Localization and Confinement

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Localization and the Covariant Laplacian

Stefan Olejnik, Mikhail Polikarpov, Sergey
Syritsyn, Alexei Kovalenko, Valentin Zakharov
and me
Niels Bohr Institutet July 2006
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• Outline
• Localization of the lowest eigenmode of the
  covariant Laplacian
  PRD 71(2005) 114507
• The density of states and the mobility edge of
  the adjoint representation spectrum
  
  hep-lat/0606008
• Localization and the Yang-Mills vacuum
  wavefunctional (in progress)

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We consider propagation of a spinless particle in
a background gauge field as a probe of the
confining - or other - non-perturbative
properties of the background field. The
natural observable is the scalar propagator
(where
is the covariant Laplacian)
but this observable is gauge covariant, not gauge
invariant. It seems that we could only extract
useful information by fixing to some gauge.
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On the other hand, we can write
where
The eigenvalue spectrum
and eigenmode densities
are gauge-invariant. Is there any sign of the
non-perturbative character of the Yang-Mills
vacuum in these quantities? What to look for?
One possibility - Localization.
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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 different,
however, from ordinary bound state formation in a
single potential well. In non-rel. quantum
mechanics, propagation of wave packets is an
interference effect among extended energy
eigenstates. No extended states means no
propagation. When the energy of the highest
localized state (the mobility edge) exceeds
the Fermi energy, the material is an insulator.
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• 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 3
  (MILC collaboration, Horvath et al).

If fermionic operators are picking up signals of
lower-dimensional substructure, is there any
relation to, e.g., center vortex sheets or
monopole worldlines?
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Our study Is localization/concentration
of low-lying modes unique to the Dirac operator
eigenmodes, or is it found in other lattice
kinetic operators, e.g. the covariant Laplacian ?
If so, is there any connection to
confinement? Is the QCD vacuum, in any
sense, an insulator?
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Signals of Localization

• The IPR
• Covariant lattice laplacian

Eigenvalue equation
Inverse Participation Ratio (IPR) of the n-th
eigenmode
where
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• 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), k1,2,,V,
with r(k) gt r(k1). Define the Remaining Norm
(RN) as
The RN is the amount of total norm (1) remaining
after counting contributions from the KltV subset
of sites with largest ?(x).
IPR and RN can be computed in any color group
representation.
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j1/2 representation
Evidence of localization
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
Is ?-independent.
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Now remove confinement by removing center
vortices. Center 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.
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Remaining Norm
unmodified lattice
vortex only
vortex removed
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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

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j 1 representation

• 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.
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ba2 constant means that localization volume is
very small, in physical units, but very large in
lattice units, at high ?.
Is the localization region surface-like, with
surface area scaling? We can rule this out just
by looking at typical eigenmodes.
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Interim Conclusions

• The low-lying eigenmodes of the covariant
  Laplacian operator are localized, but there is a
  very puzzling dependence on group representation.
• j1/2 ba4 is ?-independent. Localization
  volume is fixed in physical units, and vortex
  removal removes localization.
• j1 ba2 is ?-independent.

It is a bit mysterious why, for j1, the
localization volume seems goes to zero in
physical units, but 1 in lattice units, in the
continuum limit. Are localized states really
important? Maybe there arent enough of them to
affect anything!
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Localized spectrum for j 1
If the number of localized states is
negligible in the continuum limit, then they may
have no effect on physical observables. We need
to know two things

• What is the interval, in physical units, between
  the lowest eigenmode and the mobility edge
• What is the density of states ?(?) in this
  interval?

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We observe that if we plot the ratio
the low-lying spectrum falls on a straight-line,
with little sensitivity to lattice volume. We
fit to
and
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The results are consistent with
in the continuum limit
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The Density of States Suppose the density of
low-lying states has the form
Where s d? ?(?) 3. Then the number of
eigenmodes with eigenvalue lt ? is
Then it can be shown that under the rescaling
n(z) is independent of lattice volume V. So
- measure n(?) at various Vs and determine ?
by trial and error.
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Measurement of n(?) Calculate the first Nev
eigenvalues in Nconf independent lattices. Sort
these from lowest to highest, regardless of
eigenvalue number or configuration. Then
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Connection to a multi-critical matrix model? A
density of states ?(?) (? - ?0)2 corresponds to
that of a multicritical matrix model of degree
m1 (hermitian matrix model with a polynomial
potential and fine-tuned couplings). (Akemann,
Damgaard, Magnea and Nishigaki) We have also
found that the distribution of level spacings in
the so-called unfolded spectrum also follows a
certain distribution (the Orthogonal
Distribution) deriving from random matrix theory.
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For a free theory, the number of eigenmodes per
unit physical volume, with eigenvalues less than
some physical cutoff ?/a2 lt ? is easily shown
to be
Finite, even for a ! 0
What is the number of localized states per unit
volume in Yang-Mills?
But ?? M/a , where M 0.042 Gev, so
in the continuum limit
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So if we compute the contribution to any
observable, keeping only a finite number of
eigenmodes per unit volume, then all the
contributing states are localized in the
continuum limit. This suggests a serious
breakdown of perturbation theory for adjoint
scalar particle Greens functions, at least in
quenched approximation. Another (related)
argument Lattice propagator in quenched
approximation
Consider the contribution from finite momentum
(p/a lt P) extended states
restricted sum over extended modes
To get to physical units
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The number of eigenmodes with momentum components
not exceeding some cutoff p/a lt P cant be
greater than the number of such momenta, which is
on the order of a4 V ?Vpphys , where V is the
lattice volume, and ?Vpphys P4 is the
momentum space volume in physical units. Then
So we can estimate the magnitude
finite
1
because
1
? 0
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The upshot is that the contribution of extended
modes with (mainly) finite momentum components to
the scalar adjoint propagator vanishes in the
continuum limit, at least in quenched
approximation and in the absence of tachyon
modes. The propagator is dominated by localized
modes, suggesting a behavior, even at short
distances, much different from that of
perturbation theory.
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The Yang-Mills Vacuum Wavefunctional
One more application of the adjoint covariant
Laplacian. We are interested in computing the
ground state vacuum wavefunctional ?0A ,
satisfying
where, in D1 dimensions
Physical states must satisfy, in addition, the
Gauss Law constraint
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For an abelian theory, the solution is well known
A long time ago (1979), I suggested that at large
distance scales, the pure Yang-Mills vacuum in a
confining theory looks like
This vacuum state has the property of dimensional
reduction Computation of a spacelike loop in
D1 dimensions reduces to the calculation of a
Wilson loop in Yang-Mills theory in D Euclidean
dimensions.
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In other words

• In D2 dimensions the Wilson loop can be
  calculated analytically, and we
• know there is an area-law falloff.
• (Also Casimir Scaling - Ambjorn, Olesen,
  Peterson)
• Evidence for this sort of wavefunctional?
• Strong coupling lattice expansion (J.G.)
• Strong coupling continuum calculation in 21
  (Karabali and Nair)
• Numerical experiments (Iwasaki and J.G.)
• Can we deduce the dimensional reduction form
  analytically
• at weak couplings?

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• Suppose we ask for a wavefunctional which
• Solves the YM Schrodinger equation to O(1) in
  the coupling
• Satisfies the Gauss Law constraint exactly.
• This is simply the gauge-invariant completion of
  the abelian vacuum state

which will look at large scales like the
dimensional reduction vacuum if the kernel
is short-range. Is it? One hint
- weak localization. In 2 space dimensions,
the lore is that all eigenmodes in a stochastic
potential are localized, no matter how weak the
disorder.
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Proof (of short range kernel) by Contradiction
Look for consistency of non-confining vacuum
fluctuations with ?00A. If no confinement in
?00A, then long range
long range correlated

A-field fluctuations (perturbative vacuum)
(perturbative vacuum) So lets
compute in a background of
free-field type vacuum fluctuations, and see if
really has a long range.
We will work in 21 dimensions. The abelian
free-field vacuum is
For a first go, I will drop the transverse (T)
condition.
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Discretize the D2 spatial dimensions Let -r2
be the 2D (ordinary) Laplacian, with plane-wave
eigenstates ?n0 and eigenvalues ?n0. Then a
typical vacuum fluctuation taken from the
probability distribution ?free2 is
where the sn,i are gaussian distributed random
numbers of variance 1. Construct three copies
Aic, and then form the link variables
where a kind of lattice coupling g is introduced
to control the size of the fluctuations. From
here we derive the adjoint covariant Laplacian in
the usual way, and compute the matrix
numerically.
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We then compute the gauge-invariant quantity
and average G(x,y) over all x,y separated by
distance R. Denote the result by G(R). Then we
plot G(R) vs R at various couplings g.
20 20 lattice. Start with g0, and check the
G(R) falls like 1/R as it should.
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As the coupling increases, we begin to see an
exponential falloff on the 2020 lattice i.e. a
finite range for the vacuum kernel
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Is the short range due to a radiative mass term,
i.e.
or is it due to localization? To find out, we
repeat the calculation, subtracting out the
lowest eigenvalue (which would serve as ?m2)
small (0.01) needed for invertibility
Result Range of G(R) increases, but is still
finite! Localization seems the likely
explanation.
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Implication - Since perturbative fluctuations
are inconsistent in the ?00 state, it appears
that the simplest gauge-invariant completion of
the abelian vacuum is confining in 21
dimensions. So what about 31 dimensions?
41? In the true vacuum, does range of G(R)
(glueball mass)-1 ? Not to get carried
awaythe simplest vacuum doesnt do everything

• N-ality scaling rather than Casimir scaling for
  the asymptotic string tension (center vortices
  and all that)
• String properties what about the Luscher term?
  Roughening?

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Conclusions

• The low-lying spectrum of the covariant Laplacian
  is localized, and the degree of localization
  depends on group representation.
• For the adjoint representation, the gap between
  the lowest eigenvalue and the mobility edge
  appears to diverge in the continuum limit.
• As a result, the quenched scalar propagator would
  differ from the perturbative expression even at
  short distances.
• In D21 dimensions, solving the Yang-Mills
  Schrodinger equation to zeroth-order in coupling,
  but satisfying the Gauss Law constraint exactly,
  results in a confining vacuum state.