DIAGRAMMATIC MONTE CARLO:

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DIAGRAMMATIC MONTE CARLO From polarons to
path-integrals to skeleton technique
N. Prokofev
KITPC 5/13/14
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Classical MC
the number of variables N is constant
Quantum MC (often)
Integration variables
term order
Contribution to the answer or weight (with
differential measures!)
different terms of of the same order
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Monte Carlo (Metropolis) cycle
Accept with probability
Diagram
suggest a change
Same order diagrams
Business as usual
Updating the diagram order
Ooops
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Balance Equation
If the desired probability density
distribution of different terms in the stochastic
sum is then the updating process has to be
stationary with respect to (equilibrium
condition). Often
Flux to
Flux out of
Is the probability of proposing an update
transforming to
Detailed Balance solve equation for each pair of
updates separately
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Let us be more specific. Equation to solve
new variables
are proposed from
the normalized probability distribution
Solution
All differential measures are gone!
Efficiency rules
- try to keep - simple analytic function

ENTER
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Standard Monte Carlo setup
(depends on the model and its representation)

- configuration space
- each cnf. has a weight factor
- quantity of interest
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Statistics
Monte Carlo
states generated from probability distribution
Anything
Monte Carlo
states generated from probability distribution
Connected Feynman Anything
diagrams, e.g. for the
proper self-energy
diagram order
topology
internal variables
Answer to S. Weinbergs question
Monte Carlo
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Configuration space (diagram order, topology
and types of lines, internal variables)
This is NOT write/enumerate diagram after
diagram, compute its value, and then sum
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quasiparticle
Polaron problem


Electrons in semiconducting crystals
(electron-phonon polarons)
electron
phonons
el.-ph. interaction
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electron
phonons
el.-ph. interaction
Green function
Sum of all Feynman diagrams
Positive definite series in the
representation






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Graph-to-math correspondence
is a product of
Positive definite series in the
representation
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Diagrams for
there are also diagrams for optical
conductivity, etc.
Doing MC in the Feynman diagram configuration
space is an endless fun!
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The simplest algorithm has three updates
Insert/Delete pair (increasing/decreasing the
diagram order)
In Delete select the phonon line to be deleted at
random
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The optimal choice of
depends on the model
Frohlich polaron
1. Select anywhere on the interval
from uniform prob. density
2. Select anywhere to the left of
from prob. density
(if
reject the update)
3. Select from Gaussian prob. density
, i.e.
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New
Standard heat bath probability density
Always accepted,
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Normalization
histogram
special bin where is known exactly
Normalized histogram
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Normalization using desined bin
bin 0
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This is it! Collect statistics for
or some other corr. function. Analyze it.
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Analysing data
dispersion relation
probability of getting a bare electron
(Lehman expansion)
probability of getting two phonons in the
polaron cloud
Slope
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Standard model Hubbard model Coulomb
gas Heisenberg model Periodic Anderson Kondo
lattice models
High-energy physics High-Tc superconductors
Quantum chemistry band structure Quantum
magnetism Heavy fermion materials

- Introduced in mid 1960s or earlier
- Still not solved (just a reminder, today is
01/13/2012)
- Admit description in terms of Feynman diagrams
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Feynman Diagrams Physics of strongly correlated
many-body systems
In the absence of small parameters, are they
useful in higher orders?
Divergent series are the devil's invention...
Yes, with sign-blessing for regularized skeleton
graphs!
N.Abel, 1828
And if they are, how to handle millions and
billions of skeleton graphs?
Sample them with Diagrammatic Monte Carlo
techniques (teach computers rules of quantum
field theory)
Steven Weinberg, Physics Today, Aug. 2011
Also, it was easy to imagine any number of
quantum field theories of strong interactions
but what could anyone do with them?
Unbiased solutions beased on millions of graphs
with extrapolation to the infinite diagram order
From current strong-coupling theories based on
one lowest order skeleton graph (MF, RPA, GW,
SCBA, GG0, GG,
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Skeleton diagrams up to high-order do they make
sense for ?
NO
YES
Series diverge for large even if they
converge for
Divergent series outside of finite convergence
radius can be re-summed.
Skeleton series are not based on Many systems
remain well-defined for Lattice fermions,
quantum magnetism, resonant fermions _
regularization
Dyson Expansion in is asymptotic if for
some the system becomes pathological Continu
ous space bosons and fermions collapse to
infinite density for
Math. Statement number of skeleton graphs
asymptotic series
with zero conv. radius

Number of graphs is but because they alternate
in sign they may all cancel each other to near
zero ! Sign blessing
Asymptotic series for with zero
convergence radius are useless!
Start computing high-order diagrams!
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Re-summation of divergent series with finite
convergence radius.
Example
???? ????? ??
Define a function such that
(Gauss)
(Lindeloef)
Construct sums
and extrapolate to get
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Conventional Sign-problem vs Sign-blessing
Sign-problem
Computational complexity is exponential in system
volume
and error bars explode before a reliable
exptrapolation to can be made
(diagrams for )
Feynamn diagrams
No limit to take, selfconsistent
formulation, admit analytic results and partial
resummations.
(for )
Sign-blessing
Number of diagram of order is factorial
thus the only hope for good
series convergence properties is sign
alter- nation of diagrams leading to their
cancellation. Still,
i.e. Smaller and smaller error bars are
likely to come at exponential price (unless
convergence is exponential).
(diagrams for )
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Diagrammatic Monte Carlo in the generic many-body
setup
Feynman diagrams for free energy density

x
x
x
x
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Bold (self-consistent) Diagrammatic Monte Carlo
Diagrammatic technique for ln(Z ) diagrams
admits partial summation and

self-consistent
formulation
No need to compute all diagrams for and

Dyson Equation
Screening
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.
.
.
.
.
.
.
.
In terms of exact propagators
Dyson Equation
Screening
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More tools Build diagrams using ladders

(contact potential)
In terms of exact propagators
Dyson Equations
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Fully dressed skeleton graphs (Heidin)
.
.
Irreducible 3-point vertex
all accounted for already!
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Unpolarized system at unitarityBCS-BEC crossover
Unitary gas when
and are the only length/energy scales
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Answering Weinbergs question Equation of State
for ultracold fermions neutron matter at
unitarity
MIT group Mrtin Zwierlein, Mark Ku, Ariel
Sommer, Lawrence Cheuk, Andre Schirotzek
Uncertainty due to location of the
resonance
BDMC results
Kris Van Houcke, Felix Werner, Evgeny Kozik,
Boris Svistunov, NP
virial expansion (3d order)
Ideal Fermi gas
QMC for connected Feynman diagrams NOT particles!

Sign blessing
Sign problem
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Lattice path-integrals for bosons and spins are
diagrams of closed loops!


imaginary time
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Diagrams for
Diagrams for
imaginary time
imaginary time
lattice site
lattice site
The rest is conventional worm algorithm in
continuous time
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Simulating Bose-Hubbard model as is and
comparing to experiments

(in this example, )
Nature Physics, 6, 9981004, (2010)
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It is realistic to do about 2,000,000 or more
particles at temperatures relevant for the
experiment.
Phys. Rev. Lett. 103, 085701 (2009)
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Path-integrals in continuous space are
diagrams of closed loops too!
P
2
1
P
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Diagrams for the attractive tail in
If
and Ngtgt1 all the effort is for
something small !
Faster than conventional scheme for Ngt30,
scalable (size independent) updates with exact
account of interactions between all particles
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3D _at_ s.v.p.
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experiment
2048
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Other applications Continuous-time QMC solves
(impurity solvers)
are standard DMC schemes

Fermions with contact interaction
Rubtsov (2003)
Most efficient solvers for DMFT and DCA are based
on this approach
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Maier et al.
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Critical point from pair distribution function
Criticality
from zero of