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!

Glue vs Fabric.

Glue vs Fabric.

k

-k

k’ k’’

-k’ -k’’

Spin fluctuations = pairing bosonsGlue

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

Glue vs Fabric.

k

-k

k’ k’’

-k’ -k’’

Spin fluctuations = pairing bosonsGlue(π,0)→s±

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

Glue vs Fabric.

k

-k

k’ k’’

-k’ -k’’

Spin fluctuations = pairing bosonsGlue

Fabric: spins make the pairs

(π,0)→s±

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

Glue vs Fabric.

k

-k

k’ k’’

-k’ -k’’

Spin fluctuations = pairing bosonsGlue

Fabric: spins make the pairs

(π,0)→s±

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 be- tween 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 con- duction 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 ] .

Glue vs Fabric.

k

-k

k’ k’’

-k’ -k’’

Spin fluctuations = pairing bosonsGlue

Fabric: spins make the pairs

R lnW =

Z T

0dT 0C

0

T 0

(π,0)→s±

“Hilbert Space Spectroscopy”

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 be- tween 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 con- duction 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 ] .

Glue vs Fabric.

k

-k

k’ k’’

-k’ -k’’

Spin fluctuations = pairing bosons

~1/3 R ln(2)

Glue

Fabric: spins make the pairs

R lnW =

Z T

0dT 0C

0

T 0

(π,0)→s±

“Hilbert Space Spectroscopy”

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 be- tween 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 con- duction 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 ] .

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”

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 be- tween 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 con- duction 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 ] .

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 be- tween 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 con- duction 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.

Heavy Fermions and the Kondo Lattice 5.23

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 qq̄ 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

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cal p

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N

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j

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JH

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S[�]

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classi

cal p

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SU(N): Mesonsq̄q

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Baryonsq1q2 . . . qN

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H =�

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N

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j

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SU(N): Mesonsq̄q

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Baryonsq1q2 . . . qN

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H =�

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SU(N): Mesonsq̄q

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Baryonsq1q2 . . . qN

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

H =�

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N

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c†j�c j�S ��( j) +

JH

2N

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No Pairs !

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S[�]

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classi

cal p

ath

?

SU(N): Mesonsq̄q

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Baryonsq1q2 . . . qN

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

Cooper pairsqaq�a

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S ↵� = f †↵ f� � sgn(↵�) f †�� f�↵<latexit sha1_base64="iSWx0xA5P3fXAR2V3LsFHmFURCA=">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</latexit><latexit sha1_base64="Bqsunbgkpn6DEbxJmXZj+r6BEFM=">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</latexit><latexit sha1_base64="DWDwjoc1Gm5Gnu4uTwGx4QZSL/E=">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</latexit><latexit sha1_base64="DWDwjoc1Gm5Gnu4uTwGx4QZSL/E=">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</latexit><latexit sha1_base64="DWDwjoc1Gm5Gnu4uTwGx4QZSL/E=">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</latexit><latexit sha1_base64="b8lbGwNuLAWowhbP36ACAapjVsQ=">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</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

Heavy Fermions and the Kondo Lattice 5.25

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

(c̃†k↵, f̃ †k↵)266664✏k⌧3 V⌧1

V⌧1 ~w · ~⌧ + �Hk⌧1

3777750BBBB@c̃k↵

f̃k↵

1CCCCA +NsN

|V |2JK+ 2|�H |2JH

!(66)

wherec̃†k↵ = (c†k↵, ↵̃c�k,�↵), f̃ †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 f̃ †~⌧ f i = 0. The function �Hk = �k(cos kx �

Heavy Fermions and the Kondo Lattice 5.25

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

(c̃†k↵, f̃ †k↵)266664✏k⌧3 V⌧1

V⌧1 ~w · ~⌧ + �Hk⌧1

3777750BBBB@c̃k↵

f̃k↵

1CCCCA +NsN

|V |2JK+ 2|�H |2JH

!(66)

wherec̃†k↵ = (c†k↵, ↵̃c�k,�↵), f̃ †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 f̃ †~⌧ f i = 0. The function �Hk = �k(cos kx �

Uniform solution:

H =X

k,↵>0

(c̃†k↵, f̃†k↵)

✏k⌧3 V⌧3V⌧3 ~w · ~⌧ + �Hk⌧1

! c̃k↵f̃k↵

!+NsN

0BBBB@|V |2JK

+ 2�2

H

JH

1CCCCA

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

Heavy Fermions and the Kondo Lattice 5.25

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

(c̃†k↵, f̃ †k↵)266664✏k⌧3 V⌧1

V⌧1 ~w · ~⌧ + �Hk⌧1

3777750BBBB@c̃k↵

f̃k↵

1CCCCA +NsN

|V |2JK+ 2|�H |2JH

!(66)

wherec̃†k↵ = (c†k↵, ↵̃c�k,�↵), f̃ †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 f̃ †~⌧ f i = 0. The function �Hk = �k(cos kx �

Heavy Fermions and the Kondo Lattice 5.25

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

(c̃†k↵, f̃ †k↵)266664✏k⌧3 V⌧1

V⌧1 ~w · ~⌧ + �Hk⌧1

3777750BBBB@c̃k↵

f̃k↵

1CCCCA +NsN

|V |2JK+ 2|�H |2JH

!(66)

wherec̃†k↵ = (c†k↵, ↵̃c�k,�↵), f̃ †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 f̃ †~⌧ f i = 0. The function �Hk = �k(cos kx �

Heavy Fermions and the Kondo Lattice 5.25

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

(c̃†k↵, f̃ †k↵)266664✏k⌧3 V⌧1

V⌧1 ~w · ~⌧ + �Hk⌧1

3777750BBBB@c̃k↵

f̃k↵

1CCCCA +NsN

|V |2JK+ 2|�H |2JH

!(66)

wherec̃†k↵ = (c†k↵, ↵̃c�k,�↵), f̃ †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 f̃ †~⌧ f i = 0. The function �Hk = �k(cos kx �

Uniform solution:

H =X

k,↵>0

(c̃†k↵, f̃†k↵)

✏k⌧3 V⌧3V⌧3 ~w · ~⌧ + �Hk⌧1

! c̃k↵f̃k↵

!+NsN

0BBBB@|V |2JK

+ 2�2

H

JH

1CCCCA

<latexit 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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 operator

where 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.

Uniform solution:

H =X

k,↵>0

(c̃†k↵, f̃†k↵)

✏k⌧3 V⌧3V⌧3 ~w · ~⌧ + �Hk⌧1

! c̃k↵f̃k↵

!+NsN

0BBBB@|V |2JK

+ 2�2

H

JH

1CCCCA

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