Manybody Greens Functions

1
Many-body Greens 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

2
Many-body Greens Functions

• Space-time interpretation of Greens function
• (x,y) are space-time coordinates for the
  endpoints of the Greens function
• Greens function drawn as a solid, directed line
  from y to x
• Non-interacting Greens function Go represented
  by a single line
• Interacting Greens Function G represented by a
  double or thick single line

x
y
y
x
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Many-body Greens Functions

• Lehmann Representation (F 72 M 372) physical
  significance of G

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Many-body Greens Functions

• Lehmann Representation (physical significance of
  G)

5
Many-body Greens Functions

• Lehmann Representation (physical significance of
  G)

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Many-body Greens Functions

• Lehmann Representation (physical significance of
  G)
• Poles occur at exact N1 and N-1 particle
  energies
• Ionisation potentials and electron affinities of
  the N particle system
• Plus excitation energies of N1 and N-1 particle
  systems
• Connection to single-particle Greens function

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Many-body Greens 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

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Many-body Greens Functions

• Perturbative Expansion of Greens Function (F 83)
• Expansion of the numerator and denominator
  carried out separately
• Each is evaluated using Wicks 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)

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Many-body Greens Functions

• Fetter and Walecka notation for field operators
  (F 88)

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Many-body Greens Functions

• Nonzero contractions in numerator of MBGF

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Many-body Greens Functions

• Nonzero contractions

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Many-body Greens Functions

• Nonzero contractions in denominator of MBGF
• Disconnected diagrams are common factor in
  numerator and denominator

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Many-body Greens 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

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Many-body Greens Functions

• Rules for generating Feynman diagrams in real
  space (F 97)
• (a) Draw all topologically distinct connected
  diagrams with m interaction lines and 2m1
  directed Greens 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 represent a Greens 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) Greens 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

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Many-body Greens 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)
• Greens 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

4-momentum Conservation
Fourier Transforms
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Many-body Greens Functions

• Rules for generating Feynman diagrams in
  reciprocal space
• (a) Draw all topologically distinct connected
  diagrams with m interaction lines and 2m1
  directed Greens 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) Simple closed loops are assigned a factor
  eied Go(k,e)

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Equation of Motion for the Greens Function

• Equation of Motion for Field Operators (from
  Lecture 2)

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Equation of Motion for the Greens Function

• Equation of Motion for Field Operators

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Equation of Motion for the Greens Function

• Differentiate G wrt first time argument

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Equation of Motion for the Greens Function

• Differentiate G wrt first time argument

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Equation of Motion for the Greens Function

• Evaluate the T product using Wicks 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

(i)2v(x,r1)Go(x,r1) Go(r1,y)
(i)2v(x,r1)Go(r1,r1) Go(x,y)
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Equation of Motion for the Greens Function

• One of the next order terms in the T product
• The full expansion of the T product can be
  written exactly as

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Equation of Motion for the Greens 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)

r1
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Equation of Motion for the Greens Function

• Equation of Motion for G and the Self Energy

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Equation of Motion for the Greens Function

• Dysons Equation and the Self Energy

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Equation of Motion for the Greens Function

• Integral Equation for the Self Energy

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Equation of Motion for the Greens Function

• Dysons Equation (F 106)
• In general, S is energy-dependent
• Both first order terms in S are
  energy-independent
• Quantum Chemistry first order terms in Ho
• Condensed matter physics only direct first
  order term is in Ho
• Single-particle band gap in solids strongly
  dependent on exchange term

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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)

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Evaluation of the Single Loop Bubble

• Polarisation bubble frequency integral over b
• Integrand has poles at b e lq - id and b -a
  e lq id
• The polarisation bubble depends on q and a
• There are four possibilities for l

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Evaluation of the Single Loop Bubble

• Integral may be evaluated in either half of
  complex plane

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Evaluation of the Single Loop Bubble

• From Residue Theorem
• Exercise Obtain this result by closing the
  contour in the lower half plane

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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

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Evaluation of the Single Loop Bubble

• Self Energy

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Evaluation of the Single Loop Bubble

• Self Energy continued

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Evaluation of the Single Loop Bubble

• Real and Imaginary Parts
• Quasiparticle lifetime t diverges as energies
  approach the Fermi surface