Downfolding calculations in solids by auxiliary-field quantum Monte Carlo

We present a recent development in {\it ab initio} auxiliary-field quantum Monte Carlo (AFQMC) \footnote{S.~\ Zhang and H.~\ Krakauer, Phys. Rev. Lett. {\bf 90}, 136401 (2003)} calculations of solid systems using downfolded Hamiltonians. For a given system, the many-body downfolded Hamiltonian is expressed with respect to a truncated basis set of Kohn-Sham orbitals, which are obtained from a high-quality density-functional calculation. This approach allows many-body calculations to treat a much simpler Hamiltonian while retaining material-specific properties. Typical size of the basis set is more than an order of magnitude smaller than the original (the number of plane-waves), leading to large savings in AFQMC computation. The Hamiltonians are systematically improvable and allow one to dial, in principle, between the simplest model and the full Hamiltonian. As a by-product of this approach, pseudopotential errors can essentially be eliminated \footnote{W.~\ Purwanto, S.~\ Zhang, H.~\ Krakauer, J. Chem. Theory Comput. {\bf 9}, 4825 (2013)}. The method is demonstrated by calculating the lattice constant and bulk modulus of solids, including classic semiconductors (Si and diamond), an ionic insulator (NaCl), and metallic systems (Na and Al).