Complex Langevin calculations in QCD at finite density

Abstract

We demonstrate that the complex Langevin method (CLM) enables calculations in QCD at finite density in a parameter regime in which conventional methods, such as the density of states method and the Taylor expansion method, are not applicable due to the severe sign problem. Here we use the plaquette gauge action with β = 5.7 and four-flavor staggered fermions with degenerate quark mass ma = 0.01 and nonzero quark chemical potential μ. We confirm that a sufficient condition for correct convergence is satisfied for μ/T = 5.2 − 7.2 on a 83 × 16 lattice and μ/T = 1.6 − 9.6 on a 163 × 32 lattice. In particular, the expectation value of the quark number is found to have a plateau with respect to μ with the height of 24 for both lattices. This plateau can be understood from the Fermi distribution of quarks, and its height coincides with the degrees of freedom of a single quark with zero momentum, which is 3 (color) × 4 (flavor) × 2 (spin) = 24. Our results may be viewed as the first step towards the formation of the Fermi sphere, which plays a crucial role in color superconductivity conjectured from effective theories.

Overview of the QCD phase diagram

“…In Refs. [ 103 , 106 , 107 ], the average quark number is studied on lattices with sizes and . The authors found a plateau for as a function of the chemical potential.…”

Complex Langevin calculations in finite density QCD at large μ/T with the deformation technique

“…In order to confirm this phenomenon, we clearly need to increase the lattice size. We have already started simulations on a 8 3 × 16 lattice [36] and found that the CLM actually gives correct results even in the region of relatively large chemical potential without the deformation technique. Preliminary results for the baryon number density and the chiral condensate show a rapid change twice as we increase the chemical potential, which may be interpreted as the phase transitions to the nuclear matter and to the quark matter.…”

An overview of the QCD phase diagram at finite $T$ and $μ$

“…In Ref. [36][37][38] Complex Langevin simulations in small boxes with sizes of 8 3 × 16 and 16 3 × 8 with four flavors and a lattice spacing of 𝑎 −1 ≈ 4.7 GeV were performed. For the dependence of the quark number on the chemcial potential a plateau was observed that the authors relate to the Fermi surface and color superconductivity.…”