and Theoretisch-Physikalisches Institut, FSU Jena with Leander Dittmann, Tobias Kästner, Christian Wozar (Jena) Thomas Heinzl (Plymouth), Balázs Pozsgay (Budapest) and O Raifeartaigh Symposium Budapest 22. 24. June 2006
and
2π/q Ising models: θ x {2πk/q}, 1 k q H = J xy cos (θ x θ y ) Z q : θ x θ x + 2πn/q ferromagnetic phase: q ground states phase transition symmetric ferromagnetic d = 2 : second order q 4, first order q > 4 d = 3 : second order q 2, first order q > 2 anti-ferromagnetic phase: rich vacuum structures symmetric antiferrom: d = 3,q = 3 : second order entropy of ground states? and
entropy S B (p) = p(w)log p(w) free energy βf = inf p (β H ρ S B ) p Gibbs e βh variational characterization of (convex) effective action: ( Γ[m] = inf β H p S(p) ) e iθ(x) p = m(x) p mean : ρ(w) = x ρ x (θ x ) Γ MF [m] translational invariance: p x = p m(x) = m effective potential: Γ MF [m] = V u MF (m) ( u MF (m) = inf Kmm + ) p(θ) log p(θ) p θ and m = θ p(θ)e iθ, K = dj.
antiferromagnetic phase: translational invariance on sublattices Λ = Λ 1 Λ 2 two neighbours in different sublattices p(x) = p i m(x) = m i for x Λ i ( ) u MF (m 1,m 2 ) = 1 2 K m 1 m 2 2 + i K > K f,c > 0 m 1 = m 2 0, Z q -broken K < K a,c < 0 m 1 m 2 0, Z 2q -broken u MF (m i ), and Λ symmetric Λ ferromagn. Λ 1 Λ 2 antiferromagnetic
finite temperature gluodynamics order parameter for confinement: Polyakov loop effective action: ( ) N t e Seff[P] = DUδ P Ü, U t,ü;0 e Sw[U] 3z L 3z 3 t=0 gauge invariance: S eff = S eff [L], L Ü = TrP Ü global Z 3 center symmetry: S eff [L] = S eff [z L] and 3z 2 good ansatz for S eff?
strong coupling expansion for S eff [P] Z 3 -invariant character expansion nearest neighbour interaction S eff = λ 10 S 10 + λ 21 S 21 + λ 20 S 20 + λ 11 S 11 +... S 10 = (χ 10 (P Ü )χ 01 (P Ý ) + h.c), S 21 =... center-transformation: χ pq (zp) = z p q χ pq (P), z 3 = z z = 1 With L = TrP: leading terms S eff = (λ 10 λ 21 ) ( LÜ L Ý + h.c. ) + λ 21 ( L 2 Ü L Ý + L 2 ÝL Ü + h.c. ) and complex field with compact target space, (reduced Haar measures), close relation to 3-state Potts model
Pott-Models naive reduction to Potts: P Ü e iθü ½ centre S eff H with J = 18(λ 01 + 4λ 21 ) true for all S eff S eff is extension of Z 3 model. Conjecture (Svetitsky, Yaffe): effective finite-temperature SU(N)-gluodynamics in d dimensions = Z N spin model in d 1 dimensions. same critical exponents SU(2) and Ising (Engels et.al) same universality class (symmetric ferrom.) β/ν γ/ν ν 4d SU(2) 45 1.93 0.65 3d Ising 16 1.965 0.63 and
relevance for finite temperature SU(N) with N > 2? transition first order! phase diagrams classical analysis: minimize S eff λ 21 0.2 0.2 0.6 ferromagnetic anticenter symmetric antiferromagnetic 1.0 4 3 2 1 0 λ 10 1 and quantum fluctuations include symmetric phase new ferromagnetic anti-center phase qualitatively correct phase diagram
variational characterisation of Γ: fix χ j (P Ü ) for all χ j in S eff mean product measure and DP Ü dµ red (P Ü )p Ü (P Ü ) translational invariance on sublattices in Λ = Λ 1 Λ 2 nontrivial variational problem on two-sites most simple effective model (Polonyi) S eff = λs 10 = λ ( LÜ L Ý + h.c) Lagrangean multiplier for L i on Λ i
mean field effective potential for minimal model 2u MF (L 1,L 1,L 2,L 2) = dλ L 1 L 2 2 + v MF (L i,l i ) v MF (L,L ) = dλ L 2 + γ 0 (L,L ) γ 0 Legendre-transform of w 0 (j,j ) = log dµ red exp(jl + j L ) order parameters: and L = 1 2 (L 1 + L 2 ), M = 1 2 (L 1 L 2 ), l = L, m = M. group integral in closed form not known for SU(3)!! exp(j Tr(U)) =hypergeometric function
1.6 1.4 1.2 1.0 l 0.8 λ 10,c = 0.13433(1) l = 1.46(2) and 0.6 0.4 0.2 0.1350 0.1345 0.1340 0.1335 1.6 1.4 1.2 MF λ10 λ 10,c = 0.13721(5) l = 1.33(2) 0.25 0.20 1.0 l 0.8 0.15 ρ(l) 0.6 0.4 0.2 multicanonical MC 0.10 5 0.1378 0.1374 0.1370 λ 10
0.16 0.14 0.12 λ 10,c = 1.66667(1) and 0.10 m 8 6 4 2 0 0.1658 0.1662 0.1666 0.1670 0.1674 0.8 0.7 0.6 m 0.4 0.3 0.2 0.1 MF λ 10,c = 1.19875(5) MC 28 3, 5 10 5 λ10 0.188 0.192 0.196 0.200 0.204 λ 10 0.20 0.18 0.16 0.14 0.12 0.10 ρ(m) 8 6 4 2
Why is mean field so good? conjecture: 3 = upper crit. dimension for 3-state potts critical exponents of S AF: exponent 3-state Potts minimal S eff ν 0.664(4) 0.68(2) γ/ν 1.973(9) 1.96(2) critical exponents in mean field? finite temperature gluodynamics effective Z 3 models with compact target spaces 3-state Potts-model and universality test in unphysical region (for gluodynamics)
MC-Simulations phase diagram and transitions histograms large statistics, expensive fast algorithms! standard Metropolis: 5% to 10% accuracy multicanonical algorithm: up to 20 3 lattices near first order transitions new cluster algorithm near second order transitions: auto-correlation times down by two orders of magnitude on larger lattices comparison with mean field results for two-coupling (costy). rich phase structure: 4 different phases, second und first order transitions, tricritical points(?), mean field very good. and
0.20 0.15 0.10 MF ferrom. 2.5 2.0 1.5 and 5 λλ21 21 0 5 symmetric lr 1.0 0.10 0.15 0.20 0.15 AC 0.2 0.1 0.1 0.2 0.3 MC ferrom. λ10 antiferrom. 2.5 l r 2.5 0.10 5 λλ21 21 0 symmetric 2.0 1.5 1.0 lr 2.0 1.5 5 0.10 0.15 AC antiferrom..2.1 0.1 0.2 0.3 0.2 0.1 0.1 0.2 0.3 λ10 01
0.20 0.15 1. order 2. order and 0.10 5 λ 21 0 5 0.10 0.15 F 1 4 S 2 3 AF AC 0.2 0.1 0.1 0.2 0.3 λ 10 jenlatt, Linux cluster, 8000 MC simulations, 3000 CPUh
λ 10 = 0.13958 λ 21 = 020833 λ 10 = 0.13971 λ 21 = 019583 λ 10 = 0.13983 λ 21 = 018333 and 1 1 1 0 0 0 1 1 1 1 0 1 1 0 1 1 0 1 λ 10 = 0.13996 λ 21 = 017083 λ 10 = 0.14008 λ 21 = 015833 λ 10 = 0.14021 λ 21 = 014583 1 1 1 0 0 0 1 1 1 1 0 1 1 0 1 1: Histogramm of L, S FM, 1st 1 0 1
λ 10 = 0.18333 λ 21 = 62500 λ 10 = 0.18417 λ 21 = 62083 λ 10 = 0.18542 λ 21 = 61458 and λ 10 = 0.18667 λ 21 = 60833 λ 10 = 0.18750 λ 21 = 60417 λ 10 = 0.18958 λ 21 = 59375 2: Histogramm of L, S FM, 2nd
λ 10 = 0.18250 λ 21 = 65833 λ 10 = 0.18333 λ 21 = 66667 λ 10 = 0.18458 λ 21 = 67917 and λ 10 = 0.18583 λ 21 = 69167 λ 10 = 0.18667 λ 21 = 70000 λ 10 = 0.18875 λ 21 = 72083 3: Histogramm of L, S AC, 2nd
λ 10 = 0.19000 λ 21 = 0 λ 10 = 0.19125 λ 21 = 0 λ 10 = 0.19312 λ 21 = 0 and λ 10 = 0.19500 λ 21 = 0 λ 10 = 0.19625 λ 21 = 0 λ 10 = 0.19937 λ 21 = 0 4: Histogramm of M, S AF, 2nd
strong coupling for Polykaov-loops effective action modified MF for non-translationally invariant states new efficient cluster algorithm! rich phase structure for simple Z 3 -models mean field unexpectedly accurate (d c = 3?) calculate λ j (β) via IMC (cp. SU(2)) efficient group-theoretic Schwinger-Dyson equations! vacuum-sector of AF phase? SU(N) group integrals? is AC-phase relevant for gluodynamics? include fermions in effective Polyakov-loop dynamics. and JHEP 06 (2004) 005, PRD 72 (2005) 065005, hep-lat/0605012