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Fermi liquids and their breakdown

David Broun [email protected]

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CIFAR BRAND STANDARDS 2

CIFAR Quantum Materials Summer School 2015

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Outline

• Condensed matter and the theory of almost everything

• Fermi liquid theory and electron quasiparticles

• Heavy fermions – extreme Fermi liquids

• Quantum critical points – where Fermi liquids go to die

• Microwave spectroscopy of heavy fermions

Contents

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

1 Introduction 1

1.1 The free Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Sommerfeld free electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 The classical to quantum crossover . . . . . . . . . . . . . . . . . . . . . . . 5

2 Thermodynamics and statistical mechanics 7

2.1 Review of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Maxwell relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Review of statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Statistical mechanics of ideal quantum gases . . . . . . . . . . . . . . . . . . 11

2.4.1 Example: statistics of an impurity in a semiconductor . . . . . . . . . 13

2.4.2 Example: interacting electrons in a metal . . . . . . . . . . . . . . . . 15

3 Thermal properties of the Fermi gas 17

3.1 Specific heat of an electron gas . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 The Sommerfeld expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Specific heat of the ideal Fermi gas . . . . . . . . . . . . . . . . . . . . . . . 21

4 Review of single-particle quantum mechanics 25

4.1 Single-particle quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Many-particle quantum systems 33

5.1 Quantum mechanics of many-particle systems . . . . . . . . . . . . . . . . . 33

5.2 Second quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2.1 Occupation number representation . . . . . . . . . . . . . . . . . . . 35

5.2.2 Representation of states in second quantization . . . . . . . . . . . . 37

iii

iv CONTENTS

5.2.3 Representation of operators in second quantization . . . . . . . . . . 38

5.2.4 Field operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2.5 Examples of second quantized operators . . . . . . . . . . . . . . . . 40

6 Applications of second quantization 41

6.1 The tight-binding model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.2 The jellium model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7 The Hartree–Fock approximation 47

7.1 A model two-electron system. . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.2 The Hartree–Fock approximation . . . . . . . . . . . . . . . . . . . . . . . . 48

7.3 Hartree–Fock theory for jellium . . . . . . . . . . . . . . . . . . . . . . . . . 51

8 Screening and the random phase approximation 55

8.1 Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8.2 Thomas–Fermi theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.3 The density operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

8.4 The random phase approximation . . . . . . . . . . . . . . . . . . . . . . . . 59

8.5 Collective excitations of the electron gas . . . . . . . . . . . . . . . . . . . . 61

9 Scattering and periodic structures 63

9.1 The Born approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

9.2 Periodic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

9.2.1 Convolution as a means of replication . . . . . . . . . . . . . . . . . . 65

9.2.2 The convolution theorem . . . . . . . . . . . . . . . . . . . . . . . . . 66

9.3 The reciprocal lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

10 The nearly free electron model 69

10.1 Bloch’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

10.2 Free electron in 1–D as a Bloch wave . . . . . . . . . . . . . . . . . . . . . . 71

10.3 Nearly free electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

10.3.1 Properties of UG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

10.4 The Schrodinger equation in momentum space . . . . . . . . . . . . . . . . . 74

10.5 The periodic potential as a weak perturbation . . . . . . . . . . . . . . . . . 75

10.5.1 The nondegenerate case . . . . . . . . . . . . . . . . . . . . . . . . . 75

11 The nearly free electron model and tight binding 77

11.1 Degenerate electrons in a periodic potential . . . . . . . . . . . . . . . . . . 77

11.2 The tight binding method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

11.2.1 Introduction to tight binding . . . . . . . . . . . . . . . . . . . . . . 82

Broun – introduction to solid state physics

Broun – introduction to solid state physicsCONTENTS v

12 The tight-binding approximation 87

12.1 Degenerate tight-binding theory . . . . . . . . . . . . . . . . . . . . . . . . . 88

12.1.1 Example 1: s-like band in a face-centred cubic crystal . . . . . . . . . 91

12.1.2 Example 2: p-like bands in a face-centred cubic crystal . . . . . . . . 92

13 Energy bands and electronic structure 97

13.1 Review of the nearly free electron model . . . . . . . . . . . . . . . . . . . . 97

13.2 Free electron Fermi surfaces in 2D . . . . . . . . . . . . . . . . . . . . . . . . 100

13.3 Electrons and holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

13.4 Metals, semimetals, insulators and semiconductors . . . . . . . . . . . . . . . 102

14 Electronic structure calculations 105

14.1 Orthogonalized plane waves and pseudopotentials . . . . . . . . . . . . . . . 105

14.1.1 Example: the need for pseudopotentials . . . . . . . . . . . . . . . . . 105

14.2 Orthogonalized plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

14.3 Relativistic e↵ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

15 Density functional theory 111

15.1 The Hohenberg–Kohn theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 111

15.1.1 Proof of the Hohenberg–Kohn theorem . . . . . . . . . . . . . . . . . 112

15.1.2 Proof of the second Hohenberg–Kohn theorem . . . . . . . . . . . . . 113

15.2 Application of the Hohenberg–Kohn theorem . . . . . . . . . . . . . . . . . . 113

15.3 The Kohn–Sham equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

15.4 The local density approximation . . . . . . . . . . . . . . . . . . . . . . . . . 115

15.5 Thomas–Fermi as a density functional theory . . . . . . . . . . . . . . . . . 115

15.6 What can be calculated with density functional theory? . . . . . . . . . . . . 116

16 The dynamics of Bloch electrons I 119

16.1 Energy bands and group velocity . . . . . . . . . . . . . . . . . . . . . . . . 119

16.2 Rules of the semiclassical model . . . . . . . . . . . . . . . . . . . . . . . . . 121

16.3 The k·P method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

16.3.1 First order terms: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

16.3.2 Second order terms: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

16.4 Consequences of the semiclassical model . . . . . . . . . . . . . . . . . . . . 125

16.4.1 Electrical current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

16.4.2 Thermal current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

16.4.3 Filled bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

16.4.4 Electrons and holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

16.5 Semiclassical motion in a uniform dc electric field . . . . . . . . . . . . . . . 126

vi CONTENTS

17 The dynamics of Bloch electrons II 129

17.1 Semiclassical motion in a uniform magnetic field . . . . . . . . . . . . . . . . 129

17.1.1 The cyclotron frequency . . . . . . . . . . . . . . . . . . . . . . . . . 131

17.2 Limits of validity of the semiclassical model . . . . . . . . . . . . . . . . . . 133

17.3 Magnetic breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

18 Quantum oscillatory phenomena 139

18.1 Quantum mechanics of the orbital motion . . . . . . . . . . . . . . . . . . . 139

18.2 Degeneracy of the Landau levels . . . . . . . . . . . . . . . . . . . . . . . . . 140

18.3 Landau levels in a periodic potential . . . . . . . . . . . . . . . . . . . . . . 141

18.4 Visualizing Landau quantization . . . . . . . . . . . . . . . . . . . . . . . . . 143

18.5 Quantum oscillations as a Fermi surface probe . . . . . . . . . . . . . . . . . 144

19 Electronic structure of selected metals 149

19.1 Construction of free-electron Fermi surfaces . . . . . . . . . . . . . . . . . . 149

19.1.1 Free-electron Fermi surfaces in two dimensions . . . . . . . . . . . . . 149

19.1.2 Free-electron Fermi surfaces in three dimensions . . . . . . . . . . . . 150

19.2 The alkali metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

19.3 The noble metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

19.4 Divalent metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

19.5 The trivalent and tetravalent metals . . . . . . . . . . . . . . . . . . . . . . . 160

19.6 Transition metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

20 Strongly correlated systems & semiconductors 163

20.1 Interactions and the Hubbard U . . . . . . . . . . . . . . . . . . . . . . . . . 163

20.2 Measuring the Hubbard U . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

20.3 3d transition metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

20.4 Semiconductor structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

20.5 Semiconductor chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

20.6 Bonding, nonbonding and antibonding states . . . . . . . . . . . . . . . . . . 169

20.7 Electronic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

20.7.1 Spin–orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

20.8 Other semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Broun – introduction to solid state physicsCONTENTS vii

21 Semiconductors 175

21.1 Homogeneous semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . 175

21.1.1 Carrier density in thermal equilibrium . . . . . . . . . . . . . . . . . 175

21.1.2 The nondegenerate case . . . . . . . . . . . . . . . . . . . . . . . . . 176

21.1.3 The intrinsic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

21.1.4 The extrinsic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

21.1.5 Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

21.1.6 Population of impurity levels in thermal equilibrium . . . . . . . . . . 179

21.1.7 Thermal equilibrium carrier density . . . . . . . . . . . . . . . . . . . 180

21.1.8 Transport in nondegenerate semiconductors . . . . . . . . . . . . . . 180

21.2 Inhomogeneous semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . 181

21.2.1 The p-n junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

21.3 The p-n junction as a rectifier . . . . . . . . . . . . . . . . . . . . . . . . . . 184

22 The Boltzmann transport equation 187

22.1 The distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

22.2 The continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

22.3 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

22.4 The linearized Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . 190

22.5 The relaxation time approximation . . . . . . . . . . . . . . . . . . . . . . . 191

22.6 Transport properties of metals . . . . . . . . . . . . . . . . . . . . . . . . . . 192

22.6.1 Electrical conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 192

22.6.2 Thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

22.7 The Wiedemann–Franz law . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

23 Time-dependent perturbation theory 195

23.1 Time-dependent perturbation theory . . . . . . . . . . . . . . . . . . . . . . 195

23.2 Sudden perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

23.3 Adiabatic perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

23.4 Periodic perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

23.5 Fermi’s golden rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

23.6 The Kubo formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

24 Density response of the electron gas 203

24.1 Time and space dependent perturbations . . . . . . . . . . . . . . . . . . . . 203

24.2 Density response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

24.3 Energy loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

24.4 Screening and the dielectric function . . . . . . . . . . . . . . . . . . . . . . 207

24.5 Properties of the RPA dielectric function . . . . . . . . . . . . . . . . . . . . 208

viii CONTENTS

25 Electrons in one dimension 211

25.1 One dimensional conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

25.2 The Peierls instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

25.2.1 Static lattice distortions . . . . . . . . . . . . . . . . . . . . . . . . . 212

25.2.2 The energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

25.3 Kohn anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

25.4 Nesting of the Fermi surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

25.5 Spin–charge separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

25.6 The Luttinger–Tomonaga model . . . . . . . . . . . . . . . . . . . . . . . . . 217

26 Collective modes and response functions 221

26.1 Response functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

26.2 Collective modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

26.3 Inelastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

26.4 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

26.5 Optical conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

26.6 Oscillator strength sum rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

27 The electron spectral function 229

27.1 The Schrodinger representation . . . . . . . . . . . . . . . . . . . . . . . . . 229

27.2 The Heisenberg representation . . . . . . . . . . . . . . . . . . . . . . . . . . 229

27.3 Particles and quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

27.4 A single free particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

27.5 The spectral function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

27.6 Interacting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

27.7 Angle resolved photoemission spectroscopy . . . . . . . . . . . . . . . . . . . 235

28 Landau’s Fermi Liquid theory 239

28.1 The noninteracting Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . 239

28.2 Excited states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

28.2.1 Heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

28.2.2 Pauli spin susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . 242

28.3 Landau quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

28.4 Quasiparticle decay rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

28.5 Landau’s Fermi liquid theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

28.5.1 Interactions between quasiparticles . . . . . . . . . . . . . . . . . . . 246

28.6 Experimental consequences of Fermi liquid theory . . . . . . . . . . . . . . . 247

28.6.1 Heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

28.6.2 Spin susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

Broun – introduction to solid state physics

CONTENTS ix

29 Superconductivity I — phenomenology 251

29.1 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

29.2 Perfect conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

29.3 The Meissner–Ochsenfeld e↵ect . . . . . . . . . . . . . . . . . . . . . . . . . 252

29.4 The London theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

29.5 Flux trapping and quantization . . . . . . . . . . . . . . . . . . . . . . . . . 255

29.6 The Josephson e↵ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

30 Superconductivity II — pairing theory 259

30.1 The Cooper problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

30.2 The origin of the attractive interaction . . . . . . . . . . . . . . . . . . . . . 261

30.3 Bardeen–Cooper–Schrie↵er theory . . . . . . . . . . . . . . . . . . . . . . . . 263

30.4 The Bogoliubov transformation . . . . . . . . . . . . . . . . . . . . . . . . . 264

31 Superconductivity III — exotic pairing 267

31.1 Conventional superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . 267

31.2 Pairing glue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

31.3 Anderson’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

31.4 Unconventional pairing in specific materials . . . . . . . . . . . . . . . . . . 270

31.4.1 Cuprate superconductivity . . . . . . . . . . . . . . . . . . . . . . . . 270

31.4.2 Heavy fermion superconductivity . . . . . . . . . . . . . . . . . . . . 274

Derek Lee and Andrew Schofield, Metals without Electrons.

Condensed matter physics is concerned with very ordinary length scales.

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| �

4 states at each lattice site:

Why can’t this be solved exactly with a computer?

aa

j

i

| �⇥

| �⇥⇤| �⇥

| �

2 sites = | �⇥

| �⇥⇤| �⇥

| �

⊗ = 16

Why can’t this be solved exactly with a computer?Quantum complexity

aa

j

i

N sites: Hilbert space has 4N dimensions 416 = 4,294,967,296

Hamiltonian matrix is 416 ⨯ 416

Why can’t this be solved exactly with a computer?Quantum complexity

aa

j

i

Moore’s law lets us add one electron every 5 years

+

*12

X

i6=j

e2

4⇥�0

1|ri � rj |

+

Can we treat the Coulomb interaction in an average way?

X

i

� ~2

2mer2

riH =

i~ ⇥

⇥t|�� = H|��

one-body terms

two-body term

+ U(ri)

One-body theory

One-body theory

A.J. Schofield, Contemporary Physics 40, 95 (1999)

Non-Fermi liquids A. J. Schofield 2

can be obtained relatively simply using Fermi’s goldenrule (together with Maxwell’s equations) and I have in-cluded these for readers who would like to see wheresome of the properties are coming from.

The outline of this review is as follows. I begin witha description of Fermi-liquid theory itself. This the-ory tells us why one gets a very good description of ametal by treating it as a gas of Fermi particles (i.e. thatobey Pauli’s exclusion principle) where the interactionsare weak and relatively unimportant. The reason isthat the particles one is really describing are not theoriginal electrons but electron-like quasiparticles thatemerge from the interacting gas of electrons. Despite itsrecent failures which motivate the subject of non-Fermiliquids, it is a remarkably successful theory at describ-ing many metals including some, like UPt3, where theinteractions between the original electrons are very im-portant. However, it is seen to fail in other materialsand these are not just exceptions to a general rule butare some of the most interesting materials known. Asan example I discuss its failure in the metallic state ofthe high temperature superconductors.

I then present four examples which, from a theo-retical perspective, generate non-Fermi liquid metals.These all show physical properties which can not beunderstood in terms of weakly interacting electron-likeobjects:

• Metals close to a quantum critical point. When aphase transition happens at temperatures close toabsolute zero, the quasiparticles scatter so stronglythat they cease to behave in the way that Fermi-liquid theory would predict.

• Metals in one dimension–the Luttinger liquid. Inone dimensional metals, electrons are unstable anddecay into two separate particles (spinons andholons) that carry the electron’s spin and chargerespectively.

• Two-channel Kondo models. When two indepen-dent electrons can scatter from a magnetic impu-rity it leaves behind “half an electron”.

• Disordered Kondo models. Here the scatteringfrom disordered magnetic impurities is too strongto allow the Fermi quasiparticles to form.

While some of these ideas have been used to try and un-derstand the high temperature superconductors, I willshow that in many cases one can see the physics illus-trated by these examples in other materials. I believethat we are just seeing the tip of an iceberg of new typesof metal which will require a rather different startingpoint from the simple electron picture to understandtheir physical properties.

Figure 1: The ground state of the free Fermi gas in mo-mentum space. All the states below the Fermi surfaceare filled with both a spin-up and a spin-down elec-tron. A particle-hole excitation is made by promotingan electron from a state below the Fermi surface to anempty one above it.

2. Fermi-Liquid Theory: the electron quasi-particle

The need for a Fermi-liquid theory dates from thefirst applications of quantum mechanics to the metallicstate. There were two key problems. Classically eachelectron should contribute 3kB/2 to the specific heatcapacity of a metal—far more than is actually seen ex-perimentally. In addition, as soon as it was realizedthat the electron had a magnetic moment, there wasthe puzzle of the magnetic susceptibility which did notshow the expected Curie temperature dependence forfree moments: χ ∼ 1/T .

These puzzles were unraveled at a stroke whenPauli (Pauli 1927, Sommerfeld 1928) (apparentlyreluctantly—see Hermann et al. 1979) adopted Fermistatistics for the electron and in particular enforced theexclusion principle which now carries his name: No twoelectrons can occupy the same quantum state. In theabsence of interactions one finds the lowest energy stateof a gas of free electrons by minimizing the kinetic en-ergy subject to Pauli’s constraint. The resulting groundstate consists of a filled Fermi sea of occupied statesin momentum space with a sharp demarcation at theFermi energy ϵF and momentum pF = hkF (the Fermisurface) between these states and the higher energy un-occupied states above. The low energy excited statesare obtained simply by promoting electrons from justbelow the Fermi surface to just above it (see Fig. 1).They are uniquely labelled by the momentum and spinquantum numbers of the now empty state below theFermi energy (a hole) and the newly filled state aboveit. These are known as particle-hole excitations.

This resolves these early puzzles since only a smallfraction of the total number of electrons can take part

Independent electron modelFermi gas

Band structures of crystals Electronic properties of

semiconductors and many metals

Periodic Table of the Fermi Surfaces of Elemental Solids http://www.phys.ufl.edu/fermisurface

Ferromagnets:

Alternate Structures :

Tat-Sang Choy, Jeffery Naset , Selman Hershfield, and Christopher StantonPhysics Department, University of Florida

Seagate TechnologyJian Chen

Source of tight binding parameters (except for fcc Co ferromagnet): D.A. Papaconstantopoulos, Handbook of the band structure of elemental solids, Plenum 1986.This work is supported by NSF, AFOSR, Research Corporation, and a Sun Microsystems Academic Equipment Grant.

(15 March, 2000)

Co_fcc Co_fcc

Landau’s Fermi liquid theoryWhy we can (often) get away with treating

interactions in an average way• Originally devised for 3He: isotropic liquid of fermions

• Was later realized that it describes the normal state of most metals. (And neutron stars, atomic nuclei, etc.)

• Originally a phenomenological theory, it is now understood as a stable fixed point in RG.

• The standard model of electrons in metals

• As is particle physics, our field is characterized by the search for physics beyond the standard model.

• For example, in 1D metals are replaced by a Luttinger liquids, which exhibit spin–charge separation.

Adiabatic continuityNon-Fermi liquids A. J. Schofield 3

in the processes contributing to the specific heat andmagnetic susceptibility. The majority lie so far belowthe Fermi surface that they are energetically unable tofind the unoccupied quantum state required to mag-netize them or carry excess heat. Only the electronswithin kBT of the Fermi surface can contribute kB tothe specific heat so the specific heat grows linearly withtemperature and is small. Only electrons within µBB ofthe Fermi surface can magnetize with a moment ∼ µB

leading to a temperature independent (Pauli) suscepti-bility. Both quantities are proportional to the densityof electron states at the Fermi surface.

These new temperature dependencies exactlymatched the experiments both on metals and thenlater on the fermionic isotope of Helium - 3He (see, forexample, Wheatley 1970). But this in turn raised ques-tions. Why should a theory based on a non-interactingpicture work so well in these systems where interactionsare undoubtably important? Once interactions arepresent the problem of finding the low energy statesof the electrons becomes much harder. In addition tothe kinetic term which favours a low momentum, theenergy now contains a potential term which depends onthe relative position of all of the electrons. The energyscales of the kinetic energy and Coulomb interactionare comparable at metallic electron densities and,if that were not enough, Heisenberg’s uncertaintyprinciple prevents the simultaneously definition of themomentum and the position. How can one proceed andstill hope to retain the physics of the non-interactingelectron gas which experiment demands?

The answer provided by Landau rests on the conceptof “adiabatic continuity” (Anderson 1981): labels as-sociated with eigenstates are more robust against per-turbations than the eigenstates themselves. Consideras an example the problem of a particle in a box withimpenetrable walls illustrated in Fig. 2. In elementaryquantum mechanics one learns that the eigenstates ofthis problem consist of standing sine waves with nodesat the well walls. The eigenstates of the system can belabelled by the number of additional nodes in the wave-function with the energy increasing with the number ofnodes. Now switch on an additional weak quadraticpotential. The new eigenstates of the problem are nolonger simple sine waves but involve a mixing of all theeigenstates of the original unperturbed problem. How-ever the number of nodes still remains a good way oflabelling the eigenstates of the more complicated prob-lem. This is the essence of adiabatic continuity.

Landau applied this idea to the interacting gas ofelectrons. He imagined turning on the interactions be-tween electrons slowly, and observing how the eigen-states of the system evolved. He postulated that therewould be a one-to-one mapping of the low energy eigen-

N=4

N=3

N=2

N=1

N=0

N=4

N=3

N=2

N=1

N=0

0 1λ

Energy

Figure 2: Adiabatic continuity is illustrated in a non-interacting problem by turning on a quadratic potentialto a particle confined in box. While the energy levelsand the details of the eigenstate wavefunctions evolvesubtly , the good quantum numbers of the initial prob-lem (the number of nodes, N, in the wavefunction) arestill the appropriate description when the perturbationhas been applied.

states of the interacting electrons with the those of thenon-interacting Fermi gas. He supposed that the goodquantum numbers associated with the excitations of thenon-interacting system would remain good even afterthe interactions were fully applied. Just as Pauli’s ex-clusion principle determined the allowed labels with-out the interactions through the presence of a Fermisurface, this feature would remain even with the in-teractions. We therefore retain the picture of Fermiparticles and holes excitations carrying the same quan-tum numbers as their electron counter-parts in the freeFermi gas. These labels are not to be associated withelectrons but to ‘quasiparticles’ to remind us that thewavefunctions and energies are different from the cor-responding electron in the non-interacting problem. Itis the concept of the fermion quasiparticle that lies atthe heart of Fermi-liquid theory. It accounts of the mea-sured temperature dependences of the specific heat andPauli susceptibility since these properties only requirethe presence of a well defined Fermi surface, and arenot sensitive to whether it is electrons or quasiparticlesthat form it.

Retaining the labels of the non-interacting statemeans that the configurational entropy is unchangedin the interacting metal. [This also means that quasi-particle distribution function is unchanged from thefree particle result (see Fig. 3a).] Each quasiparticle

A.J. Schofield, Contemporary Physics 40, 95 (1999)

real particle quasi particle

real horse quasi-horse

• The weakly interacting, normal modes of the system. • The closest thing we have to real particles. • Only exist inside the system, cannot be removed.

Richard. D. Mattuck, A guide to Feynman diagrams in the many-body problem

Quasiparticles

sodium ion

Emergent particles

in solutionsodium ion

Emergent particles

in solutionsodium ion free electron+ QED vacuum polarization

(electron–positron pairs)

+

–

–

+

–

+ –

+

–

+

–

+

–

+–

+

–+

–

+

–

+

–

+

–

+

–

+

–

+

–

+

–

+

–

+

–Emergent particles

in solutionsodium ion

e/3 quasiparticles of fractional quantum Hall effect

(superposition of electrons and flux quanta)

free electron+ QED vacuum polarization

(electron–positron pairs)

+

–

–

+

–

+ –

+

–

+

–

+

–

+–

+

–+

–

+

–

+

–

+

–

+

–

+

–

+

–

+

–

+

–

+

–Emergent particles

Landau quasiparticles

electron basis quasiparticle basis

Zk

quasiparticleresidue

In FLT, the focus is on the elementary excitations, called quasiparticles. Their energies are nearly additive.

T = 0 T = 0 T = 0

Z =��he�|qpi

��2 ⇠ me

m⇤

Coleman et al., “How do Fermi liquids get heavy and die?”


Damascelli, Hussain & Shen, RMP

noninteracting

interacting


Damascelli, Hussain & Shen, RMP

noninteracting

Decay rate and T2 resistivity

conservation of momentum: p1 + p2 = p3 + p4

conservation of energy: ✏1 + ✏2 = ✏3 + ✏4

phase space for recoil ⇠ (✏1 � ✏F )2

1

⌧= a(✏1 � ✏F )

2 + b(kBT )2 ⇢ = ⇢0 +AT 2

Fermi liquid interactionsMultipole expansion of the distribution function:

spin symmetry & spin antisymmetric components;interaction parameters fS,A

`

unpolarized

heat compression spinpolarization

chargecurrent

fS0 fA

0 fS1

Fermi liquid interactionsLandau quasiparticles interact, but far less strongly than the

original particles from which they were constructed.

✏qpp� =�E

�(�np,�)= vF(p� pF) +

1

V

X

p0,�0

f(p,�;p0,�0)�np0,�0

Fermi liquid interactionsLandau quasiparticles interact, but far less strongly than the

original particles from which they were constructed.

✏qpp� =�E

�(�np,�)= vF(p� pF) +

1

V

X

p0,�0

f(p,�;p0,�0)�np0,�0

✏qpp" = vF(p� pF) +1

V

X

p0,`

n

fS` P`(cos ✓)

�

�np0" + �np0#�

+ fA` P`(cos ✓)

�

�np0" � �np0#�

o

✏qpp# = vF(p� pF) +1

V

X

p0,`

n

fS` P`(cos ✓)

�

�np0" + �np0#�

� fA` P`(cos ✓)

�

�np0" � �np0#�

o

particle density fluctuation

spin density fluctuation

Interaction energy is expressed as a multipole expansion of the particle density and spin density.

Specific heatNon-Fermi liquids A. J. Schofield 2














Classical gasU = 3

2kBTn

cV ⌘ dU

dT= 3

2kBn

Fermi gas cV = ⇡2

332kBn

T

TF⇠ m⇤

Specific heatNon-Fermi liquids A. J. Schofield 2














Classical gasU = 3

2kBTn

cV ⌘ dU

dT= 3

2kBn

Fermi gas cV = ⇡2

332kBn

T

TF⇠ m⇤

heating the Fermi distribution does not couple to any angular moments

- no FL correction other than m*

Pauli susceptibility

D( ) D( )

2 BB

D( )

2 BB

�Curie ⇠C

T�Pauli ⇠

C

T⇥ T

TF= constant

Pauli susceptibility

D( ) D( )

2 BB

D( )

2 BB

�Curie ⇠C

T�Pauli ⇠

C

T⇥ T

TF= constant

a spin polarization couples to the spin-antisymmetric monopolar moment

�FL =µ0µ2

BD(✏F)

1 + FA0

FL correction leads to Stoner instability

Classic heavy-fermion metal: UPt3

partially filled f shell in U

Extreme Fermi liquids

Kondo screening

J. Kondo, Prog. Theor. Phys. 32, 37 (1964)

conductions electrons scatter from spin-1/2 impurities

singlet formation resolves the low T singularity

kBT� ⇠ log

TK

T

J. Kondo, Prog. Theor. Phys. 32, 37 (1964)

Kondo screening

S = kB log 2

extreme Fermi liquidsHeavy fermions

Quantum oscillationsk

k

k

k

k

k

filled states= 0B

empty states

filled states

empty states

z

y

x/= 0B

y

zB

x

Christoph Bergemann

Quantum oscillations

5 10 15 20 25

-2

-1

0

(arb

. uni

ts)

B0/B

0.2 0.4 0.6 0.8 1 1.2 1.4

-2

-1

0B/B0

(arb

. uni

ts)

0 5 10 15 20DHvA Frequency (kT)

0.00

0.01

0.02

0.03

0.04

Ampl

itude

Spe

ctru

m (a

.u.)

2.9

3 kT

3.0

9 kT

6.0

9 kT

12.

58 k

T 12.

88 k

T

18.

64 k

T

15 16 17 18Field (T)

Quantum oscillations in Sr2RuO4

Bergemann et al.

Mackenzie et al.

experiments, one value of lfree (as tabulated in table 5) is sufficient to reproduce theexperimental Dingle fields for all three sheets, to within about 15–20%. There issome indication that BD might be slightly enhanced on the b and ! sheets, but at thisstage it is not clear whether this represents a real variation in lfree. We thereforeconclude that to within a reasonable degree and within our experimental resolution,the mean free path is constant in Sr2RuO4.

5.6. VisualizationThe resulting Fermi surface topography is visualized in figure 28. The numbers

in table 4 represent a refinement of an earlier parameter set published previouslyin tabular [4] and graphical [77] form.15

5.7. Consistency checks5.7.1. Resistivity anisotropy

The experimental resistivity anisotropy of "0, c="0, ab ’ 4000 (cf. table 3) hasto match the anisotropy expected from the Fermi surface geometry. Followingequation (17), one can compute the resistivity ratio from

"0, ab"0, c

¼ 2

AFS

I

FSd2kuu2Fz

ðkÞ ð41Þ

15Some corrections to those old results were necessary because the warping parameters wereinitially extracted in the circular Fermi contour approximation. The numerically more challenging fullcalculation yields the refined results presented here.

Figure 28. Visualization of the Fermi surface of Sr2RuO4. The c-axis corrugation isexaggerated by a factor of 15 for clarity.

C. Bergemann et al.688

Downloaded By: [Canadian Research Knowledge Network] At: 17:23 19 April 2011

Sr2RuO4

C. Bergemann

Heavy fermion qps in UPt3

VOLUME 60, NUMBER 15 PHYSICAL REVIEW LETTERS 11 APRIL 1988

64-

60-28:

L

o20

16-

12-

90'

gs

~'8 j~S

at& I

K) b(I M)

~g&w

a(I

a

kL 1ig~gO

0 0' 50' 0 90'FIG. 1. Variation of the fundamental dHvA frequencies

with orientation of the magnetic field in the crystallographicplanes a-b, a-c, and b-c. The frequency branches 8, p, and yare thought to arise from magnetic breakdown between orbitsdirectly responsible for the b and k branches. Harmonicbranches of a and b and fine structure of the 8 and co branchesare not displayed.

i.e., their relative positions and their slopes at eF.The dHvA magnetization was measured by a low-

frequency and low-noise field-modulation technique witha 14.5-T superconducting magnet and a 17-mK dilutionrefrigerator. The orientation of the sample was varied insitu Measurements for field orientations in the basalplane from a to b in the hexagonal (SnNi3) crystal struc-ture and from a to c and b to c were performed on twosingle-crystal disks of comparable size and purity. Thefirst was cut with its normal along the c axis, and thesecond with its normal along the a axis. Details on crys-tal preparation and on the experimental procedure maybe found in previous papers.Our main experimental results are presented in Table

I and Fig. 1. Table I is a summary of the measureddHvA frequencies and associated cyclotron masses fordirections of the magnetic field along the a and b axes.A total of ten different fundamental frequency com-ponents (or branches) are well resolved, with frequenciesranging from 4.1 to 58.5 MG. The highest frequencycorresponds to an orbit area roughly as large as onewould expect from the size of the Brillouin zone andindeed comparable to the largest extremal areas predict-ed by band-structure calculations (see below).The most remarkable of our observations is the ex-

treme magnitude of the cyclotron masses. A detailed-tudy of the temperature dependence of the dHvA am-plitudes (performed as described previously2) yields the

cyclotron-mass values listed in Table I. They range from25m, to 90m, for frequencies along the a axis and from15m, to approximately 50m, along the b axis. Althoughthese are large variations in m*, they scale roughly withfrequency, which is a common finding, and they all rep-resent enormous values compared with masses in simplemetals. The co branch has the highest cyclotron massobserved so far in any metal.The orientation dependence of the dHvA frequencies

is shown in Fig. 1 for field directions in the planesspanned by the crystallographic axes a-c, a-b (the basalplane), and b-c. Only the fundamental components aredisplayed, although second harmonics of the a and 8branches were also observed. No dHvA oscillations wereobserved for a field direction in the vicinity of the hexag-onal c axis. To some extent, this is a consequence of thegeneral tendency for all frequencies to increase rapidlyas the field orientation approaches the c axis (from eitherthe a or the b axis). Indeed, a frequency increasing rap-idly with angle implies a large curvature of the Fermisurface and usually an increasing cyclotron mass. Bothof these factors conspire to reduce the amplitude of thedHvA oscillation which may eventually fall below thelevel of detection. Nevertheless, this remains an unusualresult.It is of interest to note that fine structure was observed

on the b branch and on the ro branch. In both cases themultiple fine splitting of the dominant frequency (theone displayed in Fig. 1), resolved into several close fre-quencies at the highest fields, may be due to a field-induced exchange splitting combined with magneticbreakdown. This structure is currently under investiga-tion.It is informative to consider our dHvA results in rela-

tion to conventional energy-band models. Several calcu-lations based on the local-density approximation to theexchange-correlation potential, with the assumptionsthat the uranium f electrons are itinerant, have beenpresented and they all predict similar band structures. 3 5

Nevertheless, slight differences in the precise positions ofthe five bands found to cross the Fermi level lead to afew significant topological differences in the Fermi sur-face predicted by the various models. Figure 2(a) showsa 1 ALM section through the Fermi surface obtained byWang et al. ' using the linear muffin-tin orbital methodwith so-called combined correction terms. Their calcula-tions based on the linearized augmented plane-wave(LAPW) method and the earlier calculations of Oguchiand Freeman (see Ref. 4) and of Albers, Boring, andChristensen also lead to essentially the same Fermi sur-face, provided adjustments are made in the positions oftwo bands with respect to the Fermi level by an amountof order 5 mRy (or less), i.e., within computational accu-racy. In this way, for example, the two nested toroidalsurfaces centered on point A in the original Fermi sur-face of Oguchi and Freeman become disks as in Fig. 2.

1571

VOLUME 60, NUMBER 15 PHYSICAL REVIEW LETTERS 11 APRIL 1988

Heavy-Fermion Quasiparticles in UPt3

L. Taillefer and G. G. LonzarichCavendish Laboratory, Cambridge CB30HE, United Eingdom

(Received 21 October 1987)

The quasiparticle band structure of the heavy-fermion superconductor Upt3 has been investigated bymeans of angle-resolved measurements of the de Haas-van Alphen eff'ect, Most of the results are con-sistent with a model of 5 quasiparticle bands at the Fermi level corresponding to Fermi surfaces similarto those calculated by band theory. However, as inferred from the extremely high cyclotron masses, thequasiparticle bands are much flatter than the calculated ones. The nature of the observed quasiparticlesand their relationship to thermodynamic properties are briefly considered.

PACS numbers: 71.28.+d, 71,25.Hc, 71.25.3d, 71.25.Pi

ft ' BA(H)2tr Be

ft )f dkF 2& Ug

The intermetallic compound UPt3 exhibits thermo-dynamic properties with remarkable temperature depen-dences at low temperatures, and below 0.5 K it con-denses into an unusual superconducting state whichremains one of the outstanding enigmas in condensed-matter physics. ' In attempts to explain this low-tem-perature behavior, it has been conventional to invoke apicture of strongly renormalized quasiparticles, i.e., fer-mions with effective masses orders of magnitude largerthan the free-electron mass and having important residu-al interactions which lead to bound pair formation in theground state.To help provide a firm basis for such a quasiparticle

description, we have carried out an investigation of thede Haas-van Alphen (dHvA) effect in UPt3 which yieldsdirect evidence for the existence of heavy fermions andspecific information on the Fermi surface and cyclotronmasses which characterize them. The initial observationof the dHvA effect in UPt3 was communicated in a pre-vious paper and here we present the results of a detailedangle-resolved study, which yield unambiguous informa-tion about the quasiparticle band structure near the Fer-mi level.The information which may be inferred from the

quantum oscillatory (dHvA) magnetization M has beensummarized recently and here we shall reiterate themain points only. First, from the frequency F(H) ofeach of the several oscillatory components in M(H),measured as a function of the orientation of the magnet-ic field H, we infer the cross-sectional area A of thecorresponding extremal orbit on the Fermi surface viathe Onsager relation A(H) =(2tre/t'tc)F(H), and henceover all we obtain the dimension and topology of the Fer-mi surface. Second, from the temperature dependence ofthe amplitude of each oscillatory component, which wasfound to follow closely the behavior expected for a nor-mal Fermi liquid, we obtain directly the cyclotron effec-tive mass

TABLE I. Measured dHvA frequencies (F) and cyclotronmasses (m ) for a magnetic field applied along the a and baxes of the hexagonal crystal structure (parallel to the I K andI M directions in the reciprocal lattice, respectively). Thevalues quoted refer to a field strength of 100 k6. Note thatthe estimate of a cyclotron mass for the l branch is only ap-proximate, Also given are the identifications of the measureddHvA branches with extremal orbits on the Fermi-surfacemodel of Fig. 2. dHvA branches are labeled as in Fig. 1, andFermi-surface orbits are labeled according to their center inthe Brillouin zone (e.g. , I ) and their Fermi-surface sheet num-ber (e.g., 1). The calculated a axis results of Wang et al. (Ref.3) for F and m* are compared with the experimental values.

Branch:FS orbitF (MG)

Expt. Cale.

a axis (I K)

m*/m,Expt. Cale.

a:ML4p:L4~:r18'A 5t.'I 2co:I 3

s.4(3)6.o(4)7.3(3)14.0(3)21.O(3)sg.s(s)

10.45.28.29.124.052.8

25(3)~ ~ ~

4o(7)50(8)6o(8)90(15)

2.21.02.01.94.65.3

a:ML4BA50:A4, 5y:A4, 5y:A4, 5X:A4

b axis (I M)4. i(2)12.3(2)15.5(2)18.7(3)21.9(4)25.1(5)

is(s)3o(3)3s(7)4o(8)~ ~ S

(so)

A

where v 1,=

~H x Vi, e ~ /tl is the appropriate quasiparticle

velocity at the Fermi energy eF and the integral is overthe perimeter of A (on the cyclotron orbit). We maythink of m* as hko/vo, where ko=(A/tr)'t is an aver-age radius and I/vo is an average of the inverse of thequasiparticle velocity for the cyclotron orbit. Here weshall focus attention on these two properties, namelyA(H) and m*(H), which characterize the real part ofthe quasiparticle energy bands near the Fermi energy eF,

1570 1988 The American Physical Society

12

’

ω

δ

κ

γ

σζ

λ

0.0 2.0 4.0 6.0 8.0 10.0F (kT)

0.00

0.05

0.10

13.0 14.0 15.0 16.0B (tesla)

–0.6

–0.3

0.0

Figure 8. Typical oscillatory variation of the dHvA magnetization as seen at10mK with the applied field directed approximately 5� from the c-axis towardsthe b-axis (upper trace) and corresponding Fourier spectrum (lower trace). Thepeaks are labelled according to our assignment on the rotation plots (see figure 9).

The Greek letters in figure 9 follow [7, 27] with additions for new orbits. We have observednine new orbits in all, which we have labelled �0, ↵3, ↵

03, ↵4, ↵

04, , � 0, ⇣, ⌘ and ⌘0. Although

effective masses are not the focus of this paper, in table 3 we give the masses, and the calculatedband masses (assuming that the fully itinerant model is correct), for these new orbits.

An additional important difference from previous studies is that we have followed the �orbit all the way to the c-axis, whereas previously it had only been followed to within about 20�

of c. The significance of this is discussed below.The lines on figure 9 are the predictions of the fully itinerant model (as per figure 4), and in

the subsequent figure, figure 10, we show the data together with the predictions of the partiallylocalized model (figure 5). In the next section, we carry out a detailed comparison between thedata and the predictions of the models.

4. Discussion

The extreme angle dependence of the dc-magnetoresistance, shown in figure 6, can be explainedfollowing Taillefer et al [33] who argue that it arises from canonical !c⌧ > 1 effects. Theypoint out (a) that the large magnetoresistance seen at most angles is indicative of an open orbit,since UPt3 is compensated and therefore ought not to have a magnetoresistance unless there are

New Journal of Physics 10 (2008) 053029 (http://www.njp.org/)

12

’

ω

δ

κ

γ

σζ

λ

0.0 2.0 4.0 6.0 8.0 10.0F (kT)

0.00

0.05

0.10

13.0 14.0 15.0 16.0B (tesla)

–0.6

–0.3

0.0

Figure 8. Typical oscillatory variation of the dHvA magnetization as seen at10mK with the applied field directed approximately 5� from the c-axis towardsthe b-axis (upper trace) and corresponding Fourier spectrum (lower trace). Thepeaks are labelled according to our assignment on the rotation plots (see figure 9).

The Greek letters in figure 9 follow [7, 27] with additions for new orbits. We have observednine new orbits in all, which we have labelled �0, ↵3, ↵

03, ↵4, ↵

04, , � 0, ⇣, ⌘ and ⌘0. Although

effective masses are not the focus of this paper, in table 3 we give the masses, and the calculatedband masses (assuming that the fully itinerant model is correct), for these new orbits.

An additional important difference from previous studies is that we have followed the �orbit all the way to the c-axis, whereas previously it had only been followed to within about 20�

of c. The significance of this is discussed below.The lines on figure 9 are the predictions of the fully itinerant model (as per figure 4), and in

the subsequent figure, figure 10, we show the data together with the predictions of the partiallylocalized model (figure 5). In the next section, we carry out a detailed comparison between thedata and the predictions of the models.

4. Discussion

The extreme angle dependence of the dc-magnetoresistance, shown in figure 6, can be explainedfollowing Taillefer et al [33] who argue that it arises from canonical !c⌧ > 1 effects. Theypoint out (a) that the large magnetoresistance seen at most angles is indicative of an open orbit,since UPt3 is compensated and therefore ought not to have a magnetoresistance unless there are

New Journal of Physics 10 (2008) 053029 (http://www.njp.org/)

Taillefer and Lonzarich PRL 60, 1570 (1988)

McMullan, Julian et al., New J.Phys. 10, 053029 (2008)

• When do we expect Fermi liquid theory to breakdown?

• Power counting: kinetic vs. potential energy

Breakdown of FLT



Breakdown of FLT

Potential

V =1

4⇡✏0

e2

r⇠ 1

a⇠ n1/3



Breakdown of FLT

Potential

V =1

4⇡✏0

e2

r⇠ 1

a⇠ n1/3

Potential energy density

⇠ n

a⇠ n4/3



Breakdown of FLT

Potential

V =1

4⇡✏0

e2

r⇠ 1

a⇠ n1/3


⇠ n

a⇠ n4/3

Kinetic

T = � ~22m

r2r ⇠ 1

a2⇠ n2/3



Breakdown of FLT

Potential

V =1

4⇡✏0

e2

r⇠ 1

a⇠ n1/3


⇠ n

a⇠ n4/3

Kinetic

T = � ~22m

r2r ⇠ 1

a2⇠ n2/3

Kinetic energy density

⇠ n

a2⇠ n5/3



Breakdown of FLT

Potential

V =1

4⇡✏0

e2

r⇠ 1

a⇠ n1/3


⇠ n

a⇠ n4/3

Kinetic

T = � ~22m

r2r ⇠ 1

a2⇠ n2/3

Kinetic energy density

⇠ n

a2⇠ n5/3

Kinetic energy wins at high density – always Potential energy wins at low density – always

FLT breaks down in between.

Wigner crystallization

http://physics.aps.org/articles/v2/4

Kinetic vs. potential energy+

12

X

i6=j

e2

4⇥�0

1|ri � rj |

potential energy

X

i

p2i

2me

kinetic energy

Kinetic vs. potential energy

localization reduces potential energy at the cost

of kinetic

Stoof, Nature 415, 25 (2002).

+12

X

i6=j

e2

4⇥�0

1|ri � rj |

potential energy

X

i

p2i

2me

kinetic energy

Kinetic vs. potential energy

localization reduces potential energy at the cost

of kinetic

Stoof, Nature 415, 25 (2002).

+12

X

i6=j

e2

4⇥�0

1|ri � rj |

potential energy

X

i

p2i

2me

kinetic energy

delocalization reduces kinetic energy but introduces double

occupancy

[ri, pi] = i~

Quantum phase transitions

A. J. Schofield

QCPs in cuprates

ature T *. Second, the onset of an anomalous polarKerr rotation and neutron spin flip scatteringboth terminate at p ≈ 0.18 (12, 13), represent-ing an unidentified form of broken symmetry(which persists inside the superconductingphase for the Kerr experiment). Third, in highmagnetic fields, the sign change of the Hallcoefficient in YBa2Cu3O6+d from positive tonegative, and the anomaly in the Hall coefficientin Bi2Sr0.51La0.49CuO6+d, occur near p ≈ 0.18(11, 49), which suggests that Fermi surface re-construction from electron-like to hole-like oc-curs at this doping. Finally, p ≈ 0.18 representsthe maximum extent of incommensurate CDWorder reported in several different experiments(15, 26, 27). Although the Fermi surface recon-struction is likely related to this CDW order, itsshort correlation length and the weak dopingdependence of its onset temperature appear tobe at odds with the standard picture of long-range order collapsing to T = 0 at a QCP (50).Two scenarios immediately present themselves.In the first scenario, the suppression of super-conductivity by an applied magnetic field al-lows the CDW to transition to long-range order,as suggested by x-ray, nuclear magnetic reso-nance, and pulsed-echo ultrasound experiments(25, 26, 51). In this first scenario, we would beobserving a field-revealed QCP. In the secondscenario, CDW order is coexistent with anotherform of order that also terminates near pcrit ≈0.18. Such a coexistence is suggested by multipleexperimental results, including but not limitedto Nernst anisotropy (22), polarized neutronscattering (12), and the anomalous polar Kerreffect (13). In this second scenario, the CDW re-constructs the Fermi surface and the other hid-den form of order drives quantum criticality.Regardless of the specific mechanism, and re-gardless of whether pcrit = 0.18 is a QCP in thetraditional sense, the observation of an enhanced

effective mass coincident with the region of mostrobust superconductivity establishes the impor-tance of competing broken symmetry for high-Tc superconductivity.

REFERENCES AND NOTES

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7. P. W. Anderson et al., J. Phys. Condens. Matter 16, R755–R769(2004).

8. J. L. Tallon, J. W. Loram, Physica C 349, 53–68(2001).

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28. R. Comin et al., Science 343, 390–392 (2014).29. D. Shoenberg, Magnetic oscillations in metals (Cambridge Univ.

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ACKNOWLEDGMENTS

This work was performed at the National High Magnetic FieldLaboratory and was supported by the U.S. Department of EnergyOffice of Basic Energy Sciences “Science at 100 T” program,NSF grant DMR-1157490, the State of Florida, the Natural Scienceand Engineering Research Council of Canada, and the CanadianInstitute for Advanced Research. S.E.S. acknowledges supportfrom the Royal Society and the European Research Councilunder the European Union’s Seventh Framework Programme(FP7/2007-2013)/ERC Grant Agreement 337425. We thankS. Chakravarty, S. Kivelson, M. Le Tacon, K. A. Modic, C. Proust,A. Shekhter, and L. Taillefer for discussions; J. Baglo for sharinghis results on the effect of quenched oxygen disorder on themicrowave scattering rate in YBa2Cu3O6+d, without whichoscillations would not have been observed; and the entire 100 Toperations team at the pulsed-field facility for their supportduring the experiment. Full resistivity curves are available in thesupplementary materials. B.J.R., S.E.S., R.D.M., B.T., Z.Z., J.B.B.,and N.H. performed the high-field resistivity measurements at theNational High Magnetic Field Laboratory Pulsed Field Facility.B.J.R., J.D., R.L., D.A.B., and W.N.H. grew and prepared thesamples at the University of British Columbia. B.J.R. analyzed thedata and wrote the manuscript, with contributions from S.E.S.,R.D.M., N.H., J.D., D.A.B., and W.N.H.

SUPPLEMENTARY MATERIALSwww.sciencemag.org/content/348/6232/317/suppl/DC1Materials and MethodsSupplementary TextFigs. S1 to S15References (58–69)

15 December 2014; accepted 16 March 2015Published online 26 March 2015;10.1126/science.aaa4990

320 17 APRIL 2015 • VOL 348 ISSUE 6232 sciencemag.org SCIENCE

Fig. 4. A quantum crit-ical point near opti-mal doping.The solidblue circles correspond toTc, as defined by theresistive transition (rightaxis), atmagnetic fields of0, 15, 30, 50, 70, and 82 T[some data points takenfrom (39, 57); solid bluecurves are a guide to theeye].As themagnetic fieldis increased, the super-conducting Tc is sup-pressed. By 30 T, twoseparate domes remain,centered around p ≈ 0.08and p ≈ 0.18; by 82 T, onlythe dome at p ≈ 0.18remains.The inverse ofthe effective mass has been overlaid on this phase diagram (left axis), extrapolating to maximum massenhancement at p ≈ 0.08 and p ≈ 0.18 [white diamonds taken from (56)]. This makes explicit theconnection between effective mass enhancement and the robustness of superconductivity. Yellowsymbols parallel those in Figs. 1 to 3. Error bars are SE from regression of Eq. 1 to the data.

RESEARCH | REPORTS

Ramshaw et al., Science 348, 317 (2015)

QCPs in heavy fermions

N.D. Mathur et al., Nature 394, 39 (1998)

Custers et al., Nature 424, 524 (2003)

Mass renormalization and the optical conductivity

s1 (w

)

w

m*�1(!) =

ne2⌧⇤

m⇤1

1 + (!⌧⇤)2

Interaction effects simultaneously increase mass and renormalized lifetime.

�dc

DC conductivity

�dc =ne2⌧⇤

m⇤

Drude conductivity

DC conductivity

�dc =ne2⌧⇤

m⇤

Drude conductivity

plasma frequency lifetime

= ✏0!2p ⇥ ⌧⇤

DC conductivity

�dc =ne2⌧⇤

m⇤

Drude conductivity

~ kinetic energy density

!2p =

e2

✏0

n

m⇤


= ✏0!2p ⇥ ⌧⇤

DC conductivity

m⇤ / 1

!2p

effective mass

�dc =ne2⌧⇤

m⇤

Drude conductivity


!2p =

e2

✏0

n

m⇤


= ✏0!2p ⇥ ⌧⇤

DC conductivity

m⇤ / 1

!2p

effective mass

�dc =ne2⌧⇤

m⇤

Drude conductivity

�(!) = �1 � i�2

=ne2

m⇤1

1/⌧⇤ + i!

AC conductivity


!2p =

e2

✏0

n

m⇤


= ✏0!2p ⇥ ⌧⇤

DC conductivity

m⇤ / 1

!2p

effective mass

�dc =ne2⌧⇤

m⇤

Drude conductivity

�(!) = �1 � i�2

=ne2

m⇤1

1/⌧⇤ + i!

AC conductivity


!2p =

e2

✏0

n

m⇤!⌧⇤ =

�2

�1⇡ �

lifetime

conductivity phase angle


= ✏0!2p ⇥ ⌧⇤

DC conductivity

m⇤ / 1

!2p

effective mass

�dc =ne2⌧⇤

m⇤

Drude conductivity

�(!) = �1 � i�2

=ne2

m⇤1

1/⌧⇤ + i!

AC conductivity


!2p =

e2

✏0

n

m⇤!⌧⇤ =

�2

�1⇡ �

lifetime


plasma frequency

!2p =

�

✏0

✓i +

�1

�2

◆


= ✏0!2p ⇥ ⌧⇤

Microwave cavity perturbation

Rs = �⇥fB(T )

2⇥Xs = �⇥f0(T )

Zs = Rs + iXs

Resonator simulations by SFU undergrad Paul Carrière, using COMSOL.

quality factor: 1 to 30 million high filling factor

operates in high field

Beryllium Copper Bellows

Brass Vacuum Can

Stainless Steel Thermal Weak Link

Copper 1K Pot

0.141’’ Stainless Steel Coax Cable

Copper Enclosure

SampleRutile Resonator

Thermometer & Resistive Heater

Coupling Hole

Quartz Thermal Weak Link

Sorption Pump

Knife Edge

Coupling Loop (Coated in Epoxy)

Indium Seals

Vacuum Space

Sapphire Hot Finger

Sapphire Plate

Indium Seals

Brass Spring

Vacuum Space

1K Pot Pumping Tube

Dielectric resonator system

Microwave spectroscopy

TE011

2.91 GHz

TE013

4.82 GHz

TE021

5.57 GHz

c

a

b

d

Transverse

electric

resonant

modes

3He–4He

dilution

refrigerator

Sample

loading

interlockMicrowave

network

analyzer

Dielectric resonator

Recondensing

cryocooler

Single-

crystal

sample

Removable

sample

thermal

stage1.5 K

0.2 K0.05 K

0.2 K0.05 K

Superfluid density

Colin Truncik et al., Nature Communications 4:2477 (2013)

= +

= +

= +

metal

s-wavesuperconductor

d-wavesuperconductor

a

b

c

m*v

m*v m*v m*v

m*vm*v m*v

m*v m*v

jtot jd jp=

Ce

Co

In

0.0 0.2 0.4 0.6 0.82425262728

0.0 0.5 1.0 1.5 2.0 2.50

5

10

15

20

25

30

Temperature HKL

1êd2HwL∫wm 0s2Hmm-2 L

1êlL219.6315.9212.286.945.574.822.912.250.13

f HGHzL

0.0 0.1 0.2 0.3 0.4 0.5 0.60

2

4

6

8

Temperature HKL

DJ1íl2NHm

m-2 L

DJ1ílL2NDJ1íl2Nqp

a T1.21

a + b Ta b

Colin Truncik et al., Nature Communications 4:2477 (2013)

Superfluid density in CeCoIn5

UBe13 structure and Fermi surface

1692 K Takegahara et a1

eight Be' in 8(b): 0, 0,O; . . . ; and 96 Be" in 96 (i): 0, y , z; . . , . In the following, we set the origin of the real space coordinate at the M site. As shown in figure 1, the M-Be' system forms a simple CsC1-type structure with a lattice constant of a 3 of MBe13. Be' is surrounded by a Be" icosahedron, whereas M is surrounded by 24 Be" sitting at the corner of a snub cube. The Be'' icosahedra in the neighbourhood of half of a lattice constant are in 90" rotation around the fourfold crystal axis and thus the crystal structure is the FCC lattice. There is no set of parameters ( y , z) for which both the icosahedron and the snub cube are regular. The regularity condition of the snub cube requires y = 0.176 1 and z=0.1141. The observed values are as follows: y=O.178 and z=0.112 for CeBe13 (Bucher et a1 1975) and y=0.1763 and z=0.1150 for UBe13 (Shapiro et a1 1985). The deviation of these observed values from the values of the regularity condition of the snub cube is very small and so in the following calculations we adopt the latter values.

One of the features of this structure is that the Be' and its nearest-neighbour Be"

Figure I . Crystal structure of MBe13. Large spheres show the position of the M atoms and small spheres. with and without pattern, show the positions of Be' and Be" atoms, respectively.

Takegahara, Harima and Kasuya

U

Be

Dilute f orbitals in UBe13

Fig. 1. Perspective view of the Fermi surfaces for UBe!"

in the 1st BZ. The region of 0(k!, 0(k

"and !k

!(k

#, is cut away.

Table 1Calculated dHvA frequency F and cyclotron e!ective massmH

!for UBe

!"

Band Field orientation F(!10# Oe) mH!(m

$)

32nd !1 0 0" 1.00 3.350.62 2.860.61 2.99

!1 1 1" 0.83 3.85!1 1 0" 0.90 3.85

0.64 2.950.62 2.96

33rd !1 0 0" 2.63 3.450.95 4.350.86 1.530.14 1.39

!1 1 1" 1.94 3.571.03 2.280.29 1.790.09 0.87

!1 1 0" 0.53 2.170.38 1.240.09 0.87

than observed one (about !%"

). This di!erence is thoughtto be due to the strongly correlated electron e!ects.

The Fermi surface consists of two sheets as shown inFig. 1. The 32nd band has a closed hole surface centeredat the X point. The 33rd band forms two roughly spheri-cally shaped surfaces centered at the # point but inter-connects along the $-axis. Note that the similar bandstructure has been obtained in the LAPW band calcu-lations for UBe

!"[10], but the cross-section of intercon-

necting part along the $-axis is much smaller than ours.In Table 1, the dHvA frequencies and cyclotron e!ectivemasses are shown in high-symmetry "eld orientations.Two bands have cyclotron e!ective masses of the order of0.9m

$}4.4m

$, where m

$is the free electron mass. Rela-

tively large e!ective masses are due to the fact that thesetwo bands consist of the U f states, about 80% ofU f states.

Secondly, we show the result for LaBe!"

. La f-compo-nents are located well above the Fermi level and theFermi surfaces are mainly consist of Be p-bands, thenthey are not much in#uenced by the e!ect of spin}orbitinteraction. Moreover, Be p-bands do not seem to needa large number of basis functions. Therefore, the cal-culated band structure particularly near the vicinity ofthe Fermi level, is very similar to the previous one [2,3],which was obtained with the limited basis functions andwithout the e!ect of spin}orbit interaction. This dis-cussion is also applicable to ThBe

!"[7]. The Fermi

surface consists of six sheets; the 27th and 28th bandshave a closed hole surface centered at the # point, the31st and 32nd bands have small closed electron surfacecentered at the L point, and the 29th and 30th bands havecomplicated hole and electron surfaces, respectively.These bands have cyclotron e!ective masses of the order

of 0.08m$}1.5m

$. The calculated ! value is 6.18 mJ/

mol K&, about !!'"

of the observed value of 8.12 mJ/mol K& [11].

In ThBe!"

, the calculated cyclotron e!ective massesrange from 0.1m

$to 0.8 m

$, while the experimentally

determined masses are the order of 0.07m$}1.2m

$[7].

The calculated ! value is 6.23 mJ/mol K&, slightly smallerthan the observed value of 7.1 mJ/mol K& [11]. Thus,

K. Takegahara, H. Harima / Physica B 281&282 (2000) 764}766 765

Takegahara and Harima, Physica B 281, 764 (2000)

Small Fermi surface

UBe13 structure and Fermi surface

1692 K Takegahara et a1

eight Be' in 8(b): 0, 0,O; . . . ; and 96 Be" in 96 (i): 0, y , z; . . , . In the following, we set the origin of the real space coordinate at the M site. As shown in figure 1, the M-Be' system forms a simple CsC1-type structure with a lattice constant of a 3 of MBe13. Be' is surrounded by a Be" icosahedron, whereas M is surrounded by 24 Be" sitting at the corner of a snub cube. The Be'' icosahedra in the neighbourhood of half of a lattice constant are in 90" rotation around the fourfold crystal axis and thus the crystal structure is the FCC lattice. There is no set of parameters ( y , z) for which both the icosahedron and the snub cube are regular. The regularity condition of the snub cube requires y = 0.176 1 and z=0.1141. The observed values are as follows: y=O.178 and z=0.112 for CeBe13 (Bucher et a1 1975) and y=0.1763 and z=0.1150 for UBe13 (Shapiro et a1 1985). The deviation of these observed values from the values of the regularity condition of the snub cube is very small and so in the following calculations we adopt the latter values.

One of the features of this structure is that the Be' and its nearest-neighbour Be"

Figure I . Crystal structure of MBe13. Large spheres show the position of the M atoms and small spheres. with and without pattern, show the positions of Be' and Be" atoms, respectively.

Takegahara, Harima and Kasuya

U

Be Dilute f orbitals in UBe13

UBe13 – s/c from nFL normal state

72 EUROPHYSICS LETTERS

the 4f-derived local moments to a low-T phase with .<coherent scattering),; this crossover is marked by low-T peaks in both ,c(O, T) [91 and the negative magnetoresistivity l,c(B, T ) = = c ( ~ , r) - , c ( ~ , r) [IO]. In addition to other distinct features, like a giant thermopower exiremum [ll] or a finite residual (as T+ 0) half-width of the quasi-elastic magnetic neutron line [lz], these anomalies in ,c(B, T) are considered as good empirical measures of the characteristic temperature T* at wich the formation of a Kondo singlet starts [13, 141. At sufficiently low temperatures the latter leads, as recent theoretical work shows, to the formation of a coherent narrow band right a t the Fermi level [15]. I t consists of quasiparticle (heavy-fermion) states with essentially 4f-symmetry. Empirical .fingerprints. of this coherent-band formation are characteristic structures in the temperature dependences of both the linear-specific-heat coefficiet y ( T ) = C(T)/T [ E ] and thermopower [8, 171, and in particular, a change of sign in the magnetoresistivity +(B, r). Thus, for instance, for CeAl, measured at B = 4 T , A,c was found to change from negative to a “mal metallic,, positive value at T = 0.5 K [HI.

In the case of UBe13, earlier investigations [3, 51 revealed a negative A,@, T) down to TC-0.9K and even down to 0.5K, the superconducting transition temperature in the presence of B = 6T. In the present paper, these measurements are extended to the temperature range 50 mK d T d 4.5 K and to magnetic fields up to 10 T, thus providing information about size and sign of the magnetoresistance in UBe13 at very low temperatures. We also extended the zero-field data up to room temperature.

Measurements were done by a 4-point a.c.-method in a standard 3He-4He dilution refrigerator, in which magnetic fields up to 10T could be applied by means of a superconducting solenoid. Data were recorded continuously as a function of T at constant B- field and as a function of B at constant T. The UBe13 sample used for the resistance measurements was the same as used in [6] for determining the upper-critical-field curve BCL(T).

160 -

120-

5 C I - 80- QJ

- -

0 I I 1

1 0-‘ loo 10’ l o 2 T ( K )

0 1 2 3 T ( K )

Fig-. 1. - Resistivity of UBels as a function of temperature for different values of the external field. Inset shows 10T-data together with zero-field results up to 300K on a logarithmic temperature scale.

incoherent normal-state

transportRauchschwalbe et al.,

EPL 1, 71 (1986)

UBe13 – s/c from nFL normal state

72 EUROPHYSICS LETTERS

the 4f-derived local moments to a low-T phase with .<coherent scattering),; this crossover is marked by low-T peaks in both ,c(O, T) [91 and the negative magnetoresistivity l,c(B, T ) = = c ( ~ , r) - , c ( ~ , r) [IO]. In addition to other distinct features, like a giant thermopower exiremum [ll] or a finite residual (as T+ 0) half-width of the quasi-elastic magnetic neutron line [lz], these anomalies in ,c(B, T) are considered as good empirical measures of the characteristic temperature T* at wich the formation of a Kondo singlet starts [13, 141. At sufficiently low temperatures the latter leads, as recent theoretical work shows, to the formation of a coherent narrow band right a t the Fermi level [15]. I t consists of quasiparticle (heavy-fermion) states with essentially 4f-symmetry. Empirical .fingerprints. of this coherent-band formation are characteristic structures in the temperature dependences of both the linear-specific-heat coefficiet y ( T ) = C(T)/T [ E ] and thermopower [8, 171, and in particular, a change of sign in the magnetoresistivity +(B, r). Thus, for instance, for CeAl, measured at B = 4 T , A,c was found to change from negative to a “mal metallic,, positive value at T = 0.5 K [HI.

In the case of UBe13, earlier investigations [3, 51 revealed a negative A,@, T) down to TC-0.9K and even down to 0.5K, the superconducting transition temperature in the presence of B = 6T. In the present paper, these measurements are extended to the temperature range 50 mK d T d 4.5 K and to magnetic fields up to 10 T, thus providing information about size and sign of the magnetoresistance in UBe13 at very low temperatures. We also extended the zero-field data up to room temperature.

Measurements were done by a 4-point a.c.-method in a standard 3He-4He dilution refrigerator, in which magnetic fields up to 10T could be applied by means of a superconducting solenoid. Data were recorded continuously as a function of T at constant B- field and as a function of B at constant T. The UBe13 sample used for the resistance measurements was the same as used in [6] for determining the upper-critical-field curve BCL(T).

160 -

120-

5 C I - 80- QJ

- -

0 I I 1

1 0-‘ loo 10’ l o 2 T ( K )

0 1 2 3 T ( K )

Fig-. 1. - Resistivity of UBels as a function of temperature for different values of the external field. Inset shows 10T-data together with zero-field results up to 300K on a logarithmic temperature scale.

incoherent normal-state

transportRauchschwalbe et al.,

EPL 1, 71 (1986)

0 2 4 6 8 100

20

40

60

80

Temperature !K"

Conductivity

PhaseAngle!Degr

ees"

9 T8 T7 T6 T5 T4 T3 T2 T1 T0 T

0 2 4 6 8 100

50

100

150

Temperature !K"

Ρ!Μ#cm

"

9 T8 T7 T6 T5 T4 T3 T2 T1 T0 T

14.2 GHz resistivity


UBe13 scattering rate

0 2 4 6 8 100

50

100

150

200

250

Temperature !K"

!!K"

9 T8 T7 T6 T5 T4 T3 T2 T1 T0 T

0.0 0.5 1.0 1.5 2.0 2.5 3.00

5

10

15

20

25

30

35

Temperature !K"

!!K"

9 T8 T7 T6 T5 T4 T3 T2 T1 T0 T

coherent & field-tuneable between FL and nFL form

“optics with microwaves”

heavy electrons still forming at the onset of superconductivity

Field (T)

contours of constant effective mass

superconductingstate

superconductingstate

Bc2

Bc2

�dc =ne2⌧⇤

m⇤

�(!) =ne2

m⇤1

i! + 1/⌧⇤

UBe13 – field-tuned quantum criticality

Conclusions• Fermi liquid theory provides a standard model of

electrons in metals.

• Kinetic energy dominates potential energy at high densities, ensuring the stability of the Fermi liquid in 3D.

• Fermi liquids meet their downfall in the regime where kinetic and potential energies are finely balanced.

• The search for physics beyond the standard model should therefore start in the vicinity of a quantum phase transition.

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