The canonical partition function is related to the grand canonical one through the fugacity expansion and is known to have no sign problem. In this paper we perform the fugacity expansion by a method of the hopping parameter expansion in temporal direction for the lattice QCD: winding number expansion. The canonical partition function is constructed for Nf=2 QCD starting from gauge configurations at zero chemical potential. After derivation of the canonical partition function we calculate hadronic observables like chiral condensate and quark number density and the pressure at the real chemical potential.
We calculate the baryon chemical potential ($μ_B$) dependence of thermodynamic observables, i.e., pressure, baryon number density and susceptibility by lattice QCD using the canonical approach. We compare the results with those by the multi parameter reweighting (MPR) method; Both methods give very consistent values in the regions where errors of the MPR are under control. The canonical method gives reliable results over $μ_ B/T=3$,with $T$ being temperature. Multiple precision operations play an important roll in the evaluation of canonical partition functions.
New approach to computation of canonical partition functions in $N_f=2$ lattice QCD is presented. We compare results obtained by new method with results obtained by known method of hopping parameter expansion. We observe agreement between two methods indicating validity of the new method. We use results for the number density obtained in the confining and deconfining phases at imaginary chemical potential to determine the phase transition line at real chemical potential.
The canonical approach for finite density lattice QCD has a numerical instability. This instability makes it difficult to use the method reliably at the finite real chemical potential region. We studied this instability in detail and found that it is caused by the cancellation of significant digits. In order to reduce the effect of this cancellation, we adopt the multiple precision calculation for our discrete Fourier transformation (DFT) program, and we get the canonical partition function Zc(n,T) with required accuracy. From the obtained Zc(n,T), we calculate Lee--Yang zero distribution vary…
Finite density lattice QCD without extrapolation: Bulk thermodynamics with physical quark masses from the canonical ensemble
Wuppertal University（伍珀塔尔大学）；Pennsylvania State University（宾夕法尼亚州立大学）；ELTE Eötvös Loránd University（罗兰大学）；Jülich Supercomputing Centre, Forschungszentrum Jülich（于利希超级计算中心，于利希研究中心）；Dipartimento di Fisica, Università di Torino and INFN Torino（都灵大学物理系及都灵INFN）；MTA-ELTE Lendület ”Momentum” Strongly Interacting Matter Research Group（匈牙利科学院-罗兰大学Lendulet“动量”强相互作用物质研究组）
We clarify regions where the canonical approach works well at the finite temperature and density in the Nambu-Jona-Lasinio (NJL) and Polyakov-NJL (PNJL) models. The canonical approach is a useful method for avoiding the sign problem in lattice QCD simulations at finite density, but it involves some parameters. We find that number densities computed from the canonical approach are consistent with exact values in most of the confinement phase within the parameters, which are applicable in lattice QCD.
We study the Lee-Yang zeros in the canonical approach to search phase transition points at finite temperature and density in the Nambu-Jona-Lasinio (NJL) model as an effective model of QCD. The canonical approach is a promising method to avoid the sign problem in lattice QCD at finite density. We find that a set of Lee-Yang zeros computed with finite degrees of freedom can be extrapolated to those with infinite degrees of freedom, providing the correct phase transition point. We propose the present method as a useful method for actual lattice simulations for QCD.
We study the finite density phase transition in the lattice QCD at real chemical potential. We adopt canonical approach and the canonical partition function is constructed for Nf=2 QCD. After derivation of the canonical partition function we calculate observables like the pressure, the quark number density, its second cumulant and the chiral condensate as a function of the real chemical potential. We covered a wide range of temperature region starting from the confining low to the deconfining high temperature. We observe signals for the deconfinement and the chiral restoration phase transition…
The study of QCD phase diagram is very interesting, but we have never understood it well. This is because we face a problem at finite density in QCD. The problem is called sign problem. It causes a decrease of the calculation accuracy. This is why, we can not calculate physical quantities accurately at finite chemical potential. In this study, we try to beat the sign problem using canonical approach of finite density lattice QCD. Although it is known that the canonical approach has several numerical problems, we can reduce them and calculate thermodynamic observables accurately at finite densi…
Using GPGPU techniques and multi-precision calculation we developed the code to study QCD phase transition line in the canonical approach. The canonical approach is a powerful tool to investigate sign problem in Lattice QCD. The central part of the canonical approach is the fugacity expansion of the grand canonical partition functions. Canonical partition functions $Z_n(T)$ are coefficients of this expansion. Using various methods we study properties of $Z_n(T)$. At the last step we perform cubic spline for temperature dependence of $Z_n(T)$ at fixed $n$ and compute baryon number susceptibilit…
At finite baryon density lattice QCD first-principle calculations can not be performed due to the sign problem. In order to circumvent this problem, we use the canonical approach, which provides reliable analytical continuation from the imaginary chemical potential region to the real chemical potential region. We briefly present the canonical partition function method, describe our formulation, and show the results, obtained for two temperatures: $T/T_c = 0.93$ and $T/T_c = 0.99$ in lattice QCD with two flavors of improved Wilson fermions.
We discuss two new DoS approaches for finite density lattice QCD. The paper extends a recent presentation of the new techniques based on Wilson fermions, while here we now discuss and test the case of finite density QCD with staggered fermions. The first of our two approaches is based on the canonical formulation where observables at a fixed net quark number $N$ are obtained as Fourier moments of the vacuum expectation values at imaginary chemical potential $θ$. We treat the latter as densities which can be computed with the recently developed FFA method. The second approach is based on a dire…
We propose new way of heavy ion collisions experiment data analysis. We analyze physical parameters of fireball created in RHIC experiment based on Grand Canonical Distribution and different Lattice QCD data available at the moment. Our results on chemical potential are in agreement with previous model estimations and do not depend on Lattice setup. At same time, we found possible T(V) states of fireball and estimated the most probable temperature and volume of fireball as function of collision energy. We conclude that hadrom matter at RHIC experiment is thermalized and described by Grand Cano…
We propose and test a new approach to computation of canonical partition functions in lattice QCD at finite density. We suggest a few steps procedure. We first compute numerically the quark number density for imaginary chemical potential $iμ_{qI}$. Then we restore the grand canonical partition function for imaginary chemical potential using fitting procedure for the quark number density. Finally we compute the canonical partition functions using high precision numerical Fourier transformation. Additionally we compute the canonical partition functions using known method of the hopping parameter…
We investigate the nuclear and the quark matter at finite real chemical potential ($μ_\mathrm{R}$) and low temperature from the viewpoint of the canonical sectors constructed via the imaginary chemical potential region. Based on the large $N_\mathrm{c}$ estimation, where $N_\mathrm{c}$ is the number of color, we can discuss the confinement-deconfinement nature at finite $μ_\mathrm{R}$ from the canonical sectors. We found the expectation that the sharp change of canonical sectors at $μ_\mathrm{R} \sim M_\mathrm{B}/N_\mathrm{c}$, where $M_\mathrm{B}$ is the lowest baryon mass, is happen in the l…
QCD at fixed baryon number can be formulated in terms of transfer matrices explicitly defined in the canonical sectors. In the heavy-dense limit, the fermionic contributions to the canonical partition functions in terms of Polyakov loops and quark occupation numbers turn out to be completely factorized in space. At low temperatures and infinitely strong coupling the sign problem is reduced by orders of magnitude for any baryon number as compared to the corresponding grand-canonical ensemble. In the canonical formulation it is straighforward to integrate out the Polyakov loops in the fermionic …
The distribution of canonical determinants in QCD is determined by means of chiral perturbation theory. For a non-zero quark charge the canonical determinants take complex values. In the dilute pion gas approximation, we compute all moments of the magnitude of the canonical determinants, as well as the first nonvanishing moments of the real and imaginary parts. The non-trivial cancellation between the real and the imaginary parts of the canonical determinants is derived and the signal to noise ratio is discussed. The analytical distributions are compared to lattice data. The average density of…
We present two new suggestions for density of states (DoS) approaches to finite density lattice QCD. Both proposals are based on the recently developed and successfully tested DoS FFA technique, which is a DoS approach for bosonic systems with a complex action problem. The two different implementations of DoS FFA we suggest for QCD make use of different representations of finite density lattice QCD in terms of suitable pseudo-fermion path integrals. The first proposal is based on a pseudo-fermion representation of the grand canonical QCD partition sum, while the second is a formulation for the…
QCD at Zero Baryon Density and the Polyakov Loop Paradox
我们比较了具有四种交错夸克的巨正则配分函数和重子数B的正则配分函数在μ =0和B=0时的形式和数值模拟。我们证明了自由能量密度在热力学极限下是相等的，并表明它们可以用强子共振气体在T_T_c处很好地描述< T_c and by the free fermion gas at T>。这两个合奏之间的小差异，为热力学可观的特点解除相变，消失与晶格尺寸的增加。这些差异仅仅是由非零重子密度扇区的贡献引起的，这些扇区随着体积的增加而被指数抑制。Polyakov循环表现出不同的行为：对于所有温度和体积，它的期望值在正则公式中恰好为零，而在常用的巨正则公式中总是非零。我们澄清这个矛盾的差异，并表明，非零的Polyakov回路的期望值是由于非零triality状态，这不是物理的贡献，因为他们给零贡献的配分函数。
We study the winding number susceptibility < I^2> in a superfluid system and the topological susceptibility < Q^2> in QCD. We argue that both correlation functions exhibit similar structures, including the generation of the contact terms. We discuss the nature of the contact term in superfluid system and argue that it has exactly the same origin as in QCD, and it is related to the long distance physics which cannot be associated with conventional microscopical degrees of freedom such as phonons and rotons. We emphasize that the conceptual similarities between superfluid system and QCD may lead…
Finite Density Algorithm in Lattice QCD -- a Canonical Ensemble Approach
本文将回顾基于有限化学势的格点量子色动力学的有限密度算法，并总结其中的难点。我将提出一个正则系综方法，它从费米子行列式中投影出有限重子数扇区。对于这个算法的工作，它需要一个有效的方法来计算费米子行列式和蒙特卡罗算法，它容纳无偏估计的概率。我将报告沿这一方向取得的进展与Padé - Z $_2$估计的行列式及其实施的新开发的噪声蒙特卡罗算法。
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: $Z_G(μ,T) = \sum_n Z_C(n,T) ξ^n$, where $ξ=\exp(μ/T)$ 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 …
In the Wilson's lattice formulation of QCD, a fermionic Fock space of states can be explicitly built at each time slice using canonical creation and annihilation operators. The partition function $Z$ is then represented as the trace of the transfer matrix, and its usual functional representation as a path integral of $\exp(- S)$ can be recovered in a standard way. However, applying a Bogoliubov transformation on the canonical operators before passing to the functional formalism, we can isolate a vacuum contribution in the resulting action which depends only on the parameters of the transformat…
Canonical partition functions and Lee-Yang zeros of QCD at finite density and high temperature are studied. Recent lattice simulations have confirmed that the free energy of QCD is a quartic function of quark chemical potential at temperature slightly above pseudo-critical temperature $T_c$, as in the case with a gas of free massless fermions. We present analytic derivation of the canonical partition functions and Lee-Yang zeros for this type of free energy using the saddle point approximation. We also perform lattice QCD simulation in a canonical approach using the fugacity expansion of the f…
The chiral phase transition of the quark sector of QCD is investigated within the Hamiltonian approach in Coulomb gauge. Finite temperature T is introduced by compactifying one spatial dimension, which makes all thermodynamical quantities accessible from the ground state on the spatial manifold $\small\mathbb{R}^2 \times S^1(1/T)$. Neglecting the coupling between quarks and transversal gluons, the equations of motion of the quark sector are solved numerically and the chiral quark condensate is evaluated and compared to the results of the usual canonical approach to finite-temperature Hamiltoni…
Worldline representations were established as a powerful tool for studying bosonic lattice field theories at finite density. For fermions, however, the worldlines still may carry signs that originate from the Dirac algebra and from the Grassmann nature of the fermion fields. We show that a density of states approach can be set up to deal with this remaining sign problem, where finite density is implemented in a canonical approach by working with a fixed winding number of the fermion worldlines. We discuss the approach in detail and show first results of a numerical implementation in 2 dimensio…
