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?


2 Answers 2


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.


The most general sign problem was shown to be NP hard http://arxiv.org/abs/cond-mat/0408370, so there is no general solution in sight, and indeed no general solution expected.

This does not prevent us from solving specific sign problems, or studying models that are sign-free when written down in suitable variables (the attractive Hubbard model, for example).

However, the most interesting sign problems remain unsolved. This includes, in particular, the repulsive Hubbard model and QCD at finite baryon density.

Some interesting models can be studied using work-arounds, like DMRG or DMFT, but one should remember that these are approximate methods.

  • $\begingroup$ I would not qualify DMFT as "an approximate method that is a work around for the sign problem". DMFT is an exact solution for a lattice problem in the limit of infinite dimensions. As part of its iterative solution, an impurity solver is needed, which can use QMC, and depending on the precise characteristics of the system, this solver can face a sign problem, or not. The only reason why DMFT could be seen as a workaround to the sign problem is that the problem is less severe in the impurity problem used by DMFT than in the full lattice problem, but it is not the main purpose of the approach. $\endgroup$ Feb 20, 2017 at 13:26
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    $\begingroup$ @DominiqueGeffroy My point was simply this: Real systems obviously do not exist in an infinite dimensional space. When we apply (numerical) DMFT to a real system (like the Hubbard model) we do it because (exact) numerical methods suffer from a sign problem. $\endgroup$
    – Thomas
    Feb 21, 2017 at 14:21

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