Gaussian Quantum Monte Carlo Methods for Fermions and Bosons

We introduce a new class of quantum Monte Carlo methods, based on a Gaussian quantum operator representation of fermionic states. The methods enable first-principles dynamical or equilibrium calculations in many-body Fermi systems, and, combined with the existing Gaussian representation for bosons, provide a unified method of simulating Bose-Fermi systems. As an application relevant to the Fermi sign problem, we calculate finite-temperature properties of the two dimensional Hubbard model and the dynamics in a simple model of coherent molecular dissociation.

Figure 1

Second-order correlation function $g^{(2)}$ versus inverse temperature $\tau$ for $t_{ij}=0$, $U=2$, and $\mu =1$, for which $⟨n_{\sigma }⟩=0.5$. The solid curve gives the simulation result, and the dashed and dot-dashed lines show the estimated sampling error and deviation from the analytic result, respectively (on a $\times 1000$ scale). Calculated from 100 000 trajectories.

Figure 2

Energy $E$ per site versus inverse temperature $\tau$ for a $16\times 16$ 2D lattice for chemical potentials $\mu =2$ (solid line), $\mu =1$ (dashed line), and $\mu =0$ (dot-dashed line). Curves without crosses give the number of particles per site for $\mu =1$ (dashed line) and $\mu =0$ (dot-dashed line). $t_{ij}=1$ for nearest neighbors, $U=4$, and 50 paths initially. Dotted curves give sampling error estimate.

Figure 3

Molecular dissociation into pairs of fermionic (solid line) or bosonic (dot-dashed line) atoms. For the fermionic case, the dashed curve gives the truncated number-state calculation, and the dotted lines the estimated sampling error. In the bosonic case, the estimated sampling error is too small to be distinctly plotted on this graph. The initial state is a molecular coherent state [$N_{\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{s}}(0)=9$]. Calculated from 10 000 trajectories.