Challenges of understanding Critical and Normal Heavy Fermion

Fluids

Non Fermi Liquids: Theory and Experiment.

Piers Coleman Center for Materials Theory

Rutgers, USA

Aspen Center for Physics 14-17 June, 2014Beyond Quasiparticles: New Paradigms for Quantum Fluids

Preliminary program: "Beyond Quasiparticles: NewParadigms for Quantum Fluids", January 12 - January 17,

2014

Sunday evening, 5:00-7:00 Reception at ACPIndependent in-town group dinner -- Hickory House restaurant

Monday morning, 8:00-11:00: Non-fermi liquid metals - heavy fermionexperimentOverview and discussion leader: Piers Coleman (Rutgers University)

8.00 -8.15 am

Piers Coleman (RutgersUniversity)

Challenges of Understanding Critical andNormal Heavy Fermion Fluids

Show abstract8.15 -8.50

Collin Broholm (Johns HopkinsUniversity)

The magnetism of strongly correlatedmetals

Show abstract8.50 -9.25

Y. Matsumoto (ISSP, Universityof Tokyo)

Valence fluctuation and strange metal phasein YbAlB4

Show abstract9.25 -9.45

Coffee break

9.50 -10.25

J. D. Thompson (Los AlamosNational Laboratory)

Non-Fermi Liquid Behaviors in CeRhIn5and CeCoIn5

Show abstract10.25 -11.00

A. Yazdani (PrincetonUniversity)

Visualizing the Emergence of HeavyFermions, their Superconductivity, andHidden Orders with STM

Show abstract

Monday afternoon, 4:15-5:45: DMFT perspectives on Non-fermi liquidmetals

http://www.its.caltech.edu/~motrunch/Aspen2014_BeyondQP...

1 of 6 1/10/14, 7:47 AM

Kmetko-Smith Phase Diagram

Smith and Kmetko (1983)

Kmetko-Smith Phase Diagram

Smith and Kmetko (1983)

Rare Earth

Transition metal

Actinide

4f

3d

5fIncreasing localization

FIGURE 1. Depicting localized 4 f , 5 f and 3d atomic wavefunctions.

represented by a single, neutral spin operator

�S=h̄

2�σ

where �σ denotes the Pauli matrices of the localized electron. Localized moments de-

velop within highly localized atomic wavefunctions. The most severely localized wave-

functions in nature occur inside the partially filled 4 f shell of rare earth compounds

(Fig. 1) such as cerium (Ce) or Ytterbium (Yb). Local moment formation also occurs

in the localized 5 f levels of actinide atoms as uranium and the slightly more delocal-

ized 3d levels of first row transition metals(Fig. 1). Localized moments are the origin

of magnetism in insulators, and in metals their interaction with the mobile charge car-

riers profoundly changes the nature of the metallic state via a mechanism known as the“Kondo effect”.

In the past decade, the physics of local moment formation has also reappeared in

connection with quantum dots, where it gives rise to the Coulomb blockade phenomenon

and the non-equilibrium Kondo effect.

Kmetko-Smith Phase Diagram

Smith and Kmetko (1983)

Rare Earth

Transition metal

Actinide

4f

3d

5fIncreasing localization

FIGURE 1. Depicting localized 4 f , 5 f and 3d atomic wavefunctions.

represented by a single, neutral spin operator

�S=h̄

2�σ

where �σ denotes the Pauli matrices of the localized electron. Localized moments de-

velop within highly localized atomic wavefunctions. The most severely localized wave-

functions in nature occur inside the partially filled 4 f shell of rare earth compounds

(Fig. 1) such as cerium (Ce) or Ytterbium (Yb). Local moment formation also occurs

in the localized 5 f levels of actinide atoms as uranium and the slightly more delocal-

ized 3d levels of first row transition metals(Fig. 1). Localized moments are the origin

of magnetism in insulators, and in metals their interaction with the mobile charge car-

riers profoundly changes the nature of the metallic state via a mechanism known as the“Kondo effect”.

In the past decade, the physics of local moment formation has also reappeared in

connection with quantum dots, where it gives rise to the Coulomb blockade phenomenon

and the non-equilibrium Kondo effect.

Increasing localization

Kmetko-Smith Phase Diagram

Smith and Kmetko (1983)

Rare Earth

Transition metal

Actinide

4f

3d

5fIncreasing localization

FIGURE 1. Depicting localized 4 f , 5 f and 3d atomic wavefunctions.

represented by a single, neutral spin operator

�S=h̄

2�σ

where �σ denotes the Pauli matrices of the localized electron. Localized moments de-

velop within highly localized atomic wavefunctions. The most severely localized wave-

functions in nature occur inside the partially filled 4 f shell of rare earth compounds

(Fig. 1) such as cerium (Ce) or Ytterbium (Yb). Local moment formation also occurs

in the localized 5 f levels of actinide atoms as uranium and the slightly more delocal-

ized 3d levels of first row transition metals(Fig. 1). Localized moments are the origin

of magnetism in insulators, and in metals their interaction with the mobile charge car-

riers profoundly changes the nature of the metallic state via a mechanism known as the“Kondo effect”.

In the past decade, the physics of local moment formation has also reappeared in

connection with quantum dots, where it gives rise to the Coulomb blockade phenomenon

and the non-equilibrium Kondo effect.

Increasing localization

Kmetko-Smith Phase Diagram

Smith and Kmetko (1983)

Rare Earth

Transition metal

Actinide

4f

3d

5fIncreasing localization

FIGURE 1. Depicting localized 4 f , 5 f and 3d atomic wavefunctions.

represented by a single, neutral spin operator

�S=h̄

2�σ

where �σ denotes the Pauli matrices of the localized electron. Localized moments de-

velop within highly localized atomic wavefunctions. The most severely localized wave-

functions in nature occur inside the partially filled 4 f shell of rare earth compounds

(Fig. 1) such as cerium (Ce) or Ytterbium (Yb). Local moment formation also occurs

in the localized 5 f levels of actinide atoms as uranium and the slightly more delocal-

ized 3d levels of first row transition metals(Fig. 1). Localized moments are the origin

of magnetism in insulators, and in metals their interaction with the mobile charge car-

riers profoundly changes the nature of the metallic state via a mechanism known as the“Kondo effect”.

In the past decade, the physics of local moment formation has also reappeared in

connection with quantum dots, where it gives rise to the Coulomb blockade phenomenon

and the non-equilibrium Kondo effect.

Increasing localization

A lot of action at the brink of localization.

Heavy Fermion Primer

Spin (4f,5f): basic fabric of heavy electron physics.

2j+1

Heavy Fermion Primer

Spin (4f,5f): basic fabric of heavy electron physics.

2j+1

Heavy Fermion Primer

Spin (4f,5f): basic fabric of heavy electron physics.

2j+1 Curie

Heavy Fermion Primer

Spin (4f,5f): basic fabric of heavy electron physics.

Electron sea

2j+1 Curie

Heavy Fermion Primer

Spin (4f,5f): basic fabric of heavy electron physics.

H =X

k�

✏kc†k�ck� + J ~S · ~�(0)

Electron sea

2j+1 Curie

Heavy Fermion Primer

Spin (4f,5f): basic fabric of heavy electron physics.

H =X

k�

✏kc†k�ck� + J ~S · ~�(0)

Strong Coupling0 ∞1AFM J>0 J(T)

FIXED POINT

T>>TK T<<TKTKFIXED POINT

Weak Coupling

Electron sea

2j+1 Curie

Heavy Fermion Primer

Spin screened by conduction electrons: entangled

H =X

k�

✏kc†k�ck� + J ~S · ~�(0)

Strong Coupling0 ∞1AFM J>0 J(T)

FIXED POINT

T>>TK T<<TKTKFIXED POINT

Weak Coupling

Electron sea

2j+1 Curie

Heavy Fermion Primer

Spin screened by conduction electrons: entangled

H =X

k�

✏kc†k�ck� + J ~S · ~�(0)

Strong Coupling0 ∞1AFM J>0 J(T)

FIXED POINT

T>>TK T<<TKTKFIXED POINT

Weak Coupling

" #� # "

Electron sea

2j+1 Curie

Pauli

Heavy Fermion Primer“Kondo temperature”

Spin screened by conduction electrons: entangled

H =X

k�

✏kc†k�ck� + J ~S · ~�(0)

TK ⇠ De�1

2J⇢

Strong Coupling0 ∞1AFM J>0 J(T)

FIXED POINT

T>>TK T<<TKTKFIXED POINT

Weak Coupling

" #� # "

Heavy Fermion Primer

Heavy Fermion Primer

“Kondo Lattice”

Kondo Insulators

“Kondo Lattice”Entangled many body state of spins and electrons gives rise to many new kinds of order.

Kondo Insulators

“Kondo Lattice”Entangled many body state of spins and electrons gives rise to many new kinds of order.

Sm2.7+

B6

SmB6

1968

Kondo Insulators

Mott Phil Mag, 30,403,1974

Kondo Insulators

“Kondo Lattice”Entangled many body state of spins and electrons gives rise to many new kinds of order.

Sm2.7+

B6

SmB6

1968

Kondo Insulators

Mott Phil Mag, 30,403,1974

Kondo Insulators

“Kondo Lattice”Entangled many body state of spins and electrons gives rise to many new kinds of order.

Sm2.7+

B6

SmB6

1968

Kondo Insulators

VOLUME 74, NUMBER 9 PH YSICAL REVIEW LETTERS 27 FEBRUARY 1995

1 bar

$ P -24 kba3r

1'.:(b)

10

o 10Cl

10

33 kb10

E10

1053 kb

60 kbar66 kbar

, I

10T(K)

100

10

0.0 0.21/T(K)

0.4

FIG. 1. The electrical resistivity p as a function of tem-perature (a) and inverse temperature (b). (b) Q = 1 bar,Q = 24 kbar, = 25 kbar, = 33 kbar, A = 45 kbar, andA = 53 kbar. The solid lines in (b) are fits by the function[p(T)] ' = [po(P)] ' + (p„,(P) exp[A(P)/k&T]) ', describedin the text.

403020~ &00~10

10E, ~9

~o ~e~~1010

20 40P (kbar)

(a).

(b)I60

FIG. 2. The pressure dependences of the activation gap 5 (a)and residual carrier density no = I/R (T =H0) (b). Dashed lineindicates approximate pressure for disappearance of A. Solidlines are guides for the eye.

linearly -0.5 K/kbar from its ambient pressure value of41 K. Above 45 kbar, the resistivity is metallic and it isno longer possible to extract an activation gap.Our measurements indicate a gap instability at a critical

pressure P,. between 45 and 53 kbar, in disagreement withthe conclusions of previous workers [5,6], who foundthat 5 vanished continuously near 60 kbar. In one ofthese studies [5] the sample was of demonstrably lowerquality than our own, with a significantly smaller ambientpressure 6 = 33 K and a much smaller po —10 mA cm,both symptomatic of Sm vacancies or defects introducedin powdering [8]. Our measurements suggest that the gapinstability is a feature only of the highest quality samples,as P,. increases markedly with reduced sample quality,passing out of our experimental pressure window of180 kbar for po ~ 0.1 A cm. We further believe that thesimple activation fits used to determine 6 in both earlierexperiments were overly weighted by the temperatureindependent resistivity below -3.5 K, particularly nearP, . Figure 1(b) demonstrates that near P, the range

101 bar

10

10

24 kbare

~ Q5 kbar

(310E 33 kbar

10-

10-45 kbar

B3 kbar60 kbar

-4 '5 66 kbar10 I I ( I

0.0 0.2 0.4 0.6 0.8 1.0&/r (K')

FIG. 3. The absolute value of the Hall constant RH of SmB6as a function of inverse temperature.

of temperatures over which simple activation fits arelinear becomes increasingly limited and problematic todefine with increased pressure. In contrast, our parallelresistor formulation provides uniformly good fits over thispressure range, and consequently yield a more accuratedetermination of A.Since there is no evidence in SmB6 for a discontinuous

structural change at or below 60 kbar [9], the sudden dis-appearance of 5 suggests that it is not a simple hybridiza-tion gap, for in that case the insulator-metal transitionoccurs by band crossing and the gap is suppressed con-tinuously to zero. A valence instability can be similarlydiscounted, as high pressure x-ray absorption measure-ments [10] find that the Sm valence increases smoothlyfrom +2.6 to +2.75 between 1 bar and 60 kbar.We have used Hall effect measurements to study the

evolution of the camers in the vicinity of P, The Hallconstant RH is plotted as a function of 1/T in Fig. 3 forpressures ranging from 1 bar to 66 kbar. We find thatRH is negative for temperatures T between 1.2 and 40 Kand at all pressures, as well as independent of magneticfields as large as 18 T. As has been previously noted at1 bar [11], RH is both large and extremely temperaturedependent with a maximum at 4 K, at each pressurebecoming temperature independent below -3 K. It hasbeen proposed [12] that this temperature dependencefor RH is characteristic of Kondo lattices, rejectinga crossover from high temperature incoherent to lowtemperature coherent skew scattering. However, similarmaxima in RH(T) occur in doped semiconductors as in-gap impurity states dominate intrinsic activated processeswith reduced temperature [13].We do not address the full temperature dependence

of RH here, instead limiting our discussion to the

1630

Persistent  conduc-vity  PlateauCooleyet al (1995)

Mott Phil Mag, 30,403,1974

Kondo Insulators

“Kondo Lattice”Entangled many body state of spins and electrons gives rise to many new kinds of order.

Sm2.7+

B6

SmB6

1968

Kondo Insulators

Broken Kondo Singlets

Topological(Dzero,Sun,PC,Galitski (2010)Wolgast et al (2013).

VOLUME 74, NUMBER 9 PH YSICAL REVIEW LETTERS 27 FEBRUARY 1995

1 bar

$ P -24 kba3r

1'.:(b)

10

o 10Cl

10

33 kb10

E10

1053 kb

60 kbar66 kbar

, I

10T(K)

100

10

0.0 0.21/T(K)

0.4

FIG. 1. The electrical resistivity p as a function of tem-perature (a) and inverse temperature (b). (b) Q = 1 bar,Q = 24 kbar, = 25 kbar, = 33 kbar, A = 45 kbar, andA = 53 kbar. The solid lines in (b) are fits by the function[p(T)] ' = [po(P)] ' + (p„,(P) exp[A(P)/k&T]) ', describedin the text.

403020~ &00~10

10E, ~9

~o ~e~~1010

20 40P (kbar)

(a).

(b)I60

FIG. 2. The pressure dependences of the activation gap 5 (a)and residual carrier density no = I/R (T =H0) (b). Dashed lineindicates approximate pressure for disappearance of A. Solidlines are guides for the eye.

linearly -0.5 K/kbar from its ambient pressure value of41 K. Above 45 kbar, the resistivity is metallic and it isno longer possible to extract an activation gap.Our measurements indicate a gap instability at a critical

pressure P,. between 45 and 53 kbar, in disagreement withthe conclusions of previous workers [5,6], who foundthat 5 vanished continuously near 60 kbar. In one ofthese studies [5] the sample was of demonstrably lowerquality than our own, with a significantly smaller ambientpressure 6 = 33 K and a much smaller po —10 mA cm,both symptomatic of Sm vacancies or defects introducedin powdering [8]. Our measurements suggest that the gapinstability is a feature only of the highest quality samples,as P,. increases markedly with reduced sample quality,passing out of our experimental pressure window of180 kbar for po ~ 0.1 A cm. We further believe that thesimple activation fits used to determine 6 in both earlierexperiments were overly weighted by the temperatureindependent resistivity below -3.5 K, particularly nearP, . Figure 1(b) demonstrates that near P, the range

101 bar

10

10

24 kbare

~ Q5 kbar

(310E 33 kbar

10-

10-45 kbar

B3 kbar60 kbar

-4 '5 66 kbar10 I I ( I

0.0 0.2 0.4 0.6 0.8 1.0&/r (K')

FIG. 3. The absolute value of the Hall constant RH of SmB6as a function of inverse temperature.

of temperatures over which simple activation fits arelinear becomes increasingly limited and problematic todefine with increased pressure. In contrast, our parallelresistor formulation provides uniformly good fits over thispressure range, and consequently yield a more accuratedetermination of A.Since there is no evidence in SmB6 for a discontinuous

structural change at or below 60 kbar [9], the sudden dis-appearance of 5 suggests that it is not a simple hybridiza-tion gap, for in that case the insulator-metal transitionoccurs by band crossing and the gap is suppressed con-tinuously to zero. A valence instability can be similarlydiscounted, as high pressure x-ray absorption measure-ments [10] find that the Sm valence increases smoothlyfrom +2.6 to +2.75 between 1 bar and 60 kbar.We have used Hall effect measurements to study the

evolution of the camers in the vicinity of P, The Hallconstant RH is plotted as a function of 1/T in Fig. 3 forpressures ranging from 1 bar to 66 kbar. We find thatRH is negative for temperatures T between 1.2 and 40 Kand at all pressures, as well as independent of magneticfields as large as 18 T. As has been previously noted at1 bar [11], RH is both large and extremely temperaturedependent with a maximum at 4 K, at each pressurebecoming temperature independent below -3 K. It hasbeen proposed [12] that this temperature dependencefor RH is characteristic of Kondo lattices, rejectinga crossover from high temperature incoherent to lowtemperature coherent skew scattering. However, similarmaxima in RH(T) occur in doped semiconductors as in-gap impurity states dominate intrinsic activated processeswith reduced temperature [13].We do not address the full temperature dependence

of RH here, instead limiting our discussion to the

1630

Persistent  conduc-vity  PlateauCooleyet al (1995)

Kondo Insulators

“Kondo Lattice”Entangled many body state of spins and electrons gives rise to many new kinds of order.

Sm2.7+

B6

SmB6

1968

Kondo Insulators

Broken Kondo Singlets

Topological(Dzero,Sun,PC,Galitski (2010)Wolgast et al (2013).

11

FIG. 3: Fermi surface and dispersion maps of SmB6. (a) Fermi surface plot of SmB6

measured by 7 eV LASER source at temperature of 7 K. A small � pocket and a large X pocket

are observed. A big elliptical and a small circular shaped black dash lines around X and � points

are guide for the eyes. Inset shows a schematic plot of Fermi surface in the first Brillouin zone. (b)

Electronic dispersion map (left) and its energy distribution curves (EDCs) for � pocket. (c) same

as (b) for X band. (d) Comparison of integrated EDC for � and X band. A gap value of about 15

meV is observed in both cases.

Neupane,  arXiv:  1306.4634,    Nature(2013).

VOLUME 74, NUMBER 9 PH YSICAL REVIEW LETTERS 27 FEBRUARY 1995

1 bar

$ P -24 kba3r

1'.:(b)

10

o 10Cl

10

33 kb10

E10

1053 kb

60 kbar66 kbar

, I

10T(K)

100

10

0.0 0.21/T(K)

0.4

FIG. 1. The electrical resistivity p as a function of tem-perature (a) and inverse temperature (b). (b) Q = 1 bar,Q = 24 kbar, = 25 kbar, = 33 kbar, A = 45 kbar, andA = 53 kbar. The solid lines in (b) are fits by the function[p(T)] ' = [po(P)] ' + (p„,(P) exp[A(P)/k&T]) ', describedin the text.

403020~ &00~10

10E, ~9

~o ~e~~1010

20 40P (kbar)

(a).

(b)I60

FIG. 2. The pressure dependences of the activation gap 5 (a)and residual carrier density no = I/R (T =H0) (b). Dashed lineindicates approximate pressure for disappearance of A. Solidlines are guides for the eye.

linearly -0.5 K/kbar from its ambient pressure value of41 K. Above 45 kbar, the resistivity is metallic and it isno longer possible to extract an activation gap.Our measurements indicate a gap instability at a critical

pressure P,. between 45 and 53 kbar, in disagreement withthe conclusions of previous workers [5,6], who foundthat 5 vanished continuously near 60 kbar. In one ofthese studies [5] the sample was of demonstrably lowerquality than our own, with a significantly smaller ambientpressure 6 = 33 K and a much smaller po —10 mA cm,both symptomatic of Sm vacancies or defects introducedin powdering [8]. Our measurements suggest that the gapinstability is a feature only of the highest quality samples,as P,. increases markedly with reduced sample quality,passing out of our experimental pressure window of180 kbar for po ~ 0.1 A cm. We further believe that thesimple activation fits used to determine 6 in both earlierexperiments were overly weighted by the temperatureindependent resistivity below -3.5 K, particularly nearP, . Figure 1(b) demonstrates that near P, the range

101 bar

10

10

24 kbare

~ Q5 kbar

(310E 33 kbar

10-

10-45 kbar

B3 kbar60 kbar

-4 '5 66 kbar10 I I ( I

0.0 0.2 0.4 0.6 0.8 1.0&/r (K')

FIG. 3. The absolute value of the Hall constant RH of SmB6as a function of inverse temperature.

of temperatures over which simple activation fits arelinear becomes increasingly limited and problematic todefine with increased pressure. In contrast, our parallelresistor formulation provides uniformly good fits over thispressure range, and consequently yield a more accuratedetermination of A.Since there is no evidence in SmB6 for a discontinuous

structural change at or below 60 kbar [9], the sudden dis-appearance of 5 suggests that it is not a simple hybridiza-tion gap, for in that case the insulator-metal transitionoccurs by band crossing and the gap is suppressed con-tinuously to zero. A valence instability can be similarlydiscounted, as high pressure x-ray absorption measure-ments [10] find that the Sm valence increases smoothlyfrom +2.6 to +2.75 between 1 bar and 60 kbar.We have used Hall effect measurements to study the

evolution of the camers in the vicinity of P, The Hallconstant RH is plotted as a function of 1/T in Fig. 3 forpressures ranging from 1 bar to 66 kbar. We find thatRH is negative for temperatures T between 1.2 and 40 Kand at all pressures, as well as independent of magneticfields as large as 18 T. As has been previously noted at1 bar [11], RH is both large and extremely temperaturedependent with a maximum at 4 K, at each pressurebecoming temperature independent below -3 K. It hasbeen proposed [12] that this temperature dependencefor RH is characteristic of Kondo lattices, rejectinga crossover from high temperature incoherent to lowtemperature coherent skew scattering. However, similarmaxima in RH(T) occur in doped semiconductors as in-gap impurity states dominate intrinsic activated processeswith reduced temperature [13].We do not address the full temperature dependence

of RH here, instead limiting our discussion to the

1630

Persistent  conduc-vity  PlateauCooleyet al (1995)

Kondo Insulators

“Kondo Lattice”Entangled many body state of spins and electrons gives rise to many new kinds of order.

Sm2.7+

B6

SmB6

1968

Kondo Insulators

Broken Kondo Singlets

Topological(Dzero,Sun,PC,Galitski (2010)Wolgast et al (2013).

Collin Broholm: what happens to the correlations between d- and!f-electrons as topological Kondo insulator develops?

11

FIG. 3: Fermi surface and dispersion maps of SmB6. (a) Fermi surface plot of SmB6

measured by 7 eV LASER source at temperature of 7 K. A small � pocket and a large X pocket

are observed. A big elliptical and a small circular shaped black dash lines around X and � points

are guide for the eyes. Inset shows a schematic plot of Fermi surface in the first Brillouin zone. (b)

Electronic dispersion map (left) and its energy distribution curves (EDCs) for � pocket. (c) same

as (b) for X band. (d) Comparison of integrated EDC for � and X band. A gap value of about 15

meV is observed in both cases.

Neupane,  arXiv:  1306.4634,    Nature(2013).

VOLUME 74, NUMBER 9 PH YSICAL REVIEW LETTERS 27 FEBRUARY 1995

1 bar

$ P -24 kba3r

1'.:(b)

10

o 10Cl

10

33 kb10

E10

1053 kb

60 kbar66 kbar

, I

10T(K)

100

10

0.0 0.21/T(K)

0.4

FIG. 1. The electrical resistivity p as a function of tem-perature (a) and inverse temperature (b). (b) Q = 1 bar,Q = 24 kbar, = 25 kbar, = 33 kbar, A = 45 kbar, andA = 53 kbar. The solid lines in (b) are fits by the function[p(T)] ' = [po(P)] ' + (p„,(P) exp[A(P)/k&T]) ', describedin the text.

403020~ &00~10

10E, ~9

~o ~e~~1010

20 40P (kbar)

(a).

(b)I60

FIG. 2. The pressure dependences of the activation gap 5 (a)and residual carrier density no = I/R (T =H0) (b). Dashed lineindicates approximate pressure for disappearance of A. Solidlines are guides for the eye.

linearly -0.5 K/kbar from its ambient pressure value of41 K. Above 45 kbar, the resistivity is metallic and it isno longer possible to extract an activation gap.Our measurements indicate a gap instability at a critical

pressure P,. between 45 and 53 kbar, in disagreement withthe conclusions of previous workers [5,6], who foundthat 5 vanished continuously near 60 kbar. In one ofthese studies [5] the sample was of demonstrably lowerquality than our own, with a significantly smaller ambientpressure 6 = 33 K and a much smaller po —10 mA cm,both symptomatic of Sm vacancies or defects introducedin powdering [8]. Our measurements suggest that the gapinstability is a feature only of the highest quality samples,as P,. increases markedly with reduced sample quality,passing out of our experimental pressure window of180 kbar for po ~ 0.1 A cm. We further believe that thesimple activation fits used to determine 6 in both earlierexperiments were overly weighted by the temperatureindependent resistivity below -3.5 K, particularly nearP, . Figure 1(b) demonstrates that near P, the range

101 bar

10

10

24 kbare

~ Q5 kbar

(310E 33 kbar

10-

10-45 kbar

B3 kbar60 kbar

-4 '5 66 kbar10 I I ( I

0.0 0.2 0.4 0.6 0.8 1.0&/r (K')

FIG. 3. The absolute value of the Hall constant RH of SmB6as a function of inverse temperature.

of temperatures over which simple activation fits arelinear becomes increasingly limited and problematic todefine with increased pressure. In contrast, our parallelresistor formulation provides uniformly good fits over thispressure range, and consequently yield a more accuratedetermination of A.Since there is no evidence in SmB6 for a discontinuous

structural change at or below 60 kbar [9], the sudden dis-appearance of 5 suggests that it is not a simple hybridiza-tion gap, for in that case the insulator-metal transitionoccurs by band crossing and the gap is suppressed con-tinuously to zero. A valence instability can be similarlydiscounted, as high pressure x-ray absorption measure-ments [10] find that the Sm valence increases smoothlyfrom +2.6 to +2.75 between 1 bar and 60 kbar.We have used Hall effect measurements to study the

evolution of the camers in the vicinity of P, The Hallconstant RH is plotted as a function of 1/T in Fig. 3 forpressures ranging from 1 bar to 66 kbar. We find thatRH is negative for temperatures T between 1.2 and 40 Kand at all pressures, as well as independent of magneticfields as large as 18 T. As has been previously noted at1 bar [11], RH is both large and extremely temperaturedependent with a maximum at 4 K, at each pressurebecoming temperature independent below -3 K. It hasbeen proposed [12] that this temperature dependencefor RH is characteristic of Kondo lattices, rejectinga crossover from high temperature incoherent to lowtemperature coherent skew scattering. However, similarmaxima in RH(T) occur in doped semiconductors as in-gap impurity states dominate intrinsic activated processeswith reduced temperature [13].We do not address the full temperature dependence

of RH here, instead limiting our discussion to the

1630

Persistent  conduc-vity  PlateauCooleyet al (1995)

Kondo Insulators

“Kondo Lattice”Entangled many body state of spins and electrons gives rise to many new kinds of order.

Sm2.7+

B6

SmB6

1968

Kondo Insulators

Broken Kondo Singlets

Topological(Dzero,Sun,PC,Galitski (2010)Wolgast et al (2013).

Collin Broholm: what happens to the correlations between d- and!f-electrons as topological Kondo insulator develops?

11

FIG. 3: Fermi surface and dispersion maps of SmB6. (a) Fermi surface plot of SmB6

measured by 7 eV LASER source at temperature of 7 K. A small � pocket and a large X pocket

are observed. A big elliptical and a small circular shaped black dash lines around X and � points

are guide for the eyes. Inset shows a schematic plot of Fermi surface in the first Brillouin zone. (b)

Electronic dispersion map (left) and its energy distribution curves (EDCs) for � pocket. (c) same

as (b) for X band. (d) Comparison of integrated EDC for � and X band. A gap value of about 15

meV is observed in both cases.

Neupane,  arXiv:  1306.4634,    Nature(2013).

VOLUME 74, NUMBER 9 PH YSICAL REVIEW LETTERS 27 FEBRUARY 1995

1 bar

$ P -24 kba3r

1'.:(b)

10

o 10Cl

10

33 kb10

E10

1053 kb

60 kbar66 kbar

, I

10T(K)

100

10

0.0 0.21/T(K)

0.4

FIG. 1. The electrical resistivity p as a function of tem-perature (a) and inverse temperature (b). (b) Q = 1 bar,Q = 24 kbar, = 25 kbar, = 33 kbar, A = 45 kbar, andA = 53 kbar. The solid lines in (b) are fits by the function[p(T)] ' = [po(P)] ' + (p„,(P) exp[A(P)/k&T]) ', describedin the text.

403020~ &00~10

10E, ~9

~o ~e~~1010

20 40P (kbar)

(a).

(b)I60

FIG. 2. The pressure dependences of the activation gap 5 (a)and residual carrier density no = I/R (T =H0) (b). Dashed lineindicates approximate pressure for disappearance of A. Solidlines are guides for the eye.

linearly -0.5 K/kbar from its ambient pressure value of41 K. Above 45 kbar, the resistivity is metallic and it isno longer possible to extract an activation gap.Our measurements indicate a gap instability at a critical

pressure P,. between 45 and 53 kbar, in disagreement withthe conclusions of previous workers [5,6], who foundthat 5 vanished continuously near 60 kbar. In one ofthese studies [5] the sample was of demonstrably lowerquality than our own, with a significantly smaller ambientpressure 6 = 33 K and a much smaller po —10 mA cm,both symptomatic of Sm vacancies or defects introducedin powdering [8]. Our measurements suggest that the gapinstability is a feature only of the highest quality samples,as P,. increases markedly with reduced sample quality,passing out of our experimental pressure window of180 kbar for po ~ 0.1 A cm. We further believe that thesimple activation fits used to determine 6 in both earlierexperiments were overly weighted by the temperatureindependent resistivity below -3.5 K, particularly nearP, . Figure 1(b) demonstrates that near P, the range

101 bar

10

10

24 kbare

~ Q5 kbar

(310E 33 kbar

10-

10-45 kbar

B3 kbar60 kbar

-4 '5 66 kbar10 I I ( I

0.0 0.2 0.4 0.6 0.8 1.0&/r (K')

FIG. 3. The absolute value of the Hall constant RH of SmB6as a function of inverse temperature.

of temperatures over which simple activation fits arelinear becomes increasingly limited and problematic todefine with increased pressure. In contrast, our parallelresistor formulation provides uniformly good fits over thispressure range, and consequently yield a more accuratedetermination of A.Since there is no evidence in SmB6 for a discontinuous

structural change at or below 60 kbar [9], the sudden dis-appearance of 5 suggests that it is not a simple hybridiza-tion gap, for in that case the insulator-metal transitionoccurs by band crossing and the gap is suppressed con-tinuously to zero. A valence instability can be similarlydiscounted, as high pressure x-ray absorption measure-ments [10] find that the Sm valence increases smoothlyfrom +2.6 to +2.75 between 1 bar and 60 kbar.We have used Hall effect measurements to study the

evolution of the camers in the vicinity of P, The Hallconstant RH is plotted as a function of 1/T in Fig. 3 forpressures ranging from 1 bar to 66 kbar. We find thatRH is negative for temperatures T between 1.2 and 40 Kand at all pressures, as well as independent of magneticfields as large as 18 T. As has been previously noted at1 bar [11], RH is both large and extremely temperaturedependent with a maximum at 4 K, at each pressurebecoming temperature independent below -3 K. It hasbeen proposed [12] that this temperature dependencefor RH is characteristic of Kondo lattices, rejectinga crossover from high temperature incoherent to lowtemperature coherent skew scattering. However, similarmaxima in RH(T) occur in doped semiconductors as in-gap impurity states dominate intrinsic activated processeswith reduced temperature [13].We do not address the full temperature dependence

of RH here, instead limiting our discussion to the

1630

Persistent  conduc-vity  PlateauCooleyet al (1995)

Collin Broholm: what collective excitations result from the excitonic Kondo lattice?

Heavy Fermion Metals

“Kondo Lattice”Entangled many body state of spins and electrons gives rise to many new kinds of order.

Kondo InsulatorsTopological(Dzero,Sun,PC,Galitski (2010)Wolgast et al (2013).

Ott et al (1976)

Heavy Fermion Metals:

Heavy Fermion Metals

“Kondo Lattice”Entangled many body state of spins and electrons gives rise to many new kinds of order.

Kondo InsulatorsTopological(Dzero,Sun,PC,Galitski (2010)Wolgast et al (2013).

Ott et al (1976)

Heavy Fermion Metals:

Coherent HFs~Hole doped KIs

~TK

Heavy Fermion Metals

“Kondo Lattice”Entangled many body state of spins and electrons gives rise to many new kinds of order.

Kondo InsulatorsTopological(Dzero,Sun,PC,Galitski (2010)Wolgast et al (2013).

�(T ) = �0 + AT 2

Ott et al (1976)

Heavy Fermion Metals:

Coherent HFs~Hole doped KIs

~TK

Heavy Fermion Metals

“Kondo Lattice”Entangled many body state of spins and electrons gives rise to many new kinds of order.

Kondo InsulatorsTopological(Dzero,Sun,PC,Galitski (2010)Wolgast et al (2013).

�(T ) = �0 + AT 2

Ott et al (1976)

Heavy Fermion Metals:

Coherent HFs~Hole doped KIs

~TK

Singlet formation

J

Ncj�S�⇥ ⌘ V̄jfj⇥

Composite Fermion

Heavy Fermion Metals

“Kondo Lattice”Entangled many body state of spins and electrons gives rise to many new kinds of order.

Kondo InsulatorsTopological(Dzero,Sun,PC,Galitski (2010)Wolgast et al (2013).

�(T ) = �0 + AT 2

Ott et al (1976)

Heavy Fermion Metals:

Coherent HFs~Hole doped KIs

~TK

Singlet formation

q [2π/a,0]

Ener

gy [m

eV]

0.15 0.25 0.35 0.45 0.55 0.65

60

40

20

0

-20

-40

-60

Ali Yazdani: spectroscopically !image the composite HF?

J

Ncj�S�⇥ ⌘ V̄jfj⇥

Composite Fermion

Heavy Fermion Metals

“Kondo Lattice”Entangled many body state of spins and electrons gives rise to many new kinds of order.

Kondo InsulatorsTopological(Dzero,Sun,PC,Galitski (2010)Wolgast et al (2013).

�(T ) = �0 + AT 2

Ott et al (1976)

Heavy Fermion Metals:

Coherent HFs~Hole doped KIs

~TK

Singlet formation

q [2π/a,0]

Ener

gy [m

eV]

0.15 0.25 0.35 0.45 0.55 0.65

60

40

20

0

-20

-40

-60

Ali Yazdani: spectroscopically !image the composite HF?

J

Ncj�S�⇥ ⌘ V̄jfj⇥

Composite Fermion

Ali Yazdani: and what happens to them in a HF SC?

>

TK = D exp[�1/2J�]RKKY InteractionQuantum Criticality & SC Mott 1974, Doniach, 1977

>

TK = D exp[�1/2J�]RKKY InteractionQuantum Criticality & SC

TN ⇠ J2⇢<latexit sha1_base64="v6V64Ji4wp93YRLiwAZDuqArDCE=">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</latexit><latexit sha1_base64="v6V64Ji4wp93YRLiwAZDuqArDCE=">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</latexit>

Mott 1974, Doniach, 1977

>

TK = D exp[�1/2J�]RKKY InteractionQuantum Criticality & SC

TN ⇠ J2⇢<latexit sha1_base64="v6V64Ji4wp93YRLiwAZDuqArDCE=">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</latexit><latexit sha1_base64="v6V64Ji4wp93YRLiwAZDuqArDCE=">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</latexit>

TK ⇠ De�1/(2J⇢)<latexit sha1_base64="HYyQQZ/6g3A+hNOVbqRuFtUcsig=">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</latexit><latexit sha1_base64="HYyQQZ/6g3A+hNOVbqRuFtUcsig=">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</latexit>

Mott 1974, Doniach, 1977

>

TK = D exp[�1/2J�]

QCP

RKKY InteractionQuantum Criticality & SC

TN ⇠ J2⇢<latexit sha1_base64="v6V64Ji4wp93YRLiwAZDuqArDCE=">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</latexit><latexit sha1_base64="v6V64Ji4wp93YRLiwAZDuqArDCE=">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</latexit>

TK ⇠ De�1/(2J⇢)<latexit sha1_base64="HYyQQZ/6g3A+hNOVbqRuFtUcsig=">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</latexit><latexit sha1_base64="HYyQQZ/6g3A+hNOVbqRuFtUcsig=">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</latexit>

Mott 1974, Doniach, 1977

>

TK = D exp[�1/2J�]

QCP

1-­‐1-­‐5  Materials

RKKY Interaction

Joe Thompson

Quantum Criticality & SC

TN ⇠ J2⇢<latexit sha1_base64="v6V64Ji4wp93YRLiwAZDuqArDCE=">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</latexit><latexit sha1_base64="v6V64Ji4wp93YRLiwAZDuqArDCE=">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</latexit>

TK ⇠ De�1/(2J⇢)<latexit sha1_base64="HYyQQZ/6g3A+hNOVbqRuFtUcsig=">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</latexit><latexit sha1_base64="HYyQQZ/6g3A+hNOVbqRuFtUcsig=">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</latexit>

Mott 1974, Doniach, 1977

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TK = D exp[�1/2J�]

QCP

1-­‐1-­‐5  Materials

RKKY Interaction

Joe Thompson

Thompson: Linear Resistance due to magnetic quantum Criticality? Tanatar et al, Science 2007

WF law violation are all related, and (ii) a goodindicator for their joint occurrence is a linear-Tresistivity. Returning to our comparison withcuprates, a similar connection between r ~ Tand Z = 0 appears to exist there as well. Indeed, arecent measurement of the (azimuthal) anisotropyof the in-plane scattering rateG(f) in an overdopedcuprate (24) revealed that G ~ Tat f = 0, where theFermi surface is eventually destroyed (at lowerdoping), and G~T 2 at f = p/4, where it survives.

It is instructive to compare our findings withthe properties of other materials and theories ofquantum criticality. AT3/2 resistivity is observed inCeIn3 near the pressure-tuned QCP where its AForder vanishes (6). CeIn3 is the cubic parentcompound of tetragonal CeRhIn5 and, along withthe increase in c/a ratio, the ordering temperaturedrops from TN = 10 K in the former to TN = 3.8 Kin the latter. However, they still have comparableTSF (assuming that in CeIn3 TSF ≅ TN). CeCoIn5encounters a further stretch of the c/a ratio, andlong-range AF order is no longer stabilized.However, it can still be viewed as a layered ver-sion of CeIn3, with similar in-plane correlationsand scattering. In this sense, the T3/2 dependenceobserved in CeCoIn5 can be viewed as the result ofantiferromagnetic fluctuations that are character-istic of the parent compound. Theoretically, a T3/2

resistivity is expected for AF critical fluctuations in3D from the so-called quantum spin density wave(SDW)model (17, 25, 26). In this scenario, criticalscattering is peaked at “hot spots” connected by theAFwave vectors (25). As T→0, one would expectthe Fermi surface to remain sharp everywhere else,and thus the WF law to prevail, as found here forin-plane currents.

A T-linear resistivity is observed at thecomposition-tuned QCP of CeCu5.9Au0.1 (7)and field-tuned QCP of YbRh2Si2 (12), whereAF order is thought to disappear. [In these cases,the power law is linear in both high-symmetry

directions (15, 27).] The fact that a linear power isinconsistent with the SDW model for AF fluc-tuations in 3D prompted the proposal of a 2Dversion (28) and of an alternate theory, wherecritical scattering is local in space and thereforepresent at all wave vectors (29). These scenarioswould lead to a more extreme breakdown of FLtheory, because the Fermi surface is “hot” notonly at certain specific spots but everywhere. Itwas argued in (8) that the specific heat data onGe-doped YbRh2Si2, which shows a C/T thatexceeds the log(1/T) dependence at low temper-ature, may be an indication of such enhancedbreakdown. In CeCoIn5, the fact that it is in thedirection where r ~ T that the WF law is violatedis certainly consistent with this picture. Clearly, itwould be interesting to test the WF law inYbRh2Si2.

Bringing together our findings for T→0 andT > 0, a picture of qualitative anisotropy emerges,not present in either the SDW model or the localcriticality model, at least in their current forms.The characteristic spin fluctuation temperatureTSF vanishes at the QCP for transport along the caxis but not in the plane. As a result, the break-down of FL theory is extreme in the c direction:rc ~ T and wc ~ T down to the lowest temper-atures and the T = 0 Fermi surface is blurred, thatis, the quasiparticle Z parameter vanishes, in re-gions around the c-axis direction.

A possible origin for this anisotropic critical-ity is an anisotropic spin fluctuation spectrum.First, an AF instability is present in all threeCeMIn5 compounds (M = Co, Rh, Ir), as shownby the fact that magnetic ordering can be inducedby Cd doping (30). Second, a magnetic field doestune the magnetism. In CeRhIn5 under pressure(where it becomes in many ways more similar toCeCoIn5, e.g., by developing superconductivitywith the same Tc), a magnetic field stabilizeslong-range magnetic order (31, 32). In CeCoIn5,it is the magnetic fluctuations that are tuned by amagnetic field (21), with TSF starting at a valueequivalent to that of CeRhIn5 at high fields andthen lowered to a minimum at Hc. Third, the AFfluctuations in CeCoIn5 have strongly anisotrop-ic character (33), with magnetic moments wellcoupled in-plane but weakly coupled interplane.This is consistent with the helical ordering ofmoments in CeRhIn5, commensurate in-planeand incommensurate along the c axis. Therefore,it seems natural to link this uniaxial anisotropywith the observed anisotropy in TSF, power laws,and Z(q). What is not yet known is whether ascenario of AF critical fluctuations can indeedcause a violation of the WF law at T→0.

However, the AF scenario is not the onlycandidate for the anisotropic quantum criticalityof CeCoIn5. The “Kondo breakdown”model pro-posed recently (23, 34), a type of deconfinedQCPwhere the hybridization between conductionand f electrons goes to zero, captures some of thekey signatures—absence of magnetic order in thephase diagram, strong anisotropy, multiple ener-gy scales, and a T-linear behavior of both charge

and heat resistivities. Proximity to a Pomeranchukinstability of the Fermi surface can also causeanisotropy in electronic liquids (7). Recent cal-culations show that the transport decay rate atsuch a QCP has a linear T dependence every-where on the Fermi surface except at “cold”points, resulting in a T3/2 dependence of theresistivity (35).

References and Notes1. G. Wiedemann, R. Franz, Ann. Phys. 89, 497 (1853).2. A. Sommerfeld, Naturwissenschaften 15, 825 (1927).3. L. D. Landau, Sov. Phys. JETP 3, 920 (1957).4. Additional data, analysis, and discussion are available as

supporting material on Science Online.5. L. G. C. Rego, G. Kirczenow, Phys. Rev. B 59, 13080 (1999).6. P. Coleman, A. J. Schofield, Nature 433, 226 (2005).7. H. von Löhneysen et al., Phys. Rev. Lett. 72, 3262 (1994).8. J. Custers et al., Nature 424, 524 (2003).9. N. D. Mathur et al., Nature 394, 39 (1998).

10. S. S. Saxena et al., Nature 406, 587 (2000).11. F. Levy, I. Sheikin, B. Grenier, A. D. Huxley, Science 309,

1343 (2005).12. R. A. Borzi et al., Science 315, 214 (2007).13. J. Paglione et al., Phys. Rev. Lett. 91, 246405 (2003).14. A. Bianchi et al., Phys. Rev. Lett. 91, 257001 (2003).15. P. Gegenwart et al., Phys. Rev. Lett. 89, 056402

(2002).16. P. Coleman, J. B. Marston, A. J. Schofield, Phys. Rev. B

72, 245111 (2005).17. P. Coleman, C. Pépin, Q. Si, R. Ramazashvili, J. Phys.

Cond. Mat. 13, R723 (2001).18. A. McCollam, S. R. Julian, P. M. C. Rourke, D. Aoki,

J. Flouquet, Phys. Rev. Lett. 94, 186401 (2005).19. M. R. Norman et al., Nature 392, 157 (1998).20. A. Kanigel et al., Nat. Phys. 2, 447 (2006).21. J. Paglione et al., Phys. Rev. Lett. 97, 106606 (2006).22. J. Paglione et al., Phys. Rev. Lett. 94, 216602 (2005).23. I. Paul, C. Pépin, M. R. Norman, Phys. Rev. Lett. 98,

026402 (2007).24. M. Abdel-Jawad et al., Nat. Phys. 2, 821 (2006).25. A. J. Millis, Phys. Rev. B 48, 7183 (1993).26. T. Moriya, T. Takimoto, J. Phys. Soc. Jpn. 64, 960 (1995).27. A. Neubert et al., Physica B (Amsterdam) 230, 587 (1997).28. A. Rosch, A. Schroder, O. Stockert, H. von Lohneysen,

Phys. Rev. Lett. 79, 159 (1997).29. Q. Si, S. Rabello, K. Ingersent, J. L. Smith, Nature 413,

804 (2001).30. L. D. Pham, T. Park, S. Maquilon, J. D. Thompson, Z. Fisk,

Phys. Rev. Lett. 97, 056404 (2006).31. T. Park et al., Nature 440, 65 (2006).32. G. Knebel, D. Aoki, D. Braithwaite, B. Salce, J. Flouquet,

Phys. Rev. B 74, 020501 (2006).33. Y. Kawasaki et al., J. Phys. Soc. Jpn. 72, 2308 (2003).34. T. Senthil, M. Vojta, S. Sachdev, Phys. Rev. B 69, 035111

(2004).35. L. Dell’Anna, W. Metzner, Phys. Rev. Lett. 98, 136402

(2007).36. We thank D. G. Hawthorn, R. W. Hill, F. Ronning, and

M. Sutherland for experimental assistance, and P. C. Canfield,Y. B. Kim, C. Pépin, A. M. Tremblay, A. Rosch, and M. F.Smith for useful discussions. L.T. acknowledges support fromthe Canadian Institute for Advanced Research, a CanadaResearch Chair, the Natural Sciences and EngineeringResearch Council of Canada, the Canada Foundation forInnovation, and Le Fonds Québécois de Recherche sur laNature et la Technologie. Part of this research was carried outat the Brookhaven National Laboratory, which is operated forthe U.S. Department of Energy by Brookhaven ScienceAssociates (DE-AC02-98CH10886).

Supporting Online Materialwww.sciencemag.org/cgi/content/full/316/5829/1320/DC1Materials and MethodsFigs. S1 to S11References

2 February 2007; accepted 19 April 200710.1126/science.1140762

Fig. 3. Anisotropic quantum criticality. Electricalresistivity at the QCP (at H = 5.3, T ≅ Hc) for in-plane (ra) and inter-plane (rc) current directions.rc (T) remains linear over a 100-fold increase inmagnitude. By contrast, ra is linear only above acharacteristic fluctuation temperature TSF ≅ 4 K(arrow) (18). (Inset) Thermal resistivity (wc ≡ L0T/kc)at the QCP, for inter-plane transport. wc is perfectlylinear down to the lowest temperature.

1 JUNE 2007 VOL 316 SCIENCE www.sciencemag.org1322

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Quantum Criticality & SC

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TK ⇠ De�1/(2J⇢)<latexit sha1_base64="HYyQQZ/6g3A+hNOVbqRuFtUcsig=">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</latexit><latexit sha1_base64="HYyQQZ/6g3A+hNOVbqRuFtUcsig=">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</latexit>

Mott 1974, Doniach, 1977

>

TK = D exp[�1/2J�]

QCP

1-­‐1-­‐5  Materials

RKKY Interaction

Joe Thompson

Thompson: Linear Resistance due to magnetic quantum Criticality? Tanatar et al, Science 2007

WF law violation are all related, and (ii) a goodindicator for their joint occurrence is a linear-Tresistivity. Returning to our comparison withcuprates, a similar connection between r ~ Tand Z = 0 appears to exist there as well. Indeed, arecent measurement of the (azimuthal) anisotropyof the in-plane scattering rateG(f) in an overdopedcuprate (24) revealed that G ~ Tat f = 0, where theFermi surface is eventually destroyed (at lowerdoping), and G~T 2 at f = p/4, where it survives.

It is instructive to compare our findings withthe properties of other materials and theories ofquantum criticality. AT3/2 resistivity is observed inCeIn3 near the pressure-tuned QCP where its AForder vanishes (6). CeIn3 is the cubic parentcompound of tetragonal CeRhIn5 and, along withthe increase in c/a ratio, the ordering temperaturedrops from TN = 10 K in the former to TN = 3.8 Kin the latter. However, they still have comparableTSF (assuming that in CeIn3 TSF ≅ TN). CeCoIn5encounters a further stretch of the c/a ratio, andlong-range AF order is no longer stabilized.However, it can still be viewed as a layered ver-sion of CeIn3, with similar in-plane correlationsand scattering. In this sense, the T3/2 dependenceobserved in CeCoIn5 can be viewed as the result ofantiferromagnetic fluctuations that are character-istic of the parent compound. Theoretically, a T3/2

resistivity is expected for AF critical fluctuations in3D from the so-called quantum spin density wave(SDW)model (17, 25, 26). In this scenario, criticalscattering is peaked at “hot spots” connected by theAFwave vectors (25). As T→0, one would expectthe Fermi surface to remain sharp everywhere else,and thus the WF law to prevail, as found here forin-plane currents.

A T-linear resistivity is observed at thecomposition-tuned QCP of CeCu5.9Au0.1 (7)and field-tuned QCP of YbRh2Si2 (12), whereAF order is thought to disappear. [In these cases,the power law is linear in both high-symmetry

directions (15, 27).] The fact that a linear power isinconsistent with the SDW model for AF fluc-tuations in 3D prompted the proposal of a 2Dversion (28) and of an alternate theory, wherecritical scattering is local in space and thereforepresent at all wave vectors (29). These scenarioswould lead to a more extreme breakdown of FLtheory, because the Fermi surface is “hot” notonly at certain specific spots but everywhere. Itwas argued in (8) that the specific heat data onGe-doped YbRh2Si2, which shows a C/T thatexceeds the log(1/T) dependence at low temper-ature, may be an indication of such enhancedbreakdown. In CeCoIn5, the fact that it is in thedirection where r ~ T that the WF law is violatedis certainly consistent with this picture. Clearly, itwould be interesting to test the WF law inYbRh2Si2.

Bringing together our findings for T→0 andT > 0, a picture of qualitative anisotropy emerges,not present in either the SDW model or the localcriticality model, at least in their current forms.The characteristic spin fluctuation temperatureTSF vanishes at the QCP for transport along the caxis but not in the plane. As a result, the break-down of FL theory is extreme in the c direction:rc ~ T and wc ~ T down to the lowest temper-atures and the T = 0 Fermi surface is blurred, thatis, the quasiparticle Z parameter vanishes, in re-gions around the c-axis direction.

A possible origin for this anisotropic critical-ity is an anisotropic spin fluctuation spectrum.First, an AF instability is present in all threeCeMIn5 compounds (M = Co, Rh, Ir), as shownby the fact that magnetic ordering can be inducedby Cd doping (30). Second, a magnetic field doestune the magnetism. In CeRhIn5 under pressure(where it becomes in many ways more similar toCeCoIn5, e.g., by developing superconductivitywith the same Tc), a magnetic field stabilizeslong-range magnetic order (31, 32). In CeCoIn5,it is the magnetic fluctuations that are tuned by amagnetic field (21), with TSF starting at a valueequivalent to that of CeRhIn5 at high fields andthen lowered to a minimum at Hc. Third, the AFfluctuations in CeCoIn5 have strongly anisotrop-ic character (33), with magnetic moments wellcoupled in-plane but weakly coupled interplane.This is consistent with the helical ordering ofmoments in CeRhIn5, commensurate in-planeand incommensurate along the c axis. Therefore,it seems natural to link this uniaxial anisotropywith the observed anisotropy in TSF, power laws,and Z(q). What is not yet known is whether ascenario of AF critical fluctuations can indeedcause a violation of the WF law at T→0.

However, the AF scenario is not the onlycandidate for the anisotropic quantum criticalityof CeCoIn5. The “Kondo breakdown”model pro-posed recently (23, 34), a type of deconfinedQCPwhere the hybridization between conductionand f electrons goes to zero, captures some of thekey signatures—absence of magnetic order in thephase diagram, strong anisotropy, multiple ener-gy scales, and a T-linear behavior of both charge

and heat resistivities. Proximity to a Pomeranchukinstability of the Fermi surface can also causeanisotropy in electronic liquids (7). Recent cal-culations show that the transport decay rate atsuch a QCP has a linear T dependence every-where on the Fermi surface except at “cold”points, resulting in a T3/2 dependence of theresistivity (35).

References and Notes1. G. Wiedemann, R. Franz, Ann. Phys. 89, 497 (1853).2. A. Sommerfeld, Naturwissenschaften 15, 825 (1927).3. L. D. Landau, Sov. Phys. JETP 3, 920 (1957).4. Additional data, analysis, and discussion are available as

supporting material on Science Online.5. L. G. C. Rego, G. Kirczenow, Phys. Rev. B 59, 13080 (1999).6. P. Coleman, A. J. Schofield, Nature 433, 226 (2005).7. H. von Löhneysen et al., Phys. Rev. Lett. 72, 3262 (1994).8. J. Custers et al., Nature 424, 524 (2003).9. N. D. Mathur et al., Nature 394, 39 (1998).

10. S. S. Saxena et al., Nature 406, 587 (2000).11. F. Levy, I. Sheikin, B. Grenier, A. D. Huxley, Science 309,

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(2007).36. We thank D. G. Hawthorn, R. W. Hill, F. Ronning, and

M. Sutherland for experimental assistance, and P. C. Canfield,Y. B. Kim, C. Pépin, A. M. Tremblay, A. Rosch, and M. F.Smith for useful discussions. L.T. acknowledges support fromthe Canadian Institute for Advanced Research, a CanadaResearch Chair, the Natural Sciences and EngineeringResearch Council of Canada, the Canada Foundation forInnovation, and Le Fonds Québécois de Recherche sur laNature et la Technologie. Part of this research was carried outat the Brookhaven National Laboratory, which is operated forthe U.S. Department of Energy by Brookhaven ScienceAssociates (DE-AC02-98CH10886).

Supporting Online Materialwww.sciencemag.org/cgi/content/full/316/5829/1320/DC1Materials and MethodsFigs. S1 to S11References

2 February 2007; accepted 19 April 200710.1126/science.1140762

Fig. 3. Anisotropic quantum criticality. Electricalresistivity at the QCP (at H = 5.3, T ≅ Hc) for in-plane (ra) and inter-plane (rc) current directions.rc (T) remains linear over a 100-fold increase inmagnitude. By contrast, ra is linear only above acharacteristic fluctuation temperature TSF ≅ 4 K(arrow) (18). (Inset) Thermal resistivity (wc ≡ L0T/kc)at the QCP, for inter-plane transport. wc is perfectlylinear down to the lowest temperature.

1 JUNE 2007 VOL 316 SCIENCE www.sciencemag.org1322

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Thompson: How does a local moment time-share between SC and AFM?

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Strange Metals.

B

T

-dM/dT

Superconductivity

QCP

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Yosuke Matsumoto

Strange Metals.

B

T

-dM/dT

Superconductivity

QCP

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Y. Matsumoto et al, Science (2011)

Yosuke Matsumoto Bc < 0.1mT

Critical field, if it exists, is no bigger than the earth’s magnetic field (0.06mT)

Strange Metals.

B

T

-dM/dT

Superconductivity

QCP

�-YbAlB4

Y. Matsumoto et al, Science (2011)

Yosuke Matsumoto Bc < 0.1mT

Critical field, if it exists, is no bigger than the earth’s magnetic field (0.06mT)

Matsumoto: Strange metal phase or QCP?

Strange Metals.

B

T

-dM/dT

Superconductivity

QCP

�-YbAlB4

Y. Matsumoto et al, Science (2011)

Yosuke Matsumoto Bc < 0.1mT

Critical field, if it exists, is no bigger than the earth’s magnetic field (0.06mT)

Matsumoto: Strange metal phase or QCP?

Failed Kondo insulator? N(0) ⇠ 1pB

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Strange Metals.

B

T

-dM/dT

Superconductivity

QCP

�-YbAlB4

Y. Matsumoto et al, Science (2011)

Yosuke Matsumoto Bc < 0.1mT

Critical field, if it exists, is no bigger than the earth’s magnetic field (0.06mT)

Matsumoto: Strange metal phase or QCP?

Failed Kondo insulator? N(0) ⇠ 1pB

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V(k)~ (kx±iky)2

Double vortex in momentum space

MJMs=1/2ML=2

Aline Ramires et al, PRL (2012)

Strange Metals.

B

T

-dM/dT

Superconductivity

QCP

�-YbAlB4

Y. Matsumoto et al, Science (2011)

Yosuke Matsumoto Bc < 0.1mT

Critical field, if it exists, is no bigger than the earth’s magnetic field (0.06mT)

Matsumoto: Strange metal phase or QCP?

Failed Kondo insulator? N(0) ⇠ 1pB

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V(k)~ (kx±iky)2

Double vortex in momentum space

MJMs=1/2ML=2

Aline Ramires et al, PRL (2012)

Matsumoto: Can Topology play a role !in strange metals?

Matsumoto: Strange metal phase or quantum critical point? Matsumoto: Can Topology play a role in strange metals?

Matsumoto: Strange metal phase or quantum critical point? Matsumoto: Can Topology play a role in strange metals?

Broholm: what happens to the correlations between d- and!f-electrons as topological Kondo insulator develops?

Broholm: what collective excitations result from the excitonic Kondo lattice?

Matsumoto: Strange metal phase or quantum critical point? Matsumoto: Can Topology play a role in strange metals?

Thompson: Linear Resistance due to magnetic quantum Criticality? Thompson: How does a local moment time-share between SC and antiferromagnetism?

Broholm: what happens to the correlations between d- and!f-electrons as topological Kondo insulator develops?

Broholm: what collective excitations result from the excitonic Kondo lattice?

Matsumoto: Strange metal phase or quantum critical point? Matsumoto: Can Topology play a role in strange metals?

Thompson: Linear Resistance due to magnetic quantum Criticality? Thompson: How does a local moment time-share between SC and antiferromagnetism?

Broholm: what happens to the correlations between d- and!f-electrons as topological Kondo insulator develops?

Broholm: what collective excitations result from the excitonic Kondo lattice?

Yazdani: can one spectroscopically image the composite HF?

Yazdani: and what happens to the composite HF in a HF SC?