Lattice results for the equation of state in dense QCD-like theories

We review recent progress in Monte Carlo simulations of dense two-color QCD (QC$_2$D), focusing on the phase diagram, the equation of state, and the sound velocity in the low-temperature regime. In three-color QCD at finite density, especially at low temperatures, the notorious sign problem makes lattice Monte Carlo simulations intractable. In contrast, QC$_2$D is free from this issue due to the pseudoreality of the quark representation. Recent independent lattice studies have revealed unexpected phenomena through first-principles calculations of the phase structure and thermodynamics. A particularly notable finding is that the sound velocity exceeds the so-called conformal (holography) bound, \( c_s^2/c^2 \le 1/3 \), which had not been observed in QCD-like theories at finite temperature. In this review, we focus primarily on results from a series of works by our group~\cite{Iida:2019rah, Iida:2020emi,Iida:2022hyy, Ishiguro:2021yxr, Iida:2024irv}, along with related studies in dense QC$_2$D and three-color QCD with isospin chemical potential. We discuss the possibility and physical implications of conformal bound violation even for three-color dense QCD, together with insights from effective model analyses and recent observations of neutron stars.