# Issue in Path Integral Monte Carlo of Hooke's Helium (Potentially the “Quantum Sign Problem”)

I'm writing a code in Fortran to find the ground state energy of the following Hamiltonian in 2 spacial dimensions. $$\hat{H} = -\frac{1}{2}\nabla^{2}_{1} -\frac{1}{2}\nabla^{2}_{2} + \frac{1}{2}(r^{2}_{1}+r^{2}_{2}) + \frac{1}{|\mathbf{r}_{1}-\mathbf{r}_{2}|}.$$ The temporal propagator for the system is: $$G_{2}(\textbf{R},\textbf{R}';\Delta\tau)=<R'|e^{-2\Omega\hat{H}}|R>,\quad\Omega\equiv\Delta\tau/2$$ R is the set of all particles' positions. The local energy function is the following: $$E_{L}(\textbf{R},\textbf{R}';\Delta\tau)=\frac{\hat{H}G_{2}(\textbf{R},\textbf{R}';\Delta\tau)}{G_{2}(\textbf{R},\textbf{R}';\Delta\tau)}$$ Using $n$ beads ($n$ set of R's), we approximate the local energy further Note, the summation index loops. $$E_{L}^{(n)}=\frac{1}{n}\sum_{b=1}^{n} E_{L}(\textbf{R}_{b},\textbf{R}_{b+1};\Delta\tau),\quad \textbf{R}_{n+1}=\textbf{R}_{1}$$

The following is my pseudo code for evaluation.

1. Define the local energy.
2. Assign current time.
3. Displace all beads and their positions.
4. Using Metropolis Algorithm, accept displacement or reject it.
5. Loop on 3. and 4. many tens of times.
6. Evaluate the local energy for given position set.
7. Loop on 3., 4., and 6. many times, eg. $10^5$ to $10^7$ (necessity can vary depending on dimension of system).
8. Sum all values from 6. and divide by number of iterations on 6.
9. Repeat 2.-8. for every desired temporal node.

The energy for a 2 particle system in 2 spacial dimensions is 3, but I do not get this (instead I get somewhere around 3.2). I wrote the program with a coefficient on the interaction so I could check interaction vs no interaction. In the no interaction case one should get 2, from two 2D quantum harmonic oscillators, which I do! I'm not sure why the interaction is causing the only issue. From online readings I believe it may be related to the "quantum sign problem(?)." Below is output of the simulation with and without the interaction respectively (please ignore the red line, it's irrelevant).