : Itinerant Ferromagnetism – p. 5/34

2 Stoner–Slater theory 2.2

Electron-electron interaction

Florian Gebhard : Itinerant Ferromagnetism – p. 6/34

2 Stoner–Slater theory 2.2

Electron-electron interaction

First order perturbation theory requires the pair distribution function: nσ nσ ′ gσ ,σ ′ (r − r ′ ) gives the probability for finding a σ -electron at r when there is a σ ′ electron at r ′

Florian Gebhard : Itinerant Ferromagnetism – p. 6/34

2 Stoner–Slater theory 2.2

Electron-electron interaction

First order perturbation theory requires the pair distribution function: nσ nσ ′ gσ ,σ ′ (r − r ′ ) gives the probability for finding a σ -electron at r when there is a σ ′ electron at r ′ 1

0.8

gσσ(x)

0.6

0.4

0.2

0 0

2

4

6

8

10

x = kF | r-r’|

Florian Gebhard : Itinerant Ferromagnetism – p. 6/34

2 Stoner–Slater theory 2.2

Electron-electron interaction

First order perturbation theory requires the pair distribution function: nσ nσ ′ gσ ,σ ′ (r − r ′ ) gives the probability for finding a σ -electron at r when there is a σ ′ electron at r ′ 1

0.8

gσσ(x)

0.6

0.4

0.2

0 0

2

4

6

8

10

x = kF | r-r’|

The Coulomb interaction between electrons with like spins is smaller than the Coulomb interaction between electrons with different spins because of the Pauli or exchange hole. Energy difference: “exchange energy”

Florian Gebhard : Itinerant Ferromagnetism – p. 6/34

2 Stoner–Slater theory Result:

Florian Gebhard : Itinerant Ferromagnetism – p. 7/34

2 Stoner–Slater theory Result: tendency towards ferromagnetism (gain the exchange energy!);

Florian Gebhard : Itinerant Ferromagnetism – p. 7/34

2 Stoner–Slater theory Result: tendency towards ferromagnetism (gain the exchange energy!); counter-tendency to paramagnetism (optimize the kinetic energy!).

Florian Gebhard : Itinerant Ferromagnetism – p. 7/34

2 Stoner–Slater theory Result: tendency towards ferromagnetism (gain the exchange energy!); counter-tendency to paramagnetism (optimize the kinetic energy!). Consequence: transition at Uc with Uc Dσ (EF ) = 1

Stoner criterion

Florian Gebhard : Itinerant Ferromagnetism – p. 7/34

2 Stoner–Slater theory Result: tendency towards ferromagnetism (gain the exchange energy!); counter-tendency to paramagnetism (optimize the kinetic energy!). Consequence: transition at Uc with Uc Dσ (EF ) = 1

Stoner criterion

U measures the strength of the Coulomb interaction, Dσ (EF ) ∼ 1/W measures the importance of the kinetic energy.

Florian Gebhard : Itinerant Ferromagnetism – p. 7/34

2 Stoner–Slater theory 2.3

Praise and dispraise

Florian Gebhard : Itinerant Ferromagnetism – p. 8/34

2 Stoner–Slater theory 2.3

Praise and dispraise Exactly solvable models with Dσ (EF ) = ∞ (‘flat-band models’) show ferromagnetism at arbitrarily weak Coulomb interaction (Uc = 0+ ).

Florian Gebhard : Itinerant Ferromagnetism – p. 8/34

2 Stoner–Slater theory 2.3

Praise and dispraise Exactly solvable models with Dσ (EF ) = ∞ (‘flat-band models’) show ferromagnetism at arbitrarily weak Coulomb interaction (Uc = 0+ ). Real materials: Uc = O(W )

large energy!

Florian Gebhard : Itinerant Ferromagnetism – p. 8/34

2 Stoner–Slater theory 2.3

Praise and dispraise Exactly solvable models with Dσ (EF ) = ∞ (‘flat-band models’) show ferromagnetism at arbitrarily weak Coulomb interaction (Uc = 0+ ). Real materials: Uc = O(W )

large energy!

For U ≈ W a correlation hole between ↑ and ↓ electrons has formed: exchange and correlations are equally important.

Florian Gebhard : Itinerant Ferromagnetism – p. 8/34

2 Stoner–Slater theory 2.3

Praise and dispraise Exactly solvable models with Dσ (EF ) = ∞ (‘flat-band models’) show ferromagnetism at arbitrarily weak Coulomb interaction (Uc = 0+ ). Real materials: Uc = O(W )

large energy!

For U ≈ W a correlation hole between ↑ and ↓ electrons has formed: exchange and correlations are equally important. Conclusion: A large density of states at the Fermi energy promotes ferromagnetism (Stoner criterion). Florian Gebhard : Itinerant Ferromagnetism – p. 8/34

3 From Hund to Heisenberg

Florian Gebhard : Itinerant Ferromagnetism – p. 9/34

3 From Hund to Heisenberg 3.1

Starting point: magnetic moments in atoms

First Hund’s rule in atoms with incompletely filled shells: “Ground state has maximum spin”

Florian Gebhard : Itinerant Ferromagnetism – p. 9/34

3 From Hund to Heisenberg 3.1

Starting point: magnetic moments in atoms

First Hund’s rule in atoms with incompletely filled shells: “Ground state has maximum spin” Reason: Coulomb interaction between the electrons in the atom ˆat = H

X

(b σ ),(b σ )

U(b31σ31),(b42σ43) cˆb+1 σ1 cˆb+2 σ2 cˆb3 σ3 cˆb4 σ4

(b1 σ1 ),...,(b4 σ4 )

Florian Gebhard : Itinerant Ferromagnetism – p. 9/34

3 From Hund to Heisenberg 3.1

Starting point: magnetic moments in atoms

First Hund’s rule in atoms with incompletely filled shells: “Ground state has maximum spin” Reason: Coulomb interaction between the electrons in the atom ˆat = H

X

(b σ ),(b σ )

U(b31σ31),(b42σ43) cˆb+1 σ1 cˆb+2 σ2 cˆb3 σ3 cˆb4 σ4

(b1 σ1 ),...,(b4 σ4 )

For spherical symmetry and d electrons, the Coulomb parameters U may be expressed in terms of 3 Racah parameters A, B, C.

Florian Gebhard : Itinerant Ferromagnetism – p. 9/34

3 From Hund to Heisenberg 3.1

Starting point: magnetic moments in atoms

First Hund’s rule in atoms with incompletely filled shells: “Ground state has maximum spin” Reason: Coulomb interaction between the electrons in the atom ˆat = H

X

(b σ ),(b σ )

U(b31σ31),(b42σ43) cˆb+1 σ1 cˆb+2 σ2 cˆb3 σ3 cˆb4 σ4

(b1 σ1 ),...,(b4 σ4 )

For spherical symmetry and d electrons, the Coulomb parameters U may be expressed in terms of 3 Racah parameters A, B, C. Atomic eigenstates |Γ i ˆat |Γ i = EΓ |Γ i H The ground state |Γ i has maximum spin: atomic physics naturally provides magnetic moments.

Florian Gebhard : Itinerant Ferromagnetism – p. 9/34

3 From Hund to Heisenberg 3.2

Heisenberg theory

Florian Gebhard : Itinerant Ferromagnetism – p. 10/34

3 From Hund to Heisenberg 3.2

Heisenberg theory

Second order perturbation theory in the electron motion (“virtual hopping”) generates coupling of atomic magnetic moments. Effective theory: ˆHeisenberg = H

X

ˆ ~~Sˆ ~~ J~l,m S ~ l m

;

2 J~l,m = O(W /U ) ~

~ ~ l,m

Florian Gebhard : Itinerant Ferromagnetism – p. 10/34

3 From Hund to Heisenberg 3.2

Heisenberg theory

Second order perturbation theory in the electron motion (“virtual hopping”) generates coupling of atomic magnetic moments. Effective theory: ˆHeisenberg = H

X

ˆ ~~Sˆ ~~ J~l,m S ~ l m

;

2 J~l,m = O(W /U ) ~

~ ~ l,m

For J~l,m ~ > 0: anti-ferromagnetism for T < TNéel For J~l,m ~ < 0: ferromagnetism for T < TCurie

Florian Gebhard : Itinerant Ferromagnetism – p. 10/34

3 From Hund to Heisenberg 3.2

Heisenberg theory

Second order perturbation theory in the electron motion (“virtual hopping”) generates coupling of atomic magnetic moments. Effective theory: ˆHeisenberg = H

X

ˆ ~~Sˆ ~~ J~l,m S ~ l m

;

2 J~l,m = O(W /U ) ~

~ ~ l,m

For J~l,m ~ > 0: anti-ferromagnetism for T < TNéel For J~l,m ~ < 0: ferromagnetism for T < TCurie Physical picture: Pre-formed moments develop long-range order; For T > Tc = O(J): moments decouple, long-range order is lost but the local moments persist. Florian Gebhard : Itinerant Ferromagnetism – p. 10/34

3 From Hund to Heisenberg 3.3

Praise and dispraise

Florian Gebhard : Itinerant Ferromagnetism – p. 11/34

3 From Hund to Heisenberg 3.3

Praise and dispraise Exact description of the strong-coupling limit U ≫ W ;

Florian Gebhard : Itinerant Ferromagnetism – p. 11/34

3 From Hund to Heisenberg 3.3

Praise and dispraise Exact description of the strong-coupling limit U ≫ W ; No real motion of the electrons: insulator; Not a suitable description for transition metals!

Florian Gebhard : Itinerant Ferromagnetism – p. 11/34

3 From Hund to Heisenberg 3.3

Praise and dispraise Exact description of the strong-coupling limit U ≫ W ; No real motion of the electrons: insulator; Not a suitable description for transition metals!

3.4

Van Vleck’s “Minimum Polarity Model” (1953)

In transition metals the motion of the electrons is strongly correlated – metallic conduction, yet no charge fluctuations;

Florian Gebhard : Itinerant Ferromagnetism – p. 11/34

3 From Hund to Heisenberg 3.3

Praise and dispraise Exact description of the strong-coupling limit U ≫ W ; No real motion of the electrons: insulator; Not a suitable description for transition metals!

3.4

Van Vleck’s “Minimum Polarity Model” (1953)

In transition metals the motion of the electrons is strongly correlated – metallic conduction, yet no charge fluctuations; on long time scales: atomic situation – no charge fluctuations, local magnetic moments;

Florian Gebhard : Itinerant Ferromagnetism – p. 11/34

3 From Hund to Heisenberg 3.3

Praise and dispraise Exact description of the strong-coupling limit U ≫ W ; No real motion of the electrons: insulator; Not a suitable description for transition metals!

3.4

Van Vleck’s “Minimum Polarity Model” (1953)

In transition metals the motion of the electrons is strongly correlated – metallic conduction, yet no charge fluctuations; on long time scales: atomic situation – no charge fluctuations, local magnetic moments; coupling of moments due to electrons’ motion – long-range order at low temperatures. Florian Gebhard : Itinerant Ferromagnetism – p. 11/34

3 From Hund to Heisenberg Problem: very qualitative views – specific calculations are missing!

Florian Gebhard : Itinerant Ferromagnetism – p. 12/34

3 From Hund to Heisenberg Problem: very qualitative views – specific calculations are missing! Van Vleck, Rev. Mod. Phys. 25, 220 (1953): In summary, it may be said that the results of the present paper are rather discouraging, . . . the truth is somewhat between Stoner–Slater theory and the minimum polarity model. Unfortunately, it is much more feasible to make detailed calculations with Stoner–Slater theory than with the minimum polarity model. . . .

Florian Gebhard : Itinerant Ferromagnetism – p. 12/34

3 From Hund to Heisenberg Problem: very qualitative views – specific calculations are missing! Van Vleck, Rev. Mod. Phys. 25, 220 (1953): In summary, it may be said that the results of the present paper are rather discouraging, . . . the truth is somewhat between Stoner–Slater theory and the minimum polarity model. Unfortunately, it is much more feasible to make detailed calculations with Stoner–Slater theory than with the minimum polarity model. . . . Computational difficulties, . . . , should not obscure the recognition in principle of the situation which conforms closest to physical reality.

Florian Gebhard : Itinerant Ferromagnetism – p. 12/34

3 From Hund to Heisenberg Problem: very qualitative views – specific calculations are missing! Van Vleck, Rev. Mod. Phys. 25, 220 (1953): In summary, it may be said that the results of the present paper are rather discouraging, . . . the truth is somewhat between Stoner–Slater theory and the minimum polarity model. Unfortunately, it is much more feasible to make detailed calculations with Stoner–Slater theory than with the minimum polarity model. . . . Computational difficulties, . . . , should not obscure the recognition in principle of the situation which conforms closest to physical reality. The gist of this paper is that it would be highly desirable if good methods of computation with the minimum polarity model could be developed . . . .

Florian Gebhard : Itinerant Ferromagnetism – p. 12/34

4 Gutzwiller variational wave functions

Florian Gebhard : Itinerant Ferromagnetism – p. 13/34

4 Gutzwiller variational wave functions 4.1

Multiband Hubbard models

Combination of both (extreme) viewpoints: ˆ =H ˆ0 + H

X

ˆ~ H l,at

~ l

Florian Gebhard : Itinerant Ferromagnetism – p. 13/34

4 Gutzwiller variational wave functions 4.1

Multiband Hubbard models

Combination of both (extreme) viewpoints: ˆ =H ˆ0 + H

X

ˆ~ H l,at

~ l

Electron’s motion (starting point of Stoner–Slater theory) ˆ0 = H

X

X

′ ′ ~ ~ (bσ ),(b σ ) l,m

bσ ,b′ σ ′ + t~ cˆ~ cˆ ′ ′ ~ ~ l,m σ l,bσ m,b

=

XX ~ bσ k

~ b)dˆ+ ǫ(k, ~

k,bσ

dˆ~

k,bσ

Florian Gebhard : Itinerant Ferromagnetism – p. 13/34

4 Gutzwiller variational wave functions 4.1

Multiband Hubbard models

Combination of both (extreme) viewpoints: ˆ =H ˆ0 + H

X

ˆ~ H l,at

~ l

Electron’s motion (starting point of Stoner–Slater theory) ˆ0 = H

X

X

′ ′ ~ ~ (bσ ),(b σ ) l,m

bσ ,b′ σ ′ + t~ cˆ~ cˆ ′ ′ ~ ~ l,m σ l,bσ m,b

=

XX

~ b)dˆ+ ǫ(k, ~

~ bσ k

k,bσ

dˆ~

k,bσ

Electron’s local Coulomb interaction (starting point of Hund-Heisenberg theory) ˆ~ = H l,at

X

(b1 σ1 ),...,(b4 σ4 )

(b σ ),(b σ )

U(b31σ31),(b42σ43) cˆ~+

cˆ~+

cˆ cˆ l,b4 σ4 l,b3 σ3 ~ l,b1 σ1 l,b2 σ2 ~ Florian Gebhard : Itinerant Ferromagnetism – p. 13/34

4 Gutzwiller variational wave functions Assumptions:

Florian Gebhard : Itinerant Ferromagnetism – p. 14/34

4 Gutzwiller variational wave functions Assumptions: long-range parts of the Coulomb interaction are included in the ~ b) (screening); band structure ǫ(k,

Florian Gebhard : Itinerant Ferromagnetism – p. 14/34

4 Gutzwiller variational wave functions Assumptions: long-range parts of the Coulomb interaction are included in the ~ b) (screening); band structure ǫ(k, local U -parameters are effective couplings, not bare ones

Florian Gebhard : Itinerant Ferromagnetism – p. 14/34

4 Gutzwiller variational wave functions Assumptions: long-range parts of the Coulomb interaction are included in the ~ b) (screening); band structure ǫ(k, local U -parameters are effective couplings, not bare ones Problem:

ˆ is extremely complicated H

Florian Gebhard : Itinerant Ferromagnetism – p. 14/34

4 Gutzwiller variational wave functions Assumptions: long-range parts of the Coulomb interaction are included in the ~ b) (screening); band structure ǫ(k, local U -parameters are effective couplings, not bare ones Problem:

ˆ is extremely complicated H ‘Solution’: use approximate methods

Florian Gebhard : Itinerant Ferromagnetism – p. 14/34

4 Gutzwiller variational wave functions 4.2

Gutzwiller wave functions

Florian Gebhard : Itinerant Ferromagnetism – p. 15/34

4 Gutzwiller variational wave functions 4.2

Gutzwiller wave functions ˆat = 0: exact ground state of H ˆ =H ˆ0 is a one-particle product H state |Ψ0 i (‘Slater determinant’).

Florian Gebhard : Itinerant Ferromagnetism – p. 15/34

4 Gutzwiller variational wave functions 4.2

Gutzwiller wave functions ˆat = 0: exact ground state of H ˆ =H ˆ0 is a one-particle product H state |Ψ0 i (‘Slater determinant’). ˆ0 = 0: exact ground state of H ˆ= H eigenstates |Γ~li.

P

ˆ

~ l,at l H~

is a product of atomic

Florian Gebhard : Itinerant Ferromagnetism – p. 15/34

4 Gutzwiller variational wave functions 4.2

Gutzwiller wave functions ˆat = 0: exact ground state of H ˆ =H ˆ0 is a one-particle product H state |Ψ0 i (‘Slater determinant’). ˆ0 = 0: exact ground state of H ˆ= H eigenstates |Γ~li.

P

ˆ

~ l,at l H~

ˆ =H ˆ0 + Approximate ground state of H Gutzwiller correlated wave function

P

is a product of atomic ˆ

~ l,at : l H~

|ΨG i = PˆG |Ψ0 i

Florian Gebhard : Itinerant Ferromagnetism – p. 15/34

4 Gutzwiller variational wave functions 4.2

Gutzwiller wave functions ˆat = 0: exact ground state of H ˆ =H ˆ0 is a one-particle product H state |Ψ0 i (‘Slater determinant’). ˆ0 = 0: exact ground state of H ˆ= H eigenstates |Γ~li.

P

ˆ

~ l,at l H~

ˆ =H ˆ0 + Approximate ground state of H Gutzwiller correlated wave function

P

is a product of atomic ˆ

~ l,at : l H~

|ΨG i = PˆG |Ψ0 i Idea: the correlator PˆG reduces energetically unfavorable configurations in |Ψ0 i (suppression of charge fluctuations) Florian Gebhard : Itinerant Ferromagnetism – p. 15/34

4 Gutzwiller variational wave functions Choice of the correlator:

PˆG =

YY ~ l

Γ

ˆ ~l,Γ m

λ~

l,Γ

=

YYh ~ l

Γ

i

ˆ ~l,Γ = 1 + (λ~l,Γ − 1)m

YX ~ l

ˆ ~l,Γ λ~l,Γ m

Γ

Florian Gebhard : Itinerant Ferromagnetism – p. 16/34

4 Gutzwiller variational wave functions Choice of the correlator:

PˆG =

YY ~ l

Γ

ˆ ~l,Γ m

λ~

l,Γ

=

YYh ~ l

i

ˆ ~l,Γ = 1 + (λ~l,Γ − 1)m

Γ

YX ~ l

ˆ ~l,Γ λ~l,Γ m

Γ

The operators ˆ ~l,Γ = |Γ~lihΓ~l| m project onto the atomic eigenstate |Γ i on lattice site ~ l.

Florian Gebhard : Itinerant Ferromagnetism – p. 16/34

4 Gutzwiller variational wave functions Choice of the correlator:

PˆG =

YY ~ l

Γ

ˆ ~l,Γ m

λ~

l,Γ

=

YYh ~ l

i

ˆ ~l,Γ = 1 + (λ~l,Γ − 1)m

Γ

YX ~ l

ˆ ~l,Γ λ~l,Γ m

Γ

The operators ˆ ~l,Γ = |Γ~lihΓ~l| m project onto the atomic eigenstate |Γ i on lattice site ~ l. The quantities λ~l,Γ are real variational parameters; further parameters may be contained in |Ψ0 i, e.g., the magnetization.

Florian Gebhard : Itinerant Ferromagnetism – p. 16/34

4 Gutzwiller variational wave functions 4.3

Evaluation

Florian Gebhard : Itinerant Ferromagnetism – p. 17/34

4 Gutzwiller variational wave functions 4.3

Evaluation Task: calculate expectation values with |ΨG i; ˆ in order to determine the variational in particular, evaluate hHi parameters by minimization: E0var

ˆ Gi hΨG |H|Ψ := hΨG |ΨG i

Florian Gebhard : Itinerant Ferromagnetism – p. 17/34

4 Gutzwiller variational wave functions 4.3

Evaluation Task: calculate expectation values with |ΨG i; ˆ in order to determine the variational in particular, evaluate hHi parameters by minimization: E0var

ˆ Gi hΨG |H|Ψ := hΨG |ΨG i

This is a

difficult many-body problem! Florian Gebhard : Itinerant Ferromagnetism – p. 17/34

4 Gutzwiller variational wave functions ‘Solution’: exact evaluation in the limit Z → ∞; Z is the number of nearest neighbors (Z = 12 for fcc nickel)

Florian Gebhard : Itinerant Ferromagnetism – p. 18/34

4 Gutzwiller variational wave functions ‘Solution’: exact evaluation in the limit Z → ∞; Z is the number of nearest neighbors (Z = 12 for fcc nickel) Important steps: 1. Develop a diagrammatic perturbation theory with ~ ‘Vertices’ x~ and ‘Lines’ Peσ0 ,σ ′ (~ l, m). l,Γ1 ,Γ2

2. The expansion parameters x~l,Γ1 ,Γ2 can be chosen such that at least four lines meet at every inner vertex, there are no Hartree bubble diagrams, and the single-particle density matrices obey Peσ0 ,σ ′ (~ l, ~ l) = 0 .

(∗)

Florian Gebhard : Itinerant Ferromagnetism – p. 18/34

4 Gutzwiller variational wave functions ‘Solution’: exact evaluation in the limit Z → ∞; Z is the number of nearest neighbors (Z = 12 for fcc nickel) Important steps: 1. Develop a diagrammatic perturbation theory with ~ ‘Vertices’ x~ and ‘Lines’ Peσ0 ,σ ′ (~ l, m). l,Γ1 ,Γ2

2. The expansion parameters x~l,Γ1 ,Γ2 can be chosen such that at least four lines meet at every inner vertex, there are no Hartree bubble diagrams, and the single-particle density matrices obey Peσ0 ,σ ′ (~ l, ~ l) = 0 .

(∗)

3. In the limit Z → ∞, all skeleton diagrams collapse in position space, i.e., they have the same lattice site index. As a consequence of Eq. (*), they all vanish and not a single diagram must be calculated.

Florian Gebhard : Itinerant Ferromagnetism – p. 18/34

4 Gutzwiller variational wave functions The results remain non-trivial because of the contributions from the external vertices, e.g., the sites i~ and j~ in the one-particle ~ j). ~ density matrix Pσ ,σ ′ (i,

Florian Gebhard : Itinerant Ferromagnetism – p. 19/34

4 Gutzwiller variational wave functions The results remain non-trivial because of the contributions from the external vertices, e.g., the sites i~ and j~ in the one-particle ~ j). ~ density matrix Pσ ,σ ′ (i, Exact result for Gutzwiller wave functions for Z = ∞ Z=∞ ˆ0eff |Ψ0 i E0var = hΨ0 |H

Florian Gebhard : Itinerant Ferromagnetism – p. 19/34

4 Gutzwiller variational wave functions The results remain non-trivial because of the contributions from the external vertices, e.g., the sites i~ and j~ in the one-particle ~ j). ~ density matrix Pσ ,σ ′ (i, Exact result for Gutzwiller wave functions for Z = ∞ Z=∞ ˆ0eff |Ψ0 i E0var = hΨ0 |H

Effective single-particle Hamiltonian ˆ0eff H

=

X

X

′ ′ ~ ~ (bσ ),(b σ ) l,m

bσ ,b′ σ ′ + e cˆ~ cˆ ~ t~ ~ l,bσ m,bσ l,m

+

XX ~ l

E~l,Γ m~l,Γ

Γ

Florian Gebhard : Itinerant Ferromagnetism – p. 19/34

4 Gutzwiller variational wave functions The results remain non-trivial because of the contributions from the external vertices, e.g., the sites i~ and j~ in the one-particle ~ j). ~ density matrix Pσ ,σ ′ (i, Exact result for Gutzwiller wave functions for Z = ∞ Z=∞ ˆ0eff |Ψ0 i E0var = hΨ0 |H

Effective single-particle Hamiltonian ˆ0eff H

=

X

X

′ ′ ~ ~ (bσ ),(b σ ) l,m

bσ ,b′ σ ′ + e cˆ~ cˆ ~ t~ ~ l,bσ m,bσ l,m

+

XX ~ l

E~l,Γ m~l,Γ

Γ

Effective electron transfer matrix elements (cubic symmetry) bσ ,b′ σ ′ e t~ ~ l≠m

q q bσ ,b′ σ ′ = q~l,bσ qm,b ~ ′ σ ′ t~ ~ l≠m

Florian Gebhard : Itinerant Ferromagnetism – p. 19/34

4 Gutzwiller variational wave functions Result: we obtain a single-particle Hamiltonian with bandwidth and hybridization reduction factors q~l,bσ

Florian Gebhard : Itinerant Ferromagnetism – p. 20/34

4 Gutzwiller variational wave functions Result: we obtain a single-particle Hamiltonian with bandwidth and hybridization reduction factors q~l,bσ q~l,bσ ≥ 0 are know functions of the variational parameters, ˆ ~l,Γ |Ψ0 i . m~l,Γ = λ~2 hΨ0 |m l,Γ

Florian Gebhard : Itinerant Ferromagnetism – p. 20/34

4 Gutzwiller variational wave functions Result: we obtain a single-particle Hamiltonian with bandwidth and hybridization reduction factors q~l,bσ q~l,bσ ≥ 0 are know functions of the variational parameters, ˆ ~l,Γ |Ψ0 i . m~l,Γ = λ~2 hΨ0 |m l,Γ

4.4

Landau-Gutzwiller quasi-particles

Florian Gebhard : Itinerant Ferromagnetism – p. 20/34

4 Gutzwiller variational wave functions Result: we obtain a single-particle Hamiltonian with bandwidth and hybridization reduction factors q~l,bσ q~l,bσ ≥ 0 are know functions of the variational parameters, ˆ ~l,Γ |Ψ0 i . m~l,Γ = λ~2 hΨ0 |m l,Γ

4.4

Landau-Gutzwiller quasi-particles Interpretation scheme in density functional theory (DFT): ˆ0eff ֏ band structure ǫe(p, ~ bσ ) + comparison with experiment H

Florian Gebhard : Itinerant Ferromagnetism – p. 20/34

4 Gutzwiller variational wave functions Result: we obtain a single-particle Hamiltonian with bandwidth and hybridization reduction factors q~l,bσ q~l,bσ ≥ 0 are know functions of the variational parameters, ˆ ~l,Γ |Ψ0 i . m~l,Γ = λ~2 hΨ0 |m l,Γ

4.4

Landau-Gutzwiller quasi-particles Interpretation scheme in density functional theory (DFT): ˆ0eff ֏ band structure ǫe(p, ~ bσ ) + comparison with experiment H Landau’s idea of quasi-particles Fermi gas + hole excitation

interactions

-→

Fermi liquid + quasi-hole excitation Florian Gebhard : Itinerant Ferromagnetism – p. 20/34

4 Gutzwiller variational wave functions Realization in terms of Gutzwiller wave functions: Fermi gas |Ψ0 i : cˆp,bσ |Ψ0 i : ~

Fermi-gas ground state hole excitation

Florian Gebhard : Itinerant Ferromagnetism – p. 21/34

4 Gutzwiller variational wave functions Realization in terms of Gutzwiller wave functions: Fermi gas |Ψ0 i : cˆp,bσ |Ψ0 i : ~

Fermi-gas ground state hole excitation

Fermi liquid |ΨG i = PˆG |Ψ0 i :

Fermi-liquid ground state

~ bσ )i = PˆG cˆp,bσ |Ψ0 i : |ΨG (p, ~

quasi-hole excitation

Florian Gebhard : Itinerant Ferromagnetism – p. 21/34

4 Gutzwiller variational wave functions Realization in terms of Gutzwiller wave functions: Fermi gas |Ψ0 i :

Fermi-gas ground state

cˆp,bσ |Ψ0 i : ~

hole excitation

Fermi liquid |ΨG i = PˆG |Ψ0 i :

Fermi-liquid ground state

~ bσ )i = PˆG cˆp,bσ |Ψ0 i : |ΨG (p, ~

quasi-hole excitation

Energy of Landau-Gutzwiller quasi-particles E

QP

~ bσ ) (p,

:= Z=∞

=

ˆ G (p, ~ bσ )|H|Ψ ~ bσ )i hΨG (p, − E0var ~ bσ )|ΨG (p, ~ bσ )i hΨG (p, ~ bσ ) + µbσ − EF ǫe(p,

ˆ0eff . ~ bσ ): dispersion relation of H ǫe(p,

Florian Gebhard : Itinerant Ferromagnetism – p. 21/34

5 Generic two-band model

Florian Gebhard : Itinerant Ferromagnetism – p. 22/34

5 Generic two-band model 5.1

Model specifications two degenerate levels (b = 1, 2) as a toy model example for d(eg )-levels in cubic symmetry

Florian Gebhard : Itinerant Ferromagnetism – p. 22/34

5 Generic two-band model 5.1

Model specifications two degenerate levels (b = 1, 2) as a toy model example for d(eg )-levels in cubic symmetry ˆat H

= U

X

ˆ b,↓ + U ˆ b,↑ n n

b

+J

X σ

′

X

ˆ 2,σ ′ − J ˆ 1,σ n n

σσ′ + + cˆ2,−σ cˆ1,−σ cˆ2,σ cˆ1,σ

X

ˆ 2,σ ˆ 1,σ n n

σ

  + + + Jc cˆ1,↑ cˆ1,↓ cˆ2,↑ cˆ2,↓ + h.c.

Florian Gebhard : Itinerant Ferromagnetism – p. 22/34

5 Generic two-band model 5.1

Model specifications two degenerate levels (b = 1, 2) as a toy model example for d(eg )-levels in cubic symmetry ˆat H

= U

X

ˆ b,↓ + U ˆ b,↑ n n

+J

X

ˆ 2,σ ′ − J ˆ 1,σ n n

σσ′

b

X

′

+ + cˆ2,−σ cˆ1,−σ cˆ2,σ cˆ1,σ

σ

X

ˆ 2,σ ˆ 1,σ n n

σ

  + + + Jc cˆ1,↑ cˆ1,↓ cˆ2,↑ cˆ2,↓ + h.c.

Properties: 16 states per atom cubic symmetry: Jc = J, U − U ′ = 2J two parameters: Hubbard-U as in the one-band case, Hund’s rule coupling J Florian Gebhard : Itinerant Ferromagnetism – p. 22/34

5 Generic two-band model Two-center approximation for matrix elements t~l,m ~ between nearest and next-nearest neighbors in a simple cubic lattice à la Slater–Koster

Florian Gebhard : Itinerant Ferromagnetism – p. 23/34

5 Generic two-band model Two-center approximation for matrix elements t~l,m ~ between nearest and next-nearest neighbors in a simple cubic lattice à la Slater–Koster ‘Generic’ density of states with no ‘perfect nesting’ instabilities, no excessive peaks

Florian Gebhard : Itinerant Ferromagnetism – p. 23/34

5 Generic two-band model Two-center approximation for matrix elements t~l,m ~ between nearest and next-nearest neighbors in a simple cubic lattice à la Slater–Koster ‘Generic’ density of states with no ‘perfect nesting’ instabilities, no excessive peaks Density of states at the Fermi energy (bandwidth W = 6.6 eV)

D0 / eV-1

0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

nσ

0.8

1.0

Florian Gebhard : Itinerant Ferromagnetism – p. 23/34

5 Generic two-band model 5.2

Phase diagram (bandwidth W = 6.6 eV, filling nσ = 0.29, 0.35)

Florian Gebhard : Itinerant Ferromagnetism – p. 24/34

5 Generic two-band model Phase diagram (bandwidth W = 6.6 eV, filling nσ = 0.29, 0.35) 0.4

a)

FM

J/U

0.3

GW 0.2

PM

0.1

HF 0.4

b)

FM

0.3

GW

J/U

5.2

0.2

PM 0.1

HF 0.0 0

2

4

6

8

10

U/eV

Florian Gebhard : Itinerant Ferromagnetism – p. 24/34

5 Generic two-band model Phase diagram (bandwidth W = 6.6 eV, filling nσ = 0.29, 0.35) 0.4

a)

FM

J/U

0.3

GW 0.2

PM

0.1

HF 0.4

b)

FM

0.3

GW

J/U

5.2

0.2

PM 0.1

HF 0.0 0

2

4

6

8

10

U/eV

Stoner theory ≡ Hartree–Fock theory ferromagnetism appears at ‘moderate couplings’ – wrong; Hund’s rule coupling is irrelevant – wrong; large density of states at the Fermi energy favors ferromagnetism – correct.

Florian Gebhard : Itinerant Ferromagnetism – p. 24/34

5 Generic two-band model 0.4

a)

FM

J/U

0.3

GW 0.2

PM

0.1

HF 0.4

b)

FM

0.3

J/U

GW 0.2

PM 0.1

HF 0.0 0

2

4

6

8

10

U/eV

Florian Gebhard : Itinerant Ferromagnetism – p. 25/34

5 Generic two-band model 0.4

a)

FM

J/U

0.3

GW 0.2

PM

0.1

HF 0.4

b)

FM

0.3

J/U

GW 0.2

PM 0.1

HF 0.0 0

2

4

6

8

10

U/eV

Gutzwiller correlated electron theory ferromagnetism at UcGW > W : strong coupling phenomenon; Hund’s rule coupling is decisive for ferromagnetism; large density of states at the Fermi energy favors ferromagnetism.

Florian Gebhard : Itinerant Ferromagnetism – p. 25/34

5 Generic two-band model 5.3

Local moments

(bandwidth W = 6.6 eV; J = 0.2U , filling nσ = 0.35) 1.0 0.9

/

8

GW HF

0.8 0.7 0.6 0.5 0

3

6

9

12

U/eV

Florian Gebhard : Itinerant Ferromagnetism – p. 26/34

5 Generic two-band model 5.3

Local moments

(bandwidth W = 6.6 eV; J = 0.2U , filling nσ = 0.35) 1.0 0.9

/

8

GW HF

0.8 0.7 0.6 0.5 0

3

6

9

12

U/eV

Hartree–Fock theory: moments are formed at the transition

Florian Gebhard : Itinerant Ferromagnetism – p. 26/34

5 Generic two-band model 5.3

Local moments

(bandwidth W = 6.6 eV; J = 0.2U , filling nσ = 0.35) 1.0 0.9

/

8

GW HF

0.8 0.7 0.6 0.5 0

3

6

9

12

U/eV

Hartree–Fock theory: moments are formed at the transition Gutzwiller correlated electron theory: local moments are almost equal at the transition; idea of pre-formed moments applies. Florian Gebhard : Itinerant Ferromagnetism – p. 26/34

5 Generic two-band model Condensation energy (bandwidth W = 6.6 eV; J = 0.2U ) para Econd := E0 − E0ferro ; expectation: Econd = O (TCurie )

5.4

Florian Gebhard : Itinerant Ferromagnetism – p. 27/34

5 Generic two-band model Condensation energy (bandwidth W = 6.6 eV; J = 0.2U ) para Econd := E0 − E0ferro ; expectation: Econd = O (TCurie ) cond. energy / K

5.4

nσ=0.29

2000

nσ=0.35 1500

HF

1000

GW

500

0

0

2

4

6

8

10

12

U/eV

Florian Gebhard : Itinerant Ferromagnetism – p. 27/34

5 Generic two-band model Condensation energy (bandwidth W = 6.6 eV; J = 0.2U ) para Econd := E0 − E0ferro ; expectation: Econd = O (TCurie ) cond. energy / K

5.4

nσ=0.29

2000

nσ=0.35 1500

HF

1000

GW

500

0

0

2

4

6

8

10

12

U/eV

Hartree–Fock theory: fine tuning of U required!

Florian Gebhard : Itinerant Ferromagnetism – p. 27/34

5 Generic two-band model Condensation energy (bandwidth W = 6.6 eV; J = 0.2U ) para Econd := E0 − E0ferro ; expectation: Econd = O (TCurie ) cond. energy / K

5.4

nσ=0.29

2000

nσ=0.35 1500

HF

1000

GW

500

0

0

2

4

6

8

10

12

U/eV

Hartree–Fock theory: fine tuning of U required! Gutzwiller correlated electron theory: realistic values for all U > Uc ; not sensitive against variations of U . Florian Gebhard : Itinerant Ferromagnetism – p. 27/34

6 Results for nickel

Florian Gebhard : Itinerant Ferromagnetism – p. 28/34

6 Results for nickel 6.1

Spin density functional theory (SDFT) in trouble

Modern version of Stoner–Slater theory: SDFT

Florian Gebhard : Itinerant Ferromagnetism – p. 28/34

6 Results for nickel 6.1

Spin density functional theory (SDFT) in trouble

Modern version of Stoner–Slater theory: SDFT wrong topology of the Fermi surface SDFT: two hole ellipsoids at the X point experiment: one hole ellipsoid

Florian Gebhard : Itinerant Ferromagnetism – p. 28/34

6 Results for nickel 6.1

Spin density functional theory (SDFT) in trouble

Modern version of Stoner–Slater theory: SDFT wrong topology of the Fermi surface SDFT: two hole ellipsoids at the X point experiment: one hole ellipsoid effective mass too small

Florian Gebhard : Itinerant Ferromagnetism – p. 28/34

6 Results for nickel 6.1

Spin density functional theory (SDFT) in trouble

Modern version of Stoner–Slater theory: SDFT wrong topology of the Fermi surface SDFT: two hole ellipsoids at the X point experiment: one hole ellipsoid effective mass too small bandwidth is wrong, bands do not fit

Florian Gebhard : Itinerant Ferromagnetism – p. 28/34

6 Results for nickel 6.1

Spin density functional theory (SDFT) in trouble

Modern version of Stoner–Slater theory: SDFT wrong topology of the Fermi surface SDFT: two hole ellipsoids at the X point experiment: one hole ellipsoid effective mass too small bandwidth is wrong, bands do not fit 0 -1

-2 -3 -4

Γ

Σ→

K

X

←∆

Γ

Florian Gebhard : Itinerant Ferromagnetism – p. 28/34

6 Results for nickel 6.2

Model specifications for Gutzwiller density functional theory

Florian Gebhard : Itinerant Ferromagnetism – p. 29/34

6 Results for nickel 6.2

Model specifications for Gutzwiller density functional theory basis: 3d, 4s, 4p (9 orbitals)

Florian Gebhard : Itinerant Ferromagnetism – p. 29/34

6 Results for nickel 6.2

Model specifications for Gutzwiller density functional theory basis: 3d, 4s, 4p (9 orbitals) fit of t~l,m ~ to a paramagnetic density-functional calculation; nd = 8.8 in our calculation nd = 8.9 in DFT calculation (4sp level had to be corrected to fit the experiment)

Florian Gebhard : Itinerant Ferromagnetism – p. 29/34

6 Results for nickel 6.2

Model specifications for Gutzwiller density functional theory basis: 3d, 4s, 4p (9 orbitals) fit of t~l,m ~ to a paramagnetic density-functional calculation; nd = 8.8 in our calculation nd = 8.9 in DFT calculation (4sp level had to be corrected to fit the experiment) Our Racah parameters B/C = 4.5, as in ligand-field calculations; C = 0.4 eV (corresponds to Hund’s rule J); A = 9 eV (corresponds to Hubbard-U );

Florian Gebhard : Itinerant Ferromagnetism – p. 29/34

6 Results for nickel 6.2

Model specifications for Gutzwiller density functional theory basis: 3d, 4s, 4p (9 orbitals) fit of t~l,m ~ to a paramagnetic density-functional calculation; nd = 8.8 in our calculation nd = 8.9 in DFT calculation (4sp level had to be corrected to fit the experiment) Our Racah parameters B/C = 4.5, as in ligand-field calculations; C = 0.4 eV (corresponds to Hund’s rule J); A = 9 eV (corresponds to Hubbard-U ); A directly controls effective mass, bandwidth, magnetic moment (together with C). Florian Gebhard : Itinerant Ferromagnetism – p. 29/34

6 Results for nickel 6.3

Results (no spin-orbit coupling, cubic symmetry)

Florian Gebhard : Itinerant Ferromagnetism – p. 30/34

6 Results for nickel 6.3

Results (no spin-orbit coupling, cubic symmetry) Band structure X

3↓

0 -0.2

Γ

4↓ 1↓

-0.4

K X

(110)

0 -1

2

1↓ 4↑

1.0

1↑

3↑ 2.4

4↑ 2.2

↓

X

2↑ 1↑,↓ 5↑ 1.4 k-kΓ (Å-1)

1.6

2↓

↑

2 2

4↓

5↓

2↓

5↑ 2↑

↓

1↑ 0.8

2↓

5↑ 5↓ 3↓

-2

1↓ 1↑

3↑

1↓

↓

3

↓

-3

1↑ 1↓ 1↑

1 3↑ 1↑

2↑

12↓↑ 12 25'↓ 25'↑

2'↓ 2'↑

1↑ 1↓

-4

Γ

Σ→

K

X

←∆

Γ

Florian Gebhard : Itinerant Ferromagnetism – p. 30/34

6 Results for nickel Data (spin-only moment µs = 0.55µB , no spin-orbit coupling) Symmetry

Experiment

Gutz-DFT

SDFT

hΓ1 i

8.90±0.30

8.86

8.96[−0.11]

hX1 i

3.30±0.20

3.31[0.36]

4.37[0.20]

X2↓

0.04±0.03

0.01

−0.09

∆eg (X2 )

0.17±0.05

0.155

0.44

∆t2g (X5 )

0.33±0.04

0.38

0.56

hL2′ i

1.00±0.20

0.97[0.0]

0.24[−0.12]

hΛ3;1/3 i

0.57[0.16±0.02]

0.67[0.22]

0.90[0.42]

hΛ3;1/2 i

0.50[0.21±0.02]

0.55[0.26]

0.76[0.44]

hΛ3;2/3 i

0.35[0.25±0.02]

0.33[0.29]

0.49[0.48]

Florian Gebhard : Itinerant Ferromagnetism – p. 31/34

6 Results for nickel Data (spin-only moment µs = 0.55µB , no spin-orbit coupling) Symmetry

Experiment

Gutz-DFT

SDFT

hΓ1 i

8.90±0.30

8.86

8.96[−0.11]

hX1 i

3.30±0.20

3.31[0.36]

4.37[0.20]

X2↓

0.04±0.03

0.01

−0.09

∆eg (X2 )

0.17±0.05

0.155

0.44

∆t2g (X5 )

0.33±0.04

0.38

0.56

hL2′ i

1.00±0.20

0.97[0.0]

0.24[−0.12]

hΛ3;1/3 i

0.57[0.16±0.02]

0.67[0.22]

0.90[0.42]

hΛ3;1/2 i

0.50[0.21±0.02]

0.55[0.26]

0.76[0.44]

hΛ3;2/3 i

0.35[0.25±0.02]

0.33[0.29]

0.49[0.48]

Important details correct Fermi surface topology: one hole ellipsoid around X correct exchange splitting: small and anisotropic correct 4sp level (L2′ ): nd had to be corrected

Florian Gebhard : Itinerant Ferromagnetism – p. 31/34

6 Results for nickel Spin-orbit interaction: Gersdorf effect (1978) (ζso = 0.080 eV)

Florian Gebhard : Itinerant Ferromagnetism – p. 32/34

6 Results for nickel Spin-orbit interaction: Gersdorf effect (1978) (ζso = 0.080 eV) Gutzwiller bands with s-o coupling →

µ || (111) 0.3

[0,0,1]

[0,0,ζ]

→

µ || (001)

→

[0,0,ζ]

[ξ,0,0]

µ || (001)

→

µ || (111) [ξ,0,0]

5↓

0.2

[1,0,0]

5↓

E / eV

0.1 0.0

EF

-0.1 -0.2 -0.3

∆Z µOrb = 0.052 µB µOrb = 0.050 µB

2↓

2↓

5↑

5↑

2↑

2↑

XZ

∆Z

∆X

XX

Gersdorf scenario reproduced! Etot(111) < Etot(001)

∆X

ARPES data extrapolated data

Florian Gebhard : Itinerant Ferromagnetism – p. 32/34

6 Results for nickel Spin-orbit interaction: Fermi surface cuts

Florian Gebhard : Itinerant Ferromagnetism – p. 33/34

6 Results for nickel Spin-orbit interaction: Fermi surface cuts

We think that the experimental ‘wiggles’ are not correct. New ARPES measurements are under way . . .

Florian Gebhard : Itinerant Ferromagnetism – p. 33/34

7 Summary

Florian Gebhard : Itinerant Ferromagnetism – p. 34/34

7 Summary For nickel:

Florian Gebhard : Itinerant Ferromagnetism – p. 34/34

7 Summary For nickel: Ferromagnetism in nickel is a problem of strong electron correlations (W < Ueff ≡ A);

Florian Gebhard : Itinerant Ferromagnetism – p. 34/34

7 Summary For nickel: Ferromagnetism in nickel is a problem of strong electron correlations (W < Ueff ≡ A); Stoner–Slater theory and its modern version as spin-density functional theory are not applicable;

Florian Gebhard : Itinerant Ferromagnetism – p. 34/34

7 Summary For nickel: Ferromagnetism in nickel is a problem of strong electron correlations (W < Ueff ≡ A); Stoner–Slater theory and its modern version as spin-density functional theory are not applicable; Gutzwiller correlated electron theory provides a surprisingly good quantitative description of the quasi-particle bands.

Florian Gebhard : Itinerant Ferromagnetism – p. 34/34

7 Summary For nickel: Ferromagnetism in nickel is a problem of strong electron correlations (W < Ueff ≡ A); Stoner–Slater theory and its modern version as spin-density functional theory are not applicable; Gutzwiller correlated electron theory provides a surprisingly good quantitative description of the quasi-particle bands. General mechanism for ferromagnetism (van Vleck):

Florian Gebhard : Itinerant Ferromagnetism – p. 34/34

7 Summary For nickel: Ferromagnetism in nickel is a problem of strong electron correlations (W < Ueff ≡ A); Stoner–Slater theory and its modern version as spin-density functional theory are not applicable; Gutzwiller correlated electron theory provides a surprisingly good quantitative description of the quasi-particle bands. General mechanism for ferromagnetism (van Vleck): pre-formed moments à la Hund are coupled by the electrons’ motion through the crystal;

Florian Gebhard : Itinerant Ferromagnetism – p. 34/34

7 Summary For nickel: Ferromagnetism in nickel is a problem of strong electron correlations (W < Ueff ≡ A); Stoner–Slater theory and its modern version as spin-density functional theory are not applicable; Gutzwiller correlated electron theory provides a surprisingly good quantitative description of the quasi-particle bands. General mechanism for ferromagnetism (van Vleck): pre-formed moments à la Hund are coupled by the electrons’ motion through the crystal; they eventually order at low enough temperatures.

Florian Gebhard : Itinerant Ferromagnetism – p. 34/34