Valence Photoemission Spectroscopy and the Many-Body Problem

Nicholas S. SiricaDecember 10, 2012

The Propagator within Many-Body

Physical Interpretation: Probability Amplitude

How is a transition possible?Interactions Mix States

A few distinctions exist between the propagator of many-body and that of high energy physics

|0 ⟩→∨Ψ 0𝑁 ⟩

𝑘<𝑘𝐹

𝑐𝑘=𝑏𝑘†

𝑐𝑘†=bk

𝑘>𝑘𝐹

𝑐𝑘†=𝑎𝑘

†

𝑐𝑘=𝑎𝑘

Many-body propagator-create and annihilate excitations and holes

Photoemission SpectroscopyPhotoemission spectroscopy-experimental analogue to single particle propagator

Basic understanding of photoemission process:

Focusing on optical excitation-relate absorption to transition rate

For photoemission being a single photon single electron process

defines

under

Where taking

yields a transition rate from

Several simplifying assumptions can then be made

Ultimately defines an expression for the intensity of a photoemission spectrum

Contained with in this expression is an important quantity: the single particle spectral function

Physical interpretation-Probability which gives the distribution of spectral weight

Spectral RepresentationDoes the spectral function have anything to do with the propagator? Yes, but you have to write it in the Lehmann representation

Writing the Fourier transform

Then gives

By definition of the single particle spectral function

In separating real from imaginary components

Taking the limit

Results in

Dyson’s Equation In order to compare to spectra, we need an explicit expression for the propagator. Use definition of the propagator as a Green’s function

For the free-particle propagator

Including interactions via a potential V

Or by multiplying through by the free-particle propagator

Find a solution by way of successive iterations

Particle propagating through a many-body medium

→1

𝐸−𝐸0 (𝑘 )−Σ (𝑘 ,𝐸)

Taking gives a single-particle spectral function

QuasiparticlesInterpretation of valence photoelectron spectra nicely described in context of Fermi liquid theory

Taken to be approximate single particle states in which a strongly interacting system can be mapped onto one which is weakly interacting

FT

Zk-quasiparticle residue. A measure of the strength of interactions

Relating

Expanding the pole about the Fermi-level

Under

Possible to define quasiparticle residue in terms of an effective mass

Thanks So Much!