Approaching the sign problem by complexification Manuel Scherzer - - PowerPoint PPT Presentation
Approaching the sign problem by complexification Manuel Scherzer in collaboration with I.-O. Stamatescu, Jan M. Pawlowski, Erhard Seiler, Dens Sexty, Felix P. G. Ziegler, Stefan Bluecher, Sebastian Syrkowski, Mike Schlosser Cold Quantum
in collaboration with I.-O. Stamatescu, Jan M. Pawlowski, Erhard Seiler, Denés Sexty, Felix P. G. Ziegler, Stefan Bluecher, Sebastian Syrkowski, Mike Schlosser
May 29, 2018
What is the sign problem The Complex Langevin Method The Lefschetz Thimble Method Beyond Lefschetz Thimbles Some QCD results (with CL)
Euclidean path integral Z =
ρ(φ) = e−S(φ) is positive → probability measure. Observables via Monte Carlo simulations O = 1 Z
What if ρ(φ) = e−S(φ) ≯ 0? Nontrivial cancellations → high statistics for the tail. (1005.0539)
1 2 3
0.0 0.5 1.0
Or even ρ(φ) ∈ C? → no Monte Carlo.
Reweighting (not limited to this particular form): O =
OArg (ρ)|ρ| Arg (ρ)|ρ| denominator: Arg(ρ)|ρ| = ρ |ρ|
=
|ρ||ρ|
= Zρ Z|ρ| free energy density: f = − T
V log Z:
Zρ Z|ρ| = e− V
T ∆f
V →∞
− − − → 0 Overlap problem: Strong shift of ρ → |ρ| can lead to strong suppression of signal of observable.
Interesting Physics!
Finite density (most famous: QCD phase diagram) Real time evolution And much more in and beyond QFT!
Different ideas are needed!
For QCD Phase diagram: Taylor expansion, continuation from imaginary chemical potential Complex Langevin Lefschetz Thimbles and other path deformations Dual formulations Density of states ...
Stochastic process dx = −∂S ∂x dt + dw The corresponding Fokker-Planck equation ∂ ∂t ρ(x) = [∂x (∂x + (∂xS(x)))] ρ(x) has solution ρ(x; t → ∞) = e−S(x)
Complexify the field variable (Parisi ’83, or see 0807.1597) dz = −∂S ∂z dt + dw
dx = −Re ∂S ∂z
dy = −Im ∂S ∂z
Langevin evolution is driven by fixpoint structure
1 2
1 2
Action S(z) (and observables O) holomorphic Fokker-Planck equation for complex variable and for real and imaginary part, i.e. ρ(z; t) and P(x, y; t) CLE is correct, if: (0912.3360) Oρ(z;t) = OP(x,y;t) for all t!
Define interpolating quantity F(t, τ) =
with F(t, 0) = OP(x,y;t) F(t, t) = Oρ(z;t) Requirement: ∂τF(t, τ) = 0 (this is true if boundary terms vanish).
U(1)-one link model S(z) = iβ cos(z) O(z) = eiz Can be solved numerically (e.g. β = 0.1): O = −0.0500626i. CLE yields: O = 0. WHY?
25 50 75 100 125 150 175 200 t 0.05 0.04 0.03 0.02 0.01 0.00 Im(exp(i*x))
1 2 3 4 5 6 F(t,τ) τ t=10 t=20 t=30 t=40 t=50 t=60 t=70 t=80 t=90 t=100 t=110 t=120 t=130 t=140
F(t, τ) shows buildup of boundary effects.
t=1 t=10 t=20 t=30 t=50 t=75 t=100 t=150 t=200
2 4 0.00 0.02 0.04 0.06 0.08
Boundary terms also visible in distribution (only in y-direction, becomes x-independent at large t for this model).
Why does it go wrong? Add term that makes real axis more attractive:
1 2 3
2 4
1 2 3
2 4
Lack of stable fixed points leads to large excursions.
There are clear criteria for correctness of CLE! More complicated theories: Look at histogram of
→ one can see if CL is correct by just looking at the simulation results!
Look at integrals of the form Z =
Starting point: Complexification of the real manifold Find critical points ∂S(z)
∂z
= 0 Find Lefschetz thimbles: Steepest descent (ascent) paths that end (start) in the fixpoint ˙ z = ±∂S ∂z Can show: Im(S) is constant along thimbles
The following identity holds (1001.2933)
e−S(x)dx =
nσe−Im(S(zσ))
e−Re(S(z))dz nσ is the intersection number of the antithimble (steepest ascent path) with the original manifold, encodes topology Why does this work? Homotopy-equivalence of the
Lefschetz-thimbles
Real weight ρσ(z) = e−Re(S(z)) allows standard Monte Carlo sampling (1205.3996) O =
Oσ contains a residual sign problem, due to Jacobian, use reweighting (eq for one thimble:) O = OJ J ratio of weights via reweighting (1803.08418) Z1 Z2 =
2
Airy integral S(x) = −i
3 + αx
2 4
0.0 0.5 1.0 x
ℝ J
Makes Monte Carlo possible AND removes fluctuations
note: plot borrowed from Felix Zieglers group meeting talk
BUT: Residual sign problem due to Jacobian
S(z) = 1
2z2 + 1 4z4 + (1 + i)z
3
3
p1 3.4 x p2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2
∞ 1
z2exact = 0.73922 + 0.63009i z2numerical = 0.73922(6) + 0.63006(4)i (1803.08418)
Systematic way to apply Lefschetz Thimble Method in simple systems We are currently working on field theories
Using homotopy equivalence, other paths are possible Maryland approach: Use steepest descent to flow closer to thimbles Path optimization: Maximize average sign by making an ansatz for a manifold and optimizing the parameters (either by gradient equations or neural networks)
We look at QCD (first CLE application: 1307.7748) Lattice action S(U) = −β
1 6
µν (n)
a4
n,m∈Λ
ψ(n)D(n|m)ψ(m) D(n|m) = δa,bδα,βδn,m − κ
±4
(1 − γµ)αβUµ(n)abδn+ˆ
µ,m
with β ∼ g −2 and κ ∼ 1/m
Nf = 2 Wilson fermions (mπ ∼ 1GeV so far) Adaptive stepsize (no runaways) Gauge cooling: keep unitarity norm NU = U†U − Tr1 as small as possible (gauge transformation opposite to ∇NU)
DISCLAIMER: All plots preliminary and low statistics! Ns = 8, κ = 0.15, β = 5.9, everything in lattice units (sorry Nicolas...)
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.05 0.1 0.15 0.2 0.25 <n> 1/Nt dens mu0.0 dens mu0.05 dens mu0.1 dens mu0.15 dens mu0.2 dens mu0.25 dens mu0.3 dens mu0.35 dens mu0.4 dens mu0.45 dens mu0.5 6.56 6.58 6.6 6.62 6.64 6.66 6.68 6.7 0.05 0.1 0.15 0.2 0.25 chiral condensate 1/Nt cc mu0.0 cc mu0.05 cc mu0.1 cc mu0.15 cc mu0.2 cc mu0.25 cc mu0.3 cc mu0.35 cc mu0.4 cc mu0.45 cc mu0.5
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 polyakov loops 1/Nt Ploop mu0.0 Ploop mu0.05 Ploop mu0.1 Ploop mu0.15 Ploop mu0.2 Ploop mu0.25 Ploop mu0.3 Ploop mu0.35 Ploop mu0.4 Ploop mu0.45 Ploop mu0.5
Debunking CLE = phase quenched
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.05 0.1 0.15 0.2 0.25 polyakov loops 1/Nt Ploop mu0.0 inv Ploop mu0.0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.05 0.1 0.15 0.2 0.25 polyakov loops 1/Nt Ploop mu0.5 inv Ploop mu0.5
Sign problem leads to need for exponential growth in computation time Can be circumvented by Complex Langevin and Lefschetz Thimbles Complex Langevin: Clear criteria for whether it works Complex Langevin works in QCD in all interesting regions Outlook: Application of Lefschetz thimbles to higher dimensional manifolds → make use of symmetries
Thank you for your attention