This is a pedagogical review of the lattice study of finite density QCD, which is intended to provide the minimum necessary contents, so that the paper may be used as the first reading for a newcomer to the field and also for those working in nonlattice communities. After a brief introduction to argue why finite density QCD can be a new attractive subject, we describe fundamental formulae which are necessary for the following sections.Then we survey lattice QCD simulations in small chemical potential regions, where several prominent works have been reported recently. Next, two-color QCD calculations are discussed, where we have a chance to glance at many new features of finite density QCD, and indeed recent simulations indicated quark pair condensation and the in-medium effect. Tables of SU(3) and SU(2) lattice simulations at finite baryon density are given.In the next section, we make a survey of several related works which may be a starting point of the new development in the future, although some works do not attract much attention now. Materials are described in a pedagogical manner. Starting from a simple 2-d model, we briefly discuss a lattice analysis of NJL model. We describe a non-perturbative analytical approach, i.e., strong coupling approximation method and some results. Canonical ensemble approach instead of usual grand canonical ensample may be another route to reach high density. We examine the density of state method and show this old idea includes recently proposed factorization method. An alternative method, complex Langevin equation and an interesting model, finite isospin model, are also discussed.In the Appendix, we give several technical points which are useful in practical calculations.Contents * ) based on talks at Workshop "Thermal Field Theory", Yukawa Institute, Aug. 9. 2002 and Symposium "Towards understanding of finite density systems", JPS meeting, Sept. 14, 2002
We examine a new 2nd order integrator recently found by Omelyan et al. The integration error of the new integrator measured in the root mean square of the energy difference, ∆H 2 1/2 , is about 10 times smaller than that of the standard 2nd order leapfrog (2LF) integrator. As a result, the step size of the new integrator can be made about three times larger. Taking into account a factor 2 increase in cost, the new integrator is about 50% more efficient than the 2LF integrator. Integrating over positions first, then momenta, is slightly more advantageous than the reverse. Further parameter tuning is possible. We find that the optimal parameter for the new integrator is slightly different from the value obtained by Omelyan et al., and depends on the simulation parameters. This integrator could also be advantageous for the Trotter-Suzuki decomposition in Quantum Monte Carlo.
The 3-D Z(3) Potts model is a model for finite temperature QCD with heavy quarks. The chemical potential in QCD becomes an external magnetic field in the Potts model. Following Alford et al. , we revisit this mapping, and determine the phase diagram for an arbitrary chemical potential, real or imaginary. Analytic continuation of the phase transition line between real and imaginary chemical potential can be tested with precision. Our results show that the chemical potential weakens the heavy-quark deconfinement transition in QCD.
We present a study of the deconfinement phase transition of one-flavor QCD using the multiboson algorithm. The mass of the Wilson fermions relevant for this study is moderately large and the non-Hermitian multiboson method is a superior simulation algorithm. Finite-size scaling is studied on lattices of size 8 3 ϫ4, 12 3 ϫ4, and 16 3 ϫ4. The behaviors of the peak of the Polyakov loop susceptibility, the deconfinement ratio, and the distribution of the norm of the Polyakov loop are all characteristic of a first-order phase transition for heavy quarks. As the quark mass decreases, the first-order transition gets weaker and turns into a crossover. To investigate finite-size scaling on larger spatial lattices we use an effective action in the same universality class as QCD. This effective action is constructed by replacing the fermionic determinant with the Polyakov loop identified as the most relevant Z(3)-symmetry-breaking term. Higher-order effects are incorporated in an effective Z(3)-breaking field h, which couples to the Polyakov loop. Finite-size scaling determines the value of h where the first-order transition ends. Our analysis at the end point h ep indicates that the effective model and thus QCD are consistent with the universality class of the three-dimensional Ising model. Matching the field strength at the end point h ep to the values used in the dynamical quark simulations we estimate the end point ep of the first-order phase transition. We find ep ϳ0.08 which corresponds to a quark mass of about 1.4 GeV.
Using 1-min returns of Bitcoin prices, we investigate statistical properties and multifractality of a Bitcoin time series. We find that the 1-min return distribution is fat-tailed, and kurtosis largely deviates from the Gaussian expectation. Although for large sampling periods, kurtosis is anticipated to approach the Gaussian expectation, we find that convergence to that is very slow. Skewness is found to be negative at time scales shorter than one day and becomes consistent with zero at time scales longer than about one week. We also investigate daily volatility-asymmetry by using GARCH, GJR, and RGARCH models, and find no evidence of it. On exploring multifractality using multifractal detrended fluctuation analysis, we find that the Bitcoin time series exhibits multifractality. The sources of multifractality are investigated, confirming that both temporal correlation and the fat-tailed distribution contribute to it. The influence of "Brexit" on June 23, 2016 to GBP-USD exchange rate and Bitcoin is examined in multifractal properties. We find that, while Brexit influenced the GBP-USD exchange rate, Bitcoin was robust to Brexit.
The scaling behavior of pure gauge SU(3) in the region β = 5.85−7.60 is examined by a Monte Carlo Renormalization Group analysis. The coupling shifts induced by factor 2 blocking are measured both on 32 4 and 16 4 lattices with high statistics. A systematic deviation from naive 2-loop scaling is clearly seen. The mean field and effective coupling constant schemes explain part, but not all of the deviation. It can be accounted for by a suitable change of coupling constant, including a correction term O(g 7 ) in the 2-loop lattice β-function. Based on this improvement,M S is estimated to be 2.2(±0.1) from the analysis of the string tension σ.
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