Unadulterated spectral function of low-energy quasiparticles
Beyond Quasiparticles: New Paradigms for Quantum Fluids Aspen...
Transcript of Beyond Quasiparticles: New Paradigms for Quantum Fluids Aspen...
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
>
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
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
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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Quantum Criticality & SC Tusan Park et al, Nature (2005)
Thompson: How does a local moment time-share between SC and AFM?
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Mott 1974, Doniach, 1977
Strange Metals.
B
T
-dM/dT
Superconductivity
QCP
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Yosuke Matsumoto
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)
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
<latexit sha1_base64="CzhdWIEZ5FADMACFFUc2hPJb/sM=">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</latexit><latexit sha1_base64="CzhdWIEZ5FADMACFFUc2hPJb/sM=">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</latexit>
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
<latexit sha1_base64="CzhdWIEZ5FADMACFFUc2hPJb/sM=">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</latexit><latexit sha1_base64="CzhdWIEZ5FADMACFFUc2hPJb/sM=">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</latexit>
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?