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Fermi liquids and their breakdown
David Broun [email protected]
8 SFU BRAND GUIDELINES UNIVERSITY MASTER BRAND
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CIFAR BRAND STANDARDS 2
CIFAR Quantum Materials Summer School 2015
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
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j
� ~2
2Mjr2
Rj
nuclear kinetic energy
+12
X
i6=j
e2
4⇥�0
1|ri � rj |
electron–electron repulsion
+12
X
i6=j
ZiZje2
4⇥�0
1|Ri �Rj |
nucleus–nucleus repulsion
Theory of (nearly) everything
X
i
� ~2
2mer2
riH =
i~ ⇥
⇥t|�� = H|��
electron kinetic energy
+X
j
� ~2
2Mjr2
Rj
nuclear kinetic energy
+12
X
i6=j
e2
4⇥�0
1|ri � rj |
electron–electron repulsion
+12
X
i6=j
ZiZje2
4⇥�0
1|Ri �Rj |
nucleus–nucleus repulsion
�X
i,j
Zje2
4⇥�0
1|ri �Rj |
electron–nucleus attraction
Theory of (nearly) everything
X
i
� ~2
2mer2
riH =
i~ ⇥
⇥t|�� = H|��
+12
X
i6=j
e2
4⇥�0
1|ri � rj |
electron kinetic energy
+X
j
� ~2
2Mjr2
Rj
nuclear kinetic energy
+12
X
i6=j
ZiZje2
4⇥�0
1|Ri �Rj |
nucleus–nucleus repulsion
�X
i,j
Zje2
4⇥�0
1|ri �Rj |
electron–nucleus attraction
Freeze the nuclei (Born–Oppenheimer)
electron–electron repulsion
Theory of (nearly) everything
X
i
� ~2
2mer2
riH =
i~ ⇥
⇥t|�� = H|��
+ U(ri)
two-body term
one-body terms
+12
X
i6=j
e2
4⇥�0
1|ri � rj |
Theory of (nearly) everything
Quantum complexity
| �⇥
| �⇥⇤| �⇥
| �
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)
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
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
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?”
Landau quasiparticles
Damascelli, Hussain & Shen, RMP
noninteracting
interacting
Landau quasiparticles
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 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 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 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 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
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
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
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
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
• When do we expect Fermi liquid theory to breakdown?
• Power counting: kinetic vs. potential energy
Breakdown of FLT
Potential
V =1
4⇡✏0
e2
r⇠ 1
a⇠ n1/3
• When do we expect Fermi liquid theory to breakdown?
• Power counting: kinetic vs. potential energy
Breakdown of FLT
Potential
V =1
4⇡✏0
e2
r⇠ 1
a⇠ n1/3
Potential energy density
⇠ n
a⇠ n4/3
• When do we expect Fermi liquid theory to breakdown?
• Power counting: kinetic vs. potential energy
Breakdown of FLT
Potential
V =1
4⇡✏0
e2
r⇠ 1
a⇠ n1/3
Potential energy density
⇠ n
a⇠ n4/3
Kinetic
T = � ~22m
r2r ⇠ 1
a2⇠ n2/3
• When do we expect Fermi liquid theory to breakdown?
• Power counting: kinetic vs. potential energy
Breakdown of FLT
Potential
V =1
4⇡✏0
e2
r⇠ 1
a⇠ n1/3
Potential energy density
⇠ n
a⇠ n4/3
Kinetic
T = � ~22m
r2r ⇠ 1
a2⇠ n2/3
Kinetic energy density
⇠ n
a2⇠ n5/3
• When do we expect Fermi liquid theory to breakdown?
• Power counting: kinetic vs. potential energy
Breakdown of FLT
Potential
V =1
4⇡✏0
e2
r⇠ 1
a⇠ n1/3
Potential energy density
⇠ 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.
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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⇤
plasma frequency lifetime
= ✏0!2p ⇥ ⌧⇤
DC conductivity
m⇤ / 1
!2p
effective mass
�dc =ne2⌧⇤
m⇤
Drude conductivity
~ kinetic energy density
!2p =
e2
✏0
n
m⇤
plasma frequency lifetime
= ✏0!2p ⇥ ⌧⇤
DC conductivity
m⇤ / 1
!2p
effective mass
�dc =ne2⌧⇤
m⇤
Drude conductivity
�(!) = �1 � i�2
=ne2
m⇤1
1/⌧⇤ + i!
AC conductivity
~ kinetic energy density
!2p =
e2
✏0
n
m⇤
plasma frequency lifetime
= ✏0!2p ⇥ ⌧⇤
DC conductivity
m⇤ / 1
!2p
effective mass
�dc =ne2⌧⇤
m⇤
Drude conductivity
�(!) = �1 � i�2
=ne2
m⇤1
1/⌧⇤ + i!
AC conductivity
~ kinetic energy density
!2p =
e2
✏0
n
m⇤!⌧⇤ =
�2
�1⇡ �
lifetime
conductivity phase angle
plasma frequency lifetime
= ✏0!2p ⇥ ⌧⇤
DC conductivity
m⇤ / 1
!2p
effective mass
�dc =ne2⌧⇤
m⇤
Drude conductivity
�(!) = �1 � i�2
=ne2
m⇤1
1/⌧⇤ + i!
AC conductivity
~ kinetic energy density
!2p =
e2
✏0
n
m⇤!⌧⇤ =
�2
�1⇡ �
lifetime
conductivity phase angle
plasma frequency
!2p =
�
✏0
✓i +
�1
�2
◆
plasma frequency lifetime
= ✏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 quasi- particle (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 quasi- particle (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
conductivity phase angle
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.