# Diagrammatic Monte Carlo Method for

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15 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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Youjin

Deng

Univ. of Sci. & Tech. of China (USTC)

Umass
, Amherst

Diagrammatic Monte Carlo
Method for

the Fermi Hubbard Model

Boris
Svistunov

UMass

Nikolay

Prokof’ev

UMass

ANZMAP 2012, Lorne

Outline

Fermi
-
Hubbard Model

Diagrammatic Monte Carlo sampling

Preliminary results

Discussion

Fermi
-
Hubbard model

,,

i j i
i i
ij i i
H t a a U n n n
  
 

  
 
 
     
  
t
U

''
,,'
( )
k k k q k q p q p k
k 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
:

T
H
p
p
e
a
a
p
G
/
1
,
2
,
1
2
,
)
(
)
(
Tr
)
,
(

Feynman diagrammatic expansion:

T
H
e
a
a
G
/
1
,
2
,
1
2
)
0
(
,
0
)
(
)
(
Tr
)
,
(

k
k
k
The bare interaction vertex :

k
1

2

1

2

q
k

q
p

p
q
k
)
(
2
1

q
U
The bare Green’s function :

q
U
(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 :

+
+
+ ...

0
(,)
G p

1 2
(,)
p
 
 
(,)
G p

+


0

+

U
0

Calculate
irreducible

diagrams for

to get

0 0
G G G G
  
0 0 0
U U
    

+

0

0

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

cnf
W
-

quantity of interest

cnf cnf
cnf
cnf
cnf
A W
A
W

-

configuration space

Monte Carlo

MC
cnf
cnf
A


configurations generated
from the prob. distribution

cnf
W
{,,}
i i i
q p

Diagram order

Diagram topology

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

/4
U t

/1.5 n 0.6
t

  
/0.025/100
F
T 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
F
F
F
T
E T E E
T
n T n
 

  
 
/4
U t

/3.1 1.2
t n

  
/0.4/10
F
T t E

Comparing
DiagMC

with cluster DMFT

(DCA implementation)

!

/4
U t

/3.1 1.2
t n

  
/0.4/10
F
T t E

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

Momentum dependence of

self
-
energy

0
,
x y
T p p p
 
  
along

Discussion

Absence of large parameter

+

( ) ( )
U t t

 

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 ?

1
g

NO

Diverge for large even if

are convergent for small .

Math. Statement:

# of skeleton graphs

asymptotic series with

(n! beats any power)

3/2
2!
n
n n
 
Dyson: Expansion in

powers of g is asymptotic

if 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

1
g

N
A
1/
N
g
g
Skeleton diagrams up to high
-
order: do they make sense for ?

1
g

YES

# of graphs is

but due to
sign
-
blessing

they may compensate each
other to accuracy better
then

finite conv.

3/2
2!
n
n 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/
g
e


Divergent series outside

of

can be re
-
summed.

From strong coupling

theories based on one

lowest
-
order diagram

To accurate unbiased theories
based on millions of diagrams and
limit

N


0
0
F
k
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)

MF
T J T
 
Triangular lattice (ED=exact
diagonalization
)

1.25
T 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

1
i i
n n
 

 
Cluster DMFT / DCA methods

+

universal

-

cluster size extrapolation

Diagrammatic MC

+

universal

-

diagram
-
order extrapolation

Cluster DMFT

linear size

N

diagram order

Diagrammatic MC

D
F
L
T

 
 
 
 
Computational complexity

Is exponential :

exp{#}

for irreducible

diagrams

Computational complexity

Thank You!

Define a function

such that:

,

n N
f

a
N
1
,
1 for
n N
f n N
 
,
0 for
n N
f n N
 
Construct sums

and
extrapolate

to
get

,
0
N n n N
n
A c f

lim
N
N
A

A
0
3 9/2 9 81/4 ...?
n
n
A c

      

Example:

b
N
n
ln4
2
/
,
ln( )
,

n N
n N
n n
n N
f e
f e

N
A
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

(contact potential)

Key elements of
DiagMC

self
-
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)

+ …

+

+

+

+

+

+

=