A numerical solution to the local cohomology problem in U(1) chiral gauge theories

0.4 MB·English

by  Daisuke Kadoh

A numerical solution to the local cohomology problem in U(1) chiral gauge theories
Daisuke Kadoh1,∗ and Yoshio Kikukawa1,†
1Department of Physics, Nagoya University Nagoya 464-8602, Japan
(Dated: February 1, 2008)
We consider a numerical method to solve the local cohomology problem related to the gauge
anomaly cancellation in U(1) chiral gauge theories. In the cohomological analysis of the chiral
anomaly, it is required to carry out the differentiation and the integration of the anomaly with
respect to the continuous parameter for the interpolation of the admissible gauge fields. In our
numerical approach, the differentiation is evaluated explicitly through the rational approximation
of the overlap Dirac operator with Zolotarev optimization. The integration is performed with
a Gaussian Quadrature formula, which turns out to show rather good convergence. The Poincar´e
lemmaisreformulatedforthefinitelatticeandisimplementednumerically. Wecomputethecurrent
associated with thecohomologically trivial part of thechiral anomaly in two-dimensions and check
4 its locality properties.
0
0 PACSnumbers: 11.15.Ha,12.38.Gc.
2

n
a I. INTRODUCTION Here kµ(x) is a gauge-invariantlocal current. This local
J currentisinturnusedinthegauge-invariantconstruction
1 The construction of gauge-covariant and local lat- of the functional measure of the Weyl fermions.
2 tice Dirac operators which satisfy the Ginsparg-Wilson If one thinks of the practical computation of observ-
relation[1, 2, 3, 4, 5, 6, 7], ables in the lattice U(1) chiral gauge theories, it is re-
2 quired to compute the local current in Eq. (4) for every
v γ5D+Dγ5 =2aDγ5D, (1) admissiblegaugefield. Thepurposeofthispaperistoat-
5
tempt a numerical computation of the local current. We
2 has made it possible to introduce Weyl fermions on the
0 lattice and construct anomaly-free chiral gauge theories will compute kµ(x) in two-dimensions numerically and
1 with exact gauge invariance[8, 9, 10, 11, 12][13]. One of check its locality properties. We will also check how the
0 exact cancellation of gauge anomaly works on the finite
thecrucialstepsinthegauge-invariantconstructionisto
4 lattice.
establish the exact cancellation of the gauge anomaly at
0
a finite lattice spacing. This paper is organized as follows. In section II we
/
t In the caseof U(1) chiralgaugetheories[9][37], the ex- formulate the vector-potential-representation of the link
a
l act cancellation has been achieved through the cohomo- variables for the admissible U(1) gauge fields on the fi-
p- logical classification of the chiral anomaly[8, 35, 36][38]. nite lattice. Using this representation,we introduce one-
e TheanomalyisgivenintermsoflatticeDiracoperator[7, parameter families of the admissible fields for the in-
h 46, 47, 48, 49, 50] as terpolations. In section III we describe our numerical
: methodtocomputethebi-localcurrentwhichisthefirst-
v
i q(x)=tr{γ5(1−aD)(x,x)} (2) differentialofthechiralanomalywithrespecttothevec-
X tor potential. In section IV the Poincar´elemma is refor-
and the local field q(x) is a topological field in the sense
r mulated for a finite lattice so that we can carry out the
that it satisfies
a
cohomologicalanalysis directly on the finite lattice. The
δq(x)=0 (3) result of the cohomological analysis in two-dimensions
x is summarized. In section V we describe our numerical
X
under a localvariationofthe gaugefield. It followsfrom resultofthecomputationofthelocalcurrentkµ(x). Sec-
thispropertythattheanomalyiscohomologicallytrivial, tion VI is devoted to a summary and discussions.
e qα(x)=∂∗k (x), qα(x)= q(x)| , (4)
α µ µ U→Ueα II. ADMISSIBLE U(1) GAUGE FIELDS
Xα ON A FINITE LATTICE
forananomaly-freemultipletofWeylfermionswhichsat-
isfies the condition of the U(1) charges, Our first step is to formulate the vector-potential-
representation of the link variables associated with an
e3 =0. (5)
α admissible U(1) gauge field on a finite lattice. Such a
α
X representation has been formulated in the original co-
homological analysis in [8]. However, it is constructed
for the admissible gauge fields on the infinite lattice and
∗[email protected] theresultedvector-potentialsarenotboundedingeneral.
†[email protected] This representation therefore does not seem to be useful
2
for numerical implementations. But, as has been shown WemayregardU˜(x,µ)astheactuallocalanddynamical
in our previous paper[55], it is possible to formulate the degreesoffreedominthegiventopologicalsector. Thisis
boundedandperiodicvector-potentialrepresentationfor because the magnetic flux m is invariant with respect
µν
the admissible gauge fields on the finite lattice. to a local variation of the link field.
WesetthelatticespacingatounityandconsiderU(1) The following lemma shows that it is possible to es-
gauge fields on a finite two- or four-dimensional lattice tablish the one-to-one correspondence between U˜(x,µ)
(n = 2,4) of size L with periodic boundary conditions. and a periodic vector potential with the desired locality
L is assumed to be an even integer for simplicity. The properties on the finite lattice[55].
gaugefields onsuchalattice canbe representedthrough
periodic link fields on the infinite lattice, Lemma II There exists a periodic vector potential
A˜ (x) such that
µ
U(x,µ)∈U(1), x=(x ,x ,··· ,x )∈Zn, (6)
1 2 n
U(x+Lνˆ,µ)=U(x,µ) for all µ,ν =1,··· ,n.(7) eiA˜µ(x) =U˜(x,µ), (16)
2πm
The independent degrees of freedom are then the link ∂µA˜ν(x)−∂νA˜µ(x)=Fµν(x)− L2µν, (17)
variables at the points in the region
|A˜ (x)|≤π(1+4kxk) kxk≤L/2
µ
Γn ={x∈Zn|−L/2≤xµ <L/2}. (8) (cid:26)|A˜µ(x)|≤π(1+2L+2(n−1)L2) otherwise.
(18)
As to gauge transformations
Moreover, if A˜′ (x) is any other field with these proper-
U(x,µ)→Λ(x)U(x,µ)Λ(x+µˆ)−1, (9) µ
ties, we have
we consider only periodic functions Λ(x) ∈ U(1) which
A˜′ (x)=A˜ (x)+∂ ω(x), (19)
preserves the periodicity of the link field. µ µ µ
We impose the admissibility condition on the U(1) where the gauge function ω(x) takes values that are in-
gauge fields: teger multiples of 2π. The explicit formula of A˜ (x) in
µ
two-dimensions is given in the appendix.
|F (x)|<ǫ for all x,µ,ν, (10)
µν
As emphasized in [8], it is important to note that the
where the field tensor F (x) is defined through localitypropertiesofgaugeinvariantfieldsshouldbe the
µν
same independently of whether they are considered to
1 be functions of the link variables or the vector potential.
F (x)= lnP (x), −π <F (x)≤π, (11)
µν i µν µν Since the mapping
P (x)=U(x,µ)U(x+µˆ,ν)U(x+νˆ,µ)−1U(x,ν)−1.
µ,ν A˜ (x)→U˜(x,µ)=eiA˜µ(x) (20)
(12) µ
is manifestly local, this is immediately clear if one starts
We require this condition because it ensures that the
with a field composed from the link variables. In the
overlapDiracoperator[2,4],whichweadoptinthiswork,
other direction, one may start from a gauge invariant
is a smooth and local function of the gauge field for
localfieldφ(y)dependingthe vectorpotential. Thenthe
|1−P (x)|<1/30infour-dimensionsand|1−P (x)|<
µν µν key observation is that one is free to impose a complete
1/5in two-dimensions[6][56]. For ǫ<π/3 the admissible
axial gauge taking the point y as the origin. Around y
U(1) gauge fields on the finite lattice can be classified
the vectorpotentialislocallyconstructedfromthe given
uniquely by the magnetic fluxes m (integers indepen-
µν link field and φ(y) thus maps to a local function of the
dent of x) where
link variables residing there.
1 L−1 The vector potential A˜µ(x) represents an admissible
mµν = 2π Fµν(x+sµˆ+tνˆ). (13) fieldthrougheiA˜µ(x)×V[m](x,µ) andthe associatedfield
sX,t=0 tensor Fµν(x) = ∂µA˜ν(x) −∂νA˜µ(x) + 2πLm2µν is hence
bounded by ǫ. It is straightforward to check that this
In this respect, the following field is periodic and can be
propertyispreservedifthepotentialisscaledbyafactor
shown to have constant field tensor equal to 2πm /L2:
µν
t in the range 0 ≤ t ≤ 1, i.e. we can contract the vector
potential to zero without leaving the space of admissible
V[m](x,µ)=e−2Lπ2i[Lδx˜µ,L−1 ν>µmµνx˜ν+ ν<µmµνx˜ν], fields. Then, in the cohomological analysis of the chiral
P P (14) anomaly,wemaychooseV[m](x,µ)asthereferencegauge
where the abbreviation x˜ = x mod L has been used. field in a given magnetic flux sector and may consider
µ µ
Then any admissible U(1) gauge field in the topological theinterpolationbetweenthearbitrarygaugefieldinthe
sector with the magnetic flux m may be expressed as same sector and the reference field as follows:
µν
U(x,µ)=U˜(x,µ)V (x,µ). (15) Ut(x,µ)=eitA˜µ(x)×V[m](x,µ) t∈[0,1]. (21)
[m]
3
III. CHIRAL ANOMALY AND ITS the bi-local current, we introduce the parameter repre-
TOPOLOGICAL PROPERTIES sentation of the inverse square root of H2 as
w
H ∞ dt H
For our numerical application, we adopt the overlap w w
= . (29)
Dirac operator[2, 4] given by H2 π t2+H2
w Z−∞ w
p
1 H Then we can perform the differentiation of q(x) with re-
w
D = 1+γ5 , (22) spect to the vector potential A˜ (x) explicitly[58] as
2 Hw2! µ
p ∂q(x) 1 ∞ dt 1
where Hw is the Hermitian Wilson-Dirac operator, = − tr ×
∂A˜ (y) 2 π t2+H2
ν (cid:26)Z−∞ w
1 1
Hw =γ5 γµ2(∇µ−∇†µ)+ 2∇µ∇†µ−m0 (23) t2Vµ(y)−HwVµ(y)Hw t2+1H2(x,x) ,
(cid:18) (cid:19) w (cid:27)
(cid:0) (cid:1) (30)
(0 < m < 2). It is a smooth and local function
0
of the admissible gauge field for |1 − P (x)| < 1/30
µν where
in four-dimensions and for |1− P (x)| < 1/5 in two-
µν
dimensions[6][56]. Then the chiral anomaly, Eq. (2), is 1
V (y) = −iγ {(1−γ )δ δ U (x)
given in terms of H as µ 5 µ x,y x+µˆ,z µ
w 2
+(1+γ )δ δ U (z)−1 .
µ y,z x,z+µˆ µ
1 H
q(x)=− tr w (x,x) . (24) (31)
2 ( Hw2 ) (cid:9)
For the numerical evaluation of the differentiation,
p
Asisclearfromthisexpression,q(x)isatopologicalfield we then adopt the rational approximation of the
withrespecttoanylocalvariationoftheadmissiblegauge overlap Dirac operator[59, 60] with the Zolotarev
field, optimization[61, 62]. Namely, q(x) is approximated by
the formula with the degree N :
r
δq(x)=0. (25)
Xx q(x);−1tr h Nr bk (x,x) . (32)
This topological property of the chiral anomaly can 2 ( wk=1h2w+c2k−1 )
X
be cast into properties of a certain gauge-invariant bi-
local current. This bi-local current is defined through h is defined by H /λ where λ is the square root
w w min min
the differentiation and integration with respect to the oftheminimumoftheeigenvaluesofH2. Thedefinitions
w
continuous parameter t for the interpolation as follows: of the coefficients c and b are given in the appendix.
k k
With this approximation, Eq. (30) can be approximated
1 ∂q(x) as follows:
j (x,y)= dt . (26)
ν ∂A˜ (y)
Z0 (cid:18) ν (cid:19)A˜→tA˜ ∂q(x) 1 Nr 1
; − tr b ×
Tbih-leocoarligciunrarletnotpaoslogical field can be expressed with the ∂A˜ν(y) 2 (Xk=1 kh2w+c2k−1
1
(c v (y)−h v (y)h ) (x,x) ,
q(x)=q (x)+ j (x,y)A˜ (y), (27) 2k−1 µ w µ w h2 +c
[m] ν ν w 2k−1 (cid:27)
yX∈Γn (33)
whereq[m](x) denotes the topologicalfieldforV[m](x,µ). where vµ(y) = Vµ(y)/λmin. Finally, for the integration
Then the topological property and the gauge-invariance back with respect to the parameter for the interpolation
of q(x) imply that jν(x,y) satisfies in Eq. (26), we adopt the Gaussian Quadrature (Gauss-
Legendre) formula with N points:
←− g
j (x,y)=0, j (x,y)∂∗ =0. (28)
ν ν ν
xX∈Γn Ng ∂q(x)
j (x,y); w , (34)
These conditions provide the initial conditions for the ν Xi=1 i(cid:18)∂A˜ν(y)(cid:19)A˜→tiA˜
cohomologicalanalysis.
Forourpurpose,itisrequiredtocomputetheabovebi- where {(t ,w )|i = 1,··· ,N } is the set of the abscissas
i i g
localcurrentnumericallykeepingtheconditionsEq.(28) and weights of the degree N .
g
within a given accuracy. Our strategy is the following. Theformulae,Eqs.(33)and(34),arestillcomplicated
First of all, in order to obtain more explicit formula of for the numerical computation. But, in two-dimensions,
4
it is not demanding numerically for relatively small lat- The linear space of all these forms is denoted by Ω . An
k
ticesizestodiagonalizeH andstorealltheeigenvectors. exterior difference operator d : Ω → Ω may now be
w k k+1
Namely, we can write defined through
λ 1
hw(x,y)= λ ψλ(x)(cid:18)λmin(cid:19)ψλ†(y) (35) df(x)= k!∂µfµ1,···,µk(x)dxµdxµ1···dxµk, (40)
X
where∂ denotestheforwardnearest-neighbordifference
and then it is possible to evaluate Eqs. (33), (34) explic- µ
operator. The associated divergence operator d∗ :Ω →
itly. k
Ω is defined in the obvious way by setting d∗f =0 if
In this approximation, the gauge-invariance of the bi- k−1
f is a 0-form and
local currentand the second property of Eq. (28) can be
preserved exactly. As to the first property of Eq. (28),
1
we will see below that the choice N = 18 and N = 20 d∗f(x)= ∂∗f (x)dx ···dx (41)
r g (k−1)! µ µµ2···µk µ2 µk
gives good convergences for the admissible gauge fields
on the two-dimensional lattice of the size L = 8,10,12
in all other cases, where ∂∗ is the backward nearest-
andtheoriginaltopologicalfield,q(x),canbereproduced µ
neighbor difference operator.
through Eq. (27) within the error less than 1×10−14.
By definition, the divergence operator satisfies that
d∗2 =0 and therefore the difference equation d∗f =0 is
solvedbyallformsf =d∗g. Ithasbeenshownthatinthe
IV. COHOMOLOGICAL ANALYSIS OF
infinitelatticetheseareinfactallsolutions,anexception
CHIRAL ANOMALY ON A FINITE LATTICE
beingthe0-formswhereonehasaone-dimensionalspace
of further solutions[8]. This result is the lattice counter
A. The modified Poincar´e lemma on a finite lattice
partofthePoincar´elemmaknowninthecontinuumthe-
ory. On the finite periodic lattice the lemma does not
Our numerical cohomological analysis of the chiral
holdtrueanymore,becausethelatticeisan-dimensional
anomaly will be performed directly on a finite lattice.
torusanditscohomologygroupis nownon-trivial. How-
ForthispurposeitisrequiredtoreformulatethePoincar´e
ever, the lemma can be reformulated so that it holds
lemma on the lattice[8] for a finite lattice. For the de-
true up to exponentially small correction terms of order
tail of the proof of the lemmas, refer to our previous O(e−L/2̺) and for the form satisfying f(x) = 0,
paper[55]. x∈Γn
it holds exactly even on the finite lattice.[63] The lat-
Inthefollowingwewillconsidertensorfieldsf (x) P
µ1···µk ter result is the lattice counter part of the corollary of
on Γ that are totally anti-symmetric in the indices
n de Rham theorem known in the continuum theory. The
µ ,··· ,µ . Such tensor fields may be regarded as pe-
1 k precise statements are the following.[65]
riodic tensor fields on the infinite lattice,
Lemma IV.a (Modified Poincar´e lemma)
f (x+Lνˆ)=f (x) for all µ,ν =1,··· ,n.
µ1···µk µ1···µk Let f be a k-form which satisfies
(36)
The locality properties of such fields are assumed to be
d∗f =0 and f(x)=0 if k =0. (42)
as follows: for a certain reference point x ∈ Γ and
0 n
−L/2≤(xµ−x0µ)<L/2 (mod L), xX∈Γn
|f (x)| < C (1+kx−x kp1)e−kx−x0k/̺, Then there exist a form g ∈ Ωk+1 and a form ∆f ∈ Ωk
µ1···µk 1 0 such that
(kx−x k<L/2), (37)
0
|fµ1···µk(x)| < C2Lp2e−L/2̺, f =d∗g+∆f, |∆fµ1···µk(x)|<cLσe−L/2̺. (43)
(kx−x k≥L/2). (38)
0
Lemma IV.b (Corollary of de Rham theorem)
̺isalocalizationrangeofthetensorfield. C andp ≥0
i i
Let f be a k-form which satisfies
arecertainconstantsthatdonotdependonL. kx−x k
0
is the taxi driver distance from x to x. This local-
0
ity properties hold true for the differentials of the chiral d∗f =0 and f(x)=0. (44)
anomaly given in terms of overlap Dirac operator[2, 4] xX∈Γn
with respect to the admissible gauge fields[6][63].
Then there exist a form g ∈Ω such that
The differential forms on the finite lattice are intro- k+1
ducedasinthe continuum,following[8]. Ifwe adoptthe
f =d∗g. (45)
Einstein summation convention for tensor indices, the
general k-form on Γ is then given by
n
1 The explicit formula of g(x) in two-dimensions (n = 2)
f(x)= k!fµ1···µk(x)dxµ1···dxµk. (39) is given in the appendix.
5
B. A solution to the local cohomology problem Then we can see that q(x) is cohomologically trivial,
on the two-dimensional finite lattice
e qα(x)=∂∗k (x), (57)
α µ µ
In two dimensions, the cohomological analysis results α
X
in the following formula for q(x):
where k (x) is given explicitly as
µ
q(x)=A(x)+∂∗h (x)+∆q(x), (46)
µ µ
k (x)≡ e {h (x)+∆h (x)}α+∆k (x). (58)
µ α µ µ µ
where
α
X
A(x)=q[m](x)+φµν(x)F˜µν(x), ∂µ∗φµν(x)=0. (47) Therefore, once the bi-local current jµ(x,y) is com-
puted,thecurrentk (x)canbeobtainedbythesequence
φ (x) is obtained through the applications of the mod- µ
µν of the applications of the lemmas IV.a,b. The numeri-
ified Poincar´e lemma and the corollary of the de Rham
cal implementation of this step is straightforward using
theorem:
the explicit solutions, g(x), of the lemmas given in the
←−
j (x,y) −I−V−.→a θ (x,y)∂∗+∆j (x,y), (48) appendix.
ν νµ µ ν
←−
∆j (z,x) −I−V−.→b −2∆Φ (x)∂∗ (49)
ν νµ µ
zX∈Γ2 V. NUMERICAL RESULTS
and
We nowdescribeourresultofnumericalcomputations
1
of the local current k (x). We consider the lattice sizes
φ (x) ≡ θ (z,x)−∆Φ (x). (50) µ
µν µν µν
2 L = 8,10,12. Admissible gauge fields are generated by
zX∈Γ2 Monte Carlo simulation using the action
Intwo-dimensions,φ (x) is aconstantandassumesthe
µν
form φµν(x) = γ[m,w]ǫµν. hµ(x) is obtained through the S = 1 Fµν(x)2 =β F(cid:3)2 . (59)
application of the modified Poincar´e lemma: 4e2 1−F (x)2/ǫ2 1−F2/ǫ2
x,µν µν (cid:3) (cid:3)
X X
θ (x,y)−δ θ (z,y) −I−V−.→a ∂∗τ (x,y)
µν x,y µν λ λµν As reported in [66], the topological charge is preserved
zX∈Γ2 during the Monte Carlo updates with this type of ac-
(51) tion, even when ǫ is set to π. We adopt this option and
check the locality of the topologicalfield numerically for
and
several values of β. We consider the topological sectors
1 with m =0,1 andthe initial configurationis chosenas
h (x) ≡ τ (x,y)F˜ (y). (52) 12
µ 2 µνρ νρ V[m](x,µ) with a given m12.
yX∈Γ2 Foreachadmissiblegaugefield,wecomputethevector
∆q(x) is defined by potential A˜µ(x) formulated in section II. We also com-
pute the abscissas and weights {(t ,w )| i = 1,··· ,N }
i i g
∆q(x) ≡ ∆Φµν(x)F˜µν(x)+ ∆jν(x,y)A˜ν(y). for Gaussian Quadrature formula with the degree Ng.
By this, the discrete interpolation of the given admissi-
yX∈Γ2
ble gauge field is fixed:
(53)
For the anomaly-free multiplet satisfying the condi- U(i)(x,µ)=eitiA˜µ(x)×V (x,µ) i=1,··· ,N .
[m] g
tion e2 = 0, the anomaly part A(x) = q (x) +
α α [m] n (cid:12) (o60)
γ[m]ǫµPνF˜µν(x)cancelsuptoanexponentiallysmallterm, In table I, the abscissasand weightsf(cid:12)or the degreeNg =
20 are shown.
For each U(i)(x,µ), all eigenvalues and eigenvectors
e Aα(x)=∆A(x), (54)
α of H are computed numerically using the Householder
w
α
X method. We choose m0 = 0.9 in Hw. Then we compute
where Aα(x) = A(x)| . Since ∆A(x) = the bi-local current j (x,y) through Eqs. (33) and (34).
U→Ueα x∈Γ2 ν
0, we may apply the modified Poincar´e lemma IV.a to We can check the convergence of the above procedure
obtain P through Eq. (27) by computing the deviation
∆A(x) −I−V−.→a ∂µ∗∆kµ(x). (55) χ=maxx∈Γ2|δ(x)|, (61)
Onthe other hand, since x∈Γ2∆q(x)=0, we may also where
apply the lemma IV.b as follows:
P δ(x)=q(x)−q (x)− j (x,y)A˜ (y). (62)
∆q(x) −I−V−.→b ∂∗∆h (x). (56) [m] ν ν
µ µ yX∈Γ2
6
the lemmas. Theresultcanbeexpressedas(cf. Eq.(46)
TABLEI: TheabscissasandweightsfortheGauss-Legendre
)
formula are shown for Ng =20. t [0,1].
∈
q(x)=A(x)+∂∗h (x)+∆q(x), (63)
ti wi µ µ
1 3.435700407452558E-003 8.807003569576136E-003
where
2 1.801403636104310E-002 2.030071490019346E-002
3 4.388278587433703E-002 3.133602416705449E-002 A(x)=q[m](x)+γ[m,w]ǫµνF˜µν(x). (64)
4 8.044151408889061E-002 4.163837078835234E-002
In order to check the locality properties of the fields,
5 0.126834046769925 5.096505990862024E-002
q(x),A(x),h (x)and∆q(x),weapplyasmalllocalvari-
6 0.181973159636742 5.909726598075923E-002 µ
ation to the given admissible gauge field as U(x,µ) →
7 0.244566499024586 6.584431922458825E-002 eiηµ(x)U(x,µ) where
8 0.313146955642290 7.104805465919108E-002
9 0.386107074429177 7.458649323630189E-002 ηµ(x)=0.05×2πδx,x0δµ,1 (65)
10 0.461736739433251 7.637669356536294E-002
and compute the variations of the fields. For each varia-
11 0.538263260566749 7.637669356536294E-002
tion of the fields, δ f(x), we define
12 0.613892925570823 7.458649323630189E-002 η
13 0.686853044357710 7.104805465919108E-002 δ f(r)=max |δ f(x)| r =kx−x k (66)
η η 0
14 0.755433500975414 6.584431922458825E-002
15 0.818026840363258 5.909726598075923E-002 and see the locality pr(cid:8)operties (cid:12)(cid:12)of the fields by(cid:9)plotting
δ f(r) against r =k x − x k. For the anomaly part
16 0.873165953230075 5.096505990862024E-002 η 0
A(x), the variation of γ is considered.
17 0.919558485911109 4.163837078835234E-002 [m,w]
Theresultsareshowninfigures1,2and3forβ =3.0.
18 0.956117214125663 3.133602416705449E-002
In figure 1, the variation of the topological field q(x) is
19 0.981985963638957 2.030071490019346E-002
shown. Thelocalityofthetopologicalfieldq(x)isclearly
20 0.996564299592547 8.807003569576136E-003 seen. We can read the locality range as ̺ ≃ 0.5. The
maximum value of the field ∆q(x) is also shown in the
same figure. We can confirm that
Here the original topological fields q(x) and q (x) are
constructed by computing all eigenvalues and[mei]genvec- |∆q(x)|≃O(10−5)≃O(e−L/2̺) (L=12). (67)
tors of H for the given admissible gauge field U(x,µ)
w
Infigure2,thevariationsofthecurrenth (x)areshown.
and for the reference gauge field V (x,µ), respectively. µ
[m] It shows clearly that the current h (x) has the same lo-
In this computation,we found thatthe topologicalfields µ
cality property as the topologicalfield q(x) has and thus
have typically the values of order O(10−2) − O(10−3).
is local. In figure 3, the variations of the anomaly co-
The integer topological charge Q = q(x) = m
x∈Γ2 12 efficient γ and the field ∆q(x) are shown. We can
is reproduced within the error of order O(10−12) − [m,w]
O(10−13). In table II, we show the dPependence of χ on confirm that the gauge field dependence of γ[m,w] is in-
deed small and the same order of magnitude as the size
the degree N with N =18, L=8 and β =3.0 fixed.
g r of ∆q(x) and its variation.
The locality properties of the fields q(x), A(x), h (x)
µ
and∆q(x)areconfirmedalsofortopologicallynon-trivial
TABLE II: The convergence of the bi-local current is shown
gauge fields. See figure 4.
for various values of Ng with Nr =18,L=8,β =3.0 fixed.
We next examine the cancellation of the gauge
Ng χ anomaly. We consider the so-called 11112 model which
4 O(10−8) consists of four Left-handed Weyl fermions with unit
6 O(10−12) charge and one Right-handed Weyl fermion with charge
8 O(10−14) two. The gauge anomaly cancellation condition in two-
dimensions is satisfied as follows:
10 O(10−15)
12 O(10−15) 4
16 O(10−15) e2−(2e)2 =0. (68)
i=1
X
For the larger lattice sizes L = 10,12, we found that χ In figure 5, we plot the anomaly parts of the Left-
is less than 1×10−14 with the choice of N = 20 and handed fermions 4 × A1(x) and of the Right-handed
g
N =18. fermion A2(x), where Aα(x) = A(x)| , and the
r U→Ueα
Once the bi-localcurrentj (x,y) is computed, the co- total anomaly part Aα(x) = 4×A1(x)−A2(x) for
µ α
homological analysis of the chiral anomaly can be per- β = 3.0 and m = 0. The result is impressive. The
12
P
formednumericallybythesequenceoftheapplicationsof size of the total anomaly part is reduced to the order
7
(cid:111)
Localityofq(x) (cid:111)(cid:98)(cid:111)(cid:93)(cid:107)(cid:95)(cid:118) Localityofgamma,dq(x) (cid:101)(cid:98)(cid:111)(cid:95)(cid:107)(cid:107)(cid:95)
L=12 L=12 (cid:98)(cid:111)(cid:93)(cid:107)(cid:95)(cid:118)
(cid:46)(cid:44)(cid:47) (cid:46)(cid:44)(cid:47)
(cid:46)(cid:44)(cid:46)(cid:46)(cid:47) (cid:46)(cid:44)(cid:46)(cid:46)(cid:47)
(cid:47)(cid:46)(cid:43)(cid:51) (cid:47)(cid:46)(cid:43)(cid:51)
(cid:111)(x) gamma
d(cid:111)(x)
(cid:47)(cid:46)(cid:43)(cid:53) (cid:47)(cid:46)(cid:43)(cid:53)
(cid:47)(cid:46)(cid:43)(cid:55) (cid:47)(cid:46)(cid:43)(cid:55)
(cid:47)(cid:46)(cid:43)(cid:47)(cid:47) (cid:47)(cid:46)(cid:43)(cid:47)(cid:47)
(cid:43)(cid:48) (cid:46) (cid:48) (cid:50) (cid:52) (cid:54) (cid:47)(cid:46) (cid:47)(cid:48) (cid:47)(cid:50) (cid:43)(cid:48) (cid:46) (cid:48) (cid:50) (cid:52) (cid:54) (cid:47)(cid:46) (cid:47)(cid:48) (cid:47)(cid:50)
(cid:122)(cid:118)-x0(cid:122) (cid:122)(cid:118)-x0(cid:122)
FIG. 1: The variation of the topological field δηq(r)(filled FIG.3: Thevariationsδηγ[m,w](filleddiamond)andδη∆q(r)
circle) is plotted against r= x x . The maximum value (filledtriangle)areplottedagainstr= x x . Foraguide,
0 0
k − k k − k
of ∆q(x) (cross) is also shown. The lattice size is L = 12. the variation δηq(r) (open circle) and the maximum value of
| |
The gauge field is generated at β = 3.0 with the vanishing ∆q(x)(cross) are also shown.
| |
magnetic fluxm =0.
12
(cid:102)(cid:47)
(cid:102)(cid:48)
(cid:111) Localityofh1(x),h2(x) (cid:111)
Localityofh1(x),h2(x) (cid:102)(cid:102)(cid:47)(cid:48) Q=1,L=10 (cid:98)(cid:111)(cid:93)(cid:107)(cid:95)(cid:118)
L=12 (cid:98)(cid:111)(cid:93)(cid:107)(cid:95)(cid:118)
(cid:46)(cid:44)(cid:47)
(cid:46)(cid:44)(cid:47)
(cid:46)(cid:44)(cid:46)(cid:46)(cid:47)
(cid:46)(cid:44)(cid:46)(cid:46)(cid:47)
(cid:47)(cid:46)(cid:43)(cid:51)
h1,h2
(cid:47)(cid:46)(cid:43)(cid:51)
h1,h2
(cid:47)(cid:46)(cid:43)(cid:53)
(cid:47)(cid:46)(cid:43)(cid:53)
(cid:47)(cid:46)(cid:43)(cid:55)
(cid:47)(cid:46)(cid:43)(cid:55)
(cid:47)(cid:46)(cid:43)(cid:47)(cid:47)
(cid:47)(cid:46)(cid:43)(cid:47)(cid:47) (cid:43)(cid:48) (cid:46) (cid:48) (cid:50) (cid:52) (cid:54) (cid:47)(cid:46) (cid:47)(cid:48) (cid:47)(cid:50)
(cid:43)(cid:48) (cid:46) (cid:48) (cid:50) (cid:52) (cid:54) (cid:47)(cid:46) (cid:47)(cid:48) (cid:47)(cid:50) (cid:122)(cid:118)-x0(cid:122)
(cid:122)(cid:118)-x0(cid:122)
FIG. 4: The variations δηhµ(r) are plotted against r =
k
FIG. 2: The variations δηhµ(r) are plotted against r =k x−x0k(filledsquareandtriangle). Foraguide,thevariation
x−x0k(filledsquareandtriangle). Foraguide,thevariation δηq(r)(opencircle)andthemaximumvalueof|∆q(x)|(cross)
δηq(r)(opencircle)andthemaximumvalueof ∆q(x)(cross) are also shown. The lattice size is L=10. The gauge field is
| |
are also shown. generated at β=3.0 with themagnetic fluxm12 =1.
8
(cid:95)(cid:108)(cid:109)(cid:107)(cid:95)(cid:106)(cid:119)
(cid:95)(cid:108)(cid:109)(cid:107)(cid:95)(cid:106)(cid:119)
(cid:98)(cid:111)
(cid:50)(cid:40)(cid:111)(cid:47)
Anomalycancellation(11112) (cid:48)(cid:40)(cid:111)(cid:48) Anomalycancellation(11112) (cid:111)
onthefinitelattice(L=10)
(cid:47)
(cid:46)(cid:44)(cid:46)(cid:47)
(cid:46)(cid:44)(cid:46)(cid:47)
(cid:46)(cid:44)(cid:46)(cid:46)(cid:46)(cid:47)
(cid:46)(cid:44)(cid:46)(cid:46)(cid:46)(cid:47)
(cid:119)
(cid:95)(cid:106)(cid:119) (cid:107)(cid:95)(cid:106)
(cid:95)(cid:108)(cid:109)(cid:107) (cid:47)(cid:46)(cid:43)(cid:52) (cid:95)(cid:108)(cid:109) (cid:47)(cid:46)(cid:43)(cid:52)
(cid:47)(cid:46)(cid:43)(cid:54) (cid:47)(cid:46)(cid:43)(cid:54)
(cid:47)(cid:46)(cid:43)(cid:47)(cid:46) (cid:47)(cid:46)(cid:43)(cid:47)(cid:46)
(cid:46) (cid:48)(cid:46) (cid:50)(cid:46) (cid:52)(cid:46) (cid:54)(cid:46) (cid:47)(cid:46)(cid:46) (cid:46) (cid:48)(cid:46) (cid:50)(cid:46) (cid:52)(cid:46) (cid:54)(cid:46) (cid:47)(cid:46)(cid:46)
(cid:118)+L*t (cid:118)+L*t
FIG. 5: The anomaly parts of the Left-handedfermions 4 FIG.6: Thetotaltopologicalfield qα(x)(filledtriangle),
1(x) (open circle) and of the Right-handed fermion 2(x×) the total anomaly part α(x) (fiαlled rectangle) and the
A(cross) are plotted against the fused coordinate x+LA t. total finite volume correctαioAn P∆qα(x) (open circle) are
The total anomaly part 4 1(x) 2(x) (filled rectan×gle) plotted against the fusedPcoordinaαte x+L t.
×A −A P ×
is also plotted. The lattice size is L=10. The gauge field is
generatedatβ =3.0withthevanishingmagneticfluxm =
12
0.
Four-dimensional case is numerically demanding. The
evaluation of the differential of the topological fields
should be performed without diagonalizing H . For
O(e−L/2̺)afterthecancellationandwecanconfirmthat w
other parts, our procedure would work also in this case.
Our next step would be the numerical construc-
tion of the Weyl fermions measures and observables
∆A(x)≃O(e−L/2̺)≃O(10−4) (L=10). (69)
in the anomaly-free U(1) chiral gauge theories in two-
dimensions, keeping the gauge invariance. Work in this
This result is also confirmed in figure 6. In this figure,
direction is in progress.
the total topological field qα(x), the total anomaly
α
part Aα(x) and the total finite volume correction
∆qαα(x) are plotted. PWe see clearly that the size
α P APPENDIX A:
of ∆A(x) is the same order of magnitude as the size
P
of ∆q(x). The similar cancellation is also observed for
In this appendix, we give the explicit formulae for the
topologically non-trivial gauge fields with m =1.
12 vector-potentialrepresentationofthelinkvariablesofthe
Now the current k (x) can be computed from
µ admissible U(1) gauge field discussed in the section II.
Eqs. (55), (56) and (58). The current so obtained is
We first define
thus local. This local current can reproduce the origi-
nal topologicalchargeq(x) within the deviation of order 1
a˜ (x)≡ ln{U(x,µ)V (x,µ)−1} (A1)
O(10−15). The gauge invariance of the current is also µ i [m]
maintained within the error.
and
2πm
µν
2πn˜ (x)≡F (x)− −{∂ a˜ (x)−∂ a˜ (x)}.
VI. DISCUSSION µν µν L2 µ ν ν µ
(A2)
We have demonstrated that the cohomological anal- We also introduce a summation convention as
ysis of the chiral anomaly associated with the over-
lap fermions can be performed numerically in two- xi−1 xtii=−01f(x) (xi ≥1)
dimensional lattice with a finite volume. The resulted ′f(x)= 0P (xi =0) . (A3)
current kµ(x) is gauge invariant and local. tXi=0  −ti=1xi(−1)f(x) (xi ≤−1)
 P
9
Then, in four-dimensions, the vector-potential is de- times H is approximated by
w
fined by
n
H b
x1−1 x2−1 w =h k . (B1)
A˜4(x) = a˜4(x)+2π ′n˜14(x)+ ′2πn˜24(x)|x1=0 Hw2 wk=1h2w+c2k−1
( X
tX1=0 tX2=0 p
x3−1 h is defined by H /λ where λ is the square root
w w min min
+ ′n˜34(x)|x1=x2=0 , ofthe minimum ofthe eigenvalues ofHw2, and the coeffi-
)
tX3=0 cients ck and bk are given as follows:
x1−1 x2−1
A˜3(x) = a˜3(x)+2π ′n˜13(x)+ ′n˜23(x)|x1=0 sn2 kK′;κ′
(x4tX−1=10 L/2−1 tX2=0 ck = 1−sn(cid:16)22nk2Kn′;(cid:17)κ′ , (B2)
−δx3,L/2−1 ′ n˜34(x)|x1=x2=0, 2n Θ2(cid:0) 2kK′;(cid:1)κ′
2n
tX4=0 t3=X−L/2  λ = , (B3)
A˜ (x) = a˜ (x)+2π x1−1′n˜ (x)  kY=1Θ2 ((cid:16)2k−2n1)K′;κ(cid:17)′
2 2 12 (cid:16)n (cid:17)
(tX1=0 d0 = 2λ 1+c2k−1, (B4)
λ+1 1+c
x3−1 L/2−1 k=1 2k
−δx2,L/2−1tX3=0′t2=X−L/2n˜23(x)|x1=0,x4=0 bk = d0 nYin=−11((cc2i−c−2kc−1) ). (B5)
iQ=1,i6=k 2i−1 2k−1
x4−1 L/2−1
−δ ′ n˜ (x)| , Q
x2,L/2−1 24 x1=0 K′ is the complete elliptic integral of the first kind with
tX4=0 t2=X−L/2  modulus κ′ = 1−(λmin/λmax)2.
A˜ (x) = a˜ (x)+2π  p
1 1 

APPENDIX C: SOLUTIONS OF THE MODIFIED
x2−1 L/2−1 POINCARE´ LEMMA AND THE COROLLARY OF
−δ ′ n˜ (x)|
x1,L/2−1 12 x3=x4=0 DE RHAM THEOREM IN TWO-DIMENSIONS
tX2=0 t1=X−L/2
x3−1 L/2−1 In this appendix, we give the explicit solutions of the
−δx1,L/2−1 ′ n˜13(x)|x4=0 Poincar´elemma in two-dimensions (n=2).
tX3=0 t1=X−L/2 IV.a(0-form) and IV.b(0-form):
x4−1 L/2−1
−δx1,L/2−1tX4=0′t1=X−L/2n˜14(x). g1(x1,x2)=δx2,x02 x1 f¯(y1),
y1=Xx01−L/2
In two-dimensions, the vector potentialis defined by
x2
x1−1 g2(x1,x2)= f(x1,y2)−δy2,x02f¯(x1) ,
A˜ (x) = a˜ (x)+2π ′n˜ (x), y2=Xx02−L/2n o
2 2 12
(C1)
tX1=0
A˜ (x) = a˜ (x)
1 1
where f¯(x )= L/2−1 f(x ,y ).
x2−1 L/2−1 1 y2=−L/2 1 2
−δ ′ 2πn˜ (x).
x1,L/2−1 12 P
IV.a(1-form):
tX2=0 t1=X−L/2
(A4)
x2
g12(x1,x2)=− f1(x1,y2)−δy2,x02f¯1(x1) ,
APPENDIX B: RATIONAL APPROXIMATION y2=Xx02−L/2n o
OF THE OVERLAP DIRAC OPERATOR ∆f1(x1,x2)=δx2,x02f¯1(x1),
In this appendix, we give the formula of the rational ∆f2(x1,x2)=f2(x1,x2)|x2=x02−L/2−1, (C2)
approximation of the overlap Dirac operator with the
Zolotarev optimization. The inverse square root of Hw2 where f¯1(x1)= Ly2/=2−−1L/2f1(x1,y2).
P
10
IV.b(1-form): overlap Dirac operator. The authors are also grate-
ful to H. Suzuki for valuable discussions. Y.K. is sup-
x2 ported in part by Grant-in-Aid for Scientific Research
g12(x1,x2)=− f1(x1,y2)−δy2,x02f¯1(x1) No. 14046207. D.K. is supported in part by the Japan
y2=Xx02−L/2n o Society for Promotion of Science under the Predoctoral
x1 Research ProgramNo. 15-887.
+ f2(y1,x2)|x2=x02−L/2−1.
y1=Xx01−L/2
(C3)
ACKNOWLEDGMENTS
The authors would like to thank T.-W. Chiu for the
correspondence about the rational approximation of the
[1] P.H.GinspargandK.G.Wilson,Phys.Rev.D25,2649 the overlap Dirac operator [32,33, 34].
(1982). [14] R. Narayanan and H. Neuberger, Phys. Lett. B 302, 62
[2] H. Neuberger, Phys. Lett. B 417, 141 (1998) (1993) [arXiv:hep-lat/9212019].
[arXiv:hep-lat/9707022]. [15] R.NarayananandH.Neuberger,Nucl.Phys.B412,574
[3] P. Hasenfratz, V. Laliena and F. Niedermayer, Phys. (1994) [arXiv:hep-lat/9307006].
Lett.B 427, 125 (1998) [arXiv:hep-lat/9801021]. [16] R. Narayanan and H. Neuberger, Phys. Rev. Lett. 71,
[4] H. Neuberger, Phys. Lett. B 427, 353 (1998) 3251 (1993) [arXiv:hep-lat/9308011].
[arXiv:hep-lat/9801031]. [17] R.NarayananandH.Neuberger,Nucl.Phys.B443,305
[5] P. Hasenfratz, Nucl. Phys. B 525, 401 (1998) (1995) [arXiv:hep-th/9411108].
[arXiv:hep-lat/9802007]. [18] R. Narayanan, Nucl. Phys. Proc. Suppl. 34, 95 (1994)
[6] P.Hernandez,K.Jansen and M.Lu¨scher, Nucl.Phys.B [arXiv:hep-lat/9311014].
552, 363 (1999) [arXiv:hep-lat/9808010]. [19] H. Neuberger, Nucl. Phys. Proc. Suppl. 83, 67 (2000)
[7] M. Lu¨scher, Phys. Lett. B 428, 342 (1998) [arXiv:hep-lat/9909042].
[arXiv:hep-lat/9802011]. [20] R.NarayananandH.Neuberger,Nucl.Phys.B477,521
[8] M. Lu¨scher, Nucl. Phys. B 538, 515 (1999) (1996) [arXiv:hep-th/9603204].
[arXiv:hep-lat/9808021]. [21] P.Y.Huet,R.NarayananandH.Neuberger,Phys.Lett.
[9] M. Lu¨scher, Nucl. Phys. B 549, 295 (1999) B 380, 291 (1996) [arXiv:hep-th/9602176].
[arXiv:hep-lat/9811032]. [22] R.NarayananandJ.Nishimura,Nucl.Phys.B508,371
[10] M. Lu¨scher, Nucl. Phys. B 568, 162 (2000) (1997) [arXiv:hep-th/9703109].
[arXiv:hep-lat/9904009]. [23] Y. Kikukawaand H.Neuberger, Nucl.Phys.B 513, 735
[11] M. Lu¨scher, Nucl. Phys. Proc. Suppl. 83, 34 (2000) (1998) [arXiv:hep-lat/9707016].
[arXiv:hep-lat/9909150]. [24] R. Narayanan and H. Neuberger, Phys. Lett. B
[12] M. Lu¨scher, arXiv:hep-th/0102028. 393, 360 (1997) [Phys. Lett. B 402, 320 (1997)]
[13] Overlap formalism proposed by Narayanan and [arXiv:hep-lat/9609031].
Neuberger[14, 15, 16, 17, 18, 19, 20, 21, 22, 23] [25] Y. Kikukawa, R. Narayanan and H. Neuberger, Phys.
gives a well-defined partition function of Weyl fermions Lett. B 399, 105 (1997) [arXiv:hep-th/9701007].
on the lattice, which nicely reproduces the fermion zero [26] Y. Kikukawa, R. Narayanan and H. Neuberger, Phys.
mode and the fermion-number violating observables Rev. D 57, 1233 (1998) [arXiv:hep-lat/9705006].
(’t Hooft verteces)[24, 25, 26]. Through the recent re- [27] D. B. Kaplan, Phys. Lett. B 288, 342 (1992)
discovery of the Ginsparg-Wilson relation, the meaning [arXiv:hep-lat/9206013].
oftheoverlapformula, especially thelocality properties, [28] Y. Shamir, Nucl. Phys. B 406, 90 (1993)
becomeclearfromthepointofviewofthepath-integral. [arXiv:hep-lat/9303005].
The gauge-invariant construction by Lu¨scher[9] based [29] V.FurmanandY.Shamir,Nucl.Phys.B439,54(1995)
on the Ginsparg-Wilson relation provides a procedure [arXiv:hep-lat/9405004].
to determine the phase of the overlap formula in a [30] T. Blum and A. Soni, Phys. Rev. D 56, 174 (1997)
gauge-invariant manner for anomaly-free chiral gauge [arXiv:hep-lat/9611030].
theories. For Dirac fermions, it provides a gauge- [31] T. Blum and A. Soni, Phys. Rev. Lett. 79, 3595 (1997)
covariant and local lattice Dirac operator satisfying the [arXiv:hep-lat/9706023].
Ginsparg-Wilson relation[1, 2, 4, 6, 23]. The overlap [32] P. M. Vranas, Phys. Rev. D 57, 1415 (1998)
formula was derived from the five-dimensional approach [arXiv:hep-lat/9705023].
of domain wall fermion proposed by Kaplan[27]. In its [33] H. Neuberger, Phys. Rev. D 57, 5417 (1998)
vector-likeformalism[28,29,30,31],thelocallowenergy [arXiv:hep-lat/9710089].
effective action of the chiral mode precisely reproduces [34] Y. Kikukawaand T. Noguchi, arXiv:hep-lat/9902022.