RIKEN Accel. Prog. Rep. 50 (2017)

Ⅱ-5. Hadron Physics (Theory)

Canonical approach to ﬁnite density QCD with multiple precisioncomputation†
R. Fukuda,∗1,∗2,∗3 A. Nakamura,∗1,∗4,∗5 and S. Oka∗6Although QCD has a rich phase structure at a ﬁnite temperature and density,1) investigations based onﬁrst–principles calculations are limited in small density regions. This is because the action with a ﬁnite real quark chemical potential μq is complex ingeneral and the Monte Carlo method with the complex action does not work. This problem is called thesign problem.2) In this work, the canonical approach3)
as a method for ﬁnite density QCD is studied. In
the canonical approach, the grand canonical partition
function ZGC (μq ) at a ﬁnite real quark chemical potential is represented as the fugacity expansion withthe canonical partition function Zn as follows:
ZGC (μq ) =∞

Zn enμq /T .

(1)

n=−∞

The canonical partition function is given by the Fouriertransformation of the grand canonical partition function calculated at a purely imaginary chemical potential as follows:
2π μI 1ZGC (iμI )e−inμI /T ,(2)dZn =2π 0T
where μI is a real number. This procedure to obtain the canonical partition functions does not suﬀer
from the sign problem because the action with purely
imaginary chemical potential is real and the Monte
Carlo method works. This is a strong advantage ofthe canonical approach. However, we encounter somediﬃculties as follows. In the calculation to obtain thecanonical partition function at a large baryon number,
the Fourier transformation becomes a highly oscillating
integral. Thus, it is diﬃcult to numerically computethe integral because of cancellation of signiﬁcant digits
in the Fourier transformation. To avoid this problem,the multiple precision computation4) is adopted in thiswork. Another problem is the numerical cost for thecalculation of the grand canonical partition function
at a purely imaginary chemical potential. To performthe Fourier transformation of the grand canonical partition function, we need the fermion determinants at
many diﬀerent values of the purely imaginary chemical potential. To reduce this numerical cost, the winding number expansion based on the hopping parameter
†∗1
∗2
∗3
∗4
∗5
∗6

Condensed from the article in Phys. Rev. D. 93, 094508(2016)RIKEN Nishina Center
RIKEN BNL Research Center
Department of Physics, the University of Tokyo
Research Center for Nuclear Physics, Osaka University
School of Biomedicine, Far Eastern Federal University
Department of Physics, Rikkyo University- 126 -expansion for the Wilson fermion formalism is developed in this work. In this method, we can expand thefermion determinant det D(μ) with the complex coefﬁcients {Wn } as follows:
log det D(μ) =∞Wn enμ/T ,(3)n=−∞where μ is a complex number. Therefore, if we calculate the coeﬃcients at some quark chemical potential
once, we can obtain the fermion determinant at any
quark chemical potential.
Figure 1 shows the baryon chemical potential μB
dependence of the pressure evaluated by the canonical approach and the direct method. We directly use
the winding number expansion of Eq. 3 for the calculation of the pressure at a real baryon chemical potential
in the direct method. The upper bounds in this ﬁgure correspond to the validity range of these methods.
Therefore, we can conclude that the validity range ofthe canonical approach is wider than that of the directmethod.

10Canonical Direct

86420012345Fig. 1. The baryon chemical potential μB dependence ofthe pressure evaluated by the canonical approach andthe direct method: Δp is deﬁned as Δp = p(μB /T ) −p(0).

References
1) K. Fukushima and T. Hatsuda, Rept. Prog. Phys. 74,014001 (2011).
2) Ph. de Forcrand, PoS (LAT2009)010 (2009).
3) A. Hasenfratz and D. Toussaint, Nucl. Phys. B 371(1992), 539.
4) S. Oka for Zn –Collaboration, PoS (LAT2015)166(2015).