Properties of fermionic systems with the Path-integral ground state method

This revised manuscript is a result of the significant effort we have put into addressing each of the concerns raised by the three referees. After the first round of reviews, we are confident that our manuscript presents the subject in a clear and informative way that can be valuable for a broad audience. The several changes and additions in response to the reports contributed to improving our work regarding correctness, completeness, broadness, and readability. A list of the most substantial changes is provided below. Several minor editorial improvements were also performed and are not listed.

- Ref. 3 Weakness: In response to the main concern raised by referee 3, we are naming the method as Fixed-Node Path-Integral Ground State.
The title of the manuscript is now "Properties of fermionic systems with the Path-integral ground state method". The entire manuscript was updated accordingly. The following paragraph was added to Section 8 (Conclusions):
As a final remark, in naming the method fixed-node path-integral ground state (FN-PIGS), we follow the standard nomenclature found in the literature when the fixed-node approximation is incorporated into a given existing method. However, we believe that Density Matrix Projection (DMP) better represents how the method operates and reflects its main capabilities. In particular, results can converge to an excited state if, in the extremities of the necklace, the wave functions are orthogonal to the true ground state of the Hamiltonian.

- Ref. 1 Weakness: We made several editorial changes to illustrate our contributions better, emphasizing the novelties our work presents. The most obvious modification is in the abstract, but there are several minor additions and changes to the Introduction and the Conclusions. The abstract now reads:
We investigate strongly correlated many-body systems composed of bosons and fermions with a fully quantum treatment using the path-integral ground state method, PIGS. To account for the Fermi-Dirac statistics, we implement the fixed-node approximation into PIGS, which we then call FN-PIGS. In great detail, we discuss the pair density matrices we use to construct the full density operator in coordinate representation, a vital ingredient of the method. We consider the harmonic oscillator as a proof-of-concept and, as a platform representing quantum many-body systems, we explore helium atoms. Pure $^4$He systems demonstrate most of the features of the method. Complementarily, for pure $^3$He, the fixed-node approximation resolves the ubiquitous sign problem stemming from anti-symmetric wave functions. Finally, we investigate $^3$He-$^4$He mixtures, demonstrating the method's robustness. One of the main features of FN-PIGS is its ability to estimate any property at temperature $T = 0$ without any additional bias apart from the FN approximation; biases from long simulations are also excluded. In particular, we calculate the correlation function of pairs of equal and opposite spins and precise values of the $^3$He kinetic energy in the mixture.

- Ref. 3 Requested change 8: The referee pointed out a concern about using the expression "diagonal in coordinate space" not being clear.
We changed the nomenclature from "diagonal" and "non-diagonal" operators to "local" and "non-local" operators. Further clarifications were added to Sections 2 and 6 to elucidate what these two types of operators mean and how they are amenable to FN-PIGS.

- Ref. 3 Requested change 2: The referee observed an imprecision at the end of Section 2, Eq. 20.
We corrected the equation and explained that the expression is the same under integration after relabelling indexes.

- Ref. 3 Requested change 3: The referee requested clarification on what we meant by ``more common mathematical approaches'' in the context of density matrices (Section 4).
We included examples of such mathematical approaches and references to them (Refs. 52 through 56).

- Ref. 3 Requested change 4: The referee noticed an imprecision in our consideration of wave functions constructed from pair products (Section 4.2).
We modified the text to be more precise about in what contexts this type of wave function is well-known to be a good approach. We included examples of systems with references. Following the referee's prompt, we discussed possible improvements.

- Ref. 3 Requested change 5: The referee requested an explicit expression for the Bloch equation for partial waves that compose the relative-coordinates pair density matrix.
We included Eqs. 42 and 43, showing the Bloch equation and the solution for the free-particle case.

- Ref. 3 Requested change 6: The referee observed the behavior of the partial waves composing the relative-coordinate pair density matrix as the imaginary time interval increases (last paragraph of Section 4.3).
We included a detailed discussion about this behavior, making it clear that fewer partial waves are needed as $\delta\tau$ increases. We also added a paragraph about another point the referee raised concerning the likelihood of sampling large relative distances. This ultimately led to a comment about a fundamental aspect of path-integral-based simulations.

- Ref. 3 Requested change 7: The referee requested more information about the cross-recross error and how we monitor and mitigate it in our simulations (final part of Section 5).
We modified the text to clarify how the cross-recross error is avoided and monitored. We included a detailed discussion about the figures involved when different acceptance ratios are considered and an observation about the consequences it implies in simulation times.

- Ref. 3 Requested changes 1 and 9: The referee requested clarification about how PIGS differs from a variational approach.
We removed the phrase in Section 2.2 about PIGS not being a variational method, as we believe that was not the appropriate place to discuss it. We included a paragraph discussing a few specific aspects of the mixed estimator (Section 6.2.2), with a discussion about the zero-variance situation.

- Ref. 3 Requested change 10: The referee requested a comment on the oscillatory behavior of the energies in Figure 6.
We added the comment to the text, explaining the behavior is related to inaccuracies in the expression of the density matrix for the excited state when the image action is not considered.

- Ref. 3 Requested change 11: The referee made an observation about Section 7.2.1 and the graph shown in Figure 8.
We included a paragraph in the manuscript describing the observations in Figure 8 and discussing the observed behavior, with an observation about the limit where different estimators are expected to fully converge.

- Ref. 3 Requested change 12: The referee requested details about finite-size effects in our fermionic simulations.
We performed unsolicited simulations with larger systems to quantify the finite-size effects in calculating energies. We included paragraphs in Section 7.2.2 to discuss these effects, a justification for the number of particles we chose, and listed references in agreement with what we found.

- Unsolicited changes and additions in Section 7.2.3: We made several changes in this section to better present and discuss our results about the mixture. Several details and a few references were added. The legend of Figure 11 was changed to make the plot clear.

- Ref. 2 Requested change: The referee requested a discussion about potential applications to nuclear physics systems.
We included in our Conclusions references and a discussion about systems we believe are potential candidates to be investigated through FN-PIGS.