# Fermionic Gaussian states: an introduction to numerical approaches

### Submission summary

 As Contributors: Jacopo Surace Arxiv Link: https://arxiv.org/abs/2111.08343v2 (pdf) Code repository: https://github.com/Jacupo/F_utilities Date accepted: 2022-04-28 Date submitted: 2022-04-11 14:33 Submitted by: Surace, Jacopo Submitted to: SciPost Physics Lecture Notes Academic field: Physics Specialties: Condensed Matter Physics - Theory Condensed Matter Physics - Computational Quantum Physics Approaches: Theoretical, Computational

### Abstract

This document is meant to be a practical introduction to the analytical and numerical manipulation of Fermionic Gaussian systems. Starting from the basics, we move to relevant modern results and techniques, presenting numerical examples and studying relevant Hamiltonians, such as the transverse field Ising Hamiltonian, in detail. We finish introducing novel algorithms connecting Fermionic Guassian states with matrix product states techniques. All the numerical examples make use of the free Julia package F_utilities.

Published as SciPost Phys. Lect. Notes 54 (2022)

We really thanks the Referee 1 for their report and in-depth analysis of section 6.3. We implemented the suggested changes as follows:

• Added reference http://doi.org/10.1103/RevModPhys.80.517 and references therein as a review on the topic of entanglement in many-body systems.
• Added references https://doi.org/10.1088/1742-5468/2015/06/p06002 and https://doi.org/10.1088/1751-8121/aa6268 for the rainbow chain.

Sec.6.3:

• Added a section on the definition of MPS in the appendix A.2 and references https://doi.org/10.1016/j.aop.2014.06.013, https://doi.org/10.1103/RevModPhys.93.045003 and https://doi.org/10.1016/j.aop.2010.09.012.
• Fig. 10 and 11 are now referenced in the text (p.44). The captions have been corrected.
• The text has been updated to make it more clear, in particular we corrected to ""it makes $\vec{v}$ also an approximate eigenvalue of Λ" with the sentence "This makes $\vec{v}$ also an approximate eingenvector of Λ. In fact we know that the eigenvalues of Λ are 0 or 1, thus it is sufficient to extend the dimension of the eigenvector $\vec{v}$ adding to it N − l zeros to obtain an eigenvector of Λ." Changed "Since $\Lambda$ is a pure state" with "Since $\Lambda$ corresponds to a pure state". Changed "let $\Lambda$ be the ground state" with "Let $\Lambda$ be the correlation matrix of the ground state of a 1D local Hamiltonian."
• The notation has been uptated to make it consistent.
• The figures have been updated to make it clear that the label $m+1$ referers to the dimension of the block.
• The notation has been updated to make it consistent.
• We created new plots for bigger sizes and for a bigger value of $m$ as suggested by the referee. We updated the caption making clear that in such a situation one has an extensive scaling of the entropy with subsystem size.

Typos:

The typos have been corrected.

We gratefully thanks the Referee 2 for their careful and in-depth report. We implemented the changes suggested as follows:

1) We agree with the referee. We added a footnote with the definition of Hamming distance. 2) We agree with the referee. We moved the discussion where suggested. 3) We agree with the referee. We moved the box where suggested. 4) We thought about the possibility of defining the Pfaffian in the text. We believe that the Pfaffian is not so relevant to require its cumberson definition in the main text or in the appendix. Adding the definition of Pfaffian as a footnote is not going to help much either, as in order to define the Pfaffian one also have to introduce other elements such as the Levi-Civita symbol of groups of permutations. In the explicit examples following the introduction of the Pfaffian, some simple expressions of the Pfaffian are explicitly presented. 5) We agree with the referee. We wrote down the case for $N=2$ explicitly. 6) The variable $i$ was commented at the end of the box. Nevertheless, it was not discussed explicitly as an argument of the function. For this reason we added a comment about $i$ in the first sentence of the box as requested by the referee. 7) Added the minus sign. 8) We agree with the referee, we improved the position of the boxes. 9) Before the coding frame we wrote "in the following program we numerically compute the time evolution induced by a hopping Hamiltonian on a random translational invariant gaussian state with exponentially decaying correlation functions." The $\Gamma$ represents this state. 10) We agree with the referee. We expanded the caption of the table as requested. 11) We agree with the referee. We added more comments to the code. 12) Corrected the label of the figure. 13) We removed the word "experimental". 14) Added "Possible escape tricks exists. Perhaps one can slightly perturb the state and then compute the value of the energy after another step of the evolution, this can help in escaping from local minima. In general such tricks have to be adapted to the particular needs. " as requested. 15) Figure 11 is now called in the main text. 16) We corrected the figure numbering. 17) We agree with the referee. We merged figure 12 and 13. 18) There was a general problem with the tags of the figures. We corrected it. 19) We agree with the referee. We moved the box, as well as the code frames and the associated figures in the appropiate section. 20) We corrected the typos.

### List of changes

We really thanks the Referee 1 for their report and in-depth analysis of section 6.3. We implemented the changes suggested as follows:

+ Added reference http://doi.org/10.1103/RevModPhys.80.517 and references therein as a review on the topic of entanglement in many-body systems.
+ Added references https://doi.org/10.1088/1742-5468/2015/06/p06002 and https://doi.org/10.1088/1751-8121/aa6268 for the rainbow chain.

Sec.6.3:

+ Added a section on the definition of MPS in the appendix A.2 and references https://doi.org/10.1016/j.aop.2014.06.013, https://doi.org/10.1103/RevModPhys.93.045003 and https://doi.org/10.1016/j.aop.2010.09.012.
+ Fig. 10 and 11 are now referenced in the text (p.44). The captions have been corrected.
+ The text has been updated to make it more clear, in particular we corrected to ""it makes $\vec{v}$ also an approximate eigenvalue of Λ" with the sentence *"This makes $\vec{v}$ also an approximate eingenvector of Λ. In fact we know that the eigenvalues of Λ are 0 or 1, thus it is sufficient to extend the dimension of the eigenvector $\vec{v}$ adding to it N − l zeros to obtain an eigenvector of Λ."* Changed "Since $\Lambda$ is a pure state" with "Since $\Lambda$ corresponds to a pure state". Changed "let $\Lambda$ be the ground state" with "Let $\Lambda$ be the correlation matrix of the ground state of a 1D local Hamiltonian."
+ The notation has been uptated to make it consistent.
+ The figures have been updated to make it clear that the label $m+1$ referers to the dimension of the block.
+ The notation has been updated to make it consistent.
+ We created new plots for bigger sizes and for a bigger value of $m$ as suggested by the referee. We updated the caption making clear that in such a situation one has an extensive scaling of
the entropy with subsystem size.

Typos:

The typos have been corrected.

We really thanks the Referee 2 for their careful and in-depth report. We implemented the changes suggested as follows:

1) We agree with the referee. We added a footnote with the definition of Hamming distance.
2) We agree with the referee. We moved the discussion where suggested.
3) We agree with the referee. We moved the box where suggested.
4) We thought about the possibility of defining the Pfaffian in the text. We believe that the Pfaffian is not so relevant to require its cumberson definition in the main text or in the appendix. Adding the definition of Pfaffian as a footnote is not going to help much either, as in order to define the Pfaffian one also have to introduce other elements such as the Levi-Civita symbol of groups of permutations. In the explicit examples following the introduction of the Pfaffian, some simple expressions of the Pfaffian are explicitly presented.
5) We agree with the referee. We wrote down the case for $N=2$ explicitly.
6) The variable $i$ was commented at the end of the box. Nevertheless, it was not discussed explicitly as an argument of the function. For this reason we added a comment about $i$ in the first sentence of the box as requested by the referee.
8) We agree with the referee, we improved the position of the boxes.
9) Before the coding frame we wrote "in the following program we numerically compute the time evolution induced by a hopping Hamiltonian on a random translational invariant gaussian state with exponentially decaying correlation functions." The $\Gamma$ represents this state.
10) We agree with the referee. We expanded the caption of the table as requested.
11) We agree with the referee. We added more comments to the code.
12) Corrected the label of the figure.
13) We removed the word "experimental".
14) Added "Possible escape tricks exists. Perhaps one can slightly perturb the state and then compute the value of the energy after another step of the evolution, this can help in escaping from local minima. In general such tricks have to be adapted to the particular needs. " as requested.
15) Figure 11 is now called in the main text.
16) We corrected the figure numbering.
17) We agree with the referee. We merged figure 12 and 13.
18) There was a general problem with the tags of the figures. We corrected it.
19) We agree with the referee. We moved the box, as well as the code frames and the associated figures in the appropiate section.
20) We corrected the typos.

### Submission & Refereeing History

#### Published as SciPost Phys. Lect. Notes 54 (2022)

Resubmission 2111.08343v2 on 11 April 2022
Submission 2111.08343v1 on 28 January 2022

## Reports on this Submission

### Report

The authors have implemented all the recommendations, the manuscript can now be published.

### Requested changes

I just noticed a typo in the new Eqs. (200) and (201), the second tensor should have index $i_2$ and superscript $[2]$.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Anonymous Report 1 on 2022-4-20 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2111.08343v2, delivered 2022-04-20, doi: 10.21468/SciPost.Report.4955

### Strengths

-Clarity of the explanations
-Developed an open-source package for the numerical manipulation of Gaussian fermionic states in 1D and 2D of easy usage
-Excellent introduction of the F_utilities functions in the main text.

### Weaknesses

I do not see particular weaknesses.

### Report

In the revised version of the manuscript, the authors fixed the main issues regarding sec. 6.3/6.4 and properly implemented the other small changes that I suggested. Therefore, I recommend the manuscript for publication in SciPost Physics Lecture Notes.

### Requested changes

-

• validity: high
• significance: high
• originality: high
• clarity: top
• formatting: excellent
• grammar: excellent