Transcript Document

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/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
Štefan Olejník
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
Karl-Franzens-Universität, Graz, October 19, 2005
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
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Karl-Franzens-Universität, Graz, October 19, 2005
Part 1.
Faddeev-Popov operator, Gribov-Zwanziger
mechanism of confinement, and center vortices
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Karl-Franzens-Universität, Graz, October 19, 2005
Confinement scenario in Coulomb gauge
Classical Hamiltonian of QCD in CG:
Faddeev—Popov operator:
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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.
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A confinement condition in terms of F-P eigenstates
Color Coulomb self-energy of a color charged state:
F-P operator in SU(2):
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F-P eigenstates:
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Necessary condition for divergence of e:
To zero-th order in the gauge coupling:
To ensure confinement, one needs some mechanism of
enhancement of r(l) and F(l) at small l.
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Is the confinement condition satisfied?
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Karl-Franzens-Universität, Graz, October 19, 2005
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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.
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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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/9901020
removes the Coulomb string tension.
JG, ŠO, hep-lat/0302018
Both ensembles were brought to Coulomb gauge by
maximizing, on each time-slice,
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Institute of Physics, Slovak Academy of Sciences,
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Vortex-only configurations
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Vortex-removed configurations
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Conclusions of Part 1
1. Full configurations: the eigenvalue density and F(l) at small
l 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(l) 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 Gribovhorizon scenarios.
Part 2. Covariant Laplacians and localization
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SU(2) gauge-fundamental Higgs theory
Osterwalder, Seiler ; Fradkin, Shenker, 1979; Lang, Rebbi, Virasoro, 1981
Vortex depercolation
Vortex percolation
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“Confinement-like” phase
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“Higgs-like” phase
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Part 2.
Covariant Laplacians and localization
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Electron localization
Periodic lattice:
System with disorder:
© Peter Markoš, 2005
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Anderson model of metal-insulator transition
Density of states
delocalized states
localized states
© Peter Markoš, 2005
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
Karl-Franzens-Universität, Graz, October 19, 2005
Localization in eigenmodes of the lattice Dirac operator
Topological charge is related to zero modes of the Dirac
operator gm Dm(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).
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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.
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Signals of localization
„inverse participation ratio“
Gattringer, Göckeler, Rakow, Schaefer, Schäfer, 2001
„remaining norm“
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Inverse participation ratio
Plane wave:
Localization on a single site:
Localization in a certain
finite 4-volume b:
Extended, but lowerdimensional state:
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Fundamental representation (j=1/2)
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Scaling of the localization volume
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Karl-Franzens-Universität, Graz, October 19, 2005
Remaining norm
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Density profile
http://www.dcps.savba.sk/localization
http://lattice.itep.ru/localization
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What about higher states?
Full symmetry between lower and upper end of the spectrum:
mobility edge
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Institute of Physics, Slovak Academy of Sciences,
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mobility edge
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Adjoint representation (j=1)
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Karl-Franzens-Universität, Graz, October 19, 2005
Remaining norm
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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Bratislava, Slovakia
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„Scaling“ of the localization volume
God knows why!
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
Karl-Franzens-Universität, Graz, October 19, 2005
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Density profile
http://www.dcps.savba.sk/localization
http://lattice.itep.ru/localization
ŠO
Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
Karl-Franzens-Universität, Graz, October 19, 2005
SU(2) with fundamental Higgs field
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j=3/2 representation
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Karl-Franzens-Universität, Graz, October 19, 2005
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.
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Conclusions 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 b, localization volume fixed in physical units,
smaller in vortex-only configurations, no localization after vortex removal.
j=1: ba2 independent of b, localization disappears in the “Higgs” phase of
the model with fundamental Higgs fields coupled to gauge fields.
j=3/2: b independent of b, 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!
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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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, b!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.
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Institute of Physics, Slovak Academy of Sciences,
Bratislava, Slovakia
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