Tetsuya Takaishi

Testing and tuning symplectic integrators for the hybrid Monte Carlo algorithm in lattice QCD

We examine a new second-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, 1/2, is about 10 times smaller than that of the standard second-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.

Lattice QCD at Finite Density An Introductory Review

This is a pedagogical review of the lattice study of finite density QCD. It is intended to provide the minimum necessary content, so that it may be used as an introduction for newcomers to the field and also for those working in nonlattice areas. After a brief introduction in which we discuss the reasons that finite density QCD is an active and important subject, we present the fundamental formulae that are necessary for the treatment given in the following sections. Next, we survey lattice QCD simulational studies of system with small chemical potentials, of which there have been several prominent works reported recently. Then, two-color QCD calculations are discussed, where we are free from the notorious phase problem and have a chance to consider many new features of finite density QCD. Of special note is the result of recent simulations indicating 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 survey several related works that may represent a starting point of future development, although some of these works have not attracted much attention yet. This material is described in a pedagogical manner. Starting from a simple 2-d model, we briefly discuss a lattice analysis of the NJL model. We describe a non-perturbative analytic approach, i.e., the strong coupling approximation method and some results. The canonical ensemble approach, instead of the usual grand canonical ensemble may be another route to reach high density. We examine the density of state method and show that this old idea includes the recently proposed factorization method. An alternative method, the complex Langevin equation, and an interesting model, the finite isospin model, are also discussed. We give brief comments on a partial sum with respect to Z3 symmetry and the meron-cluster algorithm, which might solve the sign problem partially or completely. In the Appendix, we discuss several technical points that are useful in practical calculations.

Renormalization group flow of SU(3) lattice gauge theory : Numerical studies in a two coupling space

We investigate the renormalization group (RG) flow of SU(3) lattice gauge theory in a two coupling space with couplings β11 and β12 corresponding to 1×1 and 1×2 loops, respectively. Extensive numerical calculations of the RG flow are made in the fourth quadrant of this coupling space, i.e., β11>0 and β12<0 . Swendsens factor two blocking and the Schwinger–Dyson method are used to find an effective action for the blocked gauge field. The resulting renormalization group flow runs quickly towards an attractive stream which has an approximate line shape. This is a numerical evidence of a renormalized trajectory which locates close to the two coupling space. A model flow equation which incorporates a marginal coupling (asymptotic scaling term), an irrelevant coupling and a non-perturbative attraction towards the strong coupling limit reproduces qualitatively the observed features. We further examine the scaling properties of an action which is closer to the attractive stream than the currently used improved actions. It is found that this action shows excellent restoration of rotational symmetry even for coarse lattices with a∼0.3 fm.

Scaling study of pure gauge lattice QCD by Monte Carlo renormalization group method

The scaling behavior of pure gauge SU(3) in the region [beta]=5.85[minus]7.60 is examined by a Monte Carlo renormalization group analysis. The coupling shifts induced by factor 2 blocking are measured on both 32[sup 4] and 16[sup 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 [ital O]([ital g][sup 7]) in the 2-loop lattice [beta] function. Based on this improvement, [radical][sigma] /[Lambda][sub M[bar S]][sup [ital n]][sub [ital f]]=0 is estimated to be 2.2[plus minus]0.1 from the analysis of the string tension [sigma].

Fast fermion Monte Carlo

Abstract Three possibilities to speed up the Hybrid Monte Carlo algorithm are investigated. Changing the step-size adaptively brings no practical gain. On the other hand, substantial improvements result from using an approximate Hamiltonian or a preconditioned action.

Meson correlators in finite temperature lattice QCD

We analyze temporal and spatial meson correlators in quenched lattice QCD at T>0. Below Tc we observe little change in the meson properties as compared with T50. Above Tc we observe new features: chiral symmetry restoration and signals of plasma formation, but also an indication of persisting ‘‘mesonic’’ ~metastable! states and different temporal and spatial ‘‘masses’’ in the mesonic channels. This suggests a complex picture of QGP in the region (1 ‐1.5)Tc .

Response of quark condensate to the chemical potential

The responses of quark condensates to the chemical potential, as a function of temperature T and chemical potential \mu, are calculated within the Nambu--Jona-Lasinio (NJL) model. We compare our results with those from the recent lattice QCD simulations [QCD-TARO Collaboration, Nucl. Phys. B (Proc. Suppl.) 106, 462 (2002)]. The NJL model and lattice calculations show qualitatively similar behavior, and they will be complimentary ways to study hadrons at finite density. The behavior above T_c requires more elaborated analyses.

Properties of hadron screening masses at small baryonic density

Abstract The properties of hadron screening masses around the deconfinement phase transition at finite baryonic density can be studied by evaluating the Taylor coefficients with respect to the iso-scalar ( μ S = μ u = μ d ) and iso-vector ( μ V = μ u = − μ d ) chemical potentials, where μ u and μ d are u and d quark chemical potentials. We simulate 2-flavour lattice QCD with staggered fermions on a 12 2 × 24 × 6 lattice with ma = 0.05 and 0.10 and report investigations of nucleon, pseudo-scalar and vector mesons. We present new, strong evidence that in the confining phase, the screening masses at μ = 0 have significant temperature dependence, but the effect of μ S is very small. Above the critical temperature, the second derivative terms of mesons rapidly increase as contrasted to the case of baryon. We also study the responses of the screening masses to an iso-vector chemical potential and discuss some of the issues related to the properties of hadron masses at finite μ .

Choice of Integrator in the Hybrid Monte Carlo Algorithm

Abstract We study efficiency of higher order integrator schemes for the hybrid Monte Carlo (HMC) algorithm. Numerical tests are performed for Quantum Chromo Dynamics (QCD) with two flavors of Wilson fermions. We compare 2nd, 4th and 6th order integrators at various quark masses. The performance depends on both volume and quark mass. On currently accessible large lattices ( V∼24 4 ), higher order integrators can be more efficient than the 2nd order one only in heavy quark region, m q a>0.3 . Thus we conclude that for most full QCD simulations, except for heavy quark case, the usual 2nd order integrator is the best choice.

An Adaptive Markov Chain Monte Carlo Method for GARCH Model

We propose a method to construct a proposal density for the Metropolis-Hastings algorithm in Markov Chain Monte Carlo (MCMC) simulations of the GARCH model. The proposal density is constructed adaptively by using the data sampled by the MCMC method itself. It turns out that autocorrelations between the data generated with our adaptive proposal density are greatly reduced. Thus it is concluded that the adaptive construction method is very efficient and works well for the MCMC simulations of the GARCH model.