## SciPost Submission Page

# Parent Hamiltonian Reconstruction of Jastrow-Gutzwiller Wavefunctions

### by Xhek Turkeshi, Marcello Dalmonte

#### This is not the current version.

### Submission summary

As Contributors: | Marcello Dalmonte · Xhek Turkeshi |

Arxiv Link: | https://arxiv.org/abs/1909.11327v2 (pdf) |

Date submitted: | 2019-10-04 02:00 |

Submitted by: | Turkeshi, Xhek |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Quantum Physics |

Approaches: | Theoretical, Computational |

### Abstract

Variational wave functions have been a successful tool to investigate the properties of quantum spin liquids. Finding their parent Hamiltonians is of primary interest for the experimental simulation of these strongly correlated phases, and for gathering additional insights on their stability. In this work, we systematically reconstruct approximate spin-chain parent Hamiltonians for Jastrow-Gutzwiller wave functions, which share several features with quantum spin liquid wave-functions in two dimensions. Firstly, we determine the different phases encoded in the parameter space through their correlation functions and entanglement content. Secondly, we apply a recently proposed entanglement-guided method to reconstruct parent Hamiltonians to these states, which constrains the search to operators describing relativistic low-energy field theories - as expected for deconfined phases of gauge theories relevant to quantum spin liquids. The quality of the results is discussed using different quantities and comparing to exactly known parent Hamiltonians at specific points in parameter space. Our findings provide guiding principles for experimental Hamiltonian engineering of this class of states.

### Ontology / Topics

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### Submission & Refereeing History

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## Reports on this Submission

### Anonymous Report 2 on 2019-12-6 Invited Report

- Cite as: Anonymous, Report on arXiv:1909.11327v2, delivered 2019-12-06, doi: 10.21468/SciPost.Report.1374

### Report

In this work, Turkeshi and Dalmonte address the very interesting question of parent Hamiltonians for a class of Jastrow-Gutzwiller wave functions. After giving a very pedagogical introduction to JG wave functions, the authors use a Bisognano-Wichmann ansatz to reconstruct parent models via their entanglement Hamiltonians.

Over all this study is quite original (to my knowledge) and clearly deserves publications after minor revisions and perhaps some additional discussions I suggest.

Section 2.

JG functions are discussed for any values of the parameter \alpha. Perhaps the authors could say how novel their results, in particular about the extension of the critical regime. What is the status/nature of the transition claimed at \alpha_c=4.38 (no finite size scaling is provided, while the crossing in Fig. 2 Left appears at slightly larger \alpha_c)?

The interpretation of the weights Eq. (3) in terms of pseudo energies is quite interesting. As discussed in the context of participation entropies and spectroscopies (see papers by Stephan et al, Luitz, Alet... ), the logarithmic interaction has already been observed and discussed for critical XXZ ground-states (see Eq. 5). Could you please comment on this?

Regarding the correlation functions, an additional plot in log-log scale of Fig. 3 would be very helpful to better understand the scaling.

Section 3.

The way the distance between two reduced density matrices (\rho and \sigma, see Eq 33) is estimated deals with the relative entropy. Since I am more familiar with Kullback-Leibler divergences to compare how two distributions are close, I am curious to see the difference between these two estimates. I understand they are very close, but not exactly the same.

Section 4.

The framework for the Hamiltonian reconstruction, starting with a family of JG functions is based on a strong assumption at the very beginning, which takes from granted that the reduced density matrix takes the BW form Eqs. (21,22). This seems to give reliable results only for a small range of parameters, and obviously fail when Lorentz invariance is broken at \alpha\le 0. Here comes my question: why the authors did not try to compare their reconstruction approach with the simplest one based directly on the Hamiltonian itself by coupling the covariance matrix over a family of target hamiltonian (See Refs. 38,39,85). This methods appear easier since it simply requires to but a small matrix with computed correlation functions. I am curious to see the difference of results they would obtain. I would expect at least a comment on this aspect.

### Requested changes

1- Comment on the critical alpha=4.38

2- Comment on the links with participation spectroscopy regarding the weights Eq. (5)

3- Add a log-log plot of Fig. 3

4- Comment on the alternative KL divergences measures as compared to relative entropies

5- Comment on the alternative reconstruction method based on the covariance matrix.

6- Small typo page 18 |jac) -> |jas)

### Anonymous Report 1 on 2019-11-6 Invited Report

- Cite as: Anonymous, Report on arXiv:1909.11327v2, delivered 2019-11-06, doi: 10.21468/SciPost.Report.1296

### Strengths

1- The article is well written and well organised.

2- The numerical analysis is trustable.

3- The argument is timely and interesting.

### Weaknesses

1- Important references on Jastrow-Guttzwiller wave functions are missing.

2- A few more calculations would improve the quality of the work.

### Report

The work by Turkeshi and Dalmonte present a numerical analysis to obtain (almost) parent Hamiltonians for Jastrow-Gutzwiller wave functions in 1D.

I think that the paper is nice and should be published with a few corrections.

1) Since spin-liquids are mentioned in the introduction, I think that the recent works on frustrated Heisenberg models that employed Gutzwiller-projected states must be cited. among them:

*) For the j1-j2 model on the square lattice: Phys. Rev. B 88, 060402(R) (2013)

*) For the j1-j2 model on the triangular lattice: Phys. Rev. B 93, 144411 (2016)

*) For the Heisenberg and j1-j2 models on the Kagame lattice: Phys. Rev. B 84, 020407(R) (2011) and Phys. Rev. B 91, 020402(R) (2015)

*) The mapping to a classical coulomb gas was first suggested in Phys. Rev. B 73, 245116 (2006)

2) It would be nice to have a more systematic size scaling of S_{vN} shown in Fig.1 for only two values of L. Indeed, for large positive values of \alpha it saturates to log(2), while for large negative values it diverges. Is it possible to understand better the intermediate regime?

3) I am confused about Fig.3: I expected a finite correlation length for large values of \alpha: why is instead diverging?

4) I am also confused about Table 1: since h=0, there are only two free parameters in the (almost) parent Hamiltonian, namely \Delta_1 and J_1, so one of them can be fixed to 1.

5) At the beginning it is written that the Jastrow-Gutzwiller wave functions are good approximations of the (almost) parent Hamiltonian. For that, It would be useful to compare the exact ground-state energy of a given Hamiltonian (with NN or NN+NNN couplings) to the variational energy of the Jastrow-Gutzwiller state. An insightful plot would be to report the accuracy (E_ex-E_var)/E_ex as a function of the number of distances included in the (almost) parent Hamiltonian, for the optimal couplings on each case.

In summary, I think that the paper is nice and should be published after some modifications.