Qimiao SiRice University
KIAS, Oct 29, 2005
Heavy fermion metals: Global phase diagram, local quantum criticality, and experiments
S. Paschen P. Gegenwart R. KüchlerT. Lühmann S. Wirth N. OeschlerT. Cichorek K. Neumaier O. TegusO. Trovarelli C. Geibel J. A. MydoshF. Steglich P. Coleman
Lijun Zhu, Stefan Kirchner, Tae-Ho Park, Eugene Pivovarov, (Rice University)Silvio Rabello, J. L. Smith Kevin Ingersent (Univ. of Florida)Daniel Grempel (CEA-Saclay)Jianxin Zhu (Los Alamos)
Quantum Critical Point
• QCP: existence itself is conceptually simple…
• … but, can be elusive (required parameter tuning beyond practical range, order hidden, too many competing phases, 1st order along the physical axis, etc.)
• and, the nature of the QCP seems to be exceedingly rich.
• Insulating Ising magnet– LiHoF4: transverse field Ising model
• Heavy fermion magnetic metals• ‘‘Simple’’ magnetic metals
– Cr1-xVx, Sr3Ru2O7, MnSi (1st order, but …) …
• High Tc superconductors (?) • Mott transition
– V2O3, …: QCP? (magnetic ordering intervenes at low T!) – cold atoms: 2nd order?
• Frustrated magnets (?)• Field-driven BEC of magnons
Materials (possibly) showing Quantum Criticality
• Metal-insulator transition in 3D Si:P, …– Many theoretical questions remain (Finkelstein scaling theory? local moments?...)
• MIT in 2DEG of Si-MOSFETs, ... (?)– Phase diagram? (Experiments seeing a genuine metal phase?)
• Superconductor-insulator transitions in films– Phase diagram? (Intermediate metal?)
• QH-QH and QH-Insulator transitions– 2nd order?
Materials (possibly) showing Quantum Criticality
Early part of the heavy fermion field
• Heavy electron mass• Unconventional superconductivity• Kondo screening Kondo resonances
– Fermi liquid of heavy quasiparticles
On the theory front: Single-impurity: Anderson, Wilson, Nozières, Andrei, Wiegmann,
Coleman, Read & Newns, …Lattice: Varma, Doniach, Auerbach & Levin, Millis & Lee,
Rice & Ueda, …
Past decade of the heavy fermion field
• Non-Fermi Liquid Behavior
• Quantum Criticality
New focus, perhaps due to cross-fertilization w/ high Tc & other correlated systems
Heavy fermions near a magnetic QCP:
CeCu6-xAuxH. v. Löhneysen et al, PRL 1994
AF MetalTN
TN
Heavy fermions near a magnetic QCP:
CeCu6-xAux
YbRh2Si2
H. v. Löhneysen et al, PRL 1994
J. Custers et al, Nature 2003
CePd2Si2N. Mathur et al, Nature 1998
AF MetalTN
TN
TN
AF MetalSupercond.
Linearresistivity
Heavy fermions near a magnetic QCP:
– YbRh2Si
easy-plane spin-anisotropy; TK0 ≈ 25 K
– Ce(Cu1-xAux )6
Ising anisotropy; TK0 ≈ 6 K
– CePd2Si2, CeIn3 (first order?--NQR),
CeNi2Ge2, CeCu2Si2
– YbAgGe [frustrated (hexagonal) lattice]
– CeMIn5,
– URu2Si2 (?)
Kondo Lattice Model
I: RKKY interaction; AF
G=Innn/Inn etc.
Kondo Lattice Model
Bandwidth W
Kondo coupling JK
I: RKKY interaction; AF
G=Innn/Inn etc.
Kondo Lattice Model
Bandwidth W
Kondo coupling JK
Fixed I and W with I<<W, varying G and JK
I: RKKY interaction; AF
G=Innn/Inn etc.
Kondo lattices
JK
G
G ~ frustration, reduceddimensionality, etc.
Loc
al m
omen
t mag
netis
m, I
rkky
Kon
do c
oupl
ing
J K
Bandwidth W
JK
G
JK >>W>>Irkky
JK >>W>>Irkky
• xNsite tightly bound local singlets
(cf. If x were =1, Kondo insulator)
• (1-x)Nsite lone moments:
JK >>W>>Irkky
• xNsite tightly bound local singlets
(cf. If x were =1, Kondo insulator)
• (1-x)Nsite lone moments: – projection:
– (1-x)Nsite holes with U=∞
JK >>W>>Irkky
• xNsite tightly bound local singlets
(cf. If x were =1, Kondo insulator)
• (1-x)Nsite lone moments: – projection:
– (1-x)Nsite holes with U=∞
• Luttinger’s theorem:
(1-x) holes/site in the Fermi surface
(1+x) electrons/site ---- Large Fermi surface!
JK
G
PML
paramagnet, w/ Kondo screening
JK >>W>>Irkky
JK
G
JK<<Irkky<<WL
ocal
mom
ent m
agne
tism
, Irk
ky
JK<<Irkky<<W• With Ising anisotropy, the magnetic spectrum of the local moment component is gapped.
• JK is irrelevant!
JK<<Irkky<<W• With Ising anisotropy, the magnetic spectrum of the
local moment component is gapped.
• JK is irrelevant!
• Local moments stay charge neutral, and do not contribute to the electronic excitations.
Fermi surface is small
JK
G
AFS
QS, J.-X. Zhu, & D. Grempel, cond-mat /0506207
JK<<Irkky<<W
Néel, without Kondo screening
JK
G
AFS
PML
AFL
QS, J.-X. Zhu, & D. Grempel, cond-mat /0506207
I
II
Global phase diagram
Type I transition
Type II transitionHertz fixed point for T=0 SDW transition
Second order if
Destruction of Kondo screening at the magnetic QCP
Local Quantum Critical Point
• Fluctuations of the magnetic order parameter are slow at the magnetic QCP
• The slow magnetic fluctuations decohere the Kondo screening
• Kondo effect is critical, which is in addition to the critical fluctuations of magnetic order parameter
Local Quantum Critical Point
QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001)
• Anomalous spin dynamics
Destruction of Kondo effect (Eloc
* 0) at the QCP
Nature of the phases
TN
• T2 resistivity on both sides of the QCP
Nature of the phases (cont’d)
TN
• Small Fermi surface in the AF metal phase
TN
S. Araki, R. Settai, T. C. Kobayashi, H. Harima, & Y. Onuki, Phys Rev. B 64, 224417 (2001)
CeRh2Si2
• Localization of f-electrons
– Reconstruction of the Fermi surface across QCP
– m* ∞ over the entire Fermi surface as QCP
• Anomalous spin dynamics.
• Destruction of Kondo effect
– Non-Fermi liquid excitations part of the quantum-critical spectrum.
In what sense is the QCP local?
CeCu6-xAux (xc≈0.01)
H. v. Löhneysen et al, PRL 1994
AF Metal
TN
TN
Dynamical and Static Susceptibilities in CeCu5.9Au0.1
A. Schröder et al., Nature ’00; PRL ’98; O. Stockert, H. v. Löhneysen, A. Rosch, N. Pyka, & M. Loewenhaupt, PRL ’98
• E/T scaling
•Fractional exponent =0.75• =0.75 `everywhere’ in q.
INS @ q=Q E/T
q=Q
q=0
INS and M/H
T0.75
1/(q)
...
Dynamics of the quantum critical CeCu5.9Au0.1
• Frequency and temperature dependences of the dynamical spin susceptibility:– an anomalous exponent < 1 /T scaling
implying non-Gaussian fixed point
• The anomalous exponent is seen essentially `everywhere’ in the momentum space
O. Stockert, H. v. Löhneysen, A. Rosch, N. Pyka, & M. Loewenhaupt, Phys. Rev. Lett. ’98
TN
Ce(Ru1-xRhx)2Si2 (xc≈0.04)
)()(
),,T
TTQ
Q
Q
/i-1
H. Kadowaki, Y. Tabata, M. Sato, N. Aso, S. Raymond, & S. Kawarazaki, cond-mat/0504386
Ce(Ru1-xRhx)2Si2
H. Kadowaki, Y. Tabata, M. Sato, N. Aso, S. Raymond, & S. Kawarazaki, cond-mat/0504386
TN0=350mJ/K2 for x=0
C/T=0-a T1/2
• Localization of f-electrons
– Reconstruction of the Fermi surface across QCP
– m* ∞ over the entire Fermi surface as QCP
• Anomalous spin dynamics everywhere in q.
• Destruction of Kondo effect
– Non-Fermi liquid excitations part of the quantum-critical spectrum.
In what sense is the QCP local?
Hall Effect in YbRh2Si2: probing the Fermi-surface change
TN
Linear resistivity
S. Paschen, T. Lühmann, S. Wirth, P.Gegenwart, O.Trovarelli, C. Geibel, F. Steglich, P.Coleman, & QS, Nature 432, 881 (2004)
Hall Effect in YbRh2Si2
S. Paschen, T. Lühmann, S. Wirth, P.Gegenwart, O.Trovarelli, C. Geibel, F. Steglich, P.Coleman, & QS, Nature 432, 881 (2004)
Hall Effect in YbRh2Si2
Hall Effect in YbRh2Si2
Hall Effect in YbRh2Si2
• Finite T crossover width T0.5±0.1
• T=0 (extrapolation): sharp jump @ QCP
Hall Effect in YbRh2Si2
S. Paschen, T. Lühmann, S. Wirth, P.Gegenwart, O.Trovarelli, C. Geibel, F. Steglich, P.Coleman, & QS, Nature 432, 881 (2004)
• Finite T crossover width T0.5±0.1
• T=0 (extrapolation): sharp jump @ QCP
Hall Effect in YbRh2Si2
S. Paschen, T. Lühmann, S. Wirth, P.Gegenwart, O.Trovarelli, C. Geibel, F. Steglich, P.Coleman, & QS, Nature 432, 881 (2004)
H. Shishido, R. Settai, H. Harima, & Y. Onuki, JPSJ 74, 1103 (2005)_
dHvA in CeRhIn5
Divergence of the Grüneisen Ratio
L. Zhu, M. Garst, A. Rosch,and QS, Phys. Rev. Lett. ’03
xT1
zx /1with
Divergence of the Grüneisen Ratio
L. Zhu, M. Garst, A. Rosch,and QS, Phys. Rev. Lett. ’03
R. Küchler et al., Phys. Rev. Lett. ’03
Magnetic Paramagnetic
T
0 2 4-160
-80
80
160
YbRh2Si2
Grüneisen Ratio
T(K)
CeNi2Ge2
p
xT1
zx /1with
• LQCP: xloc ≈ 0.66 to 2nd order in ε-expansion for the XY case
• Cf. AF-SDW: x = 1 / z = 1
R. Küchler et al., Phys. Rev. Lett. ’03
Grüneisen exponent in Ge-doped YbRh2Si2
Spin-glass QCP in heavy fermions?
JK
Gparamagnet, w/ Kondo screening
SG, without Kondo screening
• Type I: interacting f.p.– UPdxCu5-x (?): /T scaling (M. Aronson et al ’95; D. MacLaughlin et al)– Sc1-xUxPd3 (?): /T scaling (P. Dai et al’04)
• Type II: Gaussian f.p. – Fluctuation of the spin glass order parameter
(Sachdev et al, ’95; Sengupta and Georges ‘95)
I
II
Spin-glass QCP in heavy fermions?
S. Wilson, P. Dai et al, Phys. Rev. Lett. ’05
D. Gajewski, R. Chau, and M. B. Maple, Phys. Rev. B (’00)
SUMMARY• Global phase diagram of the magnetic heavy fermion metals
• Two types of quantum critical metals
– T=0 SDW transition (Gaussian)– Locally quantum-critical: destruction of Kondo effect exactly
at the magnetic QCP (interacting)
• Evidence from magnetic dynamics, Fermi surface evolution, and thermodynamic ratio.
• Relevance to other strongly correlated metals?