Most applications of the density matrix renormalization
group (DMRG) have been to lattice models with short range interactions. But
recent developments in DMRG technology open the door to studying continuum
systems with long-range interactions in one dimension (1d). One key motivation
is simulating cold atom experiments, where it is possible to engineer
Hamiltonians of precisely this type.
We have been applying DMRG in the 1d continuum with
motivation: to investigate and improve density functional
theory (DFT). DFT has exact mathematical foundations, but in practice one must
use approximations. These approximations work incredibly well for weakly
correlated systems yet fail when correlations are strong.
Improving DFT directly for realistic 3d systems is hard
because few systems can be solved exactly. By working in the 1d continuum
instead, we can use the power of DMRG to study DFT. We can implement both the
exact DFT formalism and standard DFT approximations.
After showing how to overcome the challenges in
performing these calculations, I will discuss some of the key questions we are
investigating, for example, the ability of DFT to predict gaps of insulating