Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Eigenmodes of covariant Laplacians in SU(2) lattice gauge theory: confinement and localization J. Greensite, Š.O., D. Zwanziger, Center vortices and the Gribov horizon, JHEP 05 (2005) 070; hep-lat/ 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) ; hep-lat/

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Focus on two operators as probes of confinement 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.

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Part 1. Faddeev-Popov operator, Gribov-Zwanziger mechanism of confinement, and center vortices

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Confinement scenario in Coulomb gauge Classical Hamiltonian of QCD in CG: Faddeev—Popov operator:

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Gribov 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.

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, A confinement condition in terms of F-P eigenstates Color Coulomb self-energy of a color charged state: F-P operator in SU(2):

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, F-P eigenstates:

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, 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.

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Is the confinement condition satisfied?

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19,

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19,

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Any hint on “confiners”? Center vortices? Center vortices are identified by fixing to an adjoint gauge, and then projecting link variables to the Z N subgroup of SU(N). The excitations of the projected theory are known as P-vortices. Jeff Greensite, hep-lat/ 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.

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Two ensembles 1.“Vortex-only” configurations: 2.“Vortex-removed” configurations: Vortex removal removes the string tension, eliminates chiral symmetry breaking, sends topological charge to zero, Philippe de Forcrand, Massimo D’Elia, hep-lat/ removes the Coulomb string tension. JG, ŠO, hep-lat/ Both ensembles were brought to Coulomb gauge by maximizing, on each time-slice,

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Vortex-only configurations

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Vortex-removed configurations

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Conclusions 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

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, SU(2) gauge-fundamental Higgs theory Osterwalder, Seiler ; Fradkin, Shenker, 1979; Lang, Rebbi, Virasoro, 1981 Vortex percolation Vortex depercolation

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, “Confinement-like” phase

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19,

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, “Higgs-like” phase

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Part 2. Covariant Laplacians and localization

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Electron localization © Peter Markoš, 2005 Periodic lattice: System with disorder:

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Anderson model of metal-insulator transition © Peter Markoš, 2005 Density of states delocalized states localized states

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Localization in eigenmodes of the lattice Dirac operator 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).

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Covariant 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.

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Signals of localization „inverse participation ratio“ Gattringer, Göckeler, Rakow, Schaefer, Schäfer, 2001 „remaining norm“

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Inverse participation ratio Plane wave: Localization on a single site: Localization in a certain finite 4-volume b: Extended, but lower- dimensional state:

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Fundamental representation (j=1/2)

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19,

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Scaling of the localization volume

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Remaining norm

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Density profile

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, What about higher states? Full symmetry between lower and upper end of the spectrum: mobility edge

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Adjoint representation (j=1)

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Remaining norm

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19,

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, „Scaling“ of the localization volume God knows why!

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Density profile

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, SU(2) with fundamental Higgs field

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, j=3/2 representation

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, No localization for F-P operator 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.

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, Conclusions of Part 2 Lowest eigenstates of covariant Laplacians are localized, but exhibit rather strange dependence on the group representation: j=1/2: ba 4 independent of , localization volume fixed in physical units, smaller in vortex-only configurations, no localization after vortex removal. j=1: ba 2 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!

ŠO Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Karl-Franzens-Universität, Graz, October 19, A 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.