1. Itinerant Ferromagnetism: mechanisms and models
J. E. Gubernatis,1 C. D. Batista,1
and J. Bonča2
1Los Alamos National Laboratory
2University of Ljubljana

2. Magnetism

3. Outline Basic models Traditional mechanism
Interference (Nagaoka) mechanisms
Mixed valent mechanism: strong ferromagnetism
Relevance to experiment
Summary

4. Approach Analytic theory Numerical Simulation
Generate effective Hamiltonians
Usually, 2nd order degenerate perturbation theory
Identify physics by
Inspection
Exact solutions
Numerical simulations
Guide and interprets simulations of the original Hamiltonian
Numerical Simulation
Compute ground-state properties
Constrained-Path Monte Carlo method
Extend analytic theory

5. Standard Models f electrons
Kondo Lattice Model
Periodic Anderson Model

6. Periodic Anderson ModelV d f

7. Standard Models Anderson Heisenberg Hubbard Kondo weak Mixed valent
Infinite U
J/t >>1
0< nc <2
U/t >>1
|EF-f|<<V
0 <nd <2
<nf >=1
J/t<<1
0< nc <1
|EF-f| >> V
0< nd <1
nf=1
Heisenberg
RKKY
Segmented Band
U/t >> 1
|EF-f| ~ 0
1< n <2
Mixed valent
weak
strong

8. Standard Models
Small J/t KLM and large U/t PAM connected by a canonical transformation, when |EF-f| >>V, nf=1
Emergent symmetry: [HPAM,nif]=0.
Result: JV2/ |EF-f|
Mixed valence regime: |EF-f| 0.

9. Traditional Mechanism
Competing energy scales
TRKKY  J² N(EF)
TKondo EF exp(-1/JN(EF))
Approximate methods support this.
Kondo compensation explains
moment reduction
heavy masses.
T=0 critical point at J  EF.
Mixed valent materials are paramagnetic.
O(1)

10. Traditional Mechanism
“The fact that Kondo-like quenching of local moments appears to occur for fractional valence systems is consistent with the above ideas on the empirical ground that only when the f-level is degenerate with the d-band is the effective Schrieffer-Wolff exchange interaction likely to be strong enough to satisfy the above criterion for a non-magnetic ground state of JN(0)=O(1).”
Doniach, Physica 91B, 231 (1977).

11. Ce(M1-xXx)3 B2 CeRh3(N1-yYy)2
Quantum Monte Carlo
Bonca et al
Cornelius and Schilling, PRB 49, 3955 (1994)

12. Single Impurity Compensation… …
Compensation cloud

13. Exhaustion in strong mixed valent limit

14. Nagaoka-like mechanism for Ferromagnetism

15. Nagaoka Mechanism
Relevant for holes away from half-filing in a strongly correlated band (U/t >> 1).
Holes can lower their kinetic energy by moving through an aligned background.
Hole can cycle back to original configuration.
Ground state wavefunction results from the constructive interference of many hole-configurations.

16. Nagaoka-like Mechanism in Weak Mixed Valent Regime
Adding tf << td embeds a Hubbard model in the PAM.
When U/tf>>1, the physics of the Nagaoka mechanism applies.
In polarized regime, conduction band is a charge reservoir for localized band
Increasing pressure, converts f’s to d’s,
Decreasing pressure, converts d’s to f’s.
Holes
U= Hubbard Model
Becca and Sorella, PRL 86, 3396 (2001)

17. Periodic Anderson Model
Mixed valent regime
U/t>>1, |EF-f|~0
Observations at U=0
Two subspaces in each band.
Predominantly d or f character.
Size of cross-over region  V²/W.
Very small.
mixed
valent
regime
Localized moment regime

18. Mixed Valent Mechanism
Take U = 0,
EF  f and in lower band.
Electrons pair.
Set U  0.
Electrons in mixed valent state spread to unoccupied f states and align.
Anti-symmetric spatial part of wavefunction prevents double occupancy.
Kinetic energy cost is proportional to .

19. Mixed Valent Mechanism
A nonmagnetic state has an energy cost   to occupy upper band states needed to localize and avoid the cost of U.
Ferromagnetic state is stable if   .
TCurie  
By the uncertainty principle, a state built from these lower band f states has a restricted extension.
Not all k’s are used.

20. Numerical Consistency: 2D

21. Some Other Numerical Results
Local Moment Compensation
In the single impurity model, a singlet ground state implies
In the lattice model, it implies
In the lattice, the second term is more significant than the first.
Mainly the f electrons, not the d’s, compensate the f electrons.

22. Experimental Relevance
Ternary Ce Borides (4f).
CeRh3B2: highest TCurie (115oK) of any Ce compound with nonmagnetic elements.
Small magnetic moment.
Unusual magnetization and TCurie as a function of (chemical) pressure.
Uranium chalcogenides (5f)
UxXy , X= S, Se, or Te.
Some properties similar to Ce(Rh1-x Rux)3B2

23. Challenges Expansion case Compression case
Doping removes magnetic moments
Increases overlap
Tc decrease while M increases
Compression case
M does not increases monotonically
Specific heat peaks where M peaks

24. (LaxCe1-x)Rh3B2 Increase of M.
If CeRh3B2 is in a 4f-4d mixed valent state and EF  f, TCurie  .
With La doping,
f electron subspace increases so M increases.
Localized f moment regime reached via occupation of f states in upper band.

25. Ce(Rh1-xRux)3B2 Reduction of M. Peak in Cp
If CeRh3B2 is in a 4f-4d mixed valent state and EF  f, TCurie  .
With Ru doping, EF < f ,  increases, and eventually  ~ .
Peak in Cp
Thermal excitations will promote previously paired electrons into highly degenerate aligned states.

26. Summary
We established by analytic and numerical studies several mechanisms for itinerant ferromagnetism.
A novel mechanism operates in the PAM in the mixed valence regime.
It depends of a segmentation of non-degenerate bands and is not the RKKY interaction.
The segmentation of the bands is also relevant to the non-magnetic behavior.
Non-magnetic state is not a coherent state of Kondo compensated singlets.

27. Summary In polarized regime, we learned
Increasing pressure, converts f’s to d’s,
Decreasing pressure, converts d’s to f’s.
The implied figure of merit is |EF - f|.
If large, local moments and RKKY.
If small, less localized moments and mixed valent behavior.

28. Summary
In the unpolarized regime, the d-electrons are mainly compensating themselves; f-electrons, themselves.
In the polarized regime, more than one mechanism produces ferromagnetism.
The weakest is the RKKY, when the moments are spatially localized.
The strongest is the segmented band, when the moments are partially localized.