Heavy-dense QCD, sign optimization, and Lefschetz thimbles

We study the heavy-dense limit of QCD on the lattice with heavy quarks at high density. The effective three-dimensional theory has a sign problem which is alleviated by sign optimization where the path integration domain is deformed in complex space in a way that minimizes the phase oscillations. We simulate the theory via a hybrid Monte Carlo approach, for different volumes, both to leading order and next-to-next-to leading order in the hopping expansion, and show that sign optimization successfully mitigates the sign problem at large enough volumes where usual reweighting methods fail. Finally we show that there is a significant overlap between the complex manifold generated by sign optimization and the Lefschetz thimbles associated with the theory.

Figure 1

The average sign (left) and the Polyakov loop (right) as a function of volume computed on the real plane and sign optimized manifold, ${\mathcal{M}}_{\lambda }$, with 250 and 1000 gradient ascent steps. The dashed gray line on the right figure denotes the exact value of the Polyakov loop.

Figure 2

The average sign (left) and Polyakov loop (right) as a function of gradient ascent step for $h=0.45$ ($\kappa =0.01,\mu =0.998,N_t=100$) and $V=4^3$. The solid lines denote the exact values of ${\sigma }_{\text{SU}\left(3\right)}$ and $\langle P\rangle$.

Figure 3

The average sign, density, and Polyakov loop as a function of chemical potential for volumes $V=5^3$ and $6^3$.

Figure 4

The average sign as a function of the hopping parameter, in next-to-next-to leading order in the hopping expansion with (left) and without (right) sign optimization for different values of $\mu , V=4^3$, and $N_t=100$.

Figure 5

The density and Polyakov loop as functions of the hopping parameter, in next-to-next-to leading order in the hopping expansion with (left) and without (right) sign optimization for different values of $\mu , V=4^3$, and $N_t=100$.

Figure 6

The histogram of the eigenvalues of the Polyakov loop averaged over the volume. The green stars denote the saddle points and the dashed line represents the subspace defined by ${\alpha }_1+{\alpha }_2^{*}=0$ on which we plot the Lefschetz thimble.

Figure 7

The comparison between the surfaces generated by holomorphic gradient flow, approaching the Lefschetz thimble at large flow times, and the sign optimized manifold, plotted on the subspace defined by ${\alpha }_1+{\alpha }_2^{*}=0$. The stars denote the saddle points that lie on this subspace (see Fig. 6).

Figure 8

The parametrization of the generalized thimbles (left) and the sign optimized manifold (right) in terms of the real fields (bars). For thimbles, with increasing flow time the relevant region in the real field space that is sampled shrinks into small, disconnected regions leading to multimodal distribution in contrast to sign optimization.