Complex Langevin dynamics for nonabelian gauge theories

Transcription

1 Complex Langevin dynamics for nonabelian gauge theories Gert Aarts XQCD 14, June 2014 p. 1

2 QCD phase diagram QCD partition function Z = DUD ψdψe S YM S F = DU detde S YM at nonzero quark chemical potential [detd(µ)] = detd( µ ) fermion determinant is complex straightforward importance sampling not possible sign problem phase diagram has not yet been determined non-perturbatively XQCD 14, June 2014 p. 2

3 Outline complex Langevin: exploring a complexified field space gauge theories: from SU(N) to SL(N,C) gauge cooling status report: SU(3) + θ-term heavy dense QCD full QCD summary XQCD 14, June 2014 p. 3

4 Complex Langevin dynamics various scattered results since mid 1980s here: recent results obtained with Erhard Seiler, Dénes Sexty, Nucu Stamatescu Benjamin Jaeger Lorenzo Bongiovanni, Felipe Attanasio... some refs: [GA, IOS] [GA, ES, IOS], , [GA, FJ, ES, IOS] [ES, DS, IOS], [DS] [GA, PG, ES], [GA] [GA, LB, IOS, ES, DS] reviews: [GA], [GA, LB, IOS, ES, DS] XQCD 14, June 2014 p. 4

5 Complexified field space partition function Z = dxρ(x) with complex weight ρ(x) 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 XQCD 14, June 2014 p. 5

6 Complex Langevin dynamics does it work? for real actions: stochastic quantization Parisi & Wu 81 equivalent to path integral quantization Damgaard & Hüffel, Phys Rep 87 for complex actions: formal proof was notably absent recent progress: theoretical foundation given practical criteria for correctness formulated severe sign and Silver Blaze problems explicitly solved first results for gauge theories and even full QCD XQCD 14, June 2014 p. 6

7 Localised distributions if the action is holomorphic [log dets are an additional worry, see Kim Splittorff] and the distribution is localised, i.e. P(x,y) = 0 for y > y max [or P(x,y) 0 fast enough] then the correct result is obtained GA, ES, IOS 09 + FJ 11 applications to gauge theories... XQCD 14, June 2014 p. 7

8 Gauge theories SU(N) gauge theory: complexification to SL(N, C) links U SU(N): CL update U(n+1) = R(n)U(n) R = exp [ ( iλ a ǫka + )] ǫη a Gell-mann matrices λ a (a = 1,...N 2 1) drift: K a = D a (S B +S F ) S F = lndetm XQCD 14, June 2014 p. 8

9 Gauge theories SU(N) gauge theory: complexification to SL(N, C) links U SU(N): CL update U(n+1) = R(n)U(n) R = exp [ ( iλ a ǫka + )] ǫη a Gell-mann matrices λ a (a = 1,...N 2 1) drift: K a = D a (S B +S F ) S F = lndetm complex action: K K U SL(N,C) deviation from SU(N): unitarity norms 1 N Tr ( UU 1 ) 0 1 N Tr ( UU 1 ) 2 0 XQCD 14, June 2014 p. 8

10 Gauge theories deviation from SU(3): unitarity norm GA & IOS TrUU 1 heavy dense QCD, 4 4 lattice with β = 5.6, κ = 0.12, N f = 3 XQCD 14, June 2014 p. 9

11 Gauge theories controlled evolution: stay close to SU(N) submanifold when small chemical potential µ small non-unitary initial conditions in presence of roundoff errors XQCD 14, June 2014 p. 10

12 Gauge theories controlled evolution: stay close to SU(N) submanifold when small chemical potential µ small non-unitary initial conditions in presence of roundoff errors in practice this is not the case unitary submanifold is unstable! process will not stay close to SU(N) wrong results in practice, e.g. jumps when µ 2 crosses 0 also seen in abelian XY model XQCD 14, June 2014 p. 10

13 Gauge cooling what is the origin? can this be fixed? gauge freedom: link at site k U k Ω k U k Ω 1 k+1 Ω k = e iωk aλ a e αfk aλ a in SU(N): ω k a R in SL(N,C): ω k a C choose ω k a purely imaginary, orthogonal to SU(N) control unitarity norm = 1 N Tr ( UU 1 ) 0 gauge cooling ES, DS & IOS 12 after one gauge cooling step at site k,, linearise = α N (fk a) 2 +O(α 2 ) 0 XQCD 14, June 2014 p. 11

14 Gauge cooling what is fa k? Ω k = e αfk aλ a = α/n(fa k) choose fa k as the gradient of the unitarity norm ) fa k = 2Trλ a (U k U k U k 1 U k 1 if U SU(N): fa k = 0, = 0, no effect cooling brings the links as close as possible to SU(N) SL( N,C) SU( N) XQCD 14, June 2014 p. 12

15 Gauge cooling simple example: one-link model GA, LB, ES, DS & IOS 13 S = 1 N TrU U ΩUΩ 1 = 1 N Tr ( UU 1 ) ( f a = 2Trλ a UU U U ) note: c = TrU/N, c = TrU /N invariant under cooling cooling dynamics: = α N f2 a = 16α N TrUU [U,U ] in SU(2)/SL(2,C): = 8α ( 2 +2 ( 1 c 2) +c 2 +c 2 2 c 2) XQCD 14, June 2014 p. 13

16 Gauge cooling SU(2)/SL(2,C) one-link model = 8α ( 2 +2 ( 1 c 2) +c 2 +c 2 2 c 2) c = 1 2 TrU, c = 1 2 TrU invariant under cooling if c = c : U gauge equivalent to SU(2) matrix = 8α( +2 2c 2 ) (t) e 16α(1 c2 )t 0 if c c : U not gauge equivalent to SU(2) matrix (t) 0 = c c 2 c 2 + c 4 > 0 minimal distance from SU(2) reached exponentially fast XQCD 14, June 2014 p. 14

17 Langevin with gauge cooling complex Langevin dynamics with gauge cooling: alternate CL updates with gauge cooling updates monitor unitarity norm stay fairly close to SU(N) under investigation: SU(3) with a θ-term Lorenzo Bongiovanni, GA, ES, DS & IOS 13 + in prep heavy dense QCD GA & IOS 08, ES, DS & IOS 12 Benjamin Jaeger, Felipe Attanasio, GA, ES, DS & IOS in prep full QCD Dénes Sexty 13 + in prep XQCD 14, June 2014 p. 15

18 SU(3) Yang-Mills theory with a θ-term XQCD 14, June 2014 p. 16

19 SU(3) with aθ-term pure SU(3) Yang-Mills theory (no fermions) S = S YM iθq Q = g2 64π 2 d 4 xf a µν F a µν on the lattice: S = S W iθ L x q L (x) q L (x) = discretised lattice version θ L bare parameter, requires renormalisation lattice Q L = x q L is not topological (top. cooling) complex action for real θ L, real action for imaginary θ L imaginary θ L : real Langevin and hybrid Monte Carlo (HMC) real θ L : use complex Langevin XQCD 14, June 2014 p. 17

20 Analytical continuation Z(θ L ) = DU e S YM+iθ L Q = e Ωf(θ L) partition function and free energy even functions of θ L real θ L : i Q θl = lnz θ L = Ωχ L θ L ( 1+2b2 θ 2 L +3b 4θ 4 L +...) imaginary θ L = iθ I : Q θi = lnz θ I = Ωχ L θ I ( 1 2b2 θ 2 I +3b 4θ 4 I +...) determine topological charge susceptibility χ L and higher-order coefficients b k XQCD 14, June 2014 p. 18

21 Analytical continuation i Q θl = lnz θ L = Ωχ L θ L ( 1+2b2 θ 2 L +3b 4θ 4 L +...) no renormalisation of θ L yet! β = 6.1 Ω = 10 4 lines are fits: y = b 0 θ L ( 1±2b2 θ 2 L with b 0 = for both and b ) XQCD 14, June 2014 p. 19

22 Analytical continuation i Q θl = lnz θ L = Ωχ L θ L ( 1+2b2 θ 2 L +...) expected β dependence: drop in χ L no renormalisation of θ L yet! XQCD 14, June 2014 p. 20

23 Heavy dense QCD (and beyond) XQCD 14, June 2014 p. 21

24 Heavy dense QCD consider static quarks: fermion determinant simplifies detm = x det ) 2det ( (1+he µ/t P x 1+he µ/t Px 1 ) 2 with h = (2κ) N τ and P ( 1) (conjugate) Polyakov loops full Wilson gauge action is included nontrivial phase diagram: thermal deconfinement transition (as in pure glue) µ-driven transition at µ c ln(2κ) test case for full QCD XQCD 14, June 2014 p. 22

25 Heavy dense QCD consider static quarks: fermion determinant simplifies detm = x det ) 2det ( (1+he µ/t P x 1+he µ/t Px 1 ) 2 with h = (2κ) N τ and P ( 1) (conjugate) Polyakov loops preliminary results for Polyakov loop and density for β = 5.8 (a 0.15 fm) κ = 0.12 (µ c ln(2κ) = 1.43) volume 8 3 N τ N τ = Benjamin Jaeger XQCD 14, June 2014 p. 23

26 Heavy dense QCD (in progress) Polyakov loop XQCD 14, June 2014 p. 24

27 Heavy dense QCD (in progress) Polyakov loop XQCD 14, June 2014 p. 24

28 Heavy dense QCD (in progress) density XQCD 14, June 2014 p. 24

29 Heavy dense QCD (in progress) density XQCD 14, June 2014 p. 24

30 Heavy dense QCD (in progress) density µ c = ln(2κ) = 1.43 n sat = 12 first order transition at T = 0 (expected) see poster of Ben for more results XQCD 14, June 2014 p. 24

31 Beyond heavy dense QCD include higher-order κ corrections DS, IOS et al, in prep closer to real physics e.g. m B /3 m π /2, no immediate saturation, , nf=2, NLO, beta=5.8, kappa=0.12 vs mu/t; dens tot dens 0 dens n /T 3 vs µ at NLO (including O(κ 2 ) corrections) see poster of Nucu XQCD 14, June 2014 p. 25

32 Beyond heavy dense QCD include higher-order κ corrections DS, IOS et al, in prep closer to real physics e.g. m B /3 m π /2, no immediate saturation, fullqcd κ expansion density *4 β=5.9 κ=0.12 N F =2 µ= order of NLO corrections convergence in systematic expansion in κ 2 (DS) XQCD 14, June 2014 p. 25

33 Full QCD XQCD 14, June 2014 p. 26

34 Full QCD first application to full QCD: Dénes Sexty arxiv: fermion determinant: additional drift term in CLE requires inversion of fermion matrix stochastic inversion using conjugate gradient staggered fermions with 4 flavours (Wilson fermions in progress) monitor unitarity norm, log det, distributions,... compare with HDQCD for heavy quarks and reweighting for light quarks XQCD 14, June 2014 p. 27

35 Full QCD 1 n/n sat *6 lattice N f =4 β= µ/t HQCD ma=1 staggered ma=1 HQCD ma=4 staggered ma=4 density / density sat vs µ/t for heavier quarks and lighter quarks comparison with HDQCD Sexty 13 XQCD 14, June 2014 p. 28

36 Full QCD Polyakov loop µ/t HQCD ma=1 staggered ma=1 HQCD ma=4 staggered ma=4 8 3 *6 lattice N f =4 β=5.8 Polyakov loop vs µ/t for heavier quarks and lighter quarks comparison with HDQCD Sexty 13 XQCD 14, June 2014 p. 28

37 comparison with reweighting Full QCD Polyakov loop CLE inverse Polyakov CLE Polyakov reweighting inv. Polyakov reweighting 8 3 *4 lattice β=5.4 mass=0.05 N F = Re <exp(i 2 φ)> *4 lattice β=5.4 mass=0.05 N F =4 rew CLE µ/t µ/t Polyakov loop vs µ/t average sign for light quarks agreement until reweighting breaks down Fodor, Katz & Sexty in prep XQCD 14, June 2014 p. 28

38 Summary and outlook complex Langevin dynamics: recent progress for gauge theories better mathematical and practical understanding gauge cooling for SU(N) gauge theories work in progress for SU(3) + θ-term heavy dense QCD full QCD many things to do! XQCD 14, June 2014 p. 29