Instanton constituents in sigma models and Yang-Mills theory at finite temperature

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1 Instanton constituents in sigma models and Yang-Mills theory at finite temperature Falk Bruckmann Univ. of Regensburg Extreme QCD, North Carolina State, July 8 PRL (8) 56 [77.775] EPJ Spec.Top.5 (7) 6-88 [76.69] partly with: D. Nogradi, P. van Baal, E.-M. Ilgenfritz, B. Martemyanov, M. Müller-Preussker Falk Bruckmann Instanton constituents / 9

2 Part I: The O() sigma model a scalar field in D... S = d x ( µφ a ) a =,, : global O() symmetry... with a constraint φ a φ a = (circumvent Derrick s theorem) nontrivial properties: asymptotic freedom dynamical mass gap topology and instantons condensed matter physics and toy model for gauge theories Falk Bruckmann Instanton constituents / 9

3 Topology finite action: r : φ a const. as a mapping: φ : R { } S x S c winding number/degree: all such φ s are characterized by an integer Q = how often S c is wrapped by S x through φ here: Q = 8π d x ɛ µν ɛ abc φ a µ φ b ν φ c Z topological quantum number = invariant under small deformations of φ Falk Bruckmann Instanton constituents / 9

4 Classical solutions Bogomolnyi trick... ( µ φ a ± ɛ µν ɛ abc φ b ν φ c ) = ( µ φ a ) ± ɛ µν ɛ abc φ a µ φ b ν φ c + ( µ φ a )... and bound (integrated): S 4π Q where the equality holds iff µ φ a = ɛ µν ɛ abc φ b ν φ c selfduality equations first order (instead of second order in eqns. of motion) classical solutions: instantons = localised in both directions Falk Bruckmann Instanton constituents 4 / 9

5 Complex structure introduce complex coordinates both in space and color space: x, z = x + ix φ a u = φ + iφ N : φ a = (,, ) u = φ S : φ a = (,, ) u = self-duality equations become Cauchy-Riemann conditions on u any meromorphic function u(z) is a solution topological charge: Q = number of zeroes or poles topological charge density: q(x) = π ( + u ) u z Falk Bruckmann Instanton constituents 5 / 9

6 Charge instantons simplest functions: u(z) = λ z z u(z) = z z λ q(x) = π λ ( z z +λ ) are Q = instantons: location z, size λ Belavin-Polyakov monopole pole and zero to cover S c, one of them at infinity both, pole and zero, at finite z: u(z) = z z I z z II constituents at z = {z I, z II }? instanton quarks?! NO! same profile q(x) as above one lump conjecture: complex moduli per Q locations of constituents?! Falk Bruckmann Instanton constituents 6 / 9

7 Finite temperature = one compact direction, say: Im z = x x + β, β = /k B T instantons: use that higher charge solutions = products Q λ u(z) = z z,k Q poles k= and infinitely many copies: z,k z + k iβ, k Z a regularized u(z) is: λ u(z) = exp((z z ) π β ) small λ has residues λ at z = z + k iβ and charge over S R large λ Falk Bruckmann Instanton constituents 7 / 9

8 Boundary conditions q(x) and action density invariant under global SO() rotations rotation an SO() subgroup: φ with ω φ, u e πiω u let the fields φ and u be periodic up to that SO() subgroup: FB 7 q(x) strictly periodic novel solution: u(z + iβ) = e πiω u(z) ω [, ] u(z) = e ω(z z ) π β λ exp((z z ) π β ) has residues e πiωk λ at z = z + k iβ different orientation of the instanton copies nontrivial overlaps instanton constituents Falk Bruckmann Instanton constituents 8 / 9

9 Topological profiles ln q(x): (z =, cut off below e 5 ) λ = β λ = β λ = β - - periodic ω = / antiper for large size λ: lumps with action ω and ω = ω Falk Bruckmann Instanton constituents 9 / 9

10 Dissociation rewrite: u(z) = exp( ω(z z ) π β ) exp( ω(z z ) π β ) locations: z = z β ln λ πω, z = z + β ln λ π ω instanton size constituent distance: z z ln λ constituent size: fixed by β and ω really locations of topological lumps? YES: corrections of the second term at z = z are exp. small individual constituent: top. charge: q(x) = u(z) = exp(ωz π β ) ω analogous πω β cosh (ω Re z π β ) (static) Q = ω Falk Bruckmann Instanton constituents / 9

11 possible values for Q:,,... ω, + ω,... ω, ω,... asympt. φ φ + : N N, S S N S S N constituents alternate why instanton quarks not visible for zero temperature, i.e. on R? β : constituents large and overlap! no other scale competing with their distance generalisations: CP(N) models FB et al. in progress N constituents as expected realisation in condensed matter: cylinder of..?.. with quasi-periodic bc.s Falk Bruckmann Instanton constituents / 9

12 Fermionic zero modes gauge field description couple fermions zero modes phase-bc.s ψ(x + iβ) = e πiζ ψ(x ), evolution with ζ: Falk Bruckmann Instanton constituents FB et al. in progress / 9

13 Part II: Gauge theories pure Yang-Mills theory in (Euclidean) 4D: S = tr F µν Q = tr F µν F µν dual field strength F a µν = ɛ µνρσf a ρσ ( E a B a ) integer Q: instanton number/topological charge topology: A µ r iω µ Ω... pure gauge Q = deg(ω : S r SU(N))... winding number Falk Bruckmann Instanton constituents / 9

14 Instantons (anti)selfdual: F a ρσ = ± F a µν first order, nonlinear charge : axially symmetric ansatz and solution BPST A a µ = ηµν a x ν x + ρ trf = ρ 4 (x + ρ ) 4 η a µν {,, } size ρ localized in space and time algebraic decay, similar to O() instantons on R instanton liquid model from semiclassical path integral chiral symmetry breaking axial anomaly topological susceptibility confinement? Shuryak Falk Bruckmann Instanton constituents 4 / 9

15 Finite temperature: Calorons use higher charge solutions of same color orientation CFTW first calorons Harrington-Shepard 78 most general calorons: need ADHM formalism and Nahm transform calorons of nontrivial holonomy Kraan,van Baal; Lee, Lu 98 space-space plot of action density for SU(), intermediate holonomy lumps, almost static N c for gauge group SU(N c ), like quarks in baryons magnetic monopoles of opposite magnetic charge in fact dyons with same electric as magn. charge (selfdual) Falk Bruckmann Instanton constituents 5 / 9

16 Role of the holonomy relative gauge orientation of instanton copies in the ADHM constr. A µ periodic up to a gauge transformation e πiωσ / (cf. O()) gauge theory: compensated by time-dependent transf. e πiωσ x / introduces an asymptotic gauge field A asymptotic Polyakov loop = holonomy ( P( x) P exp i environment β dx A ) e πiωσ / P acts like a Higgs field, in the group: vev ω, direction σ monopoles have masses ω/β and ω/β, ω = ω A a= µ : power law decay (massless photon ), A a=, µ : exponential decay (massive W -bosons ) Falk Bruckmann Instanton constituents 6 / 9

17 Polyakov loop in the bulk: P( x) = ± at the monopoles Higgs field vanishes = false vacuum necessary for top. reasons Ford et al.; Reinhardt; Jahn et al. index theorem valid localisation depending on bc.s: Nye, Singer ψ(x + iβ) = e πiζ ψ(x ) (A µ still periodic) ζ { ω, ω } incl. periodic: localised at monopole ζ rest incl. antiperiodic: localised at antimonopole Garcia Perez et al. a zero in their profiles at the other monopole, topological FB calorons can be studied on the lattice by cooling Ilgenfritz et al., FB et al. physical relevance of P : conjecture: holonomy tr P deconfinement order param. trp x Falk Bruckmann Instanton constituents 7 / 9

18 Calorons and the dynamics of YM theories eff. potential at -loop: triv. holonomy favored! overruled by caloron gas contribution: Gross, Pisarski, Jaffe; Weiss Diakonov et al. minima at P = ± become unstable for low enough temperature onset of confinement gas of calorons and anticalorons put on the lattice: linearly rising interquark potential just for nontrivial holonomy! Gerhold et al. confinement from a gas of purely selfdual dyons unphysical (top. charge builds up) nevertheless interesting physical effects Diakonov, Petrov Falk Bruckmann Instanton constituents 8 / 9

19 Summary sigma models in D and YM in 4D admit instantons instanton constituents for S R and S R = finite T with fractional charges, say ω and ω in the lowest models when in compact direction periodic up to a subgroup, say e πiω.. = subgroup of global and local symmetry Yang-Mills theory: can be made periodic holonomy P caloron constituents as building blocks of semiclass. models at finite temperature: (de)confinement?! sigma models: quasi-periodic bc.s stay spin chains? skyrmion lattices? Quantum Hall effect? Falk Bruckmann Instanton constituents 9 / 9