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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Competing interactions and magnetic order in correlated electron systems

Thomas Pruschke and Robert Peters

Department of Theoretical Physics University of Göttingen

1

Actual work done by Robert Peters Computer support:

Gesellschaft für Wissenschaftliche Datenverarbeitung Göttingen Norddeutscher Verbund für Hoch- und Höchstleistungsrechnen

Financial support by: DFG through SFB 602 and Pr 298/10

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Acknowledgements

2

t2g eg Mn3+

∆CF U , U ′ , JH

S = 3

2

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

cubic environment

3

t2g eg Mn3+

∆CF U , U ′ , JH

S = 3

2

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

cubic environment + Jahn-Teller

∆JT

3

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

4

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics TMO like LaMnO3, LaTiO3, V2O3 etc.

• Correlated metals, ..., paramagnetic insulators
• Different types of magnetic and orbital ordering
• Competing interactions: High-spin/low-spin and magnetic frustration

4

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics TMO like LaMnO3, LaTiO3, V2O3 etc.

• Correlated metals, ..., paramagnetic insulators
• Different types of magnetic and orbital ordering
• Competing interactions: High-spin/low-spin and magnetic frustration

GdN and other rare-earth monopnictides

• Gd3+ in GdN ⇒ localized Gd 4f7 + Gd 5d conduction states

➡ localized Gd S=7/2 spin coupled ferromagnetically to conduction states

• Band structure theory: Semiconductor
• Experiment: Ferromagnetic semimetal

4

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

no f-electron contribution to band states

5

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics Heavy-fermion materials: Rare-earth and Actinides

• Archetype: Compounds based on Ce

➡ localized S=1/2 spin

• Band structure theory: Conventional metals
• Experimental findings: Strongly enhanced Fermi liquid parameters, several ordered phases
• Well understood: Nature of heavy Fermi liquid ⇒ hybridization between 4f and band states
• Open questions: Magnetism & supercondconductivity, quantum phase transitions

no f-electron contribution to band states

5

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics Heavy-fermion materials: Rare-earth and Actinides

• Archetype: Compounds based on Ce

➡ localized S=1/2 spin

• Band structure theory: Conventional metals
• Experimental findings: Strongly enhanced Fermi liquid parameters, several ordered phases
• Well understood: Nature of heavy Fermi liquid ⇒ hybridization between 4f and band states
• Open questions: Magnetism & supercondconductivity, quantum phase transitions

Exotic „heavy-fermion“ material: LiV2O4

• Heavy-fermion like features, origin heavily debated
• Magnetic frustration of corundum structure
• Partial localization of correlated 3d states + non-local hybridization effects

no f-electron contribution to band states

5

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

6

Unifying model for TMO, GdN ... and different flavors of heavy-fermion compounds Minimal necessary ingredients: Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

6

Unifying model for TMO, GdN ... and different flavors of heavy-fermion compounds Minimal necessary ingredients:

➡ tight-binding d-like conduction states,

including crystal-filed splittings

• ij
• αβ
• σ

tαβ

ij,σc† iασcjβσ

+

• iασ

εiασc†

iασciασ

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics H =

6

Unifying model for TMO, GdN ... and different flavors of heavy-fermion compounds Minimal necessary ingredients:

➡ tight-binding d-like conduction states,

including crystal-filed splittings

➡ Coulomb interactions among conduction states

• ij
• αβ
• σ

tαβ

ij,σc† iασcjβσ

+

• iασ

εiασc†

iασciασ

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics H = +

• i
• αβγδ

σ1σ′ 1 σ2σ′ 2

U αβγδ

σ1σ2σ′

2σ′ 2c†

iασ1c† iβσ2ciγσ′

2ciδσ′ 1 6

Unifying model for TMO, GdN ... and different flavors of heavy-fermion compounds Minimal necessary ingredients:

➡ tight-binding d-like conduction states,

including crystal-filed splittings

➡ Coulomb interactions among conduction states ➡ Localized spins

• ij
• αβ
• σ

tαβ

ij,σc† iασcjβσ

+

• iασ

εiασc†

iασciασ

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics H = +

• i
• αβγδ

σ1σ′ 1 σ2σ′ 2

U αβγδ

σ1σ2σ′

2σ′ 2c†

iασ1c† iβσ2ciγσ′

2ciδσ′ 1 6

Unifying model for TMO, GdN ... and different flavors of heavy-fermion compounds Minimal necessary ingredients:

➡ tight-binding d-like conduction states,

including crystal-filed splittings

➡ Coulomb interactions among conduction states ➡ Localized spins ➡ Exchange coupling between localized spins

and conduction states

• ij
• αβ
• σ

tαβ

ij,σc† iασcjβσ

+

• iασ

εiασc†

iασciασ

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics H =

Origin: Hund‘s exchange (J>0) or Schrieffer-Wolff exchange (J<0)

+

• i
• αβγδ

σ1σ′ 1 σ2σ′ 2

U αβγδ

σ1σ2σ′

2σ′ 2c†

iασ1c† iβσ2ciγσ′

2ciδσ′ 1

−

• i
• αβ
• σσ′

Jαβ

σσ′ c† iασ

• Si ·

τσσ′

• ciβσ′

6

Unifying model for TMO, GdN ... and different flavors of heavy-fermion compounds Minimal necessary ingredients:

➡ tight-binding d-like conduction states,

including crystal-filed splittings

➡ Coulomb interactions among conduction states ➡ Localized spins ➡ Exchange coupling between localized spins

and conduction states

Hubbard-Kondo model

• ij
• αβ
• σ

tαβ

ij,σc† iασcjβσ

+

• iασ

εiασc†

iασciασ

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics H =

Origin: Hund‘s exchange (J>0) or Schrieffer-Wolff exchange (J<0)

+

• i
• αβγδ

σ1σ′ 1 σ2σ′ 2

U αβγδ

σ1σ2σ′

2σ′ 2c†

iασ1c† iβσ2ciγσ′

2ciδσ′ 1

−

• i
• αβ
• σσ′

Jαβ

σσ′ c† iασ

• Si ·

τσσ′

• ciβσ′

6

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

7

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics Further simplifications:

• Jahn-Teller or crystal field: Low T governed by single „d“ subband at Fermi level

➡ Conduction states modelled by single-band Hubbard model

• Isotropic exchange to local „f“ spin
• Large U ⇒ TMO 3d states, small U ⇒ rare-earth 5d states; J<0 ⇒ heavy fermions

H =

• i,j,σ

tij d†

iσdjσ +

• iσ

ε(d)

σ n(d) iσ + U

• i
• n(d)

i

− 1 2 − J

• iσσ′

d†

iσ

• S(f)

i

· τσσ′

• diσ′

7

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics Further simplifications:

• Jahn-Teller or crystal field: Low T governed by single „d“ subband at Fermi level

➡ Conduction states modelled by single-band Hubbard model

• Isotropic exchange to local „f“ spin
• Large U ⇒ TMO 3d states, small U ⇒ rare-earth 5d states; J<0 ⇒ heavy fermions

Qualitative physics of model from DMFT+NRG:

• Access to interplay between different (local) correlations at T≥0, all U and J
• Magnetic properties: Different routes via U („super-exchange“) and J („RKKY“)
• Competing interactions through J and long-ranged tij

H =

• i,j,σ

tij d†

iσdjσ +

• iσ

ε(d)

σ n(d) iσ + U

• i
• n(d)

i

− 1 2 − J

• iσσ′

d†

iσ

• S(f)

i

· τσσ′

• diσ′

7

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Theorist‘s playground: The Bethe lattice

Nearest-neighbor hopping only

(Georges et al. RMP ‘96)

8

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Theorist‘s playground: The Bethe lattice

Next-nearest-neighbor hopping, simplified

(Georges et al. RMP ’96, Chitra & Kotliar, PRL ’99, Zitzler, TP et al., PRL ‘04)

8

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Theorist‘s playground: The Bethe lattice

Next-nearest-neighbor hopping, correct

(Eckstein et. al. PRB ‘05)

8

t1 Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Theorist‘s playground: The Bethe lattice

Next-nearest-neighbor hopping, correct

(Eckstein et. al. PRB ‘05)

8

t1 t2 Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Theorist‘s playground: The Bethe lattice

Next-nearest-neighbor hopping, correct

(Eckstein et. al. PRB ‘05)

8

t1 t2 Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Theorist‘s playground: The Bethe lattice

Next-nearest-neighbor hopping, correct

(Eckstein et. al. PRB ‘05)

What to expect: Half filling <n>=1 and t2=0

8

t1 t2 Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Theorist‘s playground: The Bethe lattice

Next-nearest-neighbor hopping, correct

(Eckstein et. al. PRB ‘05)

What to expect:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

t2>0?

8

t1 t2 Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Theorist‘s playground: The Bethe lattice

Next-nearest-neighbor hopping, correct

(Eckstein et. al. PRB ‘05)

What to expect: <n> < 1?

? ? ? ? ? ? ? ? ? ? ? ? ? ?

8

Z → ∞

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

• 0.8
• 0.4

0.4 0.8 ω/W 1 2 3 4 ρ

(0)(ω)⋅W

The DMFT cycle

Georges et al. RMP ’96 Pruschke et al. Adv. Phys. ‘95

9

Z → ∞ G(z) =

• dω ρ(0)(ω)

z + µ − ω

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

• 0.8
• 0.4

0.4 0.8 ω/W 1 2 3 4 ρ

(0)(ω)⋅W

The DMFT cycle

Georges et al. RMP ’96 Pruschke et al. Adv. Phys. ‘95

9

Z → ∞ G(z) =

• dω ρ(0)(ω)

z + µ − ω Γeff(ω) = ℑm

• 1

G(ω + i0+)

• Competing interactions and magnetic order

in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

• 0.8
• 0.4

0.4 0.8 ω/W 1 2 3 4 ρ

(0)(ω)⋅W

The DMFT cycle

Georges et al. RMP ’96 Pruschke et al. Adv. Phys. ‘95

• 1
• 0.5

0.5 1 ω/W 0.25 0.5 Γeff(ω)/W

9

Z → ∞ G(z) =

• dω ρ(0)(ω)

z + µ − ω Γeff(ω) = ℑm

• 1

G(ω + i0+)

• Competing interactions and magnetic order

in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

• 0.8
• 0.4

0.4 0.8 ω/W 1 2 3 4 ρ

(0)(ω)⋅W

The DMFT cycle

Georges et al. RMP ’96 Pruschke et al. Adv. Phys. ‘95

• 1
• 0.5

0.5 1 ω/W 0.25 0.5 Γeff(ω)/W

Impurity solver

(here: NRG)

9

Z → ∞ G(z) =

• dω ρ(0)(ω)

z + µ − ω Γeff(ω) = ℑm

• 1

G(ω + i0+)

• Competing interactions and magnetic order

in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

• 0.8
• 0.4

0.4 0.8 ω/W 1 2 3 4 ρ

(0)(ω)⋅W

• 0.4
• 0.2

0.2 0.4 ω/W 5 10 15 ρ(ω)⋅W

The DMFT cycle

Georges et al. RMP ’96 Pruschke et al. Adv. Phys. ‘95

• 1
• 0.5

0.5 1 ω/W 0.25 0.5 Γeff(ω)/W

Impurity solver

(here: NRG)

Green function and self-energy

9

Z → ∞ Γeff(ω) = ℑm

• 1

G(ω + i0+)

• G(z) =
• dω

ρ(0)(ω) z + µ − ω − Σ(z)

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

• 0.8
• 0.4

0.4 0.8 ω/W 1 2 3 4 ρ

(0)(ω)⋅W

• 0.4
• 0.2

0.2 0.4 ω/W 5 10 15 ρ(ω)⋅W

The DMFT cycle

Georges et al. RMP ’96 Pruschke et al. Adv. Phys. ‘95

• 1
• 0.5

0.5 1 ω/W 0.25 0.5 Γeff(ω)/W

Impurity solver

(here: NRG)

Green function and self-energy

9

Z → ∞ G(z) =

• dω

ρ(0)(ω) z + µ − ω − Σ(z)

Γeff(ω) = ℑm

• 1

G(ω + i0+) + Σ(ω + i0+)

• Competing interactions and magnetic order

in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

• 0.8
• 0.4

0.4 0.8 ω/W 1 2 3 4 ρ

(0)(ω)⋅W

• 0.4
• 0.2

0.2 0.4 ω/W 5 10 15 ρ(ω)⋅W

The DMFT cycle

Georges et al. RMP ’96 Pruschke et al. Adv. Phys. ‘95

Impurity solver

(here: NRG)

Green function and self-energy

• 1
• 0.5

0.5 1 ω/W 0.25 0.5 Γeff(ω)/W

9

Z → ∞ G(z) =

• dω

ρ(0)(ω) z + µ − ω − Σ(z)

Γeff(ω) = ℑm

• 1

G(ω + i0+) + Σ(ω + i0+)

• Competing interactions and magnetic order

in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

• 0.8
• 0.4

0.4 0.8 ω/W 1 2 3 4 ρ

(0)(ω)⋅W

The DMFT cycle

Georges et al. RMP ’96 Pruschke et al. Adv. Phys. ‘95

Impurity solver

(here: NRG)

Green function and self-energy

• 1
• 0.5

0.5 1 ω/W 0.25 0.5 Γeff(ω)/W

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 ω/W 0.2 0.4 0.6 0.8 1 1.2 ρ(ω)⋅W

9

Z → ∞ G(z) =

• dω

ρ(0)(ω) z + µ − ω − Σ(z)

Γeff(ω) = ℑm

• 1

G(ω + i0+) + Σ(ω + i0+)

• Competing interactions and magnetic order

in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

• 0.8
• 0.4

0.4 0.8 ω/W 1 2 3 4 ρ

(0)(ω)⋅W

The DMFT cycle

Georges et al. RMP ’96 Pruschke et al. Adv. Phys. ‘95

Impurity solver

(here: NRG)

Green function and self-energy

• 1
• 0.5

0.5 1 ω/W 0.25 0.5 Γeff(ω)/W

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 ω/W 0.2 0.4 0.6 0.8 1 1.2 ρ(ω)⋅W

9

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

t2=0, J≠0, Si=1/2 Kondo lattice model

Peters & TP, PRB 76, 245101 (2007)

10

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

• 0.4
• 0.2

0.2 0.4 J/W

• 0.4
• 0.2

0.2 0.4 Polarization

<sz> U=W/200 <Sz> U=W/200 <sz> U=W <Sz> U=W

Antiferromagnetic phase diagram at half filling, T=0

• Critical J for AF coupling
• Opposite polarization for

J<0 ⇒ small total moment

• Small U and J:

conduction polarization proportional to J<SI>

• Large U and J>0

Almost full polarization of conduction spins

11

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

0,2 0,4 0,6 0,8 1 0,005 0,01 0,015 0,02 T/W

Antiferromagnetic phase Ferromagnetic Phase Undetermined magnetic phase

0,2 0,4 0,6 0,8 1

• ccupation n

0,005 0,01 0,015 0,02 T/W

ferromagnetic coupling antiferromagnetic coupling

Phase diagram for finite doping, U→0, |J|=W/2

12

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

0,2 0,4 0,6 0,8 1 0,005 0,01 0,015 0,02 T/W

Antiferromagnetic phase Ferromagnetic Phase Undetermined magnetic phase

0,2 0,4 0,6 0,8 1

• ccupation n

0,005 0,01 0,015 0,02 T/W

ferromagnetic coupling antiferromagnetic coupling

0,2 0,4 0,6 0,8 1 0,1 0,2 0,3 0,4 0,5 polarization

<σz> T/W=4e-5 <Sz> T/W=4e-5 <σz> T/W=2e-3 <Sz> T/W=2e-3

0,2 0,4 0,6 0,8

• ccupation n
• 0,5
• 0,4
• 0,3
• 0,2
• 0,1

0,1 polarization

<σz> T/W=4e-5 <Sz> T/W=4e-5 <σz> T/W=4e-3 <Sz> T/W=4e-3

non convergent behaviour

Polarization of ferromagnetic solution, U→0, |J|=W/2

12

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

0,2 0,4 0,6 0,8 1 0,005 0,01 0,015 0,02 T/W

Antiferromagnetic phase Ferromagnetic Phase Undetermined magnetic phase

0,2 0,4 0,6 0,8 1

• ccupation n

0,005 0,01 0,015 0,02 T/W

ferromagnetic coupling antiferromagnetic coupling

0,2 0,4 0,6 0,8 1 0,1 0,2 0,3 0,4 0,5 polarization

<σz> T/W=4e-5 <Sz> T/W=4e-5 <σz> T/W=2e-3 <Sz> T/W=2e-3

0,2 0,4 0,6 0,8

• ccupation n
• 0,5
• 0,4
• 0,3
• 0,2
• 0,1

0,1 polarization

<σz> T/W=4e-5 <Sz> T/W=4e-5 <σz> T/W=4e-3 <Sz> T/W=4e-3

non convergent behaviour

0,2 0,4 0,6 0,8 1 0,01 0,02 0,03 T/W

antiferromagnetic phase undertermined magnetic phase ferromagnetic phase

0,2 0,4 0,6 0,8 1

• ccupation n

0,01 0,02 0,03 T/W

ferromagnetic coupling antiferromagnetic coupling

Phase diagram for finite doping, U=W/2, |J|=W/2

12

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

0,2 0,4 0,6 0,8 1 0,005 0,01 0,015 0,02 T/W

Antiferromagnetic phase Ferromagnetic Phase Undetermined magnetic phase

0,2 0,4 0,6 0,8 1

• ccupation n

0,005 0,01 0,015 0,02 T/W

ferromagnetic coupling antiferromagnetic coupling

0,2 0,4 0,6 0,8 1 0,1 0,2 0,3 0,4 0,5 polarization

<σz> T/W=4e-5 <Sz> T/W=4e-5 <σz> T/W=2e-3 <Sz> T/W=2e-3

0,2 0,4 0,6 0,8

• ccupation n
• 0,5
• 0,4
• 0,3
• 0,2
• 0,1

0,1 polarization

<σz> T/W=4e-5 <Sz> T/W=4e-5 <σz> T/W=4e-3 <Sz> T/W=4e-3

non convergent behaviour

0,2 0,4 0,6 0,8 1 0,01 0,02 0,03 T/W

antiferromagnetic phase undertermined magnetic phase ferromagnetic phase

0,2 0,4 0,6 0,8 1

• ccupation n

0,01 0,02 0,03 T/W

ferromagnetic coupling antiferromagnetic coupling

0,2 0,4 0,6 0,8 1

• ccupation n
• 0,5
• 0,4
• 0,3
• 0,2
• 0,1

0,1 polarization

<σz> T/W=4e-5 <Sz> T/W=4e-5 <σz> T/W=4e-3 <Sz> T/W=4e-3

0,2 0,4 0,6 0,8 1 0,1 0,2 0,3 0,4 0,5 polarization

<σz> T/W=4e-5 <Sz> T/W=4e-5 <σz> T/W=4e-3 <Sz> T/W=4e-3 non convergent behaviour

Polarization of ferromagnetic solution, U=W/2, |J|=W/2

12

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

0,2 0,4 0,6 0,8 1 0,005 0,01 0,015 0,02 T/W

Antiferromagnetic phase Ferromagnetic Phase Undetermined magnetic phase

0,2 0,4 0,6 0,8 1

• ccupation n

0,005 0,01 0,015 0,02 T/W

ferromagnetic coupling antiferromagnetic coupling

Phase diagram for finite doping, U→0, |J|=W/2

0,2 0,4 0,6 0,8 1 0,1 0,2 0,3 0,4 0,5 polarization

<σz> T/W=4e-5 <Sz> T/W=4e-5 <σz> T/W=2e-3 <Sz> T/W=2e-3

0,2 0,4 0,6 0,8

• ccupation n
• 0,5
• 0,4
• 0,3
• 0,2
• 0,1

0,1 polarization

<σz> T/W=4e-5 <Sz> T/W=4e-5 <σz> T/W=4e-3 <Sz> T/W=4e-3

non convergent behaviour

0,2 0,4 0,6 0,8 1 0,01 0,02 0,03 T/W

antiferromagnetic phase undertermined magnetic phase ferromagnetic phase

0,2 0,4 0,6 0,8 1

• ccupation n

0,01 0,02 0,03 T/W

ferromagnetic coupling antiferromagnetic coupling

Phase diagram for finite doping, U=W/2, |J|=W/2

0,2 0,4 0,6 0,8 1

• ccupation n
• 0,5
• 0,4
• 0,3
• 0,2
• 0,1

0,1 polarization

<σz> T/W=4e-5 <Sz> T/W=4e-5 <σz> T/W=4e-3 <Sz> T/W=4e-3

0,2 0,4 0,6 0,8 1 0,1 0,2 0,3 0,4 0,5 polarization

<σz> T/W=4e-5 <Sz> T/W=4e-5 <σz> T/W=4e-3 <Sz> T/W=4e-3 non convergent behaviour

Polarization of ferromagnetic solution Polarization of ferromagnetic solution

J>0 J<0 J>0 J<0

12

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

What is the „undetermined“ phase?

• Magnetic phase, but no convergence on AB-lattice
• Conjecture: Incommensurable spin-density wave
• Problem with DFT+NRG

No incommensurate phases detectable

• „Workaround“:

Commensurate phase with ABCD...-lattice

• Preliminary result:

Stable solution at U=0, J=W/2, <n>=0.6 with period of 14 sites

• General problem:

Only „almost“ commensurable states detectable Very tedious calculations

Lacroit & Cyrot ’79, Yunoki et al. ’98, Dagotto et al. ’98, Nagai et al. ’99, Santos & Nolting ’02, Nolting et al. ’03, Yin ’03, Kienert & Nolting ’06, Fishman et al. ‘06

5 10 Ri

• 1
• 0.5

0.5 1 polarization and occupancy

Q ≈ 5 2π

14

13

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

t2≠0, J=0 Hubbard model with frustration

Peters & TP, arXiv:0809.2689 (2008)

14

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

All you need is DOS

• 0.8
• 0.4

0.4 0.8 ω/W 1 2 3 4 ρ

(0)(ω)⋅W

t2=0 t2=3/5t1 t2=4/5t1 t2=t1

15

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Metal-insulator transition and magnetism at n=1

16

Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

1 1.1 1.2 1.3 1.4 1.5 U/W 0.002 0.004 0.006 T/W t2/t1=0 t2/t1=3/5 t2/t1=1 t2/t1=-3/5 paramagnetic metal paramagnetic insulator hystersis region

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

0.5 1 1.5 U/W

0.01 0.02 0.03 T/W

antiferromagnetic phase metal-insulator-transition

t2 t1 = 0.6

1 1.1 1.2 1.3 1.4 1.5 U/W 0.002 0.004 0.006 T/W t2/t1=0 t2/t1=3/5 t2/t1=1 t2/t1=-3/5 paramagnetic metal paramagnetic insulator hystersis region

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

0.5 1 1.5 U/W

0.01 0.02 0.03 T/W

antiferromagnetic phase metal-insulator-transition

0.5 1 1.5

U/W

0.01 0.02 0.03

T/W

antiferromgnetic phase metal-insulator-transition

t2 t1 = 0.6

t2 t1 = 0.8

1 1.1 1.2 1.3 1.4 1.5 U/W 0.002 0.004 0.006 T/W t2/t1=0 t2/t1=3/5 t2/t1=1 t2/t1=-3/5 paramagnetic metal paramagnetic insulator hystersis region

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

0.5 1 1.5 U/W

0.01 0.02 0.03 T/W

antiferromagnetic phase metal-insulator-transition

0.5 1 1.5

U/W

0.01 0.02 0.03

T/W

antiferromgnetic phase metal-insulator-transition

0.8 1 1.2

U/W

0.2 0.4 0.6 0.8 1 n-n

increasing U at T/W=1.7e-4 decreasing U at T/W=1.7e-4 increasing U at T/W=1.4e-3 decreasing U at T/W=1.4e-3

1 1.5 2 2.5

U/W 0.2 0.4 0.6 0.8 1 n-n

T/W=0.006 T/W=0.008

t2 t1 = 0.6 t2 t1 = 0.8

t2 t1 = 0.8

1 1.1 1.2 1.3 1.4 1.5 U/W 0.002 0.004 0.006 T/W t2/t1=0 t2/t1=3/5 t2/t1=1 t2/t1=-3/5 paramagnetic metal paramagnetic insulator hystersis region

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

0.5 1 1.5 U/W

0.01 0.02 0.03 T/W

antiferromagnetic phase metal-insulator-transition

0.5 1 1.5

U/W

0.01 0.02 0.03

T/W

antiferromgnetic phase metal-insulator-transition

0.75 1 1.25 1.5

U/W

0.004 0.008 0.012 T/W

increasing U decreasing U

t2 t1 = 0.6 t2 t1 = 0.8

t2 t1 = 0.8

1 1.1 1.2 1.3 1.4 1.5 U/W 0.002 0.004 0.006 T/W t2/t1=0 t2/t1=3/5 t2/t1=1 t2/t1=-3/5 paramagnetic metal paramagnetic insulator hystersis region

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

0.5 1 1.5 U/W

0.01 0.02 0.03 T/W

antiferromagnetic phase metal-insulator-transition

0.5 1 1.5

U/W

0.01 0.02 0.03

T/W

antiferromgnetic phase metal-insulator-transition

0.75 1 1.25 1.5

U/W

0.004 0.008 0.012 T/W

increasing U decreasing U

t2 t1 = 0.6 t2 t1 = 0.8 t2 t1 = 0.8

1 1.1 1.2 1.3 1.4 1.5 U/W 0.002 0.004 0.006 T/W t2/t1=0 t2/t1=3/5 t2/t1=1 t2/t1=-3/5 paramagnetic metal paramagnetic insulator hystersis region

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Very strong frustration t1≈t2

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Very strong frustration t1≈t2

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Very strong frustration t1≈t2

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Very strong frustration t1≈t2

5 10 15 20 25 30 35 40 45 50 DMFT iteration

• 1
• 0.5

0.5 1

n↑+n↓ n↑

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Very strong frustration t1≈t2

5 10 15 20 25 30 35 40 45 50 DMFT iteration

• 1
• 0.5

0.5 1

n↑+n↓ n↑

Classical spins, Z→∞: SDW type order with 120º variation

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

t1-t2 phase diagram at T=0

0.8 1 1.2 1.4 1.6 1.8 2

U/W

0.8 0.85 0.9 0.95 1

t2/t1 Néel phase

SDW phase

paramagnetic metal hysteresis region

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

t1/t2=0.98: Finite T phase diagram

1 1.2 1.4 1.6 1.8 2

U/W

0.0005 0.001 0.0015 0.002 0.0025

T/W

paramagnetic metal paramagnetic insulator

Néel phase

SDW phase

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

t1/t2=0.98: Finite T phase diagram

1 1.2 1.4 1.6 1.8 2

U/W

0.0005 0.001 0.0015 0.002 0.0025

T/W

paramagnetic metal paramagnetic insulator

Néel phase

SDW phase

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

t1/t2=0.98: Finite T phase diagram

1 1.2 1.4 1.6 1.8 2

U/W

0.0005 0.001 0.0015 0.002 0.0025

T/W

paramagnetic metal paramagnetic insulator

Néel phase

SDW phase

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

t1/t2=0.6, U/W≈1: Magnetic phases at finite doping

0.9 1 1.1

• ccupation

T/W=3e-4 T/W=1.5e-2

0.05 0.1

• µ chemical potential

0.3 0.6 0.9

n↑-n↓

Antiferromagnetism and phase separation

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

t1/t2=0.6, U/W≈1: Magnetic phases at finite doping

0.9 1 1.1

• ccupation

T/W=3e-4 T/W=1.5e-2

0.05 0.1

• µ chemical potential

0.3 0.6 0.9

n↑-n↓

Antiferromagnetism and phase separation

0.2 0.3 0.4 0.5 0.6 0.7 0.8

<n>

0.1 0.2 0.3 0.4 0.5 0.6

n↑-n↓

T/W=3e-4 T/W=1.5e-2

Ferromagnetism

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

t1/t2=0.6, U/W≈1: Magnetic phases at finite doping

0.9 1 1.1

• ccupation

T/W=3e-4 T/W=1.5e-2

0.05 0.1

• µ chemical potential

0.3 0.6 0.9

n↑-n↓

Antiferromagnetism and phase separation

0.2 0.3 0.4 0.5 0.6 0.7 0.8

<n>

0.1 0.2 0.3 0.4 0.5 0.6

n↑-n↓

T/W=3e-4 T/W=1.5e-2

Ferromagnetism

• High T: Antiferromagnetic metal
• Low T: Phase-separated Néel order
• Ferromagnetism for reasonable U

(Ulmke et al. EPJ B ’98)

• Sensitive to sign of t2 (Nagaoka!)

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

t1/t2=0.6, U/W≈1: Magnetic phase diagram at T=0

0.2 0.4 0.6 0.8 1

• ccupation n

2 4 6 8 U/W Paramagnet Ferromagnet Néel phase

S D W

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Summary:

• Complex phase diagram, extremely sensitive to U and J and t2
• Finite U: Always phase separated for t2 ≤ 0
• J>0 and t2>0: AFM, FM, phase separation and incommensurate phases
• Possible quantum critical point between FM and PM for finite J<0

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Competing interactions and magnetic order in correlated electron systems Concepts in Electron Correlations Hvar 2008 Institute for Theoretical Physics

Summary:

• Complex phase diagram, extremely sensitive to U and J and t2
• Finite U: Always phase separated for t2 ≤ 0
• J>0 and t2>0: AFM, FM, phase separation and incommensurate phases
• Possible quantum critical point between FM and PM for finite J<0

Outlook:

• Larger values of local spin plus single-ion anisotropy
• Influence of phonons: Peierls dimerization, Jahn-Teller coupling, ...

c.f. also Poster by Rok Žitko and Žitko, Peters, TP cond-mat

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