Abstract
The canonical approach, which was developed for solving the sign problem, may suffer from a new type of sign problem. In the canonical approach, the grand partition function is written as a fugacity expansion: $Z_G(\mu,T) = \sum_n Z_C(n,T) \xi^n$, where $\xi=\exp(\mu/T)$ is the fugacity, and $Z_C(n,T)$ are given as averages over a Monte Carlo update, $\langle z_n\rangle$. We show that the complex phase of $z_n$ is proportional to $n$ at each Monte Carlo step. Although $\langle z_n\rangle$ take real positive values, the values of $z_n$ fluctuate rapidly when $n$ is large, especially in the confinement phase, which gives a limit on $n$. We discuss possible remedies for this problem.
Highlights
If the fermion determinant is complex, we are in trouble
We employ method (3), in which the fermion determinant det ∆ is expanded as a series of the hopping parameter κ, and the diagrams are classified and packed with respect to the fugacity power:Nf = (det(I − κQ(μ)))Nf
Result The lattice QCD simulations reported here were performed at the Far Eastern Federal University on Vostok-1, which consists of 10 nodes (2 × Intel E5-2680 v2, 64 GB RAM; 2 × Nvidia Tesla K40X Kepler)
Summary
When μ takes a nonzero real value, the fermion determinant becomes a complex number. This is problematic, because in the Monte Carlo simulations, we generate the gluon fields with the probability. In Eq (3), the canonical partition functions, ZC, do not depend on μ, and Eq (3) works for real, imaginary, and even complex μ. When the chemical potential is pure imaginary, μ = iμI , the fermion determinant is real, and in those regions, we can construct ZC from ZG. 2. Calculation of ZC(n) In order to obtain the canonical partition functions, ZC(n, T ), first we expand the fermion determinant:. We employ method (3), in which the fermion determinant det ∆ is expanded as a series of the hopping parameter κ, and the diagrams are classified and packed with respect to the fugacity power:
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