Complex Langevin dynamics and the sign problem GGI 2012 Gert Aarts - PowerPoint PPT Presentation

Complex Langevin dynamics and the sign problem GGI 2012 Gert Aarts GGI, September 2012 – p. 1

QCD phase diagram GGI, September 2012 – p. 2

QCD phase diagram? at finite baryon chemical potential: complex weight straightforward importance sampling not possible overlap problem various possibilities: preserve overlap as best as possible use approximate methods at small µ do something radical: rewrite partition function in other dof explore field space in a different way . . . GGI, September 2012 – p. 3

Outline into complex plane reminder: real vs. complex Langevin dynamics troubled past: stability and convergence SU(3) spin model . . . . . . versus XY model Haar measure lessons? exploit freedom? GGI, September 2012 – p. 4

Overlap problem configurations differ in an essential way from those obtained at µ = 0 or with | det M | cancelation between configurations with ‘positive’ and ‘negative’ weight Re ρ( x ) dominant configurations in the path integral? x GGI, September 2012 – p. 5

Complex integrals consider simple integral � ∞ S ( x ) = ax 2 + ibx dx e − S ( x ) Z ( a, b ) = −∞ complete the square/saddle point approximation: into complex plane lesson: don’t be real(istic), be more imaginative radically different approach: complexify all degrees of freedom x → z = x + iy enlarged complexified space new directions to explore GGI, September 2012 – p. 6

Complexified field space dominant configurations in the path integral? Re ρ( x ) y ⇒ x x real and positive distribution P ( x, y ) : how to obtain it? ⇒ solution of stochastic process complex Langevin dynamics Parisi 83, Klauder 83 GGI, September 2012 – p. 7

Real Langevin dynamics dx e − S ( x ) partition function Z = � S ( x ) ∈ R Langevin equation � η ( t ) η ( t ′ ) � = 2 δ ( t − t ′ ) x = − ∂ x S ( x ) + η, ˙ associated distribution ρ ( x, t ) � � O ( x ( t ) � η = dx ρ ( x, t ) O ( x ) Langevin eq for x ( t ) ⇔ Fokker-Planck eq for ρ ( x, t ) ∂ x + S ′ ( x ) � � ρ ( x, t ) = ∂ x ˙ ρ ( x, t ) ρ ( x ) ∼ e − S ( x ) stationary solution: GGI, September 2012 – p. 8

Fokker-Planck equation stationary solution typically reached exponentially fast ∂ x + S ′ ( x ) � � ρ ( x, t ) = ∂ x ˙ ρ ( x, t ) ρ ( x, t ) = ψ ( x, t ) e − 1 2 S ( x ) write ˙ ψ ( x, t ) = − H FP ψ ( x, t ) Fokker-Planck hamiltonian: � � � � − ∂ x + 1 ∂ x + 1 H FP = Q † Q = 2 S ′ ( x ) 2 S ′ ( x ) ≥ 0 ψ ( x ) ∼ e − 1 2 S ( x ) Qψ ( x ) = 0 ⇔ 2 S ( x ) + c λ e − λt → c 0 e − 1 ψ ( x, t ) = c 0 e − 1 � 2 S ( x ) λ> 0 GGI, September 2012 – p. 9

Complex Langevin dynamics dx e − S ( x ) partition function Z = � S ( x ) ∈ C complex Langevin equation: complexify x → z = x + iy � η ( t ) η ( t ′ ) � = 2 δ ( t − t ′ ) x = − Re ∂ z S ( z ) + η ˙ y = − Im ∂ z S ( z ) ˙ S ( z ) = S ( x + iy ) associated distribution P ( x, y ; t ) � � O ( x + iy )( t ) � = dxdy P ( x, y ; t ) O ( x + iy ) Langevin eq for x ( t ) , y ( t ) ⇔ FP eq for P ( x, y ; t ) ˙ P ( x, y ; t ) = [ ∂ x ( ∂ x + Re ∂ z S ) + ∂ y Im ∂ z S ] P ( x, y ; t ) generic solutions? semi-positive FP hamiltonian? GGI, September 2012 – p. 10

Equilibrium distributions complex weight ρ ( x ) real weight P ( x, y ) main premise: � � dx ρ ( x ) O ( x ) = dxdy P ( x, y ) O ( x + iy ) if equilibrium distribution P ( x, y ) is known analytically: shift variables � � � dxdy P ( x, y ) O ( x + iy ) = dx O ( x ) dy P ( x − iy, y ) � ⇒ ρ ( x ) = dy P ( x − iy, y ) correct when P ( x, y ) is known analytically hard to verify in numerical studies! GGI, September 2012 – p. 11

Field theory Dφ e − S path integral Z = � Langevin dynamics in “fifth” time direction ∂φ ( x, t ) = − δS [ φ ] δφ ( x, t ) + η ( x, t ) ∂t Gaussian noise � η ( x, t ) η ( x ′ , t ′ ) � = 2 δ ( x − x ′ ) δ ( t − t ′ ) � η ( x, t ) � = 0 compute expectation values � φ ( x, t ) φ ( x ′ , t ) � , etc study converge as t → ∞ Parisi & Wu 81, Parisi, Klauder 83 Damgaard & H¨ uffel 87 GGI, September 2012 – p. 12

Some achievements complex Langevin dynamics can handle severe sign problems . . . . . . in thermodynamic limit describe onset at expected critical chemical potential i.e. not at phase-quenched value (Silver Blaze problem) describe phase transitions be implemented for gauge theories however, success is not guaranteed GA, Frank James, Erhard Seiler, Nucu Stamatescu (& Denes Sexty) 08-now GA & Kim Splittorff 10 GGI, September 2012 – p. 13

Troubled past 1. numerical problems: runaways, instabilities ⇒ adaptive stepsize no instabilities observed, works for SU(3) gauge theory GA, James, Seiler & Stamatescu 09 a la Ambjorn et al 86 2. theoretical status unclear ⇒ detailed analyis, identified necessary conditions GA, FJ, ES & IOS 09-12 3. convergence to wrong limit ⇒ better understood but not yet resolved in progress GGI, September 2012 – p. 14

Instabilities: heavy dense QCD adaptive time step during the evolution -4 10 -5 10 stepsize -6 10 -7 10 -8 10 0 100000 200000 300000 Langevin iteration occasionally very small stepsize required can go to longer Langevin times without problems GGI, September 2012 – p. 15

Analytical understanding consider expectation values and Fokker-Planck equations one degree of freedom x , complex action S ( x ) , ρ ( x ) ∼ e − S ( x ) � wanted: � O � ρ ( t ) = dx ρ ( x, t ) O ( x ) ∂ x + S ′ ( x ) � � ∂ t ρ ( x, t ) = ∂ x ρ ( x, t ) solved with CLE: � � O � P ( t ) = dxdy P ( x, y ; t ) O ( x + iy ) ∂ t P ( x, y ; t ) = [ ∂ x ( ∂ x − K x ) − ∂ y K y ] P ( x, y ; t ) with K x = − Re S ′ , K y = − Im S ′ question: � O � P ( t ) = � O � ρ ( t ) if P ( x, y ; 0) = ρ ( x ; 0) δ ( y ) ? GGI, September 2012 – p. 16

Analytical understanding question: � O � P ( t ) = � O � ρ ( t ) as t → ∞ ? answer: yes, use Cauchy-Riemann equations and satisfy some conditions: distribution P ( x, y ) should drop off fast enough in y direction partial integration without boundary terms possible actually O ( x + iy ) P ( x, y ) for large enough set O ( x ) ⇒ distribution should be sufficiently localized can be tested numerically via criteria for correctness � LO ( x + iy ) � = 0 with L Langevin operator 0912.3360, 1101.3270 GGI, September 2012 – p. 16

SU(3) spin model apply these ideas to 3D SU(3) spin model GA & James 11 earlier solved with complex Langevin Karsch & Wyld 85 Bilic, Gausterer & Sanielevici 88 however, no detailed tests performed ⇒ test reliability of complex Langevin using developed tools analyticity in µ 2 : from imaginary to real µ Taylor series criteria for correctness comparison with flux formulation Gattringer & Mercado 12 contrast with 3D XY model GA & James 10 GGI, September 2012 – p. 17

SU(3) spin model 3-dimensional SU(3) spin model: S = S B + S F � P x P ∗ y + P ∗ � � S B = − β x P y <xy> � e µ P x + e − µ P ∗ � � S F = − h x x SU(3) matrices: P x = Tr U x gauge action: nearest neighbour Polyakov loops (static) quarks represented by Polyakov loops complex action S ∗ ( µ ) = S ( − µ ∗ ) effective model for QCD with static quarks, centre symmetry GGI, September 2012 – p. 18

SU(3) spin model phase structure effective model for QCD with static quarks GGI, September 2012 – p. 19

SU(3) spin model � P + P ∗ � / 2 phase structure at µ = 0 : 3 µ=0, h=0.02, 10 1.5 -1 ) > < Tr(U + U 1 0.5 0 0.12 0.125 0.13 0.135 0.14 β GGI, September 2012 – p. 20

SU(3) spin model real and imaginary potential: first-order transition in β − µ 2 plane, � P + P ∗ � / 2 2 2 β=0.135 β=0.128 3 h =0.02, 10 β=0.134 β=0.126 β=0.132 β=0.124 β=0.130 β=0.120 1.5 1.5 -1 )/2 > < Tr( U+U 1 1 0.5 0.5 0 0 -1 -1 -0.5 -0.5 0 0 0.5 0.5 1 1 2 µ negative µ 2 : real Langevin — positive µ 2 : complex Langevin GGI, September 2012 – p. 21

SU(3) spin model Taylor expansion (lowest order) free energy density f ( µ ) = f (0) − ( c 1 + c 2 h ) hµ 2 + O ( µ 4 ) � n � = 2 ( c 1 + c 2 h ) hµ + O ( µ 3 ) density Polyakov loops � P � = c 1 + c 2 hµ + O ( µ 2 ) � P ∗ � = c 1 − c 2 hµ + O ( µ 2 ) in terms of c 1 = 1 c 2 = 1 � � ( P x − P ∗ P y − P ∗ � � �� � P x � µ =0 x ) y µ =0 Ω 2Ω x xy c 2 is absent in phase-quenched theory GGI, September 2012 – p. 22

SU(3) spin model start in ‘confining’ phase and increase µ density � n � = � he µ P x − he − µ P ∗ x � : no Silver Blaze region 1.5 0.02 3 β=0.125, h =0.02, 10 0.015 1 0.01 < n> 0.005 0 0.5 0 0.4 0.8 1.2 full phase quenched 0 0 0.5 1 1.5 2 2.5 3 3.5 µ inset: lines from first-order Taylor expansion GGI, September 2012 – p. 23

SU(3) spin model start in ‘confining’ phase and increase µ splitting between � P � and � P ∗ � : no Silver Blaze region 2 3 β =0.125, h =0.02, 10 <Tr U > <Tr U > 1.5 -1 > -1 > -1 > <Tr U <Tr U <Tr U >, <Tr U 1 0.4 0.5 0.2 0 0 0.5 1 1.5 0 0 0.5 1 1.5 2 2.5 3 3.5 µ inset: lines from first-order Taylor expansion GGI, September 2012 – p. 24