() Continuous Time Monte Carlo for SC-QCD. QCD in the Strong Coupling Limit

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1 Continuous Time Monte Carlo for QCD in the Strong Coupling Limit with Philippe de Forcrand, ETH Zürich/CERN XQCD, San Carlos, Mexico / 35

2 Outline Motivation for Strong Coupling Lattice QCD Motivation for Continuous Time Limit of SC-LQCD Continuous Time Partition Function and Algorithm Crosschecks with Analytic Results for U() Results on the U(3) Transition and SU(3) Phase Diagram Conclusion 2 / 35

3 Motivation for SC-LQCD The QCD (µ, T ) phase diagram: T [MeV ] C LH IC, RH Cross 0 FA IR over CP & ent n em tio fin nsi con Tra er De iral st ord Ch 0 Hadronic Matter Vacuum Quark Gluon Plasma 00 Early Universe 200 Nuclear Matter Baryonic Crystal? Neutron Stars 0 Color Superconductor? B [GeV ] rich phase structure conjectured chiral and deconfinement transition QGP at high temperatures exotic matter at high density 3 / 35

4 Motivation for SC-LQCD The QCD (µ, T ) phase diagram: T [MeV ] QCD has a severe sign problem for finite chemical potential µ: fermions anti-commute: Cross 00 γ5 (i p/ +m+µγ0 )γ5 = (i p/ +m+µγ0 ) Quark Gluon Plasma over the fermion determinant det M(µ) becomes complex! 0 e Sf = det M(µ) = det M( µ ) Hadronic Matter Vacuum 0 Nuclear Matter B [GeV ] because of the sign problem: very little is known only recently: agreement on crossover temperature Tc at zero baryon chemical potential little hope that it can be circumvented: - Taylor expansion, - imaginary µ with analytic continuation, - reweighting method are all limited to small µ/t. 3 / 35

5 Motivation for SC-LQCD SC-QCD Partition Function Motivation for SC-LQCD (2) Conventional Monte Carlo with staggered fermions: based on partition function Z Z(T, V, µ) = [du] det M[U, µ]e SG [U] M[U, µ] = mq + X ηµ (x ) e µδµ0 Uµ (x ) e µδµ0 Uµ (x µ ), 2 x,µ ηµ (x ) = ( ) Σν<µ xν /T = N a = N a / 2 V /3 =N a 4 / 35

6 Motivation for SC-LQCD SC-QCD Partition Function Motivation for SC-LQCD (2) Conventional Monte Carlo with staggered fermions: based on partition function Z Z(T, V, µ) = µ]e SG [U] [du] det M[U, X M[U, µ] = mq + 2 ηµ (x ) e µδµ0 Uµ (x ) e µδµ0 Uµ (x µ ), x,µ ηµ (x ) = ( )Σν<µ xν /T = N a = N a / 2 V /3 =N a Instead: look at QCD in a regime where the sign problem can be made mild: strong coupling by changing the nature of integrations variables sampled no sampling of {U} (no HMC)! 4 / 35

7 Motivation for SC-LQCD SC-QCD Partition Function What is Strong Coupling Lattice QCD? Staggered QCD in the strong coupling limit: start frompthe -flavor staggered QCD Lagrangian in Euclidean time: P LQCD = x χ 2 ν ην (x ) e µδν0 Uν (x ) e µδν0 Uν (x ν ) + mq χ P 2Nβc trup + trup + O(a2 ) P 5 / 35

8 Motivation for SC-LQCD SC-QCD Partition Function What is Strong Coupling Lattice QCD? Staggered QCD in the strong coupling limit: start from the -flavor staggered QCD Lagrangian in Euclidean time: P P LQCD = x χ 2 µ ηµ (x ) e µδµ0 Uµ (x ) e µδµ0 Uµ (x µ ) + mq χ ((( P ( (( (tru tru + O(a2 ) ( P + ( P P ( ((( 2Nβc g, β = take limit of infinite gauge coupling: 2Nc g2 0 allows to integrate out the gauge fields completely, as link integration factorizes: Z [du] = YZ duµ (x ) x,µ 5 / 35

9 Motivation for SC-LQCD SC-QCD Partition Function What is Strong Coupling Lattice QCD? Staggered QCD in the strong coupling limit: start from the -flavor staggered QCD Lagrangian in Euclidean time: P P LQCD = x χ 2 µ ηµ (x ) e µδµ0 Uµ (x ) e µδµ0 Uµ (x µ ) + mq χ ((( P ( (( (tru tru + O(a2 ) ( P + ( P P ( ((( 2Nβc g, β = take limit of infinite gauge coupling: 2Nc g2 0 allows to integrate out the gauge fields completely, as link integration factorizes: Z [du] = YZ duµ (x ) x,µ Drawback: strong coupling limit is converse to asymptotic freedom: lattice spacing becomes very large as a(β) exp( β/4nc b0 ), no continuum limit degrees of freedom in SC-QCD live on a crystal fermions have no spin 5 / 35

10 Motivation for SC-LQCD SC-QCD Partition Function Why Strong Coupling Lattice QCD? -flavor strong coupling QCD might appear crude, but exhibits confinement (only color singlet degrees of freedom) and chiral symmetry breaking/restoration: UA (): χ(x ) 7 e iη5 (x )θa χ(x ), χ (x ) 7 χ (x )e iη5 (x )θa, d η5 (x ) = ( )Σν=0 xν is spontaneously broken below Tc, hence its phase diagram might be similar to the QCD phase diagram SC-LQCD as effective theory for nuclear matter: derive nuclear interactions between hadrons from (Lattice) QCD transition at high density is the nuclear transition 6 / 35

11 Motivation for SC-LQCD SC-QCD Partition Function Why Strong Coupling Lattice QCD? -flavor strong coupling QCD might appear crude, but exhibits confinement (only color singlet degrees of freedom) and chiral symmetry breaking/restoration: UA (): χ(x ) 7 e iη5 (x )θa χ(x ), χ (x ) 7 χ (x )e iη5 (x )θa, d η5 (x ) = ( )Σν=0 xν is spontaneously broken below Tc, hence its phase diagram might be similar to the QCD phase diagram SC-LQCD as effective theory for nuclear matter: derive nuclear interactions between hadrons from (Lattice) QCD transition at high density is the nuclear transition Just another effective model for QCD? yes and no: think of SC-LQCD rather as a -parameter deformation of QCD 6 / 35

12 Motivation for SC-LQCD SC-QCD Partition Function SC-QCD Partition Function () One-link integral z(x, y = x + µ ) for SU(Nc ): Z duµ (x ) exp ηµ (x ) χ (x )Uµ (x )χ(y ) χ (y )Uµ (x )χ(x ) SU(Nc ) = Nc X (Nc k)! Nc!k! ηµ (x )γ δ0,µ 2 M(x )M(y ) k Nc + κ ρ(x, y ) B (x )B(y ) + ( ρ(y, x )) Nc B (y )B(x ) k=0 7 / 35

13 Motivation for SC-LQCD SC-QCD Partition Function SC-QCD Partition Function () One-link integral z(x, y = x + µ ) for SU(Nc ): Z duµ (x ) exp ηµ (x ) χ (x )Uµ (x )χ(y ) χ (y )Uµ (x )χ(x ) SU(Nc ) = Nc X (Nc k)! Nc!k! ηµ (x )γ δ0,µ 2 M(x )M(y ) k Nc + κ ρ(x, y ) B (x )B(y ) + ( ρ(y, x )) Nc B (y )B(x ) k=0 Last step: integrate out the fermion fields, using R = : dξd ξ ξξ Z Y [dχa (x )d χ a (x )]e 2amq χ (x )χ(x ) (χ (x )χ(x ))k = a Nc! (2amq )nx nx! 7 / 35

14 Motivation for SC-LQCD SC-QCD Partition Function SC-QCD Partition Function () One-link integral z(x, y = x + µ ) for SU(Nc ): Z duµ (x ) exp ηµ (x ) χ (x )Uµ (x )χ(y ) χ (y )Uµ (x )χ(x ) SU(Nc ) = Nc X (Nc k)! Nc!k! ηµ (x )γ δ0,µ 2 M(x )M(y ) k Nc + κ ρ(x, y ) B (x )B(y ) + ( ρ(y, x )) Nc B (y )B(x ) k=0 Only confined, colorless degrees of freedom remain after link integration (Nc = 3): Mesons M(x ) = χ (x )χ(x ), represented by monomers and dimers (non-oriented meson hoppings): n(x ) {0,,..., 3} Baryons B(x ) = 3 i...i3 χi (x )... χi3 (x ) form self-avoiding oriented loops: kµ (x ) {0,,..., 3} ρ(x, y ) = ηµ (x ) γ exp(±at µ)δµ 0 + ( δµ 0 ) B (x )B(y ) {0, } Note: baryons transform under gauge trafo Ω U(3) as B(x ) B(x ) det Ω not gauge invariant, only SU(3) contains baryons (κ = ), U(3) purely mesonic (κ = 0) 7 / 35

15 Motivation for SC-LQCD SC-QCD Partition Function SC-QCD Partition Function (2) Exact rewriting of SC-QCD partition function (no approximation!): X Z(mq, µ, Nτ ) = Y (3 kb )! Y 3! {k,n,l} b=(x,µ ) 3!kb! x nx! (2amq )nx Y w (l, µ) l Grassmann constraint: color neutral states at each site nx + X kµ = 3 x Nσ 3 Nτ µ =±0,...±d weight for baryon loop l (sign σ(l)):! w (l, µ) = Y 3! σ(l)γ 3N0 e 3Nτ rl at µ x l 8 / 35

16 Motivation for Continuous Time SC-LQCD SC-LQCD at finite temperature How to vary temperature? at = /Nτ discrete steps atc ' discrete steps too coarse need to introduce an anisotropy γ in the Dirac couplings Should we expect a/at = γ, as suggested in weak coupling? c +)(Nc +2) No: meanfield predicts a/at = γ 2, since γc2 = Nτ (d )(N 6(Nc +3) sensible, Nτ -independent definition of the temperature: at ' γ2 Nτ Moreover: SC-QCD partition function is a function of γ 2 : Z(mq, µ, γ, Nτ ) = X Y (3 kb )! {k,n,l} b=(x,µ ) 3!kb! γ 2kb δ0 µ Y 3! x nx! (2amq )nx Y w (l, µ) l However: precise correspondence between a/at and γ 2 not known 9 / 35

17 Motivation for Continuous Time SC-LQCD Motivation for Continuous Time: Strategy for unambiguous answer: the continuous Euclidean time limit (CT-limit): Nτ, γ, γ 2 /Nτ fixed same as in analytic studies: at = 0, at = β R 0 / 35

18 Motivation for Continuous Time SC-LQCD Motivation for Continuous Time: Strategy for unambiguous answer: the continuous Euclidean time limit (CT-limit): Nτ, γ, γ 2 /Nτ fixed same as in analytic studies: at = 0, at = β R Several advantages of continuous Euclidean time approach: ambiguities arising from the functional dependence of observables on the anisotropy parameter will be circumvented no need to perform the continuum Nτ allows to estimate critical temperatures more precisely, with a faster algorithm baryons become static in the CT-limit, the sign problem is completely absent! 0 / 35

19 Motivation for Continuous Time SC-LQCD Motivation for Continuous Time: Strategy for unambiguous answer: the continuous Euclidean time limit (CT-limit): Nτ, γ, γ 2 /Nτ fixed same as in analytic studies: at = 0, at = β R Several advantages of continuous Euclidean time approach: ambiguities arising from the functional dependence of observables on the anisotropy parameter will be circumvented no need to perform the continuum Nτ allows to estimate critical temperatures more precisely, with a faster algorithm baryons become static in the CT-limit, the sign problem is completely absent! Aim to show in this talk: results obtained with an algorithm in continuous time improve on systematics CT-methods interesting in themselves, SC-QCD a first QFT-like application 0 / 35

20 Motivation for Continuous Time SC-LQCD Example of discretization errors For precise determination of transitions temperatures: need to go to large Nτ : Nτ = 6 has still large discretization errors convergence non-monotonic, and only reached for Nτ 32 3 Comparison of chiral susceptibility for 6 xnτ lattices 0000 Nτ continuous Nτ continuous at at / 35

21 Continuous Time Partition Function Origin of Continuous Time Monte Carlo Continuous time (CT) algorithms have been introduced in Quantum Monte Carlo: First proposed for the Heisenberg quantum anti-ferromagnet (B. B. Beard & U. J. Wiese, 996): persue a world-line formulation (i. e. rewrite the partition function in terms of transition/decay probabilities) from Phys. Rev. Lett. 77 (996) / 35

22 Continuous Time Partition Function Origin of Continuous Time Monte Carlo Continuous time (CT) algorithms have been introduced in Quantum Monte Carlo: First proposed for the Heisenberg quantum anti-ferromagnet (B. B. Beard & U. J. Wiese, 996): persue a world-line formulation (i. e. rewrite the partition function in terms of transition/decay probabilities) 2 make time direction continuous, transitions may occur at any time from Phys. Rev. Lett. 77 (996) / 35

23 Continuous Time Partition Function Origin of Continuous Time Monte Carlo Continuous time (CT) algorithms have been introduced in Quantum Monte Carlo: First proposed for the Heisenberg quantum anti-ferromagnet (B. B. Beard & U. J. Wiese, 996): persue a world-line formulation (i. e. rewrite the partition function in terms of transition/decay probabilities) 2 make time direction continuous, transitions may occur at any time from Phys. Rev. Lett. 77 (996) 532 Today: rich literature for quantum impurity systems (see review by Gull et al 200) but continuous time methods not yet applied to quantum field theories 2 / 35

24 Continuous Time Partition Function Continuous Time Partition Function () Key observation: multiple spatial dimers are suppressed with powers of γ 2 : double/triple spatial dimers become resolved into single dimers as at ' a/γ single spatial dimers survive (constant density) baryonic spat. hops are suppressed with γ 7 discrete time no spatial 3-dimers no spatial 2- and 3-dimers continuous time Time χ / Nτ / 35

25 Continuous Time Partition Function Continuous Time Partition Function () Key observation: multiple spatial dimers are suppressed with powers of γ 2 : T A double/triple spatial dimers become resolved into single dimers as at ' a/γ 2 0 single spatial dimers survive (constant density) L B baryonic spat. hops are suppressed with γ Partition function can be decomposed into spatial and temporal parts, i. e.: in spatial dimers (meson hoppings) and two types of temporal intervals in between: dashed ( ) and solid intervals ( ) the weight of a configuration only depends on the kind and number of vertices at which spatial hoppings are attached to solid/dashed lines, L -vertices of weight vl = and T -vertices of weight vt = 2/ 3, where a solid line emits a spatial dimer. B Time A B T L B 3 / 35

26 Continuous Time Partition Function Continuous Time Partition Function (2) Partition function (restricted to chiral limit mq = 0, where monomers are absent) in limit of large γ 2, Nτ : Z(γ, Nτ ) ' γ 3V Y X Y {k} x VM = X e 3nB µ/t 3 b=(x,i ) Y (3 kb )! Y 0 b0 =(x,0 ) 3!kb0! Y vl nl (x ) vt nt (x ) x VM {k,nb } γ 2kb γ γ, e 3σ(x )µ/t x VB VM VB = Nσ 3 with VM, VB the subvolume of mesonic/baryonic sites and nb the baryon number typical 2-dimensional configurations in discrete and continuous time at the same temperature no multiple dimers, only static baryons in continuous time 4 / 35

27 Continuous Time Partition Function Continuous Time Partition Function (3) Continuous time Monte Carlo based on diagrammatic expansion in a parameter: here: expansion in powers of γ 2, i. e. in total number of spatial hoppings k integrate over all spatial hopping positions Nτ, resulting in an expansion in inverse temperature β = Nτ /γ 2 : Z(β) = X (β/2)2k X k 2N k! n (C) N (C) vtt C Γk with configurations C defined up to shifts of spatial hoppings which leave the time ordering unchanged (Γ0 the space of all dimer-network topologies) N (C) non-trivial combinatorical factors, governed by relative order of L-/T-vertices 5 / 35

28 Continuous Time Partition Function Continuous Time Partition Function (4) Validity of expression for Z(β) demonstrated by reweighting:.6 Reweighted rew. around β=0.66 rew. around β=0.5 rew. around β=0.25 data energy number of spatial hoppings: k = V β P3 hki i i= (mesonic) energy: he i = V hk0 i = VNc /2 k 6 / 35

29 Continuous Time Partition Function A "Hopping parameter" expansion for massless U(3) In principle, diagrammatic expansion possible, but not feasible label spatial emissions with time index enumerate labelings which preserve time ordering A'A' A'B' = A'B' B'B' = B'B' B'B' ACA ADA AEA AEA AFA = AFA BCB BDB BEB BEB BFB BFB BCB BDB BEB BEB BFB BFB and more Graphs, generated by (A+B+B)(C+D+E+E+F+F+G+G)(A+B+B) A2+A2-A2+A2- B2+B2-B2+B2- B2+B2-B2+B2- and more Graphs, generated by (A2++B2++B2+)(A2-+B2-+B2-)(A2++B2++B2+)(A2-+B2-+B2-) 7 / 35

30 Continuous Time Partition Function Exponential Distribution of Interval Lengths The hopping times are uniformly distributed and according to the statistics of a Poisson process: weight of a configuration does not depend on location of spatial dimers the dashed/solid interval lengths are exponentially distributed: P( β) = exp( λ β), β [0, β = /at ] λ is the decay constant for spatial dimer emissions with baryons present, it is space-dependent, in d spatial dimensions: λ = dm (x )/4, dm (x ) = 2d X nb (x + µ ) µ =±i with dm (x ) the number of mesonic neighbors at x in a purely mesonic system (i. e. U(3) gauge group) in d dimensions, dm = 2d 8 / 35

31 Continuous Time Partition Function Poisson Distribution of Interval Lengths Histogram for number of polymer sites N (either static meson or baryon, i.e. sites with no spat. dimers attached) is almost Poisson-like (V = Nσ 3 ): V N V λeff N/T e e λeff /T, N g(n, T ) = < N/V >= exp( λeff /T ).8 T=0.8, L: Histogram available sites for baryons λeff < λ for low temperatures, λeff > λ for high temperatures uniformity also implies (approximate) Poisson process even for higher-dimensional lattices sites with periodic boundary conditions 9 / 35

32 Continuous Time Partition Function Distribution of interval lengths Even/odd decomposition apparent (remnant of staggered fermions), odd intervals favored, λ = 6/4 found to be confirmed from fit to set of smallest t/at Nτ= /at # intervals t / at 20 / 35

33 Continuous Time Algorithm Algorithms for SC-LQCD Algorithms for SC-LQCD First: Monte Carlo with MDP Algorithm (Karsch & Mütter Nucl. Physs B3 (989)) trades dimers for monomer pairs, polymers for baryon loops suffers from critical slowing down (local update) 2 / 35

34 Continuous Time Algorithm Algorithms for SC-LQCD Algorithms for SC-LQCD First: Monte Carlo with MDP Algorithm (Karsch & Mütter Nucl. Physs B3 (989)) trades dimers for monomer pairs, polymers for baryon loops suffers from critical slowing down (local update) Better: Worm Algorithm! invented by Prokof ev & Svistunov, Phys. Lett. A238 (998) applied to Ising model: rewrite in terms of P partition Q function Pbonds +Jσ σ Vd Z = {σx } hxy i e x y which satisfy constraint = (cosh J) {bxy =0,} Σy bxy = 0 mod 2 (tanh J)Σbxy x worm algorithm enlarges space of sampled configurations by relaxing constraint adapted for SC-QCD by Adams & Chandrasekharan, Nucl. Phys. B662 (2003) 2 / 35

35 Continuous Time Algorithm Algorithms for SC-LQCD Algorithmic Details () Idea of worm algorithm: Put a worm s head and tail at a lattice site as source and sink for monomers and move the worm s head through the lattice until it closes again sample monomer 2-point correlation function during Worm evolution:,x2 )) G(x, x2 ) = Nc O(G(x KCP obtain monomer (chiral) susceptibility χ = V P x,x2 G(x, x2 ) worm update is a guided random walk sample closed path configurations on which to measure other observables, global update 22 / 35

36 Continuous Time Algorithm Algorithms for SC-LQCD Algorithmic Details () Idea of worm algorithm: Put a worm s head and tail at a lattice site as source and sink for monomers and move the worm s head through the lattice until it closes again sample monomer 2-point correlation function during Worm evolution:,x2 )) G(x, x2 ) = Nc O(G(x KCP obtain monomer (chiral) susceptibility χ = V P x,x2 G(x, x2 ) worm update is a guided random walk sample closed path configurations on which to measure other observables, global update Worm for SC-LQCD: two version on the market: diffusion type algorithm (not suited for continuous time) directed path algorithm (well suited for continuous time), no backtracking 22 / 35

37 Continuous Time Algorithm Algorithms for SC-LQCD Algorithmic Details (2) Decompose the partition function and the lattice into active sites and passive sites: Worm tail and all sites of same parity (checkerboard) are active, those with opposite parity passive. Updates proceed in three parts: 2 3 T MDP (2 monomers dimer) Mesonic Worm (move dimers around) Baryonic Worm (undirected "polymers" directed baryon loops, change contour) 23 / 35

38 Continuous Time Algorithm Algorithms for SC-LQCD Algorithmic Details (2) Restrict here to Mesonic Worm: choose tail position x with probability Px = n(x )/Nc 2 (Active Site) choose a dimer touching x to be removed with probability Pµ = kµ /Nc 3 (Passive Site) choose a direction ρ touching site y = x + µ in which to add a dimer, with heatbath W probability Pρ = WD ρ (y ) with Wρ = γ 2δρ,±0, WD = 4 P ρ Wρ close the worm with probability nx by removing a P = Nc k ρ monomer, else continue with step (2) T H 23 / 35

39 Continuous Time Algorithm Algorithms for SC-LQCD Algorithmic Details (2) Restrict here to Mesonic Worm: choose tail position x with probability Px = n(x )/Nc 2 (Active Site) choose a dimer touching x to be removed with probability Pµ = kµ /Nc 3 (Passive Site) choose a direction ρ touching site y = x + µ in which to add a dimer, with heatbath W probability Pρ = WD ρ (y ) with Wρ = γ 2δρ,±0, WD = 4 P ρ Wρ close the worm with probability nx by removing a P = Nc k ρ monomer, else continue with step (2) T H 23 / 35

40 Continuous Time Algorithm Algorithms for SC-LQCD Algorithmic Details (2) Restrict here to Mesonic Worm for U(3): choose tail position x with probability Px = n(x )/Nc 2 (Active Site) choose a dimer touching x to be removed with probability Pµ = kµ /Nc 3 (Passive Site) choose a direction ρ touching site y = x + µ in which to add a dimer, with heatbath W probability Pρ = WD ρ (y ) with Wρ = γ 2δρ,±0, WD = 4 P ρ Wρ close the worm with probability nx by removing a P = Nc k ρ monomer, else continue with step (2) T H 23 / 35

41 Continuous Time Algorithm Algorithms for SC-LQCD Algorithmic Details (3) Continuous version deviates in several aspects: CT algorithm proposed here is derived from directed path Worm algorithm (Adams & Chandrasekharan, 2003) emission/absorption event decomposition instead of active/passive site decomposition idea of continuous time Worm in SC-QCD: use the probability distribution e λdt for spatial dimer emission after time interval dt absorption events ps = k t p t = k t k t p t = k t k t k t p t = k t k t k t a a remove spatial dimer choose temporal direction emission events p t =0.5 p t =e 2d / 4 p t =0.5 e e ps = e 2d / 4 choose temporal direction emit spatial dimer incoming direction outgoing direction solid/dashed lines 24 / 35

42 Continuous Time Algorithm Algorithms for SC-LQCD Generalization to arbitrary U(Ncc) baryons only become static for SU(Nc ) with Nc 3 Nc = 2: diquark loops introduce sign problem U() U(2) U(3) U(4) 25 / 35

43 Continuous Time Algorithm Algorithms for SC-LQCD Comparison with analytic results for U() We solved the U() system on 2x Nτ latties analytically: w (k) = 0.45 k β 2 /cosh(β), k even U() at=β w(nhop) = /Nhop! βnhop 2k k! () Continuous Time 30 40Monte Carlo 50 for SC-QCD / 35

44 Continuous Time Algorithm Algorithms for SC-LQCD Comparison with analytic results for U() We solved the U() system on 2x Nτ latties analytically: Nτ χ(nτ, γ ) = tanh acsch(γ 2 ) χ(t ) = p + γ 4 Nτ + tanh acsch(γ 2 ) 2! tanh (/2T ) ( + tanh (/2T )) 2.2. γ Nτ cont. time χ Nτ at Monte Carlo results on chiral susceptibility in perfect agreement with analytic results 27 / 35

45 Continuous Time Algorithm Algorithms for SC-LQCD The U(3) chiral transition temperature Determination of the critical temperature with O(2) critical exponents in chiral limit χ has no peak determine atc via data collaps rescaled chiral susceptibility, intersections between different volumes: Nτ=6 0 Nσ.895 Tc Tc+m Nσ χ/nσγ/ν /ν min Nσ Tc(4,8)=.89358(9) Tc(8,6)=.8865(0) Tc(6,32)=.88550(6) Tc(32,64)=.88549(6) at 28 / 35

46 Continuous Time Algorithm Algorithms for SC-LQCD The U(3) chiral transition temperature Determination of the critical temperature with O(2) critical exponents in chiral limit χ has no peak determine atc via data collaps rescaled chiral susceptibility, intersections between different volumes: Nσ χ/nσγ/ν Tc(4,8)=.8933() Tc(8,6)=.8854() Tc(6,32)=.8846() Tc(32,64)=.8847() at 28 / 35

47 Continuous Time Algorithm Algorithms for SC-LQCD Continuum Limit of U(3) transition temperature Precise determination of the extrapolation, including Nτ = 64: fit ansatz for pseudo-critical temperatures: Tpc (Nτ ) = Tc +a/nτ +b/nτ 2 also exhibits non-monotonic behavior, a > 0, b < 0: continuous time results 29 / 35

48 Results on the SU(3) phase diagram SC-QCD Phase Diagram () Comparison of SU(3) phase diagram with mean field predictions (Y. Nishida, 2004); similar qualitative behavior re-entrance seen, but in contrast to mean field below TCP atc 0.8 Continuous Time: 2nd order tricritical point st order Mean Field: 2nd order tricritical point st order aµ / 35

49 Results on the SU(3) phase diagram SC-QCD Phase Diagram (2) Comparison of phase diagram with Nτ = 4 data (M. Fromm, 200): CT-data compared with Nτ = 4 data for identification aµ = γ 2 at µ behavior at low µ agrees well, location of TCP agrees within errors re-entrance region is seen at higher values of µc.4.2 atc 0.8 Continuous Time: 2nd order tricritical point st order Discrete Time: 2nd order tricritical point st order aµ / 35

50 Results on the SU(3) phase diagram SC-QCD with Imaginary chemical potential CT-simulations with imaginary chemical potenital iµ: straight forward for 3iµ/T π/2, resummation needed for 3iµ/T > π/2 at 3iµ/T = π/2: recover Tc =.884 of U(3) transition at 3iµ/T = π: Roberge Weiss transition, (aµc, atc ) = (3.38(2), 3.23(3)) unexpected (compared to QCD): coefficients of polyn. fit have opposite signs atc fit result in interval [-0.25,0]: c=-0.73(7), c2=.68(35) 0.5 Tc(µc)=Tc(0)(+c (µ/t)2+ c2(µ/t)4) (3µc/ πtc) / 35

51 Results on the SU(3) phase diagram Conclusions Main messages: first time a QFT is formulated with continuous time, preserving gauge invariance systematic errors/ambiguities for a/at = f (γ 2 ) eliminated bonus: no sign problem Prospects: unequal time correlations: analytic continuation well posed CT partition function: new formulation for analytic treatment phase diagram in the (ρ, T ) plane include (small) monomer densities to obtain results away from the chiral limit 33 / 35

52 Results on the SU(3) phase diagram Conclusions Main messages: first time a QFT is formulated with continuous time, preserving gauge invariance systematic errors/ambiguities for a/at = f (γ 2 ) eliminated bonus: no sign problem Prospects: unequal time correlations: analytic continuation well posed CT partition function: new formulation for analytic treatment phase diagram in the (ρ, T ) plane include (small) monomer densities to obtain results away from the chiral limit Thank you for your attention! 33 / 35

53 Results on the SU(3) phase diagram Appendix Why Study Strong Coupling QCD on the Lattice? Possible relation to (L)QCD phase diagram =, N =4 nd 2 2nd order order st order st order /m B /m B tastes): =, N =2 =, N =2 staggered T /m B T /m B =, N =4 (four 34 / 35