Antoine Georges Strong Correlations From Hund's Rule Coupling

Strong Correlations
From Hund’s Rule Coupling
Antoine Georges
Bariloche, November 2011

The Platters said:
« Only U can do
make all this world
seem right… »
… Take-home message of this talk:
« Not only U, also J H » !
Friedrich Hund
1896-1997

Based mainly on two papers:
Special thanks to:
PRL (in print)
Jernej Mravlje ; Luca ‘de Medici ; Markus Aichhorn; Michel Ferrero

I - A model study: 3 degenerate bands
w/ same bandwidth 2D
occupied by N electrons per site

N=1 electron N=2 electrons
Quasiparticle
weight Z
vs. U/D
N=3 electrons

Key points
* At ½ filling (3 el. In 3 orbitals) Mott critical
U c is strongly reduced upon increasing J
cf. Bunemann, Weber, Gebhard et al.; Han, Jarrell, Cox PRB1998
• IN CONTRAST, away from ½ filling (e.g.
N=1,2,4,5) U c is strongly ENHANCED
cf. L. ‘de Medici PRB 83, 205112 (2011)
• At the same time, quasiparticle coherence
scale (~ZD) is strongly SUPPRESSED as
J is increased

As a result, for generic filling…
(i.e. not ½ filling
and not 1 electron or hole)
e.g. for 2 or 4 electrons
in degenerate t 2g bands
… J is « Janus-faced » :
it has two ANTAGONISTIC
effects

Suppresses coherence scale « Bad metal » behaviour, small Z
Increases U c Enhances range of metallic state

Intensity map of quasiparticle weight Z (J/U=0.15)
revealing wide range of « bad metal » behaviour for N=2,4
cf pioneering papers by P.Werner et al « spin-freezing » PRL2008, and
K.Haule & G.Kotliar, New J. Physics 2009 in the context of pnictides

J- dependence of U c : estimate from the
atomic limit

Reduction of coherence scale:
effect of J on Kondo screening
cf. Okada & Yosida, Prog. Theor.Phys 1973; Yanase & Yamada JPSJ1997;
Pruschke & Bulla, EPJ 2005; Nevidomskii & Coleman PRL2010
Ground-state degeneracy is reduced by J
« more quantum » : T K goes down
Toy model: N spin-1/2 coupled to K channels (exact screening)

Suppression of T K due to J :
NRG evidence
Pruschke & Bulla, EPJ 2005
Mravlje et al. 2011

DMFT is an ideal tool
to handle these effects:
- Has atomic physics and
multiplets right from the start
- Describes screening
of local atomic multiplets
by electron hopping

A theoretical framework
for strongly-correlated materials
starting from ATOMS
instead of an electron gas picture
« Dynamical Mean-Field Theory »
Correlated electrons in large dimensions:
W.Metzner & D.Vollhardt, 1989 (Aachen)
Dynamical Mean-Field Theory:
A.G. & G.Kotliar, 1991 (Princeton/Rutgers)

Dynamical Mean-Field Theory:
viewing a material as an (ensemble of) atoms
coupled to a self-consistent medium!
Solid: crystal lattice of atoms
Atom
Effective
Medium

A simple example: the Hubbard model#
Focus on a given lattice site:
Atom can be in 4 possible configurations:
Describe ``history’’ of fluctuations between those configurations

Drawing a map of early
transition-metal oxides
(both 3d and 4d)
with Hund’s rule coupling
as guidance

Correlation effects in 4d oxides due to J, not to Mott physics
(except when strong splitting between orbitals)
3d oxides: U/D ~ 4
4d oxides: U/D ~ 2
Focus in the following on:
Technetium and Ruthenate oxides

``Atsushi Fujimori’s map of RMO 3 perovskites’’
J.Phys Chem Sol. 53 (1992) 1595
YTiO 3
LaTiO 3
CaVO 3
SrVO 3
Partially filled d-shells… and yet often insulators

-II- Record-high Neel temperature in
Technetium oxide (SrTcO 3 )
- contrast to SrMnO 3

Mechanism: SrTcO 3 (T N ~1000K)
very close to Mott transition
(In contrast SrMnO 3 is a more localized insulator, T N ~ 260 K)
Mravlje,
Aichhorn, A.G.
arXiv:1108.1168
LDA+DMFT

- III -
Sr 2 RuO 4 : a Hund’s coupling
correlated metal
(J.Mravlje et al., PRL 2011)

Sr 2 RuO 4 : the `Helium 3’ of
transition-metal oxides !
• Huge high-quality crystals !
• Has been investigated with basically all
techniques in the experimentalist’s toolbox
• 4d-row structural analogue of La 2 CuO 4
• Beautiful review articles:
- A.Mackenzie and Y.Maeno
RMP 75, 657 (2003)
- Bergemann, Adv. Phys. 52, 639 (2003)
[Focus on dHvA quantum oscillations]

Yet, outstanding puzzles about this
compound remain…
• A 4d material
expect not very large U (< 3eV)
- Yet, effective mass enhancement (vs. band/LDA value)
as large as ~ 5
- Strong orbital dependence
- Low Fermi-liquid coherence scale:
- T 2 law obeyed only below ~ 30K
- Quasiparticles suppressed above ~ 130K, at
which interlayer resistivity crosses over from
metallic to insulating

Basic electronic structure :
• 4 electrons in t 2g shell
• d xy orbital yields a quasi 2D
band γ-sheet of FS
- d xz , (resp. d yz ) bands with directional
hopping along x (resp. y) α,β sheets

Fermi surface of
Sr 2 RuO 4
and populations from dHvA:
FS from photoemission:
β-sheet
(0.9 electrons)
α-sheet
0.2 hole
k y
FS from quantum oscillations:
k x
γ-sheet (xy)
1.3 electrons

Bands (LDA)
~ 1.5 eV
(xz,yz)
~ 3.6 eV
(xy band)
But kinetic energies of all bands comparable

Effective masses
- Rather large !
- Strongly orbital dependent.
- The widest band has the largest eff. mass enhancement !

Dynamical Mean-Field Theory
calculations (using full LDA bandstructure):
- effective masses -
- Increase of effective mass as J is increased
- Orbital differentiation: xy heavier
- Comparable mass enhancement
would require U=5eV at J=0 !
For
U=2.3 eV

Quasiparticle coherence scale
Inverse QP lifetime / T:
Fermi Liquid @ low-T
T-linear above ~ 400 K
Reaches ~kT at ~ 70K
Low-T calculations made possible by recent progress in
DMFT technology: CT-QMC solver of P. Werner et al.

Comparison to ARPES

Self-energies vs. ARPES

Qualitative understanding: some hints
from local picture (effective quantum impurity).
4-electron subspace splits into:
T=2,S=0: 5 states
T=0,S=0: 1 state
T=1,S=1: 9 states
At J=0: all degenerate, 15-dim rep. of SU(6) symmetry
Very high Kondo/coherence scale ~ fraction of an eV
For non-zero J: SU(2)*SU(2), 9-fold ground-state multiplet
Exactly screened local model Fermi liquid
Suppressed T K
J becomes active when larger than
coherence scale of J=0 model

Orbital dependence:
why is xy more correlated ?
Hund’s coupling lowers the inter-orbital coupling:
Individual fermiology of orbitals becomes relevant
xy `effective hybridization strength’ to environment is
smaller (by at least a factor of 2) than xz, yz
Initial LDA eff. Hybridization:
Proximity of van Hove singularity
for xy makes DOS larger and
Strongly p-h asymmetric
Both effects reduce
hybridization

Another important class of materials for
which Hund’s coupling- induced
correlations are key:
IRON-based superconductors
pnictides & chalcogenides
see previous talk by G.Kotliar
Haule and Kotliar, New J. Phys 11 (2009) 025021
+ several recent articles
See also: Aichhorn et al. PRB 2010 (FeSe)
Werner et al. arXiv:1107.3128 (BaFe2As2)

From model calculations to real
materials: the happy marriage of
DMFT with electronic structure
(DFT) methods
- an interdisciplinary effort
started in ~ 1998 and still continuing
today
Anisimov, Kotliar, Poteryaev et al., 1997
Lichtenstein & Katsnelson, 1998

Recent algorithmic breakthroughs
entering a new age for DMFT approaches
(and extensions) …!
Continuous-time quantum Monte Carlo (CT-QMC)
*Rubtsov 2005 Interaction expansion(CT-INT)
*P. Werner, M.Troyer et al (ETH) 2006
Hybridization expansion(CT-HYB)
*Gull/Parcollet 2008 Auxiliary field (CT-AUX)
* Submatrix updates (Gull et al. 2010) convergence in
cluster size reached for some parameter regime (full k-
dependence resolved)

Electronic structure with DMFT:
The FLAPW (Wien2k) / Wannier route
The TRIQS project: first release 01-11-11 !
Toolbox for Research on Interacting Quantum Systems
Project Leaders: Olivier Parcollet (CEA-IPhT), M.Ferrero (EP)
Key developers of Wien2TRIQS: M.Aichhorn, L.Pourovskii,
C.Martins, V. Vildosola, with also S.Biermann, A.Georges
Self-contained library of codes,
including CT-QMC `impurity’ (DMFT) solver
and Wien2TRIQS interface for DFT+DMFT calculations

To be presented
(M.Ferrero, M.Aichhorn)
together with introduction
to LDA+DMFT
at the….

Conclusions
• Other pathways to strong correlations than Mott-
Hubbard
• Hund’s coupling crucial, especially for 4d
oxides but also other materials (iron-SC)
• 5d oxides: spin-orbit !
• Splitting of orbital degeneracy also a key
relevant parameter
The DMFT framework
helped by recent algorithmic breakthroughs,
can handle these issues in a realistic
way for specific materials.