Fermion bag approach to lattice field theories

We propose a new approach to the fermion sign problem in systems where there is a coupling $U$ such that when it is infinite the fermions are paired into bosons, and there is no fermion permutation sign to worry about. We argue that as $U$ becomes finite, fermions are liberated but are naturally confined to regions which we refer to as fermion bags. The fermion sign problem is then confined to these bags and may be solved using the determinantal trick. In the parameter regime where the fermion bags are small and their typical size does not grow with the system size, construction of Monte Carlo methods that are far more efficient than conventional algorithms should be possible. In the region where the fermion bags grow with system size, the fermion bag approach continues to provide an alternative approach to the problem but may lose its main advantage in terms of efficiency. The fermion bag approach also provides new insights and solutions to sign problems. A natural solution to the “silver blaze problem” also emerges. Using the three-dimensional massless lattice Thirring model as an example, we introduce the fermion bag approach and demonstrate some of these features. We compute the critical exponents at the quantum phase transition and find $\nu =0.87(2)$ and $\eta =0.62(2)$.

Figure 1

An illustration of a “fermion bag” configuration as discussed in the text.

Figure 2

An illustration of a fermion bag configuration with one temporal winding bag and two nontemporal winding bags.

Figure 3

Plot of the average number of fermion bags $⟨N_B⟩$ of size $S$ as a function of the $S$ for $L=8$, 12, 16 and 20 at $U=1.3$. The solid line is an exponential fit as discussed in the text. The inset shows the same plot ($L=8$ data has been omitted) scaled with the volume. The data collapse shows that the density of fermion bags of a given size does not depend on the volume.

Figure 4

Plot of the average distribution of fermion bags $⟨N_B⟩$ of size $S$ as a function of the $S$ for $U=0.1$, 0.25, 0.5, 0.8, 1.3 and 1.5 for $L=8$. We note that the distribution changes qualitatively between large and small $U$.

Figure 5

Plot of the typical bag size as a function of $L$ for three different values of $U$. The solid line is a fit to the form $A{L}^3$. Note the $L$ axis is shown on a logarithmic scale.

Figure 6

Plots of $2\chi /L^3$, $⟨q_{\chi }^2⟩/4L$ and $⟨Q_f^2⟩$ as a function of the lattice size for various values of $U$ is shown in the left figures. The same quantity is plotted as a function of $U$ for $L=8$, 12 on the right.

Figure 7

Plots of $\chi$, $⟨q_{\chi }^2⟩$ and $⟨Q_f^2⟩$ as a function of the lattice size for small values of $U$. The solid lines are fits to the expected forms given in Eqs. (31a, 31b, 31c)

Figure 8

Plots of $⟨q_{\chi }^2⟩$ and $\chi /L^{2-\eta }$ as a function of $U$ for $L=8$, 12, 16, and 20. The solid lines show the combined fit of all the data to Eq. (32) as discussed in the text. Based on the fits we find the critical point to be $U=0.2604(5)$, $\nu =0.87(2)$ and $\eta =0.62(2)$.

Figure 9

A sketch of a fermion bag configuration in a model described in the text that contains the physics of BCS-BEC crossover. The solid lines represent spin-up and spin-down fermions strongly paired into a spin-zero boson. The blobs represent regions where fermions can be liberated and become essentially free.