In this se tion we dis uss the great breakthrough asso iated with the fa t
that itwas realized that there is an alternative viewon hiral symmetry on
the latti e, i.e. that the Nielsen-Ninomiya theorem an be over ome in an
elegantway.
1.4.1 Ginsparg-Wilson relation
In 1982 (i.e. only one year after establishing the Nielsen-Ninomiya
theo-rem), Ginsparg and Wilson, basing on renormalization group
transforma-tions,showed [34℄thataremnantof hiralsymmetryispresentonthelatti e
without the doublermodes, if the orresponding Dira operator
D ˆ
obeys anequation now alledthe Ginsparg-Wilsonrelation:
γ 5 D + ˆ ˆ Dγ 5 = a ˆ Dγ 5 D. ˆ
(1.70)It is a modi ation of the anti ommutation relation(1.52) and the term on
the right-handside vanishesinthe ontinuum limithen e, inthislimitthe
standard hiral symmetry relation(1.52) is regained.
However, for many years it has not been realized that the
Ginsparg-Wilson relation allows one to dene hiral symmetry also on the latti e,
i.e. at non-vanishinglatti e spa ing. It lasted until around 1997 before the
Ginsparg-Wilson relation was redis overed. First, P. Hasenfratz realized
that a kind of latti e fermions alled the xed point fermions satises this
relation [35, 36℄ and shortly afterwards a similar observation was made by
Neuberger regarding the overlap formalism [37, 38℄. Moreover, Lüs her [29℄
found that the Ginsparg-Wilsonrelationleads toa non-standardrealization
oflatti e hiralsymmetry. Thefermiona tionisinvariantunderthefollowing
hiral rotation:
ψ → e iθγ 5
“ 1− a ˆ 2 D ”
ψ, ¯ ψ → ¯ ψe iθγ 5
“ 1− a ˆ 2 D ”
.
(1.71)Inthe ontinuumlimitthistransformationis(1.26)with
u i = 1
. Toprovetheinvarian e of the massless a tion with respe t to the above transformation,
intera tionsandtheexpli itviolationofisospinsymmetrybydierentupanddownquark
masses. InLatti eQCDwithmass-degeneratequarksthe hargedandneutralpionmasses
should beequal.
the left- and right-handed proje tors of fermion elds [11℄ with a modied
γ 5
-matrixγ ˆ 5 = γ 5 (1 − a ˆ D)
:P ˆ ± = 1 ± ˆγ 5
2 .
(1.72)Thusdened proje torshavethe sameproperties asthe standard ontinuum
proje tors andhen e one an de ompose the fermion part ofthe Lagrangian
into left- and right-handed massless parts and a symmetry breaking mass
term that mixes the left- and right-handed omponents. Taking the
proper-ties ofthe latti eproje tors
P ˆ ±
intoa ount,onends thatthe masstermisof the form
m ¯ ψ
1 − a ˆ 2 D
ψ
, whi h meansthat themassive Ginsparg-Wilson Dira operatorD(m) ˆ
that orresponds tothe massless operatorD ˆ
reads:D(m) = ˆ ˆ D + m 1 − a ˆ D 2
!
=
1 − am
2 ˆ D + m.
(1.73)Sin etheGinsparg-Wilsonrelationisanon-standardrealizationof hiral
symmetry,the onditionsoftheNielsen-Ninomiyatheoremdonotapply and
one anhave hiralsymmetry withoutthe doublers,whi hwas onsidered to
beimpossible for many years.
Moreover, itwasalsoshownbyHasenfratz,Laliena,Niedermayer[39℄and
in adierentway by Lüs her [29℄ thatthe Ginsparg-Wilsonrelationimplies
thatthe axialanomalyis orre tlyreprodu ed. Thea tionisinvariantunder
the transformation (1.71), but the fermioni measure
D ¯ ψDψ
is not itsJa obian
J
is non-trivial:J = exp[−2iθ
Tr(γ 5 (1 − a ˆ D/2))]
and it an alsobe expressed as
J = exp[−2iθQ top ]
, whereQ top
is the topologi al harge, to be dis ussed later. This issue was further elu idated by Fujikawa [40℄, whostudied the ontinuum limitof this Ja obian.
Furthermore, a onsequen e of the Ginsparg-Wilson relation is that
fer-mionsare prote tedfromadditivemass renormalizationandmixingbetween
four-fermion operators in dierent hiral representations (Hasenfratz [36℄)
and there an beno
O(a)
latti e artefa ts (Niedermayer [30℄).Inthenextse tionwedis ussoneofthesolutionsofthe Ginsparg-Wilson
equation, dening the so- alled overlap operator.
1.4.2 Overlap fermions
As wehave already stated, for many years it has not been realizedthat the
Ginsparg-Wilsonrelationprovidesauseful (fromthesimulationalviewpoint)
tions of this equationhave been known. In 1997, Neuberger[37, 38℄found a
parti ularlysimpleformofalatti eDira operatorthatobeysthe
Ginsparg-Wilson relation. It is now usually referred to as overlap fermions and the
massless overlap Dira operatoris given by:
D ˆ
ov(0) = 1 a
1 − A(A † A) −1/2
,
(1.74)where:
A = 1 + s − a ˆ D
Wilson(0)
(1.75)and
s
is a parameter whi h satises|s| < 1
and an be used to optimizelo ality properties. Note that instead of
D ˆ
Wilson(0)
, one ould use in thekernel operator
A
any massless latti e Dira operator that is lo al and hasnodoublermodes[30℄. Moreover, ifthe operatorusedin
A
itselfsatisestheGinsparg-Wilsonrelation,itwillbejustreprodu edbyeq. (1.74),sin ethen
A † A = 1
.The massiveoperator is given, a ording to (1.73),by:
D ov (m) =
1 + s − am 2
D ov (0) + m,
(1.76)where
m
is the bare overlap quark mass.Afterthe overlap operatorwas proposed,it was essential toshowthat it
is lo al. Else, this attitude would lead to ausality violations and render it
useless. The denition (1.74) in ludesthe highly non-lo al term
(A † A) −1/2
,whi h raises doubts about lo ality. Algebrai ally, stri t lo ality (or
ultra-lo ality) would mean that the Dira operator matrix element
D(x, y) ˆ
on-ne ting sites
x
andy
of the latti e is non-zero only if the distan e betweenx
andy
is smaller thansome spe ied smallvalue and alsothat this matrixelement dependsongauge linksonlyin somesmall neighbourhoodofsites
x
and
y
[41℄. These properties are true for the Wilson-Dira operator (hen e, itisasparsematrix),but theoverlapDira operatorhasnon-zero entriesforall pairs of latti esites and thusit is lear that itis not stri tly lo al.
However, stri t lo ality for a Dira operator is not really needed. It is
enoughthattheDira operatorfallsoexponentially,i.e. wehave
(suppress-ing the Dira and olorindi es):
|| ˆ D(x, y)|| ≤ Ce −ρ||x−y|| ,
(1.77)for some onstants
C
andρ
, where|| · ||
is the distan e between sitesx
andy
, e.g. the taxi-driver distan e||x − y|| = P
µ |x i − y i |
. If su h inequalityholds, it means that the intera tion range in physi al units
1/ρ
tends to 0(the de ay rate in latti e units
aρ
does not depend on the latti e spa ing)asone approa hes the ontinuumlimitand inthe ontinuum onehas alo al
eld theory, as desired[11℄.
A thorough analyti al and numeri al investigationof the lo ality of the
overlap Dira operator was performed by Hernandez, Jansen and Lüs her
[41℄, who showed that this operator is lo al under very general onditions,
i.e. for a wide range of bare oupling onstants.
1.4.3 Other kinds of hiral fermions
Apart from overlap fermions, there a few kinds of latti e fermions that also
preserve hiral symmetry. In this subse tion, we shortly dis uss a few of
them.
1.4.3.1 Domain wall fermions
Closely related (mathemati allyequivalent) to overlap fermions are domain
wall fermions, introdu edby Kaplan[42℄ and Shamir[43℄ in 1992 and 1993,
respe tively. The general idea of this approa h is to introdu e an auxiliary
(non-physi al) fth dimension and onsider massive Dira fermions with a
spa e-dependent mass in the shape of a domain wall. Kaplan showed that
su h theory has a zero mode with denite hirality lo alizedon the domain
wall and from the pointof view of the 4-dimensional theory this zero mode
is a hiral fermion. The way that this formulation ir umvents the
Nielsen-Ninomiya theorem is that translational invarian e in the 5-dimensional
sys-temisbroken(by thespa e-dependentmassterm),butitisstill onserved in
the 4-dimensionalphysi alworld [44℄. Ifthe fthdimension isinnite,there
does not exist a doubler mode. But in the ase of a nite fth dimension
(whi h is of ourse always true in a latti e simulation),an extra zero mode
of opposite hirality appears on a se ond domain wall. However, both zero
modes have an exponentially smalloverlap and hen e an not ommuni ate
if theirseparation islarge enough. What ismore,itwas alsoshown that the
anomaly stru ture is orre t both in the innite and nite fth dimension
ase. A rst investigationof these properties was performedby Jansen [45℄,
shortly after the birth of the idea of domain wallfermions.
Aftertheoverlapformalismwasinvented,Neubergeralsoshowed[46℄that
domain wall fermions with innite fthdimension are equivalent to overlap
fermions. Therefore, at nite fth dimension, they an be regarded as an
approximationto overlap fermions.
In pra ti al simulations, the domain wall formalism is now widely used
in a dynami al setup (e.g. by the RBC/UKQCD Collaboration [47℄) or in
staggered fermions in the sea se tor [48℄). However, the size of the fth
dimension is usually taken in the range 8-16, whi h means that the hiral
symmetryisonlyapproximateandthisentailsadditivemassrenormalization
of the quark mass, i.e. a shift away from zero of the bare quark mass for
whi h one has a vanishing pion mass [10℄. The value of this shift is usually
referred to asthe residualmass.
The mainadvantage of domainwall fermions withrespe t to the Wilson
fermions(and othernon- hirallysymmetri formulations)isthat hiral
sym-metry breaking by the domain wall fermions is rather mild and is believed
to be ontrollable. Their main disadvantage is that one needs to simulate
a 5-dimensional theory, instead of a 4-dimensional one, and hen e the
om-putational ost is higher by a fa tor of the order of the size of the fth
dimension.
Withrespe ttooverlapfermions,anadvantageisthatthis omputational
ost is still mu h smaller than the one for overlap, atthe pri e, however, of
not havingexa t hiral symmetry, but only anapproximationto it.
1.4.3.2 Creutz fermions
A dierent approa h to ir umvent the Nielsen-Ninomiya theorem is to
re-stri t oneself to the minimal number of doubler modes, i.e. to two modes
of opposite hirality. This was pointed out in the 1980s by Karsten [49℄
and Wil zek [50℄. Re ently, this idea reemerged in the work of Creutz [51℄,
who wasmotivatedbythe ele troni stru tureof graphene (whi hisbuilt of
two-dimensionallayers of graphite). The low-energyex itations ingraphene
are des ribed byatwo-dimensionalDira equationformasslessfermionsand
are hen e hiral. Furthermore, hirality is a hieved exa tly in the way that
involves the minimum number of fermion modes required by the
Nielsen-Ninomiya theorem, i.e. they are minimally-doubled. Creutz showed how
to generalize these properties to four dimensions. Creutz's idea was soon
elaboratedon by Bori i[52℄, who derived a moregeneralform ofthe a tion.
Creutzfermionsexhibitanexa t
N f = 2
avour ontinuum hiralsymme-try, whi h implies also that the leadingdis retization errors are of
O(a 2 )
10,and they are stri tly lo al. These are very appealing properties, sin e they
imply that one ould simulate hiral fermions without the high
omputa-tional ost ofoverlap fermions. However, Creutz fermionsbreak anumberof
dis rete symmetries, su h as parity, harge onjugation and time ree tion
[53℄. Therefore, toapproa h the ontinuum limitinthe intera ting ase one
10
An expli ittestof thispropertywillbedis ussedin Chapter2.
in the Symanzik ee tive a tion and this would make the pra ti al
simula-tions with Creutz fermions very di ult. However, a preliminary quen hed
test wasperformedby Bori i[54℄, who omputed thepion mass andfound a
behaviour onsistentwiththe predi tionsof hiral perturbationtheory. This
ledhimtoa on lusionthat Creutz fermionsare stillworth exploringin the
future, despitethefa tthattheybreakimportantdis retesymmetries. Also,
the on lusion by Bedaque et al. [53℄ was that for ertain values of the
pa-rameters, the minimally-doubled fermion a tions may exhibit non-standard
symmetries, that ouldeliminaterelevantoperatorsoftheSymanzikee tive
theory and hen emoderate the problemof dis rete symmetriesbreaking.
The expressions for the Dira operator for Creutz fermions (by whi h
we will mean both fermions related to Creutz's original idea and Bori i's
generalization) will be dis ussed in Chapter 2, only in the ontext of a test
of their ontinuum-limits alingat tree-level of perturbation theory [55, 56℄.
1.4.4 Topology on the latti e
TheQCDva uumhasanon-trivialtopologi alstru ture,whi hhasmany
im-portant impli ations for hadron properties. Forexample, we have remarked
in Se tion 1.1 that the mass of the avour singlet
η ′
meson is related tothe topologi alu tuations of the QCD va uum. This is aninherently
non-perturbativephenomenon andhen eseems tobewell-suitedtobeaddressed
by Latti eQCD al ulations.
Letusstart withthe eld-theoreti aldenitionof thetopologi al harge:
Q top = 1 32π 2
Z
d 4 x ǫ µνρσ
Tr(F µν (x), F ρσ (x)) ≡ Z
d 4 x q(x),
(1.78)where
q(x)
is alled the topologi al harge density [57℄. Gauge eld ong-urations that have a non-zero and integer topologi al harge are e.g.super-positionsof instantons [58℄ and anti-instantons,whi hare lassi al solutions
of the Eu lidean eld equations.
Thetopologi al harge an berelatedtothenumberof hiralzero modes
of the massless Dira operatorvia the Atiyah-Singer index theorem [59℄:
Q top = Q index ≡ N − − N + ,
(1.79)where
N ±
denotesthenumberofzeromodesinthepositive/negative hirality se tor andQ index
is alled the index of the Dira operator.If one wants to ompute the topologi al harge of a given gauge eld
ongurationonthe latti e,one an,in prin iple,use the dis retizedversion
[60℄. This an be over ome by applying smearing on gauge ongurations,
e.g. APE smearing[61℄, whi hmovesthe topologi al harge losertointeger
values, but it an also destroy small topologi al obje ts and thus lead to
in orre t values of the harge.
Su h problems are avoided if one uses the index theorem and omputes
the topologi al harge as the index of the massless Dira operator. For this
to be possible, one has to employ a Dira operator that an have hiral
zero modes (at any value of the latti e spa ing), i.e. eigenstates with zero
eigenvalue,whi hhavedenite hirality(thatistheyarealsoeigenstatesof
γ 5
witheigenvalue
±1
). ThismeansthatthemasslessDira operatormustobey(latti e) hiral symmetry, e.g. it an be the overlap Dira operator, whi h
will beused to ompute topologi al harge infurther part of this work.
Sin ethe QCDpath integralissymmetri with respe t tothesign of the
topologi al harge, wehave
hQ top i = 0
. However, anon-trivialquantity that one an ompute is relatedto the u tuations of the topologi al harge andis alled topologi alsus eptibility. In the ontinuum, itis dened by:
χ top = Z
d 4 xhq(x)q(0)i,
(1.80)whi hon the latti e be omes
χ top = hQ 2 index i
V ,
(1.81)where
V
is the latti e volume.Ithasbeenmentionedbeforethatthetopologi alsus eptibility isrelated
via the Witten-Veneziano formula(1.27) to the mass of the
η ′
meson.Phe-nomenologi ally, this formula implies a value of
χ top ≈ (180
MeV) 4
. It isworth to emphasize that this value agrees rather wellwith several quen hed
latti e omputations,e.g. [62℄.