On the validity of the complex Langevin method near the deconfining phase transition in QCD at finite density

A bstract In our previous paper [1], we found that the complex Langevin (CL) method works for QCD at finite density on the 16 3 × 32 lattice in the low-temperature high-density regime within the range μ / T = 1.6 – 9.6 with μ and T being the quark chemical potential and the temperature, which enabled us to see a clear trend towards the formation of the Fermi sphere. Here we investigate the validity of the CL method on the 24 3 × 12 lattice in the deconfined phase near the deconfinement phase transition. As before, we use four-flavor staggered fermions and judge the validity using the criterion based on the probability distribution of the drift term. The spatial extent is L = (1.3 − 2.7 fm) &gt; $$ {\Lambda}_{\textrm{LQCD}}^{-1} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Λ</mml:mi> <mml:mtext>LQCD</mml:mtext> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> ∼ 1 fm, in contrast to our previous study with L &lt; $$ {\Lambda}_{\textrm{LQCD}}^{-1} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Λ</mml:mi> <mml:mtext>LQCD</mml:mtext> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> . We find that the CL method works in a broad region up to μ / T = 4.8, while it starts to fail as we approach the phase boundary due to the singular drift problem, which can be understood qualitatively by extending the Banks-Casher relation to the case at finite density.