Instanton constituents in sigma modelsand Yang-Mills theory at finite temperature

Falk BruckmannUniv. of Regensburg

Extreme QCD, North Carolina State, July 2008

PRL 100 (2008) 051602 [0707.0775]EPJ Spec.Top.152 (2007) 61-88 [0706.2269]

partly with: D. Nogradi, P. van Baal, E.-M. Ilgenfritz,B. Martemyanov, M. Müller-Preussker

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Part I: The O(3) sigma model

a scalar field in 2D . . .

S =

∫d2x

12(∂µφ

a)2 a = 1,2,3 : global O(3) symmetry

. . . with a constraint

φaφa = 1 (circumvent Derrick’s theorem)

nontrivial properties:

asymptotic freedomdynamical mass gaptopology and instantons

condensed matter physics and toy model for gauge theories

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Topology

finite action:

r →∞ : φa → const.

as a mapping:φ : R2 ∪ ∞ ' S2

x −→ S2c

winding number/degree: all such φ’s are characterized by an integer Q= how often S2

c is wrapped by S2x through φ

here:Q =

18π

∫d2x εµνεabcφ

a∂µφb∂νφ

c ∈ Z

topological quantum number = invariant under small deformations of φ

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Classical solutions

Bogomolnyi trick. . .

(∂µφa ± εµνεabcφ

b∂νφc)2 = (∂µφ

a)2 ± 2εµνεabcφa∂µφ

b∂νφc + (∂µφ

a)2

. . . and bound (integrated):

S ≥ 4π|Q|

where the equality holds iff

∂µφa = ∓ εµνεabcφ

b∂νφc ‘selfduality equations’

first order (instead of second order in eqns. of motion)

classical solutions:instantons = localised in both directions

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Complex structure

introduce complex coordinates both in space and color space:

x1,2 → z = x1 + ix2

φa → u =φ1 + iφ2

1− φ3N : φa = (0,0,1) u = ∞S : φa = (0,0,−1) u = 0

⇒ 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) =1π

1(1 + |u|2)2

∣∣∣∣∂u∂z

∣∣∣∣2

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Charge 1 instantons• simplest functions:

u(z) = λz−z0

u(z) = z−z0λ

q(x) = 1π

λ2

(|z−z0|2+λ2)2

Belavin-Polyakov monopoleare Q = 1 instantons: location z0, size λ

1 pole and 1 zero to cover S2c , one of them at infinity

• both, pole and zero, at finite z:

u(z) =z − zI

z − zII

constituents at z = zI , zII? ; ‘instanton quarks’?!

NO! same profile q(x) as above ⇒ one lump

conjecture: 2 complex moduli per Q ; locations of 2 constituents?!Falk Bruckmann Instanton constituents 6 / 19

Finite temperature= one compact direction, say: Im z = x2 ∼ x2 + β, β = 1/kBT• instantons:

use that higher charge solutions = products

u(z) =Q∏

k=1

λ

z − z0,kQ poles

and infinitely many copies: z0,k ≡ z0 + k · iβ, k ∈ Z

• a regularized u(z) is:

u(z) =λ

exp((z − z0)2πβ )− 1

has residues λ at z = z0 + k · iβand charge 1 over S1 · R1

small λ large λ

-4-3

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Boundary conditionsq(x) and action density invariant under global SO(3) rotations

an SO(2) subgroup: φ→

rotationwith ω

1

φ, u → e 2πiωu

• let the fields φ and u be periodic up to that SO(2) subgroup: FB ’07

u(z + iβ) = e 2πiωu(z) ω ∈ [0,1]

q(x) strictly periodic

• novel solution:

u(z) =e ω(z−z0)

2πβ · λ

exp((z − z0)2πβ )− 1

has residues e 2πiωkλ at z = z0 + k · iβ

⇒ ‘different orientation’ of the instanton copies⇒ nontrivial overlaps ⇒ instanton constituents

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Topological profilesln q(x): (z0 = 0, cut off below e−5)

λ = β λ = 10β λ = 100β

periodic

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ω = 1/3

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antiper.

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⇒ for large size λ: 2 lumps with action ω and ω = 1− ωFalk Bruckmann Instanton constituents 9 / 19

‘Dissociation’• rewrite:

u(z) =1

exp(−ω(z − z1)2πβ )− exp(ω(z − z2)

2πβ )

locations: z1 = z0 − β ln λ2πω , z2 = z0 + β ln λ

2πω

instanton size → constituent distance: z2 − z1 ∼ lnλconstituent size: fixed by β and ω

• really locations of topological lumps?YES: corrections of the second term at z = z1 are exp. small

• individual constituent:

u(z) = exp(ωz2πβ

) ω analogous

top. charge:

q(x) =πω2

β2 cosh2(ω Re z 2πβ )

(static) Q = ω

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• possible values for Q: 0,1, . . . ω,1 + ω, . . . 1− ω,2− ω, . . .

asympt. φ−∞ → φ+∞: N→ N, S→ S N→ S S→ Nconstituents alternate

• why instanton quarks not visible for zero temperature, i.e. on R2?

β →∞: constituents large and overlap!no other scale competing with their distance

• generalisations: CP(N) models FB et al. in progress

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46 N constituents as expected

• realisation in condensed matter:cylinder of ..?.. with quasi-periodic bc.s

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Fermionic zero modesgauge field description ⇒ couple fermions ⇒ zero modesphase-bc.s ψ(x0 + iβ) = e 2πiζψ(x0), evolution with ζ: FB et al. in progress

Falk Bruckmann Instanton constituents 12 / 19

Part II: Gauge theories

pure Yang-Mills theory in (Euclidean) 4D:

S =

∫12

tr F 2µν ≥ |Q| = |

∫12

tr Fµν Fµν |

dual field strength F aµν =

12εµνρσF a

ρσ (~Ea ~Ba)

integer Q: instanton number/topological charge

topology:

Aµr→∞→ iΩ−1∂µΩ . . . pure gauge

Q = deg(Ω : S3r→∞ → SU(N)) . . . winding number

Falk Bruckmann Instanton constituents 13 / 19

Instantons

(anti)selfdual: F aρσ = ±F a

µν first order, nonlinear

charge 1: axially symmetric ansatz and solution BPST

Aaµ = ηa

µν

2xν

x2 + ρ2 trF 2 =ρ4

(x2 + ρ2)4 ηaµν ∈ −1,0,1

size ρlocalized in space and timealgebraic decay, similar to O(3) instantons on R2

instanton liquid model from semiclassical path integral Shuryak

chiral symmetry breakingaxial anomalytopological susceptibilityconfinement?

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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 ofaction density for SU(2),intermediate holonomy

⇒ 2 lumps, almost staticNc for gauge group SU(Nc), like quarks in baryons

⇒magnetic monopoles of opposite magnetic chargein fact dyons with same electric as magn. charge (selfdual)

Falk Bruckmann Instanton constituents 15 / 19

Role of the holonomyrelative gauge orientation of instanton copies in the ADHM constr.

⇒ Aµ periodic up to a gauge transformation e 2πiωσ3/2 (cf. O(3))

gauge theory: compensated by time-dependent transf. e 2πiωσ3x0/2

⇒ introduces an asymptotic gauge field A0

⇒ asymptotic Polyakov loop = holonomy

P(~x) ≡ P exp(

i∫ β

0dx0 A0

)→ e 2πiωσ3/2 ≡ P∞

‘environment’

acts like a Higgs field, in the group: vev ω, direction σ3

•monopoles have masses ω/β and ω/β, ω = 1− ω

• Aa=3µ : power law decay (massless ‘photon’),

• Aa=1,2µ : exponential decay (massive ‘W -bosons’)

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• Polyakov loop in the bulk: P(~x) = ±12 at the monopolesHiggs field vanishes = ‘false vacuum’necessary for top. reasons Ford et al.; Reinhardt; Jahn et al.

• index theorem valid Nye, Singer

localisation depending on bc.s:

ψ(x0 + iβ) = e2πiζψ(x0) (Aµ still periodic)

ζ ∈ −ω2 ,

ω2 incl. periodic: localised at monopole Garcia Perez et al.

ζ ∈ rest incl. antiperiodic: localised at antimonopole

a zero in their profiles at the ‘other’ monopole, topological FB

• calorons can be studied on the lattice by cooling Ilgenfritz et al. ’02, FB et al.

• physical relevance of P∞:conjecture: holonomy trP∞ deconfinement order param. 〈trP〉x

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Calorons and the dynamics of YM theories

• eff. potential at 1-loop: triv. holonomy favored! Gross, Pisarski, Jaffe; Weiss

⇒ overruled by caloron gas contribution: Diakonov et al.

⇒minima at P = ±12 become unstable for low enough temperature⇒ onset of confinement

• gas of calorons and anticalorons put on the lattice: Gerhold et al.

⇒ linearly rising interquark potential just for nontrivial holonomy!

• confinement from a gas of purely selfdual dyons Diakonov, Petrov

⇒ unphysical (top. charge builds up)⇒ nevertheless interesting physical effects

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Summary

sigma models in 2D and YM in 4D admit instantons

instanton constituents for S1 × R1 and S1 × R3 = finite T

with fractional charges, say ω and 1− ω in the lowest models

when in compact direction periodic up to a subgroup, say e 2πiω..

= 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 19 / 19