INT, 05/18/2011

MC sampling of skeleton Feynman diagrams:Road to solution for interacting fermions/spins?

Nikolay Prokofiev, Umass, Amherst

Boris SvistunovUMass

work done in collaboration with

Kris van HouckeUMass, U. Gent

Evgeny Kozik ETH

Felix WernerUMass, ENS

+ proof from Nature

MIT group: Mrtin Zwierlein, Mark Ku, Ariel Sommer,

Lawrence Cheuk, Andre Schirotzek

Feynman diagrams have become our everyday’s language. “Particle A scattersoff particle B by exchanging a particle C … “

Feynman Diagrams: graphical representation for the high-order perturbation theory

int0 H HH

/

0 0 0/ ...

H Tn n

n

H Tn n

n

AeA A AB AC

e

Diagrammatic technique: admits partial resummation and self-consistent formulation

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

+ + + ...0 ( , )G p

1 2( , )p

G U

+ Dyson Equation:( , )G p

U +U

(0) + U

Screening:

Ladders:(contact potential)

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

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

23 31 G

G

3

1 UG

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?

Feynman Diagrams

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

1g

NO

Diverge for large even if are convergent for small .

Math. Statement: # of skeleton graphs

asymptotic series withzero conv. radius

(n! beats any power)

3/22 !nn n

Dyson: Expansion in powers of g is asymptoticif for some (e.g. complex) g one finds pathological behavior.

Electron gas:

Bosons:

[collapse to infinite density]

e i e

U U

Asymptotic series for with zero convergence radius

1g

NA

1/ N

gg

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

1g

YES

# of graphs is

but due to sign-blessingthey may compensate each other to accuracy better then leading to finite conv. radius

3/22 !nn n

1/ !n

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.

g1/ge

Divergent series far outside of convergence radius can

be re-summed.

From strong couplingtheories based on onelowest-order diagram

To accurate unbiased theories based on billions of diagrams and limit N

Define a function such that:

, n Nf , n Nf

aN

1 , 1 for n Nf n N

, 0 for n Nf n N

Construct sums and extrapolate to get ,0

N n n Nn

A c f

lim NNA

A

0

3 9 / 2 9 81/ 4 ...nn

A c

Example: бред какой то

Re-summation of divergent series with finite convergence radius.

bN

n

ln 3

2 /,

( / ) ln( / ),

n Nn N

n N n Nn N

f e

f e

NA

1/ N

Lindeloef

Gauss

{ , , }i i iq p

Diagram order

Diagram topology

MC update

MC

update

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

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

Computational complexity is factorial : !N

Resonant Fermions:

0rr

( )V r

( )r

1/3 ~ /Fn k

0 0Fk r Universal results in the zero-range, , and thermodynamic limit

Unitary gas: . SFk a

2

2

2

( )2

23

( 0, 0)

( , )

4( , 0)

p

m

p

m

C m r

p C e

p n em

Useful ‘bold’ relations:

all ladder diagrams

(0)

... ( , ) p k

,FG k

† †(0,0) (0,0) ( , ) ( , ) ,r t r t Skeleton graphsbased on ,G

2 2 /B kC

controls contributing diagram orders

resummation andextrapolation for

density

Unitary gas EOS (full story in previous talks)(in the universal & thermodynamic limit with quantifiable error bars) 0 0Fk r

Goulko, Wingate ‘10

3 3 3'( ), / (2 / 3 ) / ( )T T Tn z x P T E V T z x (calculated independently and cross-checked for universality)

2

21 ( )MF k

Mean-field behavior:

Critical point from pair distribution function

(2 )

(2 )

1/ 1

1/ 1

k k

k

Criticality:

0.038

21 ( )

AB

k

( , 0)k

2 (2 )

( )C

A

from( )C

2.25

0.160(5)C

F

T

E

0.152(7)C

F

T

E

Burovski et. al ’06, Kozik et. al ‘08

0.171(5)C

F

T

E

Goulko & Wingate ‘10

1

2C

F

T

E

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