Spectral sums for Dirac operator and Polyakov loops A. Wipf Theoretisch-Physikalisches Institut, FSU Jena with Franziska Synatschke, Kurt Langfeld Christian Wozar Phys. Rev. D75 (2007) 114003 and arxiv:0803.0271 [hep-lat] St.Goar, 17th March 2009 Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 1 / 31
1 Spectral Sums vs. Polyakov Loop on the Lattice 2 Spectral sums for the continuum theory 3 Numerical investigations for SU(2) 4 Conclusions Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 2 / 31
related results by: Bilgici, Bruckmann, Gattringer, Hagen, Soldner,... cp. Christof Gattringer talk cp. poster Synatschke and Wozar Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 3 / 31
Center transformations, twisting multiply all U 0 (x 0, Ü) in one time-slice with center elements z {U µ (x)} { z U µ (x)} twisted configuration on V = N τ N 3 s C xx winds n-times around periodic time direction: x zu 0 (x) x 0 W Cxx z n W Cxx Polyakov loop: cp Ü zp Ü Yang-Mills action invariant Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 4 / 31
6000 4000 2000 0 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 Im 0.5 Re histogram of P = trp Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 5 / 31
Spectral sums for Dirac operator D with nearest neighbour interaction: Wilson, staggered,... γ 5 -hermiticity non-real eigenvalues come in pairs λ p,λ p spectral problem Dψ p,l = λ p ψ p,l, ψ p,l normalized matrix functions f (D), spectral resolution in position basis x tr c,s f (D) x = n D p=1 f (λ p ) p(x), p(x) spectral space-time density p(x) = l ψ p,l (x) 2 p(ü) = x 0 p(x 0, Ü), p x p(x) = deg(λ p ) Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 6 / 31
twisting: U z U = D z D, λ p z λ p, p(x) z p(x) spectral problems for twisted z D center averaged sums S f (x) = zk zk p(x)f ( z k λ p ) k p S f = S f (x) = zk Tr(z k D) x k Gattringer formula for choice f (λ) = λ Nτ : S(Ü) = k S = k z k zk p(ü) ( ) z Nτ k λ p = κ P(Ü) p z k Tr (z k D) Nτ = κl any {U µ (x)} L spatial average of P(Ü) = trp(ü) Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 7 / 31
Polyakov loop spectral data of D Confinement chiral symmetry breaking but: S(x) dominated by large eigenvalues: partial sum S n = k z k n ( zk ) Nτ λ p (assume p = 1) p related problem: sick continuum limit N τ generalized partial sums S f,n (x) = k z k n p=1 zk p(x)f ( z k λ p ) n nd Sf (x) space-averages S f,n Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 8 / 31
Propagator spectral sums f (λ) = λ s = Σ ( s) Zeta-function, Σ Nt = S 0 10 3-1 -2-3 Σ( 1),rot -4-5 -6-7 0 0.2 0.4 Lrot 0.6 0.8 1 The expectation value of Σ 1,rot as function of L rot Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 9 / 31
Gaussian spectral sums f (λ) = e λ2 = G heat kernel 0.5 10 3 0.0-0.5-1.0 Grot -1.5-2.0-2.5-3.0 0 0.2 0.4 0.6 0.8 1 Lrot The expectation value of G rot as function of L rot Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 10 / 31
partial Gaussian sums G n converge quickly G, lowest 3% sufficient 0.0 10 3-0.5-1.0 Grot n 5.200 5.475 5.615 6.000-1.5-2.0-2.5 0 10 20 30 40 50 60 70 80 90 100 % of lowest eigenmodes small 4 3 3 lattice, n D = 2304, β c = 5.49 Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 11 / 31
Spectral sums for the continuum theory Euclidean theory on 4d torus T 4, volume V = β L 3 (anti)periodic fields in time direction with period β = 1/k B T periodic (mod gauge transformations) in spatial directions Dirac operator D = iγ µ D µ + im, D µ = µ ia µ, A µ = A a µ λ a spectral problem Dψ p,l = λ p ψ p,l, ψ p,l (x) normalized on torus gauge transformation g A µ = g(a µ + i µ )g 1, g ψ(x) = g(x)ψ(x) Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 12 / 31
g periodic invariant λ p and spectral density p(x) = l ψ p,l (x) 2, p(ü) = β 0 dx 0 (x 0, Ü) center transformation = transformation with non-periodic g(x): g(x 0 + β, Ü) = z g(x 0, Ü), z Z g A µ still periodic, but g ψ(x 0 + β, Ü) = z g ψ(x 0, Ü) twisted Diracoperator z D A Dg A = gd A g 1 g ψ not eigenfunctions of z D for center transformation spectrum changes if z ½ twisting: A g A = D z D, λ p z λ p, p(x) z p(x) Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 13 / 31
spectral sums in continuum: weighted center average of x f (D) x n S f (x) = lim S f,n(x), S f,n (x) = z zk p(x)f n k ( z k λ p ) p=1 k S f = d d x S f (x) = zk Trf (z k D) k all spectral sums are order parameters for center symmetry S f = k z k Trf (z k D) A g A k zk Trf (zkz D) z kz=z k = z k z k Trf ( z k D ) = zs f similarly S f (x) z S f (x) Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 14 / 31
Spectral sums and Polyakov loops S f (x) gauge invariant function of W Cx L β neglect loops winding around the spatial directions S f (x) transforms under twists as dressed Polyakov loops P CÜ S f (Ü) = C Ü P CÜ F CÜ (W CÜ ) F CÜ gauge- and center invariant center U(1) contractable W CÜ and combinations PC Ü P C Ü. on lattice or t Hooft s constant field strength configurations on torus: S f (Ü) L β const P(Ü) lattice: hopping parameter expansion; continuum: explicit calulation Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 15 / 31
Constant field strength configurations Here: Schwinger model for SU(2) on T 4 see arxiv:0803.0271 [hep-lat] boundary conditions ψ(x 0 + β,x 1 ) = ψ(x 0,x 1 ), ψ(x 0,x 1 + L) = e iγ(x) ψ(x 0,x 1 ) γ = 2πq x 0 /β, q instanton number minimal action instanton configuration (A 1 = 0) A 0 = Bx 1 + 2π β h = F 01 B, P(Ü) = e 2πih iφ x 1/L U(1) twist with g = e 2πiαx 0/β = z = e 2πiα, g A 0 = Bx 1 + 2π β (h + α) Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 16 / 31
spectral problem for Dm=0 2 reduces to harmonic oscillator on circle elliptic functions groundstate: θ-function; excited states (y = x 1 + L q (l h 1 2 )) ψ p,l (x) e 2πilx 0/β s= H p ( B (y + sl) ) ξ 0 (y + sl)e 2πi sqx 0/β eigenvalues of D 2 twist-independent { 0 degeneracy: q µ p = 2pB degeneracy: 2q. center average p(x) = 1 0 dαe 2πiα α p(x) (z = e 2πiα ) Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 17 / 31
p(ü) proportional to P(Ü): p(ü) dx 0 p(x) = q L P(Ü) L p(πqτ)e πqτ/2, τ = β L general spectral sum for Schwinger-model instanton (L 1 0) S f (Ü) = q L P(Ü) f (µ p ) { L p (πqτ) + L p 1 (πqτ) } e πqτ/2 p=0 all spectral sums are proportional to P(Ü) convergence for which f? (cp. Banks-Casher f (µ) µ 1/2 ) speed of convercence depends on f Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 18 / 31
Gaussian spectral sum G for f (D) = exp(td 2 ): G t (Ü) = q L coth(tb) exp ( πqτ 2 coth(tb) ) P(Ü) large volumes: 2πq = BV = Φ fixed G t (Ü) β 2πt e β2 /4t P(Ü) small-t and large-t asymptotics βl tq G t (Ü) 1 q L tb e πqτ/(2tb) P(Ü) for t 0 1 L (q 2qe 2tB )e πqτ/2 P(Ü) for t asymptotic expansion for small t: all a n = 0 large t: zero-mode contribution + O(e µ 1t ) Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 19 / 31
zero-mode subtracted heat kernel G t(x) = zk zk p(x)e t zkµ = G t (Ü) 0(x) p:µ p>0 k exponentiell fall-off, vanishing asymptotic expansion Mellin transform exist for all s Σ ( 2s) (x) = 1 Γ(s) spectral sum for f (D) = 1/D 2 Σ ( 2) (Ü) = p,µ p>0 ( k z k 0 dt t s 1 G t(x) 1 ) zk p(ü) z kµp = β { γ + log(πqτ) e πqτ Γ(0,πqτ) } e πqτ/2 P(Ü) 4π Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 20 / 31
1 τ q = 0.01 0.8 Σ ( 2) n 0.6 0.4 0.2 τ q = 0.03 τ q = 0.1 0 0 100 200 300 400 500 600 n The partial sums Σ 2 n as function of n Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 21 / 31
On the convergence of spectral sums zero-mode subtracted heat kernel K (t,x) = K(t,x) 0(x), K(t,x) = x e td2 x = p e tµp p(x) large and small-t asymptotics K (t,x) e tµ1 1(x) for t K(t,x) t d/2{ N } a n (x)t n + O(t N+1 ) for t 0 n=0 a n gauge-invariant function of F µν and its covariant derivatives a n center-invariant, Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 22 / 31
Mellin transform ζ D (s,x) = x 1 (D 2 ) s x = 1 Γ(s) dt t s 1 K (t,x) d even: simple poles at s = d 2,...,1,residues a 0(x),...,a d 1(x) 2 a k center-invariant Σ ( 2s) (x) = 1 Γ(s) dt t s 1 G (t,x) = k z k ζ z k D (s,x) no poles in s spectral sums Σ 2s convergent s important: first sum over k and only afterwards over p! γ 5 symmetry for m = 0 S D 2s(x) = 1 2 ( 1 + ( 1) 2s ) S (D 2 ) s(x) Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 23 / 31
Numerical investigations for SU(2) static quark potential V (r) = T log C(r), C(r) = P(Ü)P(Ü + r 3 ) insert Gattringers result P(Ü) = S(Ü) cancellation of huge contributions (convergence proof) must include all eigenfunction use IR-dominated spectral sum, e.g. G n correlation measure ω n = P(Ü) Sf,n (Ü) Ü P 2 (Ü) Ü S 2 f,n (Ü) Ü average over Ü for fixed configuration Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 24 / 31
1 standard µ=1/3 µ=infinity 0.5 ω n 0-0.5 0 0.2 0.4 0.6 0.8 1 n/n p ω n for S n (Ü), G n (Ü) as function of n/n D, 6 4 lattice Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 25 / 31
use G n (Ü) instead of S n (Ü): measure ω n almost constant simulations: staggered fermions, SU(2) G n (x) := 1 8 n ( p(x)e λ2 p /µ2 z p(x)e z λ 2/µ2) p p=1 improved action (rotational invariance, scaling) S = β [ ] γ 1 P µν (x) + γ 2 P µν (2) (x) µ>ν,x β = 1.35, γ 1 =.0348, γ 2 = 0.10121, σa 2 = 0.1244(7) Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 26 / 31
compare V (r) with V G n (r) = T log C G n (r), C G n (r) = G n(ü)g n (Ü + r 3 ) simulation parameters β σa 2 lattice T/T c configurations 1.35 0.1244(7) 12 3 6 0.7 8658 1.35 0.1244(7) 12 3 4 1.0 12000 1.35 0.1244(7) 12 3 2 2.1 12000 Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 27 / 31
6 4 mode sum N t =6 Polyakov line N t =6 mode sum N t =2 Polyakov line N t =2 V(r) / T 2 0-2 0 1 2 3 4 5 6 7 r/a potential from P and G n, N t = 6 confined, N t = 2 deconfined Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 28 / 31
6 V(r) / T 4 2 mode sum N t =6 mode sum N t =4 Polyakov line N t =4 1/r fit 0-2 0 1 2 3 4 5 6 7 8 9 10 r/a potential from P and G n for T T c (N t = 4) Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 29 / 31
mode sum G n (Ü) in a 20 3 spatial hypercube. L = 3.2 fm smooth texture at scale 0.3 fm Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 30 / 31
Conclusions, remarks all spectral sums define order parameters for center symmetry spectral sum approach can be formulated for continuum theories first sum over center elements and afterwards over the eigenvalues spectral sums exist for almost all f (D) reconstructed Polyakov loop locally from spectral sums on lattice same construction for continuum theory and constant F µν beyond constant field strength? relation to Banks-Casher in continuum (CSB confinement) Andreas Wipf (FSU Jena) Spectral sums for Dirac operator and Polyakov loops 31 / 31