Can Stochastic Quantization Evade the Sign Problem? The Relativistic Bose Gas at Finite Chemical Potential

A nonperturbative study of field theories with a complex action, such as QCD at finite baryon density, is difficult due to the sign problem. We show that the relativistic Bose gas at finite chemical potential has a sign and “silver blaze” problem, similar to QCD. We then apply stochastic quantization and complex Langevin dynamics to study this theory with nonperturbative lattice simulations. Independence of chemical potential at small and a transition to a condensed phase at large chemical potential are found. Lattices of size $N^4$, with $N=4$, 6, 8, 10, are used. We show that the sign problem is severe, however, we find that it has no negative effect using this approach. This improves the prospects of applying stochastic quantization to QCD at nonzero density.

Figure 1

Real part of the density, $\mathrm{Re}⟨n⟩$, as a function of chemical potential for lattices of size $N^4$, with $N=4$, 6, 8, 10. The parameters are $m=\lambda =1$ and stepsize $ϵ=5\times {10}^{-5}$. The inset shows a blowup around the transition. In the thermodynamic limit, the density vanishes below the critical chemical potential.

Figure 2

Real part of the square of the field modulus, $\mathrm{Re}⟨|\varphi {|}^2⟩$, as a function of $\mu$. The inset shows a blowup at smaller $\mu$.

Figure 3

Density in the phase-quenched theory, $⟨n{⟩}_{\mathrm{pq}}$, as a function of $\mu$. The inset shows a blowup at smaller $\mu$.

Figure 4

Average phase factor in the phase-quenched theory, $\mathrm{Re}⟨e^{i\phi }{⟩}_{\mathrm{pq}}$, as a function of $\mu$, indicating the severeness of the sign problem in the thermodynamic limit.