instanton constituents in sigma models and yang …...instanton constituents in sigma models and...

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

Falk Bruckmann Instanton constituents 1 / 19

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

Falk Bruckmann Instanton constituents 2 / 19

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

Falk Bruckmann Instanton constituents 3 / 19

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

Falk Bruckmann Instanton constituents 4 / 19

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

Falk Bruckmann Instanton constituents 5 / 19

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

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Falk Bruckmann Instanton constituents 7 / 19

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

Falk Bruckmann Instanton constituents 8 / 19

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

Falk Bruckmann Instanton constituents 10 / 19

• 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?

Falk Bruckmann Instanton constituents 14 / 19

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’)

Falk Bruckmann Instanton constituents 16 / 19

• 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

Falk Bruckmann Instanton constituents 17 / 19

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

Falk Bruckmann Instanton constituents 18 / 19

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