# Topology, Random Matrix Theory and the spectrum of the Wilson Dirac operator

Article (PDF Available) · December 2011with27 Reads
Source: arXiv
Abstract
We study the spectrum of the hermitian Wilson Dirac operator in the epsilon-regime of QCD in the quenched approximation and compare it to predictions from Wilson Random Matrix Theory. Using the distributions of single eigenvalues in the microscopic limit and for specific topological charge sectors, we examine the possibility of extracting estimates of the low energy constants which parametrise the lattice artefacts in Wilson chiral perturbation theory. The topological charge of the field configurations is obtained from a field theoretical definition as well as from the flow of eigenvalues of the hermitian Wilson Dirac operator, and we determine the extent to which the two are correlated.

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Topology, Random Matrix Theory and the spectrum
of the Wilson Dirac operator
Albert Deuzeman
, Urs Wenger and Jaïr Wuilloud
Albert Einstein Center for Fundamental Physics
Institute for Theoretical Physics
University of Bern
Switzerland
E-mail: deuzeman@itp.unibe.ch, jair@itp.unibe.ch, wenger@itp.unibe.ch
We study the spectrum of the hermitian Wilson Dirac operator in the ε-regime of QCD in the
quenched approximation and compare it to predictions from Wilson Random Matrix Theory. Us-
ing the distributions of single eigenvalues in the microscopic limit and for speciﬁc topological
charge sectors, we examine the possibility of extracting estimates of the low energy constants
which parametrise the lattice artefacts in Wilson chiral perturbation theory. The topological
charge of the ﬁeld conﬁgurations is obtained from a ﬁeld theoretical deﬁnition as well as from the
ﬂow of eigenvalues of the hermitian Wilson Dirac operator, and we determine the extent to which
the two are correlated.
XXIX International Symposium on Lattice Field Theory
July 10 – 16, 2011
Squaw Valley, Lake Tahoe, California
Speaker.
c
Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/
arXiv:1112.5160v1 [hep-lat] 21 Dec 2011
Page 1
Topology, Random Matrix Theory and the Wilson Dirac operator spectrum Albert Deuzeman
1. Introduction
It has been known for some time that the low lying eigenvalues of the QCD Dirac operator are
reproduced in the microscopic limit by the eigenvalues of large random matrices with the correct
anti-hermitian symmetry structure [1, 2]. The connection between random matrix theory (RMT)
and chiral perturbation theory (χPT) has been established within the so-called ε-regime of QCD,
where pion masses are small enough that their wavelength becomes of the order of the ﬁnite size of
the lattice. If this condition is enforced, ﬁnite volume effects are noticeable even when the volume
is taken to inﬁnity. Pion momenta are suppressed and, to leading order, pion ﬁelds are constant
over the whole lattice while the effect of spontaneous chiral symmetry breaking is captured in a
single low energy constant (LEC), the quark condensate Σ. From the Banks-Casher relation it is
known that Σ can be deﬁned through the spectral density of the Dirac operator at the origin, so this
provides the connection between the spectrum of the Dirac operator and the one from RMT in the
microscopic regime.
So far, attempts to obtain physical parameters from numerical simulations of QCD in the ε-
regime have been restricted to the use of lattice Dirac operators with an exact chiral symmetry [3,
4, 5]. Recent progress has shown, however, that a sensible analysis can also be performed using
the standard Wilson Dirac operator, for which the chiral symmetry is explicitly broken. This is
made possible in the framework of Wilson χPT that includes the effects of the lattice discretisation
and that has recently been formulated in the ε-regime [6, 7]. The Lagrangian of this effective
low-energy theory,
L (U) =
1
2
mΣTr(U +U
) a
2
W
6
Tr(U +U
)
2
a
2
W
7
Tr(U U
)
2
a
2
W
8
Tr(U
2
+U
2
),
contains on top of the continuum Lagrangian three additional operators which parametrise the
lattice artefacts to leading order in a using the LECs W
6,7,8
. Chiral RMT, too, can be extended to
include the effects of these additional operators, leading to what is known as Wilson RMT [8, 9].
There is a one-to-one correspondence between the new parameters in both frameworks, so we will
use the Wilson χPT nomenclature for both.
Simulations of QCD, when taken towards the chiral limit at ﬁxed lattice spacing, are likely to
run into numerical instabilities due to the onset of ε-regime dynamics. To avoid such problems,
one needs a thorough understanding of the complicated way the spectrum depends on the different
scales like the quark masses, the lattice volume and the lattice spacing. In this context, analytic
results concerning the spectral density of the Wilson Dirac operator obtained in [8, 9, 10] have
already provided important insights. These proceedings report on a feasibility study, comparing
the Wilson RMT description to results from quenched QCD simulations in the ε-regime. Our
primary interest is whether the effective description can indeed reproduce the important features
of the Wilson Dirac spectrum that we measure, and we show that this seems to be the case. A
secondary goal is to determine the extent to which one can extract information on the LECs from
this setup. Eventually, the results could also have an impact on (Wilson) χPT analysis of dynamical
simulations outside of the ε-regime, since the same LECs occur in the regular regimes of χPT.
2
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Topology, Random Matrix Theory and the Wilson Dirac operator spectrum Albert Deuzeman
2. Deﬁnitions of topological charge
The QCD partition function in the ε-regime naturally separates into contributions from differ-
ent topological charge sectors [11]. When constructing Wilson χPT in the ε-regime, the presence
of the different sectors can be parametrised through a phase factor in the partition function,
Z =
ν
Z
U
dU det[U]
ν
exp(V L (U)),
decomposing it in a manner akin to a Fourier transform [8]. As a consequence of this construc-
tion the study of the low lying eigenvalues in the Dirac operator spectrum can be done separately
for each of the charge sectors, but this obviously requires good control over the topological prop-
erties of the system. However, at ﬁnite lattice spacing the topological charge can not be deﬁned
Figure 1: Sample of a typical eigenvalue ﬂow calculation
for a volume of 24
3
× 24 at a gauge coupling of β = 6.2
using thirty levels of HYP smearing. The region around
the critical mass m
c
where zeros are expected to occur is
indicated in blue, while the eigenmode drawn in red shows
a crossing eigenvalue.
uniquely and this creates an ambigu-
ity in the assignment of conﬁgurations
to the different charge sectors. The
straightforward deﬁnition from the ﬁeld
strength tensor ν
FT
R
V
F
µν
˜
F
µν
is at-
tractive, because it can easily be cal-
culated even on large lattices, but for
the current purpose, the natural deﬁni-
tion of the charge is through the chi-
ral overlap index. One way to cal-
culate it is provided by the eigenvalue
ﬂow method [12, 13] which counts the
number of real modes of D
W
weighted
with the signs of their chiralities. To be
speciﬁc, the procedure determines those
values of the mass m for which the her-
mitian operator D
5
(m) satisﬁes
D
5
(m)ψ = γ
5
(D
W
+ m)ψ = 0,
hence the zero modes of D
5
(m) correspond to the real eigenvalues of D
W
. Moreover, perturbation
theory shows that the slope of an eigenvalue as a function of m is given by the chirality of the
eigenmode, hence the ﬂow of each eigenvalue can be traced as a smooth function of the mass, as
illustrated in ﬁgure 1, and the net number of crossings then yields the chiral index. In practice,
one needs to choose a cut-off m
cut
beyond which no further physically relevant zeros are assumed
to exist. Obviously, this has an inﬂuence on the value of the index. Sufﬁciently close to the
continuum limit, however, any choice of m
cut
< m
c
' 0 will produce the same value for the index.
The approach to this limit can in fact be improved by HYP smearing, and for our current exploratory
studies we rather aggressively use thirty levels of HYP smearing.
3
Page 3
Topology, Random Matrix Theory and the Wilson Dirac operator spectrum Albert Deuzeman
Figure 2: A 2D histogram showing the
distribution of the topological charge as
deﬁned through a ﬁeld theoretical deﬁni-
tion versus the charge deﬁned through the
eigenvalue ﬂow on a 20
3
× 20 lattice at
β = 6.2.
A comparison between the ﬁeld theoretic deﬁni-
tion ν
FT
and the one from the eigenﬂow ν
EF
is shown in
ﬁgure 2 for conﬁgurations generated using the Wilson
gauge action at β = 6.2 on a volume of 20
3
× 20. Both
deﬁnitions agree for 96.4% of the conﬁgurations, while
the deviations are almost always limited to ν = 1.
The table included in ﬁgure 3 gives estimates for the
fraction of deviating results over a range of parame-
ters. For relatively ﬁne lattices and moderate volumes,
the ﬁeld theoretical charge deﬁnition and the overlap
index agree quite well. The agreement deteriorates as
we go to larger volumes, but the effect is mild and jus-
tiﬁes the use of the ﬁeld theoretic deﬁnition on lattices
for which explicit eigenvalue ﬂow calculations are too
expensive. Of course this comes at the cost of intro-
ducing a systematic error due to the occasional misin-
terpretation of the charge. To estimate the size of that
error, one can mimic the effect of misassigning charges
by mixing measurements from two separate RMT cal-
culations at identical parameters but in different charge
sectors. The results of such an exercise are included in ﬁgure 3. While the impact of mixing the
sectors is discernible, the impact is small enough to have limited impact on the precision of our
current preliminary ﬁts, even at large mixing ratios of up to 0.2.
Parameters frac
ν
FT
6=ν
EF
β = 5.9, 14
3
× 16 0.065
β = 6.2, 14
3
× 16 0.008
β = 6.2, 20
3
× 20 0.036
β = 6.2, 24
3
× 24 0.096
Figure 3: Left: Table of the estimated fraction of conﬁgurations that produce a different result for the
topological charge according to the ﬁeld theoretical and eigenvalue ﬂow deﬁnitions. Right: RMT simulations
of the spectral density of the Wilson Dirac operator at ν = 0, displaying the effect of mixing in 0% (black),
10% (blue) and 20% (red) conﬁgurations with charge ν = 1.
3. Wilson RMT and the Wilson Dirac operator spectrum of quenched QCD
With these preliminaries handled, we turn to the ﬁtting of our lattice data. One prerequisite
for a sensible comparison is the availability of eigenvalues small enough to be within the ε-regime.
When simulating at a Wilson gauge coupling of β = 6.2, the smallest volume producing a reason-
able number of eigenvalues without apparent bulk effects, but with a mass gap, turned out to be
4
Page 4
Topology, Random Matrix Theory and the Wilson Dirac operator spectrum Albert Deuzeman
24
3
× 24. At this volume the fraction of misassigned conﬁgurations is a manageable ten percent,
so we use the ﬁeld teoretical deﬁnition of the topological charge instead of the Wilson ﬂow charge.
Figure 4: Distributions of the 12 lowest lying eigenvalues of the hermitian Wilson Dirac operator in the
sectors ν = 0 (left) and ν = 1 (right) together with ﬁts including the effects of the operator W
8
. Results are
for a volume of 24
3
×24 at β = 6.2. The gray lines in the right panel indicate the effects of varying the value
of W
8
by ±10%.
In ﬁgure 4 we show two samples of the distribution of the 12 lowest lying eigenvalues of the
Wilson Dirac operator in the charge sectors ν = 0 and 1 for a volume of 24
3
× 24 at β = 6.2.
The spectrum does not appear to exhibit the undulating pattern that can, for example, be seen
in the RMT data of ﬁgure 3. If this was down to statistics only, we would require an increase
in statistics by an order of magnitude. However, rather than using the formulae of [9] for the
full spectrum, RMT can be used to extract distributions for the single eigenvalues. This provides
additional information which can be used in the ﬁtting procedure [14].
We therefore implemented a Monte Carlo spectrum calculation using Wilson RMT. The spec-
tra were then used to ﬁt the histograms of the separate eigenvalues, the results of which are also
shown in ﬁgure 4. To estimate the precision that can be reached in the determination of the LECs
by such an approach, we vary the value of W
8
. The right panel of ﬁgure 4 displays curves showing
the effect of a ten percent change in W
8
in gray. The impact of such a change in W
8
quickly dis-
sipates beyond the lowest eigenvalue, however, even a small 10% variation in W
8
produces a clear
effect on the spectrum, and we conclude that the value of W
8
can be constrained to within at least
ten percent using this procedure.
A complicating factor, however, is the potential presence of W
6
and W
7
in the effective theory.
While the assumption of their suppression is not unreasonable, they may in fact provide a possible
explanation for the observed lack of structure in the spectra. In our RMT setup they can straightfor-
wardly be included in the ﬁts and we refer to [9, 14] for details on the implementation in the Wilson
RMT. We ﬁnd (cf. ﬁgure 5) that the optimal ﬁt to single eigenvalue distributions including now the
effects from W
6
,W
7
and W
8
indeed leads to an absence of structure in the aggregate spectrum. At
this point, however, the effects of the various LECs are hard to disentangle and the uncertainties
5
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Topology, Random Matrix Theory and the Wilson Dirac operator spectrum Albert Deuzeman
Figure 5: Fits to the low lying eigenvalues of the Wilson Dirac operator in the charge sector ν = 1, including
the effects of all three LECs W
6,7,8
. Results are for a volume of 24
3
× 24 at β = 6.2.
on each of the parameters is considerably larger than when just taking W
8
into account. Since the
separate eigenvalues show a varying sensitivity to the different LECs, performing such ﬁts at larger
volumes, where more eigenvalues are available within the ε-regime, may help pinning down the
various predictions.
4. Conclusions
Recent developments which include lattice artefacts into the RMT allow for detailed predic-
tions of the low lying eigenvalue spectrum of the Wilson Dirac operator in the ε-regime of QCD
and provides a method for determining low energy constants of Wilson chiral perturbation theory.
In these proceedings, we have reported on a preliminary study concerning the practical applica-
bility of these predictions and we examined the sensitivity to the LECs. A potentially important
source of uncertainty for these ﬁts lies in the correct separation of conﬁgurations into topological
sectors. We have studied the use of a ﬁeld theoretical deﬁnition of the topological charge as a
predictor for the overlap index which is the natural choice in connection with the Wilson Dirac
operator spectrum. A high degree of correlation between the two deﬁnitions is found, although the
ﬁeld theoretical approach becomes unreliable as the volume is increased. If the LECs W
6
and W
7
are assumed to be zero, we ﬁnd that W
8
can be constrained to within about ten percent. For this
purpose, ﬁts to the separate eigenvalues, currently only available from direct RMT Monte Carlo
calculations, appear to be the most powerful tool. The particular pattern of deviations seen in the
separate eigenvalues, however, seems to point to a non-negligible contribution from the additional
LECs W
6
and W
7
. For a more elaborate and more quantitative analysis along the lines described
here (and consistently using the topological charge as determined from the eigenvalue ﬂow), we
refer to our detailed results presented in [14]. Furthermore, we would also like to point to the work
by Heller et al. [15] who presented the results of a similar study at this conference [16].
6
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Topology, Random Matrix Theory and the Wilson Dirac operator spectrum Albert Deuzeman
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• ##### Low-lying Dirac operator eigenvalues, lattice effects and random matrix theory
[Show abstract] [Hide abstract] ABSTRACT: Recently, random matrix theory predictions for the distribution of low-lying Dirac operator eigenvalues have been extended to include lattice effects for both staggered and Wilson fermions. We computed low-lying eigenvalues for the Hermitian Wilson-Dirac operator and for improved staggered fermions on several quenched ensembles with size $\approx 1.5$ fm. Comparisons to the expectations from RMT with lattice effects included are made. Wilson RMT describes our Wilson data nicely. For improved staggered fermions we find strong indications that taste breaking effects on the low-lying spectrum disappear in the continuum limit, as expected from staggered RMT.
Article · Dec 2011 · Journal of Physics A Mathematical and Theoretical
• ##### Mixing of orthogonal and skew-orthogonal polynomials and its relation to Wilson RMT
[Show abstract] [Hide abstract] ABSTRACT: The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such a random matrix ensemble. Although the chiral Gaussian unitary ensemble as well as the Gaussian unitary ensemble are associated to the Dyson index $\beta=2$ the intermediate ensembles exhibit a mixing of orthogonal polynomials and skew-orthogonal polynomials. We consider the Hermitian as well as the non-Hermitian Wilson random matrix and derive the corresponding polynomials, their recursion relations, Christoffel-Darboux-like formulas, Rodrigues formulas and representations as random matrix averages in a unifying way. With help of these results we derive the unquenched $k$-point correlation function of the Hermitian and then non-Hermitian Wilson random matrix in terms of two flavour partition functions only. This representation is due to a Pfaffian factorization drastically simplifying the expressions for numerical applications. It also serves as a good starting point for studying the Wilson-Dirac operator in the $\epsilon$-regime of lattice quantum chromodynamics.
Full-text · Article · Feb 2012
• ##### The Effect of the Low Energy Constants on the Spectral Properties of the Wilson Dirac Operator
[Show abstract] [Hide abstract] ABSTRACT: We summarize recent analytical results obtained for lattice artifacts of the non-Hermitian Wilson Dirac operator. Hereby we discuss the effect of all three low energy constants. In particular we study the limit of small lattice spacing and also consider the regime of large lattice spacing which is closely related to the mean field limit. Thereby we extract simple relations between measurable quantities like the average number of additional real modes and the low energy constants. These relations may improve the fitting for the low energy constants.
Full-text · Article · Oct 2013 · Journal of Physics A Mathematical and Theoretical