Perturbative Quantum Monte Carlo Method for Nuclear Physics

While first order perturbation theory is routinely used in quantum Monte Carlo (QMC) calculations, higher-order terms present significant numerical challenges. We present a new approach for computing perturbative corrections in projection QMC calculations. We demonstrate the method by computing nuclear ground state energies up to second order for a realistic chiral interaction. We calculate the binding energies of several light nuclei up to ${}^{16}\mathrm{O}$ by expanding the Hamiltonian around the Wigner SU(4) limit and find good agreement with data. In contrast to the natural ordering of the perturbative series, we find remarkably large second-order energy corrections. This occurs because the perturbing interactions break the symmetries of the unperturbed Hamiltonian. Our method is free from the sign problem and can be applied to QMC calculations for many-body systems in nuclear physics, condensed matter physics, ultracold atoms, and quantum chemistry.

Figure 1

ptQMC binding energies as functions of the projection time $\tau$ compared with nonperturbative results. The circles (red), down triangles (green), and diamonds (blue) denote the energies at the zeroth, first, and second orders, respectively. The squares (black) represent the exact results calculated with sparse matrix multiplications for ${}^3\mathrm{H}$ and full nonperturbative QMC for ${}^4\mathrm{He}$ and ${}^{16}\mathrm{O}$, respectively. Each group of results are fitted with a sum of exponential functions (dashed lines). The red bars mark the experimental binding energies.

Figure 2

(a),(b) The dashed and dash-dotted lines denote the binding energies of ${}^3\mathrm{H}$ (a) and ${}^4\mathrm{He}$ (b) as functions of the small parameter $\lambda$ in first- and second-order ptQMC calculations, respectively. The black squares are the exact results. (c) The first- and second-order ptQMC calculations for ${}^{16}\mathrm{O}$, starting from three different zeroth-order interactions $V_0$, $1.1V_0$ ($†$), and $1.2V_0$ ($‡$). (d) Schematic plot for a perturbative calculation. The zeroth-order wave functions $|{\mathrm{\Psi }}_0⟩$ and $|{\mathrm{\Psi }}_0^{\prime }⟩$ are confined in a subspace corresponding to an IR of SU(4).