Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Eigenmodes...

Štefan OlejníkŠtefan Olejník Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia

Eigenmodes of covariant Laplacians in Eigenmodes of covariant Laplacians in SU(2) lattice gauge theory: confinement SU(2) lattice gauge theory: confinement

and localizationand localization

• J. Greensite, Š.O., D. Zwanziger, Center vortices and the Gribov horizon, JHEP 05 (2005) 070; hep-lat/0407032

• J. Greensite, Š.O., M. Polikarpov, S. Syritsyn, V. Zakharov, Localized eigenmodes of covariant Laplacians in the Yang–Mills vacuum, Phys. Rev. D 71 (2005) 114507; hep-lat/0504008

ŠOŠO Institute of Physics, Slovak Academy of Institute of Physics, Slovak Academy of Sciences, Bratislava, SlovakiaSciences, Bratislava, Slovakia

2Karl-Franzens-Universität, Graz, October 19, 2005Karl-Franzens-Universität, Graz, October 19, 2005

Focus on two operators as probes of Focus on two operators as probes of confinementconfinement

Faddeev-Popov operator in Coulomb gauge:

Covariant Laplacian operator:

We looked at their lowest eigenmodes on a lattice, hoping that

their properties are sensitive to confining disorder,

they can provide some information on the nature of configurations responsible for confinement,

the structure of eigenmodes can reveal the dimensionality of underlying structures in the QCD vacuum.



PartPart 1. 1. FaddeevFaddeev--Popov operator, Gribov-Popov operator, Gribov-Zwanziger mechanism of confinement, Zwanziger mechanism of confinement, and center vorticesand center vortices



Confinement scenario in Coulomb gaugeConfinement scenario in Coulomb gauge

Classical Hamiltonian of QCD in CG:

Faddeev—Popov operator:



Gribov ambiguity and Gribov copiesGribov ambiguity and Gribov copies

Gribov region: set of transverse fields, for which the F-P operator is positive; local minima of I.Gribov horizon: boundary of the Gribov region.Fundamental modular region: absolute minima of I.GR and FMR are bounded and convex.Gribov horizon confinement scenario: the dimension of configuration space is large, most configurations are located close to the horizon. This enhances the energy at large separations and leads to confinement.



A confinement condition in terms of F-P A confinement condition in terms of F-P eigenstateseigenstates

Color Coulomb self-energy of a color charged state:

F-P operator in SU(2):



F-P eigenstates:



Necessary condition for divergence of :

To zero-th order in the gauge coupling:

To ensure confinement, one needs some mechanism of enhancement of () and F() at small .



Is the confinement condition satisfied?Is the confinement condition satisfied?



Any hint on “confiners”? Center vortices?Any hint on “confiners”? Center vortices?

Center vortices are identified by fixing to an adjoint gauge, and then projecting link variables to the ZN subgroup of SU(N). The excitations of the projected theory are known as P-vortices.

Jeff Greensite, hep-lat/0301023

Direct maximal center gauge in SU(2): One fixes to the maximum of

and center projects

Center dominance plus a lot of further evidence that center vortices alone reproduce much of confinement physics.



Two ensemblesTwo ensembles

1. “Vortex-only” configurations:2. “Vortex-removed” configurations:

Vortex removalremoves the string tension,eliminates chiral symmetry breaking,sends topological charge to zero,

Philippe de Forcrand, Massimo D’Elia, hep-lat/9901020

removes the Coulomb string tension.JG, ŠO, hep-lat/0302018

Both ensembles were brought to Coulomb gauge by maximizing, on each time-slice,



Vortex-only configurationsVortex-only configurations



Vortex-removed configurationsVortex-removed configurations



Conclusions of Part 1Conclusions of Part 1

1. Full configurations: the eigenvalue density and F() at small consistent with divergent Coulomb self-energy of a color charged state.

2. Vortex-only configurations: vortex content of configurations responsible for the enhancement of both the eigenvalue density and F() near zero.

3. Vortex-removed configurations: a small perturbation of the zero-field limit.

Support for the Gribov-horizon scenario.Firm connection between center-vortex and Gribov-horizon scenarios.

Part 2. Covariant Laplacians and localization



SU(2) gauge-fundamental Higgs theorySU(2) gauge-fundamental Higgs theory

Osterwalder, Seiler ; Fradkin, Shenker, 1979; Lang, Rebbi, Virasoro, 1981

Vortex percolation

Vortex depercolation



““Confinement-like” phaseConfinement-like” phase



““Higgs-like” phaseHiggs-like” phase



PartPart 22.. Covariant Laplacians and localizationCovariant Laplacians and localization



Electron localizationElectron localization

© Peter Markoš, 2005

Periodic lattice: System with disorder:



AndersoAndersonn model model of metal-insulator transition of metal-insulator transition

© Peter Markoš, 2005

Density of states

delocalized states

localized states



Localization in eigenmodes of the lattice Dirac Localization in eigenmodes of the lattice Dirac operatoroperator

Topological charge is related to zero modes of the Dirac operator D(A) (Atiyah–Singer index theorem):

The chiral condensate is given by the density of near-zero eigenmodes of the Dirac operator (Banks-Casher relation):

BUT: which Dirac operator? – problems with chiral symmetry and „doublers“. Results with different operators not always the same.Let’s have a look at simpler operators with no problems with chiral symmetry and there are no doublers (which can be studied with computer power at our disposal).



Covariant LaplacianCovariant Laplacian

It is not the square of the Dirac operator (only for the free theory).It determines the propagation of a spinless color-charged particle in the background of a vacuum gauge-field configuration.Let’s consider the non-relativistic situation and study the propagation of a single particle, either a boson or a fermion, in a random potential. Now suppose that all eigenmodes of the Hamiltonian are localized. This means that the particle cannot propagate, because in quantum mechanics propagation is actually an interference effect among extended eigenstates.



Signals of localizationSignals of localization

„inverse participation ratio“

Gattringer, Göckeler, Rakow, Schaefer, Schäfer, 2001

„remaining norm“



Inverse participation ratioInverse participation ratio

Plane wave:

Localization on a single site:

Localization in a certain finite 4-volume b:

Extended, but lower-dimensional state:



FundamentalFundamental repre representationsentation ( (jj=1/2)=1/2)



Scaling of the localization volumeScaling of the localization volume



Remaining normRemaining norm



Density profileDensity profile

http://www.dcps.savba.sk/localizationhttp://lattice.itep.ru/localization



What about higher statesWhat about higher states??

Full symmetry between lower and upper end of the spectrum:

mobility edge mobility edge



Adjoint representationAdjoint representation ( (jj=1)=1)



Remaining normRemaining norm



„„ScalingScaling“ “ of the of the llocalization volumeocalization volume

God knows why!



Density profileDensity profile

http://www.dcps.savba.sk/localizationhttp://lattice.itep.ru/localization



SU(2) with fundamental Higgs fieldSU(2) with fundamental Higgs field



jj=3/2=3/2 representation representation



No localization for No localization for FF--P operP operaatortor

Localization of low-lying eigenmodes implies short-range correlation (McKane, Stone, 1981).Color Coulomb potential is long range, therefore one would expect no localization in low-lying states of the F-P operator.



Conclusions of Part 2Conclusions of Part 2

Lowest eigenstates of covariant Laplacians are localized, but exhibit rather strange dependence on the group representation:

j=1/2: ba4 independent of , localization volume fixed in physical units, smaller in vortex-only configurations, no localization after vortex removal.j=1: ba2 independent of , localization disappears in the “Higgs” phase of the model with fundamental Higgs fields coupled to gauge fields. j=3/2: b independent of , localization independent of presence/absence of vortices or of the phase of the gauge-Higgs model.

It seems that localization of eigenstates of the covariant Laplacian in j=1/2 and 1 could be related to confinement, but not for j=3/2 (and higher?) representation.Eigenstates of the Faddeev-Popov operator in Coulomb gauge are not localized (which is consistent with the fact that the color Coulomb potential is long-range).

We may have touched something interesting, but we don’t understand it!



A speculation: QCD vacuum as insulatorA speculation: QCD vacuum as insulator

In disordered solids there is a conductor-insulator transition when the mobility edge crosses the Fermi energy. Could it be that in QCD, in physical units,

in the continuum, !1, limit? Then all eigenmodes of the kinetic operator, with finite eigenvalues, are localized in the continuum limit. This would mean that the vacuum is an “insulator” of some sort, at least for scalar particles.This possibility is under investigation.

Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Eigenmodes...

Documents

Transcript of Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Eigenmodes...