Youjin DengUniv. of Sci. & Tech. of China (USTC)

Adjunct: Umass, Amherst

Diagrammatic Monte Carlo Method for the Fermi Hubbard Model

Boris SvistunovUMass

Nikolay Prokof’evUMass

ANZMAP 2012, Lorne

Outline

• Fermi-Hubbard Model• Diagrammatic Monte Carlo sampling• Preliminary results• Discussion

Fermi-Hubbard model

†

, ,

i j ii iij i i

H t a a U n n n

t U

† † †' '

, , '

( ) k k k q k q p q p kk kpq

H a a U a a a a

momentum representation:

Hamiltonian

Rich Physics: Ferromagnetism Anti-ferromagnetism

Metal-insulator transition

Superconductivity

? Many important questions still remain open.

Feynman’s diagrammatic expansion

Quantity to be calculated:

The full Green’s function:

TH

pp eaapG /1,2,12, )()(Tr),(

Feynman diagrammatic expansion:

THeaaG /

1,2,12)0(

,0)()(Tr),( kkk

The bare interaction vertex :

k1 2

1

2

qk

qp p

q

k

)( 21 qU

The bare Green’s function :

qU

(0)2 3( , )G k

(0)

4 5( , )G p

A fifth order example:

+

+ …+ + +

+0 ( , )G p

=0 ( , )G p + +

Full Green’s function is expanded as :

Boldification:

Calculate irreducible diagrams for to get G

Dyson Equation :

The bare Ladder :

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

1 2( , )p ( , )G p

+

0 + U

0

Calculate irreducible diagrams for to get

0 0G G G G

0 0 0U U

+

0 0The bold Ladder :0 0

Two-line irreducible Diagrams:

Self-consistent iteration

,G Diagrammatic expansion

Dyson’s equation

,

Why not sample the diagrams by Monte Carlo?

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

Diagrammatic expansion

Monte Carlo sampling

Standard Monte Carlo setup:

- each cnf. has a weight factor cnfW

- quantity of interest

cnf cnfcnf

cnfcnf

A WA

W

- configuration space

Monte Carlo

MC

cnfcnf

A configurations generated from the prob. distribution cnfW

{ , , }i i iq p

Diagram order

Diagram topology

MC update

MC update

MC u

pdat

e

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

/ 4U t / 1.5 n 0.6t

/ 0.025 /100FT t E 2D Fermi-Hubbard model in the Fermi-liquid regime

Preliminary results

N: cutoff for diagram order

Series converge fast

Fermi –liquid regime was reached

2'

'

2

2 2

( ) (0) ( )6

( ) (0)6

F FF

F

TE T E E

Tn T n

/ 4U t / 3.1 1.2t n

/ 0.4 /10FT t E

Comparing DiagMC with cluster DMFT(DCA implementation)

!

/ 4U t / 3.1 1.2t n

/ 0.4 /10FT t E

2D Fermi-Hubbard model in the Fermi-liquid regime

Momentum dependence of self-energy

0 , x yT p p p along

Discussion

• Absence of large parameter

+

( ) ( )U t t The ladder interaction:

Trick to suppress statistical fluctuation

+ 0

1

Define a “fake” function:

+

• Does the general idea work?

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 coupling : recall the BCS answer (one lowest-order diagram)

- Regularization techniques

g

1/ge

Divergent series outside of finite convergence radius

can be re-summed.

From strong couplingtheories based on onelowest-order diagram

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

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

• Proven examples

Resonant Fermi gas:Nature Phys. 8, 366 (2012)

Square and Triangular lattice spin-1/2 Heisenberg model test:arXiv:1211.3631

Square lattice (“exact”=lattice PIMC)

MFT J T

Triangular lattice (ED=exact diagonalization)

1.25T J

Sign-problem

Variational methods+ universal- often reliable only at T=0- systematic errors- finite-size extrapolation

Determinant MC+ “solves” case - CPU expensive - not universal- finite-size extrapolation

1i in n Cluster DMFT / DCA methods+ universal- cluster size extrapolation

Diagrammatic MC+ universal- diagram-order extrapolation

Cluster DMFT

linear size

N diagram order

Diagrammatic MC

DF LT

Computational complexityIs exponential :exp{# }

for irreducible diagrams

• Computational complexity

Thank You!

Define a function such that:

, 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:

bN

n

ln 4

2 /,

ln( ),

n Nn N

n nn N

f e

f e

NA

1/ N

(Lindeloef)

(Gauss)

Key elements of DiagMC resummation technique

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

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

1 2( , )p

G U

+ Dyson Equation:( , )G p

U +U

Screening:

Irreducible 3-point vertex: 3

3 1 U

G

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

(0) + U

Ladders:(contact potential)

Key elements of DiagMCself-consistent formulation

What is DiagMC

MC sampling Feyman Diagrammatic series:• Use MC to do integration• Use MC to sample diagrams of different order and/or

different topology

What is the purpose?• Solve strongly correlated quantum system(Fermion,

spin and Boson, Popov-Fedotov trick)

+ …+ + ++ ++=