1. DIAGRAMMATIC MONTE CARLO:
N. Prokof’ev
Advancing Research in Basic Science and Mathematics
Kourovka, 2018

3. Complexity of an interacting fermion problem
Ask the relevant physical question “What does it take to obtain some quantity A with accuracy ?”
in the thermodynamic limit (TL)
[macroscopically large system]
Think of experiment!
Without specifying a particular method of getting A , it makes no sense to talk about fermioninc sign problem.
2. Theoretically, we need to define the computational complexity problem (CCP) as
The method is said to have CCP if the CPU time, , required to compute with accuracy diverges faster than any polynomial function of The problem is considered solved if
Why thermodynamic limit? In a system with particles the ultimate scaling is always by the central-limit theorem (if this scaling can be reached in practice; say for )

4. Sing Problem for methods based on Z
when is not always positive
with
If we do not know the answer even approximately
If we can live that long and then accuracy is , but we need
Ultimately, accuracy is limited by finite-size corrections that can remain too large in some cases.
Compromise: systematic bias (variational, AFQMC, …)

5. CCP is solved! Let is a convergent series with convergence radius .
Simple observation
Let is a convergent series with convergence radius
Define finite sums:
Then , is such that
Even if it takes with , to compute , the computation time required
to obtain with any specified relative accuracy  is subject to polynomial scaling
CCP is solved!
When accuracy gain is exponential in one should not  consider exponential scaling   as
a problem preventing us from computing A with  high (and systematically improvable) accuracy

6. Diag.MC for connected Feynman diagrams vs CCP for fermions
explicitly in the thermodynamic limit (connected diagrams)
Sign-blessing I: Feynman diagrammatic expansions converge only if same order diagrams cancel each other
(for lattice fermions at finite T)
Sign-blessing II: Sum of all connected topologies carry some properties of fermionic determinants
(sums of all connected and disconnected graphs; n! terms in n3 operations)
For convergent series with
For fermions, all order-N connected graphs can be computed in time [R. Rossi, PRL ‘17]
CCP is solved by Diag.MC:

7. we just don’t know yet how exactly in many cases …
Personal viewpoint (at least at the level of standard thermodynamics):
If it can be measured experimentally it can be calculated theoretically
on classical computers with the same accuracy …
we just don’t know yet how exactly in many cases …

8. 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:
Calculate irreducible diagrams for , , … to get , , …. from Dyson equations

9. . . . . . . . . One possible scheme: G2W- expansion Dyson Equation:
In terms of “exact” propagators
Dyson Equation:
Screening:

10. More tools: Build diagrams using ladders:
(contact potential)
In terms of “exact” propagators
Dyson Equations:

11. The only triple-couting (at the lowest order)
More tools: Build diagrams using three ladders:
(contact potential)
The only triple-couting (at the lowest order)

12. . . Fully dressed skeleton graphs (Heidin):
Irreducible 3-point vertex:
all accounted for already!

13. Bare or “dressed” series or skeleton sequences?
Pros. for bare:
- Taylor series  theory of analytic functions & resummation techniques
[work outside of convergence radius]
in diagrams are analytic; no self-consistent loops
Pros. for dressed/skeleton:
- Same graphs as in bare, just a much smaller set  # of irreducible graphs at order n
is smaller by a factor of about 1/n for each “dressed” channel: ,
One can simulate higher orders faster
are readily available from tables  as fast and easy, as intrinsic
- Bare series for long-range interactions are ill-defined, e.g. for
the 2d-order bubble diagram involves Need to screen, or sum
up all bubble chains.

14. Pros. for dressed/skeleton:
- Different expansion parameter: consider a dilute gas with large V. Define
a pair propagator based on ladder diagrams, and arrive
at the techniques where expension is in powers of density, not V .
Con. for skeleton:
- Skeleton sequence for divergent bare series may converge to the wrong
answer!
Solution for this Con.
- dress lines only partially by low-order bare or skeleton graphs
- do Taylor series for the rest using an auxiliary parameter

15. Shifted action tools
Original setup:
Bare series produce Taylor series in and the physical answer corresponds to
Introduce
Unphysical, in general, with formally arbitrary set of functions , but if we demand
then at we will recover the physical action identically Proceed with
Taylor series expansion in but it’s a different series because of counter-terms
- If are based on diagrams, then we expand on top of partially dressed G
- If are based on diagrams, then we deal with skeleton dressing
- This setup can be generalized (with the help of Hubbard-Stratanovich transformation)
to partial or skeleton dressing of interactions, ladders, etc.

16. in the functional space, and  =0 may not be even physical
When series diverge or convergence is slow
Recall
Complex plane
Complex plane
pole
pole
branch cut
branch cut
This is just a cartoon analogy, in shifted action we change the “origin of expansion”
in the functional space, and  =0 may not be even physical

17. When series diverge or convergence is slow
Complex U plane
Complex z plane
Conformal mapping:
Generalization = Pade
Hypergeometric functions
Borel, conformal Borel, never-heard-of-method …

18. When convergence radius is zero & series are asymptotic (unitary ultra-cold Fermi gas)
Quantitative understanding of series divergent behavior
saddle-point in g
method for large exp.
saddle-point in g and
bosonic field (after
Hubbard-Stratanovich
transformation) for large
exponent
L. N. Lipatov, Sov. Phys. JETP 45, 216 (1977)]
Can be generalized for partially summed and skeleton series
Suggests efficient resummation technique & uniqueness of Z(g)
“designer” conformal Borel transforms series from asymptotic
to convergent!
R. Rossi, K. van Houcke, F. Werner ‘yesterday

19. If you can
compute enough orders (say n~10; billions of skeleton graphs) and
know what to do with them, then any interacting Fermi system can be solved accurately

20. Sign alternating statistics
Sample or sum?
(Diagrammatic series for fermions converge, despite having about n! of them at order n, only because they cancel each other )
Sample
Sum
exact answer !
Definitely sum!

21. Spending CPU on performing “smart” global updates
Let with , then making local updates is not optimal because getting an uncorrelated set will require updates   
``Learn’’ how propose global updates
with accaptance ratio
``Heat bath’’ idea + ``self-learning’’ algoritms
Type A updates:
Interpret the second ratio as the relative weight of configurations when proposing a change.

22. … Monte Carlo Markov-chain cycle: Accept/Reject propose Local update
propose
Local update
Accept/Reject …
Large number L >> n of fast local updates
One of the ways to make a multi-dimensional stochastic proposal, one variable at a time.
If and and is simple enough, then the CPU cost is close to none!
The only thing left is to optimize to be such that (or at least >>1/n )
Within some variational ansatz for minimize
“Learn” your on trial runs

23. Never did it myself yet  (or know of any data for connected Feynman graphs), so …
In real space one may try to measure the gyration radius
where is the graph’s “center of mass”,
and measure the asymptotic dependence
when
One possible form is
Within the Rg ball control
Dipole, quadrupole, etc. fluct.

24. Applications:
“Yes, we can, but …”

25. When series converge: 2D Fermi-Hubbard model at
- Free energy density
R. Rossi, PRL ‘17
Six to five digit (depending on quantity) accuracy for a finite-T answer!

26. When series converge
T-independent
in Fermi-liquid

27. Conclusions/perspectives
we need a different scheme
For U>4 .and. n>0.7
SIMONS COLLABORATION ON MANY ELECTRON PROBLEM
Any dispersion relation for the same U and n

28. What did we know with meaningful error barsFM ? ? ?
Stripes?
PHASE SEPARATION? ? ?
AFM ? ?

30. Cooper instability in the BCS regime
is temperature (and frequency) independent in the limit
Fermi-liquid form at
determines

31. Protocol:
Compute Fermi surface parameters ; verify their T-independence
Transform
Solve for eigenvectors and eigenfunctions
Predicted divergence (standard weak-coupling BCS)
To determine compute correction (future work):
is -independent, to leading order in small g

32. Hlubina, PRB
59, 9600 (1999)
up to
up to

33. G&T-matrix self-consistent skeleton formulation (contact potential)
BOLD DIAGRAMMATIC CALCULATION:
Define “bare” pair propagator :
(exact solution of the two-body problem)
Skeleton diagrams
In terms of “exact” or “fully dressed” propagators &
Dyson Equations:

35. G&T-matrix self-consistent formulation (contact potential)
BOLD DIAGRAMMATIC CALCULATION:

37. An the “roller-coaster” story goes as …
When series converge
Graphene-type system
Dispersion relation Dirac points
(dimensionless coupling)
in suspended graphene
An the “roller-coaster” story goes as …

38. Adding next order diagrams
Fock diagram
Gonzalez, Guinea, andVozmediano ‘94
In RG language using or
asymptotically free Dirac liquid
renormalized dispersion
Adding next order diagrams
Mishchenko ‘07; Vafek and Case ‘08
Possibility of an unstable fixed point at
Dirac liquid  insuator

39. Upgrading 2d-order calculation to RPA:
But the value of is not small and the answer cannot be trusted!
Upgrading 2d-order calculation to RPA:
Gonza´lez, Guinea, and Vozmediano ‘99;
Das Sarma, Hwang, and Tse ‘07;
Polini et al. ‘07; Son ‘07;
Foster and Aleiner ‘08;
Kotov, Uchoa, and Castro Neto ‘09
From to
= …
sreened int.
is now such that
Dirac liquid is back! (by adding some but not all graphs)
But for the answer cannot be trusted either!

40. Direct numerical simulation:
Hybrid Deperminant Quantum Monte Carlo
(no sign-problem in the particle/hole symmetric case; L/a=24)
Coulomb law after on-site U
Hands and Strouthos ‘08
Drut and La¨hde ’09
Dirac liquid  insulator for
But … experiments find Dirac liquid in suspended graphene!
Fermi velocity (density dependence)
Elias, Gorbachev, Mayorov, Morozov,
Zhukov, Blake, Ponomarenko, Grigorieva,
Novoselov, Guinea, Geim, Mukhin ’11-’12
Siegel, Park, Hwang, Deslippe, Fedorov,
Louie, and Lanzara, ‘11
ARPES

41. Dirac liquid in suspended graphene is back alive!
Hybrid Deperminant Quantum Monte Carlo
Ulybyshev, Buividovich,
Katsnelson, and Polikarpov ’12-’13
Smith and von Smekal ‘14).
Dirac liquid in suspended graphene is back alive!
Status of previous work
Short-range correlations are extremely important in graphene; they prevent one from
revealing the long-range Coulomb effects in finite systems.
No solution for RG flow of (beyond uv RG cutoff scale)

42. BDMC approach
1. Eliminate Mott-insulator instability by making potential
flat at short-range  can study larger values of
2. We can afford electrons with
and RG flow up to L=256
Flat!
The flow is always to weaker coupling
changes are small

43. Flowgram idea
Kuklov, Matsumoto, NP, Svistunov, Troyer ‘08
Single parameter flow assumption/universality (at large distances) :
with universal flow functions
All dependence on microscopic parameters is in or
If for two systems A and B we get then the flows for and
should be the same. In terms of selecting a different value of is equivalent to a horizontal shift
If true, expect data collapse and a smooth “master” curve; if false, observe derivative jumps

45. Match
and obtain a smooth RG flow curve spanning 15 orders of magnitude! (from simulations of 14 different microscopic Hamiltonians)
Finally, increase the maximum diagram order in the skeleton expansion to account for vertex corrections.
Dirac liquid in a graphene-type system is
stable against strong Coulomb interaction

46. E.g. quantum Heisenberg model
Popov-Fedotov trick
Two-body spin-dependent interaction
with complex
Standard diagrammatic technique
(with “flat-band” bare dispersion)
Compute diagram after diagram for, say,
proper self-energy and polarization …
Reliable results down to temperature T=J/6 for pyrochlore lattice

47. ? How the SU(2) cooperative paramagnet looks like?
Heisenberg model on pyrochlore lattice (3D Kagome)
Cooperative paramagnet
Emergent
spin-ice
________ ? ?
Ising spin-ice
How the SU(2) cooperative
paramagnet looks like?
U(1) spin liquid
Y. Huang, K. Chen, Y. Deng, NP, and B. Svistunov ‘16

48. + Classical-to-Quantum correspondence ---“fingerprint” identification of quantum states)
Let Then for any there exists such that
for any distance R

49. Classical-to-Quantum correspondence is accurate at the level of 0
Classical-to-Quantum correspondence is accurate at the level of 0.3% (but not exact)

50. When series diverge or convergence is slow: 2D Fermi-Hubbard model
(optimize for best convergence in the atomic limit)
Divergent series
Expansion order
Expansion order
W. Wu, M. Ferrero, A. Georges, E.Kozik, arXiv:

51. When series diverge or convergence is slow: 2D Fermi-Hubbard model
Use to study pseudogap physics at
Nodal vs anti-nodal dichotomy. What scattering processes dominate at different parts of FS?

52. Solution of the non-equilibrium transport through Kondo impurity
When series diverge or convergence is slow
Solution of the non-equilibrium transport through Kondo impurity

53. When the convergence radius is zero: unitary Fermi gas
Fermions with strong “zero”-range potential leading to large s-wave scattering length
Unitary limit:
In this case and are
the only length/energy scales
Skeleton diagrammatic series are asymptotic with leading divergence ,
but are subject to the series resummation technique

54. Real thing, not a cartoon!
tiny 10-6 fraction

55. Equation of state for the unitary Fermi gas
(superfluid transition is at
R. Rossi, F. Werner, K. van Houcke (private comm. ‘16)

56. Conclusions:
1. For convergent and subject to re-summation series, Diag.MC solves the computational complexity problem ( )  correlated fermionic systems can be addressed with systematically improvable accuracy. What we lack is theoretical understanding of analytic properties of the model properties.
2. If item 1 above is understood, then the possibilities are unlimited !
- Hubbard model (need better formulations near half-filling -> second fermionization,
dual fermions, parquet, …
- Resonant fermions (develop DiagMC for superfluid states, explore polarized gases, move
away from unitarity, …)
- frustrated magnets (unique method for the cooperative paramagnet regime (can do pyrochlore), but
need to explore alternative formulations for T<<J, develop SU(N) schemes, …)
- interacting topological materials (ready to explore stability bounds and phase diagrams)
- Real materials? In progress (Simons Collaboration on the Many Electron Problem)

57. + + Lattice path-integrals for particles and
imaginary time
Lattice path-integrals for particles and
spins are “diagrams” of closed loops
formed by occupation number trajectories

58. + Diagrams for Diagrams for
imaginary time
imaginary time
lattice site
lattice site
The rest is conventional worm algorithm in continuous time

59. Find the ... differences
Simulations can handle up to particles at exper. relevant temperature

60. No fit comparison of time-of-flight images (momentum distributions)
S. Trotzky, L. Pollet, F. Gerbier,
U. Schnorrberger, I. Bloch, NP
B. Svistunov, M. Troyer, ‘10

61. Path-integrals in continuous space are “diagrams” of closed loops too!2 1 P

62. with exact account of interactions between all particles
Diagrams for the attractive tail in :
If and N>>1 all the effort is for something small ! P
statistical interpretation
ignore : stat. weight 1
Account for : stat. weight p 2 1 P
Faster than conventional scheme for N>30, scalable (size independent) updates
with exact account of interactions between all particles

63. s.v.p. 64
experiment
2048

64. Any math. representation of the problem Diagrammatic expansions
Any math. representation
of the problem
Diagrammatic expansions
Positive-definite
Sign-blessed
Positive-definite
Sign-blessed
Polarons
Path integrals for bosins/spins
Continuous time (CT) Determinant DiagMC Impurity solvers, …
Connected diagrams DiagMC (for bare series expansions) Rossi arXiv:

65. p-pairing dominates
only d-pairing

66. Compensation trick: Original representation Identical representation
(under integration)
Arbitrary auxiliary function with
“Design” it to compensate