MC sampling of skeleton Feynman diagrams:
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INT, 05/18/2011 - PowerPoint PPT Presentation

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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

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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

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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 … “

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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 …

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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?

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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

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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

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Re-summation of divergent series with finite convergence radius.

Example: бред какойто


Define a function

such that:


Construct sums and extrapolate to get

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Configuration space = radius.(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

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Resonant Fermions: radius.

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

Unitary gas: .

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Skeleton graphs radius.

based on

all ladder diagrams

Useful ‘bold’ relations:

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resummation and radius.

extrapolation for


controls contributing diagram orders

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Unitary gas EOS (full story previous talks)

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

Goulko, Wingate ‘10

(calculated independently and

cross-checked for universality)

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Critical point from pair distribution function radius.

Mean-field behavior:



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Burovski et. al ’06, Kozik et. al ‘08 radius.

Goulko & Wingate ‘10

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Conclusions/perspectives radius.

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