Fermions at Finite Density in $2+1$ Dimensions with Sign-Optimized Manifolds

We present Monte Carlo calculations of the thermodynamics of the ($2+1$)-dimensional Thirring model at finite density. We bypass the sign problem by deforming the domain of integration of the path integral into complex space in such a way as to maximize the average sign within a parameterized family of manifolds. We present results for lattice sizes up to $10^3$ and we find that at high densities and/or temperatures the chiral condensate is abruptly reduced.

Figure 1

$⟨\overline{\psi }\psi ⟩$ (left) and $⟨\sigma ⟩$ (right) as a function of $\mu$ for $\beta \times 6^2$ lattices. Notice the increase of $⟨\sigma ⟩$ for large values of $\mu$ as expected from the discussion in the text. The black points are $⟨\sigma ⟩$ for simulations on $(S^1{)}^N$ on a $\beta =10$ lattice.

Figure 2

Left: $⟨\overline{\psi }\psi ⟩$ as a function of $\mu$ for $\beta =10$ at two different volumes: $L^2={10}^2$ and $6^2$. A sharpening of the chiral transition can be seen as the volume is increased. Right: $⟨\overline{\psi }\psi ⟩$ as a function of $T$ and $\mu$ for a spatial volume of size $6^2$. The thick central band indicates the location of $⟨\overline{\psi }\psi {⟩}_{\mu ,T}=0.5⟨\overline{\psi }\psi {⟩}_0$ and its width represents the statistical error. The peripheral thin lines indicate $⟨\overline{\psi }\psi {⟩}_{\mu ,T}=(0.5\pm 0.05)⟨\overline{\psi }\psi {⟩}_0$ to help gauge the sharpness of the transition.