Solution to sign problems in models of interacting fermions and quantum spins

We show that solutions to fermion sign problems that are found in the formulation where the path integral is expanded in powers of the interaction in continuous time can be extended to systems involving fermions interacting with dynamical quantum spins. While these sign problems seem unsolvable in the auxiliary field approach, solutions emerge in the world-line representation of quantum spins. Combining this idea with meron-cluster methods, we are able to further extend the class of models that are solvable. We demonstrate these solutions to sign problems by considering several examples of strongly correlated systems that contain the physics of semimetals, insulators, superfluidity, and antiferromagnetism.

Figure 1

Illustration of a hard-core boson world-line configuration in one spatial dimension. The spatial sites are numbered 1–6, with odd sites colored red and even sites colored blue. Among the world lines a cross indicates the absence of the boson and a solid line indicates its presence. Insertions of $S_x$ create or annihilate hard-core bosons. Closed circles indicate a creation event, while open circles indicate the annihilation event. On each site the $S_x$ operators come in pairs due to temporal periodicity of the world lines.

Figure 2

World-line diagram for Heisenberg model. Again, the spatial sites are numbered 1–6 with red dots for odd sites and blue dots for even sites.

Figure 3

The top row shows four nonzero matrix elements that result from an insertion of $\frac{1}{2}+S^x$ on a site. All four have the same weight $h/2$. The closed circle represents spin up and the open circle represents spin down. The bottom row shows four nonzero matrix elements that result from an insertion of $\frac{1}{4}-{\vec{S}}_i·{\vec{S}}_j$ on a bond connecting neighboring sites. All four have weights with the same magnitude $J/2$. The off-diagonal terms have negative signs.

Figure 4

Illustration of a cluster configuration that emerges uniquely from a given configuration $[n,\{d\},l,\{k\left\}\right]$ of operator insertions. Open lines that end on two different sublattices turn out to be a meron cluster, shown as dashed lines in the figure. The spin trace $G_s$ vanishes if the cluster configuration contains a meron.