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Lattice QCD at finite temperature and density

Heng-Tong Ding

Abstract

I review recent lattice results on strongly interacting matter under extreme conditions, with emphasis on the finite-temperature QCD transition at $μ_B=0$, its approach toward the chiral limit and the fate of the $U_A(1)$ anomaly, as well as recent constraints on the QCD phase boundary and the possible critical endpoint at $μ_B>0$. I also discuss selected advances in lattice methods and in QCD thermodynamics under external conditions, in particular strong magnetic fields, isospin chemical potential, rotation, acceleration, and quark spin polarization.

Lattice QCD at finite temperature and density

Abstract

I review recent lattice results on strongly interacting matter under extreme conditions, with emphasis on the finite-temperature QCD transition at , its approach toward the chiral limit and the fate of the anomaly, as well as recent constraints on the QCD phase boundary and the possible critical endpoint at . I also discuss selected advances in lattice methods and in QCD thermodynamics under external conditions, in particular strong magnetic fields, isospin chemical potential, rotation, acceleration, and quark spin polarization.
Paper Structure (32 sections, 3 equations, 10 figures)

This paper contains 32 sections, 3 equations, 10 figures.

Table of Contents

  1. Introduction
  2. QCD thermodynamics at $\mu_B=0$
  3. Chiral crossover at the physical point from different discretization schemes
  4. Symmetry restoration and the $U_A(1)$ anomaly
  5. Macroscopic manifestations: correlators, screening masses, and topology
  6. Physical-point benchmark from topological susceptibility $\chi_t(T)$.
  7. Toward the chiral limit: screening masses in $N_f=2$ MDWF.
  8. $N_f$ and $M_\pi$ systematics.
  9. Dirac eigenvalue spectrum: microscopic mechanism and quantitative diagnostics
  10. Cohen's gap criterion and constraints from analyticity.
  11. A useful parametrization of $\rho(\lambda)$.
  12. Mass-derivative method and evidence for $\rho\propto m_\ell^2$ with a $\delta$-like IR peak.
  13. Infrared peak structures and eigenmode geometry.
  14. Universality, scaling, and a generalized Banks--Casher relation
  15. Suppressing regular terms: improved order parameter.
  16. ...and 17 more sections

Figures (10)

  • Figure 1: Screening-mass restoration patterns toward the chiral limit in $N_f=2$ MDWF simulations Aoki:2025mue. $N_f=2$ MDWF screening masses toward the chiral limit: V--AV (left), PS--S (middle) and X--T splittings (right).
  • Figure 2: Quark-mass derivatives of the Dirac eigenvalue spectrum $\rho(\lambda,m_\ell)$ in $N_f=2+1$ QCD at $T\simeq205~\mathrm{MeV}$. The near-coincidence of $(\partial\rho/\partial m_\ell)/m_\ell$ with $\partial^2\rho/\partial m_\ell^2$, together with $\partial^3\rho/\partial m_\ell^3\to 0$ within uncertainties, indicates $\rho(\lambda\!\to\!0,m_\ell)\propto m_\ell^2$ in the infrared. The infrared peak sharpens as the lattice spacing is reduced, suggesting an emerging $\delta(\lambda)$-like near-zero-mode contribution. Plots are taken from Ref. Ding:2020xlj.
  • Figure 3: Dirac eigenvalue spectrum $\rho(\lambda,m_\ell)$ in $N_f=2+1$ QCD. Left: Obtained using stout staggered fermion at T=230 MeV Alexandru:2024tel. Right: Obtained using overlap fermions at T=170 MeV Kotov:2025lat.
  • Figure 4: From left to right: $P_1$, $P_2$, and rescaled $P_1$ and $P_2$ with corresponding macroscopic critical behavior. Figures are taken from Ding:2023oxy.
  • Figure 5: Status of lattice constraints and recent estimates for the CEP location. Left: exclusion region from Wuppertal--Budapest, including the bound of no CEP for $\mu_B<450~\mathrm{MeV}$ at $2\sigma$Borsanyi:2025dyp. Middle: Wuppertal--Budapest Lee--Yang-edge analysis, yielding an $84\%$ probability for no CEP below $T_{\rm CEP}<103~\mathrm{MeV}$Adam:2025phc. Right: Parma--Bielefeld Lee--Yang-edge extrapolation from imaginary-$\mu_B$ simulations Clarke:2024ugt.
  • ...and 5 more figures