### Diagrammatic Monte Carlo simulation of the Fermi

```Diagrammatic Monte Carlo Method for
the Fermi Hubbard Model
Youjin Deng
Univ. of Sci. & Tech. of China (USTC)
Nikolay Prokof’ev
UMass
Boris Svistunov
UMass
ANZMAP 2012, Lorne
Outline
•
•
•
•
Fermi-Hubbard Model
Diagrammatic Monte Carlo sampling
Preliminary results
Discussion
Fermi-Hubbard model
Hamiltonian
H  t

ij ,
ai a†j  U  ni ni    ni
i ,
i
momentum representation:
H   ( k   )ak† ak 
k ,
Rich Physics:
 U a
q
kpq ,
†
k q
a†p q 'a p 'ak
'
Ferromagnetism
Anti-ferromagnetism
Metal-insulator transition
Superconductivity
? Many important questions still remain open.
t
U
Feynman’s diagrammatic expansion
Quantity to be calculated:
The full Green’s function:
G, ( p,  2   1 )  Tr a, p ( 2 ) a, p ( 1 ) e  H / T 


Feynman diagrammatic expansion:
The bare Green’s function :
G( 0,) (k ,  2   1 )  Tr a,k ( 2 ) a,k ( 1 ) e  H 0 / T 


1
k
2
k
The bare interaction vertex :
1
k q
q
Uq  (1  2 )
p
2
pq
Uq
A fifth order example:
G(0) (k, 2  3 )
G(0) ( p, 4  5 )
Full Green’s function is expanded as :
G0 ( p, ) =
+
G0 ( p, )
+
+
+
+
+
+
+…
Boldification:
Calculate irreducible diagrams for
G( p, )

Dyson Equation
G0 ( p, )
 to get G
( p,1   2 )
+

G  G0  G0 G :
Calculate irreducible diagrams for
+
+ ...

+
 to get 
0  U  U 00 :
  0  0
0

+
U

:

0
+
0
0


Two-line irreducible Diagrams:
Self-consistent iteration
G, 
Diagrammatic
expansion
Dyson’s
equation
, 
Monte Carlo sampling
Diagrammatic
expansion
Why not sample the
diagrams by Monte Carlo?
Configuration space = (diagram order, topology and types of lines, internal variables)
Standard Monte Carlo setup:
- configuration space
- each cnf. has a weight factor
Wcnf
A W

W
cnf
- quantity of interest
A
cnf
cnf
cnf
cnf
Monte Carlo
MC
 Acnf 
cnf
configurations generated
from the prob. distribution
Wcnf
Diagram order
{qi , i , pi }
Diagram topology
This is NOT: write diagram after diagram, compute its value, sum
Preliminary results
U /t  4
2D Fermi-Hubbard model in the Fermi-liquid regime
 / t  1.5  n  0.6
T / t  0.025 EF /100
N: cutoff for diagram order
Series converge fast
Fermi –liquid regime was reached
E (T )  E (0)  (  F   F' EF )
n(T )  n(0)  
'
F
 2T 2
6
 2T 2
6
U /t  4
 / t  3.1  n  1.2
T / t  0.4 EF /10
Comparing DiagMC with cluster DMFT
(DCA implementation)
!
2D Fermi-Hubbard model in the Fermi-liquid regime
U /t  4
 / t  3.1  n  1.2
T / t  0.4 EF /10
Momentum dependence of
self-energy
 0  T , p  along
px  py
Discussion
• Absence of large parameter

+
U (t )  (t )
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 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:
Bosons:
Math. Statement:
# of skeleton graphs
 2n n3/2 n !
asymptotic series with
(n! beats any power)
e  ie
U  U
[collapse to infinite density]
AN
g 1
Asymptotic series for
1/ N
Skeleton diagrams up to high-order: do they make sense for g  1 ?
YES
Divergent series outside
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
coupling g: recall the BCS answer
(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.
  e1/ g
- Regularization techniques
From strong coupling
theories based on one
lowest-order diagram
To accurate unbiased theories
based on millions of diagrams and
limit N  
• Proven examples
Resonant Fermi gas:
Nature Phys. 8, 366 (2012)
Universal results in the zero-range,
kF r0  0,
and thermodynamic limit
Square and Triangular lattice spin-1/2 Heisenberg model test:
arXiv:1211.3631
Square lattice (“exact”=lattice PIMC)
T  J  TMF
Triangular lattice (ED=exact diagonalization)
T  1.25 J
• Computational complexity
Sign-problem
Variational methods
Determinant MC
+ “solves” ni  ni   1 case
+ universal
- often reliable only at T=0
- systematic errors
- finite-size extrapolation
- CPU expensive
- not universal
- finite-size extrapolation
Cluster
DMFT /MC
DCA methods
Diagrammatic
+ universal
- diagram-order
cluster size extrapolation
extrapolation
Computational complexity
Is exponential : exp{#  }
Cluster DMFT
 F
T
 
 D
L

linear size
Diagrammatic MC
N
diagram order
for irreducible
diagrams
Thank You!
Key elements of DiagMC
resummation technique

Example:
A   cn  3  9 / 2  9  81/ 4  ...  ?
n 0
Define a function
such that:
f n, N
f n, N
Nb
 1 for n  N 1
f n, N  e
f n , N  e  n ln( n )
fn, N  0 for n  N
Na
n

Construct sums
 n2 / N
AN
AN   cn f n , N and extrapolate Nlim

to get
A
n 0
AN
ln 4
1/ N
(Gauss)
(Lindeloef)
Key elements of DiagMC
self-consistent formulation
 , , … to getG , U , …. from Dyson equations
Calculate irreducible diagrams for
G( p, )

Dyson Equation:
+
G0 ( p, )
(
p,1   2 ) +

Screening:
+
+ ...

+
U
U
G
Irreducible 3-point vertex:
3 
 1 
3
More tools: (naturally incorporating Dynamic mean-field theory solutions)

(contact potential)
 (0)
+
U

U
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)
+…
```