Complex Langevin Dynamics in 1+1D QCD at finite densities SIGN - PowerPoint PPT Presentation

Complex Langevin Dynamics in 1+1D QCD at finite densities SIGN workshop Sebastian Schmalzbauer supervisor: Jacques Bloch September 29 th , 2015

1 1 Motivation • QCD phase diagram • many other systems also plagued by sign problem ⇒ find new methods to cure it • But why low dimensional QCD? • sign problem already present • study viability of the complex Langevin method • can compare with analytical results (0+1D) or other methods (1+1D) 1 / 22

1 1 QCD at finite µ : Partition Function • partition function (1 flavour) after integrating over fermions � D [ U ] det D [ U ] e − S G [ U ] Z = • staggered Dirac operator at chemical potential µ : d − 1 � � � η k ,ν U k ,ν e a µδ ν, 0 δ k +ˆ ν, l − U − 1 ν,ν e − a µδ ν, 0 δ k − ˆ D k , l = m δ k , l + , ν, l k +ˆ 2 a ν = 0 with quark mass m , staggered phase η = ± 1, antiperiodic boundary conditions in time direction and U ∈ SL ( 3 , ❈ ) • chiral symmetry not explicitly broken for m = 0 • µ = 0 det D ∈ ❘ µ � = 0 det D ∈ ❈ ⇒ importance sampling not possible 2 / 22

1 1 Complex Langevin Dynamics • Langevin equation for Gell-Mann representation ( λ a ) � ( dU x ,µ ) U − 1 x ,µ = − λ a ( D a , x ,µ S ( U ) dt + dw a , x ,µ ) , a with independent Wiener increments dw a , x ,µ and group derivative D a , x ,µ S ( U ) = ∂ α S ( e i αλ a U x ,µ ) | α = 0 • discrete time evolution = SL ( 3 , ❈ ) rotation � a ( ǫ K a , x ,µ + √ ǫη a , x ,µ ) U x ,ν x ,ν = e i λ a ′ U • drift K a , x ,µ different for Euler, Runge-Kutta schemes 1 • gaussian noise � η a , x ,µ � = 0 � η a , x ,µ η b , y ,ν � = 2 δ ab δ xy δ µν 1 Chang ’87, Batrouni et al. ’85, Bali et al. ’13 3 / 22

1 1 Equivalence to Fokker-Planck Equation • FP: real fields with complex probability � � d x O ( x ) ρ ( x ; t ) = d x d y O ( x + iy ) P ( x , y ; t ) CL: complex fields with real probability expect correct expectation values as long as 2 • solution of FPE asymptotes to correct probability distribution det D e S G t →∞ ρ ( x , t ) lim → • boundary terms in partial integration step vanish • sufficient falloff of probability distribution • singular drifts (det D = 0) suppressed enough ⇒ gauge cooling 2 Aarts, James, Seiler, Stamatescu ’10 ’11 Nagata, Nishimura, Shimasaki ’15 4 / 22

1 1 0+1D Gauge Cooling • Dirac determinant can be reduced to � � e µ/ T P + e − µ/ T P − 1 + 2 cosh ( µ c / T ) det ( D ) ∝ det , with Polyakov loop P = Π t U t and effective mass a µ c = arsinh ( am ) • reduce unitarity norm � � � � − 1 − 2 � P † P + P † P ||U|| = tr x ,µ P → GPG − 1 via SL ( 3 , ❈ ) gauge trafos • diagonalizing P = maximal cooling 5 / 22

1 1 Equivalence to Eigenvalue Representation • diagonalizing P � = working in eigenvalue representation   e i φ 1 0 0   e i φ 2 P = 0 0 φ 1 , φ 2 ∈ ❈   e − i φ 1 − i φ 2 0 0 • gauge cooling not required • additional term in the action (and hence in the drift) S = − log J − log det D , arising from the Haar measure � φ 1 − φ 2 � � 2 φ 1 + φ 2 � � φ 1 + 2 φ 2 � J ( φ 1 , φ 2 ) = sin 2 sin 2 sin 2 2 2 2 6 / 22

1 1 0+1D: Results quark density n q chiral condensate Σ 3 0.4 analytic 0.35 uncooled 2.5 chiral condensate Σ quark density n q cooled 0.3 2 0.25 analytic 1.5 uncooled 0.2 cooled 0.15 1 0.1 0.5 0.05 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 chemical potential µ/T chemical potential µ/T 20 20 | Δ n q |/ σ 15 15 | ΔΣ |/ σ 10 10 5 5 3 σ 3 σ 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 • uncooled results results have significant deviation 3 • cooled results look good, but not perfect for all µ • Polyakov loop looks fine for all µ 3 analytic results: Bilic, Demeterfi ’88 7 / 22

1 1 General Effects of SL ( 3 , ❈ ) Gauge Trafos • measurement of observables are invariant O ( P ) = O ( GPG − 1 ) • drift term K a and gauge cooling step commute K a ( GPG − 1 ) λ a = K a ( P ) G λ a G − 1 • noise distribution η a used in the CL step is not invariant 4 η a λ a � = η a G λ a G − 1 ⇒ SL ( 3 , ❈ ) gauge trafos and CL step do not commute ⇒ apply gauge cooling: different trajectories 4 SU(3) gauge trafos leave the noise distributions invariant 8 / 22

1 1 0+1D: Effects of Gauge Cooling effect on det D distribution: • compressed in imaginary direction • avoids the origin 40 40 30 30 20 20 10 10 Im(det D) ⇒ 0 0 10 10 20 20 30 30 40 40 20 0 20 40 60 80 100 120 20 0 20 40 60 80 100 120 Re(det D) Re(det D) effect on observable distribution: broad ⇔ narrow distribution 10 1 10 0 Σ = 0.1010(4) uncooled tr P = 0.647(1) uncooled Σ = 0.08315(9) cooled tr P = 0.6587(8) cooled 10 0 10 -1 Σ = 0.0831 tr P = 0.6590 distribution distribution 10 -1 10 -2 10 -2 10 -3 10 -3 10 -4 10 -4 10 -5 10 -5 -4 -2 0 2 4 -6 -4 -2 0 2 4 6 9 / 22 chiral condensate Σ Polyakov loop tr P

1 1 From 0+1D to dD • lattice volume V = N t N d − 1 s • size of Dirac operator increases: 3 V × 3 V • time to compute D − 1 : ∝ V 3 • general gauge trafos GUG − 1 G x U x ,ν G − 1 → x +ˆ ν ⇒ gauge cooling via diagonalizing no longer works! • can include gauge action S G ⇒ extra drift term • we worked in 1+1D with strong coupling on 4 × 4 lattices 10 / 22

1 1 Gauge Cooling in Higher Dimensions � � � � − 1 − 2 • unitarity norm ||U|| = � U † U † x ,µ U x ,µ + x ,µ tr x ,µ U x ,µ • minimize via SL ( 3 , ❈ ) gauge trafos: G x U x ,µ G − 1 U x ,µ → x +ˆ µ using steepest descent 10 0 10 0 10 -2 10 -1 unitarity norm ||U|| unitarity norm ||U|| 10 -4 10 -2 10 -6 10 -3 cooled 10 -8 µ=0 uncooled 10 -4 10 -10 µ=0.11 uncooled 10 -5 10 -12 µ=0.25 uncooled µ=0.25 cooled 10 -6 10 -14 µ=0.11 cooled 10 -16 10 -7 0 100000 200000 300000 400000 500000 0 50000 100000 150000 200000 250000 300000 Langevin time Langevin time ⇒ even for µ = 0 we need distance from SU(3) depends gauge cooling on µ 11 / 22

1 1 Convergence with Stepsize • interested in continuum solution ǫ → 0 µ = 0 . 07 µ = 0 . 25 2 2.4 subsets: 2.02(2) chiral condensate Σ chiral condensate Σ 1.8 2.2 convergence ∝ ε subsets: 2.22(1) 1.6 convergence ∝ ε 2 1.4 Euler uncooled Euler uncooled 1.2 1.8 Runge-Kutta uncooled Runge-Kutta uncooled Euler cooled Euler cooled 1 1.6 Runge-Kutta cooled Runge-Kutta cooled 0.8 10 -4 10 -3 10 -2 10 -4 10 -3 10 -2 stepsize ε stepsize ε • uncooled simulation is stable, but converges to a wrong limit • for some µ even the cooled simulation does not converge correctly (benchmark: subset method 5 ) 5 J. Bloch, F. Bruckmann, T. Wettig, ’12 ’14 12 / 22

1 1 1+1D Results: Chiral Condensate 2.5 m=0.1 m=0.5 uncooled cooled subsets 2 m=1.0 chiral condensate Σ 1.5 m=2.0 1 m=0.01 0.5 0 0 0.5 1 1.5 2 chemical potential µ 13 / 22

1 1 1+1D Results: Quark Density 3.5 m=0.1 m=0.5 m=1.0 m=2.0 3 2.5 quark density n q 2 1.5 1 0.5 uncooled cooled subsets 0 0 0.5 1 1.5 2 chemical potential µ 14 / 22

1 1 1+1D results: Polyakov Loop 0.7 m=0.1 m=0.5 m=1.0 m=2.0 0.6 0.5 Polyakov loop tr P 0.4 uncooled cooled 0.3 subsets 0.2 0.1 0 -0.1 0 0.5 1 1.5 2 2.5 chemical potential µ 15 / 22

1 1 Effect of Gauge Cooling on det D (m=0.1) 1e 7 1e 7 1.0 1.0 0.5 0.5 Im(det D) ⇒ µ = 0 . 07 0.0 0.0 0.5 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.5 0.5 0.0 0.5 1.0 1.5 Re(det D) 1e 7 Re(det D) 1e 7 1e 7 1e 7 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 Im(det D) ⇒ µ = 0 . 25 0.0 0.0 0.5 0.5 1.0 1.0 1.5 1.5 2.0 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 Re(det D) 1e 7 Re(det D) 1e 7 ⇒ cooling not enough in some µ ranges for light quarks! 16 / 22

1 1 Effect of Gauge cooling on det D (m=2.0) 1e17 1e17 1.0 1.0 0.5 0.5 Im(det D) ⇒ µ = 1 . 25 0.0 0.0 0.5 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.5 0.5 0.0 0.5 1.0 1.5 Re(det D) 1e17 Re(det D) 1e17 1e17 1e17 2 2 1 1 Im(det D) ⇒ µ = 1 . 50 0 0 1 1 2 2 0.5 0.0 0.5 1.0 1.5 2.0 2.5 0.5 0.0 0.5 1.0 1.5 2.0 2.5 Re(det D) 1e18 Re(det D) 1e18 ⇒ cooling works in all µ ranges for heavy quarks! 17 / 22