I've been reading about the Monte Carlo sign problem, and I am a little confused about its current status. Specifically, after reading this post
When is the "minus sign problem" in quantum simulations an obstacle?
I am confused on whether or not we can simulate "fermi hamiltonians away from special symmetry points". For example, in this article, the sign problem is solved for "a class of lattice field theories involving massless fermions". However, in this paper by Ceperley and Wagner, they discuss a first principles Monte Carlo to correlated electron systems, and make no discussion of the former solution. In this paper by Ferris, an unbiased Monte Carlo is introduced that "has been shown
to mitigate the sign problem given a sufficiently large bond dimension". Finally, there is also Majorana Monte Carlo, which uses the Majorana representation to simulate "a class of spinless fermion models on bipartite lattices at half filling and with arbitrary range of (unfrustrated) interactions".
So my question is this: the sign problem in Monte Carlo seems to be partially solved at this point. Not only can we simulate simple fermionic systems, but recent progress in the field has led us to understand more complex and varied models. Therefore, at this point in time, what are the limitations of Monte Carlo in simulating fermionic systems? That is, what can tensor network techniques like DMRG and PEPS do that QMC can't?
Best Answer
I know that in Full Configurational Interaction Quantum Monte Carlo(FCIQMC), where they start from the Schrödinger equation and sample the full configurational space with integer walkers, there is a spontaneous symmetry breaking between the $\Psi$ and $-\Psi$ after a sufficient number of walkers are spawned into the configurational space. It treats the strongly correlated systems quite well. For a brief introduction, you can have a look at the first link and also this paper.
Unlike in Diffusion Quantum Monte Carlo where you need to use fixed-node approximation to fix the sign of the wavefunction, FCIQMC provides a "phase transition" way to tackle the sign problem. However, FCIQMC scales exponentially with the system size, that's the major problem in this method.
There is also this coupled cluster theory, which I am now studying, but, for now, I cannot say any useful information about it. I will probably update in the future. The only thing I know is that it has comparable performance in including the correlations, but scales less severely as the system size as FCIQMC.