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.
Highlights
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
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
An alternative criterion for correct convergence has been discussed from the viewpoint of the boundary terms [31, 32], which appear in the original argument [20, 21] for justifying the CLM based on the continuous Langevin-time formulation
Summary
If the histogram of this quantity falls off exponentially or faster, we can trust the results This can be violated either by the excursion problem or by the singular drift problem as we mentioned in the Introduction. That recent work [49] on 2D U(1) theory with a θ term suggests that the CLM works even if the unitarity norm becomes large as far as the drift histogram falls off fast.. In such a way that the unitarity norm is minimized Adding this procedure after each step of the Langevin-time evolution does not spoil the argument for justification as is shown in refs. The gauge cooling keeps all the link variables as close to unitary matrices as possible during the Langevin-time evolution. These two quantities (2.14) and (2.15) are calculated by the so-called noisy estimator using 20 noise vectors
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