We determine the scale setting function and the pseudo-critical temperature on the lattice in $N_f=2$ two-color QCD using the Iwasaki gauge and Wilson fermion actions. Although two-color QCD does not correspond to the real world, it is very useful as
a good testing ground for three-color QCD. The scale setting function gives the relative lattice spacings of simulations performed at different values of the bare coupling. It is a necessary tool for taking the continuum limit. Firstly, we measure the meson spectra for various combinations of ($beta,kappa$) and find a line of constant physics in $beta$--$kappa$ plane. Next, we determine the scale setting function via $w_0$ scale in the gradient flow method. Furthermore, we estimate the pseudo-critical temperature at zero chemical potential from the chiral susceptibility. Combining these results, we can discuss the QCD phase diagram in which both axes are given by dimensionless quantities, namely, the temperature normalized by the pseudo-critical temperature on the lattice and the chemical potential normalized by the pseudoscalar meson mass. It makes it easy to compare among several lattice studies and also makes it possible to compare theoretical analyses and lattice studies in the continuum limit.
We delineate equilibrium phase structure and topological charge distribution of dense two-colour QCD at low temperature by using a lattice simulation with two-flavour Wilson fermions that has a chemical potential $mu$ and a diquark source $j$ incorpo
rated. We systematically measure the diquark condensate, the Polyakov loop, the quark number density and the chiral condensate with improved accuracy and $jto0$ extrapolation over earlier publications; the known qualitative features of the low temperature phase diagram, which is composed of the hadronic, Bose-Einstein condensed (BEC) and BCS phases, are reproduced. In addition, we newly find that around the boundary between the hadronic and BEC phases, nonzero quark number density occurs even in the hadronic phase in contrast to the prediction of the chiral perturbation theory (ChPT), while the diquark condensate approaches zero in a manner that is consistent with the ChPT prediction. At the highest $mu$, which is of order the inverse of the lattice spacing, all the above observables change drastically, which implies a lattice artifact. Finally, at temperature of order $0.45T_c$, where $T_c$ is the chiral transition temperature at zero chemical potential, the topological susceptibility is calculated from a gradient-flow method and found to be almost constant for all the values of $mu$ ranging from the hadronic to BCS phase. This is a contrast to the case of $0.89T_c$ in which the topological susceptibility becomes small as the hadronic phase changes into the quark-gluon plasma phase.
The chemical potential ($mu$) dependence of the topological susceptibility with two-color two-flavor QCD is studied. We find that at temperature $T approx T_c /2$, where $T_c$ denotes the critical temperature at zero chemical potential, the topologic
al susceptibility is almost constant throughout $0 leq amu lesssim 1.0$, while at $Tapprox T_c$, it decreases significantly from the $mu=0$ value in a high $mu$ regime. In this work, we perform the simulation for $mu/T le 16$, which covers even the low temperature and the high chemical potential regime. In this regime, we introduce a diquark source term, which is characterized by $j$, into the action. We also show our results for the phase diagram in a low temperature regime ($Tapprox T_c/2$), which is obtained after taking the $j to 0$ limit of the diquark condensate and the Polyakov loop.
The effects of fluctuations are discussed around the phase boundary of the inhomogeneous chiral transition between the inhomogeneous chiral phase and the chiral-restored phase. The particular roles of thermal and quantum fluctuations are elucidated a
nd a continuity of their effects across the phase boundary is suggested. In addition, it is argued that anomalies in the thermodynamic quantities should have phenomenological implications for the inhomogeneous chiral transition. Some common features for other phase transitions, such as those from the normal to the inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov state in superconductivity, are also emphasized.
We investigate inhomogeneous chiral condensates, such as the so-called dual chiral density wave of dense quark matter, under an external magnetic field at finite real and imaginary chemical potentials. In a model-independent manner, we find that anal
ytic continuation from imaginary to real chemical potential is not possible due to the singularity induced by inhomogeneous chiral condensates at zero chemical potential. From the discussion on the non-analyticity and methods used in lattice QCD simulations, e.g., Taylor expansion, and the analytic continuation with an imaginary chemical potential, it turns out that information on an inhomogeneous chiral condensed phase is missed in the lattice simulations at finite baryon chemical potentials unless the non-analyticity at zero chemical potential is correctly considered. We also discuss an exceptional case without such non-analyticity at zero chemical potential.
