The thermodynamics of massless ideal gas of overlap quarks has been
investigated numerically for both zero and nonzero baryon chemical potential
$\mu$. While the parameter M has been shown to be irrelevant in the continuum
limit, it is shown numerically that the continuum limit can be reached with
relatively coarser lattices for certain range of M. Numerical limitation of the
existing method of introduction of chemical potential in the overlap formalism
is discussed. We have also studied the energy density of free domain wall
fermions in the absence of $\mu$ and estimated the extent of lattice in the
fifth dimension $L_5$ for which the overlap results are recovered.
Interestingly, this value of $L_5$ is also minimum for the same range of M
found in the overlap case.Comment: 5 pages, 4 figures, to appear in the proceedings of the 20th
  International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions:
  Quark Matter 2008 (QM2008

Banerjee D Gavai R V Sharma S

R V Gavai

Sayantan Sharma

English

arXiv

Crossref

Towards QCD thermodynamics using exact chiral symmetry on a lattice

The thermodynamics of massless ideal gas of overlap quarks has been investigated numerically for both zero and nonzero baryon chemical potential μ. While the parameter M has been shown to be irrelevant in the continuum limit, it is shown numerically that the continuum limit can be reached with relatively coarser lattices for certain ranges of M. Numerical limitation of the existing method of introduction of chemical potential in the overlap formalism is discussed. We have also studied the energy density of free domain wall fermions in the absence of μ and estimated the extent of lattice in the fifth dimension L5 for which the overlap results are recovered. Interestingly, this value of L5 is also minimum for the same range of M found in the overlap case

Gavai, R. V.

Sharma, Sayantan

Publications of the IAS Fellows

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Towards QCD thermodynamics using exact chiral
symmetry on lattice
R V Gavai and Sayantan Sharma
Department of Theoretical Physics, Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400005, India.
E-mail: gavai@tifr.res.in,ssharma@theory.tifr.res.in
PACS numbers: 11.15.Ha, 12.38.Mh, 12.38.Gc
Abstract. The thermodynamics of massless ideal gas of overlap quarks has been
investigated numerically for both zero and nonzero baryon chemical potential µ. While
the parameter M has been shown to be irrelevant in the continuum limit, it is shown
numerically that the continuum limit can be reached with relatively coarser lattices
for certain ranges of M . Numerical limitation of the existing method of introduction
of chemical potential in the overlap formalism is discussed. We have also studied the
energy density of free domain wall fermions in the absence of µ and estimated the
extent of lattice in the fifth dimension L5 for which the overlap results are recovered.
Interestingly, this value of L5 is also minimum for the same range of M found in the
overlap case.
Towards QCD thermodynamics using exact chiral symmetry on lattice 2
1. Introduction
The phase diagram of Quantum Chromodynamics(QCD) has been an important subject
of study in the recent years and lattice gauge theory has emerged as a major tool for
studying it. It is believed that spontaneously broken chiral symmetry is restored at high
temperatures so it is very important to study the QCD thermodynamics with fermions
having exact chiral symmetry on lattice. The most popular among such fermions are
the overlap and the domain wall fermions. The overlap operator is highly non-local,
involving inversion and square root of Dirac matrices, making it computationally very
expensive to simulate. The domain wall fermions on the other hand are necessarily
defined on a 5-D lattice so their computational cost is more than the standard 4-D
fermions as the Wilson and staggerred, but is still less than the overlap. For the
above mentioned practical reasons, we investigate numerically whether the irrelevant
parameterM in such fermion operators can be tuned optimally to recover the continuum
values of various thermodynamic quantities for free fermions with the smallest possible
lattice sizes. This would give us an estimate of the optimum lattice parameters for full
QCD computations with such fermions. In section 2, we compute the energy density of
overlap and domain wall fermions in absence of chemical potential. It was shown[1] that
the domain wall operator reduces to the overlap when the extent of lattice in the fifth
dimension, L5 → ∞. We try to find out at what L5, the overlap results are obtained.
In the subsequent sections the energy density in the presence of chemical potential µ
and the susceptibility expressions are computed and their approach to the continuum
for different values of M are studied.
2. Energy density in absence of chemical potential
2.1. Overlap fermions
The overlap Dirac operator[2] for massless quarks defined on a N3 × NT lattice with
spacing a and a4 in the spatial and temporal directions respectively, is given by,
Dov = 1 + γ5sgn(γ5DW ) , (1)
where sgn denotes the sign function and
DW (x, y) = −M δx,y −
a
a4
[δx−4ˆ,y
1 + γ4
2
+
1− γ4
2
δx+4ˆ,y]
−
3∑
i=1
[δx−iˆ,y
1 + γi
2
+
1− γi
2
δx+iˆ,y] + [3 +
a
a4
]δx,y. (2)
The sign function of the matrix γ5DW can be written as the sign of its eigenvalues[3].
These eigenvalues are positive and greater than zero in this case so the sign function is
well-defined. The overlap operator hence can be diagonalised in the momentum space
in terms of the variables
hi = − sin api , h4 = −
a
a4
sin a4p4 , h5 = M−
3∑
i=1
(1−cos api)−
a
a4
(1−cos a4p4) . (3)
Towards QCD thermodynamics using exact chiral symmetry on lattice 3
 0
 0.2
 0.4
 0.6
 0.8
 1
 1.2
 1.4
 1.6
 1.8
 2
 0  5  10  15  20  25
ε/ε
SB
NT
ζ= 4 M=1.00M=1.45
M=1.50
M=1.55
M=1.60
M=1.65
 0.2
 0.4
 0.6
 0.8
 1
 1.2
 1.4
 1.6
 1.8
 0  5  10  15  20  25
ε/ε
SB
NT
ζ= 4
L5=20
18
16
14
12
10
6
Figure 1. (a)The variation of ǫ/ǫSB with M for fixed ζ = 4 for overlap fermions and
(b)The domain wall fermion results matches with the overlap for L5 ≥ 14 with ζ = 4
and M = 1.55.
The energy density can be obtained as
ǫ =
T 2
V
∂ lnZ(V, T )
∂T
∣∣∣∣∣
V
= −
1
N3a3NT
(
∂ln detDov
∂a4
)
a
=
2
N3a4NT
∑
pj ,p4
h2(1− cos ap4)− h5h4 sin ap4
h2(h2 + h25)
(
√
h2 + h25 + h5) , (4)
where a = a4 has been chosen after performing the a4-derivative. The same can be
computed numerically keeping the aspect ratio ζ = N/NT fixed. The lattice expression
so obtained was fitted to the ansatz A + B/N4T for a range of values of M. The higher
order NT dependent correction terms in the energy density expression due to finite
size effects manifest themselves as an effective coefficient B(M). We seek those values
of M for which B(M) approaches 7π2/60, its value in the continuum limit. The zero
temperature part A, was subtracted from the energy density. The resultant ǫ divided
by continuum value ǫSB was plotted as a function of NT . From figure 1(a) we observe
that the M-dependence is quite pronounced and for 1.55 < M < 1.60 the continuum
limit is reached for the smallest possible lattice size(NT ∼ 12). For the oft-favoured
choice of M = 1, the convergence to the continuum is seen to be very slow. The energy
density can be evaluated analytically by converting the sum over Matsubara frequencies
ap4 = ω = (2m + 1)π/NT where m are integers, to contour integrals[4]. It has been
shown that no M-dependence of the energy density exist in the continuum[4] but for
observing that we need lattice sizes much greater than NT = 32.
2.2. Domain wall fermions
The domain wall fermions[5] are defined on a 5-D Euclidian lattice in presence of a
domain wall of height M. Taking the lattice spacing in the fifth direction a5 → 0 and
Towards QCD thermodynamics using exact chiral symmetry on lattice 4
the number of sites N5 →∞ for fixed L5 = N5a5, the effective domain wall operator is
DDW = (1 +
mq
2M
) + (1−
mq
2M
)γ5 tanh(
L5
2
γ5DW ) . (5)
Here mq is the bare quark mass and L5 is in lattice units. In the momentum space, using
h1, h2, h3, h4 and h5 as given in (3), the energy density for massless quarks(mq = 0) is
given by
εa4 =
∑
pj ,p4
4 sinh[ sL5
2
]((−h4h5α + h
2γ) cosh[ sL5
2
] + (h4h5α + (h
2
5 + s
2)γ + 2h5s
2p))
sN3NT (h2 + (s2 + h25) cosh[2sL5]
cosh(3sL5
2
)− 2s(h25p+ h5γ + (h
2p+ h4α + 2h5(h5p+ γ)) cosh[sL5]) sinh[
sL5
2
])
−2h5s sinh[2sL5])
, (6)
where
α = −h4 , γ = 1− cos((2m+ 1)π/NT ) , (7)
h2 = h21 + h
2
2 + h
2
3 + h
2
4 , s
2 = h2 + h25 , (8)
p =
(− sin2 ap4 + h5(1− cos ap4))(− tanh
L5s
2
+ L5s
2
sech2 L5s
2
)
s2 tanh L5s
2
. (9)
Following the same fitting procedure mentioned in the previous section and choos-
ing M = 1.55 the NT dependent part of energy density ǫ was plotted for various values
of L5. Clearly from figure 1(b) we observe that for L5 ≥ 14, the energy density converges
to the continuum case at NT ≥ 12. L5 ∼ 14 is thus the optimum lattice size in the fifth
dimension to converge to the overlap results.
3. Non-zero chemical potential
In the presence of chemical potential µˆ = µa there is a critical value of µˆc beyond
which the sign function is not always defined[4]. Hence we work with µˆ < µˆc, where the
argument of sign function is positive definite and the calculation follows analogously as
in section 2.1. The chemical potential can be incorporated as[6] eµˆ and e−µˆ multiplying
1±γ4 terms respectively in (2) so as to cancel µˆ
2/a2 terms on the lattice[7]. The energy
density expression on the lattice has the same form as in (4) but with r = µ/T = µˆNT
kept constant and h4 and h5 changed to[3]:
h4 = −
a
a4
sin(a4p4−iµˆ) , h5 = M−
3∑
j=1
(1−cos apj)−
a
a4
(1−cos(a4p4−iµˆ)) . (10)
The energy density in presence of µˆ has been analytically shown to have the correct
continuum limit[4]. The two observables of interest here are the change in the energy
density, ∆ǫ(µ, T ) = ǫ(µ, T )−ǫ(0, T ) and the quark number susceptibility at µˆ = 0 which
in the continuum limit, are given by
∆ǫ(µ, T )
T 4
=
r4
4π2
+
r2
2
& χ(0, T ) = χSB =
T 2
3
(11)
As shown in figure 2, both these quantities also approach the continuum limit faster for
the same optimum range of M.
Towards QCD thermodynamics using exact chiral symmetry on lattice 5
 0
 0.05
 0.1
 0.15
 0.2
 0.25
 0.3
 0  5  10  15  20  25  30  35
∆ε
/T
4
NT
ζ= 4 M=1.45M=1.50
M=1.55
M=1.60
M=1.65
0.127
 0.22
 0.24
 0.26
 0.28
 0.3
 0.32
 0.34
 0.36
 15  20  25  30  35  40  45  50  55  60  65
χ(
0)/
χ S
B
NT
ζ= 5
M=1.30
M=1.55
M=1.60
M=1.65
M=1.70
M=1.75
M=1.85
Figure 2. (a)Comparison of the lattice ∆ǫ/T 4 with that for the continuum for r = 0.5.
(b)The variation of χ(0)/χSB vs NT for different M and ζ = 5.
4. Conclusion
The range 1.55 < M < 1.60 is best suited for the computation of different
thermodynamic quantities of free overlap and domain wall fermions on the lattice as the
continuum results are obtained on the smallest possible lattice size. This range seem
to be quite universal even in the presence of µˆ. The L5 needed to obtain the overlap
results starting from the domain wall formalism is also minimum for this range of M. It
is anticipated that this range would remain roughly same even in the presence of gauge
fields and thus this work justifies the use of M > 1 for faster computations even in full
QCD with such fermions.
Acknowledgements
It is a pleasure to thank our collaborator Debasish Banerjee. S.S would like to
acknowledge the Council of Scientific and Industrial Research(CSIR) for financial
support through the Shyama Prasad Mukherjee(SPM) fellowship.
References
[1] H. Neuberger, Phys. Rev. D57, 5417 (1998)
[2] R. Narayanan and H. Neuberger, Phys. Rev. Lett. 71, 3251 (1993); H. Neuberger, Phys. Lett. B
417, 141 (1998).
[3] C. Gattringer and L. Liptak, Phys. Rev. D76, 054502 (2007).
[4] D. Banerjee, R. V. Gavai and S. Sharma , e-Print arXiv:0803.3925.
[5] D. B. Kaplan, Phys. Lett. B288, 342 (1992).
[6] J. Bloch and T. Wettig, Phys. Rev. Lett. 97, 012003 (2006).
[7] P. Hasenfratz and F. Karsch, Phys. Lett. B125, 308 (1983).


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