Simulating QCD at nonzero baryon density to all orders in the hopping parameter expansion

Progress in simulating QCD at nonzero baryon density requires, among other things, substantial numerical effort. Here we propose two different expansions to all orders in the hopping parameter, preserving the full Yang-Mills action, which are much cheaper to simulate. We carry out simulations using complex Langevin dynamics, both in the hopping expansions and in the full theory, for two flavors of Wilson fermions, and agreement is seen at sufficiently high order in the expansion. These results provide support for the use of complex Langevin dynamics to study QCD at nonzero density, both in the full and the expanded theory, and for the convergence of the latter.

Figure 1

Dependence of the quark density (in lattice units) on the order of the truncation of the $\kappa$ and ${\kappa }_s$ expansions, for $\mu =0.7$ and 1.1, on a $4^4$ lattice with $\beta =5.9$, $\kappa =0.12$, and $N_f=2$. The region where the $\kappa$ expansion breaks down for $\mu =1.1$ is indicated. The lines show the result for full QCD. Saturation density is $n_{\text{sat}}=2N_c{N}_f=12$.

Figure 2

As in Fig. 1, for the chiral condensate (top) and the spatial plaquette (bottom).

Figure 3

As in Fig. 1, with the density at $\kappa =0.14$ on a $4^4$ lattice (top) and at $\kappa =0.12$ on a $8^4$ lattice (bottom), in both cases for the ${\kappa }_s$ expansion and full QCD.