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APPENDIX LIST OF SYMBOLS

n the following table of symbols, the many transitory symbols that I have no application beyond their immediate relevance to the deri• vation involved have not been included for the sake of clarity. Even so, many symbols have had to be used for more than one purpose. Where possible I have kept to the standard usage, but many of the symbols still need to be taken in the context of the equation. For example, Z is used for both the ionic charge number and the renormalization constant, but since the former always appears with the electronic charge e, the danger of confusion is small. As an indication the chapter number has been given if a symbol has a definition restricted to that chapter. Throughout, the following standard notation has been used to indicate the possible meanings of a symbol (0). o An operator. o The matrix form of a function. o A vector. 0+ The Hermitian conjugate. 0-1 The inverse. 6 The time derivative. (0) The expectation value. °lm The matrix element (1101 m). Heisenberg to Schrodinger representation transformation operator. A(k,w) Spectral weight function. A(1,2) The two- propagator [G(1,2)G(1,2)]. An(x) A known function of x (Chap. 5). Ax(t) Fourier coefficient of the electromagnetic potential for the wave vector kx and polarization A. A(r,t) Electromagnetic vector potential. a Interatomic separation of on the linear chain.

311 APPENDIX

lth components of the expansion of the Landau parameters in Legendre polynomials. Creation and annihilation () operators for the normal mode k. Creation and annihilation operators in the "thermal" Heisenberg representation. Annihilation and creation operators corresponding to the condensate state. Coefficient of the state of wave vector k, band index n. Landau interaction parameters. Spectral weight function. Annihilation and creation operators for the state j. Annihilation and creation operators for the plasmon excitation of wave vector k. Velocity of . annihilation and creation operators for the state m.

D(rl ,tl ,r2,t2) Green's function (propagator).

Do(rv t 1,r2, t2) Bare phonon Green's function. E of a system. Ek Energy of the jth state of the M-particle system (j = 0 corresponds to the ground state). Energy of the state n. Eigenvalue of the particle i in the state ~~(r) (Chap. 1). Real and imaginary parts of the energy (Chap. 5). Et/> Energy of the N-particle system for a given external potential ¢(r). (Chap. 12). Ground-state energy of the interacting system (Chap. 5). E(k) Energy of the isolated Landau quasiparticle state k in the presence of other quasiparticle (Chap. 5). E(k) Energy of the state k.

312 LIST OF SYMBOLS

E(k) Energy eigenvalue for the state corresponding to the transformed operators &jJk (Chap. 13). Energy of the isolated Landau quasiparticle state k (Chaps. 5 and 9). E(v) Energy of the N-particle system as a function of the velocity of the observer (Chap. 5). E(r,t) . E[n] The energy as a functional of the density. ExJn] The exchange and correlation energy functional. e Electronic charge. e Polarization vector. F Number of closed fermion loops in a diagram (Chap. 8). F Free energy (Chaps. 10, 12).

F(rI1Tl1r 2,T2) Anomalous Green's function (Chap. 13). f(k,k') Landau quasiparticle interaction function. r(kF,8),f(kF,8) The direct and exchange parts of the quasiparticle interaction. f(r) Arbitrary function (Chap. 2). G Reciprocal lattice vector (Chap. 9). Expansion coefficients for the Green's function (Chap. 2). Anomalous Green's functions. The normal (Le., nonpairing) part of the two- particle Green's function (Chap. 13). G(k,w) Momentum-energy dependent Green's function. G(r,r',t,t') Time-dependent Green's function (Chap. 2). G(r,r',E) Energy-dependent Green's function (Chap. 2). G(x,x',w) Energy-dependent Green's function. G[n] The functional which contains the kinetic, exchange, and correlation contributions to the total energy (Chap. 12).

Gc(X l1 t 1,X2l t2) Condensate part of the Green's function.

Gix1,tI1 X2,t2) Real-time thermal Green's function.

G:(X1,tI1 X2l t2) Advanced real-time thermal Green's function.

G:(Xl1 t 1,X2,t2) Retarded real-time thermal Green's function. GO(Xl,tI1X2,t2) Noninteracting Green's function.

313 APPENDIX

Green's function corresponding to the single• particle Hamiltonian without the Hartree potential (Chap. 7). nth order approximation to the single-particle Green's function.

G2( xI,t I,X2f t2,X3, t 3,X4, t4) Two-particle Green's function. G±(r,r,t - t') Advanced (+) and retarded (-) Green's function (Chap. 2). Noncondensate part of the Green's function (Chap. 11). G-I(Xl1tl,X2ft2) Inverse of the single-particle Green's function. G'(XI,tl,X2,t2) The matrix Green's function for a system with a condensate (Chap. 11).

G(k,w n) Momentum-energy dependent thermal Green's function. g (X, r ,x',r') Thermal Green's function. go(x,r,x',T') Noninteracting thermal Green's function. go(XI ,X2fWn) Noninteracting energy-dependent thermal Green's function. The spin-up electron propagator (Chap. 13). g The strength of the effective electron-electron interaction (Chap. 13). Energy functions (not functionals) of the electron density. The energy density functional. Hamiltonian. Hamiltonian describing the background neutralizing charge (Chap. 11). Hamiltonian containing the coulomb repulsion (Chap. 12).

Hel Electron Hamiltonian (Chaps. 11 and 12).

Hel- Electron-ion interaction Hamiltonian (Chap. 11). HINT Hamiltonian describing the interaction between the N-particle system and the added electron (Chap. 11). Electron-phonon interaction Hamiltonian (Chap. 12). Ion Hamiltonian.

314 LIST OF SYMBOLS

Coupling Hamiltonian between localized and de localized states (Chap. 12). Hamiltonian for the N-particle system (Chap. 11). Hamiltonian for the delocalized states (Chap. 12). HPh Phonon Hamiltonian (Chap. 12). Ho Single-particle Hamiltonian. HI Hamiltonian for one particle (Chap. 11). Ho(xl) Single-particle Hamiltonian. H(r,t) Magnetic field (Chap. 3). H[n1 Hamiltonian as a functional of the electron density. H[cf>1 Hamiltonian as a function of the external potential. 71 Hamiltonian density. 71 Magnetic field (Chap. 12). 71/ An effective internal magnetic field (Chap. 12).

71 0 External magnetic field (Chap. 12). h Plank's constant. 1(1,2,3,4) T-matrix for electron-electron scattering. i(x) Single-particle operator. Jo(r) Zero-order Bessel function (Chap. 9). d(x) Operator density. K Effective Spring constant for internuclear forces (Chap. 3). Grand canonical Hamiltonian. Soluble part of the grand canonical hamiltonian. Eigenvalue of the grand canonical hamiltonian for the state n. The normal (i.e., nonpairing) part of the grand canonical Hamiltonian (Chap. 13). Variable conjugate to x [= (k, spin)]. Boltzmann constant. Fermi wave vector, i.e., wave vector of state at the Fermi level. k Wave vector of a plane-wave state. L Lagrangian. .L Lagrangian denSity.

315 APPENDIX

M Atomic mass. M Occupation number of the q = 2kF phonon in the one-dimensional (Chap. 12). M(k,j,q) Matrix element appearing in the response function of an . m Electron mass. m* Effective m'ass of an electron. N Number of atoms in the linear chain (Chap. 3). N Electron density (Chap. 11). N(E) at energy E. N. Number eigenvalue of the state I n) (Chap. 10). n Number of not in the condensate (Chap. 11). n Order number of a diagram (Chap. 8). n(k) Number of particles in the eigenstate k. n(r) Density of . fl(r) Density operator. n[¢] The electron density as a functional of the external potential. Number operator for the normal mode of wave vector k. Number operator for the state 1,0". no(k) Ground-state occupation factor (Chap. 5). o General operator. OH(q,t) Heisenberg representation operator. O[(q,t) Interaction representation operator. OK(q,T) Operator in the "thermal" Heisenberg representation. Os(q,t) Schrodinger representation operator. P(q,w) Energy-momentum dependent polarization propagator.

P( X II t I1X21 t2) Polarization propagator. P(Xl1TllX21T2) Thermal response function. P{} Permutation operator which produces all possible permutation of the labels; thus, P{1/Ia1/l~1/I.,} = 1/Ia1/l~1/I., + 1/Ia1/l.,1/I~ + 1/I.,1/Ia1/l~ + t/;'Yt/;~t/;a + t/;~t/;'Yt/;a + t/;~t/;at/;'Y' Probability of finding a system in the state In} (Chap. 10).

316 LIST OF SYMBOLS

PO(X1,t1,X2,t2) First-order polarization propagator. p Momentum operator. P Momentum operator (single-particle). Momentum of ith (Chap. 3). Fourier transform of Pi and momentum coordinate of the normal mode of wave vector k (Chap. 3). Particle momentum. Momentum of the ith atom (Chap. 11). Momentum coordinate for the normal mode (k,A). Q(k,k') The transformed interaction matrix element in a paired electron system (Chap. 13). q Generalized position variable. qi Shift of ith atom from its equilibrium position (Chap. 3). Position variable of the normal mode of wave vector k. The Fourier transform of qi (Chap. 3). Position coordinate for the normal mode (k,A) (Chap. 11). q Wave vector variable. R(k,k') As Q(k,k') (Chap. 13). R(r1,t1,r2,t2) The plasmon Green's function (or propagator). RA Operator occurring in the electron electromagnetic field interaction (Chap. 4). Atomic position vector. Position vector describing the equilibrium position of the ith atom in a lattice. Electron position vector for the nth electron. 5 Entropy of a system (Chap. 10). 5 Operator used to explicitly remove the effects of a perturbation from the system variables (Chap. 7). 5(k,k') As Q(k,k') Chap 13). 5(p,w) Denominator of the anomalous Green's function (Chap. 11). 5(1,2,3,4) Summation of the effects of repeated electron• hole scattering (Chap. 8). A canonical transformation variable (Chap. 12).

317 APPENDIX

SA,j Operator occurring in the electron electromagnetic field interaction (Chap. 4). So( 1,2,3,4) Lowest-order electron-hole scattering function (Chap. 8). T Temperature. T(k,k/) As Q(k,k/) (Chap. 13). T[] Time ordering operator. T[n] Kinetic energy functional (Chap. 12). Tr[] Trace operator. Tc Critical temperature. TJ] Ordering operator. t Time variable. U(r) Potential function. Uo(r) Applied potential (Chap. 1).

U(Rj - Rj ) Two-body interaction (Chap. 11).

U(rv t1,r2,t2) Effective two-body interaction including phonon effects (Chap. 11). Contribution to the energy of a state from multiple occupancy (Chap. 12). Uk Canonical (Bogoliubov) transformation coefficient (Chap. 13). V Volume of the system (Chap. 11). V(r) The potential energy of a particle at a position r. VI Imaginary part of the potential (Chap. 5). Vk,1 Matrix element for the between localized 1 and de localized k states (Chap. 12). vt). Electron-phonon scattering matrix element (Chap. 11). VR Real part of the potential (Chap. 5). V •.(k) Exchange potential of state k (Chap. 5). V:x(r) Local Hartree-Fock exchange potential for the state (Chap. 1). V• .(r,r') Hartree-Fock exchange potential. VH(r,t) Hartree potential. VW(x,t) nth order approximation to the Hartree potential (Chap. 7).

VINT(rl • . . rN) The interaction caused by the coulomb potential between the electrons situated at position

vectors r1,r21 ••• ,rN'

318 LIST OF SYMBOLS

Effective potential for state k (Chap. 9). The effective scattering matrix element produced by an experimental probe (Chap. 13).

V1s(r), V2S(r) Atomic Hartree potentials (Chap. 1). v(q) Momentum-dependent coulomb interaction. v(x,x') Coulomb interaction. Vk Canonical transformation coefficient (Chap. 13). Vo Fermi velocity. V Velocity variable.

Vk Velocity contribution of state k (Chap. 5). W(q,w) Energy- and momentum-dependent screened interaction. W(XV tl,X2,t2) Screened coulomb interaction. W(XI/TI,X2,T2) Thermal screened interaction. WO(XI,tvX2,t2) First-order screened coulomb interaction. X Total number of systems in a grand canonical ensemble (Chap. 10). Xn Number of systems in a grand canonical ensemble which are in the state n. x A general spatial + spin variable [= (r,a)]. z Partition function (Chap. 10). z Spectral weight or renormalization constant. z Ionic charge. a Spin index. a Magnetic field coupling parameter (Chap. 12). A perturbation matrix element (Chap. 12). Annihilation and creation operators for the paired states in the Bogoliubov transformation (Chap. 13). Expansion coefficient for a general wave function in terms of the eigenstates (Chaps. 4, 5). ~lm u.l.j A function containing the details of the electron interaction arising from the operator RI. (Chap. 4). Spin index. Thermodynamic variable (= 1/ kBT). Annihilation and creation operators for the

319 APPENDIX

paired states in the Bogoliubov transformation. RIm ~Ai A function containing implicitly the details of the electron-phonon interaction arising from the operator SA'i. r(k,w,q,O) Energy-momentum dependent vertex function. r(Xl,t1,X2I t2,X3,t3) Vertex function. r(X1,T l,X2,T2IX3,T3) Thermal vertex function. rst: ] Subset of the complete set of eigenstates of the system symmetrized to conform to the Pauli principle. 'Y Electron-phonon coupling constant (Chap. 11). 'Y(q) Electron-phonon coupling matrix element (Chap. 12). 'Yk,n An expansion coefficient (Chap. 12). 'Y(k,l,m) Matrix element in the anharmonic linear chain expansion (Chap. 3). 'Y(r) Electron-phonon coupling function (Chap. 11).

'Y'(r1,t1,r2lt2) Screened electron-phonon coupling function (Chap. 11). Electron-plasmon coupling constant. Gap parameter (or order parameter) (Chap. 12). An infinitesimal (Chap. 9). Change in energy from the ground-state energy due to changes in the distribution function (Chap. 5). an(k) Deviation from the ground-state occupation for the quasiparticle state k (Chap. 5). aE(k) Contribution to the energy of the quasiparticle state k from the self-energy (Chap. 9). Superconducting gap parameter (Chap. 13). The mean pairing potential (Chap. 13). The Fermi level approximation to the gap parameter (Chap. 13). oHEJ(p The contribution to the Hamiltonian of a system arising from an experimental probe (Chap. 13). o(r - r') Dirac delta function. On," Kronecker delta function. E Infinitesimal energy (Chap. 2).

320 LIST OF SYMBOLS

Coefficient for the first anharmonic term in the interatomic potential (Chap. 3). 8(k) Quasiparticle energy (Chap. 9). E(q) Dielectric response function (Chap. 1). E(q,W) Energy- and momentum-dependent dielectric response function.

E(X1,tl ,X:ut2) Dielectric response function. E(O) Static long-range dielectric response function (classical dielectric constant) (Chap. 9). An approximation to the Lindhard response function for the electron (Chap. 9).

E/ Energy of the state [. EM(q,W) Metallic response function (Chap. 9). EM(j,k) jth excitation energy of M-particle state with momentum hk (Chap. 6). Excitation of the (N ± I)-particle system (Chap. 6). The local exchange and correlation energy density function (Chap. 12).

EO Permittivity of free space. EI(qIW),E2(q,W) Real and imaginary parts of the dielectric response function (Chap. 9). Inverse of the dielectic response function. Polarization vector for the normal mode kA (Chap. 11). 11 An infinitesimal. 8 Angle between interacting new Fermi surface (Chap. 5). 8(q) Step function: 8(q) = 0, q < 0; 8(q) = 1, q > O. A Normal-mode label (includes wave vector and polarization) (Chap. 3). Spin index. Strength of the contact interaction potential (Chap. 13). Thomas-Fermi wave vector (Chaps. 1 and 9). Chemical potential (Fermi energy). An atomic magnetic moment (Chap. 12). Permeability of free space.

321 APPENDIX

~(x,t) Part of the condensate Green's function (Chap. 11). ~(x,t) Condensate part of the field operator (Chap. 11). ~o(x) Condensate wave function (Chap. 11). 1r(r,t) Generalized momentum function. P Statistical density matrix operator (Chap. 10). p(r) Density operator (Chap. 11). p(x,t) Charge density (Chap. 7). Pk Fourier transform of the density operator (Chap. 11). Pk(E) Density of state of wave vector k (Chap. 5). Density matrix operator for state I n} (Chap. 10). ~(q,w) Energy- and momentum-dependent self-energy. ~(Xl,X2'W) Energy-dependent self-energy. ~(Xl,tl,X2,t2) The self-energy. ~(Xl,Tl,X2/T2) Thermal self-energy. ~c.H.(k,w) The coulomb hole (or coulomb correlation) part of the self-energy. ~H-F(k,w) The Hartree-Fock approximation to the self• energy. nth order approximation to the self-energy (Chap. 7). ~s.dk,w) The screened exchange part of the self-energy. ~1(q,W)'~2(q,W) Real and imaginary parts of the self-energy (Chap. 7). Components of the matrix self-energy used in a Bose system with a condensate (Chap. 11). Matrix self-energy for a system with a condensate (Chap. 11). Spin variable. Fourier transform of the momentum operator for a set of particles (Chap. 11). T "Thermal" Heisenberg representation variable (imaginary time variable) (Chap. 10). T Time difference variable (zero-temperature formulation). General basis state for the n-particle system (Chap. 4).

322 LIST OF SYMBOLS

(r,t) Plasmon field operator. cI>/(r) A localized state (Chap. 12). cjJm Eigenfunction of the single-particle Hamiltonian. ¢(r) Phonon field operator. cjJ(r,t) Generalized position function [conjugate to ll'(r,t)] (Chap. 3). cjJ(x,t) External potential applied to the N-particle system, used in deriving the iterative solution to the Green's function equation of motion. cjJ(X,T) Perturbation used in deriving the iterative solution to the thermal Green's function equations of motion. Phonon field operator in the Heisenberg representation. cjJp(x) A general eigenstate (Chap. 10). X(X,t) Noncondensate part of the field operator (Chap. 11). Xk A Bogoliubov transformation variable (Chap. 13).

'It(r1,r2' ... ,rN ) The total wave function for the N-particle system. Schrodinger representation wave function (Chap. 4). Heisenberg representation wave function (Chap. 4). Interaction representation wave function (Chap. 4). The field operators corresponding to spin-up and spin-down electrons (Chap. 13). Single-particle wave function. The subscript denotes the state concerned. Field operator in the "thermal" Heisenberg representation. ~(q,T) Field operator. n An energy or frequency variable. n Probability of a configuration of the grand canonical ensemble (Chap. 10). Volume of the system (Chaps. 1,13).

323 APPENDIX

Renormalized phonon energy (Chap. 12). An energy variable. Oscillation frequency of the normal mode of wave vector k (Chap. 3). Energy of the normal mode k,A (Chap. 11). Energy variable for the thermal functions conjugate to T (Chap. 10). Plasmon energy (long-wavelength limit). Plasmon energy at wave vector q (Chap. 11). The "bare" phonon energy, i.e., the phonon energy of the ion system in which the electrons do not respond (Chap. 11). The energy of a phonon with wave vector q (Chap. 11). Energy interval defining the energy range of the electron-electron interaction (Chap. 13). I· .. ) A wave function in the Dirac notation. Details of the wave functions are contained inside the bracket. IM,j) jth state of the M-particle system (Chaps. 6, 11). IM,j,k) jth state of the M-particle system with momentum hk (Chap. 6). IN) Ground state of the N-particle system. IN,

324 INDEX

Adiabatic approximation, 72, 135 Correlation, 13, 15-17 Anderson-Hubbard model, 270-276 energy, 20-21, 22 Anticommutator,77 hole, 15-17,23, 191 and Bogoluibov transformation, 296 self-energy, 191-192 Coulomb hole, 15, 16, 191-193 and image potential, 212-215 Band Coulomb repulsion, 271-274 conduction, 203, 209-211 Coupling constant index, 203 electron phonon, 246, 282, valence, 203, 209-211, 270, 283 302 Bandgap, 203-205,208,271, 283-286 electron plasmon, 252 variation of, 209-211 renormalized, 247, 249 Bethe-Salpeter equation, 178, 291 Critical temperature, 286 Bogoluibov transformation, 259-299 quasiparticles, 299-301 Boltzmann equation, 108-110 Density functional, 266-268 Born-Oppenheimer approximation, 3 Density matrix, 39, 219-220 Boson, 76, 79 statistical 220, 222-223 field operators, 86 Density of states, 13, 18, 300 Green's functions, 114,239-262 Density operator, 241 Landau theory, 103 Diagrams Bose-Einstein distribution function, 224 bubble, 167, 177 Brillouin zone, 204-206 condensate 256-259 Burstein shift, 210-211 definitions, 158 evaluation of, 174-177 irreducible, 167-170,258 Canonical transformation, 276-279 and iteration solution, 162-166 and , 296-299 ladder, 178, 179,290-292 Chemical potential, 118,225,261-262 momentum-energy, 171-173 See also: Fermi energy reducible, 167-170 Commutator, 46, 52, 76 single particle, 36-38 and canonical transformation, 277 tadpole, 171 and field operators, 86 See also: individual functions; Green's Condensate, 253-262 function; Polarization, Self• eigenfunction, 255 energy, etc. field operator, 254 Dielectric function, 21, 144-146 Green's function, 254-256 complex, 186 operators, 255-256 crystal,202-205 as particle source and sink, 258 Drude,188 Configuration, statistical, 221-222 energy-momentum, 153 , 180,290-295 high-frequency, 188

325 INDEX

Dielectric function (cont.) Equation of motion imaginary part, 187 field operators, 87, 126, 304 insulator, 202-206 Green's function, 126-128, 135,253, inverse, 145, 202 302, 304-306 of jellum solid, 186-189 Heisenberg, 69-70, 80 limiting forms, 188 iteration solution, 139-140, 147-148, Lindhard,186-189,203-204 162-166 matrix elements, 202-203 phonon, 245 Penn model, 205-206 plasmon, 251 real part, 186 for superconductivity, 304-306 , 202-206 thermal Heisenberg, 225-226, 304 static, 192 Ergodoic theorem, 99 Thomas-Fermi, 23,188 Exchange See also: Interaction dynamic, 191 Distribution function, 101, 103, 109, screened, 20, 192 224 Slater, 14 Dyson equation, 32, 33, 150, 169 Exchange energy, 100 and condensate, 257-260 EXChange hole, 15-17 electron-electron interaction, 291 Excitation, 187 electron-hole interaction, 178-179 , 179 matrix form, 260, 306 Exclusion principle, 11, 100 Nambu form, 306 Expectation values, 39, 67, 68 and density matrix, 219-220 Effective mass and Green's function, 39-40, 125-126, Hartree-Fock, 100-101 225 Landau quasiparticle, 104-108 of number operator, 224 Eigenstate, 74 in occupation number representation, approximate, 96 78 electron photon, 83 representation independant form, 220 See also: State; Wavefunction in statistical systems, 220, 223-225 Eigenvalue and thermal Green's functions, 226, and Green's function, 38, 116-120 232-233 for linear chain, 48 for quasiparticles in superconductor, Fermi-Dirac distribution function, 224 299,307 Fermi energy, 10, 18, 99, 273 Electromagnetic field, 55-58, 81-85 and Green's function, 118,237 Electron-hole pair, 178-179, 203 Fermi surface, 106, 205 creation of, 187 instability of, 294 Elementary excitations, 59, 79 , 76, 99 Energy and Green's function, 114, 115-116 complex, 95 and Landau theory, 103 excitation, 101, 102, 118-119, 154,234 and Lehman representation, 121- ground state, 101, 116,266-267 122 quasiparticle state, 102-103 , 280-281 See also: Eigenvalue Field Energy functional, 267 driving, 281 Energy loss, 187 external, 281 Energy transform, 117, 152, 230 internal, 282 Entropy, 222 magnetic, 280

326 INDEX

Field operators, 85-88 Green's function (cant.) of condensate, 254 single-particle, 27-40, 251 in interaction representation, 136 advanced, 34 phonon, 245 eigenfunction expansion, 29 plasmon, 251 Helmholtz equation, 28 Free energy, 223, 285 matrix form, 31 Functional, 266-270 perturbation theory, 30-33 energy density, 267 retarded, 34 and time development, 35, 39 Gap equation, 299, 307-308 time dependence, 33-36 Gell-Mann and Lowe theorem, 137 Ground state Grand canonical ensemble, 221, 223, condensate, 255 234 as function of external potential, 265- Green's function 266 many-body Hartree-Fock, 7, 8 advanced, 234-237 Heisenberg, 136 anomalous, 259-261, 305-306 in Landau theory, 101-103 condensate, 254-257 linear chain, 48, 50 definition for finite temperature, local density method, 265-270 226,230-233,233-237 wave function, 104-105 definition for zero temperature, zero point energy, 49 113-115, 153 See also: Vacuum state diagram, 158, 161 Dyson equation, 257, 306 Hamiltonian energy structure, 116-120,234 Anderson-Hubbard, 274 equation of motion, 126-128, 135, and Bogoluibov transformation, 296- 227 297,300-301 first-order solution, 142, 159,195- boson, 277 196 electron, 241, 244 interaction representation, 136 of electromagnetic field, 56 inverse, 139, 141, 153 of electron moving in electromagnetic iteration equations, 147-148 field, 81 , 195-196, 198 electron-phonon, 241, 282-284 Lehman representation, 120-124, electron-photon, 83 153-154,196-197,234-237 grand canonical, 224, 225, 296 momentum representation, 153 interacting , 126, 265 noninteracting, 127, 140, 147-148, interaction, 85 158,161,164,184,231-232,236- ion, 240 237,251,256-257 linear chain, 44 periodicity at finite temperature, non-Hermitian, 96 228-230 noninteracting, separable, 1 phonon, 245-250 phonon, 241, 243 plasmon, 251-252 plasmon, 242-243 retarded, 234-237 of a solid, 2-4 and single-particle Green's for superconductivity, 302 function, 115-116 Hamiltonian density, 52-53 thermal, 226-237 Hamilton's equations, 51, 109 two-particle, 128, 129, 132,292-293, Hartree potential, 6, 9, 99, 130, 138, 148, 302-303 183-184,254,269

327 INDEX

Hartree theory, 5-10, 302-307 Lagrangian density, 52 and Green's function solution, 130, Landau theory, 101-110, 199-201 148, 149 Linear chain, 44-50 jellium solution, 9-10 anharmonic,61 lithium atom, 7-8 eigenvalues, 48 and relaxation, 8 ground state, 48 self-consistency of solution, 6, 8, 10 Hamiltonian, 44, 47 and , 183-184, 197 harmonic oscillator form, 46 Hartree-Fock theory, 10-15, 100 ladder operators, 48 failure of, 13 oscillation frequency, 45 and Green's function solution, 129- Local density theory, 268-270 130, 141-142 and linearization of interaction, 302-303 Magnetic dipoles, 280 See also: Exchange; Hartree theory Magnetic moment, 280-281 Helmholtz equation, 28 Mean field theory, 279-282 Hole, 114, 161, 178-179,209-211 Momentum conjugate, 53 Insulators, 201-211 conservation, 62,120-121 Interaction and Cooper pairs, 294 contact, 291, 302 of electromagnetic field, 56, 58 Coulomb, 2, 5 of a field system, 53 diagrams, 153, 163-164, 173,246-248 in interacting systems, 122 electron with electromagnetic field, operator, 52-53 81-85 transform, 121, 152, 230-231 electron-hole, 178-179 electron-ion, 244 Nambu equations, 306 and electron pairing, 303-304 Normal mode electron-phonon, 244, 246, 284, 299 in crystal, 240 electron-photon, 85 for electromagnetic field, 56 electron-plasmon, 251 for general field system, 54 exchange, 12-13 for linear chain, 44-45 ion-ion, 44, 240, 243 in jellium, 189 and Landau quasiparticles, 102-104, Operator 108, 198-201 annihilation and creation, 60, 74-81, 85,244 overscreening, 253 density, 125 by phonon propagation, 161 and phonon exchange, 246, 252-253 density matrix, 220 field, 85-88 and plasmon exchange, 252-253 screened,20, 145, 147-149, 150-151, general operator (in terms of field operator), 86-87 153,246-250,253 surface, 212-214 in Heisenberg representation, 69, 72 in interaction representation, 71, 72 two-body, 254 ladder, 48-50, 55, 57 See also: Potential in linear chain, 46 momentum, 53, 58 Lagrange's equation, 53 non-Hermitian, 46 Lagrangian number, 50, 77, 224 for electromagnetic field, 55 in occupation number representation, for particle, 51 77-79

328 INDEX

Operator (cont.) Polarization in Schrooinger representation, 68 electromagnetic wave, 55, 82 time ordering, 71, 114 phonon, 240 velocity, 104-106 Polarization propagator Order parameter, 279-282 definition, 145-146 Oscillation strength: see Spectral weight diagram, 158, 161, 173 Overscreening, 253 and electron-hole interaction, 178- 179 Pairing, 253, 295, 296, 307 first-order solution, 148, 159 Cooper, 290-295 in insulators, 201 effect on the interaction, 302-304 for jellium solid, 184-186, 198 and matrix elements, 300-301 momentum-energy, 153, 173 Partition function, 223 in one dimension, 283 Peierls transition, 282-286 perturbation expansion, 167, 177-178 Permutation operator, 11 Poles Perturbation theory and excitation energy, 38,119,234- anharmonic linear chain, 61-63 235 by canonical transform, 276-279 plasmon, 188-189 diagramatic illustration of, 36-38, Potential 158-166 complex, 94-95 and elementary excitations, 62 delta function, 36 failure of, 289-290 and denSity functional, 226-269 functional derivative solution, 135- effective local, 211-212, 269, 303- 140 304 in interaction representation, 71-72 and electron pairing, 303-304, 305- and selective summations, 177-180 308 and single-particle Green's functions, exchange, 12,99, 130,270,303 30-33 Hartree, 6, 9, 99, 130, 138, 148, 183- for thermal Green's function, 227- 184,254,269 228 image, 212-214 Wigner-Brillouin,31 nonlocal, 12 transitions, 279-282 optical, 214 Phonon, 240-241, 246 at surface, 212-214 emmission and absorption of, 91-93 Propagator: see Green's function equation of motion, 245 exchange of, 246 Quasiparticles, 93, 100 field operator for, 245 Bogoluibov,296-301,307 frequency of, 240,250,282-283 energy of, 102, 191, 199 Green's function, 245 interaction of, 102 Hamiltonian, 241 in jellium solid, 196-197 scattering of, 62-63 Landau, 101-110, 199-201 Photon, 83-85, 161 and Lehman representation, 123 Plasmon, 162, 212 lifetime of, 124, 196-197 emission of, 194-195 peak, 196, 198-199,206 exchange of, 252 state, 207 Green's function, 251, 252 time dependence of (decay), 124 Hamiltonian for, 242-243 oscillation frequency of, 188, 189 of electromagnetic field, 55-58 and screened interaction, 189 of general field, 51, 54-55 surface, 212, 214 at linear chain, 44-50

329 INDEX

Reciprocal lattice vector, 202 Self-energy (cont.) Relaxation, 8 real part, 190-192 Renormalization and screened interaction, 147, 150 and Green's function, 124, 198 second-order solution, 164-166 of interaction, 248 and spectral weight functions, 154 of phonon, 247-248,283 at surface, 211-215 See also: Spectral weight thermal, 228, 231, 233 Representation time-dependent, 131-133, 138-139, Heisenberg, 68-70, 225 141-144,254 interaction, 70-72 Lehman, 120-124, 153-155, 196-197, Semiconductor, 202-211 234-237 band gap, 203-211 occupation number, 73-80 dielectric function, 202-206 Schrodinger, 68, 70 impurity in, 274-276 thermal Heisenberg, 225-226 Slater determinant, 11, 73 thermal interaction, 226, 228 Solid Response function: see Dielectric jellium,4,184-199 function insulator, 201-211 two-band model of, 205, 207 Schrodinger equation, 1, 68 Wigner model of, 16 and field operators, 87 Sommerfeld model, 5, 10, 18 Screened interaction Spectral weight, 124, 198 complex energy structure of, 189 Spectral weight function, 122-124, 153- diagrams for, 158, 161, 162-164 155,196-197,200,235-237 for insulators, 201-207 Spin for jellium, 189 and electron pairing, 294 momentum-energy, 153, 173 and localized states, 271 and selective summation, 150-151 summation in diagrams, 174-176 at surfaces, 212-213 State Screening, 15, 143-147 approximate, 96 Self-energy Bloch, 202, 244, 271 and band gap, 208-211 bound,38,179,294 calculated, 196-197 in interaction representation, 73-77 complex, 153-154, 193 linear chain, 50 and condensate, 256-259 localized,270-276 and Coulomb correlation hole, 191- single particle, 91-93, 99, 296-270 192,208 vacuum, 60, 74,75 diagrams, 158, 162, 172-173 See also: Eigenstate; Ground state; and dynamic exchange, 190, 192,208- Wavefunction 209 Stirling's approximation, 221 first-order solution, 149, 159, 164 Statistical mechanics, 99, 221-224 imaginary part, 193-195, 197 Superconductivity, 180, 289-309 insulator, 205-209 Bardeen -Cooper-Schrieffer irreducible, 258 Hamiltonian, 302 jellium, 190-199 Bogoluibov transformation, 259-299 and Landau interaction, 199-200 gap equation, 299-300, 305, 307-308 matrix, 260 Green's function solution, 301-309 matrix elements, 207-208 Nambu formulation, 306 momentum-energy, 153, 172-173 quasiparticle, 296-299 and plasmon emission, 194-195 Surfaces, 211-215

330 INDEX

Thomas-Fermi theory, 17-23 Wave quantization, 43-66 dielectric functions, 22, 188 Wavefunction equation, 19 approximate, 97 interaction potentials, 20 basis, 74 and self-energy limits, 191-193 Bloch,207 wavevector, 19, 187, 210 ground state, 104-105 Thermal ordering operator, 226 Hartree,6 Time ordering operator, 71, 114- Hartree-Fock, 11 115 Heisenberg, 68 Translational invariance, 152 interaction representation, 70-77 Translational symmetry, 202 many-particle, 73 perturbation expansion for, 30 Vacuum state, 60, 74, 278-279, 300 product, 6, 10 Variational principle, 10, 267-268 occupation number representation, Vector potential, 55, 81 73-75 Velocity quasiparticle, 207 expectation value of, 106 Schrodinger representation, 68, 87 Fermi,187 single-particle, 2, 12, 73 operator for, 104-106 Wick's theorem, 133, 166, 174 Vertex function, 147-149 Wigner Brillouin perturbation theory, 31 diagrams for, 158, 162-163, 172 momentum-energy, 152, 172 Zero point energy, 49, 58

331