Density-Matrix Mean-Field Theory

Dear Editor,

We write to resubmit our revised manuscript titled “Density-Matrix Mean-Field Theory” for your consideration in SciPost Physics.

In response to the concerns raised in the previous round of review, a primary focus was placed on evaluating the accuracy of our method. Taking the advice from the referees, we have incorporated a benchmark calculation using the state-of the-art technique density-matrix renormalization group (DMRG). Additionally, we have expanded our manuscript to include more comprehensive and quantitative comparisons with both DMRG results and results reported in previous literatures, as suggested by one of the referees. We find a good alignment between our results and those obtained through DMRG, as well as with previously reported results, which provides more robust support for the accuracy and reliability of our proposed method.

Moreover, we have expanded our discussions by comparing our method not only to dynamical mean-field theory (DMFT) and DMRG, but also to linked-cluster expansion (LCE) and cluster variation method (CVM). These addresses suggestions made by referee 3 to connect our method to alternative cluster-based approaches. Furthermore, we have also included an Appendix that provides detailed implementation guidelines for our method applied to the antiferromagnetic Heisenberg model on triangular lattices (AFHTL). This addition directly addresses the concerns raised by referee 1.

In addition, we have expanded our discussions about the Affleck-Kennedy-Lieb-Tasaki (AKLT) model and the AFHTL. Regarding the AKLT model, while we acknowledge its simplicity, the capability of our method to detect the Haldane phase with topological order while the conventional Néel order vanishes provides valuable insights beyond known mean-field methods. Furthermore, while we have presented our results for the special case of β=1/3, our conclusions are still valid for the generic range of 0<β<1/3, where the topological phase persists, since our calculations do not rely on special properties at β=1/3. Additionally, our work is not restricted to the demonstrative examples discussed in the manuscript. It can serve as an efficient tool for studying various quantum orders, including, for example, stripy phases in $J_1$-$J_2$ Heisenberg model and dimerized phases. Providing references to relevant works will enhance the understanding of readers with broader interests.

In conclusion, this revised manuscript comprehensively addresses all essential points raised by the referees, and provides substantial evidence supporting the accuracy and efficacy of our method. We are confident that these enhancements align with the high standards of your flagship journal, SciPost Physics.

We appreciate your editorial efforts and thank you for considering our manuscript for publication in your esteemed journal.

Sincerely,

Junyi Zhang,
on behalf of the authors.

1. To address the referees’ major concern about the accuracy of our method, we have conducted additional benchmark calculations for the antiferromagnetic Heisenberg model on triangular lattices (AFHTL) using the state-of-the-art density-matrix renormalization group (DMRG) technique. The comparison between our results and those obtained through DMRG, as well as with previously reported results, shows good alignment, which provides robust support for the accuracy and reliability of our proposed method.
a) Page 12-Page 14, since the last paragraph on Page 12 through the end of Sec. 3.
We have expanded Sec. 3.2 to include detailed DMRG calculations for the AFHTL and relevant discussions. This includes descriptions of the system geometry used in DMRG calculations and the magnetization curve obtained. Additionally, we have provided quantitative comparisons between the results of DMRG and our method, density-matrix mean-field theory (DMMFT). We have also compared our results to those from previous reports suggested by Referee 2.
b) Page 13, Fig. 3.
we have included a new figure (Fig. 3) illustrating the system geometry used in the DMRG calculations, along with overlaid magnetization curves calculated using DMMFT, conventional mean-field theory, and DMRG.

2. As suggested by Referee 3, we have discussed the connection and difference of our method to other cluster-based methods, the cluster variation method (CVM) and the linked-cluster expansion (LCE)
a) Page 14 – Page 17, Sec. 4.1.
We have reorganized Section 4.1 to provide a more structured comparison between DMMFT and other methods. The original Section 4.1 has been split into two subsections: 4.1.1 and 4.1.2, dedicated to comparisons with DMRG and DMFT, respectively. Additionally, two new subsections, 4.1.3 and 4.1.4, have been added to compare DMMFT with CVM and LCE, respectively. These additions provide a more comprehensive analysis of the similarities and differences between our method and alternative cluster-based approaches.
b) Page 3, the third paragraph.
We have also updated the third paragraph on Page 3 to introduce the new discussions related to CVM and LCE in Section 4.1. This introductory paragraph motivates the connection of our method to alternative cluster-based methods, providing readers with a clear understanding of the context for the subsequent discussions.

3. In response to Referee 3’s inquiry regarding the translational symmetry, we have added a new paragraph to discuss this issue explicitly. Particularly, we have provided a detailed protocol for imposing lattice symmetries, including translational symmetries, with Eq. 25 in the revised manuscript. This addition ensures a consistent treatment of symmetries throughout our method.
a) Page 16, last paragraph of 4.1.2.

4To address Referee 1’s request “a simple illustration of it on some simple toy model would help to understand it better”, we have included a new appendix section, Append. B. This appendix provides a detailed description of the implementation of DMMFT for the AFHTL, along with a step-by-step pseudocode. This addition aims to enhance the understanding of our method through a more accessible example.
a) Page 20-23, Apped. B.

5. We appreciate Referee 2 for bringing to our attention some relevant previous studies that were previously overlooked. These studies have been properly cited in our revised manuscript (Ref. 47-50). Additionally, we have provided precise citations to describe the ground states of the AKLT model, as requested by Referee 2. We have expanded our discussion to justify the non-trivial insights provided by the AKLT model and to explain the relevance of the $J_1$-$J_2$ Heisenberg model to our method and to readers with broader interests.
a) Page 3, last paragraph, Line 7.
We have updated the references regarding the AFHTL according to Referee 2’s suggestion, now citing Ref. 47-50 in the revised manuscript.
b) Page 8, the introductory paragraphs of Sec. 3.
We have expanded the discussion to justify the significance of the AKLT model and to explain the relevance of the $J_1$-$J_2$ Heisenberg model. This ensures that our readers understand the broader context and importance of these models in relation to our method.
c) Page 9, third to the last paragraph.
We have revised statement concerning the ground state properties of the AKLT model as “The AKLT model is an exactly solvable system, showcasing non-trivial topological order. Notably, it features spin-1/2 (fractional to spin-1) edge states and exhibits a 4-fold degeneracy of the ground states for an open chain [43], which persists even for β < 1/3 deviating from its integrable point, provided the gap does not close [60, 68].”, where we have provided citations (Ref. 43, 60, 68) to support the statement. The statement about the nature of the ground states of the AKLT model has been made by AKLT in their original paper, here cited as Ref. 43. Moreover, it is important to recognize that the properties of these topological states do not depend on the special point of β = 1/3; instead, they persist within the range of 0 < β < 1/3, supporting the existence of topological ground states. This aspect justifies our choice of the AKLT model and ensures the generality of our discussions. The statement regarding the persistence of topological states can be found and is cited as Ref. 60 and 68.

6. We conducted a thorough review of the references and made necessary corrections to ensure accurate formatting of all chemical compounds, names of individuals, and model names. Specifically, we have overridden the default compilation bib style to rectify misprinted titles, such as "XXZ," and names like "Néel" in the titles, ensuring their proper formatting.