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nils-uni@bluemer.name
Prof. Dr. Nils Blümer |
|
Attention: these pages are mostly outdated! For current information on Nils Blümer, see web pages at the KU.
Last changed:
12-Sep-13
Mott-Hubbard Metal-Insulator Transition and Optical Conductivity in High Dimensions
by Nils Blümer
(Shaker Verlag, ISBN 3-8322-2320-7)
Index
Red/bold page numbers: main entry, blue/italic page numbers: figures.
Ab initio 1, 251f, 280 Absorption 177, 235 Absorption coefficient 179 Abstract models 1, 5, 184, 199 Action in elm. field 188, 305 single-site 17, 19, 254 Adiabatic approximation 7 Adiabatic switching on 307 Akima spline 85 Aluminum 181 Analytic continuation 19, 23, 25, 160 Analyticity 159, 178f, 229, 297 Anderson localization 2, 40, 70, 283 Angle, bond 199, 246, 291 Angular momentum 281, 284 Annealing method 162 Annihilation operator 8 Antiferromagnetism (AF) 10, 12, 67, 70f, 139, 162, 207, 247f, 252, 255, 259, 277, 279, 284 suppression of 38 Asymmetry of DOS 27, 39, 41, 64, 71, 139, 141, 174, 222 parameter for 43, 62 Atomic limit 44, 86, 251 Atomic radii, effective 69 Atomic sphere approximation (ASA) 251 Autocorrelation 22, 25, 153 Auxiliary (binary) field 20 Average over spins, bands 264
Band eg 247, 266 t2g 247 degeneracy 246 edge 34, 280 index 184 insulator 246 structure 251 width 69, 256, 268 Basis 203, 251, 256 Bayes' theorem 26 Beryllium 181 Bessel function 127, 194 Bethe ansatz 10, 18 Bethe lattice 27, 39, 40, 175, 199 fully frustrated 71 optical conductivity for 198-211, 242 redefined 53, 213, 234, 243 stacked 210, 204-211, 236 with NNN hopping 45 Bethe-Peierls approximation 40f Bipartite lattice 14, 32, 41, 291 Bloch functions 1, 8 Bond 199 Bravais lattice 189, 209, 291 Brillouin zone 1, 32, 285 Broadening (Gaussian, Lorentzian) 259, 261 Bryan's method 157 Bubble diagram 193, 196, 207, 212, 217 Bulk properties 178, 277
Causality 177 Cayley tree 40, 41 Cellular/cluster dynamical mean-field theory (CDMFT) 287 Central limit theorem 14, 21 Chain of bonds 204 Charge density wave (CDW) 283 Chemical potential 257, 261 Chrome 68 Clausius-Clapeyron equation 123 Cluster finite 254 periodic 287 Coefficient functions 59 Coexistence points 110, 113 Coexistence region 68, 79, 88, 108-113, 144, 173, 279 boundaries of 89, 100, 102, 109f dependence on Δτ 89, 113 ground state 112 Coherence 176, 205, 207, 216-219, 234, 236, 243 Coherent potential approximation (CPA) 254 Commutator 209, 207-209 Composition 247 Compressibility 79 Computational cost (of QMC) 20, 23, 106 Conductivity static (dc) 175, 207 tensor 177 transverse 223, 225, 243 Connectivity 40 Conserving approximation 11, 73, 79 Constrained LDA 252f Controlled approaches 11 Convergence 65, 91, 96, 145, 151, 153 factor 78 Coordination number 11f Core electrons 7, 180 Core ions 7 Correlation 1, 248, 264, 266, 280 short-range 288 Correlation function 25, 192, 206, 212, 218, 243, 308 Corundum structure 69, 69 Coulomb gauge 6, 186, 306 Coulomb interaction 1, 5, 9, 249 instantaneous 6 nearest-neighbor 283 on-site see Hubbard interaction Covariance matrix 155, 156f Creation operator 8 Critical end point 70, 74, 109, 112, 146, 153, 174 exponent 174 points 151 slowing down 146, 150-155 Criticality 146, 174 Crossing point 109 Crossover 71f, 93, 109 Current 176 density (operator) 184, 188, 190, 205 diamagnetic 186, 192 paramagnetic 186 total 186 vertex 175, 192, 206 Curvature 95
d electrons 281 d-wave pairing 10 Default model 26, 161, 160-163, 226, 262 Degeneracy 10, 255, 281 Density (operator) 27, 181, 249, 307 Density distribution 249f Density functional theory (DFT) 249-255, 280 Density matrix renormalization group (DMRG) 289 Density of states (DOS) 14, 27, 42, 64, 175, 226, 255, 289 ''tanh'' 54 2-particle 14, 38 asymmetric model 43, 43, 62 fcc 36f, 37 for t-t' hopping 33-35 gapped 59 general construction of 58, 60, 61, 47-64 hypercubic 33 hyperdiamond 292, 294, 295 interacting see Spectral funktion LDA (for La1-xSrxTiO3) 256, 269 Lorentzian 18 moments of 43 partial 61, 62 semi-elliptic (Bethe) 18, 42, 67, 176, 213, 242, 280, 297 Density-density interaction 253 Derivative 24, 124, 195, 297 Diamond lattice 291 Dielectric function 177, 179, 179, 182 transverse vs. longitudinal 178 Differential equation 121, 123 linearized 116, 124, 132, 173 solution of 133 Dimension(ality) 10 critical 30 finite 59, 214, 217, 219-222, 268, 279, 284 limit of infinite 11, 175, 193, 243 Dipole 259 Direction 199 Discontinuity 73 Discretization (of imaginary time) 19, 159, 261-265, 280, 297-304 Disorder 40, 46, 64, 69, 184, 211-213, 236, 243, 254, 260, 273, 275, 277, 283 Dispersion 1, 27f, 47, 192, 197, 293 Doping 69, 243, 247, 255, 276 Double counting 253 Double occupancy 9, 73, 79, 80, 91, 93, 94, 98, 104, 113-121, 121, 128, 136, 137, 147-154, 262, 304 fit for insulator 99, 99 fit for metal 115 PSCT estimate for metal 127 Downhill scheme 144 Drude peak/weight 175, 181, 184, 191, 234f, 244, 273 Dynamical cluster approximation (DCA) 193, 254, 286-288 Dynamical mean-field theory (DMFT) 2, 11, 11-18, 73, 254, 279, 284, 297 methods for solution 18 Dyson equation 16, 227, 254 for Bethe lattice 42 numerical inversion of 228, 269
Easy axis 284 Edge 40 Effective interaction 2, 7, 252 Effective overlap 247 Einstein phonon 284 Electric field 176, 305 Electromagnetic field 259, 305-308 Electron charge 186 Electron-phonon interaction 244, 283 Electronic energy loss spectroscopy (EELS) 175, 179, 182f, 273 Emery model 283 Energy 61, 77, 95, 98, 100-108, 121, 130, 249, 255 eigenstate 201, 203 fit for insulator 97, 97, 98 functional 249f in ground state 128 linearity in U 95f, 100, 107 resolution 81 scale 27 Ensemble of systems 6, 73 Entropy 73, 105, 121, 123, 126, 174, 279 Shannon-Jaynes 26 Ergodic 6, 22 Errors 90, 159, 162, 261-265, 299 Euclidean length 29, 48 Exact diagonalization (ED) 18, 80, 81, 111, 129, 131, 141, 243, 289 Exchange interaction 253 Exchange-correlation energy 250 Excitation 96, 251 Expectation value 31, 186, 195, 307 Experimental results 70, 68-71, 181-183, 273-275 Extrapolation 89, 93, 96, 104, 178, 263
f electrons 281 f-sum rule 179-181, 191, 201, 220, 241, 241, 242, 243f, 280 dependency on model 184 for Bravais lattice 190 for hc lattice 189 partial 181, 183, 234 within the DMFT 195-198, 211 Face-centered cubic (fcc) lattice 32, 34, 36, 219, 220 Fermi energy 257, 268 Fermi golden rule 259 Fermi liquid 76, 129, 170, 205, 230 Fermi surface 32, 281, 285 Fermi velocity 51, 185, 201, 215, 219, 223, 226, 239, 268, 269, 271, 292f averaged squared 51, 55, 56 Fermion propagator 15 Ferromagnetism 10, 62, 255, 283 Filling control of 266 half 279 Filling-induced MIT 67 First-order phase transition 70, 74, 79, 104, 173, 279 absence of 72 Fixed point 73, 144, 173 Flat-band systems 10, 52 Floating-zone method 259 Fluctuation matrix 146, 151 Fluctuation-exchange approximation (FLEX) 18, 254 Fluctuations 153 Four-fermion term 20 Fourier transformation (FT) 17, 19, 82, 84, 87f, 159, 177, 262, 297-304 comparison of FT schemes 87 Free energy 73, 121, 126, 134, 173 direct evaluation of 144-146 functional 74, 74, 142 gradient of 142-144, 145, 152 Frequency (grid) 157, 255 Frustration 27, 38, 46, 64, 71 full 3, 67, 279 partial 138-142, 174, 222-225, 279 Full-potential DFT 251 Fully connected model 43, 46 Functional 249 Functional integral 17, 20
Gap 96, 157, 159, 231, 247, 251 Gauge transformation 306 General dispersion formalism 51, 64, 176, 197, 213-216, 234, 243, 276 Ginzburg-Landau functional 73, 142f, 146 Ginzburg-Landau theory fit to 147, 148, 149, 150, 147-150 Gradient field 125 Graph 40 Grassmann variable 17 Green function 15, 22, 41, 143, 175, 192, 203, 207, 212, 254, 287, 297-304 analytic properties of 84 derivatives of 25, 163 for real frequencies 227 high-frequency expansion of 44 momentum dependent 78, 193 of imaginary time 136, 155, 160, 166f, 263, 298 on imaginary axis 80 useful relations for 24 Ground state 173, 249, 279 Gutzwiller wave function 80
Half-filled Hubbard model 71, 242 Hall resistance 222f Hamiltonian 186, 210, 249 electronic 7 for LDA+DMFT 253 full nonrelativistic 6 multi-band Hubbard 282 Hartree interaction 8, 13, 30, 250 Hartree-Fock (HF) theory 10, 251 Heisenberg model/interaction 10, 12, 282 Hermite polynomials 49, 50 High-frequency expansion 298 High-temperature expansion 19 Hilbert transformation 75 Histogram 158 Historical development 110 Holstein-Hubbard model 283 Homogeneous 7, 176, 180 Honeycomb lattice 291, 294, 296 Hopping amplitude 9, 27, 53, 175, 199, 252, 282, 287, 295 decay of 59 even range 65 for Bethe DOS 55 in elm. field 188 real and isotropic 48 Hopping range 27, 65, 141, 268 extended 30, 38, 45, 195 truncation of 57 Hopping vector 47 Hubbard bands (upper, lower) 257f Hubbard interaction 27, 122, 184, 279, 282 estimate from LDA 253, 256 Hubbard model 2 ''pure'' 9f general 8 single-band 5, 9, 245, 279 Hubbard-I approximation 254 Hubbard-Stratonovich transformation 20 Hund's rule couplings 282 Husimi cactus 40, 41 Hybridization 17, 69, 254, 260, 277 Hybridization function 143, 146 Hypercubic lattice 13, 29, 32, 71, 188, 192, 197f, 219, 243, 279, 284 Hyperdiamond (hd) lattice 222, 291-296 Hyperplane 210, 294 Hypersphere 284 Hypersurface 73 Hysteresis 73, 81
Image reconstruction 25 Imaginary time/axis 19, 82, 155, 255, 297 Importance sampling 122 Impurities 244 Impurity problem 18 Index of diffraction 177, 179 Insulator 2, 96-99, 178, 239, 247, 279 Interaction induced MIT 67 Interaction picture 307 Interband excitation 184, 272 Ionic radius 246 Ising model/interaction 13, 40, 282 Isostructural transition 70 Isotropic/anisotropic 7, 28, 47, 177, 210, 213, 217, 243, 291, 293, 296 Iterated perturbation theory (IPT) 18, 71, 79, 80f, 82, 110, 127, 147, 147, 174, 243, 255, 258, 303 Iteration scheme (in DMFT) 75, 144
Jacobian 228, 269 Jellium model 180, 249f
Kernel, fermion 23, 24, 157 Kinetic energy 1, 9, 47, 78, 94, 95, 201, 205, 211, 249-251, 289 in elm. field 188 Kink 95 Kohn-Sham equations 250, 252 Kondo interaction 11, 282 Kondo temperature 257 Kramers-Kronig trafo/analysis 178, 180, 227, 230, 269 Kubo formalism 186-191, 243
Lagrange parameter 26, 157, 252 Lagrangian 305 Landau Fermi-liquid theory 2, 126 Landau theory 142-150 Landau-Ginzburg theory 104 Lanthanum strontium titanate 3, 157, 246-248, 276, 280, 283 results for 255-275 structure of 246, 255 Lattice 64, 175 bipartite 71 distortion 69 impact within DMFT 27 Laue function 16 LDA++ 254 LDA+DMFT method 3, 245, 251-255, 276, 280 LDA+DMFT(QMC) 255, 277 LDA+U method 252 Least squares fit 25, 96, 100 Level number 40, 199, 201 Lieb-Mattis theorem 10 Lifetime 259 Likelihood function 158 Limit of infinite dimensions 11 Linear augmented plane wave (LAPW) 251 Linear muffin-tin orbital (LMTO) 251 Linear response 176, 306-308 Lithium 181 Local 12, 193, 215, 269 density approximation (LDA) 248, 250 propagator, effective 17 spin density approximation (LSDA) 250 Locator formalism 211 Long-wavelength limit 177, 180, 187f Loops 40, 64, 198, 209, 207-209 self-avoiding 43f Lorentz force 305 Lorentz oscillator model 178 Lorentzian fit 235 Low-density limit 14 Low-temperature asymptotics 125-131, 133, 173 of MIT line 126 Luttinger theorem 76, 166
Magnetic field 223, 249, 305 Magnetic moment 247, 247 Manganites 11, 243, 282 Markov process 21 Mass renormalization 76, 94, 181 Material-specific calculation 245 Matsubara frequencies 16, 82, 297 sum over 78 Maximum entropy method 25, 155-165, 174, 226, 257, 262, 264, 276 classic 26 historic 26 schemes ''a'', ''b'' 265 Maxwell construction 137 Mean-field theory 11, 17, 73, 185 Mean-free path, effective 46, 205, 259 Measurement (within QMC) 91 Mesoscopic systems 6 Message passing interface (MPI) 92 Metal-insulator transition (MIT) 2, 12, 67-175, 244, 247, 276, 279f Metallic 2, 100, 279 Metamagnetic 284 Metastable solution/phase 74 Metropolis transition rule 22 Microscopic model 1 Mid-infrared peak 234 Minimal lattice model 5 Minimum 61, 74 Minus-sign problem 21, 23, 282 Model extensions 281 Model Hamiltonian approaches 1 Momentum 1, 8, 177, 184, 199, 284-286 canonical 306 conservation 239, 286 distribution function 195, 241, 242 parameter 39, 39 summation 223 transfer 285 Monte Carlo 21, 28, 219 for BZ integrals 60, 64 importance sampling 21f simple 21 Mott transition 67 Muffin tin orbital (MTO) 251 Multi-band case 304
Néel temperature 71, 72, 139, 140, 247, 247f Natural boundary conditions 85 Nearest-neighbor (NN) pairs 9, 27 Nernst's law 174, 279 Nesting direction 285 New York metric 30 Newton scheme 228, 269 Next-nearest neighbors (NNN) 27, 41, 45, 139 Non-crossing approximation (NCA) 18, 243, 255, 258 Nonequivalent ions (in compound) 11, 282 Nongeneric momenta 14, 61, 241, 284-286 Noninteracting electrons 1, 8, 27, 185, 188, 195, 207, 240, 298 Nonperturbative approaches 2, 11f, 279 Norm of vector 29f, 284 Normalization 229 Numerical procedure 226-230 Numerical renormalization group (NRG) 18, 82, 110, 129, 131, 171, 172 Nyquist's theorem/frequency 24, 84, 86, 161, 264, 297, 299
Observable 17, 93, 173 Occupancy (density) 9 Occupation number formalism 8 Octahedra 246 On-site energy 252, 282 Optical conductivity 3, 23, 28, 182, 185, 175-244, 255, 262, 271, 272, 273, 274, 268-276, 280, 291 for Bravais lattices 190 for continuum systems 187 for disordered systems 213 for hc lattices 189, 194 for stacked lattices 207, 211 in terms of polarization 191 in the limit d→∞ 192-198 numerical results for 233-241 Orbital degrees of freedom 71, 254, 281 Orbital order 252 Orbitals eg, t2g 281 Order parameter 69, 74, 104, 136 Orthorhomic distortion 246f, 260 Oscillations 164, 299 Outlier 231 Oversampled data 25, 156 Oversampling 85, 90, 299
Padé approximants 25 Pair hopping 282f Parallelization 92 Paramagnetic insulator (PI) 67 Paramagnetic metal (PM) 67 Parameter space 73 Parity 192 Partial integration 195 Particle number 249 Particle-hole transformation 32 Partition function 17, 20, 22, 122 Patch 287 Pauli exclusion principle 1 Pauli matrices 282, 284 Pearson's coefficient 156 Peierls phase factor 188 Perfect nesting 32, 292 Periodic boundary conditions 288 Periodically stacked lattices 209-211, 217 Perovskite structure 246 Perron's theorem 42 Perturbation 306 Perturbation expansion, diagrammatic 15, 30, 192 Phase diagram 3, 72, 80f, 83, 109f, 135, 155, 163, 247, 279 coexistence 108-113 comparison 138 construction of 93-142 final 134, 174 Phase separation 283 Phase shift 177 Phase transition 131, 162 characterization of 73-79 order of 73, 140 Phenomenological theory 1 Phonons 184, 275, 283 Photo current 259 Photo electron 259 Photoemission spectroscopy (PES) 23, 245, 248, 255, 258, 260f, 267, 276, 280 inverse 267 Photon 177, 258 Plasma frequency 179-181, 187 valence electron 181f Plasma oscillation 179 Polarization 7, 176, 181, 190 Polaron 275 Potential energy 1, 94, 95 Pressure 69, 123 Primitive lattice vector 291 Prior probability 162 Probability distribution 21, 26, 158, 285f Program 280 Projection 294f Projective self-consistent technique (PSCT) 82, 111, 126f, 129 Pseudogap 79, 170, 222
Quantum chemistry 250 Quantum Monte-Carlo 2, 18, 173, 226, 242, 250, 255, 279f, 282, 297 algorithm 19-23, 82-92 numerical effort 84 Quasiparticle 2, 25, 104, 170 peak 76, 79, 257f weight 76, 77, 93, 94, 129, 130f, 133, 173, 191, 257, 276, 299, 303
Random dispersion approximation (RDA) 15, 81, 112, 288 Random number generator 92 Realistic modeling 245, 280 Recursion relation 49 Reflectivity 175, 178, 179, 182 Relativistic effects 6, 180, 284, 306 Relaxation 75, 228, 269 Renormalization group (RG) 10 Renormalized perturbation expansion (RPE) 39, 44 Resistivity 71, 274f Resolution 259 Root-finding algorithm 54, 228
Scalar potential 305 Scaling 12f, 31, 47, 210 Scenarios for MIT 79, 89, 139 Scheme 17, 31, 40f, 69, 75, 139f, 142, 145, 179, 184f, 199, 205f, 208-210, 227, 246, 287f, 294 Screening 253 Second order perturbation theory 127 response theory 259 Second quantization 8 Self energy 16, 76, 193, 254, 264, 297-299 for real frequencies 226, 229, 231, 232, 230-233, 270 locality of 15 model 298 momentum dependent 286 on imaginary axis 87f, 300-303 second moment of 136 Self-consistency condition, modified 136 Self-consistency cycle in DMFT 17, 17, 75, 75, 82, 288 Self-interaction correction (SIC) 251f Self-retracing path 212 Semimetal 294 Shielding 7 Shoulder 170 Single-band assumption 27, 281 Single-impurity Anderson model (SIAM) 11, 16, 19 Single-particle problem 249 Singular behavior 74, 146 Singular value decomposition (SVD) 157 Small parameter 11 Smoothing trick 86, 159, 262, 301 Smoothness 85 Soft mode 146 Solids 1 Specific heat 78, 96, 123, 173, 247 linear coefficient of 126, 129, 133, 135 Spectral function 23, 155, 163, 168f, 171f, 174, 226, 257f, 261, 265f, 276 boundaries for 161 moments of 25, 163 momentum-resolved 230 of self-energy 231 Spectral representation 23, 308 Spectrum see Spectral funktion Spherical model 13 Spin models 12 operator 282 order 252 Spin-orbit interaction 284 Spline interpolation 85, 297 Square-root singularity 34, 43 Stability 104, 173 Stable solution/phase 74 Stacked planes 199 State (initial, final) 259 Statistical physics 6 Stirling formula 52 Strong coupling 10 Strongly correlated electron systems 2, 5, 67, 245, 280 Struve function 127 Sublattice 217 Sudden approximation 258 Sum rule 174 for self-energy 232, 233 Supercell LDA 252f Supercomputer, vector 92 Superconductivity 2, 10, 79, 246, 283, 288 Surface 178, 259f Susceptibilities 28, 147, 187, 247, 276 Sweeps (warmup, measurement) 22, 90 number of 23, 153 Symmetry 177, 187, 247, 281f, 288, 291, 293, 295 System of units 177
t-J model 10 t-t' hopping 36, 221, 222-225, 269 Tails (of spectra) 257 Taxi cab metric 30 Temperature 27, 121, 255, 258, 260, 265, 275, 277, 279 inverse 17 Tensor 295 Thermal average 17 Thermalization of Ising field 90 Thermodynamic (in)consistency 11, 79 Thermodynamic limit 12, 200, 249, 289 Thermodynamic MIT line 68, 111, 123, 121-138, 173 slope of 123 Thermogravimetry 259 Thermopower 222 Thomas-Fermi approximation 251 Thomas-Reiche-Kuhn oscillator sum rule 179 Tight-binding linear muffin-tin orbital (TB-LMTO) 251f, 256 Tight-binding model 252 Time slice 78, 84, 263 Titanium 68 Tolerance factor 246 Trace (in simulation time) 151, 152-154 Transformation function 49, 53, 197 for Bethe DOS 54, 55, 57 monotonic 52 truncated 52 Transformation technique 137, 146 Transition metal compound/oxide 2, 248, 246-248, 280 early 256 Transmission experiments 28, 51, 177 Transport characteristics 214-216, 219-222, 224f integrals 223 properties 175, 210, 269, 279, 295 Tree (in graph theory) 40, 198 Trigonal 281 Trotter-Suzuki decomposition 19, 164 Truncation error 58, 78 Two-particle properties 28, 289
Ultraviolet photoemission spectroscopy (UVS) 258 Umklapp scattering 244, 286 Units 255 Unperturbed system 176 Unstable solution 136
Vacancy, octahedral 69 Valence electrons 1, 7 Van-Hove singularities 32, 215, 219 Vanadium 68 Vanadium oxide 68f, 70, 279 Variance 31 Variation principle 249 Varma-Schmitt-Rink-Abrahams model 283 Vector potential 305 Vertex 79 corrections 176, 243f function 175, 192
Walker (for QMC) 90 Wannier functions 8, 188, 199 Wannier representation 9 Weak coupling 10 Weiss mean-field approximation 12, 17 Wick's theorem 20, 78 Wigner-Seitz radius 256 World line 305
X-ray absorption edge 7 X-ray absorption spectroscopy (XAS) 23, 267, 280 X-ray photoemission spectroscopy (XPS) 258
1/Z expansion/corrections 13, 18, 175
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