Determinant of a new fermionic action on a lattice - (II)

6 Pages·0.24 MB·English

by  A. Takami

Determinant of a new fermionic action on a lattice - (II)
A. Takami, T. Hashimoto, M. Horibe, and A. Hayashi
Department of Applied Physics, Fukui University, Fukui 910
We investigate the fermion determinant of a new action on a (1+D)-dimensional lattice for
D
SU(2)gaugegroups. Thisactionpossessesthediscretechiralsymmetryandprovides2 -component
fermions. We also comment on the numerical results on fermion determinants in the (1+D)-
dimensional SU(3) gauge fields.
PACS number(s): 11.15.Ha
I. INTRODUCTION
1
0
0 As is well known the lattice formulation of fermions has extra physical particles or breaks the chiral symmetry.
2 This is unavoidable under a few plausible assumptions [1]. Several methods have been proposed to deal with this
difficulty. Wilson’s formulation [2], which is one of the most popular schemes, eliminates the unwanted particles
n
with an additional term which vanishes in the naive continuum limit. However, this formulation sacrifices the chiral
a
J symmetry. An alternative scheme is the staggered fermion formulation proposed by Kogut and Susskind [3]. This

6 schemepreservesthediscretechiralsymmetryandinthispointthestaggeredfermionhasanadvantageovertheWilson
1 fermion. But, the staggered formulation describes a theory with 21+2D degenerate quark flavours (21+D components)
in (1+D) dimensions, while there is no restrictiononthe flavournumber inthe Wilsonformulation. Recently, it has
3
been shown that lattice fermionic actions with the Ginsparg-Wilson relation [4] have an exact chiral symmetry and
v
are free fromrestrictionon the flavournumber. But, these actions cannot be ”ultralocal”[5], which makes numerical
6
0 simulations complicated.
0 In the recentpapers [6,7],we proposeda new type offermionic actionona (1+D)-dimensionallattice. The action
3 is ultralocal and has discrete chiral symmetry. On the Euclidean lattice the minimal number of fermion components
0 is 2D, which should be compared with 21+D of the staggered fermion. When dynamical fermions are included, the
0 numerical feasibility relies on the reality and positivity of the fermion determinant. In the previous paper [8] we
0
investigated,analyticallyandnumerically,the fermiondeterminantof ournew actioninthe (1+1)-dimensionalU(1)
/
t lattice gaugetheory. We showedthe realityofourfermiondeterminantunder the conditionfixing the globalphaseof
a
l link variables along the temporal direction. By a similar discussion to the U(1) gauge group, we could also find the
-
reality and the positivity of our fermion determinant in the (1+1)-dimensional SU(N) lattice gauge theory.
p
e In this paper we analytically show that our fermion determinant with the SU(2) gauge fields is realand positive in
h (1+D)dimensions. Wealsocommentonthenumericalresultsofthefermiondeterminantinthe(1+D)-dimensional
: SU(3) gauge fields, and discuss the effectiveness of our new action for SU(2) and SU(3) lattice gauge theories.
v
i
X
r II. NEW FERMIONIC ACTION
a
In the recent paper [7], we proposed a new fermionic action on the Euclidean lattice. Though the action keeps the
discretechiralsymmetrylikethestaggeredfermionaction,thefermionfieldhas2D componentsin(1+D)dimensions.
In this section we briefly sketch our formalism for later convenience.
The action can be written with a fermion matrix Λ as
S = ψ†Λ ψ , (2.1)
f n nm m
Xn,m
where the summation is over lattice points and spinor indices, and our fermion matrix is defined by
Λ=1−S†U . (2.2)
0 E
Here U is the Euclidean time evolution operator and S is the unit shift operator defined by
E µ
S ψ(x0,x1,...,xµ,...,xD)=ψ(x0,x1,...,xµ+1,...,xD) (µ=0,1,...,D). (2.3)
µ
We require that the propagator has no extra poles and find that U has the form
E
1
D
r
U =1− E iX S −S† +(1−Y ) S −2+S† , (2.4)
E 2 i i i i i i
Xi=1 n (cid:16) (cid:17) (cid:16) (cid:17)o
where r is the ratio of the temporal lattice constant to the spatial one. The spinor matrices X’s and Y’s should
E
satisfy the following algebra:
2
{X ,X } = δ ,
i j ij
 rE
 {Xi,Yj} = 0, 1 (2.5)
{Y ,Y } = 2 δ +1 ,
i j ij
(cid:18)r (cid:19)
 E
where i and j run from 1 to D. The matrix 2(δ /r +1) is positive definite for any positive r , therefore X’s and
ij E E
Y’s can be assumed hermitian,
X† =X , Y† =Y . (2.6)
i i i i
The matrices X’s and Y’s can be expressed by the Clifford algebra:
Γ† =Γ , {Γ ,Γ }=2δ (n,m=1,...,2D) (2.7)
n n n m nm
in several ways. One is
D
1
X = Γ , Y = α Γ , (2.8)
i rr i i ij D+j
E Xj=1
as was used in the previous paper. Another one is
1 1
X = Γ , Y = Γ +Γ , (2.9)
i rr i i rr D+i 2D+1
E E
where Γ is
2D+1
Γ =(−i)DΓ Γ ···Γ . (2.10)
2D+1 1 2 2D
The latter is more convenient than the former for later use. The dimension of the irreducible representation for Γ’s
is 2D and accordingly ψ has 2D components.
The interaction of the fermion with gauge fields is introduced by replacing the unit shift operators by covariant
ones:
S →S (x)≡U S , (2.11)
µ µ x,x+µˆ µ
where µˆ is the unit vector along the µ’th direction, and U is a link variable connecting sites x and y.
x,y
The fermion matrix Eq.(2.2) and the time evolution operator Eq.(2.4) become
Λ(x)=1−S†(x)U (x), (2.12)
0 E
and
D
r
U (x)=1− E iX S (x)−S†(x) +(1−Y ) S (x)−2+S†(x) . (2.13)
E 2 i i i i i i
Xi=1 n (cid:16) (cid:17) (cid:16) (cid:17)o
2
III. FERMION DETERMINANT FOR SU(2) CASE
In this section we analytically study the determinant of our fermion matrix in SU(2) gauge fields.
First, in the (1+1)-dimensional case, the complex conjugation of U (x) is
E
r
U∗(x)=1− E −iX∗ S∗(x)+S∗†(x) +(1−Y∗) S∗(x)−2+S∗†(x) , (3.1)
E 2 1 1 1 1 1 1
n (cid:16) (cid:17) (cid:16) (cid:17)o
where we can write
3
S (x)=α (x)1+i α (x)τ , (3.2)
1 0 i i
Xi=1
sincethelink variablesinS (x)areSU(2)gaugegroupelements. Hereα (x)andα (x)arerealanddependonlattice
1 0 i
points and τ are the Pauli-matrices:
1,2,3
0 1 0 −i 1 0
τ = , τ = , τ = . (3.3)
1 (cid:18)1 0 (cid:19) 2 (cid:18) i 0 (cid:19) 3 (cid:18) 0 −1(cid:19)
Then we have
S (x)τ =τ S∗(x). (3.4)
1 2 2 1
By the same discussion for other unit shift operators S† and S†, we also have
0 1
S†(x)τ =τ S∗†(x), S†(x)τ =τ S∗†(x). (3.5)
0 2 2 0 1 2 2 1
If we can find the matrix Γ such that
X∗Γ = −ΓX ,
1 1 (3.6)
(cid:26) Y1∗Γ = ΓY1,
it is easily shown that
∗
(Γ⊗τ ) S†(x)U (x) (Γ⊗τ )=S†(x)U (x). (3.7)
2 0 E 2 0 E
(cid:16) (cid:17)
For example, we make the following choice:
1 1
X = τ , Y = τ +τ , (3.8)
1 rr 1 1 rr 2 3
E E
the matrix Γ acting on two components fermi fields defined by
Γ=τ (3.9)
3
satisfiesEq.(3.6). TheEq.(3.7)impliesthatifλissomeeigenvalueofourfermionmatrixΛ(x),thenλ∗ isaneigenvalue
of Λ∗(x) and thus also of Λ(x). Therefore eigenvalues of Λ(x) are either real or come in complex conjugate pairs.
From the above discussion we can prove the reality of our fermion determinant for the SU(2) gauge groups.
Next we show its positivity. We define
Γ′ =(Γ⊗τ )K, (3.10)
2
where K is complex-conjugation operator. We find
Γ′Γ′ =(τ ⊗τ )K(τ ⊗τ )K =(τ ⊗τ )(τ ⊗(−τ ))=−1, (3.11)
3 2 3 2 3 2 3 2
and from the relation Eq.(3.7) we can show
[Γ′,Λ(x)]=0. (3.12)
For a real eigenvalue λ of Λ(x) and the eigenvector v for this eigenvalue, from Eq.(3.12) we obtain
R R
3
Λ(x)Γ′v =Γ′Λ(x)v =Γ′λ v =λ Γ′v . (3.13)
R R R R R R
Suppose Γ′v =cv , then we find
R R
Γ′2v =Γ′cv =c∗Γ′v =|c|2v , (3.14)
R R R R
which is inconsistent with Eq.(3.11), so that Γ′v is different eigenvector for the same eigenvalue. Therefore the
R
eigenvalues on real axis are degenerate in pairs and the determinant of Λ(x) is positive.
The above proof of the positivity of our fermion determinant for the SU(2) group can be expanded to higher
dimensions. We canmakeafundamentalrepresentationforthe Cliffordalgebrawith2D elementsΓ (n=1,...,2D)
n
using direct products of the Pauli-matrices:
Γ = τ ⊗τ ⊗···⊗τ
1 1 3 3
D−1
Γ2 = 1⊗τ|1⊗τ3{z⊗···}⊗τ3
D−2
.. | {z }
.
Γ = 1⊗···⊗1⊗τ ⊗τ ⊗···⊗τ
i 1 3 3
i−1 D−i
.. | {z } | {z }
.
Γ = 1⊗···⊗1⊗τ
D 1
D−1 (3.15)
ΓD+1 = |τ2⊗{τz3⊗·}··⊗τ3
D−1
ΓD+2 = 1⊗τ|2⊗τ3{z⊗···}⊗τ3
D−2
.. | {z }
.
Γ = 1⊗···⊗1⊗τ ⊗τ ⊗···⊗τ
D+i 2 3 3
i−1 D−i
.. | {z } | {z }
.
Γ = 1⊗···⊗1⊗τ
2D 2
D−1
| {z }
where i runs from 1 to D. It can be easily checked that Γ ’s satisfy the relation
n
{Γ ,Γ }=2δ (n,m=1,...,2D). (3.16)
n m nm
Moreover,we can see that the matrix Γ is real and the matrix Γ is pure imaginary:
i D+i
Γ∗ =Γ , Γ∗ =−Γ (i=1,...,D). (3.17)
i i D+i D+i
From the anti-commutation relation, we find the hermite matrix Γ which anti-commutes with all Γ ’s:
2D+1 n
Γ =(−i)DΓ Γ ···Γ
2D+1 1 2 2D
=τ ⊗τ ⊗···⊗τ . (3.18)
3 3 3
Clearly, the matrix Γ is hermitian and the square of this matrix is equal to the unit matrix. Thus, we have
2D+1
Γ Γ∗Γ = −Γ ,
2D+1 i 2D+1 i (3.19)
Γ Γ∗ Γ = Γ .
2D+1 D+i 2D+1 D+i
In (1+D) dimensions, the relation Eq.(3.6) is rewritten as follows:
X∗Γ = −ΓX ,
i i (3.20)
(cid:26) Yi∗Γ = ΓYi.
4
8
4
0
−4
−8
−7 −3 1 5 9
FIG.1. The spectrum of Λ(x) in the complex planeon a 4×4×4 lattice in SU(2) gauge group.
Then, for the representation of Eq.(2.9),
1 1
X = Γ , Y = Γ +Γ , (3.21)
i rr i i rr D+i 2D+1
E E
we find
Γ=Γ . (3.22)
2D+1
SincetheeigenvaluesofΛ(x)alwaysconsistofcomplexconjugatepairsanddegeneratedonesonrealaxis,weconclude
the determinant of our fermion matrix is positive in SU(2) gauge fields in any dimensions.
Now we show a numerical evidence. Fig.1 shows the spectrum of our fermion matrix in a typical background
configuration of link variables for SU(2) gauge group in (1 + 2) dimensions. We find that their distribution is
symmetric with respect to the real axis as expected. Similarly in (1+3) dimensions we can numerically confirm the
symmetry with respect to the real axis, and the positivity of the determinant.
IV. DISCUSSION AND SUMMARY
Inthepreviouspaper[8]wereportedanalyticalandnumericalresultsonthefermiondeterminantofournewaction
in (1+1) dimensions. In the case of U(1) gauge group, we were faced with the problem of convergence in numerical
simulations. The cause of the poorness of the convergence is that the summation of the det(1 −S†(x)U (x)) =
0 E
det(1−eiΘS˜†(x)U (x)) over arbitrary phase angle Θ is canceled out accidentally. The element eiΘ comes from the
0 E
one sided time difference operator S†(x) with θ (x) replaced by θ (x)+Θ, i.e. S†(x) = eiΘS˜†(x), where θ (x) is
0 0 0 0 0 0
definedbyU =eiθ0(x) [8]. Thereforewemusthavecontrolofthe phaseangleΘinordertogetgoodconvergence
x,x+ˆ0
in the (1+1)-dimensional U(1) gauge theory. In fact we analytically showedthat our fermion determinant is real for
all configurations and positive for most configurations under the T-condition (θ (x) = 0), which corresponds to the
0
temporal gauge condition on the infinite lattice, or the GT-condition ( θ (x)=nπ : n=even), which is achieved
x 0
by a gauge transformation on the infinite lattice. It was also verifiedPnumerically. On the other hand we got good
convergence without any conditions in (1+1)-dimensional SU(N) case, because the element like eiΘ does not belong
to the SU(N) group.
5
(a) SU(2) (b) SU(3)
1.5x10182 3x10268
1x10182 2x10268
5x10181 1x10268
0 0
−5x10181 −1x10268
−1x10182 −2x10268
−1.5x10182 −3x10268
0 5x10181 1x10182 1.5x10182 0 1x10268 2x10268 3x10268
FIG.2. Thedistributioninthecomplexplaneofourfermiondeterminantsina)SU(2)andb)SU(3)foreachconfigurations
of 2000 Monte Carlo iterations after getting good equilibrium, i.e. after 2000 iterations, on a 4×4×4 lattice at β =2.0.
Theabovediscussionisapplicabletohigherdimensionstoacertainextent. InSU(2)groupourfermiondeterminant
is analytically shown real and positive in any dimensions. In Fig.2(a) we give the numerical evidence that the
determinant is real and positive in (1+2) dimensions. In the case of SU(3) group, we cannot prove the reality of
the determinant. But fromFig.2(b) we see that the distributionof the determinantis concentratednearthe realaxis
without any conditions and the phase angle of the determinant is small. We have obtained similar results in (1+3)
dimensions. When the phase angle of the fermion determinant is small enough, we can neglect the phase factor and
make use of |detΛ(x)| instead of detΛ(x). In the above numerical simulations link variables are updated by the
Metropolis method and determinants are calculated by the LU decomposition. So there are no systematic errors in
the determinants.
In conclusion, we believe that our new fermionic action is a profitable formulation for the numerical simulations of
SU(2) and SU(3) lattice gauge theory.
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York,1977).
[3] L. Susskind,Phys. Rev.D 16, 3031 (1977).
[4] P. H. Ginsparg and K.G. Wilson, Phys. Rev.D 25, 2649 (1982).
[5] I. Horv´ath,Phys. Rev.Lett. 81, 4063 (1998); W.Bietenholz, hep-lat/9901005.
[6] A. Hayashi, T. Hashimoto, M. Horibe, and H.Yamamoto, Phys.Rev. D 55, 2987 (1997).
[7] M. Horibe, T. Hashimoto, A. Hayashi,and H.Yamamoto, Phys.Rev. D 56, 6006 (1997).
[8] A. Takami, T. Hashimoto, M. Horibe and A.Hayashi, hep-lat/0001011.
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