heavy fermion physics: a 21st century julich, perspective · 2015. 9. 22. · piers coleman:...

Julich, 21 Sept, 2015

Piers Coleman: Rutgers Center for Materials Theory, USA

Heavy Fermion Physics: a 21st Century perspective

Quantum Criticality & Strange Metals

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.

Topological Kondo Insulators

CeCoIn5':'2.5'K'Superconductor

Heavy Fermion Superconductivity

2

FIG. 1: a. Upper critical field Hc2 of the superconduct-ing state in URu2Si2 determined from the onset of resistiv-ity at ⇡ 30 mK. An example trace is shown in the inset.b. Schematic representation of the angle-dependent magneticquantum oscillations adapted from Fig. 18 of reference [22],with the indices of the spin zeroes indicated. In order to showthe oscillatory behavior, the sign of the amplitude is negatedon crossing each spin zero.

fermion condensate [20] for all orientations of the mag-netic field � the exception being a narrow range of angleswithin ⇠ 10� of the [100] axis in Fig. 2 (likely associatedwith the dominant role of diamagnetic screening currentsonce g⇤

e↵

is strongly suppressed [19]).A further key observation is that the field orientation-

dependence of g⇤e↵

in Fig. 2 is very di↵erent from theusual isotropic case of g⇤ ⇡ 2 for band electrons (dottedline), indicating the spin susceptibility of the quasipar-ticles in URu

2

Si2

to di↵er along the two distinct crys-talline axes. Since the Zeeman splitting of the quasi-particles is given by the projection M · H of the spin

magnetizationM = ⇢µ

2B2

(g2a

cos ✓, 0, g2c

sin ✓)H alongH =H(cos ✓, 0, sin ✓) [where ⇢ is the electronic density-of-

states], setting M · H = ⇢µBg

⇤eff

2

H defines an e↵ectiveg-factor

g⇤e↵

=qg2c

sin2 ✓ + g2a

cos2 ✓ (3)

that (in the case of a strong anisotropy) traces the form

FIG. 2: Polar plot of the field orientation-dependence of g⇤e↵estimated using equations (1) and (2) represented by openand closed circles respectively. Also shown, is a fit (solid line)to equation (3) to g⇤e↵ , and the isotropic g⇤ ⇡ 2 (dotted line)expected for conventional band electrons. In Fig. 1a we as-sumeHc2 ⇡ Hp. In extracting g⇤e↵ from the index assignmentsof g⇤e↵(m

⇤/me↵) in Fig. 1b, the weakly angle-dependent m⇤

is interpolated from the measured values in reference [22].

of a figure of ‘8.’ A fit to equation (3) in Fig. 2 (solidline) yields g

c

= 2.65 ± 0.05 and ga

= 0.0 ± 0.1, implying

a large anisotropy in the spin susceptibility �c

�a=

�gc

ga

�2

.

To obtain a lower bound for the anistropy, we plot ge↵

(circles) in Fig. 3 extracted from quantum oscillation ex-periments [22] versus sin ✓ (in the vicinity of the cusp inFig. 2) together with the prediction (lines) for di↵erent

values of �a

�b=

�gc

ga

�2

made using equation (3). The ob-servation of a spin zero in Fig. 1 at angles as small as 3�

implies a lower bound �a

�b& 1000. A smaller anisotropy

would be expected to lead to the observation of fewer spinzeroes and nonlinearity in the plot with an upturn in g

e↵

at small values of sin ✓ (as shown in the simulations).

A large anisotropy in the magnetic susceptibility is thebehavior expected for local magnetic moments of largeangular momenta whose confinement within a crystallattice gives rise to an Ising anisotropy. Kondo cou-pling provides the means by which such an anisotropycan be transferred to itinerant electrons [8]. In the caseof an isolated magnetic impurity (i.e. an isolated mag-netic moment), Kondo singlets can be considered the re-sult of an antiferromagnetic coupling between the impu-rity and conduction electron states expanded as partialwaves of the same angular momenta [26]. A Fermi liquidcomposed of ‘composite heavy quasiparticles’ with heavye↵ective masses and local angular momentum quantumnumbers is one of the anticipated outcomes in a latticeof moments should such partial states overlap and sat-isfy Bloch’s theorem at low temperatures [27, 28], as ap-pears to be the case in URu

2

Si2

. The finding of a largeanisotropic impurity susceptibility (�c

�a⇠ 140) in the di-

URu2Si2

Hidden Order

Piers Coleman: Rutgers Center for Materials Theory, USA

Heavy Fermion Physics: a 21st Century perspective

Collaborators.Q. Si RiceR. Ramazashvili ToulouseC. Pepin CEA, SaclayAline Ramires ETH

Rebecca Flint Iowa StatePremi Chandra Rutgers

Andriy Nevidomskyy RiceAlexei Tsvelik Brookhaven NLHai Young Kee U. TorontoNatan Andrei RutgersOnur Erten Rutgers

Maxim Dzero Kent StateVictor Galitski U. Maryland Kai Sun U. Michigan

3

Experimentalists:

H. von Lohneysen KarlsruheG. Aeppli ETH, Zurich A. Schröder Kent StateS. Nakatsuji ISSPG. Lonzarich CambridgeS. Paschen ViennaJ. Thompson Los AlamosJ. Allen U. MichiganZ. Fisk UC IrvineF Steglich Dresden/Zhejiang

Notes: “Introduction to Many Body Physics”, Ch 8,15-16”, PC, CUP to be published (2015).

“Heavy Fermions: electrons at the edge of magnetism.” Wiley encyclopedia of magnetism. PC. cond-mat/0612006.

“I2CAM-FAPERJ Lectures on Heavy Fermion Physics”, (X=I, II, III) http://physics.rutgers.edu/~coleman/talks/RIO13_X.pdf

General reading:

A. Hewson, “Kondo effect to heavy fermions”, CUP, (1993). “The Theory of Quantum Liquids”, Nozieres and Pines (Perseus 1999).

http://harpsichord.rutgers.edu/~coleman/talks/wurzburg2011.pdf

1. Trends in the periodic table.

2. Introduction: Heavy Fermions and the Kondo Lattice.

3. Kondo Insulators: the simplest heavy fermions.

4. Large N expansion for the Kondo Lattice

5. Heavy Fermion Superconductivity

6. Topological Kondo Insulators

7. Co-existing magnetism and the Kondo Effect.

Outline of the Topics

Please ask questions!

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=h

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.

Smith and Kmetko (1983)

Increasing localization


Rare Earth

Transition metal

Actinide

4f

3d




�S=h

2�σ












Mott Mechanism.Anderson U (Anderson 1959)

• No double occupancy: strongly correlated• Residual valence fluctuations induce AFM Superexchange.

U

Smith and Kmetko (1983)



Many things are possible at the brink of magnetism.

Rare Earth

Transition metal

Actinide

4f

3d




�S=h

2�σ












Mott Mechanism.Anderson U (Anderson 1959)

U

Diversity of new ground-states on the brink of localization.

f-electron systems: 4f Ce, Yb systems 5f U, Np, Pu systems.

Cuprates Tc=11-92K

T(K)0 200 400 600 800 1000

0.0

0.2

0.4

0.6

0.8

1.0

Takagi et al, PRL ‘92

PrFxO1-xFeAs Z.A. Ren et.al, Beijing, (08)

Iron based sc Tc= 6 - 53 ++ ? K

HF 115s Tc=0.2 -18.5 K

Local

NpAl2Pd5

d-electron systems: e.g Pnictides, Cuprate SC.

Diversity of new ground-states on the brink of localization.

f-electron systems: 4f Ce, Yb systems 5f U, Np, Pu systems.

Cuprates Tc=11-92K

T(K)0 200 400 600 800 1000

0.0

0.2

0.4

0.6

0.8

1.0

Takagi et al, PRL ‘92

PrFxO1-xFeAs Z.A. Ren et.al, Beijing, (08)

Iron based sc Tc= 6 - 53 ++ ? K

HF 115s Tc=0.2 -18.5 K

Local

NpAl2Pd5

d-electron systems: e.g Pnictides, Cuprate SC.

A new era of mysteries

Electron sea

2j+1

Curie

Heavy Fermions + Kondo

Spin (4f,5f): “quark” of heavy electron physics.

“Scales to Strong Coupling”

H =X

k�

✏kc†k�ck� + J ~S · ~�(0)

J. Kondo, 1962

5.4 Piers Coleman

Fig. 2: (a) In isolation, the localized atomic states of an atom form a stable, sharp excitationlying below the continuum. (b) In a crystal, the 2 j+1 fold degenerate state splits into multiplets,typically forming a low lying Kramers doublet. (c) The inverse of the Curie-Weiss susceptibilityof local moments ��1 is a linear function of temperature, intersecting zero at T = �✓.

predicted by Philip W. Anderson [4, 5], which results from high energy valence fluctuations.Jun Kondo [6] first analyzed the e↵ect of this scattering, showing that as the temperature islowered, the e↵ective strength of the interaction grows logarithmically, according to

J ! J(T ) = J + 2J2⇢ lnDT

(4)

where ⇢ is the density of states of the conduction sea (per spin) and D is the band-width. Thegrowth of this interaction enabled Kondo to understand why in many metals at low temperatures,the resistance starts to rise as the temperature is lowered, giving rise to resistance minimum.

Fig. 3: (a) Schematic temperature-field phase diagram of the Kondo e↵ect. At fields and tem-peratures large compared with the Kondo temperature TK, the local moment is unscreened witha Curie susceptibility. At temperatures and fields small compared with TK, the local moment isscreened, forming an elastic scattering center within a Landau Fermi liquid with a Pauli sus-ceptibility � ⇠ 1

TK. (b) Schematic susceptibility curve for the Kondo e↵ect, showing cross-over

from Curie susceptibility at high temperatures to Pauli susceptibility at temperatures below theKondo temperature TK. (c) Specific heat curve for the Kondo e↵ect. Since the total area is thefull spin entropy R ln 2 and the width is of order TK, the height must be of order � ⇠ R ln 2/TK.This sets the scale for the zero temperature specific heat coe�cient.

Today, we understand this logarithmic correction as a renormalization of the Kondo couplingconstant, resulting from fact that as the temperature is lowered, more and more high frequency

Chapter 16. c⃝Piers Coleman 2010

where the density of conduction electron states ρ(ϵ) is taken to be constant. The Poor Man’s renor-malization procedure follows the evolution of J(D) that results from reducing D by progressivelyintegrating out the electron states at the edge of the conduction band. In the Poor Man’s procedure,the band-width is not rescaled to its original size after each renormalization, which avoids the needto renormalize the electron operators so that instead of Eq. (16.81), H(D′) = HL.

To carry out the renormalization procedure, we integrate out the high-energy spin fluctuationsusing the t-matrix formulation for the induced interaction Hint, derived in the last section. Formally,the induced interaction is given by

δHintab =

12[Tab(Ea) + Tab(Eb)]

where

Tab(E) =∑

λ∈|H⟩

⎡⎢⎢⎢⎢⎢⎢⎣H(I)aλH

(I)λb

E − EHλ

⎤⎥⎥⎥⎥⎥⎥⎦

where the energy of state |λ⟩ lies in the range [D′,D]. There are two possible intermediate statesthat can be produced by the action of H(I) on a one-electron state: (I) either the electron state isscattered directly, or (II) a virtual electron hole-pair is created in the intermediate state. In process(I), the T-matrix can be represented by the Feynman diagram

’"’’"

kk’’

"

#$ %k’

for which the T-matrix for scattering into a high energy electron state is

T (I)(E)k′βσ′;kασ =

∑

ϵk′′ ∈[D−δD,D]

[1

E − ϵk′′

]J2(σaσb)βα(S aS b)σ′σ

≈ J2ρδD[

1E − D

](σaσb)βα(S aS b)σ′σ (16.102)

In process (II),

k$

’’"" ’"

%k’

k’’#

the formation of a particle-hole pair involves a conduction electron line that crosses itself, leadingto a negative sign. Notice how the spin operators of the conduction sea and antiferromagnet reversetheir relative order in process II, so that the T-matrix for scattering into a high-energy hole-state isgiven by

T (II)(E)k′βσ′;kασ = −∑

ϵk′′ ∈[−D,−D+δD]

[1

E − (ϵk + ϵk′ − ϵk′′)

]J2(σbσa)βα(S aS b)σ′σ

38

⇡ J2⇥�D

D(⇤a⇤b)⇥�(SaSb)⇤0⇤




δHintab =


where

Tab(E) =∑

λ∈|H⟩

⎡⎢⎢⎢⎢⎢⎢⎣H(I)aλH

(I)λb

E − EHλ

⎤⎥⎥⎥⎥⎥⎥⎦


’"’’"

kk’’

"

#$ %k’



∑


[1

E − ϵk′′


≈ J2ρδD[

1E − D


In process (II),

k$

’’"" ’"

%k’

k’’#



ϵk′′ ∈[−D,−D+δD]

[1

E − (ϵk + ϵk′ − ϵk′′)


38

⇡ �J2⇥�D

D(⇤b⇤a)⇥�(SaSb)⇤0⇤

1

10

0.1

1<latexit sha1_base64="kHbZkymBEKdJYVegr3MhQQtg3Qo=">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</latexit><latexit sha1_base64="kHbZkymBEKdJYVegr3MhQQtg3Qo=">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</latexit>

J

Wea

k co

uplin

gSt

rong

cou

plin

g

Electron sea

2j+1

Curie

Heavy Fermions + Kondo “Kondo temperature”

Spin (4f,5f): “quark” of heavy electron physics.

TK = WpJ⇢e�

12J⇢

“Scales to Strong Coupling”

T

H =X

k�

✏kc†k�ck� + J ~S · ~�(0)

J. Kondo, 1962

5.4 Piers Coleman

Fig. 2: (a) In isolation, the localized atomic states of an atom form a stable, sharp excitationlying below the continuum. (b) In a crystal, the 2 j+1 fold degenerate state splits into multiplets,typically forming a low lying Kramers doublet. (c) The inverse of the Curie-Weiss susceptibilityof local moments ��1 is a linear function of temperature, intersecting zero at T = �✓.

predicted by Philip W. Anderson [4, 5], which results from high energy valence fluctuations.Jun Kondo [6] first analyzed the e↵ect of this scattering, showing that as the temperature islowered, the e↵ective strength of the interaction grows logarithmically, according to

J ! J(T ) = J + 2J2⇢ lnDT

(4)

where ⇢ is the density of states of the conduction sea (per spin) and D is the band-width. Thegrowth of this interaction enabled Kondo to understand why in many metals at low temperatures,the resistance starts to rise as the temperature is lowered, giving rise to resistance minimum.

Fig. 3: (a) Schematic temperature-field phase diagram of the Kondo e↵ect. At fields and tem-peratures large compared with the Kondo temperature TK, the local moment is unscreened witha Curie susceptibility. At temperatures and fields small compared with TK, the local moment isscreened, forming an elastic scattering center within a Landau Fermi liquid with a Pauli sus-ceptibility � ⇠ 1

TK. (b) Schematic susceptibility curve for the Kondo e↵ect, showing cross-over

from Curie susceptibility at high temperatures to Pauli susceptibility at temperatures below theKondo temperature TK. (c) Specific heat curve for the Kondo e↵ect. Since the total area is thefull spin entropy R ln 2 and the width is of order TK, the height must be of order � ⇠ R ln 2/TK.This sets the scale for the zero temperature specific heat coe�cient.

Today, we understand this logarithmic correction as a renormalization of the Kondo couplingconstant, resulting from fact that as the temperature is lowered, more and more high frequency

“Kondo Resistance Minimum”

1

10

0.1

1<latexit sha1_base64="kHbZkymBEKdJYVegr3MhQQtg3Qo=">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</latexit><latexit sha1_base64="kHbZkymBEKdJYVegr3MhQQtg3Qo=">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</latexit>

J

Wea

k co

uplin

gSt

rong

cou

plin

g

Electron sea

2j+1

Curie

Pauli


Spin screened by conduction electrons: entangled

TK = WpJ⇢e�

12J⇢

H =X

k�

✏kc†k�ck� + J ~S · ~�(0)

J. Kondo, 1962

Heavy Fermions and the Kondo Lattice 5.5

quantum spin fluctuations become coherent, and these strengthen the Kondo interaction. Thee↵ect is closely analogous to the growth of the strong-interaction between quarks, and likequarks, the local moment in the Kondo e↵ect is asymptotically free at high energies. However,as you can see from the above equation, once the temperature becomes of order

TK ⇠ D exp"� 1

2J⇢

#

the correction becomes as large as the original perturbation, and at lower temperatures, theKondo interaction can no longer be treated perturbatively. In fact, non-perturbative methodstell us that this interaction scales to strong coupling at low energies, causing electrons in theconduction sea to magnetically screen the local moment to form an inert Kondo singlet denotedby

|GS i = 1p2

(| *#i � | +"i) , (5)

where the thick arrow refers to the spin state of the local moment and the thin arrow refers to thespin state of a bound-electron at the site of the local moment. The key features of the impurityKondo e↵ect are are:

• The electron fluid surrounding the Kondo singlet forms a Fermi liquid, with a Pauli sus-ceptibility � ⇠ 1/TK .

• The local moment is a kind of qubit which entangles with the conduction sea to forma singlet. As the temperature T is raised, the entanglement entropy converts to thermalentropy, given by the integral of the specific heat coe�cient,

S (T ) =Z T

0dT 0

CV(T 0)T 0.

Since the total area under the curve, S (T ! 1) = R ln 2 per mole is the high tempera-ture spin entropy, and since the characteristic width is the Kondo temperature, it followsthat the the characteristic zero temperature specific heat coe�cient must be of order theinverse Kondo temperature: � = CV

T (T ! 0) ⇠ R ln 2TK

. (See Fig. 3 b)

• The only scale in the physics is TK . For example, the resistivity created by magneticscattering o↵ the impurity has a universal temperature dependence

R(T )RU= ni�

TTK

!(6)

where ni is the concentration of magnetic impurities, �(x) is a universal function and ⇢U

is the unit of unitary resistance (basically resistance with a scattering rate of order theFermi energy),

RU =2ne2

⇡m⇢(7)

Experiment confirms that the resistivity in the Kondo e↵ect can indeed be scaled onto asingle curve that fits forms derived from the Kondo model (see Fig 4).




δHintab =


where

Tab(E) =∑

λ∈|H⟩

⎡⎢⎢⎢⎢⎢⎢⎣H(I)aλH

(I)λb

E − EHλ

⎤⎥⎥⎥⎥⎥⎥⎦


’"’’"

kk’’

"

#$ %k’



∑


[1

E − ϵk′′


≈ J2ρδD[

1E − D


In process (II),

k$

’’"" ’"

%k’

k’’#



ϵk′′ ∈[−D,−D+δD]

[1

E − (ϵk + ϵk′ − ϵk′′)


38

⇡ J2⇥�D

D(⇤a⇤b)⇥�(SaSb)⇤0⇤




δHintab =


where

Tab(E) =∑

λ∈|H⟩

⎡⎢⎢⎢⎢⎢⎢⎣H(I)aλH

(I)λb

E − EHλ

⎤⎥⎥⎥⎥⎥⎥⎦


’"’’"

kk’’

"

#$ %k’



∑


[1

E − ϵk′′


≈ J2ρδD[

1E − D


In process (II),

k$

’’"" ’"

%k’

k’’#



ϵk′′ ∈[−D,−D+δD]

[1

E − (ϵk + ϵk′ − ϵk′′)


38

⇡ �J2⇥�D

D(⇤b⇤a)⇥�(SaSb)⇤0⇤T I

T II

c⃝2010 Piers Coleman Chapter 16.

= −J2ρδD[

1E − D


where we have assumed that the energies ϵk and ϵk′ are negligible compared with D. Adding (Eq.16.102) and (Eq. 16.103) gives

δHintk′βσ′;kασ = T I + T II = −

J2ρδDD

[σa,σb]βαS aS b

=J2ρδDDσβαS σ′σ. (16.104)

In this way we see that the virtual emission of a high energy electron and hole generates an antifer-romagnetic correction to the original Kondo coupling constant

J(D′) = J(D) + 2J2ρδDD

High frequency spin fluctuations thus antiscreen the antiferrromagnetic interaction. If we introducethe coupling constant g = ρJ, we see that it satisfies

∂g∂ lnD

= β(g) = −2g2 + O(g3).

This is an example of a negative β function: a signature of an interaction which is weak at highfrequencies, but which grows as the energy scale is reduced. The local moment coupled to theconduction sea is said to be asymptotically free. The solution to this scaling equation is

g(D′) =go

1 − 2go ln(D/D′)(16.105)

and if we introduce the scaleTK = D exp

[−12go

](16.106)

we see that this can be written2g(D′) =

1ln(D′/TK)

This is an example of a running coupling constant- a coupling constant whose strength depends onthe scale at which it is measured. (See Fig. 16.14).

Were we to take this equation literally, we would say that g diverges at the scale D′ = TK . Thisinterpretation is too literal, because the above scaling equation has only been calculated to order g2,nevertheless, this result does show us that the Kondo interaction can only be treated perturbativelyat energy scales large compared with the Kondo temperature. We also see that once we have writtenthe coupling constant in terms of the Kondo temperature, all reference to the original cut-off energyscale vanishes from the expression. This cut-off independence of the problem is an indication thatthe physics of the Kondo problem does not depend on the high energy details of the model: there isonly one relevant energy scale, the Kondo temperature.

39


= −J2ρδD[

1E − D




J2ρδDD

[σa,σb]βαS aS b





∂g∂ lnD

= β(g) = −2g2 + O(g3).


g(D′) =go

1 − 2go ln(D/D′)(16.105)


[−12go

](16.106)


1ln(D′/TK)



39

(g = �J)


= −J2ρδD[

1E − D




J2ρδDD

[σa,σb]βαS aS b





∂g∂ lnD

= β(g) = −2g2 + O(g3).


g(D′) =go

1 − 2go ln(D/D′)(16.105)


[−12go

](16.106)


1ln(D′/TK)



39

(SaSb)�0�

.2

2i�abc⇥c⇥�z }| {

S=0.5 R ln2

Linear S. Heat

Spin entanglement entropy

Electron sea

2j+1

Curie

Pauli


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

Spin screened by conduction electrons: entangled

TK = WpJ⇢e�

12J⇢



TK ⇠ D exp"� 1

2J⇢

#


|GS i = 1p2

(| *#i � | +"i) , (5)




S (T ) =Z T

0dT 0

CV(T 0)T 0.


T (T ! 0) ⇠ R ln 2TK

. (See Fig. 3 b)


R(T )RU= ni�

TTK

!(6)



RU =2ne2

⇡m⇢(7)


Heavy Fermion Primer

S=0.5 R ln2

Linear S. Heat

Spin entanglement entropy

5.6 Piers Coleman

• The scattering o↵ the Kondo singlet is resonantly confined to a narrow region of orderTK , called the Kondo or Abriksov-Suhl resonance.

Fig. 4: Temperature dependence of resistivity associated with scattering from an impurityspin from [7, 8]. The resistivity saturates at the unitarity limit at low temperatures, due to theformation of the Kondo resonance. Adapted from [7].

1.3 The Kondo lattice

In heavy fermion material, containing a lattice of local moments, the Kondo e↵ect developscoherence. In a single impurity, a Kondo singlet scatters electrons without conserving momen-tum, giving rise to a huge build-up of resistivity at low temperatures. However, in a lattice, withtranslational symmetry, this same elastic scattering now conserves momentum, and this leads tocoherent scattering o↵ the Kondo singlets. In the simplest heavy fermion metals, this leads to adramatic reduction in the resistivity at temperatures below the Kondo temperature.As a simple example, consider CeCu6 a classic heavy fermion metal. Naively, CeCu6 is justa copper alloy, in which 14% of the copper atoms are replaced by cerium, yet this modestreplacement radically alters the metal. In this material, it actually proves possible to follow thedevelopment of coherence from the dilute single ion Kondo limit, to the dense Kondo lattice, byforming the alloy La1�xCexCu6. Lanthanum is iso-electronic to cerium, but has an empty f-shell,so the limit x! 0 corresponds to the dilute Kondo limit, and in this limit the resistivity followsthe classic Kondo curve. However, as the concentration of cerium increases, the resistivitycurve starts to develop a coherence maximum, an in the concentrated limit drops to zero with acharacteristic T 2 dependence of a Landau Fermi liquid (see Fig. 6).CeCu6 displays the following classic features of a heavy fermion metal:

• A Curie-Weiss susceptibility � ⇠ (T + ✓)�1 at high temperatures.

• A paramagnetic spin susceptibility � ⇠ cons at low temperatures.



TK ⇠ D exp"� 1

2J⇢

#


|GS i = 1p2

(| *#i � | +"i) , (5)




S (T ) =Z T

0dT 0

CV(T 0)T 0.


T (T ! 0) ⇠ R ln 2TK

. (See Fig. 3 b)


R(T )RU= ni�

TTK

!(6)



RU =2ne2

⇡m⇢(7)


Universality

Heavy Fermion Primer

“Kondo Lattice”

H =X

k�

✏kc†k�ck� + J

X

j

~Sj · ~�(j)<latexit sha1_base64="fEWzijHG+GggQCC7z85CqyBFMwk=">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</latexit><latexit sha1_base64="fEWzijHG+GggQCC7z85CqyBFMwk=">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</latexit>

Kondo Lattice Model (Kasuya, 1951)

DONIACH’S Hypothesis. Doniach (1977)

>

TK = D exp[�1/2J�]

QCP

Large Fermi surface of composite Fermions

+

+

++

+

++

+

++

+

++ + +

Kondo Lattice Model (Kasuya, 1951)

DONIACH’S Hypothesis. Doniach (1977)

>

TK = D exp[�1/2J�]

QCP

Large Fermi surface of composite Fermions

+

+

++

+

++

+

++

+

++ + +

?

The main result ... is that there should be a second-order transition at zero temperature, as the exchange is varied, between an antiferromagnetic ground state for weak J and a Kondo-like state in which the local moments are quenched.

T(K)

electricalresistivity

50

150

250

200

100 200 300

ρ m (!

Ωcm

)

UBe13

CeCu6 x 2.5

10

20

ρ m (!

Ωcm

)

0 0.5 1.00

CeCu6

100

00

Ce3Bi4Pt3x 1/2

CeRhIn5 x 4

T2

incoherent scattering

“Kondo Lattice”

H =X

k�


X

j


100

300

200

0.01 0.1 1 10 100

ρ m (!

Ωcm

/Ce)

CexLa1-xCu6x=0.0940.29

0.50.730.9

0.991.0

T(K)

Impurity

Lattice

Coherent Heavy Fermions�(T ) = �0 + AT 2

Entangled spins and electrons → Heavy Fermion Metals

� = LimT⇥0

�CV

T

⇥=

⇥2k2B

3N(0)�.

Ep =p2

2m⇤ , N⇤(0) =m⇤pF⇡2~3

Heavy Fermions: magnetically polarizable Landau Fermi liquids.

eg Cu vs CeCu6 (copper, spin doped) γCu ~ 1 mJ/mol/K2, ϒ[CeCu6] ~ 1000 mJ/mol/K2,

m*/me ~ 1000

W =⌅

�= 3

�µB

2⇤kB

⇥2 11 + F a

0

CeCu6

Cu

⇥ =µ2

BN�(0)1 + F a

0

T(K)


50

150

250

200

100 200 300

ρ m (!

Ωcm

)

UBe13

CeCu6 x 2.5

10

20

ρ m (!

Ωcm

)

0 0.5 1.00

CeCu6

100

00

Ce3Bi4Pt3x 1/2

CeRhIn5 x 4

T2


“Kondo Lattice”

H =X

k�


X

j



Ott, Fisk, Smith 1983

Entangled spins and electrons → Heavy Fermion Metals→ New kinds of superconductor

T(K)


50

150

250

200

100 200 300

ρ m (!

Ωcm

)

UBe13

CeCu6 x 2.5

10

20

ρ m (!

Ωcm

)

0 0.5 1.00

CeCu6

100

00

Ce3Bi4Pt3x 1/2

CeRhIn5 x 4

T2


“Kondo Lattice”

H =X

k�


X

j



Entangled spins and electrons → AFM/Superconductivity

SC anomaly at Tc reveals only a strong sensitivity to imper-fections, for example due to a mismatch between incommen-surate magnetic ordering !AFI" and SC or due to additionalnodes in the gap function of the SC state caused by thecrossing of the FS with the magnetic Brillouin zone.27 Directevidences on the inhomogenous superconducting transitionbelow pc

! are given by the large discrepancy between Tc de-tected by resistivity Tc

!,8 ac susceptibility Tc",11 and the

present specific heat measurements TcC with a sequence

Tc!#Tc

"#TcC. Up to 1.5 GPa, the resistivity anomaly at Tc

!

may not be a bulk property; a similar case is reported forCeIrIn5.3,28 By contrast, the specific heat measurementclearly shows a sharp and very large superconductinganomaly at p=2.17 GPa indicating a pure superconductingground state, basically gapped. At p=2.07 GPa no extra AFtransition can be detected below Tc. Above pc

!, an eventualdomain of a coexistence of AF and SC will be extremelynarrow in pressure and experimentally very difficult to pointout.

The !p ,T" phase diagram of CeRhIn5 in zero field is sum-marized in Fig. 2 !data from Ref. 11 have been included".The low temperature specific heat measurements clearlyshow the interplay of AF and SC in the pressure range from1.6–1.9 GPa. A first order transition seems to emerge atpc

!=1.95 GPa, where Tc#TN. A linear p extrapolation ofTN to zero temperature indicates that TN may be fullysuppressed near the pressure of the maximum of Tc atpc#2.4±0.1 GPa and the maxima of some of the effectivemasses.12 If Tc#TN, the ground state in zero field seemspurely superconducting with d-wave symmetry10 as inCeCoIn5.4,5 The opening of a superconducting gap on largeparts of the main FS leads to the suppression of the magneticordering.

Figure 3 shows the specific heat for various fields H $ab atdifferent pressures below and above pc

!. The !H ,T" phasediagrams obtained from these data are displayed in Fig. 4. At1.2 GPa, in the normal AF state, three magnetic transitionsappear for H#3 T. By comparison to p=0 results21 the mag-netic phase diagram is only weakly p dependent %see Fig.

4!a"&. For p=2.07 GPa the superconducting transition at 0 Tis still rather broad, but above Hc2 another phase transition atTM appears in the normal state %Fig. 4!b"&. At 2.2 GPa thetransition has a width less than 0.1 K at H=0. For H#4 T,the additional transition develops on cooling already insidethe superconducting phase and survives entering in thenormal phase. No second transition can be detected belowH#4 T. For H$4 T two well separated anomalies insidethe superconducting state become obvious at 2.41 GPa,almost as observed in UPt3.29 For higher pressures!p=2.73 GPa", only a unique superconducting phasepersists.

The assignment of the superconducting anomaly undermagnetic field is unambiguously given by the form of theHc2!T" curve, also in comparison to previous resistivitydata.30 There are strong indications that the observed secondanomaly is associated to AF. This transition at TM is almostH independent, at least for H#4 T, as the antiferromagnetictransition at p=0 and p=1.2 GPa. The inset of Fig. 2 showsthe p dependence of the crossing temperature correspondingto Tc!H"'TM!H" for p# pc

!. Its extrapolation to zero is ob-tained for p#2.6 GPa, slightly higher than pc. That can in-dicate the enhancement of TN when AF and SC coexist inthis mixed state. In the simplest model, weak coupling in aclean limit, the p dependence of the effective mass m! of thequasiparticles can be estimated from the initial slopeHc2! =dHc2 /dT% !m!"2Tc.32 For p=2.07 GPa we findHc2! =20 T/K, near 2.41 GPa Hc2! increases to 33 T/K anddecreases to 25 T/K for 2.73 GPa while Tc is almostunchanged.33 This, as well as the size of the specific heatjump at Tc, indicates that m! has its maximum near 2.4 GPain agreement with the expected variation of m! at pc for anantiferromagnetic QCP.35

The new phase presumably with AF and SC appears forp# pc

! only above some critical field of the order H#4 T.31

FIG. 1. !Color online" Temperature dependence of the ac spe-cific heat divided by temperature for different pressures. Arrowsindicate the superconducting transition temperatures Tc !↑" and theNéel temperature TN !↓", respectively. Below pc

!#1.95 GPa the su-perconducting anomaly is very small. For p slightly above pc

! a nicesuperconducting anomaly appears. !Data are normalized to 1 at4 K."

FIG. 2. !Color online" !p ,T" phase diagram of CeRhIn5 fromspecific heat !!, !", susceptibility !", Ref. 11", and resistivitymeasurements !&, Ref. 8". Below 1.5 GPa CeRhIn5 orders in anincommensurable structure AFI, the hatched area indicates an inho-mogeneous superconducting state, and AFI+SC corresponds to theregion where SC appears in the specific heat experiment in themagnetically ordered state below TN. A pure superconducting stateSC is realized above pc

!. The vertical line marks a possible firstorder transition from AFI+SC to SC. The inset shows the extrapo-lation of TN to zero in the absence of SC. !"" indicates the tem-perature where TM!H" crosses Tc!H", and corresponds to TN if SC issuppressed !see also Fig. 4".

KNEBEL et al. PHYSICAL REVIEW B 74, 020501!R" !2006"

RAPID COMMUNICATIONS

020501-2

Knebel et al,

Custers et al, (2003)

YbRh2Si2 : Field tuned quantum criticality.

23

Kondo insulators: HistoryMenth,)Buehler)and)Geballe)(PRL)22,295,)1969))Aeppli)and)Fisk)(Comments)CMP)16,)155,)1992)

Simplest)Kondo)LaFceSimplest)Kondo)LaFce

Sm2.7+

B6

SmB6

24

Menth,)Buehler)and)Geballe)(PRL)22,295,)1969))Aeppli)and)Fisk)(Comments)CMP)16,)155,)1992)

Simplest)Kondo)LaFceSimplest)Kondo)LaFce

Hybridization picture.

Mott Phil Mag, 30,403,1974

kEf

E(k)

unitary operator with an associated quantum number, the “Kramers index”K (25). The Kramers

index, K = (−1)2J of a quantum state of total angular momentum J defines the phase factor

acquired by its wavefunction after two successive time-reversals, Θ2|ψ⟩ = K|ψ⟩ = |ψ2π⟩. An

integer spin state |α⟩ is unchanged by a 2π rotation, so |α2π⟩ = +|α⟩ and K = 1. However,

conduction electrons with half-integer spin states, |kσ⟩, where k is momentum and σ is the spin

component, change sign, |kσ2π⟩ = −|kσ⟩, so K = −1.

While conventional magnetism breaks time-reversal symmetry, it is invariant under dou-

ble reversals Θ2 so the Kramers index is conserved. However in URu2Si2, the hybridization

between integer and half-integer spin states requires a quasiparticle mixing term of the form

H = (|kσ⟩Vσα(k)⟨α| + H.c) in the low energy fixed point Hamiltonian that does not conserve

the Kramers index. After two successive time-reversals

|kσ⟩Vσα(k)⟨α| → |kσ2π⟩V 2πσα (k)⟨α2π| = −|kσ⟩V 2π

σα (k)⟨α|. (1)

Since the microscopic Hamiltonian is time-reversal invariant, it follows that Vσα(k) = −V 2πσα (k);

the hybridization thus breaks time-reversal symmetry in a fundamentally new way, playing the

role of an order parameter that, like a spinor, reverses under 2π rotations. The resulting “hastatic

(Latin: spear) order”, is a state of matter that breaks both single and double time-reversal sym-

metry and is thus distinct from conventional magnetism.

Indirect support for time-reversal symmetry-breaking in the hidden order phase of URu2Si2 is

provided by recent magnetometry measurements that indicate the development of an anisotropic

basal-plane spin susceptibility, χxy, at the hidden order transition (16). As noted elsewhere (9),

χxy is a conduction electron response to scattering off the hidden order (c.f. Fig. 2.), leading to

a scattering matrix of the form

t(k) = (σx + σy)d(k) (2)

where d(k) is the scattering amplitude. This scattering matrix has been linked to a spin nematic

4

Maple + Wohlleben, 1972

Allen and Martin, 1979

“In SmB6 and high-pressure SmS a very small gap separates occupied from unoccupied states, this in our view being due to hybridization of 4f and 4d bands.” Mott 1974

Kondo insulators: History

H = JX

~�(j) · ~Sj � tX

(i,j)

(c†i�cj� +H.c)

J >> tStrong coupling Kondo Lattice

ne = nspinsKondo insulator

Hole doping: mobile heavy holes ne = nspins � �

-t

� 3J/2

~ 8t

2

✓vFS

(2⇡)D

◆= 2� � = nspins + ne

FS sum rule counts spins as charged qp.

3J/2k

Ek~ 8t

2� �

But what about weak coupling: J << t?

cf Cox, Pang, Jarell (96) PC, Kee, Andrei, Tsvelik (98)

Single FS, two channels.

⇥(x, �)

S[�]

�, x

How can we tame the wild Quantum fluctuations?

Z =

Z

Fieldse� S[ ]

<latexit sha1_base64="q7QOxGOxUDdnX1g7AVPfQ8tn4LE=">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</latexit><latexit sha1_base64="q7QOxGOxUDdnX1g7AVPfQ8tn4LE=">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</latexit>

Path Integral

c†�( j) =1�V

�

kck�eik·R j

H =�

k

�kc†k�ck� +JN

�

j

�†a( j)�a( j)S ba( j)



⇥(x, �)

S[�]

�, x

1N� �eff

Large N expansion.

�! (�N

2,N

2)

<latexit sha1_base64="aM0c6e/JcHMzL8WRDspV6Eo6nXo=">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</latexit><latexit sha1_base64="aM0c6e/JcHMzL8WRDspV6Eo6nXo=">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</latexit>

� 2 (�1

2,1

2)

<latexit sha1_base64="/Ni1jXDK+FhtPlxy6n5a4r73Blg=">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</latexit><latexit sha1_base64="/Ni1jXDK+FhtPlxy6n5a4r73Blg=">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</latexit>

Z =

Z

Fieldse�S[ ]

1/N

<latexit sha1_base64="LA2kaxXsEvJf3TFvCk2P4+RydVw=">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</latexit><latexit sha1_base64="LA2kaxXsEvJf3TFvCk2P4+RydVw=">AAADBXicjZFNbxMxEIad5aMlfKXAjYtFhcSFkPQAvSBVICFOUAShFfE28npnNyZe27K9bCJrbxz4NZwQV/4CV/4IV/BuitSsQGIkS6PnHc/Y8yZacOtGox+96Nz5Cxe3ti/1L1+5eu36YOfGW6tKw2DClFDmOKEWBJcwcdwJONYGaJEIOEoWTxv96AMYy5V841Ya4oLmkmecURfQbPAek7aJr+bcQf3uMSZcupkn80Qt/TMOIrV1jftw4knKrRZ0Zd1KAL7vSWYo86+nRFse13784LTTS0NlDvWLjcYhZoPd8XDUBv53sntwC7VxONvpfSSpYmUB0jFBrZ2OR9rFnhrHmYC6T0oLmrIFzWEaUkkLsLFvx9b4biApzpQJRzrc0rM3PC2sXRVJqCyom9uu1sC/adPSZfux51KXDiRbD8pKgZ3CzYZxyg0wJ1Yhoczw8FbM5jTsygUf+kRCxVRRUJkSQz1pltX+5QxPc39CUprnYDaFZOE9STK8qDtcr7nu8uWaLzu8bOpLTY1RVUdKqyClqpJ/xP7/+TbZGz4cjl4F/56s/UPb6Da6g+6hMXqEDtBzdIgmiKHv6Cf61UPRp+hz9CX6ui6Neqd3bqKNiL79Bn/6/vo=</latexit>

c†�( j) =1�V

�

kck�eik·R j

H =�

k


�

j

�†a( j)�a( j)S ba( j)



c†�( j) =1�V

�

kck�eik·R j

H =�

k


�

j

�†a( j)�a( j)S ba( j)

⇥(x, �)

S[�]

�, x

Large N expansion.

Z =

Z

Fieldse�S[ ]

1/N

<latexit sha1_base64="LA2kaxXsEvJf3TFvCk2P4+RydVw=">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</latexit><latexit sha1_base64="LA2kaxXsEvJf3TFvCk2P4+RydVw=">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</latexit>

classi

cal p

ath

1N� �eff

S ba = f †b fa �n f

N�ab

HI �JN

�f †j�c j�

� �c†j� f��

� V�c†j� f��+�

f †j�c j��

V + NVVJ

H =�

k�

�kc†k�ck� +�

j

HI(j)

HI(j) = � J

N

�c†j⇥fj⇥

⇥ �f†

j�cj�

⇥

c†j� =1⇥Ns

�

k

c†k�e�ik·⌃Rj

�gA†A⇥ A†V + V A +V V

g

HI(j)� HI [V, j] = Vj

�c†j�fj�

⇥+

�f†

j�cj�

⇥Vj + N

VjVj

J.

Large N Approach.Read and Newns ’83.

Constraint nf = Q=qN all terms extensive in N

Large N Approach

Z =�D[V,�]

=Tr�Texp(� R �

0 H[V,⇥]d⌅)⇥

� ⇣✏ ��D[c, f ] exp

⇧

⌥��

0

⇤

↵

k⇤

c†k⇤⇥⌅ ck⇤ +↵

j⇤

f†j⇤⇥⌅fj⇤ + H[V,�]

⌅

⌦

⌃

�

H[V,⇥] =�

k�


j

(HI [Vj , j] + ⇥j [nf (j)�Q]) ,

HI [V, j] = Vj

�c†j�fj�

⇥+

�f†

j�cj�

⇥Vj + N

VjVj

J.

U(1) constraint: note nf =Q =(qN)

Read and Newns ’83.

Extensive in N

H[V,⇥] =�

k�


j

(HI [Vj , j] + ⇥j [nf (j)�Q]) ,

HI [V, j] = Vj

�c†j�fj�

⇥+

�f†

j�cj�

⇥Vj + N

VjVj

J.

Z = Tre��HMF T , (N �⇥)

V V

Large N Approach.Read and Newns ’83.

Vj = V at each site

Detailed calcn.


17.1.4 Mean-field theory of the Kondo Lattice

Diagonalization of the Hamiltonian

We can now make the bold jump from the single impurity problem, to the lattice. Most of themethods described in the last subsection generalize very naturally from the impurity to the lattice:the main difficulty is to understand the underlying physics. The mean-field Hamiltonian for thelattice[49? ] takes the form

HMFT =∑

kσ

ϵkc†kσckσ +

∑

j,α

(f † jαψ jαVo + Voψ† jβ f jβ + λo f † jα f jα

)+ Nn

(VoVoJ− λoq

),

where n is the number of sites in the lattice. Notice, before we begin, that the composite f-stateat each site of the lattice is entirely local, in that hybridization occurs at one site only. Were thecomposite f-state to be in any way non-local, we would expect that the hybridization of one f-state would involve conduction electrons at different sites. We begin by rewriting the mean fieldHamiltonian in momentum space, as follows

HMFT =∑

kσ

(c†kσ, f

†kσ

) ( ϵk VoVo λo

) (ckσfkσ

)+ Nn

(VoVoJ− λoq

)

where

f †kσ =1√n

∑

jf † jσeik·R j

is the Fourier transform of the f−electron field. The absence of k− dependence in the hybridizationis evident that each composite f−electron is spatially local. This Hamiltonian can be diagonalizedin the form

HMFT =∑

kσ

(a†kσ, b

†kσ

) (Ek+ 00 Ek−

) (akσbkσ

)+ Nn

(VoVoJ− λoq

)

where a†kσ and b†kσ are linear combinations of c

†kσ and f

†kσ, playing the role of “quasiparticle op-

erators” of the theory and the momentum state eigenvalues Ek± of this Hamiltonian are determinedby the condition

Det[Ek±1 −

(ϵk VoVo λo

)]= 0,

which gives

Ek± =ϵk + λo

2±⎡⎢⎢⎢⎢⎢⎣

(ϵk − λo2

)2+ |Vo|2

⎤⎥⎥⎥⎥⎥⎦

12

(17.44)

are the energies of the upper and lower bands. The dispersion described by these energies is shownin Fig. 17.8 . A number of points can be made about this dispersion:

75

28 Heavy electrons

18.6 Mean-field theory of the Kondo Lattice

18.6.1 Diagonalization of the Hamiltonian

We can now make the jump from the single impurity problem to the lattice. The virtue of the large N methodis that while approximate, it can be readily scaled up to the lattice. We’ll now recompute the effective actionfor the lattice, using equation 18.69. Let us assume that the hybridization and constraint fields at the saddlepoint are uniform, with Vj = V and λ j = λ at every site. Infact, even if we start with a Vj = Ve−iφ j witha different phase at each site, we can always use the phase φ j using the Read Newns gauge transformation(18.56) to absorb the additional phase onto the f-electron field. We then have a translationally invariant mean-field Hamiltonian. We begin by rewriting the mean field Hamiltonian in momentum space as follows

HMFT =∑

kσ

(c†kσ, f †kσ

)h(k)︷!!!︸︸!!!︷(

ϵk VV λ

) (ckσfkσ

)+ NNs

( |V |2J− λq

)(18.111)

=∑

kσψ†kσ h(k) ψkσ + NNs

( |V |2J− λq

).

Here, f †kσ =1√Ns

∑j f † jσeik·R j is the Fourier transform of the f−electron field and we have introduced the

two component notation

ψkσ =

(ckσfkσ

), ψ†kσ =

(c†kσ, f †,kσ

), h(k) =

(ϵk VV λ

). (18.112)

We should think about HMFT as a renormalized Hamiltonian, describing the low energy quasiparticles,moving through a self-consistently determined array of resonant scattering centers. Later, we will see that thef-electron operators are composite objects, formed as bound-states between spins and conduction electrons.

The mean-field Hamiltonian can be diagonalized in the form

HMFT =∑

kσ

(a†kσ, b†kσ

) (Ek+ 00 Ek−

) (akσbkσ

)+ Nn

(VVJ− λq

). (18.113)

Here a†kσ = ukc†kσ+ vk f †kσ and b†kσ = −vkc†kσ+uk f †kσ are linear combinations of c†kσ and f †kσ, playingthe role of “quasiparticle operators” with corresponding energy eigenvalues

Det[E±k 1 −

(ϵk VV λ

)]= (Ek± − ϵk)(Ek± − λ) − |V |2 = 0, (18.114)

or

Ek± =ϵk + λ

2±

[( ϵk − λ2

)2+ |V |2

] 12

, (18.115)

and eigenvectors taking the BCS form

{ukvk

}=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣12± (ϵk − λ)/2

2√(

ϵk−λ2

)2+ |V |2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

12

. (18.116)

28 Heavy electrons




HMFT =∑

kσ

(c†kσ, f †kσ

)h(k)︷!!!︸︸!!!︷(

ϵk VV λ

) (ckσfkσ

)+ NNs

( |V |2J− λq

)(18.111)

=∑


( |V |2J− λq

).




ψkσ =

(ckσfkσ

), ψ†kσ =

(c†kσ, f †,kσ

), h(k) =

(ϵk VV λ

). (18.112)



HMFT =∑

kσ

(a†kσ, b†kσ

) (Ek+ 00 Ek−

) (akσbkσ

)+ Nn

(VVJ− λq

). (18.113)


Det[E±k 1 −

(ϵk VV λ

)]= (Ek± − ϵk)(Ek± − λ) − |V |2 = 0, (18.114)

or

Ek± =ϵk + λ

2±

[( ϵk − λ2

)2+ |V |2

] 12

, (18.115)


{ukvk

}=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣12± (ϵk − λ)/2

2√(

ϵk−λ2

)2+ |V |2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

12

. (18.116)

28 Heavy electrons




HMFT =∑

kσ

(c†kσ, f †kσ

)h(k)︷!!!︸︸!!!︷(

ϵk VV λ

) (ckσfkσ

)+ NNs

( |V |2J− λq

)(18.111)

=∑


( |V |2J− λq

).




ψkσ =

(ckσfkσ

), ψ†kσ =

(c†kσ, f †,kσ

), h(k) =

(ϵk VV λ

). (18.112)



HMFT =∑

kσ

(a†kσ, b†kσ

) (Ek+ 00 Ek−

) (akσbkσ

)+ Nn

(VVJ− λq

). (18.113)


Det[E±k 1 −

(ϵk VV λ

)]= (Ek± − ϵk)(Ek± − λ) − |V |2 = 0, (18.114)

or

Ek± =ϵk + λ

2±

[( ϵk − λ2

)2+ |V |2

] 12

, (18.115)


{ukvk

}=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣12± (ϵk − λ)/2

2√(

ϵk−λ2

)2+ |V |2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

12

. (18.116)

28 Heavy electrons




HMFT =∑

kσ

(c†kσ, f †kσ

)h(k)︷!!!︸︸!!!︷(

ϵk VV λ

) (ckσfkσ

)+ NNs

( |V |2J− λq

)(18.111)

=∑


( |V |2J− λq

).




ψkσ =

(ckσfkσ

), ψ†kσ =

(c†kσ, f †,kσ

), h(k) =

(ϵk VV λ

). (18.112)



HMFT =∑

kσ

(a†kσ, b†kσ

) (Ek+ 00 Ek−

) (akσbkσ

)+ Nn

(VVJ− λq

). (18.113)


Det[E±k 1 −

(ϵk VV λ

)]= (Ek± − ϵk)(Ek± − λ) − |V |2 = 0, (18.114)

or

Ek± =ϵk + λ

2±

[( ϵk − λ2

)2+ |V |2

] 12

, (18.115)


{ukvk

}=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣12± (ϵk − λ)/2

2√(

ϵk−λ2

)2+ |V |2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

12

. (18.116)

Detailed calcn.

28 Heavy electrons




HMFT =∑

kσ

(c†kσ, f †kσ

)h(k)︷!!!︸︸!!!︷(

ϵk VV λ

) (ckσfkσ

)+ NNs

( |V |2J− λq

)(18.111)

=∑


( |V |2J− λq

).




ψkσ =

(ckσfkσ

), ψ†kσ =

(c†kσ, f †,kσ

), h(k) =

(ϵk VV λ

). (18.112)



HMFT =∑

kσ

(a†kσ, b†kσ

) (Ek+ 00 Ek−

) (akσbkσ

)+ Nn

(VVJ− λq

). (18.113)


Det[E±k 1 −

(ϵk VV λ

)]= (Ek± − ϵk)(Ek± − λ) − |V |2 = 0, (18.114)

or

Ek± =ϵk + λ

2±

[( ϵk − λ2

)2+ |V |2

] 12

, (18.115)


{ukvk

}=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣12± (ϵk − λ)/2

2√(

ϵk−λ2

)2+ |V |2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

12

. (18.116)

28 Heavy electrons




HMFT =∑

kσ

(c†kσ, f †kσ

)h(k)︷!!!︸︸!!!︷(

ϵk VV λ

) (ckσfkσ

)+ NNs

( |V |2J− λq

)(18.111)

=∑


( |V |2J− λq

).




ψkσ =

(ckσfkσ

), ψ†kσ =

(c†kσ, f †,kσ

), h(k) =

(ϵk VV λ

). (18.112)



HMFT =∑

kσ

(a†kσ, b†kσ

) (Ek+ 00 Ek−

) (akσbkσ

)+ Nn

(VVJ− λq

). (18.113)


Det[E±k 1 −

(ϵk VV λ

)]= (Ek± − ϵk)(Ek± − λ) − |V |2 = 0, (18.114)

or

Ek± =ϵk + λ

2±

[( ϵk − λ2

)2+ |V |2

] 12

, (18.115)


{ukvk

}=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣12± (ϵk − λ)/2

2√(

ϵk−λ2

)2+ |V |2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

12

. (18.116)

28 Heavy electrons




HMFT =∑

kσ

(c†kσ, f †kσ

)h(k)︷!!!︸︸!!!︷(

ϵk VV λ

) (ckσfkσ

)+ NNs

( |V |2J− λq

)(18.111)

=∑


( |V |2J− λq

).




ψkσ =

(ckσfkσ

), ψ†kσ =

(c†kσ, f †,kσ

), h(k) =

(ϵk VV λ

). (18.112)



HMFT =∑

kσ

(a†kσ, b†kσ

) (Ek+ 00 Ek−

) (akσbkσ

)+ Nn

(VVJ− λq

). (18.113)


Det[E±k 1 −

(ϵk VV λ

)]= (Ek± − ϵk)(Ek± − λ) − |V |2 = 0, (18.114)

or

Ek± =ϵk + λ

2±

[( ϵk − λ2

)2+ |V |2

] 12

, (18.115)


{ukvk

}=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣12± (ϵk − λ)/2

2√(

ϵk−λ2

)2+ |V |2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

12

. (18.116)

30 Heavy electrons

enlarged Fermi surface volume now counts the total number of occupied quasiparticle states

Ntot = ⟨∑

kλσ

nkλσ⟩ = ⟨n f + nc⟩ (18.118)

where nkλσ = a†kλσakλσ is the number operator for the quasiparticles and nc is the total number ofconduction electrons. This means

Ntot = NVFS a3

(2π)3 = Q + nc, (18.119)

where a3 is the volume of the unit cell. This is rather remarkable, for the expansion of the Fermi surfaceimplies an increased charge density in the Fermi sea. Since charge is conserved, we are forced to con-clude there is a compensating +Q|e| charge density per unit cell provided by the Kondo singlets formedat each site, as illustrated in Fig. 18.10.

• We can construct the mean-field ground-state from the quasiparticle operators as follows:

|MF⟩ =∏

|k|<kFσ

b†kσ|0⟩ =∏

|k|<kFσ

(−vkckσ + uk f †kσ)|0⟩. (18.120)

However, this state only satisfies the constraint on the average. We can improve it by imposing theconstraint, forming a “Gutzwiller” wavefunction[?, ?, ?]

|GW⟩ = PQ

∏

|k|<kFσ

(−vkckσ + uk f †kσ)|0⟩, (18.121)

where, using (18.48)

PQ =∏

j

PQ( j) =∫ 2π

0

∏

j

dα j

2πei

∑j α j(n f j−Q). (18.122)

The action of the constraint gives rise to a highly incompressible Fermi liquid, in which the compress-ibility is far smaller than the density of states.

18.6.2 Mean Field Free Energy and Saddle point

Let us now use the results of the last section to calculate the mean-field free energy FMFT and determine,self-consistently the parameters λ and V which set the scales of the Kondo lattice. Using 18.69 we obtain

FMF = −NT∑

k,iωr

Tr ln[

−G−1k (iωr)︷!!!!!!!!!!!!!︸︸!!!!!!!!!!!!!︷

−iωr +

(ϵk VV λ

)]+Ns

(N|V |2

J− λQ

), (18.123)

whereNs is the number of sites in the lattice. Note that translational invariance means that momentum is con-served and the Green’s function is diagonal in momentum, so we can re-write the trace over the momentum asa sum over k. Let us remind ourselves of the steps taken between (18.69) and (18.70 ). We begin by re-writingthe trace of the logarithm as a determinant, which we then factorize in terms of the energy eigenvalues,

Tr ln[−iωr1 +

(ϵk VV λ

)]= ln det

[−z1 +

(ϵk VV λ

)]= ln

[ (Ek+−iωr)(Ek−−iωr)︷!!!!!!!!!!!!!!!!!!!!!!!!!︸︸!!!!!!!!!!!!!!!!!!!!!!!!!︷(ϵk − iωr)(λ − iωr) − V2

]

=∑

n=±ln(Ekn − iωr). (18.124)

30 Heavy electrons


Ntot = ⟨∑

kλσ

nkλσ⟩ = ⟨n f + nc⟩ (18.118)


Ntot = NVFS a3

(2π)3 = Q + nc, (18.119)



|MF⟩ =∏

|k|<kFσ

b†kσ|0⟩ =∏

|k|<kFσ

(−vkckσ + uk f †kσ)|0⟩. (18.120)


|GW⟩ = PQ

∏

|k|<kFσ

(−vkckσ + uk f †kσ)|0⟩, (18.121)


PQ =∏

j

PQ( j) =∫ 2π

0

∏

j

dα j

2πei

∑j α j(n f j−Q). (18.122)




FMF = −NT∑

k,iωr

Tr ln[

−G−1k (iωr)︷!!!!!!!!!!!!!︸︸!!!!!!!!!!!!!︷

−iωr +

(ϵk VV λ

)]+Ns

(N|V |2

J− λQ

), (18.123)


Tr ln[−iωr1 +

(ϵk VV λ

)]= ln det

[−z1 +

(ϵk VV λ

)]= ln


]

=∑

n=±ln(Ekn − iωr). (18.124)

30 Heavy electrons


Ntot = ⟨∑

kλσ

nkλσ⟩ = ⟨n f + nc⟩ (18.118)


Ntot = NVFS a3

(2π)3 = Q + nc, (18.119)



|MF⟩ =∏

|k|<kFσ

b†kσ|0⟩ =∏

|k|<kFσ

(−vkckσ + uk f †kσ)|0⟩. (18.120)


|GW⟩ = PQ

∏

|k|<kFσ

(−vkckσ + uk f †kσ)|0⟩, (18.121)


PQ =∏

j

PQ( j) =∫ 2π

0

∏

j

dα j

2πei

∑j α j(n f j−Q). (18.122)




FMF = −NT∑

k,iωr

Tr ln[

−G−1k (iωr)︷!!!!!!!!!!!!!︸︸!!!!!!!!!!!!!︷

−iωr +

(ϵk VV λ

)]+Ns

(N|V |2

J− λQ

), (18.123)


Tr ln[−iωr1 +

(ϵk VV λ

)]= ln det

[−z1 +

(ϵk VV λ

)]= ln


]

=∑

n=±ln(Ekn − iωr). (18.124)

28 Heavy electrons




HMFT =∑

kσ

(c†kσ, f †kσ

)h(k)︷!!!︸︸!!!︷(

ϵk VV λ

) (ckσfkσ

)+ NNs

( |V |2J− λq

)(18.111)

=∑


( |V |2J− λq

).




ψkσ =

(ckσfkσ

), ψ†kσ =

(c†kσ, f †,kσ

), h(k) =

(ϵk VV λ

). (18.112)



HMFT =∑

kσ

(a†kσ, b†kσ

) (Ek+ 00 Ek−

) (akσbkσ

)+ Nn

(VVJ− λq

). (18.113)


Det[E±k 1 −

(ϵk VV λ

)]= (Ek± − ϵk)(Ek± − λ) − |V |2 = 0, (18.114)

or

Ek± =ϵk + λ

2±

[( ϵk − λ2

)2+ |V |2

] 12

, (18.115)


{ukvk

}=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣12± (ϵk − λ)/2

2√(

ϵk−λ2

)2+ |V |2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

12

. (18.116)

Detailed calcn.

28 Heavy electrons




HMFT =∑

kσ

(c†kσ, f †kσ

)h(k)︷!!!︸︸!!!︷(

ϵk VV λ

) (ckσfkσ

)+ NNs

( |V |2J− λq

)(18.111)

=∑


( |V |2J− λq

).




ψkσ =

(ckσfkσ

), ψ†kσ =

(c†kσ, f †,kσ

), h(k) =

(ϵk VV λ

). (18.112)



HMFT =∑

kσ

(a†kσ, b†kσ

) (Ek+ 00 Ek−

) (akσbkσ

)+ Nn

(VVJ− λq

). (18.113)


Det[E±k 1 −

(ϵk VV λ

)]= (Ek± − ϵk)(Ek± − λ) − |V |2 = 0, (18.114)

or

Ek± =ϵk + λ

2±

[( ϵk − λ2

)2+ |V |2

] 12

, (18.115)


{ukvk

}=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣12± (ϵk − λ)/2

2√(

ϵk−λ2

)2+ |V |2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

12

. (18.116)

28 Heavy electrons




HMFT =∑

kσ

(c†kσ, f †kσ

)h(k)︷!!!︸︸!!!︷(

ϵk VV λ

) (ckσfkσ

)+ NNs

( |V |2J− λq

)(18.111)

=∑


( |V |2J− λq

).




ψkσ =

(ckσfkσ

), ψ†kσ =

(c†kσ, f †,kσ

), h(k) =

(ϵk VV λ

). (18.112)



HMFT =∑

kσ

(a†kσ, b†kσ

) (Ek+ 00 Ek−

) (akσbkσ

)+ Nn

(VVJ− λq

). (18.113)


Det[E±k 1 −

(ϵk VV λ

)]= (Ek± − ϵk)(Ek± − λ) − |V |2 = 0, (18.114)

or

Ek± =ϵk + λ

2±

[( ϵk − λ2

)2+ |V |2

] 12

, (18.115)


{ukvk

}=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣12± (ϵk − λ)/2

2√(

ϵk−λ2

)2+ |V |2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

12

. (18.116)

28 Heavy electrons




HMFT =∑

kσ

(c†kσ, f †kσ

)h(k)︷!!!︸︸!!!︷(

ϵk VV λ

) (ckσfkσ

)+ NNs

( |V |2J− λq

)(18.111)

=∑


( |V |2J− λq

).




ψkσ =

(ckσfkσ

), ψ†kσ =

(c†kσ, f †,kσ

), h(k) =

(ϵk VV λ

). (18.112)



HMFT =∑

kσ

(a†kσ, b†kσ

) (Ek+ 00 Ek−

) (akσbkσ

)+ Nn

(VVJ− λq

). (18.113)


Det[E±k 1 −

(ϵk VV λ

)]= (Ek± − ϵk)(Ek± − λ) − |V |2 = 0, (18.114)

or

Ek± =ϵk + λ

2±

[( ϵk − λ2

)2+ |V |2

] 12

, (18.115)


{ukvk

}=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣12± (ϵk − λ)/2

2√(

ϵk−λ2

)2+ |V |2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

12

. (18.116)

30 Heavy electrons


Ntot = ⟨∑

kλσ

nkλσ⟩ = ⟨n f + nc⟩ (18.118)


Ntot = NVFS a3

(2π)3 = Q + nc, (18.119)



|MF⟩ =∏

|k|<kFσ

b†kσ|0⟩ =∏

|k|<kFσ

(−vkckσ + uk f †kσ)|0⟩. (18.120)


|GW⟩ = PQ

∏

|k|<kFσ

(−vkckσ + uk f †kσ)|0⟩, (18.121)


PQ =∏

j

PQ( j) =∫ 2π

0

∏

j

dα j

2πei

∑j α j(n f j−Q). (18.122)




FMF = −NT∑

k,iωr

Tr ln[

−G−1k (iωr)︷!!!!!!!!!!!!!︸︸!!!!!!!!!!!!!︷

−iωr +

(ϵk VV λ

)]+Ns

(N|V |2

J− λQ

), (18.123)


Tr ln[−iωr1 +

(ϵk VV λ

)]= ln det

[−z1 +

(ϵk VV λ

)]= ln


]

=∑

n=±ln(Ekn − iωr). (18.124)

30 Heavy electrons


Ntot = ⟨∑

kλσ

nkλσ⟩ = ⟨n f + nc⟩ (18.118)


Ntot = NVFS a3

(2π)3 = Q + nc, (18.119)



|MF⟩ =∏

|k|<kFσ

b†kσ|0⟩ =∏

|k|<kFσ

(−vkckσ + uk f †kσ)|0⟩. (18.120)


|GW⟩ = PQ

∏

|k|<kFσ

(−vkckσ + uk f †kσ)|0⟩, (18.121)


PQ =∏

j

PQ( j) =∫ 2π

0

∏

j

dα j

2πei

∑j α j(n f j−Q). (18.122)




FMF = −NT∑

k,iωr

Tr ln[

−G−1k (iωr)︷!!!!!!!!!!!!!︸︸!!!!!!!!!!!!!︷

−iωr +

(ϵk VV λ

)]+Ns

(N|V |2

J− λQ

), (18.123)


Tr ln[−iωr1 +

(ϵk VV λ

)]= ln det

[−z1 +

(ϵk VV λ

)]= ln


]

=∑

n=±ln(Ekn − iωr). (18.124)

30 Heavy electrons


Ntot = ⟨∑

kλσ

nkλσ⟩ = ⟨n f + nc⟩ (18.118)


Ntot = NVFS a3

(2π)3 = Q + nc, (18.119)



|MF⟩ =∏

|k|<kFσ

b†kσ|0⟩ =∏

|k|<kFσ

(−vkckσ + uk f †kσ)|0⟩. (18.120)


|GW⟩ = PQ

∏

|k|<kFσ

(−vkckσ + uk f †kσ)|0⟩, (18.121)


PQ =∏

j

PQ( j) =∫ 2π

0

∏

j

dα j

2πei

∑j α j(n f j−Q). (18.122)




FMF = −NT∑

k,iωr

Tr ln[

−G−1k (iωr)︷!!!!!!!!!!!!!︸︸!!!!!!!!!!!!!︷

−iωr +

(ϵk VV λ

)]+Ns

(N|V |2

J− λQ

), (18.123)


Tr ln[−iωr1 +

(ϵk VV λ

)]= ln det

[−z1 +

(ϵk VV λ

)]= ln


]

=∑

n=±ln(Ekn − iωr). (18.124)

28 Heavy electrons




HMFT =∑

kσ

(c†kσ, f †kσ

)h(k)︷!!!︸︸!!!︷(

ϵk VV λ

) (ckσfkσ

)+ NNs

( |V |2J− λq

)(18.111)

=∑


( |V |2J− λq

).




ψkσ =

(ckσfkσ

), ψ†kσ =

(c†kσ, f †,kσ

), h(k) =

(ϵk VV λ

). (18.112)



HMFT =∑

kσ

(a†kσ, b†kσ

) (Ek+ 00 Ek−

) (akσbkσ

)+ Nn

(VVJ− λq

). (18.113)


Det[E±k 1 −

(ϵk VV λ

)]= (Ek± − ϵk)(Ek± − λ) − |V |2 = 0, (18.114)

or

Ek± =ϵk + λ

2±

[( ϵk − λ2

)2+ |V |2

] 12

, (18.115)


{ukvk

}=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣12± (ϵk − λ)/2

2√(

ϵk−λ2

)2+ |V |2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

12

. (18.116)31 Mean-field theory of the Kondo Lattice

!Fig. 18.10 (a) High temperature state: small Fermi surface with a background of spins; (b)Lowtemperature state where large Fermi surface develops against a background ofpositive charge. Each spin “ionizes” into Q heavy electrons, leaving behind abackground of Kondo singlets, each with charge +Qe.

Next, by carrying out the summation over Matsubara frequencies, using the result −T∑

iωr ln(Ekn − iωr) =−T ln(1 + e−βEkn ), we obtain

FN= −T

∑

k,±ln

[1 + e−βEk±

]+Ns

(V2

J− λq

). (18.125)

Let us discuss the ground-state, in which only the lower-band contributes to the Free energy. As T → 0,we can replace −T ln(1+ e−βEk )→ θ(−Ek)Ek, so the ground-state energy E0 = F(T = 0) involves an integralover the occupied states of the lower band:

Eo

NNs=

∫ 0

−∞dEρ∗(E)E +

(V2

J− λq

)(18.126)

where we have introduced the density of heavy electron states ρ∗(E) =∑

k,± δ(E −E(±)k ). Now by (18.114) the

relationship between the energy E of the heavy electrons and the energy ϵ of the conduction electrons is

E = ϵ +V2

E − λ .

As we sum over momenta k within a given energy shell, there is a one-to-one correspondence between eachconduction electron state and each quasiparticle state, so we can write ρ∗(E)dE = ρ(ϵ)dϵ, where the densityof heavy electron states

ρ∗(E) = ρdϵdE= ρ

(1 +

V2

(E − λ)2

). (18.127)

Here we have approximated the underlying conduction electron density of states by a constant ρ = 1/(2D).The originally flat conduction electron density of states is now replaced by a “hybridization gap”, flanked by

Detailed calcn.


bands. The new Fermi surface volume now counts the total number of particles. To see thisnote that

Ntot = ⟨∑

kλσnkλσ⟩ = ⟨n f + nc⟩

where nkλσ = a†kλσakλσ is the number operator for the quasiparticles and nc is the totalnumber of conduction electrons. This means

Ntot = NVFS(2π)3

= Q + nc.

This expansion of the Fermi surface is a direct manifestation of the creation of new states bythe Kondo effect. It is perhaps worth stressing that these new states would form, even if the local moments were nuclearIn other words, they are electronic states that have only depend on the rotational degrees offreedom of the local moments.

The Free energy of this system is then

FN= −T

∑

k,±

ln[1 + e−βEk±

]+ ns(VVJ− λq)

Let us discuss the ground-state energy, Eo- the limiting T → 0 of this expression. We can write thisin the form

EoNns=

∫ 0

−∞ρ∗(E)E +

(VVJ− λq)

where we have introduced the density of heavy electron states ρ∗(E) =∑k,± δ(E − E

(±)k). Now

the relationship between the energy of the heavy electrons (E) and the energy of the conductionelectrons (ϵ) is given by

E = ϵ +V2

E − λso that the density of heavy electron states related to the conduction electron density of states ρ by


(1 +

V2

(E − λ)2

)(17.46)

The originally flat conduction electron density of states is now replaced by a “hybridizationgap”, flanked by two sharp peaks of width approximately πρV2 ∼ TK . With this information, wecan carry out the integral over the energies, to obtain

EoNns=D2ρ2+

∫ 0

−DdEρVV

E(E − λ)2

+

(VVJ− λq)

(17.47)

where we have assumed that the upper band is empty, and the lower band is partially filled. If weimpose the constraint ∂F∂λ = ⟨n f ⟩ − Q = 0 we obtain

∆

πλ− q = 0

77



Ntot = ⟨∑



Ntot = NVFS(2π)3

= Q + nc.



FN= −T

∑

k,±

ln[1 + e−βEk±

]+ ns(VVJ− λq)


EoNns=

∫ 0

−∞ρ∗(E)E +

(VVJ− λq)


(±)k). Now


E = ϵ +V2



(1 +

V2

(E − λ)2

)(17.46)


EoNns=D2ρ2+

∫ 0

−DdEρVV

E(E − λ)2

+

(VVJ− λq)

(17.47)


∆

πλ− q = 0

77

31 Mean-field theory of the Kondo Lattice




FN= −T

∑

k,±ln

[1 + e−βEk±

]+Ns

(V2

J− λq

). (18.125)


Eo

NNs=

∫ 0

−∞dEρ∗(E)E +

(V2

J− λq

)(18.126)


k,± δ(E −E(±)k ). Now by (18.114) the


E = ϵ +V2

E − λ .



(1 +

V2

(E − λ)2

). (18.127)


31 Mean-field theory of the Kondo Lattice




FN= −T

∑

k,±ln

[1 + e−βEk±

]+Ns

(V2

J− λq

). (18.125)


Eo

NNs=

∫ 0

−∞dEρ∗(E)E +

(V2

J− λq

)(18.126)


k,± δ(E −E(±)k ). Now by (18.114) the


E = ϵ +V2

E − λ .



(1 +

V2

(E − λ)2

). (18.127)


28 Heavy electrons




HMFT =∑

kσ

(c†kσ, f †kσ

)h(k)︷!!!︸︸!!!︷(

ϵk VV λ

) (ckσfkσ

)+ NNs

( |V |2J− λq

)(18.111)

=∑


( |V |2J− λq

).




ψkσ =

(ckσfkσ

), ψ†kσ =

(c†kσ, f †,kσ

), h(k) =

(ϵk VV λ

). (18.112)



HMFT =∑

kσ

(a†kσ, b†kσ

) (Ek+ 00 Ek−

) (akσbkσ

)+ Nn

(VVJ− λq

). (18.113)


Det[E±k 1 −

(ϵk VV λ

)]= (Ek± − ϵk)(Ek± − λ) − |V |2 = 0, (18.114)

or

Ek± =ϵk + λ

2±

[( ϵk − λ2

)2+ |V |2

] 12

, (18.115)


{ukvk

}=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣12± (ϵk − λ)/2

2√(

ϵk−λ2

)2+ |V |2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

12

. (18.116)

32 Heavy electrons

two sharp peaks of width approximately πρV2 ∼ TK (Fig. 18.9). Note that the lower band-width is loweredby an amount −V2/D. With this information, we can carry out the integral over the energies, to obtain

Eo

NNs= ρ

∫ 0

−D−V2/DdEE

(1 +

V2

(E − λ)2

)+

(V2

J− λq

)(18.128)

where we have assumed that the upper band is empty, and the lower band is partially filled. Carrying out theintegral we obtain

Eo

NNs= −ρ

2

(D +

V2

D

)2

+∆

π

∫ 0

−DdE

(1

E − λ +λ

(E − λ)2

)+

(V2

J− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(V2

J− λq

)(18.129)

where we have replaced ∆ = πρV2 and have dropped terms of order O(∆2/D). We can rearrange this expres-sion, absorbing the band-width D and Kondo coupling constant into a single Kondo temperature TK = De−

1Jρ

as followsE0

NNs= −D2ρ

2+∆

πln

( λD

)+

(πρV2

πρJ− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(∆

πρJ− λq

)

= −D2ρ

2+∆

πln

(λ

De−1Jρ

)− λq

= −D2ρ

2+∆

πln

(λ

TK

)− λq. (18.130)

This describes the energy of a whole scaling trajectory of Kondo lattice models with different J(D) and cuttoffD, but fixed Kondo temperature. If we impose the constraint ∂E0

∂λ = ⟨n f ⟩ − Q = 0 we obtain ∆πλ − q = 0, so

Eo(V)NNs

=∆

πln

(∆

πqeTK

)− D2ρ

2, (∆ = πρ|V |2) (18.131)

Let us pause for a moment to consider this energy functional qualitatively. There are two points to be made

!Fig. 18.11 Mexican hat potential for the Kondo Lattice, evaluated at constant ⟨n f ⟩ = Q as afunction of a complex hybridization V = |V |eiφ

32 Heavy electrons


Eo

NNs= ρ

∫ 0

−D−V2/DdEE

(1 +

V2

(E − λ)2

)+

(V2

J− λq

)(18.128)


Eo

NNs= −ρ

2

(D +

V2

D

)2

+∆

π

∫ 0

−DdE

(1

E − λ +λ

(E − λ)2

)+

(V2

J− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(V2

J− λq

)(18.129)


1Jρ

as followsE0

NNs= −D2ρ

2+∆

πln

( λD

)+

(πρV2

πρJ− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(∆

πρJ− λq

)

= −D2ρ

2+∆

πln

(λ

De−1Jρ

)− λq

= −D2ρ

2+∆

πln

(λ

TK

)− λq. (18.130)



Eo(V)NNs

=∆

πln

(∆

πqeTK

)− D2ρ

2, (∆ = πρ|V |2) (18.131)



32 Heavy electrons


Eo

NNs= ρ

∫ 0

−D−V2/DdEE

(1 +

V2

(E − λ)2

)+

(V2

J− λq

)(18.128)


Eo

NNs= −ρ

2

(D +

V2

D

)2

+∆

π

∫ 0

−DdE

(1

E − λ +λ

(E − λ)2

)+

(V2

J− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(V2

J− λq

)(18.129)


1Jρ

as followsE0

NNs= −D2ρ

2+∆

πln

( λD

)+

(πρV2

πρJ− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(∆

πρJ− λq

)

= −D2ρ

2+∆

πln

(λ

De−1Jρ

)− λq

= −D2ρ

2+∆

πln

(λ

TK

)− λq. (18.130)



Eo(V)NNs

=∆

πln

(∆

πqeTK

)− D2ρ

2, (∆ = πρ|V |2) (18.131)





Ntot = ⟨∑



Ntot = NVFS(2π)3

= Q + nc.



FN= −T

∑

k,±

ln[1 + e−βEk±

]+ ns(VVJ− λq)


EoNns=

∫ 0

−∞ρ∗(E)E +

(VVJ− λq)


(±)k). Now


E = ϵ +V2



(1 +

V2

(E − λ)2

)(17.46)


EoNns=D2ρ2+

∫ 0

−DdEρVV

E(E − λ)2

+

(VVJ− λq)

(17.47)


∆

πλ− q = 0

77


so that the ground-state energy can be written

EoNns=∆

πln(∆eπqTK

)(17.48)

where TK = De−1Jρ as before.

Let us pause for a moment to consider this energy functional qualitatively. The Free energysurface has the form of “Mexican Hat” at low temperatures. The minimum of this functional willthen determine a familiy of saddle point values V = Voeiθ, where θ can have any value. If wedifferentiate the ground-state energy with respect to V2, we obtain

0 =1πln(∆e2

πqTK

)

or∆ =πqe2TK

confirming that ∆ ∼ TK .

Composite Nature of the heavy quasiparticle in the Kondo lattice.

We now turn to discuss the nature of the heavy quasiparticles in the Kondo lattice. Clearly, at anoperational level, the composite f−electrons are formed in the same way as in the impurity model,but at each site, i.e

1NΓαβ( j, t)ψ jα(t) −→

(VJ

)f jα(t)

This composite object admixes with conduction electrons at a single site- site j. The bound-stateamplitude in this expression can be written

−VoJ=1N⟨ f †βψβ⟩ (17.49)

To evaluate the contributions to this sum, it is useful to notice that the condition ∂E/∂V = 0 can bewritten

1N∂E∂Vo

= 0 =VoJ+1N⟨ f †βψβ⟩

=VoJ+ Vo

∫ 0

−DdEρ

E(E − λ)2

(17.50)

where we have used (17.47) to evaluate the derivative. From this we see that we can write

VoJ= −Vo

∫ 0

−DdEρ(1

E − λ+

λ

(E − λ)2

)

= −Voρ ln[λeD

](17.51)

78

32 Heavy electrons


Eo

NNs= ρ

∫ 0

−D−V2/DdEE

(1 +

V2

(E − λ)2

)+

(V2

J− λq

)(18.128)


Eo

NNs= −ρ

2

(D +

V2

D

)2

+∆

π

∫ 0

−DdE

(1

E − λ +λ

(E − λ)2

)+

(V2

J− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(V2

J− λq

)(18.129)


1Jρ

as followsE0

NNs= −D2ρ

2+∆

πln

( λD

)+

(πρV2

πρJ− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(∆

πρJ− λq

)

= −D2ρ

2+∆

πln

(λ

De−1Jρ

)− λq

= −D2ρ

2+∆

πln

(λ

TK

)− λq. (18.130)



Eo(V)NNs

=∆

πln

(∆

πqeTK

)− D2ρ

2, (∆ = πρ|V |2) (18.131)



32 Heavy electrons


Eo

NNs= ρ

∫ 0

−D−V2/DdEE

(1 +

V2

(E − λ)2

)+

(V2

J− λq

)(18.128)


Eo

NNs= −ρ

2

(D +

V2

D

)2

+∆

π

∫ 0

−DdE

(1

E − λ +λ

(E − λ)2

)+

(V2

J− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(V2

J− λq

)(18.129)


1Jρ

as followsE0

NNs= −D2ρ

2+∆

πln

( λD

)+

(πρV2

πρJ− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(∆

πρJ− λq

)

= −D2ρ

2+∆

πln

(λ

De−1Jρ

)− λq

= −D2ρ

2+∆

πln

(λ

TK

)− λq. (18.130)



Eo(V)NNs

=∆

πln

(∆

πqeTK

)− D2ρ

2, (∆ = πρ|V |2) (18.131)



Detailed calcn.

32 Heavy electrons


Eo

NNs= ρ

∫ 0

−D−V2/DdEE

(1 +

V2

(E − λ)2

)+

(V2

J− λq

)(18.128)


Eo

NNs= −ρ

2

(D +

V2

D

)2

+∆

π

∫ 0

−DdE

(1

E − λ +λ

(E − λ)2

)+

(V2

J− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(V2

J− λq

)(18.129)


1Jρ

as followsE0

NNs= −D2ρ

2+∆

πln

( λD

)+

(πρV2

πρJ− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(∆

πρJ− λq

)

= −D2ρ

2+∆

πln

(λ

De−1Jρ

)− λq

= −D2ρ

2+∆

πln

(λ

TK

)− λq. (18.130)



Eo(V)NNs

=∆

πln

(∆

πqeTK

)− D2ρ

2, (∆ = πρ|V |2) (18.131)



32 Heavy electrons


Eo

NNs= ρ

∫ 0

−D−V2/DdEE

(1 +

V2

(E − λ)2

)+

(V2

J− λq

)(18.128)


Eo

NNs= −ρ

2

(D +

V2

D

)2

+∆

π

∫ 0

−DdE

(1

E − λ +λ

(E − λ)2

)+

(V2

J− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(V2

J− λq

)(18.129)


1Jρ

as followsE0

NNs= −D2ρ

2+∆

πln

( λD

)+

(πρV2

πρJ− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(∆

πρJ− λq

)

= −D2ρ

2+∆

πln

(λ

De−1Jρ

)− λq

= −D2ρ

2+∆

πln

(λ

TK

)− λq. (18.130)



Eo(V)NNs

=∆

πln

(∆

πqeTK

)− D2ρ

2, (∆ = πρ|V |2) (18.131)



32 Heavy electrons


Eo

NNs= ρ

∫ 0

−D−V2/DdEE

(1 +

V2

(E − λ)2

)+

(V2

J− λq

)(18.128)


Eo

NNs= −ρ

2

(D +

V2

D

)2

+∆

π

∫ 0

−DdE

(1

E − λ +λ

(E − λ)2

)+

(V2

J− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(V2

J− λq

)(18.129)


1Jρ

as followsE0

NNs= −D2ρ

2+∆

πln

( λD

)+

(πρV2

πρJ− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(∆

πρJ− λq

)

= −D2ρ

2+∆

πln

(λ

De−1Jρ

)− λq

= −D2ρ

2+∆

πln

(λ

TK

)− λq. (18.130)



Eo(V)NNs

=∆

πln

(∆

πqeTK

)− D2ρ

2, (∆ = πρ|V |2) (18.131)



�E0

��= 0

<latexit sha1_base64="XDFS1geKQLjnexgzRc7NX8Y9jyk=">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</latexit><latexit sha1_base64="XDFS1geKQLjnexgzRc7NX8Y9jyk=">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</latexit>

32 Heavy electrons


Eo

NNs= ρ

∫ 0

−D−V2/DdEE

(1 +

V2

(E − λ)2

)+

(V2

J− λq

)(18.128)


Eo

NNs= −ρ

2

(D +

V2

D

)2

+∆

π

∫ 0

−DdE

(1

E − λ +λ

(E − λ)2

)+

(V2

J− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(V2

J− λq

)(18.129)


1Jρ

as followsE0

NNs= −D2ρ

2+∆

πln

( λD

)+

(πρV2

πρJ− λq

)

= −D2ρ

2+∆

πln

( λD

)+

(∆

πρJ− λq

)

= −D2ρ

2+∆

πln

(λ

De−1Jρ

)− λq

= −D2ρ

2+∆

πln

(λ

TK

)− λq. (18.130)



Eo(V)NNs

=∆

πln

(∆

πqeTK

)− D2ρ

2, (∆ = πρ|V |2) (18.131)



Heavy electron = (electron x spinflip)

Singlet formation

Coherence and composite fermions

•The large N approach to the Kondo lattice. Spin x conduction = composite fermion

Composite FermionJ

Nc†j�S�� V f†

j�<latexit sha1_base64="4Hu+csdI3pdMYgaAk82J82Ji8+g=">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</latexit><latexit sha1_base64="4Hu+csdI3pdMYgaAk82J82Ji8+g=">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</latexit>

36 Heavy electrons

electrons that span decades of energy up to a cutoff, beit the Debye energy ωD in superconductivity or the(much larger) bandwidth D in the Kondo effect [32, 33].

To follow this analogy in greater depth, recall that in the path integral the Kondo interaction factorizes as

JN

c†βS αβcα −→ V(c†α fα

)+

(f †αcα

)V + N

VVJ, (18.143)

so by comparing the right and left hand side, we see that the composite operators S βαcβ and c†βS αβ behaveas a single fermion denoted by the contractions:

1N

∑

β

S βαcβ =(

VJ

)fα,

1N

∑

β

c†βS αβ =(V

J

)f †α, (18.144)

Composite Fermion

Physically, this means that the spins bind high energy electrons, transforming themselves into compositeswhich then hybridize with the conduction electrons. The resulting “heavy fermions” can be thought of asmoments ionized in the magnetically polar electron fluid to form mobile, negatively charged heavy electronswhile leaving behind a positively charged “Kondo singlet”.

Microscopically, the many body amplitude to scatter an electron off a local moment develops a bound-statepole, which for large N we can denote by the diagrams:

Γ ≡O(1)

V V+

O(1/N)

The leading diagram describes a kind of “condensation” of the hybridization field; the second and higherterms describe the smaller O(1/N) fluctuations around the mean-field theory.

The temporal correlations between spin-flips and conduction electrons extend over a finite time, describedby the contraction

1N

∑

β

cβ(τ)S βα(τ′) = g(τ − τ′) fα(τ′). (18.145)

Here the spin-flip correlation function g(τ − τ′) is an analogue of the Gor’kov function, extending out to acoherence time τK ∼ !/TK . Notice that in contrast to the Cooper pair, this composite object is a fermion andthus requires a distinct operator fα for its expression. The Fourier (Laplace) decomposition of g(τ) describesthe Spectral distribution of electrons and spin-flips inside the composite f-electron which we may calculateas follows:

1N

∑

β

cβ(τ)S βα(τ′) =1N

∑

β

cβ(τ) f †β(τ′) fα(τ′)

=1N

∑

β

⟨Tcβ(τ) f †β(τ′)⟩ fα(τ′)

5.22 Piers Coleman

1N

X

�

S �↵c� = VJ

!f↵,

1N

X

�

c†�S ↵� =✓V

J

◆f †↵, (58)

Composite Fermion

Physically, this means that the spins bind high energy electrons, transforming themselves intocomposites which then hybridize with the conduction electrons. The resulting “heavy fermions”can be thought of as moments ionized in the magnetically polar electron fluid to form mobile,negatively charged heavy electrons while leaving behind a positively charged “Kondo singlet”.Microscopically, the many body amplitude to scatter an electron o↵ a local moment develops abound-state pole, which for large N we can denote by the diagrams:

� ⌘O(1)

V V+

O(1/N)+ . . .

The leading diagram describes a kind of “condensation” of the hybridization field; the secondand higher terms describe the smaller O(1/N) fluctuations around the mean-field theory.By analogy with superconductivity, we can associate a wavefunction associated with the tem-poral correlations between spin-flips and conduction electrons, as follows

1N

X

�

c�(⌧)S �↵(⌧0) = g(⌧ � ⌧0) f↵(⌧0). (59)

where the spin-flip correlation function g(⌧�⌧0) is an analogue of the Gor’kov function, extend-ing over a coherence time ⌧K ⇠ ~/TK . Notice that in contrast to the Cooper pair, this compositeobject is a fermion and thus requires a distinct operator f↵ for its expression.

4 Heavy Fermion Superconductivity

We now take a brief look at heavy fermion superconductivity. There are a wide variety ofheavy electron superconductors, almost all of which are nodal superconductors, in which thepairing force derives from the interplay of magnetism and electron motion. In the heavyfermion compounds, as in many other strongly correlated electron systems superconductiv-ity frequently develops at the border of magnetism, near the quantum critical point where themagnetic transition temperature has been suppressed to zero. In some of them, such as UPt3

(Tc=0.5K) [36] the superconductivity develops out of a well-developed heavy Fermi liquid,and in these cases, we can consider the superconductor to be paired by magnetic fluctuationswithin a well-formed heavy Fermi liquid. However, in many other superconductors, such asUBe13(Tc=1K) [37, 38], the 115 superconductors CeCoIn5 (Tc=2.3K) [39], CeRhIn5 underpressure (Tc=2K) [17], NpAl2Pd5(Tc=4.5K) [40] and PuCoGa5 (Tc=18.5K) [41, 42], the su-perconducting transition temperature is comparable with the Kondo temperature. In many of

Glue vs Fabric.

k

-k

k’ k’’

-k’ -k’’

Spin fluctuations = pairing bosons

~1/3 R ln(2)

SPIN Hilbert space BUILDS the pairs.

Glue

Fabric: spins make the pairs

R lnW =

Z T

0dT 0C

0

T 0

(π,0)→s±

“Hilbert Space Spectroscopy” How?

Anderson: RVB (1987); Coleman Andrei (1989)Emery & Kivelson: composite pairs (1993)

k-k k’Eliashberg Approach (cf B. Keimer et al)

4058 P Coleman and N Andrei

Energy

ESL-TK Kondo-stobilised spin l iquid

Figure 1. Energy diagram illustrating how the energy of a spin liquid can be lowered below that of an antiferromagnetic state by Kondo compensation.

states [ 6 ] , and x may be large due to frustrationt. A combination of these two factors will then tend to suppress development of conventional local moment magnetism.

A useful way to visualise the formation of a Kondo-stabilised spin liquid is to use Anderson’s resonating valence bond picture [7] (figure 2). A pure spin liquid is visualised by linking pairs o f f spins together into singlets or valence bonds. Spin exchange between sites causes the ends of the valence bonds to resonate throughout the spin system forming a sort of ‘quantum spaghetti’. When we introduce Kondo coupling to the conduction electrons, the ends of the valence bonds occasionally link up with a conduction electron lying within an energy TK of the Fermi level, resonantly scattering the electrons close to the Fermi energy. Typically, the number of conduction electrons within this energy is far smaller than the number of f spins, and in keeping with the Nozieres exhaustion principle, most of the valence bonds must stay within the spin liquid.

Conduction e-

f spins

L e - I b i

Figure 2. Illustrating how Kondo compensation of a spin liquid results in an escape of the valence bonds into the conduction sea, generating singlet pairs of conduction electrons, thereby inducing a pairing component to the resonant Kondo scattering of conduction electrons.

Occasionally however, spin exchange will occur between two valence bonds that link conduction electrons to f moments, causing the momentary escape of one valence bond entirely into the conduction sea. Such brief excursions of valence bonds into the conduction sea will produce resonant singlet pairing amongst low-energy conduction electrons, and as we shall see, this generates superconductivity in the heavy fermion system.

In this paper we examine this hypothesis within a new path integral formalism, using a lattice model for heavy fermions that contains both RKKY and Kondo interactions.

t In the 2D cuprate superconductors we believe a similar effect may also be taking place, where in this case TK should be replaced by JK and a is very close to unity. See [ 7 ] .

~ 1/3 R log 2

Pauli

Curie

UPt3

Pauli paramagnetic by 30K

Frings et al. J. Magn. Magn. Mater. 31, 240(1983) Brison et al. J. Low Temp. Phys. 95, 145(1994)

CeCoIn5

Curie

No Pauli paramagnetismTc = 2.3K

Shishido et al. JPSJ 71, 162 (2002) Petrovic et al. J.Phys Condens. Matter 13 337 (2001)

Tc = 0.5K

115 Materials.


these materials, the entropy of condensation

S c =

Z Tc

0

CV

TdT (60)

can be as large as (1/3)R ln 2 per rare earth ion, indicating that the spin is, in some-way entan-gling with the conduction electrons to build the condensate. In this situation, we need to be ableto consider the Kondo e↵ect and superconductivity on an equal footing.

Fig. 12: (a) Phase diagram of 115 compounds CeMIn5, adapted from [43], showing magneticand superconducting phases as a function of alloy concentration. (b) Sketch of specific heat co-e�cient of CeCoIn5, (with nuclear Schottky contribution subtracted), showing the large entropyof condensation associated with the superconducting state. (After Petrovic et al 2001 [39]).

4.1 Symplectic spins and SP (N).

Although the SU(N) large N expansion provides a very useful description of the normal stateof heavy fermion metals and Kondo insulators, there is strangely, no superconducting solution.This short-coming lies in the very structure of the S U(N) group. S U(N) is perfectly tailoredto particle physics, where the physical excitations - the mesons and baryons appear as colorsinglets, with the meson a a qq quark-antiquark singlet while the baryon is an N-quark singletq1q2 . . . qN , (where of course N = 3 in reality). In electronic condensed matter, the mesonbecomes a particle-hole pair, but there are no two-particle singlets in S U(N) beyond N = 2. Theorigin of this failure can be traced back to the absence of a consistent definition of time-reversalsymmetry in S U(N) for N > 2. This means that singlet Cooper pairs and superconductivity cannot develop at the large N limit.A solution to this problem which grew out an approach developed by Read and Sachdev [44] forfrustrated magnetism, is to use the symplectic group S P(N), where N must be an even number

⇥(x, �)

S[�]

�, x

1N� �eff

classi

cal p

ath

?

SU(N): Mesonsqq

<latexit sha1_base64="MvB6FhftxnuN9A35V5IdjOI76bo=">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</latexit><latexit sha1_base64="MvB6FhftxnuN9A35V5IdjOI76bo=">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</latexit>

Baryonsq1q2 . . . qN

<latexit sha1_base64="vINqq/aPxq1OaMY5s6mWvmI9Jtw=">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</latexit><latexit sha1_base64="vINqq/aPxq1OaMY5s6mWvmI9Jtw=">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</latexit>

SP(N): qq<latexit sha1_base64="MvB6FhftxnuN9A35V5IdjOI76bo=">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</latexit><latexit sha1_base64="MvB6FhftxnuN9A35V5IdjOI76bo=">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</latexit>

Cooper pairsqaq�a

<latexit sha1_base64="sRua9RaeYT+mKuQ9djNVZpxggI8=">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</latexit><latexit sha1_base64="sRua9RaeYT+mKuQ9djNVZpxggI8=">AAACwHicjZFLb9QwEMe94VWW1xa4cbGoKnFhleUAHCt64VgQSyttltXEmaTW+lXbYbtYuXHj03CFT8K3wUmK1EYgMdJI499/xmPP5EZw59P01yi5dv3GzVs7t8d37t67/2Cy+/Cj07VlOGdaaHuSg0PBFc499wJPjEWQucDjfH3Y6sef0Tqu1Qe/NbiUUClecgY+otVkP+vuCO+xaOjZCujFubKIqiXhOTR0NdmbTdPO6L+DvYPHpLOj1e7oa1ZoVktUnglwbjFLjV8GsJ4zgc04qx0aYGuocBFDBRLdMnS9G7ofSUFLbaMrTzt6uSKAdG4r85gpwZ+6odbCv2mL2pevl4ErU3tUrG9U1oJ6Tdvh0IJbZF5sYwDM8vhWyk7BAvNxhONM4YZpKUEVmYUQXVXdXy7xogqfsgKqCu1VIV+HkOUlXTcDbnpuhvy85+cDXrf5tQFr9WYgFZsoFXqj/ojj/9vb/MX05TR9F/f3pt8f2SFPyFPyjMzIK3JA3pIjMieMfCPfyQ/yMzlMeKKTsz41GV3UPCJXLPnyG+1w4UI=</latexit>

“Symplectic Large N” R. Flint and PC ’08

H =�

k��kc†k�ck� +

JK

N

�

j

c†j�c j�S ��( j) +JH

2N

�

(i, j)

S ��(i)S ��( j).

No Pairs !

SP(N) Large N Approach. H = Hc + HK + HRKKY


4.2 Superconductivity in the Kondo Heisenberg Model

Let us take a look at the way this works in a nearest neighbor “Kondo Heisenberg model” [47],

H = Hc + HK + HM. (63)

Here Hc =P

k� ✏kc†k�ck� describes the conduction sea, whereas HK and HM are the Kondo andHeisenberg (RKKY) interactions, respectively. These take the form

HK =JK

N

X

j

c† j↵c j�S �↵( j)! � JK

N

X

i, j

⇣(c† j↵ f j↵)( f † j�c j�) + ↵�(c† j↵ f † j�↵)( f j��c j�)

⌘

HM =JH

2N

X

(i, j)

S ↵�( j)S �↵( j)! � JH

N

X

j

h( f †i↵ f j↵)( f † j� fi�) + ↵�( f †i↵ f † j�↵)( f j�� fi�)

i(64)

where we’ve introduced the notation ↵ = Sgn(↵) and have shown how the interactions areexpanded in into particle-hole and particle-particle channels. Notice how the interactions areequally divided between particle-hole and particle-particle channels. When we carry out theHubbard Stratonovich decoupling, in each of these terms, we obtain

HK !X

j

hc† j↵

⇣Vj f j↵ + ↵�

Kj f † j�↵

⌘+ H.c

i+ N

0BBBBB@|Vj|2 + |�K

j |2

JK

1CCCCCA

HH !X

(i, j)

hti j f †i↵ f j↵ + �i j↵ f †i↵ f † j�↵ + H.c

i+ N

" |ti j|2 + |�i j|2JH

#(65)

At each site, we can always rotate the f-electrons in particle-hole space to remove the “Kondopairing” component and set �K

j = 0, but the pairing terms in the Heisenberg component cannot be eliminated. This mean-field theory describes a kind of Kondo stabilized spin-liquid [47].The physical picture is as follows: in practice, a spin-liquid is unstable to magnetism, but itshappy co-existence with the Kondo e↵ect brings its energy below that of the antiferromagnet.The hybridization of the f with the conduction sea converts the spinons of the spin-liquid intocharged fermions. The ti j terms describe various kind of exotic density waves. The �i j termsnow describe pairing amongst the composite fermions.To develop a simple theory of the superconducting state, we restrict our attention to uniform,static saddle points, dropping the ti j. Lets look at the resulting mean-field theory. In two dimen-sions, this becomes

H =X

k,↵>0

(c†k↵, f †k↵)266664✏k⌧3 V⌧1

V⌧1 ~w · ~⌧ + �Hk⌧1

3777750BBBB@ck↵

fk↵

1CCCCA +NsN

|V |2JK+ 2|�H |2JH

!(66)

wherec†k↵ = (c†k↵, ↵c�k,�↵), f †k↵ = ( f †k↵, ↵ f�k,�↵) (67)

are Nambu spinors for the conduction and f-electrons. The vector ~Wof Lagrange multiplierscouples to the isospin of the f-electrons: stationarity of the Free energy with respect to thisvariable imposes the mean-field constraint that h f †~⌧ f i = 0. The function �Hk = �k(cos kx �




H = Hc + HK + HM. (63)

Here Hc =P


HK =JK

N

X

j

c† j↵c j�S �↵( j)! � JK

N

X

i, j


⌘

HM =JH

2N

X

(i, j)

S ↵�( j)S �↵( j)! � JH

N

X

j


i(64)


HK !X

j

hc† j↵

⇣Vj f j↵ + ↵�

Kj f † j�↵

⌘+ H.c

i+ N


j |2

JK

1CCCCCA

HH !X

(i, j)


i+ N

" |ti j|2 + |�i j|2JH

#(65)



H =X

k,↵>0

(c†k↵, f †k↵)266664✏k⌧3 V⌧1

V⌧1 ~w · ~⌧ + �Hk⌧1

3777750BBBB@ck↵

fk↵

1CCCCA +NsN

|V |2JK+ 2|�H |2JH

!(66)






H = Hc + HK + HM. (63)

Here Hc =P


HK =JK

N

X

j

c† j↵c j�S �↵( j)! � JK

N

X

i, j


⌘

HM =JH

2N

X

(i, j)

S ↵�( j)S �↵( j)! � JH

N

X

j


i(64)


HK !X

j

hc† j↵

⇣Vj f j↵ + ↵�

Kj f † j�↵

⌘+ H.c

i+ N


j |2

JK

1CCCCCA

HH !X

(i, j)


i+ N

" |ti j|2 + |�i j|2JH

#(65)



H =X

k,↵>0

(c†k↵, f †k↵)266664✏k⌧3 V⌧1

V⌧1 ~w · ~⌧ + �Hk⌧1

3777750BBBB@ck↵

fk↵

1CCCCA +NsN

|V |2JK+ 2|�H |2JH

!(66)






H = Hc + HK + HM. (63)

Here Hc =P


HK =JK

N

X

j

c† j↵c j�S �↵( j)! � JK

N

X

i, j


⌘

HM =JH

2N

X

(i, j)

S ↵�( j)S �↵( j)! � JH

N

X

j


i(64)


HK !X

j

hc† j↵

⇣Vj f j↵ + ↵�

Kj f † j�↵

⌘+ H.c

i+ N


j |2

JK

1CCCCCA

HH !X

(i, j)


i+ N

" |ti j|2 + |�i j|2JH

#(65)



H =X

k,↵>0

(c†k↵, f †k↵)266664✏k⌧3 V⌧1

V⌧1 ~w · ~⌧ + �Hk⌧1

3777750BBBB@ck↵

fk↵

1CCCCA +NsN

|V |2JK+ 2|�H |2JH

!(66)



Uniform solution:

SP(N) Large N Approach. H = Hc + HK + HRKKY

5.24 Piers Coleman

[45, 46]. This little-known group is a subgroup of S U(N). In fact for N = 2, S U(2) = S P(2)are identical, but they diverge for higher N. For example, S U(4) has 15 generators, but itssymplectic sub-group S P(4) has only 10. At large N, S P(N) has approximately half the numberof generators of S U(N). The symplectic property of the group allows it to consistently treattime-reversal symmetry of spins and it also allows the formation of two-particle singlets for anyN.One of the interesting aspects of S P(N) spin operators, is their relationship to pair operators.Consider S P(2) ⌘ S U(2): the pair operator is † = f †" f †# and since this operator is a sin-glet, it commutes with the spin operators, [ , ~S ] = [ †, ~S ] = 0 which, since and † arethe generators of particle-hole transformations, implies that the S U(2) spin operator is particle-hole symmetric. It is this feature that is preserved by the S P(N) group, all the way out toN ! 1. In fact, we can use this fact to write down an S P(N) spins as follows: an S U(N)spin is given by SS U(N)

↵� = f †↵ f�. Under a particle hole transformation f↵ ! Sgn(↵) f�↵. If wetake the particle-hold transform of the S U(N) spin and add it to itself we obtain an S P(N) spin,

S ↵� = f †↵ f� + Sgn(↵�) f�� f †�↵, (61)

Symplectic Spin operatorwhere the values of the spin indices are ↵, � 2 {±1/2, . . . ,±N/2}. This spin operator commuteswith the three isospin variables

⌧3 = nf � N/2, ⌧+ =X

↵>0

f †↵ f †�↵, ⌧� =X

↵>0

f�↵ f↵. (62)

With these local symmetries, the spin is continuous invariant under SU (2) particle-hole rota-tions f↵ ! u f↵ + vSgn↵ f †�↵, where |u2| + |v2| = 1, as you can verify. To define an irreduciblerepresentation of the spin, we also have to impose a constraint on the Hilbert space, which in itssimplest form is ⌧3 = ⌧± = 0, equivalent to Q = N/2 in the S U(N) approach. In other-words,the s-wave part of the f-pairing must vanish identically.

Fig. 13: Phase diagram for the two-dimensional Kondo Heisenberg model, derived in theS P(N) large N approach, adapted from [47], courtesy Rebecca Flint.




H = Hc + HK + HM. (63)

Here Hc =P


HK =JK

N

X

j

c† j↵c j�S �↵( j)! � JK

N

X

i, j


⌘

HM =JH

2N

X

(i, j)

S ↵�( j)S �↵( j)! � JH

N

X

j


i(64)


HK !X

j

hc† j↵

⇣Vj f j↵ + ↵�

Kj f † j�↵

⌘+ H.c

i+ N


j |2

JK

1CCCCCA

HH !X

(i, j)


i+ N

" |ti j|2 + |�i j|2JH

#(65)



H =X

k,↵>0

(c†k↵, f †k↵)266664✏k⌧3 V⌧1

V⌧1 ~w · ~⌧ + �Hk⌧1

3777750BBBB@ck↵

fk↵

1CCCCA +NsN

|V |2JK+ 2|�H |2JH

!(66)



Uniform solution:

49

eI

e+

e

h

Conventional band insulator: adiabatic continuation of the vacuum.

Vacuum

ConvenMonal)band)insulator

50

eI

e+

Topological insulator

Vacuum

Topological)“insulator”)(Berry)ConnecMon)“twisted”)

Gap must close at interface betweentwo different vacua

e

h

Metallic surfaces.

: adiabatically disconnected from thevacuum.

Dirac cone surface states.

Figure 3

Showing (a) topologically trivial band insulator with Z2

= +1 (b) band-crossing of even and oddparity states at an odd number of high symmetry points leads to a topological insulator withZ2

= �1. Each band crossing generates a Dirac cone of spin-momentum locked surface states.

2.1. Topology meets strong correlation

In 2010, Maxim Dzero, Kai Sun, Victor Galitski and Piers Coleman (16) proposed that

Kondo insulators can form strongly interacting versions of the Z2

topological insulator.

The key points motivating this idea were that:

• The spin orbit coupling of f-electrons in a Kondo insulator, of the order of 0.5eV, is

much larger than the characteristic 10meV gap of a Kondo insulator, making these

essentially infinite spin orbit coupled systems, ideal candidates for spin-orbit driven

topological order.

• f-states are odd-parity, whereas the predominantly d-band conduction bands that

hybridize with them are even parity, so that each time there is a band-crossing between

the two, the Z2

index changes sign, leading to a topological insulator.

The TKI proposal provides an appealing potential resolution of a long-standing mystery

in the Kondo insulator SmB6

, which for more than thirty years, had been known to exhibit

a low temperature resistivity plateau (54, 55) (see Fig. 7), which could be naturally under-

stood as a consequence of topologically protected surface states (16, 56). In 2012, teams at

the University of Michigan (17) and the University of California, Irvine (18), confirmed the

existence of robust surface states in SmB6

. Most recently 2014 (57) Xu et al. have detected

the spin-polarized structure of the surface states in these materials that tentatively confirm

their topological character (see discussion in section 4.2).

8 Maxim Dzero et al.

Band Crossing of odd and even parity states Yields a Z2 Topological Insulator (Fu, Kane, Mele, 2007)


Figure 3












topological order.



the two, the Z2













Z2= 1 Z2= -1

Topological Texture of Berry Connection

Band Crossing of odd and even parity states Yields a Z2 Topological Insulator (Fu, Kane, Mele, 2007)


Figure 3












topological order.



the two, the Z2













Z2 =Y

i

�(�i)

Fu)and)Kane) ))))))))(2008))):Parity)equaMon.)

Z2= -1Z2= 1

Topological Texture of Berry Connection

Are Kondo insulators topological?

Dzero, Sun, Galitski, PC Phys. Rev. Lett. 104, 106408 (2010)

Band Theory SmB6: T. Takimoto, J. Phys. Soc. Jpn. 80, 123710 (2011).

Maxim Dzero, Kai Sun, Piers Coleman and Victor Galitski, Phys. Rev. B 85 , 045130-045140 (2012).

Victor Alexandrov, Maxim Dzero and Piers Coleman PRL (2013).

Gutzwiller + Band Theory F. Lu, J. Zhao, H. Weng, Z. Fang and X. Dai, Phys. Rev. Lett. 110, 096401 (2013).

Figure 4

(a) If we ignore the e↵ects of topology in a conventional Kondo insulator, the interaction can beturned on adiabatically. When the interactions are turned on, the lower band is pushed into theupper band. Two bands of the same parity will always repel one-another and will not cross whenthe interactions are turned on. (b) When interactions are turned on in a topological insulator,they can lead to band-crossing and a topological phase transition. Here, interactions cause anf-band to push up into a d-band. Since the two bands have opposite parity, they do not hybridizeat the high symmetry point so band-crossing occurs, leading to a topological phase transition.

hybridization vanishes high symmetry points, and this opens up the possibility that interac-

tions will induce band-crossing, changing the topology of the ground-state. For example, let

us assume that in the non-interacting limit, the ALM Hamiltonian is topologically trivial,

with a completely filled band of f-states (See Fig. 4 b). For a system with time-reversal

and inversion symmetry, the Z2

topological invariant ⌫ = 0, 1 is determined by the parity

operator eigenvalues (15). For the ALM and taking into account (19) it is simply given by

Z2

= (�1)⌫ =8Y

m=1

sign[✏c

(km

)� ✏f

(km

)], (20)

where km

is a momentum at one of the eight high-symmetry points of the 3D Brillouin zone

and ✏c,f

(k) is the dispersion of the conduction and the f -electrons correspondingly. As we

switch the interaction Uf

adiabatically the conduction d-band and f-bands will renormalize,

with the f-level moving upwards relative to the conduction bands due to their stronger

www.annualreviews.org

•Topological Kondo Insulators 11

R. Martin & J. Allen, J. Applied Physics, 50,7561 (1979)

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(Plateau

Many Body Delocalization

Many Body Localized

Three crossings: THREE DIRAC CONESON SURFACE. 54

at the � point. Away from the � point, the eg

orbitals split into two Kramers doublets,

the lower one dipping down at the X point, where it dives through the the 4f bands.

Hybridization between the two bands forces 4f states from the valence to the conduction

band, forming heavy 4f electron band pockets at the X points. Once the d-band crosses

through the f-band at the three X points, so long as there are no other crossings, the

resulting non-interacting band-structure is innevitably topological, independently of the

details of the f-multiplets (See Fig. ??).

Figure 6

Schematic illustration of the band-crossing between d- and f- states at the X point in SmB6

. (a)Bands uncrossed. The filled 4f6 band of f-electrons is a conventional insulator. (b) Bands crossed:the d-band cuts beneath the f-band at the X-point, displacing an odd parity f-state from thevalence band to the conduction band. The resulting (�1)3 sign reversal in the Z

2

index gives riseto a topological insulator.

Infact, in the cubic environment, the six J = 5/2 4f orbitals of the Samarium split into

a �7

doublet and a �8

quartet. LDA studies (? ? ) suggest that the physics of the 4f

orbitals is governed by valence fluctuations involving electrons of the �8

quartet and the

conduction eg

states, e� + 4f5(�(↵)

8

) ⌦ 4f6. The �(↵)

8

(↵ = 1, 2) quartet consists of the

following combination of orbitals: |�(1)

8

i =q

5

6

��± 5

2

↵+q

1

6

��⌥ 3

2

↵, |�(2)

8

i =��± 1

2

↵. This then

leads to a simple physical picture in which the �8

quartet of f -states hybridizes with an eg

quartet of d-states to form a Kondo insulator.

In 2011, Takimoto (? ) introduced a tight-binding model for SmB6

in which the

hopping amplitudes in the Hamiltonian (??) are non-zero for nearest- and next-nearest-

neighbors, while hybridization involves the nearest-neighbor overlap integrals only. The

values of the hopping amplitudes were adjusted to fit the LDA band structure results, while

the e↵ect of interactions between the f -electrons is modelled as a renormalization of the bare

f -energy level and the hybridization. In Takimoto’s model, a singlet d-like orbital inverts

with an f -like orbital at the X point of the bulk Brillouin zone, while the remaining two

bands remain inert. This band inversion at the X points implies the existence of three Dirac

cones on the surface: one at the surface � point and two at the X points. Interestingly,

the corresponding Fermi velocities for the electrons at the � point are the same, while the

Fermi velocities at the X are strongly anisotropic. (? )











Figure 6



2



a �7

doublet and a �8



quartet and the

conduction eg

states, e� + 4f5(�(↵)

8

) ⌦ 4f6. The �(↵)

8



8

i =q

5

6

��± 5

2

↵+q

1

6

��⌥ 3

2

↵, |�(2)

8

i =��± 1

2

↵. This then





in which the














Figure 3












topological order.



the two, the Z2













Features)of)the)new)modelDzero et al, Annual Reviews of Condensed Matter Physics (2016), arXiv 1506.05635

55Three crossings: THREE DIRAC CONES

ON SURFACE.









Figure 6



2



a �7

doublet and a �8



quartet and the

conduction eg

states, e� + 4f5(�(↵)

8

) ⌦ 4f6. The �(↵)

8



8

i =q

5

6

��± 5

2

↵+q

1

6

��⌥ 3

2

↵, |�(2)

8

i =��± 1

2

↵. This then





in which the














Figure 3












topological order.



the two, the Z2













Hybridization of f (P=+) and d (P=-) vanishes at X point.

Features)of)the)new)modelDzero et al, Annual Reviews of Condensed Matter Physics (2016), arXiv 1506.05635

=3

H(k) =�

�k V sk · ��V sk · ��

f k

�

sk = (sin k

x

, sin k

y

, sin k

z

) � k

Like He-3B: an adaptive insulator.

=3

H(k) =�

�k V sk · ��V sk · ��

f k

�

sk = (sin k

x

, sin k

y

, sin k

z

) � k

d(k) = ks(k) = k

XV��(k) = Vsk · ��

56


a special significance for Kondo insulators, which contain odd parity f-electrons hybridizingwith even parity d-electrons. Each time an f-electron crosses through the band-gap, exchangingwith a conduction d-state, this changes the Z2 index, making it highly likely that certain Kondoinsulators are topological. The oldest known Kondo insulator SmB6, discovered almost 50 yearsago was well known to possess a mysterious low temperature conductivity plateau [60,61], andthe idea that this system might be a topological Kondo insulator provided an exciting wayof explaining this old mystery. The recent observation of robust [62, 63] conducting surfacestates in the oldest Kondo insulator SmB6supports one of the key elements of this prediction,prompting a revival of interest in Kondo insulators as a new route for studying the interplay ofstrong interactions and topological order.SmB6 is really a mixed valent system, which takes us a little beyond the scope of this lec-ture. One of the other issues with SmB6, is that its local crystal field configuration is likelyto be a �8 quartet state [64], rather than a Kramers doublet. Nevertheless, key elements of itsputative topological Kondo insulating state are nicely illustrated by a spin-orbit coupled Kondo-Heisenberg model, describing the interaction of Kramer’s doublet f-states with a d-band. Themodel is essentially identical with (Eq. 63)

H =X

k�

✏k †

k�ck� + JK

X

j

† j↵ j�S �↵( j) + JH

X

i, j

S ↵�(i)S �↵( j) (69)

with an important modification that takes into account the large spin-orbit coupling and theodd-parity of the f-states. This forces the local Wannier states j↵ that exchange spin with thelocal moment to be odd parity combinations of nearest neigbour conduction electrons, given by

† j↵ =X

i,�

c†i��↵(Ri � R j) (70)

We’ll consider a simplified model with the form factor

�(R) =

8>><>>:�iR · ~�2 , R 2 n.n

0 otherwise(71)

This form factor describes the spin-orbit mixing between states with orbital angular momentuml di↵ering by one, such as f and d or p and s orbitals. The odd-parity of the form-factor�(R) = ��(�R) derives from the odd-parity f - orbitals, while the prefactor �i ensures thatthe hybridization is invariant under time-reversal. The Fourier transform of this Form factor,�(k) =

PR�(R)eik·R is then

�(k) = ~sk · ~� (72)

where the s-vector ~sk = (sin k1, sin k2, sin k3) is the periodic equivalent of the unit momentumvector k. Notice how ~s(�i) = 0 vanishes at the high symmetry points.The resulting mean-field Hamiltonian takes the form

HT KI =X

k

†kh(k) k +Ns

" V2

JK+

3t2

JH� �Q

!#(73)



H =X

k�

✏k †

k�ck� + JK

X

j

† j↵ j�S �↵( j) + JH

X

i, j

S ↵�(i)S �↵( j) (69)


† j↵ =X

i,�

c†i��↵(Ri � R j) (70)


�(R) =

8>><>>:�iR · ~�2 , R 2 n.n

0 otherwise(71)



�(k) = ~sk · ~� (72)


HT KI =X

k

†kh(k) k +Ns

" V2

JK+

3t2

JH� �Q

!#(73)



H =X

k�

✏k †

k�ck� + JK

X

j

† j↵ j�S �↵( j) + JH

X

i, j

S ↵�(i)S �↵( j) (69)


† j↵ =X

i,�

c†i��↵(Ri � R j) (70)


�(R) =

8>><>>:�iR · ~�2 , R 2 n.n

0 otherwise(71)



�(k) = ~sk · ~� (72)


HT KI =X

k

†kh(k) k +Ns

" V2

JK+

3t2

JH� �Q

!#(73)



H =X

k�

✏k †

k�ck� + JK

X

j

† j↵ j�S �↵( j) + JH

X

i, j

S ↵�(i)S �↵( j) (69)


† j↵ =X

i,�

c†i��↵(Ri � R j) (70)


�(R) =

8>><>>:�iR · ~�2 , R 2 n.n

0 otherwise(71)



�(k) = ~sk · ~� (72)


HT KI =X

k

†kh(k) k +Ns

" V2

JK+

3t2

JH� �Q

!#(73)

5.28 Piers Coleman

where †k = (c†k�, f †k�) and

h(k) =0BBBB@

✏k V~� · ~sk

V~� · ~sk ✏ f k

1CCCCA (74)

while ✏ f k = 2t f (cx+cy+cz)+� (cl ⌘ cos kl) is the dispersion of the f-state resulting from a mean-field decoupling of the intersite Heisenberg coupling in the particle-hole channel. For small k,the hybridization in Hamiltonian h(k) takes the form V~� · k, a form which closely resemblesthe topologically non-trivial triplet p-wave gap structure of superfluid He-3B. Like He-3B, thehybridization only develops at low temperatures, making SmB6 an adaptive insulator.

Fig. 14: (a) When the d-band is above the filled f-band, a trivial insulator is formed. (b) Whenthe d-band crosses the f-band at the three X-points, the Z2 parity changes sign, giving rise to atopological insulator.

Let us for the moment treat h(k) as a rigid band structure. Suppose the f-band were initiallycompletely filled, with a completely empty d-band above it. (See Fig. 14 a). This situationcorresponds to a conventional band insulator with Z2 = +1 Next, let us lower the d-conductionband until the two bands cross at a high symmetry point, causing the gap to close, and thento re-open. We know, from dHvA studies of the iso-electronic material LaB6 [65] (whoseband-structure is identical to SmB6 but lacks the magnetic f-electrons), from ARPES studies[?, 66–68], that in SmB6, the d-band crosses through the Fermi surface at at the three X points.Once the d-band is lowered through the f-band around the three X points, the odd-parity f-statesat the X point move up into the conduction band, to be replaced by even-parity d-states. Thischanges the sign of Z2 ! (�1)3 = �1, producing a topological ground-state. Moreover, sincethere are three crossing, we expect there to be three spin-polarized surface Dirac cones.We end by noting that at the time of writing, our understanding of the physics SmB6 is inrapid flux on both the experimental and theoretical front. Spin resolved ARPES [69] mea-surements have detected the presence of spin-textures in the surface Fermi surfaces around thesurface X point, a strong sign of topologically protected surface states. Two recent theoreti-

5.28 Piers Coleman

where †k = (c†k�, f †k�) and

h(k) =0BBBB@

✏k V~� · ~sk

V~� · ~sk ✏ f k

1CCCCA (74)

while ✏ f k = 2t f (cx+cy+cz)+� (cl ⌘ cos kl) is the dispersion of the f-state resulting from a mean-field decoupling of the intersite Heisenberg coupling in the particle-hole channel. For small k,the hybridization in Hamiltonian h(k) takes the form V~� · k, a form which closely resemblesthe topologically non-trivial triplet p-wave gap structure of superfluid He-3B. Like He-3B, thehybridization only develops at low temperatures, making SmB6 an adaptive insulator.

Fig. 14: (a) When the d-band is above the filled f-band, a trivial insulator is formed. (b) Whenthe d-band crosses the f-band at the three X-points, the Z2 parity changes sign, giving rise to atopological insulator.

Let us for the moment treat h(k) as a rigid band structure. Suppose the f-band were initiallycompletely filled, with a completely empty d-band above it. (See Fig. 14 a). This situationcorresponds to a conventional band insulator with Z2 = +1 Next, let us lower the d-conductionband until the two bands cross at a high symmetry point, causing the gap to close, and thento re-open. We know, from dHvA studies of the iso-electronic material LaB6 [65] (whoseband-structure is identical to SmB6 but lacks the magnetic f-electrons), from ARPES studies[?, 66–68], that in SmB6, the d-band crosses through the Fermi surface at at the three X points.Once the d-band is lowered through the f-band around the three X points, the odd-parity f-statesat the X point move up into the conduction band, to be replaced by even-parity d-states. Thischanges the sign of Z2 ! (�1)3 = �1, producing a topological ground-state. Moreover, sincethere are three crossing, we expect there to be three spin-polarized surface Dirac cones.We end by noting that at the time of writing, our understanding of the physics SmB6 is inrapid flux on both the experimental and theoretical front. Spin resolved ARPES [69] mea-surements have detected the presence of spin-textures in the surface Fermi surfaces around thesurface X point, a strong sign of topologically protected surface states. Two recent theoreti-



H =X

k�

✏k †

k�ck� + JK

X

j

† j↵ j�S �↵( j) + JH

X

i, j

S ↵�(i)S �↵( j) (69)


† j↵ =X

i,�

c†i��↵(Ri � R j) (70)


�(R) =

8>><>>:�iR · ~�2 , R 2 n.n

0 otherwise(71)



�(k) = ~sk · ~� (72)


HT KI =X

k

†kh(k) k +Ns

" V2

JK+

3t2

JH� �Q

!#(73)







7. AFM meets the Kondo Effect.



Custers et al, (2003)

YbRh2Si2 : Field tuned quantum criticality.

Magnetism meets Kondo

V. FRITSCH et al. PHYSICAL REVIEW B 89, 054416 (2014)

FIG. 1. (Color online) (a) View of the ab plane of CePdAl. TheCe and Pd(2) atoms form a plane at z = 0; the Al and Pd(1) atomsform a plane at z = 1/2. (b) View of the ab plane, showing oneof the three symmetry-related magnetic structures as derived fromRef. [25]. Only Ce atoms are shown for clarity. (c) Kagome latticefor comparison.

simulations [33]. Unfortunately, a model incorporating theinterplane coupling J⊥ to be compared with the experimentallydetermined magnetic structure does not exist up to now. Inview of the incommensurate z component τ , even long-rangeinteractions might have to be taken into account. Previous workon CePdAl showed that TN can be suppressed by hydrostaticpressure [28] or partial substitution of Pd by Ni [34,35],suggesting the possibility for a QCP.

II. EXPERIMENTAL DETAILS

Polycrystalline samples of CePd1−xNixAl were preparedby arc-melting appropriate amounts of the pure elements Ce(Ref. [36]), Pd(99.95), Ni(99.95), Al(99.999) under argonatmosphere with titanium gettering. To achieve homogeneity,the samples were remelted several times. The total weightloss after preparation did not exceed 0.5%. The sampleswere investigated in the as-cast state since annealing maycause a structural change [37]. They were characterized bypowder x-ray diffraction, revealing the single-phase ZrNiAlstructure (P 62m) of the parent compounds. Atom absorptionspectroscopy was used to determine the actual Ni concen-trations x that are quoted throughout this paper. The latticeconstants a and c and the unit-cell volume V approximatelyfollow Vegard’s law in the concentration range investigated(x < 0.15). Specific-heat measurements were performed in thetemperature range 0.05 ! T " 2.5 K using the standard heat-pulse technique. A Physical Properties Measurement System(PPMS, Quantum Design) was used to obtain data at highertemperatures for some samples. The dc magnetic susceptibilityχ was measured at 0.1 T in the zero-field-cooled field-heated

FIG. 2. (Color online) (a) Specific heat C of CePd1−xNixAlplotted as C/T versus log T . (b) Specific heat C of CePd0.856Ni0.144Alplotted as C/T versus T

12 .

mode in a vibrating sample magnetometer (VSM, OxfordInstruments). A sample-dependent residual background con-tribution χ0 ≈ 2 × 10−4 µB/T f.u. independent of T wassubtracted from the data. Below 20 K, χ0 corresponds to <1%of the total susceptibility χ .

III. RESULTS

The specific heat C is shown as C/T vs log T in Fig. 2(a).The pure CePdAl compound exhibits a sharp anomaly at theNeel temperature TN = 2.7 K in agreement with literaturedata [38]. The anomaly broadens and moves to lower T withincreasing Ni content x indicating a suppression of the antifer-romagnetic (AF) transition. For x = 0.144, the C/T vs log Tdata follow a straight line in Fig. 2(a) over almost two decadesof temperature in the range 0.05 ! T ! 3 K, i.e., C/T =a log(T0/T ), with a = 0.705 J/mol K2 and T0 = 11.7 K.The non-Fermi-liquid behavior in the form of a logarithmicdivergence of C/T versus T extends over nearly two ordersof magnitude in T (0.05–3 K). The HMM model [12–14]predicts for 2D AF fluctuations a logarithmic dependence ofC/T near the QCP as observed for CePd1−xNixAl, while theHMM prediction for 3D antiferromagnets, C/T ∝ γ − a

√T

for T → 0, is clearly not compatible with our data, as can beseen from the plot of C/T vs

√T in Fig. 2(b).

We use the specific-heat data to obtain TN as the tem-perature where C(T ) is a maximum for low x. For largerNi concentrations, however, the small specific-heat anomaly

054416-2

V. FRITSCH et al. PHYSICAL REVIEW B 89, 054416 (2014)

FIG. 1. (Color online) (a) View of the ab plane of CePdAl. TheCe and Pd(2) atoms form a plane at z = 0; the Al and Pd(1) atomsform a plane at z = 1/2. (b) View of the ab plane, showing oneof the three symmetry-related magnetic structures as derived fromRef. [25]. Only Ce atoms are shown for clarity. (c) Kagome latticefor comparison.

simulations [33]. Unfortunately, a model incorporating theinterplane coupling J⊥ to be compared with the experimentallydetermined magnetic structure does not exist up to now. Inview of the incommensurate z component τ , even long-rangeinteractions might have to be taken into account. Previous workon CePdAl showed that TN can be suppressed by hydrostaticpressure [28] or partial substitution of Pd by Ni [34,35],suggesting the possibility for a QCP.

II. EXPERIMENTAL DETAILS

Polycrystalline samples of CePd1−xNixAl were preparedby arc-melting appropriate amounts of the pure elements Ce(Ref. [36]), Pd(99.95), Ni(99.95), Al(99.999) under argonatmosphere with titanium gettering. To achieve homogeneity,the samples were remelted several times. The total weightloss after preparation did not exceed 0.5%. The sampleswere investigated in the as-cast state since annealing maycause a structural change [37]. They were characterized bypowder x-ray diffraction, revealing the single-phase ZrNiAlstructure (P 62m) of the parent compounds. Atom absorptionspectroscopy was used to determine the actual Ni concen-trations x that are quoted throughout this paper. The latticeconstants a and c and the unit-cell volume V approximatelyfollow Vegard’s law in the concentration range investigated(x < 0.15). Specific-heat measurements were performed in thetemperature range 0.05 ! T " 2.5 K using the standard heat-pulse technique. A Physical Properties Measurement System(PPMS, Quantum Design) was used to obtain data at highertemperatures for some samples. The dc magnetic susceptibilityχ was measured at 0.1 T in the zero-field-cooled field-heated

FIG. 2. (Color online) (a) Specific heat C of CePd1−xNixAlplotted as C/T versus log T . (b) Specific heat C of CePd0.856Ni0.144Alplotted as C/T versus T

12 .

mode in a vibrating sample magnetometer (VSM, OxfordInstruments). A sample-dependent residual background con-tribution χ0 ≈ 2 × 10−4 µB/T f.u. independent of T wassubtracted from the data. Below 20 K, χ0 corresponds to <1%of the total susceptibility χ .

III. RESULTS

The specific heat C is shown as C/T vs log T in Fig. 2(a).The pure CePdAl compound exhibits a sharp anomaly at theNeel temperature TN = 2.7 K in agreement with literaturedata [38]. The anomaly broadens and moves to lower T withincreasing Ni content x indicating a suppression of the antifer-romagnetic (AF) transition. For x = 0.144, the C/T vs log Tdata follow a straight line in Fig. 2(a) over almost two decadesof temperature in the range 0.05 ! T ! 3 K, i.e., C/T =a log(T0/T ), with a = 0.705 J/mol K2 and T0 = 11.7 K.The non-Fermi-liquid behavior in the form of a logarithmicdivergence of C/T versus T extends over nearly two ordersof magnitude in T (0.05–3 K). The HMM model [12–14]predicts for 2D AF fluctuations a logarithmic dependence ofC/T near the QCP as observed for CePd1−xNixAl, while theHMM prediction for 3D antiferromagnets, C/T ∝ γ − a

√T

for T → 0, is clearly not compatible with our data, as can beseen from the plot of C/T vs

√T in Fig. 2(b).

We use the specific-heat data to obtain TN as the tem-perature where C(T ) is a maximum for low x. For largerNi concentrations, however, the small specific-heat anomaly

054416-2

V. Frisch et al, PRB 89, 054416 (2014),

1. CePdAlInhomogenious Kondo/AFM

2. CeRh2In5

Pure KondoLM + KondoMagnetism

• Local quantum criticality (Si, Ingersent, Smith, Rabello, Nature 2001): Spin is the critical mode, Fluctuations critical in time.

Requires a two dimensional spin fluid

New Ideast

H

Si, Ingersent

•Two fluid scenario.

D. Pines Z. Fisk S. Nakatsuji Y. Yang

Nature (2008), PRL (2004)

Pure KondoLM + KondoMagnetism

•Supersymmetry? Coleman, Pepin, Tsvelik (1999)Ramires Coleman (2014)

Description of unconventional QCP requiresnew formalism.

Spin = B Spin = B,F Spin = F

Strange Metal = Unbroken Susy?

Why Supersymmetric Spins?

Gan, Coleman and Andrei, PRL (1992) Coleman, Pepin, Tsvelik, PRB (2000)

HFLAFM

TFL

T*

QCP

+

Schwinger Boson

Abrikosov Fermion

SUSY

Unusual QCP:

Coexistance:

2-fluid picture:

Knebel, arXiv 2009

Nakatsuji, Pines and Fisk, PRL (2004)

How to describe the generic HF phase diagram in its entirety?

C/T ~ log Tρ ~ Τ(α<2)

Heavy Fermion Systems: Why Supersymmetric Spins?

Bosonic Fermionic

Gan, Coleman and Andrei, 1992 Coleman, Pepin, Tsvelik, 2000

Supersymmetric Spin

AFMHFL

NFL

SCSC+AFM

Control parameter

T

ResultsWithin a static mean field solution the free energy have the following closed form:

The energy will be minimized by different representations in different

areas of the phase diagram

✦ Unusual critical behavior;

✦ 2nd order transition F ➔ F+B;

✦ Fermionic modes go soft;

0.25 0.5 0.750

2

4

6

nf + nbN

pJHêTK

2nd order1st order

F

B

F+B

✦ F+B Phase ➔ Coexistence;

Thank You!

heavy fermion physics: a 21st century julich, perspective · 2015. 9. 22. · piers coleman:...

Documents