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
Road to solution for interacting fermions/spins?
Nikolay Prokofiev, Umass, Amherst
work done in collaboration with
+ proof from
Kris van Houcke
UMass, U. Gent
MIT group: Mrtin Zwierlein,
Mark Ku, Ariel Sommer,
Lawrence Cheuk, Andre Schirotzek
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
More tools: (naturally incorporating Dynamic mean-field theory solutions)
Higher “level”: diagrams based on effective objects (ladders), irreducible 3-point vertex …
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.
[collapse to infinite density]
# 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
# of graphs is
but due to sign-blessing
they may compensate each other to accuracy better then leading to finite conv. radius
Divergent series far outside
of convergence radius can
From strong coupling
theories based on one
To accurate unbiased theories based on billions of diagrams and limit
Example: бред какойто
Define a function
Construct sums and extrapolate to get
Configuration space = radius.(diagram order, topology and types of lines, internal variables)
Computational complexity is factorial :
This is NOT: write diagram after diagram, compute its value, sum
Resonant Fermions: radius.
Universal results in the zero-range, , and thermodynamic limit
Unitary gas: .
Skeleton graphs radius.
all ladder diagrams
Useful ‘bold’ relations:
resummation and radius.
controls contributing diagram orders
Unitary gas EOS (full story radius.in previous talks)
(in the universal & thermodynamic limit with quantifiable error bars)
Goulko, Wingate ‘10
(calculated independently and
cross-checked for universality)
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)