Folie 1

1
Experimental Tutorial on Quantum Criticality
Philipp Gegenwart Max-Planck Institute for
Chemical Physics of Solids, Dresden, Germany

• Reviews on quantum criticality in strongly
  correlated electron systemsE.g.
• G.R. Stewart, Rev. Mod. Phys. 73, 797 (2001).
• H. v. Löhneysen, A. Rosch, M. Vojta, P. Wölfle,
  cond-mat/0606317
• Outline of this talk
• Introduction
• Quantum criticality in some antiferromagnetic HF
  systems (mainly those studied in Dresden)
• Ferromagnetic quantum criticality

Second part
2
Collaborators
T. Westerkamp, J.-G. Donath, F. Weickert, J.
Custers, R. Küchler, Y. Tokiwa, T. Radu, J.
Ferstl, C. Krellner, O. Trovarelli, C. Geibel, G.
Sparn, S. Paschen, J.A. Mydosh, F. Steglich K.
Neumaier1, E.-W. Scheidt2, G.R. Stewart3, A.P.
Mackenzie4, R.S. Perry4,5,Y. Maeno5, K. Ishida5,
E.D. Bauer6, J.L. Sarrao6, J. Sereni7, M. Garst8,
Q. Si9, C. Pépin10 P. Coleman11 1Walther
Meissner Institute, Garching, Germany 2Augsburg
University, Germany 3University of Florida,
Gainesville FL, USA 4St. Andrews University,
Scotland 5Kyoto University, Japan 6Los Alamos
National Laboratory, USA 7CNEA Bariloche,
Argentina 8University of Minnesota,
Minneapolis, USA 9Rice University, Texas,
USA 10CEA-Saclay, France 11Rutgers University,
USA
3
f-electron based Heavy Fermion systems

• Lattice of certain f-electrons (most Ce, Yb or
  U) in metallic environment
• La3 4f0, Ce3 4f1 (J 5/2), Yb3 4f13 (J
  7/2), Lu3 4f14 (6s25d1,l3)
• partially filled inner 4f/5f shells ? localized
  magnetic moment
• CEF splitting ? effective S1/2

4
Microscopic model Kondo effect
(Jun Kondo 63)
J hybridization between local moments and
conduction el.? AF coupling J lt 0
local moment
conduction el
?
TK characteristic Kondo-temperature
Kondo- minimum
TK
T5
lnT
T lt TK formation of a bound state between local
spin and conduction electron spin ? local spin
singlet
5
Anderson Impurity Model
cond.-el
f-el
hybridizationVsf
on-site Coulombrepulsion Uff
Formation of an (Abrikosov-Suhl) resonance at EF
of width kBT ? extremely high N(EF) ? heavy
fermions
6
Landau Fermi liquid
Lev Landau 57
Excitations of system with strongly interacting
electrons
11correspondence
Freeelectron gas
7
Magnetic instability in Heavy Fermion systems
Doniach 1977
8
Itinerant (conventional) scenario
Moriya, Hertz, Millis, Lonzarich,
OP fluctuations in space and time AF z2 (deff
dz)

• Heavy quasiparticles stay intact at QCP,
  scattering off critical SDW ? NFL
• unconventional quantum criticality (Coleman,
  Pépin, Senthil, Si)
• Internal structure of heavy quasiparticles
  important ? 4f-electrons localize
• Energy scales beyond those associated with
  slowing down of OP fluctuations

9
CeCu6-xAux
CeCu6-xAux xc0.1 inelastic neutron
scattering O. Stockert et al., PRL 80 (1998)
critical fluctuations quasi-2D ! A. Schröder et
al., Nature 407 (2000)
non-Curie-Weiss behavior q-independent ? local !!
10
Grüneisen ratio analysis
Thermal expansion ? 1/V ?S/?p ? V-1
dV/dT Specific heat C/T ?S/?T
Itinerant theory ? T??z T-1(L. Zhu, M.
Garst, A. Rosch, Q. Si, PRL 2003)
11
Experimental classification
unconventional CeCu6-xAux YbRh2Si2
conventional CeNi2Ge2 CeIn3-xSnx CeCu2Si2 CeCoIn5

12
CeNi2Ge2 very clean system close to zero-field
QCP
TK 30 K, paramagnetic ground state
P. Gegenwart, F. Kromer, M. Lang, G. Sparn, C.
Geibel, F. Steglich, Phys. Rev. Lett. 82, 1293
(1999) See also F.M. Grosche, P. Agarwal, S.R.
Julian, N.J. Wilson, R.K.W. Haselwimmer, S.J.S.
Lister, N.D. Mathur, F.V. Carter, S.S. Saxena,
G.G. Lonzarich, J. Phys. Cond. Matt. 12 (2000)
L533L540
13
CeNi2Ge2 thermal expansion
aT1/2bT
R. Küchler, N. Oeschler, P. Gegenwart, T.
Cichorek, K. Neumaier, O. Tegus, C. Geibel, J.A.
Mydosh, F. Steglich, L. Zhu, Q. Si, Phys. Rev.
Lett. 91, 066405 (2003)
14
CeNi2Ge2 specific heat
R. Küchler et al., PRL 91, 066405 (2003). T.
Cichorek et al., Acta. Phys. Pol. B34, 371 (2003).
15
CeNi2Ge2 Grüneisen ratio
critical components ?cr?(T)-bT CcrC(T)-?T
?cr Vmol/?T ??cr/Ccr

• ?cr(T) T-1/(?z)
• prediction ? ½, z 2 ? x 1
• observations in accordance with itinerant
  scenario
• INS no hints for 2D critical fluct.
• Remaining problem
• QCP not identified (would require negative
  pressure)

?cr 1/Txwith x1 (-0.1 / 0.05)
16
Cubic CeIn3-xSnx
N.D. Mathur et al., Nature 394 (1998)

• Increase of J by Sn substitution
• Volume change subdominant
• TN can be traced down to 20 mK !

CeIn3
R. Küchler, P. Gegenwart, J. Custers, O.
Stockert, N. Caroca-Canales, C. Geibel, J.
Sereni, F. Steglich, PRL 96, 256403 (2006)
17
CeIn3-xSnx

• Thermodynamics in accordance with 3D-SDW
  scenario
• Electrical resistivity ?(T) ?0 AT,
  however large ?0 !

R. Küchler, P. Gegenwart, J. Custers, O.
Stockert, N. Caroca-Canales, C. Geibel, J.
Sereni, F. Steglich, PRL 96, 256403 (2006)
18
CeCu6-xMx
C/T log T (universal!)
H.v. Löhneysen et al., PRL 1994, 1996 A. Rosch et
al., PRL 1997 O. Stockert et al., PRL 1998
2D-SDW scenario ?

• A. Schröder et al., Nature 2000
• E/T scaling in ?(q,?)
• ?(q) T??(q)?0.75 for all q
• locally critical scenario
• could we disprove 2D-SDW scenario
  thermodynamically?

19
CeCu6-xAgx
QCP
AF
E.-W. Scheidt et al., Physica B 321, 133 (2002).
20
CeCu5.8Ag0.2
R. Küchler, P. Gegenwart, K. Heuser, E.-W.
Scheidt, G.R. Stewart and F. Steglich, Phys. Rev.
Lett. 93, 096402 (2004).
21
CeCu5.8Ag0.2
Incompatible with itinerant scenario!
R. Küchler et al., Phys. Rev. Lett. 93, 096402
(2004)
22
YbRh2Si2 a clean system very close to a QCP
P. Gegenwart et al., PRL 89, 056402 (2002).
23
YbRh2(Si0.95Ge0.05)2
J. Custers et al., Nature 424, 524 (2003)
24
Stronger than logarithmic mass divergence
?0
YbRh2(Si.95Ge.05)2
b?1/3
b
J. Custers et al., Nature 424, 524 (2003)
25
Thermal expansion and Grüneisen ratio
R. Küchler et al., PRL 91, 066405 (2003)
Prediction ?cr(T) T-1/(?z) (L. Zhu, M. Garst,
A. Rosch, Q. Si, PRL 2003) ? ½, z2 (AF) ? x
1 ? ½, z3 (FM) ? x ?
26
AF and FM critical fluctuations
B // c
P. Gegenwart, J. Custers, Y. Tokiwa, C. Geibel,
F. Steglich, Phys. Rev. Lett. 94, 076402 (2005).
27
Pauli-susceptibility
P. Gegenwart et al., PRL 2005
28
29Si NMR on YbRh2Si2
K. Ishida et al. Phys. Rev. Lett 89, 107202
(2002)
Knight shift K ?(q0) ?bulk Saturation in FL
state at B gt Bc Spin-lattice relaxation
rate 1/T1T q-average of ?(q,?) At B gt 0.15 T
Koringa relation S ? 1/T1TK2 holds with
dominating q0 fluct. B ? 0.15 T disparate
behavior ? Competing AF (q?0) and FM (q0)
fluctuations ? ?(q,?) has a two component
spectrum
29
Comparison YbRh2Si2 vs CeCu5.9Au0.1
Spin-Ising symmetry
Easy-plane symmetry
YRS
AF and FM quantum critical fluct.
30
Hall effect evolution
S. Paschen et al., Nature 432 (2004) 881
Large change of ?H though tiny ?ordered! SDW
continuous evolution of ?H
31
Thermodynamic evidence for multiple energy scales
at QCP
Fermi surface change ? clear signatures in
thermodynamics Multiple energy scales at QCP
P. Gegenwart et al., cond-mat/0604571.
32
Conclusions of part 1

• There exist HF systems which display itinerant
  (conventional) quantum critical behavior
  CeNi2Ge2, CeIn3-xSnx,
• YbRh2Si2 incompatible with itinerant scenario
• Stronger than logarithmic mass divergence
• Grüneisen ratio divergence T?0.7
• Hall effect change
• Multiple energy scales vanish at quantum
  critical point
• QC fluctuations have a very strong FM component
• Divergence of bulk susceptibility
• Highly enhanced SW ratio, small Korringa ratio,
  A/?02 scaling
• Relation to spin anisotropy (easy-plane)?

33
Metallic ferromagnetic QCPs ?
Itinerant ferromagnets QPT becomes generically
first-order at low-T
Experiments on ZrZn2, MnSi, UGe2, M. Uhlarz, C.
Pfleiderer, S.M. Hayden, PRL 04 D. Belitz and
T.R. Kirkpatrick, PRL 99

• New route towards FM quantum criticality
  metamagnetic QC(E)P e.g. in URu2Si2, Sr3Ru2O7,
• What happens if disorder broadens the first-order
  QPT?

34
Layered perovskite ruthenates Srn1RunO3n1
n1 unconventional superconductor n2
strongly enhanced paramagnet (SWR 10)
metamagnetic transition! n3 itinerant
el. Ferromagnet (Tc 105 K) n? itinerant el.
Ferromagnet (Tc 160 K)
35
Field angle phase diagram on second-generation
samples(RRR 80)
S.A. Grigera et al. PRB 67, 214427 (2003)
Evidence for QC fluctuations Diverging A(H) at
Hc (S.A. Grigera et al, Science 2001)
36
Thermal expansion
Calculation for itinerant metamagnetic QCEP
P. Gegenwart, F. Weickert, M. Garst, R.S. Perry,
Y. Maeno,Phys. Rev. Lett. 96, 136402 (2006)
37
Behavior consistent with 2D QCEP scenario
P. Gegenwart, F. Weickert, M. Garst, R.S. Perry,
Y. Maeno, Phys. Rev. Lett. 96, 136402 (2006)
38
Thermal expansion on Sr3Ru2O7
Compatible with underlying2D QCEP at Hc 7.85
T ?0 marks accumulation points of entropy
39
Fine-structure near 8 Tesla
Dominant elastic scattering ? Formation of
domains!

T (K)
2.1
0.1
0.3
r(mWcm)
1.6
0.6

0.9
1.2
1.2
6.5
7.0
7.5
8.0
8.5
9.0
B (T)
S.A. Grigera, P. Gegenwart, R.A. Borzi, F.
Weickert, A.J. Schofield, R.S. Perry, T. Tayama,
T. Sakakibara, Y. Maeno, A.G. Green and A.P.
Mackenzie, SCIENCE 306 (2004), 1154.
40
Thermodynamic analysis of fine-structure

• No clear phase transitions
• Signatures of quantum criticality survive in QC
  regime
• also 1/(T1T)1/T _at_7.9T down to 0.3K!! (Ishida
  group)
• 3) First-order transitions haveslopes pointing
  away from bounded state
• Clausius-Clapyeron

• Enhanced entropy in bounded regime!

41
Conclusion Sr3Ru2O7

• Quantum criticality in accordance with itinerant
  scenario for metamagentic quantum critical end
  point (d2)
• Fine-structure close to 8 Tesla due to domain
  formation
• Formation of symmetry-broken phase
  (Pomeranchuk instability)? Unlikely because of
  enhanced entropy
• Real-space phase separation? (C.
  Honerkamp, PRB 2005)

42
Smeared Ferromagnetic Quantum Phase Transition
Theoretical prediction FM QPT generically
first order at T 0 D. Belitz et al, PRL 1999
Sharp QPT can be destroyed by disorder
exponential tail T. Vojta, PRL 2003
M. Uhlarz et al, PRL 2004
43
The Alloy CePd1-xRhx

• Orthorhombic CrB structure
• CePd is ferromagnetic with TC 6.6 K
• CeRh has an intermediate valent
• ground state

• High T measurements suggested
• quantum critical point
• (dotted red line)
• Detailed low T investigation tail

44
AC Susceptibility in the Tail Region
Crossover transition for x gt 0.6 indicated by
sharp cusps in cAC down to mK temperatures Frequ
ency dependence at low frequencies and high
sensitivity on tiny magnetic DC fields
no long range order
Maxima of c(T) in phase diagram
c(T) in DC field
45
Spin Glass-like Behavior

• Frequency shift
• (e.g. x0.85 DTC/TCD log(n) of 5)
• Spin glass-like behavior

No maximum in specific heat but NFL behavior for
x 0.85
46
Grüneisen parameter shows no divergence
47
Kondo Cluster Glass

• Strong increase of TK for x 0.6 indicated by
  Weiss temperature qP,
• evolution of entropy and lattice parameters

Possible reason for spin glass-like state
Variation of TK for Ce ions depending on Rh or
Pd nearest neighbors leading distribution
of local Kondo temperatures Kondo
cluster glass
48
Conclusion Outlook

• Classification of different types of QCPs in HF
  systems (conventional vs unconventional)
• Importance of frustration in the spin
  interaction?
• Role of disorder? e.g. smearing of sharp 1st
  order trans.