Philipp GegenwartMax-Planck Institute for Chemical Physics of Solids, Dresden, Germany

Experimental Tutorial on Quantum Criticality

Reviews on quantum criticality in strongly correlated electron systems: in strongly correlated electron systems:

E.g.E.g.

• G.R. Stewart, Rev. Mod. Phys. 73, 797 (2001).G.R. Stewart, Rev. Mod. Phys. 73, 797 (2001).

• H. v. Löhneysen, A. Rosch, M. Vojta, P. Wölfle, cond-mat/0606317H. v. Löhneysen, A. Rosch, M. Vojta, P. Wölfle, cond-mat/0606317

Outline of Outline of this talk::

• IntroductionIntroduction

• Quantum criticality in some antiferromagnetic HF systemsQuantum criticality in some antiferromagnetic HF systems

(mainly those studied in Dresden)(mainly those studied in Dresden)

• Ferromagnetic quantum criticalityFerromagnetic quantum criticality

First partFirst part

Second partSecond part

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

Collaborators

• 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

f-electron based Heavy Fermion systems

T

T* ~ 5 – 50 K

localized moments+

conduction electrons

moments boundin

spin singlets

Microscopic model: Kondo effect (Jun Kondo ´63)

sSJH sd

local moment conduction el

J: hybridization

between local moments

and conduction el.

AF coupling J < 0

lnT

Kondo-minimum

TK

T5

TK: characteristic

„Kondo“-temperature

T < TK: formation of a bound state

between local spin and conduction

electron spin local spin singlet

Anderson Impurity Model

Usffs HHHHH cond.-

elf-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

Freeelectron gas

1:1correspondence

Magnetic instability in Heavy Fermion systems

Fermi-surface:

Doniach 1977

Itinerant (conventional) scenario

Moriya, Hertz, Millis, Lonzarich, …

g

T

TN

gc

TK

NFL

FLSDW 2/3

00 )/ln(/

32

TT

TTTTC

dd

OP fluctuations in space and timeAF: z=2 (deff = d+z)

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: 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

0T0.75

H/T

1/(q)

T0.75

non-Curie-Weiss behavior

q-independent local !!

CeCu6-xAux

FLAF

= p, x, B

NFL

T

Thermal expansion = –1/V ∂S/∂p = V-1 dV/dT

Specific heat: C/T = ∂S/∂T

p

E

EVTST

pS

VC molp

T

mol

*

*

1

/

/1~

! QCP at

Itinerant theory: ~ Tz ~ T-1

(L. Zhu, M. Garst, A. Rosch, Q. Si, PRL 2003)

Grüneisen ratio analysis

1

2

3

4

5

6

7

8

9

10

Resolution: < 0.01Ål/l = 10-10 (l = 5 mm)for T 20 mK, B 20 Tesla

Experimental classification:

conventionalconventional

CeNi2Ge2

CeIn3-xSnx

CeCuCeCu22SiSi22

CeCoInCeCoIn55

……

unconventionalunconventional

CeCuCeCu6-x6-xAuAuxx

YbRh2Si2

……

CeNi2Ge2: very clean system close to zero-field QCP

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

0 1 2 3T (K)

0.3

0.5

0.5

2.8

3

(cm)

1.51.5

1.401.40

1.371.37

CeNi2Ge2CeNi2Ge2

B (T)B (T)

~ T1/2 ~ T1/2

~ T~ T

TK = 30 K, paramagnetic ground state

0 1 2 3 4 5 60

2

4

6

8

10

12

14

16

II a

II c

CeNi2Ge

2

(1

0-6K

-1)

T (K)

0 2 40

5

10

-1 0 2 4 6

0

5

10

/ T

(10

-6K

-2)

T (K)

~ aT1/2+bT

CeNi2Ge2: thermal expansion

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)

~ aT1/2+b In accordance with prediction of itinerant theory

30 T

dT

T

C

0 1 2 3 4 50.2

0.3

0.4

0.5

B (T) 0

CeNi2Ge

2

C

/ T

(Jm

ol-1

K-2)

T (K)

0 1 2 3 4 50.2

0.3

0.4

0.5

B (T) 0 2

CeNi2Ge

2

C

/ T

(Jm

ol-1

K-2)

T (K)

for T 0

0 1 2 3 4 50.2

0.3

0.4

0.5

0 2 4

100

150

B (T) 0 2

CeNi2Ge

2

C

/ T

(Jm

ol-1

K-2)

T (K)

T (K)

CeNi2Ge2: specific heat

R. Küchler et al., PRL 91, 066405 (2003).

T. Cichorek et al., Acta. Phys. Pol. B34, 371 (2003).

CeNi2Ge2: Grüneisen ratio

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)

critical components: cr=(T)−bT

Ccr=C(T)−T

cr = Vmol/T cr/Ccr

0.1 1 5

100

1000

T (K)

cr

0.1 1 5

100

1000

T (K)

cr

cr ~ 1/Tx

with x=1 (−0.1 / +0.05)

Cubic CeIn3-xSnx

N.D. Mathur et al., Nature 394 (1998)

CeIn3

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

• Increase of Increase of JJ by Sn substitution by Sn substitution

• Volume change subdominantVolume change subdominant

• TTNN can be traced down to 20 mK ! can be traced down to 20 mK !

CeIn3-xSnx

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

• Thermodynamics in accordance with 3D-SDW scenarioThermodynamics in accordance with 3D-SDW scenario

• Electrical resistivity: Electrical resistivity: ((TT) = ) = 00 + + AA’’TT, however: large , however: large 00 ! !

CeCu6-xMx

C/T ~ log T(universal!)

H.v. Löhneysen et al., PRL 1994, 1996A. Rosch et al., PRL 1997O. 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

0.1 1 50

1

2

3

4

0 0.5 1.00.0

0.5

1.0

CeCu6-x

Agx

x

TN (

K)

CeCu6-x

Agx

x 0.2 0.3 0.4 0.48 0.8

C /

T (

J/m

ole

K2 )

T (K)

E.-W. Scheidt et al., Physica B 321, 133 (2002).

AF

QCP

CeCu5.8Ag0.2

0.05 0.1 1 30.5

1.0

1.5

2.0

2.5

3.0

3.5

0.1 1 60

10

20

30

40

50

60

70

ba

/ T

(10

-6K2)

T (K)

B (T) 0 1.5 3 4 5 8

Cel /

T (

J / m

ole

K2 )

T (K)

CeCu5.8

Ag0.2

1

2

3

R. Küchler, P. Gegenwart, K. Heuser, E.-W. Scheidt, G.R. Stewart and F. Steglich, Phys. Rev. Lett. 93, 096402 (2004).

CeCu5.8Ag0.2

R. Küchler et al., Phys. Rev. Lett. 93, 096402 (2004)

0 1 2 330

60

90

120

150

CeCu5.8

Ag0.2

T (K)

0.1 1 430

60

90

120

Incompatible

with itinerant

scenario!

YbRh2Si2: a clean system very close to a QCP

P. Gegenwart et al., PRL 89, 056402 (2002).

0.0 0.5 1.0 1.5 2.0 2.50

50

100

150

T*

LFL

NFL

TN

AF

T (

mK

)

B (T)

11 B c Bc

0.02 0.1 1 20

1

2

3

TN

0.8

B c

0.4

0.2

0.1

B (T) 0 0.025 0.05

YbRh2 (Si

0.95 Ge

0.05 )

2

C

el /

T (

J m

ol -

1 K

-2 )

T (K)

0.02 0.1 1 20

1

2

3

TN

B c

B (T) 0

YbRh2 (Si

0.95 Ge

0.05 )

2

C

el /

T (

J m

ol -

1 K

-2 )

T (K)

0.02 0.1 1 20

1

2

3

TN

B c

B (T) 0 0.025

YbRh2 (Si

0.95 Ge

0.05 )

2

C

el /

T (

J m

ol -

1 K

-2 )

T (K)

=Bc

C/T ~ T-1/3

0(b)

J. Custers et al., Nature 424, 524 (2003)

YbRh2(Si0.95Ge0.05)2

0.02 0.1 1 100

1

2

a

0 (J

mol

-1 K

-2 )

(B - Bc

) (T)

Stronger than logarithmic mass divergence

~b1/3

b=

0 YbRh2(Si.95Ge.05)2

• stronger than logarithmic mass

divergence incompatible with

itinerant theory

• T/b scaling

FLAF

NFL

T

1

2

J. Custers et al., Nature 424, 524 (2003)

Thermal expansion and Grüneisen ratio

0.01 0.1 1 100

1

2

3

4

Cel /

T (

J /

K2 m

ol)

T (K)

0

5

10

15

20

25

YbRh2(Si

0.95Ge

0.05)2

/

T (

10-6

K-2)

0.1 110

100

x = 1

x = 0.7

cr

T (K)

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, z=2 (AF) x = 1

= ½, z=3 (FM) x = ⅔

0.01 0.1 1 100

2

4

6

8

10

0.0 0.1 0.2 0.3 0.4 0.50.1

0.2

0.3

~ T0.6

~ T2

0

B (T)

0.03

0.05

0.065

0.1

0.15

0.4

YbRh2(Si

0.95Ge

0.05)

2

(10-6

m3 m

ol-1)

T (K)

B (T) 0 0.03 0.05 0.065

1(1

06 mol

m-3)

T (K)

AF and FM critical fluctuations

P. Gegenwart, J. Custers, Y. Tokiwa, C. Geibel, F. Steglich, Phys. Rev. Lett. 94, 076402 (2005).

B // c

Pauli-susceptibility

P. Gegenwart et al., PRL 2005

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 (q0) and FM

(q=0) fluctuations

’’(q,) has a two component

spectrum

Comparison: YbRh2Si2 vs CeCu5.9Au0.1

q

q

q

q

Q

Q

0

CeCu5.9Au0.1

YbRh2Si2

AF and FM quantum critical fluct.

YRS

Spin-Ising symmetry

Easy-plane symmetry

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

Fermi surface change clear

signatures in thermodynamics

Multiple energy scales at QCP

P. Gegenwart et al., cond-mat/0604571.

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 ~ T0.7

- 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)

020

4060

80100

0

200

400

600

800

1000

1200

1400

5

6

78

Field

[tesla

]

Tem

pera

ture

[mK

]

angle from ab [degrees]S.A. Grigera et al. PRB 67, 214427 (2003)

QCEP @ 8 T // c-axis

Evidence for QC fluctuations: Diverging A(H) at Hc (S.A. Grigera et al, Science 2001)

Thermal expansion

P. Gegenwart, F. Weickert, M. Garst, R.S. Perry, Y. Maeno,Phys. Rev. Lett. 96, 136402 (2006)

cHH

c

mol

HHhdP

dH

h

S

P

SV

c

~,with Calculation for itinerant metamagnetic QCEP

Behavior consistent with 2D QCEP scenario

P. Gegenwart, F. Weickert, M. Garst, R.S. Perry, Y. Maeno, Phys. Rev. Lett. 96, 136402 (2006)

Thermal expansion on Sr3Ru2O7

Compatible with underlying

2D QCEP at Hc = 7.85 T

=0 marks accumulation

points of entropy

6.5 7.0 7.5 8.0 8.5 9.0

1.2

1.6

2.1 T (K)

0.1

0.3

0.6

0.9

1.2

0.2

0.4 0.5

0.7 0.8

1.0 1.1

1.3

cm)

B (T)

Dominant elastic scattering Formation of domains!

Fine-structure near 8 Tesla

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.

7.6 7.8 8.0 8.20.0

0.5

1.0

Field (tesla)

T (

K)

Thermodynamic analysis of fine-structure

1) No clear phase transitions2) 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:

dT

dHMS c

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

• Formation of symmetry-broken

phase (Pomeranchuk instability)?

Unlikely because of enhanced entropy

Real-space

phase separation?

(C. Honerkamp, PRB 2005)

liquid

gastwo-phase

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

Orthorhombic CrB structure

CePd is ferromagnetic with TC = 6.6 K CeRh has an intermediate valent ground state

c

Ce

Pd,Rh

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

TC

(K

)

x

Cp,max

M '-ac "-ac

CePd1-x

Rhx

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

0.0 0.5 1.0 1.5 2.0

0.5

1.0

1.5

2.0

2.5

x = 0.8, = 113 Hz

single crystal

CePd1-xRhx

' (1

0-6

m3

/mo

l)

T (K)

AC Susceptibility in the Tail Region

0.0 0.5 1.0 1.5 2.0

0.5

1.0

1.5

2.0

2.5

x = 0.8, = 113 Hz

single crystal

CePd1-xRhx

' (1

0-6

m3

/mo

l)

T (K)

B = 0 mTB = 5 mTB = 10 mTB = 15 mT

0.75 0.80 0.85 0.90 0.95 1.000

100

200

300

400

500 CePd1-xRhx

T (m

K)

Rh content x

single crystals polycrystals

= 13 Hz

Crossover transition for x > 0.6indicated by sharp cusps in AC‘ down to mK temperatures

Frequency dependence at low frequencies and high sensitivity on tinymagnetic DC fields no long range order

Maxima of ‘(T) in phase diagram‘(T) in DC field

Spin Glass-like Behavior

0.15 0.20 0.25 0.30 0.35

2.0

2.5

3.0

' (1

0-6m

3/m

ol)

T (K)

13 Hz113 Hz1113 Hz

x=0.85

CePd1-xRhxsingle crystal

0.1 0.4 1 100.0

0.2

0.4

0.6

0.8

1.0

1.2

x = 0.80 x = 0.85 x = 0.87 x = 0.90

CePd1–x

Rhx

C/T

(J/

(mo

l K2 ))

T (K)

Frequency shift (e.g. x=0.85: TC/[TC log()] of 5%)

Spin glass-like behavior

No maximum in specific heatbut NFL behavior for x ≥ 0.85

0.1 0.4 1 100.0

0.2

0.4

0.6

0.8

1.0

1.2

x = 0.85

CePd1–x

Rhx

C/T

(J/

(mo

l K2 ))

T (K)

Grüneisen parameter shows no divergence

”Kondo Cluster Glass“

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

Possible reason forspin glass-like state:

Variation of TK for Ce ionsdepending on Rh or Pdnearest neighborsleading distribution oflocal Kondo temperatures

”Kondo cluster glass“

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

6

7

xcr

300

30

CePd1-x

Rhx

TC

(K)

x (Rh conc)

TC from

M(T)

'ac

- He3

C

max

'ac

- He3/He4

100

P (K

)

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.