Confining vacua and Q-state Potts models with Q

Confining vacua and Q-state Potts models withQ<1

F. Gliozzi

DFT & INFN, Universitá di Torino

Regensburg, August, 1 , 2007

F. Gliozzi ( DFT & INFN, Universitá di Torino ) Confinement and Potts with Q<1 Regensburg, August, 1 , 2007 1 / 23

Plan of the talk

1 Introduction

2 Abelian dominance versus center dominance

3 The gauge duals of Q-state Potts models

4 An intriguing vacuum at Q<1

5 Conclusion


Introduction

Introduction


Introduction

❋ Magnetic monopoles and center vortices are widely believed to bethe most important degrees of freedom for confinement in YangMills theories

❋ Plausibility arguments suggest that magnetic monopolecondensates (’t Hooft and Mandelstam) or percolation of centervortices imply area law for large Wilson loop

❋ In most YM models the phase with magnetic monopolecondensation coincides with that where center vortices percolate,thus it is not clear which of these two properties is most directlyinvolved in producing confinement

❋ in this talk it is pointed out that there is a class of 3D gaugemodels, which can be though of as duals of Q-state Potts modelswith Q < 1, where the magnetic monopole condensation is notassociated to the percolation of center vortices.

❋ In these models a confining vacuum can be observed only whenboth confining mechanisms are present.


Abelian dominance versus center dominance

Condensation of Monopoles& Percolation of center vortices



Magnetic monopole condensation

❄ Abelian dominance: After a suitable abelian gauge projection, theunbroken U(1)N−1 symmetry of the full SU(N) gauge groupdescribes the relevant IR degrees of freedom of the vacuum

❄ The vacuum turns out to be a dual superconductor characterisedby the condensation of a (composite) magnetic monopole field

➫ The associated dual Meissner effect squeezes the colour flux tubein a string-like structure

➫ As a consequence, large Wilson loops decay with an area law



Percolation of center vortices (or spaghetti vacuum)

❄ Center dominance: The center-projected link elements carry mostof the information about the confining properties of the full theory

❄ In the confining vacuum the center vortices, i.e. the 2D (or 1D)structures carrying flux in the center of the gauge group form aninfinite network of intersecting surfaces (in 4D gauge theories) orlines (in 3D theories)

➫ The random fluctuations in the number of such vortices linked to aWilson loop explains the area-law


The gauge duals of Q-state Potts models

The gauge duals of 3D Q-state Potts models



Q-state Potts models

= Spin models defined by the Hamiltonian on a cubic lattice Λ

H = −∑〈i j〉

δσi σj , (σ = 1, 2 . . . Q)

➫ Its global symmetry is the permutation group of Q elements SQ

➫ In 3D is dual to a gauge model with gauge symmetry SQ

❋ The properties of the gauge theory can be read directly in the spin(or disorder parameter) formulation

❋ In these models the implementation of the confining mechanisms(monopole condensation & center vortices percolation) isparticularly simple



For a microscopic description of monopole condensation and centervortices percolation resort to the Fortuin Kasteleyn (FK) randomcluster representation:

Z ≡∑{σ}

e−β H =∑G⊆Λ

vbGQcG ,

➫ v = eβ − 1,

➫ G = spanning subgraphs of Λ.

➫ bG = number of links of G(active bonds –)

➫ cG number of connectedcomponents (FK clusters).



Wilson loops❋ The Wilson operators Wγ , are associated to arbitrary loops γ of

the dual lattice Λ̃ and their values on a graph G of active bondsare set by the following rule

➊ Wγ(G) = 1 if no cluster of G istopologically linked to γ;

➋ Wγ(G) = 0 otherwise

➫ linking of W depends only on closedpaths=center vortices

➫ The area law falloff requires aninfinite network of center vortices

⇓hence the formation of an infinite,percolating FK cluster= magneticmonopole condensate

➯ 〈Wγ〉 =PG⊆Λ Wγ(G) vbG QcG

Z

W=1

W=0



Area law and the structure of FK clusters

Confinement⇒

infinite network of loops

⇓ ( ⇑ ?)Magnetic monopole condensate

≡infinite FK cluster

❋ Magnetic condensation in the gauge model ⇔ spontaneoussymmetry breaking of the Q-state Potts model.

❋ Why such a condensation should imply confinement at amicroscopic level?

❋ In the cases studied so far [Q=4,Q=3, Q=2 (Ising), Q=1 (Randompercolation)] the formation of the infinite FK cluster is alwaysassociated to an infinite network of loops

➫ when Q≥ 1 the vacuum with monopole condensate is endowedwith a infinite network of center vortices


An intriguing vacuum at Q<1

A new vacuum at Q<1



A new vacuum at Q<1

❋ In the Potts models Q acts as cluster fugacity

➫ as Q decreases , so does the number of clusters cG

❋ Two kinds of

active bonds��

��

��

��

➊ bond linking twootherwise disconnectedclusters (bridge)

➋ bond closing a loop in acluster

��

��

��

��

➫ If the fraction of active bonds is kept constant, the number ofactive bonds of type ➋ is reduced when Q decreases

2D Example: in a square lattice at criticality the fraction of activebonds is 1

2 for any Q. Those belonging to a loop are√

Q2(1+

√Q)



The new vacuum at Q<1

➫ When 0 ≤ Q < 1 the formation of an infinite, percolating FKcluster does not imply the presence of an infinite network ofclosed paths (i.e. center vortices)

➫ there is a new vacuum separating the trivial, non confiningvacuum (no monopole condensation) from the usual confining one(network of percolating center vortices)

➫ In such a vacuum the magnetic monopoles condense withoutproducing confinement: there the center vortices form a dilute gasof loops (rather than an infinite network) which cannot give rise tothe area law decay of large Wilson loops



Phase diagram of gauge Q-state Potts (schematic)

β

0 1 2

non −confiningmagneticcondensate

confining vacuum

the string tension

vanishing of

non confining vacuumsymmetric

transitionII order bulk

:

of center vorticesmagnetic condensate & infinite network

Q



A local Monte Carlo algorithm for Q<1 Potts model① draw a uniformly distributed random number 0 ≤ r` ≤ 1 for each

link ` of the lattice

② if r` < p = 1− e−β put a bond in `

③ if p ≤ r` ≤ pQ (1−p)+p put a bond only if ` is a bridge

④ if r` > pQ (1−p)+p erase any bond in `

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

p0p

Q(1−p)+p

anyanyany any

old

configurat.

configurat.

updated

1rl



A dilute gas of loops at Q<1

Q= 120 at criticality (2D)

Q= 1 at criticality (2D)

projection on loop subgraph

projection on loop subgraph



Numerical simulations at Q= 110

��

��

β

0 1 2

the string tension

vanishing of

transitionII order bulk

:

Q

0.065

0.051

σ > 0

string tension from large Wilson loops

monopole massthrough two−point

correlator

p=1−exp(−β)



The percolation threshold from the vanishing of thecorrelation length

pt− p

( pt −p) νm=M + ....

m pt 0.0500(18)=

ν = 2.50(9) 0.1

0.01



Scaling of the string tension

σ

p =

ν=

0.063(1)

0.83(1)

p−p

σ=S(p−p )2ν

o

o

o1/10

σ

σ

Q=

0.0001

0.001

0.01

0.001 0.01 0.1

Q=1/10 p o> p t ν σ << ν

ν=σν;t

p=o

pQ > 1


Conclusion

Conclusion


Conclusion

➀ The gauge duals of 3D Q<1 Potts models have a vacuum in whichthe magnetic monopole condensate does not imply confinement,being the string tension σ vanishing

➁ There is also a confining vacuum characterised by a magneticmonopole condensate and an infinite network of center vortices

➂ The transition between these two vacua does not involve a bulkphase transition

➃ The scaling properties of the string tension defines presumably anew, unexpected universality class.


Confining vacua and Q-state Potts models with Q

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