Localized Eigenmodes of the Covariant Lattice Laplacian

Stefan Olejnik, Mikhail Polikarpov, Sergey Syritsyn, Valentin Zakharov and J.G.

“Understanding Confinement” Castle Ringberg

May 2005

hep-lat/0504008

Very old story: electron propagation in a periodic potential. Bloch waves (extended, plane-wave-like states).

Old story (Anderson, 1958): Disorder in the potential can disrupt Bloch waves; low-lying energy eigenstates become exponentially localized. This is an interference effect due to multiple scattering, rather than ordinary bound state formation in a single potential well.

When the energy of the highest localized state (the ``mobility edge’’) exceeds the Fermi energy, the material is an insulator.

Recently, it was found that in Yang-Mills theory:

• Wilson-Dirac fermions have a low-lying spectrum of localized eigenmodes in certain regions of the phase diagram (Golterman & Shamir)

• Low-lying modes of the Asqtad fermion operator, although extended, seem to concentrate on lattice sub-volumes of dimensionality < 4 (MILC collaboration).

If fermionic operators are picking up signals of lower-dimensional substructure, is there any relation to, e.g., center vortex sheets or monopole worldlines?

Are there indications of lattice-scale 2-brane structures, along the lines suggested by Zakharov?

Previous numerical findings: infinite action density 2-branes at P-vortex locations. (Gubarev et al.)

Our study…

Is localization/concentration unique to Dirac operator eigenmodes, or is it found in other lattice kinetic operators, e.g. the Faddeev-Popov and covariant Laplacian operators?

If so, is there any connection to confinement?

A. The IPR

Covariant lattice laplacian

Eigenvalue equation

Inverse Participation Ratio (IPR) of the n-th eigenmode

where

Signals of Localization

Opposite tendencies:

• Extended eigenmodes. ¼ 1 / V, IPR ¼ 1

• Localized eigenmodes. ¼ 0 except in a region of volume b ,

B. Remaining Norm

Sort (x) into a 1-dim array r(k), k=1,2,…,V, with r(k) > r(k+1). Define the Remaining Norm (RN) as

The RN is the amount of total norm (=1) remaining after counting contributions from the K<V subset of sites with largest (x).

IPR and RN can be computed in any color group representation.

Log-log plot of IPR-A vs lattice length L, at various . The lines are a fit to IPR = A + L4/b .

Log-log plot of IPR-A vs physical volume V=(La)4. The fact that all points fall on the same line means that the localization volume in physical units

Evidence of localization

Is -independent.

j=1/2 representation

Center Vortices

Center vortices are surfacelike (D=4) objects in the QCD vacuum which can be topologically linked to closed loops. Creation of a center vortex linked to a Wilson loop multiplies the loop by a center element of the gauge group.

The center vortex theory of confinement holds that the area law falloff of a Wilson loops is due to vacuum fluctuations in the number of vortices linking the loop.

In 1997, methods were devised for located center vortices in lattice configurations, and also for removing them.

There has since been lots of work on vortices in the lattice community.

Reviews: J.G., hep-lat/0301023,

Michael Engelhardt, http:// lqcd.fnal.gov/lattice04 (video)

Kurt Langfeld, this meeting

Locating Center Vortices

Fix the SU(N) gauge symmetry leaving a residual ZN center symmetry, e.g. maximal center gauge maximizing

This is followed by center projection of each link to the nearest center element, e.g. for SU(2)

This maps SU(2) configurations into Z2 configurations, whose excitations are thin center vortices known as P-vortices. The idea is that the P-vortices roughly locate the middle of thick center vortices in the unprojected configuration.

Numerical Findings:

The sign of a large Wilson loop is strongly correlated with the P-vortex linking number;

P-Vortex density scales according to asymptotic freedom;

SU(2) action at vortex locations is much larger than the vacuum average;

P-vortices account for the entire string tension.

Removing center vortices from the unprojected configuration

• removes the string tension

• eliminates chiral symmetry breaking

• takes the topological charge to zero

Vortex removal is a minimal change – only the action at P-vortex plaquettes is changed, and the density of those plaquettes drops exponentially with . How is localization affected?

Somewhat greater localization in center-projected (“vortex-only”) configurations.

bphys is reduced.

No localization at all in vortex-removed configurations.

Remaining Norm:

unmodified lattice vortex only

vortex removed

Scaling of the extension of the localization volume

Conclusions so far:

Low-lying modes are localized

Localization is correlated with confinement/center vortices

Localization volume scales

Not all eigenmodes are localized, only the low and high-lying modes. The bulk of states are extended.

unmodified lattice vortex only

If (x) is an eigenmode, so is .

From this fact, one can show that

mobility edgemobility edgemobility edge

We repeat everything for links in the adjoint representation, and find

much sharper localization

scaling appropriate to 2-surfaces, not 4-volumes, i.e.

seems to be -independent.

Exponential localization is lost in the Higgs phase.

j=1 representation

ba2 constant means that localization volume is very small, in physical units, but very large in lattice units, at high .

We do not understand this.

Is the localization region surface-like, with surface area scaling?

We can rule this out just by looking at typical eigenmodes.

The de Forcrand-D’Elia trick for removing vortices doesn’t work in the adjoint representation. Instead we work in a gauge-Higgs theory, and look for localization across the transition. For an SU(2), the action can be written as

Higgs (confinement-like) phase distinguished by the broken (symmetric) realization of a remnant global gauge symmetry in Coulomb gauge (Olejnik, Zwanziger, J.G.).

Center vortices cease to percolate in the Higgs phase.

There is a transition to the Higgs phase, at = 2.1, around = 1.9 .

Compute IPRs at = 2.1, =0.7 (confinement-like phase)

= 2.1, = 1.2 (Higgs phase)

Transition at = 2.1, ¼ 0.9

Exponential localization is lost in the Higgs phase.

(Same thing happens also in the fundamental representation.)

An even greater degree of localization

No scaling! b is -independent, localization volume is O(a4).

Vortex removal has no effect on localization.

Gauge-higgs transition has very little effect.

Conclusion: In this case, localization must be due to lattice-scale (i.e. perturbative) vacuum fluctuations.

j=3/2 representation

Green’s function

Gauge-invariant correlator

IF K(x-y) long range, e.g. » 1/|x-y|4, then as V ! 1

Localization --> Short Range Correlators

The divergence

can only come from near-zero eigenmodes, assuming min=0.

But if the low-lying eigenmodes are localized, then a long-range correlation from those modes is not possible. Remove those modes and the integral is finite, so a long-range correlation can’t come from the higher modes either.

Conclusion: Localization of low-lying eigenmodes implies short-range correlation. (McKane and Stone)

The Faddeev-Popov Operator

The Faddeev-Popov operator in Coulomb gauge looks very similar to the covariant Laplacian (in D=3 dimensions)

( D(A) is the covariant derivative )

So one might expect that the low-lying eigenvalues are localized.On the other hand, the Coulomb energy of a given charge distribution is

where

so if M-1 is short-range, the Coulomb potential / K(x,y) is short range.

But in fact, we know that asymptotically

Olejnik & J.G

where coul ¼ 3 , so it must be that the low-lying eigenmodes of the F-P operator are not localized. This is an interesting check…

Low IPR’s even at =2.1. No apparent localization -- in agreement withCoulomb confinement -- despite the similarity to the covariant Laplacian.

The low-lying eigenmodes of the covariant Laplacian operator are localized, but there is a very puzzling dependence on group representation.

j=1/2 : ba4 is -independent. Localization volume is fixed in physical units, and vortex removal removes localization.

j=1 : ba2 is -independent. Localization disappears in the Higgs phase of a gauge-Higgs theory.

j=3/2 : b is -independent. Localization almost unaffected by vortex removal, transition to the Higgs phase.

For j=1/2, 1, it seems that localization is associated with confinement. This doesn’t seem to be the case for j=3/2. It is also mysterious why, for j=1, thelocalization volume goes to zero in physical units, but 1 in lattice units, in the continuum limit.

There is something going on here that we don’t understand!

Conclusions