Entanglement and the fermion sign problem in auxiliary field quantum Monte Carlo simulations

Quantum Monte Carlo simulations of fermions are hampered by the notorious sign problem whose most striking manifestation is an exponential growth of sampling errors with the number of particles. With the sign problem known to be an NP-hard problem and any generic solution thus highly elusive, the Monte Carlo sampling of interacting many-fermion systems is commonly thought to be restricted to a small class of model systems for which a sign-free basis has been identified. Here we demonstrate that entanglement measures, in particular the so-called Rényi entropies, can intrinsically exhibit a certain robustness against the sign problem in auxiliary-field quantum Monte Carlo approaches and possibly allow for the identification of global ground-state properties via their scaling behavior even in the presence of a strong sign problem. We corroborate these findings via numerical simulations of fermionic quantum phase transitions of spinless fermions on the honeycomb lattice at and below half filling.

Figure 1

Upper panel: Sign-ignoring entanglement entropy $S_2^{\mathrm{abs}}\left(L\right)$ for spinless fermions on the honeycomb lattice at half filling subject to a nearest-neighbor repulsive interaction $V$. The latter drives a transition from a Dirac semimetal for $V<V_c\approx 1.356\left(1\right)$ (indicated by the vertical gray line) to a charge density wave for $V>V_c$. The sign-ignoring data is from projector auxiliary-field QMC calculations using a real-valued Hubbard-Stratonovich transformation at $\theta =40$. Sign-free reference data from Majorana-QMC simulations at $\theta =20$ is indicated by the gray dots. Lower panel: The average sign ${〈\sigma 〉}_{\mathrm{abs}}$ indicating the strength of the sign problem.

Figure 2

Scaling of the sign-ignoring entanglement entropy with system size of a spinless fermion system on the honeycomb lattice with a small nearest-neighbor repulsive interaction $V/t=0.1$. The system exhibits semimetallic states with a Dirac cone at half filling and a nodal line at one-third filling. Data is from projector auxiliary-field QMC calculations using a complex-valued Hubbard-Stratonovich transformation at $\theta =1$, for which there is a severe sign problem with ${〈\sigma 〉}_{\mathrm{abs}}\approx 0$ for any finite $V$ and all but the smallest system sizes. Error bars are smaller than the symbol sizes.

Figure 3

Upper panel: Sign-ignoring entanglement entropy $S_2^{\mathrm{abs}}\left(L\right)/L$ for spinless fermions on the honeycomb lattice at one-third filling subject to a nearest-neighbor repulsive interaction $V$. The numerical data is obtained from projector auxiliary-field QMC calculations using a complex-valued Hubbard-Stratonovich transformation at $\theta =20$, for which there is a severe sign problem for all coupling parameters and system sizes as indicated by the vanishing average sign ${〈\sigma 〉}_{\mathrm{abs}}$ shown in the lower panel. The gray bar indicates our best estimate for the region in which the logarithmic contribution vanishes and the system starts to exhibit pure area law behavior.

Figure 4

Sign-ignoring entanglement entropy $S_2^{\mathrm{abs}}\left(L\right)$ (upper panel) and expectation values of the two signs $〈{\sigma }_0〉$ and $〈{\sigma }_1〉$ (lower panel) for spinless fermions on a $2\times 3\times 3$ honeycomb lattice at half filling at $V=1.0$. Results were obtained using project auxiliary-field QMC calculations using a complex Hubbard-Stratonovich transformation and plotted against the projection time $\theta$.

Figure 5

Geometries of the honeycomb lattice and its bipartitions into subsystems $A$ (shaded in blue) and $B$ used in (a) Fig. 1 and (b) Fig. 2 of the main text, respectively.

Figure 6

Scaling of free spinless fermions on the honeycomb lattice at half and one-third filling using the same geometries as described in the previous section.