Heavy Fermions: a DMFT Perspective
G. Kotliar
Work with Kristjan Haule and Jihoon Shim at Rutgers
University.
Supported by the National Science Foundation.
July 27th 2008, Ohio State University
DMFT: trick to sum an infinite diagrams
Lattice Model, i, j,k,l site indices
1
[G, ]  TrLn[G0  ]  Tr[G ]  [G ]
[G ]  Sum 2PI graphs with G lines andU vertices


[G ,U ]
G
1
(G0  )  G
  DMFT [Gii, ii ]
1
1
=
[Gij  0, ij  0, i  j ]
1
(G0  ii [Gii ])  Gii
DMFT sums an infinite number of graphs. One can provide a on
perturbative definition for the sum, useful tricks for carrying wi
th high precision, a simple picture in terms of impurity models
But how accurate is it ?
Ulrich Schneider’s talk
But…..one band model, relatively high temperatures, how else can
we test (and therefore improve ) the method ?
A different (tried and true) approach, compare against experiment in a
wide set of materials, explore chemical trends.
Heavy Fermions: intermetallics containing 4f ele
ments Cerium, and 5f elements Uranium. Broad s
pd bands + atomic f open shells.
How do we know that the electrons
are heavy ?
Heavy Fermion Metals
-1 (emu/mol)-1
300
CeAl3
200
UBe13
100
0
0
100
T(K)
200
A Very Selected Class of HF
LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G
. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997).
Treat the local correlations of the f shell using DMFT. Treat the
non local correlations and the spd bands using LDA.
æ0
ö
0
÷
÷
S = ççç
÷
çè0 S ff (w)÷
ø
t (k ) ®
1
G ( k , i ) 
i  H [k ]  (i )
æH [k ]spd ,sps
çç
ççè H [k ] f ,spd
ö
H [k ]spd , f ÷
÷
÷
H [k ] ff ÷
ø
Determine energy and and  self consistently from extremizin
g a functional Chitra and Kotliar (2001) . Savrasov and Kotliar (2
001) Full self consistent implementation
Gdft[r ] ¾ ¾
® Glda +
t[Gloc, r ,U ]
dmf
12
LDA+DMFT CeIrIn5 (115)
Local f spectral function vs T
In
X
Ce
•At 300K, only Hubbard bands
SO
•At low T, very narrow q.p. peak
(width ~3meV)
•SO coupling splits q.p.: +-0.28eV
•Redistribution of weight up to very high
frequency
J. H. Shim, KH, and G. Kotliar
Science 318, 1618 (2007).
Buildup of coherence
Very slow crossover!
coherent spectral weight
Buildup of coherence in single impurity case
coherence pea
k
T
TK
scattering rate
T*
Slow crossover compared to AIM
Crossover around 50K
Optical conductivity in LDA+DMFT
Shim, HK Gotliar Science (2007)
D. Basov et.al.
K. Burch et.al.
•At 300K very broad Drude peak (e-e scattering, spd lifetime~0.1eV)
•At 10K:
•very narrow Drude peak
•First MI peak at 0.03eV~250cm-1
•Second MI peak at 0.07eV~600cm-1
eV
Structure Property Relation: Ce115’s
J. Shim et.
Optics and Multiple hybridization gaps al. Science
non-f spectra
10K
300K
In
Ce
In
•Larger gap due to hybridization with out of
plane In
•Smaller gap due to hybridization with inplane In
Conclusions Ce 115’s
•Accounts for many of the observed features of Ce ba
sed heavy fermions.
•Crossover is slower than in single impurity because of
the self consistency condition feedback.
•Structure Property Relation. Out of plane In site contr
ols hybridization. Confirmed by NMR.
•Predictions for ARPES currently being tested.
•Validates renormalized band theory at T=0 , explains
why it is not a good guide to experiments at most
temperatures.
•Accounts for Co 3d –Rh 4d -Ir 5d (Haule et. al. 2009)
A Very Selected Class of HF
Hidden Order in URu2Si2 dark matter problem.
U
Ru
Si
T. T. M. Palstra et.al. PRL 55, 2727 (1985
) D. A. Bonn et al. PRL (1988).
“Adiabatic continuity” between HO & AFM
phase
•Similar T0 and TN
•Almost identical thermodynamic
quantities (jump in Cv)
E. Hassinger et.al. PRL 77, 115117 (2008)
Two Broken Symmetry Solutions
Hidden Orde
r
LMA
K. Haule and GK
DMFT excitonic order
parameter
Order parameter:
Different orientation gives different
phases: “adiabatic continuity” explained.
Hexadecapole order testable by resonant X-ray
In the atomic limit:
DMFT “STM” URu2Si2 T=20 K
Ru
Si
Si
U
Fano lineshape:
q~1.24, ~6.8meV, very similar to exp Dav
Orbitally resolved DOS
Simplified toy model phase
diagram mean field theory
Mean field
Exp. by E. Hassinger et.al. PRL 77, 115117 (20
08)
A Hidden Order The CMT dark matter problem. (A. Schofield)
U
Ru
Si
URu2Si2: T. T. M. Palstra, A. A. Menovsky, J. van den Berg, A. J. Dirkmaat, P. H. Kes
, G. J. Nieuwenhuys and J. A. Mydosh Physical Review Letters 55, 2727 (1985)
C5f / T (mJ/K2mol)
Neutron Scattering. Specific heat vs.
magnetic Bragg-peak intensity. Tc’s.
500
URu2Si2
400
Smag ~ 0.2 R ln 2
300
Tc
200
100
To
Intensity (arb.unit)
0
1
mord ~ 0.01 - 0.04 mB
Q = (1,0,0)
Mason
Fåk
Honma
0
0
5
10
15
T (K)
20
25
Type-I AF xc ~ 100 Å
x ~ 300 Å
Hidden order
•Moment is tiny
(likely small admixture of AFM phase)
•Large loss of entropy can not be reconciled
with small moment
• Other primary symmetry breaking.
WHICH?
Small Effect at T0. Resistivity decreases as T decrea
se.
ThCr2Si2 bct - type ( I4/mmm )
URu2Si2
U
Ru
Si
a = 4.127 (Å)
c = 9.570 (Å)
Heavy fermion at high T,low T HO + SC
500
I // a
r (mW cm )
400
300
To ~ 17.5 K
200
100
01
T.T.M. Palstra et al.(1985)
W. Schlabitz et al.(1986)
M.B. Maple et al.(1986)
I // c
Tc ~ 1.2 K
10
100
T (K)
1000
Pseudo-gap opens at Tc. URu2Si2 measured through optical co
nductivity, D. A. Bonn et al. PRL (1988).
Hall effect as function of temperature in different external fields, Y.
S. Oh et al. PRL 98, 016401(2007). Fermi surface reconstruction in
zero and small fields. Very large fields polarized Fermi liquid.
Bernal et. al. PRL,2002.
P. Chandra P. Coleman et. al. (orbital antifer
romagnetism) time reversal symmetry breaki
ng.
Si NMR Spectra
0.10
0.10
See however the later experiments in
unstrained samples that do not show the
effect.
T = 20 K
Intensity (AU)
Intensity (AU)
0.08
0.06
Fit
Data
0.05
0.00
0.04
T = 4.2
-40
-20
0
K
0.02
20
40
60
Frequency (kHz)
URu2Si2
0.00
-60
-40
-20
0
20
40
Frequency (kHz)
60
80
Some Proposals for hidden order in the literature
•Lev. P. Gorkov: 1996:
-Three spin correlators.
• Chandra et al., Nature’02
- Incommensurate Orbital Antiferromagnetism
• Mineev & Zhitomirsky, PRB ’05
- SDW with tiny moment.
• Varma & Zhu, PRL’06
- Helical Order, Pomeranchuk instability of the Fermi surface ?
• Elgazaar, & Oppeneer, Nature Materials’08
- DFT: with weak antiferromagnetic order parameter
• Santini and Amoretti PRL 04
-Quadrupolar ordering.
• Fazekas and Kiss PRB 07
-Octupolar ordering.
Neutron scattering under hydrostatic pressu
re
H. Amitsuka, M. Sato, N.
Metoki, M. Yokoyama, K.
Kuwahara, T. Sakakibara,
H. Morimoto, S. Kawarazaki,
Y. Miyako, and JAM
PRL 83 (1999) 5114
URu2Si2 Stress in ab plane
Large moment when stress in ab plane
No moment when stress in c plane
M Yokoyama, JPSJ 71, Supl 264 (2002).
Further Japanese work showed that NMR in unstrained samples did not broad
en below T0
P – T phase diagram
H. Amitsuka et al.,JMM
M
310, 214(
2007).
LMAF
Little change in bulk properties with const. P when crossings into HO(T0)
or LMAF(TN) phases, e.g. opening of similar gaps: Adiabatic Continuity.
Phase diagram T vs P based upon resistivity and calorimetric
experiments under pressure.
E. Hassinger et al. PRB 77, 115117(2008).
Similar to Amitsuka’s T – P phase diagram
“Adiabatic continuity” between HO & AFM
phase
•Similar T0 and TN
•Almost identical thermodynamic
quantities (jump in Cv)
E. Hassinger et.al. PRL 77, 115117 (2008)
ARPES does not agree with LDA
J.D. Denlinger et.al., 2001
Comparison of low-field bulk properties - pure vs.
4%Rh
Y.S.Oh, K.H.Kim, N. Harrison, H.Amitsuka & JAM, JMMM 310,855(2007).
Effects of Rh Dopiong. U(Ru1-x,Rhx)2Si2
T
HO state in URu2Si2 develops a gap in FS below 17K
Rh doping removes HO state to make HF groundstates
Tcoh~56 K
THO=17.5 K
HF
HO
TC1.5 K
HO+AF
0K
0.0
0.04
Rh x
M. Jaime et al. PRL (2002)
N. Harrison PRL (2003)
K. H. Kim et al., PRL (2003)
K. H. Kim et al., PRL (2004)
Comments concerning Hidden Order
URu2Si2, at high temperatures is not too different from a garden
variety heavy fermion.
ARPES does not agree with LDA at 30 K.
HO can be totally destroyed by H and Rh-x
HO converts to LMAF by P thru a first order line.
HO and LMAF are remarkably similar (“Mydosh’s adiabatic contin
uity”)
HO opens some form of a gap in optics.
HO likely involves an electronic topological transition
[Hall Effect, also Nernst]
HO exhibits two INS modes:(100)@2meV and (1.400)@5meV of longi
tudinal fluctuations/excitations.
HO (but not LAMF) turns into superconductivity at 1.7 K.
Heavy Femions Early Theoretical
Work.
Early work : variational wave functionsVarma, C. M. and Y
afet, Y., Phys. Rev. B13, 295 (1975);
SlaveBosons MFT , 1/ N , A. Auerbach and K. Levin. Phys. R
ev. Lett. 57 (1986), p. 877
A. Millis and P.A. Lee. Phys. Rev. B 35, 3394 (1987)
Simple description of high and low T regimes.
Very simple Renormalized band theory at T=0.
G.Kotliar A. Ruckenstein extensions to finite U
Early workGeneralized Anderson Lattice Model



f
f i† f j  U  ni ni  
i,

i
 V
ij
i , j  ,



f
 m ij )(ci† c j  c †j ci )
f i† c j  c.c.  HALM
i
 V
i , j  ,
i , j  ,
ij
f i† f j   i [b † ibi   f i† f i  1] 
i,

 (t
ij

 (t
i , j  ,
ij
 m ij )(ci† c j  c†j ci )
f i† bic j  c.c.  HALM
•High temperature
Ce-4f local moments
•Low temperature –
Itinerant heavy bands
6
Dynamical Mean Field Theory. Cavity Construction.
-
å
J ij Si S j - h å Si
i, j

 (t
i , j  ,
D (w) =
ij
H MF = - heff So
i
†
H Anderson
† Imp   (V c0 A +c.c). 
 m ij )(c c j  c j ci ), U  ni ni
†
i
 A A  



†
,
mc0† c0  Uc0†c0c0† c0
,
i
å
a
*
Va Va
w - ea
A()
b
b
b
¶
c
(
t
)[
+ m- D (t - t ')]cos (t ') + U ò no­ no¯
òò
¶
t
0 0
0
†
os
10
Glatt (iwn , k )[D ] =
1
[iwn + m+ t (k ) - S imp (iwn )[D ]]
A() D ( w)
Gimp (iwn ) =
mi = th[b å J ij m j + h ]
å
k
1
[- D (iwn ) +
1
- t (k )]
Gimp (iwn )[D ]
j
A. Georges, G. Kotliar (1992)
Gimp (iwn )[D ] =
å
k
G latt (iwn , k )[D ]
11
Dynamical Mean Field Theory




Exact in the limit of large coordination (Metzner an
d Vollhardt 89) .
Can treat arbitrary broken symmetry solutions delta
is site , spin, orbital, etc. dependent.
Extension to real materials (Anisimov and Kotliar 19
97, Kotliar et. al. RMP 2006).
DMFT equations are still hard to analyze and solve.
| 0 > ,|­ > ,|¯> ,|­ ¯>
®
| LSJM J g... >
t (k )
®
æH [k ]spd ,sps
çç
ççè H [k ] f ,spd
DFT+DMFT
ö
H [k ]spd , f ÷
÷
÷
H [k ] ff 22 ÷
ø
12
DMFT “STM” URu2Si2 T=20 K
Ru
Si
Si
U
Fano lineshape:
q~1.24, ~6.8meV, very similar to exp Dav
At T=20 small effects on spd larger
gapping of the f’s. PES-DMFT
Partial DOS
Ground state atomic multiplet of f2
configuration in tetragonal field
High T
Low T
J=4
Only 35K!
|state>==|J=4,Jz>
DMFT allows two broken symmetry states
at low T. Look for two sublattice structure.
Density matrix for U 5f state
the J=5/2 subspace
Large moment phase:
J=5/2
Moment free phase:
tetragonal symmetry broken->
these terms nonzero
J=5/2
Valence histogram point of view.
The DMFT density matrix has mostly weight in two singlet f^2 configuratio
ns
Definitely f^2, “Kondo “ limit, J=4, two low lying singlets
Test via photoemission [Denlinger and Allen ]
Therefore there are two singlests relevant at low energies but they are not
non Kramer doublets. Conspiracy between cubic crystal field splittings and
tetragonal splittings bring these two states close.
This is why URu2Si2 is sort of unique.
.
DMFT excitonic order
parameter
Order parameter:
Different orientation gives different
phases: “adiabatic continuity” explained.
Hexadecapole order testable by resonant X-ray
In the atomic limit:
Simplified toy model phase
diagram mean field theory
Mean field
Exp. by E. Hassinger et.al. PRL 77, 115117 (20
08)
Arrested Kondo effect
On resonance
•DFT f-core: goof descriptio
n of bands 30meV away from
EF
•DFT f-valence: many f-bands at
EF, substantial disagreement with
ARPES & DMFT
DMFT:
very narrow region of f-spectral
weight ±10meV around EF appears
below T*~70K
Below 35K, partial gap starts to open->singlet to singlet Kondo effect
At low temperature, full gap in f’s (not spd’s).

200meV
DMFT A(k,) vs ARPES
Off resonance
Very good agreement,
except at X point
Surface origin of pocket at X point
LDA+DMFT - Si-terminated surface slab
LDA+DMFT - bulk
Surface Slab Calculation
Z

X

• No hole-pocket at the X-point.
Z

X

• Hole pocket surface state appears at X-point!
Layer resolved
spectra
Fermi surface nesting,
reconstruction below Tc
T>T0
Nesting 0.6a* and 1.4a*
T<T0
Conclusions URu2Si2
•5f^2
configuration in the Kondo limit
•Hidden order has hexadecapole character
•Simple connection between the LMAF and the HO
state.
•Fermi surface reconstruction at low temperature
•Absorbtion of f degrees of fredom at very low ene
rgies is arrested by a (small) crystal field splitting
•D. Cox original guess for UBe13 was (almost) right
for URu2Si2.
•DMFT results should be confronted carefully wit
h experiments.
Conclusion: some general com
ments.
•DMFT
approach. Can now start from the material.
•Can start from high energies, high temperatures, where
the method (I believe ) is essentially exact, far from critic
al points, provided that one starts from the right “refere
nce frame”.
•Still need better tools to analyze and solve the DMFT eq
uations.
•Still need simpler approaches to rationalize simpler limit
.
•Validates some aspects of slave boson mean field theorie
s, modifies quantitatively and sometimes qualitatively the
answers.
•At
lower temperatures, one has to study different
broken symmetry states.
•At lower temperatures, one has to study
different broken symmetry states.
•Compare
•Beyond
free energies, draw phase diagram
DMFT: Write effective low energy theories that
•match the different regions of the phase diagram.
•Close contact with experiments.
•Many materials are being tried, methods are being refined
•Contemplating material design using correlated electron
systems.
Conclusions
•DMFT tools can be used to understand/pred
ict
properties of correlated materials
•Kondo effect in URu2Si2 is arrested
bellow crystal field splitting energy. Gives room to
ordered states, either AFM state or orbital order.
•AFM state and hidden order state have the same or
der parameter: mixing between atomic singlet states
.
•Orientation of the order parameter decided which st
ate is stabilized.
•Mystery of URu2Si2 hidden order solved
.
• Strong 2nd layer Ru contribution to G=Z equivalence
Si
Ru
Surface State(s) origin
• SS hole band distinctly
originates from the top Si atom
• Some U5f weight pulled into
Ru surface bands
Si
A(k,), first 4 layers
U
• Very little 3rd layer Si contribution to the SS hole band
CeMIn5 M=Co, Ir, Rh
Ir
 CeRhIn5: TN=3.8 K;   450 mJ/molK2
In
Ce
 CeCoIn5: Tc=2.3 K;   1000 mJ/molK2;  C
eIrIn5: Tc=0.4 K;   750 mJ/molK2
out of plane
in-plane
Phase diagram of 115’s
Why CeIrIn5?
•Ir atom is less correlated than
Co or Rh (5d / 3d or 4d)
•CeIrIn5 is more itinerant(coherent)
than Co (further away from QCP)
Generalized Anderson Lattice
Model



f
f i† f j  U  ni ni  
i,

i
 V
i , j  ,
ij
 (t
i , j  ,
ij
 m ij )(ci† c j  c †j ci )
f i† c j  c.c.  HALM
•High temperature
Ce-4f local moments
C+ff+
•Low temperature –
Itinerant heavy bands
6
Angle integrated photoemission
Expt Fujimori et al., PRB 73,
224517 (2006) P.R B 67, 144
507 (2003).
Experimental resolution ~30meV
Surface sensitivity at 122 ev ,
theory predicts 3meV broad band
Theory: LDA+DMFT, impurity solvers
SUNCA and CTQMC Shim Haule and G
K (2007)
Uranium Heavy Fermions
Relevance of the Kondo effect.
Dan Cox, 5f^2 configuration +crystal fields select a ground which is a non
Kramers doublet.
Multichannel Quadrupolar Kondo effect in U . Non Fermi liquid.
Early work of D. Cox. U Phys. Rev. Lett. 59, 1240 (1987)
Other possibilities, magnetic Kondo effect when U is f^3
When a heavy Fermi liquid is formed, what is the volume of the Fermi
surface? Luttinger theorem: it contains nf+ ncond electrons. Mod 2.
For f^1 configuration ( Cerium ) the Fermi surface expands as the tem
perature is reduced. The T=0 Fermi surface is well approximated by th
e LDA Fermi surface.
Is the true (experimental) Fermi surface of f^2 compounds close to th
e LDA Fermi surface as well ?
URu2Si2
Heavy Fermion Problem (more
general).
o
Intermetallic compounds.
o
Bare (high energy) degrees of freedom: open shell ions , i.
e. Ce, U and conduction electrons.
o
Low energy degrees of freedom. Quasiparticles composed
of those degrees of freedom sometimes form a heavy Ferm
liquid. M*/M ~50-1000.
o
Large variety of ground states, superconducting, magnetic
, etc.