Slide1 l.jpg
This presentation is the property of its rightful owner.
Sponsored Links
1 / 17

INT, 05/18/2011 PowerPoint PPT Presentation

  • Updated On :
  • Presentation posted in: General

MC sampling of skeleton Feynman diagrams: Road to solution for interacting fermions/spins?. Nikolay Prokofiev, Umass, Amherst. work done in collaboration with . + proof from Nature. Boris Svistunov UMass. Kris van Houcke UMass, U. Gent. Evgeny Kozik ETH. Felix Werner

Download Presentation

INT, 05/18/2011

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

MC sampling of skeleton Feynman diagrams:

Road to solution for interacting fermions/spins?

Nikolay Prokofiev, Umass, Amherst

work done in collaboration with

+ proof from


Boris Svistunov


Kris van Houcke

UMass, U. Gent

Evgeny Kozik


Felix Werner

UMass, ENS

MIT group: Mrtin Zwierlein,

Mark Ku, Ariel Sommer,

Lawrence Cheuk, Andre Schirotzek

INT, 05/18/2011

Feynman Diagrams:

graphical representation for the high-order perturbation theory

Feynman diagrams have become our

everyday’s language. “Particle A scatters

off particle B by exchanging a particle C … “

Diagrammatic technique: admits partial resummation and self-consistent formulation

Calculate irreducible diagrams for , , … to get , , …. from Dyson equations

Dyson Equation:



(contact potential)

More tools: (naturally incorporating Dynamic mean-field theory solutions)

Higher “level”: diagrams based on effective objects (ladders), irreducible 3-point vertex …

Feynman Diagrams

Physics of strongly correlated many-body systems, i.e. no small parameters:

Are they useful in higher orders?

And if they are, how one can handle billions of skeleton graphs?

Skeleton diagrams up to high-order: do they make sense for ?


Dyson: Expansion in

powers of g is asymptotic

if for some (e.g. complex) g one finds pathological behavior.

Electron gas:


[collapse to infinite density]

Math. Statement:

# of skeleton graphs

asymptotic series with

zero conv. radius

(n! beats any power)

Diverge for large even if

are convergent for small .

Asymptotic series for

with zero convergence radius

Skeleton diagrams up to high-order: do they make sense for ?


# of graphs is

but due to sign-blessing

they may compensate each other to accuracy better then leading to finite conv. radius

  • Dyson:

  • Does not apply to the resonant Fermi gas and the Fermi-Hubbard model at finite T.

  • not known if it applies to skeleton graphs which are NOT series in bare :

  • e.g. the BCS theory answer

  • (lowest-order diagrams)

  • - Regularization techniques are available.

Divergent series far outside

of convergence radius can

be re-summed.

From strong coupling

theories based on one

lowest-order diagram

To accurate unbiased theories based on billions of diagrams and limit

Re-summation of divergent series with finite convergence radius.

Example: бред какойто


Define a function

such that:


Construct sums and extrapolate to get

Configuration space = (diagram order, topology and types of lines, internal variables)

Diagram order

MC update

MC update

Computational complexity is factorial :

Diagram topology

This is NOT: write diagram after diagram, compute its value, sum

Resonant Fermions:

Universal results in the zero-range, , and thermodynamic limit

Unitary gas: .

Skeleton graphs

based on

all ladder diagrams

Useful ‘bold’ relations:

resummation and

extrapolation for


controls contributing diagram orders

Unitary gas EOS (full story in previous talks)

(in the universal & thermodynamic limit with quantifiable error bars)

Goulko, Wingate ‘10

(calculated independently and

cross-checked for universality)

Critical point from pair distribution function

Mean-field behavior:



Burovski et. al ’06, Kozik et. al ‘08

Goulko & Wingate ‘10


Diag.MC for skeleton graphs works all the way to the critical point

Phase diagrams for strongly correlated states can be done, generically

Res. Fermions: population imbalance, mass imbalance, etc

Fermi-Hubbard model (any filling)

Coulomb gas

Frustrated magnetism

Cut one line – interpret the rest as self-energy for this line:

  • Login