youjin deng univ. of sci. & tech. of china (ustc) adjunct : umass , amherst
DESCRIPTION
Diagrammatic Monte Carlo Method for the Fermi Hubbard Model. Youjin Deng Univ. of Sci. & Tech. of China (USTC) Adjunct : Umass , Amherst. Nikolay Prokof’ev UMass. Boris Svistunov UMass. ANZMAP 2012, Lorne. Outline. Fermi-Hubbard Model Diagrammatic Monte Carlo sampling - PowerPoint PPT PresentationTRANSCRIPT
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)
+ …+ + ++ ++=