Many-body Green’s Functions Propagating electron or hole interacts with other e-/h+ Interactions modify (renormalize) electron or hole energies Interactions produce finite lifetimes for electrons/holes (quasi-particles) Spectral function consists of quasi-particle peaks plus ‘background’ Quasi-particles well defined close to Fermi energy MBGF defined by

Many-body Green’s Functions Space-time interpretation of Green’s function (x,y) are space-time coordinates for the endpoints of the Green’s function Green’s function drawn as a solid, directed line from y to x Non-interacting Green’s function Go represented by a single line Interacting Green’s Function G represented by a double or thick single line y x time Add particle Remove particle t > t’ t’ t’ > t t Go(x,y) x,t y,t’ G(x,y) x,t y,t’ x y

Many-body Green’s Functions Lehmann Representation (F 72 M 372) physical significance of G

Many-body Green’s Functions Lehmann Representation (physical significance of G)

Many-body Green’s Functions Lehmann Representation (physical significance of G)

Many-body Green’s Functions Lehmann Representation (physical significance of G) Poles occur at exact N+1 and N-1 particle energies Ionisation potentials and electron affinities of the N particle system Plus excitation energies of N+1 and N-1 particle systems Connection to single-particle Green’s function

Many-body Green’s Functions Gell-Mann and Low Theorem (F 61, 83) Expectation value of Heisenberg operator over exact ground state expressed in terms of evolution operators and the operator in question in interaction picture and ground state of non-interacting system

Many-body Green’s Functions Perturbative Expansion of Green’s Function (F 83) Expansion of the numerator and denominator carried out separately Each is evaluated using Wick’s Theorem Denominator is a factor of the numerator Only certain classes of (connected) contractions of the numerator survive Overall sign of contraction determined by number of neighbour permutations n = 0 term is just Go(x,y) x, y are compound space and time coordinates i.e. x ≡ (x, y, z, tx)

Many-body Green’s Functions Fetter and Walecka notation for field operators (F 88)

Many-body Green’s Functions Nonzero contractions in numerator of MBGF (6) ) ( ψ ˆ ' (5) (4) (3) (2) (1) y x r + (-1)3 (i)3v(r,r’)Go(r’,r) Go(r,r’) Go(x,y) (-1)4(i)3v(r,r’)Go(r,r) Go(r’,r’) Go(x,y) (-1)5(i)3v(r,r’)Go(x,r) Go(r’,r’) Go(r,y) (-1)4(i)3v(r,r’)Go(r’,r) Go(x,r’) Go(r,y) (-1)6(i)3v(r,r’)Go(x,r) Go(r,r’) Go(r’,y) (-1)7(i)3v(r,r’)Go(r,r) Go(x,r’) Go(r’,y)

Many-body Green’s Functions x r r’ (1) (2) (3) (4) (5) (6) Nonzero contractions -(i)3v(r,r’)Go(r’,r) Go(r,r’) Go(x,y) (1) +(i)3v(r,r’)Go(r,r) Go(r’,r’) Go(x,y) (2) -(i)3v(r,r’)Go(x,r) Go(r’,r’) Go(r,y) (3) +(i)3v(r,r’)Go(r’,r) Go(x,r’) Go(r,y) (4) +(i)3v(r,r’)Go(x,r) Go(r,r’) Go(r’,y) (5) -(i)3v(r,r’)Go(r,r) Go(x,r’) Go(r’,y) (6)

Many-body Green’s Functions Nonzero contractions in denominator of MBGF Disconnected diagrams are common factor in numerator and denominator (-1)3(i)2v(r,r’)Go(r’,r) Go(r,r’) (-1)4(i)2v(r,r’)Go(r,r) Go(r’,r’) r r’ (7) r r’ (8) Numerator = [ 1 + + + … ] x [ + + + … ] Denominator = 1 + + + …

Many-body Green’s Functions Expansion in connected diagrams Some diagrams differ in interchange of dummy variables These appear m! ways so m! term cancels Terms with simple closed loop contain time ordered product with equal times These arise from contraction of Hamiltonian where adjoint operator is on left Terms interpreted as iG(x, y) = + + + …

Many-body Green’s Functions Rules for generating Feynman diagrams in real space and time (F 97) (a) Draw all topologically distinct connected diagrams with m interaction lines and 2m+1 directed Green’s functions. Fermion lines run continuously from y to x or close on themselves (Fermion loops) (b) Label each vertex with a space-time point x = (r,t) (c) Each line represents a Green’s function, Go(x,y), running from y to x (d) Each wavy line represents an unretarded Coulomb interaction (e) Integrate internal variables over all space and time (f) Overall sign determined as (-1)F where F is the number of Fermion loops (g) Assign a factor (i)m to each mth order term (h) Green’s functions with equal time arguments should be interpreted as G(r,r’,t,t+) where t+ is infinitesimally ahead of t Exercise: Find the 10 second order diagrams using these rules

Many-body Green’s Functions Feynman diagrams in reciprocal space For periodic systems it is convenient to work in momentum space Choose a translationally invariant system (homogeneous electron gas) Green’s function depends on x-y, not x,y G(x,y) and the Coulomb potential, V, are written as Fourier transforms 4-momentum is conserved at vertices q1 q2 q3 Fourier Transforms 4-momentum Conservation

Many-body Green’s Functions Rules for generating Feynman diagrams in reciprocal space (a) Draw all topologically distinct connected diagrams with m interaction lines and 2m+1 directed Green’s functions. Fermion lines run continuously from y to x or close on themselves (Fermion loops) (b) Assign a direction to each interaction (c) Assign a directed 4-momentum to each line (d) Conserve 4-momentum at each vertex (e) Each interaction corresponds to a factor v(q) (f) Integrate over the m internal 4-momenta (g) Affix a factor (i)m/(2p)4m(-1)F (h) A closed loop or a line that is linked by a single interaction is assigned a factor eied Go(k,e)

Equation of Motion for the Green’s Function Equation of Motion for Field Operators (from Lecture 2)

Equation of Motion for the Green’s Function Equation of Motion for Field Operators

Equation of Motion for the Green’s Function Differentiate G wrt first time argument

Equation of Motion for the Green’s Function Differentiate G wrt first time argument

Equation of Motion for the Green’s Function Evaluate the T product using Wick’s Theorem Lowest order terms Diagram (9) is the Hartree-Fock exchange potential x Go(r1,y) Diagram (10) is the Hartree potential x Go(x,y) Diagram (9) is conventionally the first term in the self-energy Diagram (10) is included in Ho in condensed matter physics (9) y r1 x (i)2v(x,r1)Go(x,r1) Go(r1,y) x y r1 (10) (i)2v(x,r1)Go(r1,r1) Go(x,y)

Equation of Motion for the Green’s Function One of the next order terms in the T product The full expansion of the T product can be written exactly as (i)3v(1,2) v(x,r1)Go(1,x) Go(r1,2) Go(2,r1) Go(1,y) (11) Go(1,y) y 1 x S(x,1) 2 r1

Equation of Motion for the Green’s Function The proper self-energy S* (F 105, M 181) The self-energy has two arguments and hence two ‘external ends’ All other arguments are integrated out Proper self-energy terms cannot be cut in two by cutting a single Go First order proper self-energy terms S*(1) Hartree-Fock exchange term Hartree (Coulomb) term Exercise: Find all proper self-energy terms at second order S*(2) (9) x’ x x x’ (10) r1

Equation of Motion for the Green’s Function Equation of Motion for G and the Self Energy

Equation of Motion for the Green’s Function Dyson’s Equation and the Self Energy

Equation of Motion for the Green’s Function Integral Equation for the Self Energy

Equation of Motion for the Green’s Function Dyson’s Equation (F 106) In general, S* is energy-dependent and non-Hermitian Both first order terms in S are energy-independent Quantum Chemistry: first order self energy terms included in Ho Condensed matter physics: only ‘direct’ first order term is in Ho Single-particle band gap in solids strongly dependent on ‘exchange’ term G(x,y) = = + + + … S(x’,x’’)= + + …

Evaluation of the Single Loop Bubble One of the 10 second order diagrams for the self energy The first energy dependent term in the self-energy Evaluate for homogeneous electron gas (M 170) a+b, ℓ+q b, ℓ w-a, k-q a, q a+b, ℓ+q b, ℓ

Evaluation of the Single Loop Bubble Polarisation bubble: frequency integral over b Integrand has poles at b = e ℓ - id and b = -a + e ℓ+q + id The polarisation bubble depends on q and a There are four possibilities for ℓ and q x y

Evaluation of the Single Loop Bubble Integral may be evaluated in either half of complex plane x y

Evaluation of the Single Loop Bubble From Residue Theorem Exercise: Obtain this result by closing the contour in the lower half plane

Evaluation of the Single Loop Bubble Polarisation bubble: continued For Both poles in same half plane Close contour in other half plane to obtain zero in each case Exercise: For Show that And that

Evaluation of the Single Loop Bubble w-a, k-q a, q a+b, ℓ+q Self Energy

Evaluation of the Single Loop Bubble Self Energy: continued

Evaluation of the Single Loop Bubble Real and Imaginary Parts Quasiparticle lifetime t diverges as energies approach the Fermi surface