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 Kris van Houcke
UMass
UMass, U. Gent
Evgeny Kozik
ETH
Felix Werner
UMass, ENS
INT, 05/18/2011
MIT group: Mrtin Zwierlein,
Mark Ku, Ariel Sommer,
Lawrence Cheuk, Andre Schirotzek
Feynman Diagrams:
graphical representation for the high-order perturbation theory
H  H0  Hint



Ae  H /T  n
n
A
n
n
e
 H /T
n
Feynman diagrams have become our
everyday’s language. “Particle A scatters
off particle B by exchanging a particle C … “
n
 A 0  AB
0
 AC
0
 ...
Diagrammatic technique: admits partial resummation and self-consistent formulation
Calculate irreducible diagrams for
G( p, )

Dyson Equation:
+
Gp
( ,)
0
,  , … to get G , U ,

(,


)
p
+
1
2

Screening:
(contact potential)
+ ...

+

 (0)
+
U
U
Ladders:
…. from Dyson equations
+

U
More tools: (naturally incorporating Dynamic mean-field theory solutions)
Higher “level”: diagrams based on effective objects (ladders), irreducible 3-point vertex …
 3  1   3G 2
 
G
G
 1 
3
U
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 g  1 ?
NO
Diverge for large g even if
are convergent for small g .
Dyson: Expansion in
powers of g is asymptotic
if for some (e.g. complex) g
one finds pathological
behavior.
Electron gas:
e  ie
Bosons:
U  U
Math. Statement:
# of skeleton graphs
 2n n3/2 n !
asymptotic series with
zero conv. radius
(n! beats any power)
[collapse to infinite density]
AN
Asymptotic series for g  1
with zero convergence radius
1/ N
Skeleton diagrams up to high-order: do they make sense for g  1 ?
YES
Divergent series far outside
of convergence radius can
be re-summed.
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 g :
e.g. the BCS theory answer
  e1/ g
(lowest-order diagrams)
- Regularization techniques
available.
From strong coupling
theories based on one
lowest-order diagram
# of graphs is
 2n n3/2 n !
but due to sign-blessing
they may compensate
each other to accuracy
better then 1 / n ! leading
to finite conv. radius
are
To accurate unbiased theories
based on billions of diagrams
and N   limit
Re-summation of divergent series with finite convergence radius.

A   cn  3  9 / 2  9  81/ 4  ... 
Example:
бред какой то
n 0
f
Define a function n, N
such that:
fn, N  1 for n  N
Gauss
f n, N
f n, N  e
Nb
1
 n2 / N
f n, N  e ( n / N )ln( n / N )
fn, N  0 for n  N
Lindeloef
Na
n

Construct sums
AN   cn f n , N
n 0
and extrapolate
lim AN
N 
AN
ln 3
1/ N
to get
A
Configuration space = (diagram order, topology and types of lines, internal variables)
Diagram order
{qi , i , pi }
Diagram topology
This is NOT: write diagram after diagram, compute its value, sum
Computational complexity is factorial :
N!
Resonant Fermions:
V (r )
 (r )
r0
r
n1/3 ~ kF / 
Universal results in the zero-range,
kF r0  0,
Unitary gas: .kF aS
and thermodynamic limit

   † (0,0) † (0,0) (r , t ) (r , t ) , (0) 
Skeleton graphs
based on G ,
all ladder diagrams
G
k
,
F





(
p

k
,)

.
.
.
Useful ‘bold’ relations:
C  m 2 (r  0,  0)
( p,   )  Ce
( p,  0)   n
 (   )
2C  B / k 2
p2
2m
4
e
3
m
 
p2
2m
resummation and
extrapolation for
density
controls contributing diagram orders
Unitary gas EOS (full story in previous talks)
(in the universal kF r0  0 & thermodynamic limit with quantifiable error bars)
Goulko, Wingate ‘10
n3T  z '( x),P3T / T  (2E / 3V )3T / T  z( x)
(calculated independently and
cross-checked for universality)
Critical point from pair distribution function
Mean-field behavior:
 MF
2

1  ( k ) 2
Criticality:
(k ,   0)
  1/ k (2 ) k
1
  1/  (2 ) k
1
  0.038
A

B
2 
1  ( k )
( )C
  2.25
from
   
A   2    (2 )
  (  )C  
1
TC TC

 0.160(5)
EF EF 2
TC
 0.152(7)
EF
Burovski et. al ’06, Kozik et. al ‘08
TC
 0.171(5)
EF
Goulko & Wingate ‘10
Conclusions/perspectives
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
…
G

G
Cut one line – interpret the rest as self-energy for this line:
G
G
 
G 
G
