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 systems:
E.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
First part
(mainly those studied in Dresden)
• Ferromagnetic quantum criticality
Second part
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
3University
of Florida, Gainesville FL, USA
2Augsburg
4St.
Andrews University, Scotland
5Kyoto
University, Japan
6Los
7CNEA
Bariloche, Argentina
8University
9Rice
University, Texas, USA
11Rutgers
University, USA
University, Germany
Alamos National Laboratory, USA
of Minnesota, Minneapolis, USA
10CEA-Saclay,
France
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,l=3)
• partially filled inner 4f/5f shells  localized magnetic moment
• CEF splitting  effective S=1/2
T
localized moments
+
conduction electrons
T* ~ 5 – 50 K
moments bound
in
spin singlets
Microscopic model: Kondo effect
(Jun Kondo ´63)
H sd  J  S  s
local moment
conduction el
J: hybridization
between local moments
and conduction el.
 AF coupling J < 0

TK: characteristic
„Kondo“-temperature
Kondominimum
TK
T5
lnT
T < TK: formation of a bound state
between local spin and conduction
electron spin  local spin singlet
Anderson Impurity Model
H  H s  H f  H sf  H U
cond.el
f-el
hybridization
Vsf
on-site
Coulomb
repulsion Uff
Formation of an (Abrikosov-Suhl) resonance at EF of width kBT*
 extremely high N(EF)  heavy fermions
Landau Fermi liquid
Lev Landau ´57
Excitations of system
with strongly
interacting electrons
1:1
correspondence
 Definition of quasiparticles near EF with
renormalized mass m*
 specific heat:
C/T   
k B2
3 2
k F  m* ~
 electrical resistivity:   aT 2 mit a ~

a
2
 const . = 10-5 cm(molK/mJ)2
1
T*
1
T *2
Free
electron gas
Magnetic instability in Heavy Fermion systems
T
TKondo
Doniach 1977
local 4f
AF ordering
Heavy
fermion
behavior
Jc
Fermi-surface:
itinerant 4f
TRKKY
J
Itinerant (conventional) scenario
Moriya, Hertz, Millis, Lonzarich, …
TK
T
TN
OP fluctuations in space and time
AF: z=2 (deff = d+z)
NFL
C /T
SDW
FL
gc
g

d 2
d 3
ln( T0 / T )  0   T
T
T 3/ 2
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
CeCu6-xAux
CeCu6-xAux: xc=0.1 inelastic neutron scattering
O. Stockert et al., PRL 80 (1998): critical fluctuations quasi-2D !
A. Schröder et al., Nature 407 (2000):
E/T
S(q,)T0.75
1/(q)
0T0.75
non-Curie-Weiss behavior
q-independent  local !!
T0.75
H/T
Grüneisen ratio analysis
T
1
2
NFL
3
4
5
6
AF
7
FL
8
 = p, x, B
9
10
Thermal expansion  = –1/V ∂S/∂p
 = V-1 dV/dT
Specific heat: C/T = ∂S/∂T
1 S / p T
1 E *
~ 

C
Vmol T S / T  p Vmol E * p

Resolution: < 0.01Å
l/l = 10-10 (l = 5 mm)
for T  20 mK, B  20 Tesla
  at QCP !
Itinerant theory:  ~ Tz ~ T-1
(L. Zhu, M. Garst, A. Rosch, Q. Si, PRL 2003)
Experimental classification:
conventional
CeNi2Ge2
CeIn3-xSnx
CeCu2Si2
CeCoIn5
…
unconventional
CeCu6-xAux
YbRh2Si2
…
CeNi2Ge2: very clean system close to zero-field QCP
TK = 30 K, paramagnetic ground state
CeNi2Ge2
CeNi2Ge2
(cm)
B (T)
 ~ T2

1.5
1.40
1.37
0.5
0.5
0
1
2
T (K)
3
B (T)
2.8
0.3
15
~ T
3
0.4
0.3
10 1/2
~T
0.2
5
0
a
b
0.1
 ~ T
0
1
2
T (K)
3 0
a (cmK-2)

5
10
B (T)
15
0
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) L533–L540
CeNi2Ge2: thermal expansion
16
-6
14
12
2
4
6
CeNi2Ge2
10
~ aT1/2+b
5
10
-6
In accordance with
prediction of itinerant
theory
5
10
 II c
-1
 (10 K )
0
-2
 / T (10 K )
-1
8
0
0
0
2
4 T (K)
 II a
6
4
2
0
~ aT1/2+bT
0
1
2
3
4
5
6
T (K)
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)
CeNi2Ge2: specific heat
0.5
B (T)
0
2
0.4
  for T  0

100
-1
-2
C / T (Jmol K )
150
0
2
T (K)
4
0.3
C
d
   T  3
0
T
T
CeNi2Ge2
0.2
0
1
2
3
T (K)
4
5
R. Küchler et al., PRL 91, 066405 (2003).
T. Cichorek et al., Acta. Phys. Pol. B34, 371 (2003).
CeNi2Ge2: Grüneisen ratio
critical components:
cr=(T)−bT
Ccr=C(T)−T
cr
= Vmol/T
cr/Ccr
cr ~ 1/Tx
with x=1 (−0.1 / +0.05)
1000
prediction:
 = ½, z = 2  x = 1
cr

cr(T) ~ T−1/(z)
 observations in accordance with
itinerant scenario
INS: no hints for 2D critical fluct.
Remaining problem:
QCP not identified (would require
100
negative pressure)
0.1
1
T (K) 5
Cubic CeIn3-xSnx
N.D. Mathur et al., Nature 394 (1998)
CeIn3
• Increase of J by Sn substitution
• Volume change subdominant
• TN can be traced down to 20 mK !
10
CeIn3-xSnx
T(K)
TN
5
AF
0
TI
0
T* LFL
0.5
x
1
R. Küchler, P. Gegenwart, J. Custers, O. Stockert, N. Caroca-Canales, C. Geibel, J.
Sereni, F. Steglich, PRL 96, 256403 (2006)
CeIn3-xSnx
T
0.310.01
100
100
cr
-
CeIn3-xSnx

CeIn2.35Sn0.65
100
20
0.1
1
 = 1.1  0.1
10
50
x
0.65
0.7
0.8
0.55
0.85
300 K
10
0
0.0
0.1
1
T (K)
1
T (K)
0.5
0.1
1
T (K)
10
• Thermodynamics in accordance with 3D-SDW scenario
• Electrical resistivity: (T) = 0 + A’T, however: large 0 !
R. Küchler, P. Gegenwart, J. Custers, O. Stockert, N. Caroca-Canales, C. Geibel, J.
Sereni, F. Steglich, PRL 96, 256403 (2006)
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?
CeCu6-xAgx
4
CeCu6-xAgx
x
0.2
0.3
0.4
0.48
0.8
2
C / T (J/mole K )
3
1.0
CeCu6-xAgx
TN (K)
2
0.5
1
QCP
AF
0.0
0
0.5 x 1.0
0
0.1
T (K)
1
E.-W. Scheidt et al., Physica B 321, 133 (2002).
5
CeCu5.8Ag0.2
3.5
70
B (T)
0
1.5
3
4
5
8
2
2.5
2.0
60
1
2
3

50
40
CeCu5.8Ag0.2
-6
30
 / T (10 K )
2
Cel / T (J / mole K )
3.0
1.5
20
1.0
10
a
0.5
0.05 0.1
b
0
T (K)
1
3
0.1
T (K)
1
6
R. Küchler, P. Gegenwart, K. Heuser, E.-W. Scheidt, G.R. Stewart and F. Steglich,
Phys. Rev. Lett. 93, 096402 (2004).
CeCu5.8Ag0.2
150

CeCu5.8Ag0.2
120
120
Incompatible
with itinerant
scenario!
90
60
90
30
0.1
1
4
60
30
0
1
2
T (K)
R. Küchler et al., Phys. Rev. Lett. 93, 096402 (2004)
3
YbRh2Si2: a clean system very close to a QCP
4
1.6
YbRh2Si2
1.4
 (cm)
TN
c
1.2
2
Yb
1.0
0.0
0.1
0.2
0.3
0.4
Rh
0.5
Si
T (K)
1
a
TN
150
0
0.00
0.05
0.10
0.15
0.20
T (K)
0.25
100
T (mK)
2
C/T (J/K mol)
3
NFL
0.30
TN
11 B  c
Bc
50
AF
0
0.0
P. Gegenwart et al., PRL 89, 056402 (2002).
T*
LFL
0.5
1.0
1.5
B (T)
2.0
2.5
YbRh2(Si0.95Ge0.05)2
YbRh2 (Si0.95 Ge0.05 )2
TN
B c
K )
3
Cel / T (J mol
-1
-2
B (T)
0
0.025
0.05
2
0.1
1
0(b)
=Bc
C/T ~ T-1/3
0.2
0.4
0.8
0
0.02
0.1
1
T (K)
2
J. Custers et al., Nature 424, 524 (2003)
Stronger than logarithmic mass divergence
0
YbRh2(Si.95Ge.05)2
T
2
-2
0 (J mol K )
NFL
1
-1
~b1/3
AF
1
FL
2
• stronger than logarithmic mass
divergence incompatible with
itinerant theory
• T/b scaling
a
0
0.02

0.1
b= (B - Bc )
1
10
(T)
J. Custers et al., Nature 424, 524 (2003)
Thermal expansion and Grüneisen ratio
4
25
cr
R. Küchler et al.,
PRL 91, 066405 (2003)
x = 0.7
20
x=1
-6
10
2
15
-2
3
2
Cel / T (J / K mol)
100
 / T (10 K )

0.1
T (K)
1
10
1
5 Prediction: cr(T) ~ T−1/(z)
(L. Zhu, M. Garst, A. Rosch, Q. Si, PRL
YbRh2(Si0.95Ge0.05)2
0
0.01
0.1
2003)
1
T (K)
0  = ½, z=2 (AF)  x = 1
10  = ½, z=3 (FM)  x = ⅔
AF and FM critical fluctuations
YbRh2(Si0.95Ge0.05)2
10
6
0.03
B (T)
0
0.03
0.05
0.065
0.2
1
-1
 (10 m mol )
8
0.6
~T
-3
B (T)
0
 (10 mol m )
0.3
0.05
6
-6
3
0.1
0.0 0.1 0.2 0.3 0.4 0.5 T (K)
0.065
4
2
~T
0.1
2
0.15
0.4
0
0.01
B // c
0.1
1
10
T (K)
P. Gegenwart, J. Custers, Y. Tokiwa, C. Geibel, F. Steglich, Phys. Rev. Lett. 94, 076402 (2005).
Pauli-susceptibility
7
6
4
20
-6
K 
-1
30
0 (10 m mol )
5
3
~ (BBc)
YbRh2(Si0.95Ge0.05)2
0.6
3
10
2
0
SommerfeldWilson
0.1
B  Bc (T)
1
0
0.01
0.1
1
B  Bc (T)
P. Gegenwart et al., PRL 2005
1
10
29Si
– NMR on YbRh2Si2
K. Ishida et al. Phys. Rev. Lett 89,
107202 (2002):
Knight shift K ~ ’(q=0) ~ bulk
Saturation in FL state at B > Bc
Spin-lattice relaxation rate
1/T1T ~ q-average of ’’(q,)
At B > 0.15 T:
Koringa –relation S  1/T1TK2
holds with dominating q=0 fluct.
B  0.15 T: disparate behavior
 Competing AF (q0) and FM
(q=0) fluctuations
 ’’(q,) has a two component
spectrum
Comparison: YbRh2Si2 vs CeCu5.9Au0.1
Spin-Ising symmetry
q
CeCu5.9Au0.1
q
Q
q
Easy-plane symmetry
YbRh2Si2
-3
0T
1
AF and FM quantum critical fluct.

6
Q
YRS
 (10 mol m )
0
q
b
0
0
1
2
0.6
0.6
T 3 (K )
Hall effect evolution
S. Paschen et al., Nature 432 (2004) 881:
P. Coleman, C. Pépin, Q. Si, R. Ramazashvili,
J. Phys. Condes. Matter 13 R723 (2001).
Large change of H though tiny ordered!
SDW: continuous evolution of H
Thermodynamic evidence for multiple energy scales at QCP
Magnetization
Magnetostriction
4.0
4
YbRh2(Si0.95Ge0.05)2
YbRh2Si2
0.20
0
T (K)
0.13
0.2
0.3
0.5
0.8
0
0
0
3
0.15
0.10
T (K)
0.09
0.23
0.54
T
0.05
-6
M (B/Yb)
-6
1
-1
3.5
2
1
0.1 T
3.0
a
0.00
0.0
0.2
0.4
0.6
0.8
b
0.0
0.2
B (T)
0
0
5
10
15
0.4
Phase diagram
0.8
T*
0.6
T (K)
Multiple energy scales at QCP
0.4
0.2
TLFL
TN
P. Gegenwart et al., cond-mat/0604571.
0.0
0
2
0.6
T (K)
20
H/ Hc
Fermi surface change  clear
signatures in thermodynamics
 (10 m mol )
3
[110] (10 T )
Susceptibility
0.25
4
H/ Hc
0.8
2.5
1.0
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)?
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
1) New route towards FM quantum criticality: metamagnetic QC(E)P e.g. in
URu2Si2, Sr3Ru2O7, …
2) What happens if disorder broadens the first-order QPT?
Layered perovskite ruthenates Srn+1RunO3n+1
n=1: unconventional superconductor
n=2: strongly enhanced paramagnet
(SWR = 10)
metamagnetic transition!
n=3: itinerant el. Ferromagnet
(Tc = 105 K)
n=: itinerant el. Ferromagnet
(Tc = 160 K)
Field angle phase diagram on “second-generation” samples
(RRR ~ 80)
1400
1200
1000
800
QCEP @ 8 T // c-axis
600
400
8
200
0 0
7
20
40
6
60
80
5
100
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)
Thermal expansion
10
Sr3Ru2O7
H // c
5
6
7
7.5
7.8
8.2
8.5
10
-6
-1
c (10 K )
0H (T)
0
-5
0
1
2
3
4
5
T (K)
Calculation for itinerant metamagnetic QCEP
 S 
 S 
 Vmol      
 P 
 h 
 dH 
with    c 
, h ~ H  Hc
 dP  H  H c
P. Gegenwart, F. Weickert, M. Garst, R.S. Perry, Y. Maeno,
Phys. Rev. Lett. 96, 136402 (2006)
Behavior consistent with 2D QCEP scenario
-5
10
-5
10
-6
10
-7
10
-8
4/3
0H (T)
10
10
10
5
6
6.5
6.8
7
7.2
7.3
7.4
7.45
7.5
7.6
7.65
7.7
-6
-7
0H (T)
8.07
8.15
8.2
8.35
8.5
8.8
9
9.5
10
b
a
-8
-3
-2
-1
0
1
2
10 10 10 10 10 10 10
3
3
10
-2
10
-1
0
1
2
10 10 10 10
7/3
T 0|HHc|
P. Gegenwart, F. Weickert, M. Garst, R.S. Perry, Y. Maeno, Phys. Rev. Lett. 96, 136402 (2006)
3
/T 0|HHc|
/T 0|HHc|
4/3
10
Thermal expansion on Sr3Ru2O7
30
Sr3Ru2O7
T (K)
0.2
1.2
-6
-2
c/T (10 K )
20
10
~ | H  H c|
4/3
0
a
~ | H  H c|
-10
T (K)
3
2
1
b
>0
max
=0
min
max
2
=0 marks accumulation
points of entropy
4/3
<0
~ |HHc|
2/3
2
d /dT
0
5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0
0H (T)
Compatible with underlying
2D QCEP at Hc = 7.85 T
Fine-structure near 8 Tesla
Dominant elastic scattering  Formation of domains!
T (K)
2.1
cm)
T (K)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.0
0.5
0.0
7.6
1.6
7.8
8.0
8.2
Field (tesla)
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.
Thermodynamic analysis of fine-structure
T (K)
10
0H (T)
1.0
5
0.0
-6
1
c (10 K )
0.5
7.8
7.9
8.0
8.1
H (T)
0
7.8
7.82
7.84
7.87
7.9
8.07
8.15
0
a
-5
0H (T)
b
7.52
7.77
7.82
7.92
8.02
8.07
8.5
d/dT (cm/K)
3
2
1
0
0.0
0.5
1.0
1.5
T (K)
2.0
2.5
3.0
1) No clear phase transitions
2) Signatures of quantum criticality
survive in QC regime
also: 1/(T1T)~1/T @7.9T down to
0.3K!! (Ishida group)
3) First-order transitions have
slopes pointing away from
bounded state
Clausius-Clapyeron:
dH c
S  M 
dT
 Enhanced entropy in bounded
regime!
Conclusion Sr3Ru2O7
• Quantum criticality in accordance with itinerant scenario for
metamagentic quantum critical end point (d=2)
• Fine-structure close to 8 Tesla due to domain formation
 Real-space
phase separation?
1.0
T (K)
• Formation of symmetry-broken
phase (Pomeranchuk instability)?
Unlikely because of enhanced entropy
0.5
liquid
0.0
7.6
(C. Honerkamp, PRB 2005)
7.8
8.0
Field (tesla)
two- gas
phase
8.2
Smeared Ferromagnetic Quantum Phase Transition
Theoretical prediction:
FM QPT generically
first order at T = 0
[D. Belitz et al, PRL 1999]
QCEP
Sharp QPT can be
destroyed by disorder
exponential tail
[T. Vojta, PRL 2003]
[M. Uhlarz et al, PRL 2004 ]
The Alloy CePd1-xRhx
Ce
 Orthorhombic CrB structure
 CePd is ferromagnetic with TC = 6.6 K
 CeRh has an intermediate valent
ground state
Cp,max
TC (K)
6
M
'-ac
"-ac
4

FM
CePd1-xRhx
0.2
c
 High T measurements suggested
quantum critical point
(dotted red line)
2
0
0.0
Pd,Rh
 Detailed low T investigation: tail
0.4
0.6
x
0.8
1.0
AC Susceptibility in the Tail Region
Crossover transition for x > 0.6
indicated by sharp cusps in AC‘ down to mK temperatures
Frequency dependence at low frequencies and high sensitivity on tiny
magnetic DC fields
no long range order
‘(T) in DC field
Maxima of ‘(T) in phase diagram
2.5
500
CePd1-xRhx
x = 0.8, = 113 Hz
400
single crystals
polycrystals
1.5
T (mK)
' (10-6m3/mol)
2.0
CePd1-xRhx
B = 0 mT
B = 5 mT
B = 10 mT
B = 15 mT
1.0
 = 13 Hz
300
200
100
0.5
0.0
single crystal
0.5
1.0
T (K)
1.5
2.0
0
0.75
0.80
0.85
0.90
Rh content x
0.95
1.00
Spin Glass-like Behavior
 Frequency shift
(e.g. x=0.85: TC/[TC log()] of 5%)
2.5

-6 3
' (10 m /mol)
 Spin glass-like behavior
3.0
1.2
CePd1–xRhx
2
C/T (J/(mol K ))
1.0
0.6
single crystal
x=0.85
2.0
x = 0.80
x = 0.85
x = 0.87
x = 0.90
0.8
CePd1-xRhx
0.15
13 Hz
113 Hz
1113 Hz
0.20
0.25
0.30
0.35
T (K)
0.4
No maximum in specific heat
but NFL behavior for x ≥ 0.85
0.2
0.0
0.1
0.4
1
T (K)
10
Grüneisen parameter shows no divergence
thermExpan/Gruene87_90
80
60

40
20
CePd1-xRhx
0.90
0.87
0.80
0
-20
0
2
4
T [K]
6
”Kondo Cluster Glass“
 Strong increase of TK for x ≥ 0.6 indicated by Weiss temperature P,
evolution of entropy and lattice parameters
7
6
CePd1-xRhx
TC from
3
M(T)
3
'ac- He
2
Cmax
0
0.0
100

3
1
300
4
'ac- He /He
xcr
30
0.2
0.4
0.6
x (Rh conc)
0.8
1.0
P (K)
TC (K)
5
4
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“
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.