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, Denés Sexty, Felix P. G. Ziegler, Stefan Bluecher, Sebastian Syrkowski, Mike Schlosser Cold Quantum Coffee May 29, 2018

Contents What is the sign problem The Complex Langevin Method The Lefschetz Thimble Method Beyond Lefschetz Thimbles Some QCD results (with CL)

The path integral measure and probability Euclidean path integral � D φ e − S ( φ ) Z = ρ ( φ ) = e − S ( φ ) is positive → probability measure. Observables via Monte Carlo simulations �O� = 1 � O e − S Z

What is the sign problem? What if ρ ( φ ) = e − S ( φ ) ≯ 0? Nontrivial cancellations → high statistics for the tail. (1005.0539) 1.0 0.5 0.0 - 0.5 - 1.0 - 3 - 2 - 1 0 1 2 3 Or even ρ ( φ ) ∈ C ? → no Monte Carlo .

Reweighting Reweighting (not limited to this particular form): �O Arg ( ρ ) � | ρ | � � O ρ O Arg ( ρ ) | ρ | �O� = ρ = Arg ( ρ ) | ρ | = � � � Arg ( ρ ) � | ρ | denominator: � ρ ρ � | ρ | | ρ | � = Z ρ � Arg ( ρ ) � | ρ | = = � | ρ | | ρ | Z | ρ | | ρ | free energy density: f = − T V log Z : Z ρ V →∞ = e − V T ∆ f − − − → 0 Z | ρ | Overlap problem: Strong shift of ρ → | ρ | can lead to strong suppression of signal of observable.

Who cares? Interesting Physics! Finite density (most famous: QCD phase diagram) Real time evolution And much more in and beyond QFT!

What now? 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 ...

Complex Langevin

Langevin Stochastic process dx = − ∂ S ∂ x dt + dw The corresponding Fokker-Planck equation ∂ ∂ t ρ ( x ) = [ ∂ x ( ∂ x + ( ∂ x S ( x )))] ρ ( x ) has solution ρ ( x ; t → ∞ ) = e − S ( x )

Complex Langevin Complexify the field variable (Parisi ’83, or see 0807.1597) dz = − ∂ S ∂ z dt + dw or � ∂ S � dx = − Re dt + dw ∂ z � ∂ S � dy = − Im dt ∂ z

What happens? Langevin evolution is driven by fixpoint structure 2 1 0 - 1 - 2 - 2 - 1 0 1 2

When does it work? 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) �O� ρ ( z ; t ) = �O� P ( x , y ; t ) for all t !

When does it work? Define interpolating quantity � F ( t , τ ) = P ( x , y ; t − τ ) O ( x + iy ; τ ) dxdy with F ( t , 0 ) = �O� P ( x , y ; t ) F ( t , t ) = �O� ρ ( z ; t ) Requirement: ∂ τ F ( t , τ ) = 0 (this is true if boundary terms vanish).

Example U ( 1 ) -one link model S ( z ) = i β cos ( z ) O ( z ) = e iz Can be solved numerically (e.g. β = 0 . 1): �O� = − 0 . 0500626 i . CLE yields: �O� = 0. WHY? 0.00 0.01 Im(exp(i*x)) 0.02 0.03 0.04 0.05 0 25 50 75 100 125 150 175 200 t

Example 0 t=10 -0.005 t=20 t=30 -0.01 t=40 t=50 -0.015 t=60 -0.02 t=70 t=80 -0.025 F(t, τ ) t=90 t=100 -0.03 t=110 -0.035 t=120 t=130 -0.04 t=140 -0.045 -0.05 -0.055 0 1 2 3 4 5 6 τ F ( t , τ ) shows buildup of boundary effects.

Example 0.08 t = 1 t = 10 0.06 t = 20 t = 30 t = 50 0.04 t = 75 t = 100 t = 150 0.02 t = 200 0.00 - 4 - 2 0 2 4 Boundary terms also visible in distribution (only in y -direction, becomes x -independent at large t for this model).

Example Why does it go wrong? Add term that makes real axis more attractive: 4 4 2 2 0 0 - 2 - 2 - 4 - 4 - 3 - 2 - 1 0 1 2 3 - 3 - 2 - 1 0 1 2 3 Lack of stable fixed points leads to large excursions.

Summary pt 1/3 There are clear criteria for correctness of CLE! More complicated theories: Look at histogram of observables instead of solution of Fokker-Planck → one can see if CL is correct by just looking at the simulation results!

Lefschetz Thimbles

The Lefschetz Thimble Method � e − s Look at integrals of the form Z = Starting point: Complexification of the real manifold Find critical points ∂ S ( z ) = 0 ∂ z 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 Lefschetz Thimble Method The following identity holds (1001.2933) � � � e − S ( x ) dx = n σ e − Im ( S ( z σ )) e − Re ( S ( z )) dz R J σ σ n σ is the intersection number of the antithimble (steepest ascent path) with the original manifold, encodes topology Why does this work? Homotopy-equivalence of the original manifold with the union of (contributing) Lefschetz-thimbles

The Lefschetz Thimble Method Real weight ρ σ ( z ) = e − Re ( S ( z )) allows standard Monte Carlo sampling (1205.3996) σ n σ e − iIm ( S ( z σ )) Z σ �O� σ � �O� = � σ n σ e − iIm ( S ( z σ )) Z σ �O� σ contains a residual sign problem, due to Jacobian, use reweighting (eq for one thimble:) �O� = �O J � � J � ratio of weights via reweighting (1803.08418) Z 1 e Re ( S 2 − S 1 ) � � = 2 Z 2

ℝ Why does this help? � � x 3 Airy integral S ( x ) = − i 3 + α x 1.0 0.5 0.0 x J - 0.5 - 1.0 - 4 - 2 0 2 4 Makes Monte Carlo possible AND removes fluctuations note: plot borrowed from Felix Zieglers group meeting talk BUT: Residual sign problem due to Jacobian

∞ Example 2 z 2 + 1 4 z 4 + ( 1 + i ) z S ( z ) = 1 3 1.2 1 p1 1.0 3.4 x p2 0.8 0 0 - ∞ 0.6 0.4 0.2 - 3 0.0 - 3 0 3 0.0 0.2 0.4 0.6 0.8 1.0 � z 2 � exact = 0 . 73922 + 0 . 63009 i � z 2 � numerical = 0 . 73922 ( 6 ) + 0 . 63006 ( 4 ) i (1803.08418)

Summary pt 2/3 Systematic way to apply Lefschetz Thimble Method in simple systems We are currently working on field theories

Beyond Lefschetz Thimbles 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)

QCD

Complex Langevin and QCD We look at QCD (first CLE application: 1307.7748) Lattice action � 1 � � � Tr U µν ( n ) + Tr U − 1 � � S ( U ) = − β µν ( n ) − 1 6 n ∈ Λ µ<ν � a 4 � + ψ ( n ) D ( n | m ) ψ ( m ) flavours n , m ∈ Λ ± 4 � D ( n | m ) = δ a , b δ α,β δ n , m − κ ( 1 − γ µ ) αβ U µ ( n ) ab δ n +ˆ µ, m µ = ± 1 with β ∼ g − 2 and κ ∼ 1 / m

Complex Langevin and QCD N f = 2 Wilson fermions ( m π ∼ 1GeV so far) Adaptive stepsize (no runaways) Gauge cooling: keep unitarity norm N U = U † U − Tr 1 as small as possible (gauge transformation opposite to ∇ N U )

Some results DISCLAIMER: All plots preliminary and low statistics! N s = 8, κ = 0 . 15, β = 5 . 9, everything in lattice units (sorry Nicolas...) 0.4 6.7 dens mu0.0 0.35 dens mu0.05 dens mu0.1 6.68 dens mu0.15 0.3 dens mu0.2 6.66 dens mu0.25 chiral condensate 0.25 dens mu0.3 cc mu0.0 dens mu0.35 6.64 0.2 <n> dens mu0.4 cc mu0.05 cc mu0.1 dens mu0.45 0.15 6.62 cc mu0.15 dens mu0.5 cc mu0.2 0.1 cc mu0.25 6.6 cc mu0.3 0.05 cc mu0.35 cc mu0.4 6.58 0 cc mu0.45 cc mu0.5 -0.05 6.56 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25 1/Nt 1/Nt 0.14 Ploop mu0.0 Ploop mu0.05 0.12 Ploop mu0.1 Ploop mu0.15 0.1 Ploop mu0.2 Ploop mu0.25 polyakov loops 0.08 Ploop mu0.3 Ploop mu0.35 Ploop mu0.4 0.06 Ploop mu0.45 Ploop mu0.5 0.04 0.02 0 -0.02 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 1/Nt

Some results Debunking CLE = phase quenched 0.35 0.45 Ploop mu0.0 Ploop mu0.5 inv Ploop mu0.0 0.4 inv Ploop mu0.5 0.3 0.35 0.25 0.3 polyakov loops polyakov loops 0.2 0.25 0.15 0.2 0.15 0.1 0.1 0.05 0.05 0 0 -0.05 -0.05 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25 1/Nt 1/Nt

Summary pt 3/3 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

THE END Thank you for your attention