1. Effects of electronic correlations in iron and iron pnictides
A. A. Katanin
In collaboration with: A. Poteryaev, P. Igoshev, A. Efremov, S. Skornyakov, V. Anisimov
Institute of Metal Physics, Ekaterinburg, Russia
Special thanks to Yu. N. Gornostyrev for stimulating discussions

2. Iron properties a - bcc, g - fcc, e - hcp a  g Ts =1185 K
Mikhaylushkin, PRL 99, (2007)
a-iron: TC = K, meff =3.13mB
g-iron: qCW =-3450K, meff =7.47mB
a - bcc, g - fcc, e - hcp
a  g Ts =1185 K
Arajs. J.Appl.Phys. 31, 986 (1960)
Parsons, Phil.Mag. 3, 1174 (1959)

3. Itinerant approach (Stoner theory)
𝐼𝑁 𝐸 𝐹 =1
Large DOS implies ferromagnetism, provided that other magnetic
or charge instabilities are less important
- Too large magnetic transition temperatures, no CW-law
Moriya theory: paramagnons
Reasonable magnetic transition temperatures, CW law
Local moment approaches (e.g. Heisenberg model)
CW law

4. Rhodes-Wollfarth diagram
a-Iron (almost) fulfills Rhodes-Wollfarth criterion
(pc/ps  1)
Proposals for iron:
Local moments are formed by eg electrons (Goodenough, 1960)
95% d-electron localization (Stearns, 1973)
Local moments are formed from the vH singularity eg states (Irkhin, Katsnelson, Trefilov, 1993)

5. The magnetism of iron
a-Iron shows features of both, itinerant (fractional magnetic moment) and localized (Curie-Weiss law with large Curie constant) systems
Can one decide unbiasely (ab-initio), which states are localized (if any) ?
What is the correct physical picture for decribing local magnetic moments in an itinerant system?
Itinerant (Stoner and Moriya
theory)
Local moments (Heisenberg model)
Mixed (Shubin s-d(f)
= FM Kondo model)
How the local moments (if they exist) influence magnetic properties?
What is the similarity and differences between magnetism of a- and g- iron?

6. Dynamical Mean Field Theory
Energy-dependent effective medium theory
The self-energy of the embedded atom coincides
with that of the solid (lattice model), which is
approximated as a k-independent quantity
A. Georges et al., RMP 68, 13 (1996)

7. Spin-polarized LDA+DMFT
U = 2.3 eV, J = 0.9 eV
Magnetic moment (3.13)
Critical temperature K (1043K)
Lichtenstein, Katsnelson, Kotliar, PRL 87, (2001)

8. a (Bcc) iron: band structureeg
t2g
t2g и eg states are qualitatively different and weakly hybridized
A. Katanin et al., PRB 81, (2010)
Correlations can “decide”, which of them become local

9. a-iron: orbitally-resolved self-energy
Imaginary frequencies
Comparison to MIT:
t2g states - quasi-particles
Linear for the Fermi liquid
Divergent for an insulator
eg states - non-quasiparticle!
Bulla et al., PRB 64, (2001)
A. Katanin et al., PRB 81, (2010)

10. Self-energy and spectral functions at the real frequency axis
Real frequencies
a-Fe
Comparison to MIT:
From: Bulla et al., PRB 64, (2001)

11. How to see local moments: local spin correlation function
J=0.9
J=0
S(0)S()
Local moments are stable when
Fulfilled at the conventional Mott transition. Can it be fulfilled in the metallic phase ? t
A. Katanin et al., PRB 81, (2010)

12. Fourier transform of spin correlation function

13. Fourier transform of spin correlation function
Local moments formed out of eg states do exist in iron!

14. Which form of 𝜒 𝑙𝑜𝑐 𝜔 one can expect for the system with local moments?
Broaden delta-symbol:
(𝜔≪𝐽)
g is the damping of local collective excitations
For a-iron:
𝜇 eff =3.3 𝜇 𝐵
𝛾≈𝑇/2
There is an interval in densities where spin correlation
function depends weakly on temperature

15. Curie law for local susceptibilityeg
t2g
Total
neg =2*1.30=2.6=3-0.4=(up-up-down)-0.4 nt2g =3*1.46=4.4=3+1.4=(up up down)+1.4=1+3.4=(up)+3.4
Коллективизировано 3/7=42%
local moment
p(eg) = 0.56
p(t2g) = 0.45
p(total)=1.22
agrees with the
experimental data (known also after A.Liechtenstein, M. Katsnelson, and G. Kotliar, PRL 2001)

16. Effective model
The local moments are coupled via RKKY-type of exchange:
RKKY type
(similar to s-d Shubin-Vonsovskii model).
The theoretical approaches, similar to those for s-d model can be used

17. (fcc) iron
TN≈100K
Which physical picture (local moment, itinerant) is suitable to describe g-iron ?
What is the prefered magnetic state for the g iron at low T (and why)?

18. LDA DOS
The peak in eg band is shifted by 0.5eV downwards with respect to the Fermi level

19. (fcc) iron P. A. Igoshev et al., PRB 88, 155120 (2013)
More itinerant than a-iron ?

20. DOS with correlations

21. Static local susceptibility
P. A. Igoshev, A. Efremov, A. Poteryaev, A. K., and V. Anisimov,
PRB 88, (2013)

22. Dynamic local susceptibility

23. Size of local moment

24. Magnetic state: Itinerant picture
QX=(0,0,2) SDW2
Comparison of energies in LDA approach Shallcross et al., PRB 73, (2006)

25. Magnetic state: Heisenberg model picture
A. N. Ignatenko, A.A. Katanin, V.Yu.Irkhin, JETP Letters 87, 555 (2008)
For stability of (0,0,2p) state one needs J1>0, J2<0.

26. The polarization bubble, low T
LDA
LDA+DMFT k m m'
k+q
T=290К
2p(1,0,0)
2p(1/2,1/2,1/2)
2p(1,1/2,0)

27. Experimental magnetic structure
q = (2p/a) (1, 0.127, 0)
Tsunoda, J.Phys.: Cond.Matt. 1, (1989)
Naono and Tsunoda, J.Phys.: Cond.Matt. 16, 7723 (2004)

28. Colorcoding: red – eg, green – t2g, blue – s+p
Fermi surface nesting
Colorcoding: red – eg, green – t2g, blue – s+p
(0,x,2p) state is supported by the Fermi surface geometry – an evidence for itinerant nature of magnetism

29. The polarization bubble, high T
LDA
LDA+DMFT
T=1290К

30. Uniform susceptibilityk m m' k 1/
From high-temperature part: 𝜇 𝑒𝑓𝑓 =4.07 𝜇 𝐵 ( 𝑝 𝐶 ≅3/2)

31. (fcc) iron
𝜇 𝐶𝑊 =7. 7 𝜇 𝐵
The experimental value of the Curie constant is reproduced by the theory, although the absolute value of paramagnetic Curie temperature appears too large
Strong frustration!
Nonlocal correlations are important

32. Magnetic exchange in -iron
𝜒 𝐪 = 𝜒 0 1− 𝐽 𝐪 𝜒 0
𝜒 𝐪 = 𝜒 irr (𝐪) 1−Γ 𝜒 irr (𝐪) # 1 2 3 4 5 6 7 8
J z/2,K
-669
173
-449 17
-25
-123
-116 29
𝐽 𝐪 =− [𝜒 irr (𝐪)] −1 +𝐶
𝐽 𝟎 =−2500𝐾
𝐽 𝐐 =1200𝐾
The Neel temperature is much larger than the experimental one, similar to the result of the Stoner theory:
Paramagnons
Frustration, i.e. degeneracy of spin susceptibility in different directions

33. Local spin susceptibility of Ni
A. S. Belozerov, I. A. Leonov, and V. I. Anisimov, PRB 2013

34. Iron pnictide LaFeAsO Antiferromagnetic fluctuations Superconductivity
Itinerant system in the normal state
Effect of electronic correlations?
Possibility of local moment formation?

35. Density of states 387K 580К 1160К xy 0.142 0.242 0.454 xz, yz 0.131
0.163
0.306
3z2-r2
0.054
0.092
0.228
x2-y2
0.053
0.101
0.334
Damped qp states
Electronic correlations
qp states
dxz, dyz, dxy states can be
more localized
No qp states

36. Local susceptibility 387K 580К 1160К xy -0.142 -0.242 -0.454 xz, yz
-0.131
-0.163
-0.306
3z2-r2
-0.054
-0.092
-0.228
x2-y2
-0.053
-0.101
-0.334

37. Spin correlation functions
The situation is similar to g-iron, i.e. local moments may exist
at large T only, and, therefore,
seem to have no effect on
superconductivity

38. Orbital-selective uniform susceptibility
Local fluctuations are responsible for the part of linear-dependent
term in (T)
S. L. Skornyakov, A. Katanin, and V. I. Anisimov, PRL ’ 2011

39. Summary
The peculiarities of electronic properties (flat bands, peaks of density of states) near the FL may lead to the formation of local moments;
Analysis of orbitally-resolved static and dynamic local susceptibilities proves to be helpful in classification of different substances regarding the degree of local moment formation
In alfa-iron:
The existence of local moments is observed within the LDA+DMFT approach
The formation of local moments is governed by Hund interaction
In gamma-iron:
Local moments are formed at high T>1000K, where this substance exist in nature, but not at low-T (in contrast to alfa-iron); the low-temperature magnetism appears to be more itinerant
Antiferromagnetism is provided by nesting of the Fermi surface

40. Conclusions Thank you for attention ! In the iron pnictide:
Electronic correlations are important, but, similarly to g-iron, local moments may be formed at large T only
Different orbitals give diverse contribution to magnetic properties
Linear behavior of uniform susceptibility is (at least partly) due to peaks of density of states near the Fermi level
In the iron pnictide:
Thank you for attention !

42. Spin correlation functions

45. Spectral functions
Damped qp states
qp states
No qp states

47. Effective model and diagram technique
(similar to s-d Shubin-Vonsovskii model).
Treat eg electrons within DMFT and t2g electrons perturbatively
Simplest way is to decouple an interaction and integrate out t2g electrons
“bare” quadratic term
quartic interaction
Kind of boson (“4”) model, containing both, itinerant and local
moment degrees of freedom

48. Diagram technique: perspective

49. The dynamic susceptibility
“Moriya”
correction
bare
bare
RKKY
Influence of itinerant
electrons on local moment
degrees of freedom
Exchange integrals and magnetic properties can be extracted

50. Local moments in transition metals
Two different approaches to magnetism of transition metals (and explaining Curie-Weiss behavior):
Itinerant (Stoner, Moriya, …)
Local moment (Heisenberg, …)
Local moments in transition metals
Since they are (good) metals, at first glance no ‘true’
local moments are formed
However, under some conditions the formation
of (orbital-selective) local moments is possible:
Weak hybridization between different states (e.g.
t2g and eg)
Presence of Hund exchange interaction
Specific shape of the density of states
Can one unify these approaches
(one band: Moriya, degenerate bands: Hubbard, …)
More importantly: what is the ‘adequate’ (‘appropriate’) effective model, describing magnetic properties of transition metals ?

51. Local moments in transition metals
Since they are (good) metals, at first glance no ‘true’
local moments are formed
However, under some conditions the formation
of (orbital-selective) local moments is possible:
Weak hybridization between different states (e.g.
t2g and eg)
Presence of Hund exchange interaction
Specific shape of the density of states

52. Dependence on imaginary frequency

53. Paramagnetic LDA+DMFT
U = 2.3 eV, J = 0.9 eV, T = 1120 K
t2g states
eg states
Weakly correlated compound ?!?!?!?

54. t2g и eg состояния качественно различны и слабо гибридизованы
Важно учесть влияние
электронных корреляций

55. U dependence
J = 0.9 eV, b = 10 eV-1

56. Stability with temperature

57. Weak itinerant magnets
Saturation magnetic moment is small
The thermodynamic properties are detrmined by paramagnons; Hertz-Moriya-Millis theory: for ferromagnets (d=3, z=3) the bosonic mean-field (Moriya) theory is sufficient to describe qualitatively thermodynamic properties even close to QCP.
“paramagnon”
Curie-Weiss-like susceptibility

58. Frustration in Heisenberg FCC model

59. Polarization bubble m m' eg
t2g
t2g-e2g

60. G. Stollhoff, 2007

61. Diagram technique: perspective

62. Spin correlation function at different U
S(0)S() eg
almost flat !
t2g 

63. Weak itinerant magnets
Saturation magnetic moment is small
The thermodynamic properties are detrmined by paramagnons; Hertz-Moriya-Millis theory: for ferromagnets (d=3, z=3) the bosonic mean-field (Moriya) theory is sufficient to describe qualitatively thermodynamic properties even close to QCP.
“paramagnon”
Curie-Weiss-like susceptibility

64. Effective model it loc (similar to s-d Shubin-Vonsovskii model).
Treat itinerant electrons perturbatively:
introduce effective bosons for an interaction between itinerant electrons and integrate out itinerant fermionic degrees of freedom
“bare” quadratic term
quartic interaction

65. The dynamic susceptibility
“Moriya”
correction
bare
bare
RKKY
Influence of itinerant
electrons on local moment
degrees of freedom
Exchange integrals and magnetic properties can be extracted

66. Return to a-iron

67. Return to a-iron How do we recover RKKY exchange for a-iron? Assume:
𝜒 irr (𝐪) = 1/𝐼+ 𝜒 ′ irr (𝐪)
𝜒′ irr ≪1/𝐼
𝐽 𝐪 =− (𝜒 irr ) −1 +𝐶= 𝐶 ′ +𝐼2 𝜒 ′ irr (𝐪)
I ~ 1 eV – extracted in this way, in agreement with performed analysis and band structure calculations

68. Size of local moment

69. Orbitally-resolved DOS
LDA
U = 4 eV, b = 10 eV-1
a-Iron can be viewed as a system in the vicinity of an orbital-selective Mott transition (OSMT)

70. Ratio of moments

71. The size of the instantaneous and effective moment

72. Magnetic exchange: r r' L(S)DA formula:
(A. I. Liechtenstein, M. I. Katsnelson, et al.)
Requires a ‘reference magnetic state’ to calculate exchange integrals:
Reference state is needed to introduce magnetic moment in an itinerant approach
In which cases one can avoid use of the ‘reference state’ ?
Example: (one-band) Hubbard model at half filling
due to metal-insulator transition the electrons are localized, Jij=4t2/U

73. NM FM
(2,0,0)
,,)
FM, bcc