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Density-Matrix Mean-Field Theory

by Junyi Zhang, Zhengqian Cheng

Submission summary

Authors (as registered SciPost users): Junyi Zhang
Submission information
Preprint Link:  (pdf)
Date submitted: 2024-01-23 03:21
Submitted by: Zhang, Junyi
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


Mean-field theories (MFTs) have proven to be efficient tools for exploring various phases of matter, complementing alternative methods that are more precise but also more computationally demanding. Conventional mean-field theories (MFTs) often fall short in capturing quantum fluctuations, which restricts their applicability to systems characterized by strong quantum fluctuations. In this article, we propose a novel mean-field theory, density-matrix mean-field theory (DMMFT). DMMFT constructs effective Hamiltonians, incorporating quantum environments shaped by entanglements quantified by the reduced density matrices. Therefore, it offers a systematic and unbiased approach to account for effects of fluctuations and entanglements in quantum ordered phases. As demonstrative examples, we show that DMMFT can not only quantitatively evaluate the renormalization of order parameters induced by quantum fluctuations but can even detect the topological order of quantum phases. Additionally, we discuss the extensions of DMMFT for systems at finite temperatures and those with disorders. Our work provides a novel and efficient approach to explore phases exhibiting unconventional quantum orders, which can be particularly beneficial for investigating frustrated spin systems in high spatial dimensions.

Current status:
Awaiting resubmission

Reports on this Submission

Anonymous Report 3 on 2024-3-4 (Invited Report)


The manuscript by Zhang and Cheng discusses a novel mean-field theory for interacting quantum systems, the so-called DMMT approach. The writing presents the formalism and discusses its relation to comparable schemes, such as DMFT, DMET and DMRG.
The text ist well written, the method becomes accessible and the generally revealed physics appears sound. Main idea appears to separate the full quantum-lattice problem into the more manageable problem of individual clusters building up the lattice under consideration. That idea is heavily used on various levels of interacting lattice problems. In quantum theory, the linked cluster expansion (LCE) approach is e.g. well known for that picture. In classical statistical mechnics, the cluster variation method (CVM) takes on that role. However, both techniques are not mentioned in the present context.
Thus, aside from the comparisons to DMFT, DMET and DMRG, the authors should also connect their approach to LCE and CVM. In addition, the crucial role of cluster nestings (i.e. overlapping clusters) and the issue of translational invariance (e.g. being a key obstacle in the various flavors of cluster-DMFT) is only scarcely touched. This aspect should be discussed more clearly and concrete statements in that respect are needed.
If the authors take the aforementioned points of criticism into account in a revised version of their manuscript, support for publication may be granted.

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Anonymous Report 2 on 2024-3-3 (Contributed Report)


The manuscript makes an interesting proposal, namely a new "Density-Matrix Mean-Field Theory" (DMMFT). However, whether or not this merits publication in SciPost Physics (see for the Acceptance criteria) depends on the performance of the method. In this respect, chapter 3 of the manuscript is essential, but here I am not convinced [yet].

The application in section 3.1 to the Affleck-Kennedy-Lieb-Tasaki Model, i.e., the spin-1 bilinear biquadratic chain Eq. (18) with $\beta = \frac13$ is fairly simple and thus not very deep. Nevertheless, the statement "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 4-fold degeneracy of the ground states for an open chain." would need to be substantiated with references. In fact, it would probably be possible to also investigate the case $\beta \ne \frac13$, but in this case there are further (numerical) investigations that would also need to be cited.

This renders section 3.2 about the antiferromagnetic spin-1/2 Heisenberg model on the triangular lattice central. Here, the real benchmark would not be conventional mean-field theory, but numerical results that are available in the literature. In fact, I would consider a quantitative comparison more important than the generic comparison with alternative methods that is presented in section 4.1. I also note that in the related list of references [31-47] that are cited in the Introduction, not all relevant ones are mentioned, and not all cited are indeed relevant (e.g., some of these references concern only the zero-field case that is not really relevant here, and some concern the $J_1$-$J_2$ model, where only the special case $J_2=0$ is relevant here). Previous publications on the XXZ model that might be relevant include A. Honecker, J. Schulenburg, and J. Richter, J. Phys.: Condens. Matter 16, S749 (2004) and D. Sellmann, X.-F. Zhang, and S. Eggert, Phys. Rev. B 91, 081104(R) (2015) as well as J. Richter, J. Schulenburg, A. Honecker, Lect. Notes Phys. 645, 85 (2004) (specifically Sec. 2.6.2 loc. cit.) and T. Sakai and H. Nakano, Phys. Rev. B 83, 100405(R) (2011) for the Heisenberg model. There may be more recent relevant references, but there definitely are older ones; the latter are cited in the publications mentioned here.

I note in passing that there is spurious lowercasing in the titles of the references. For example, the "XXZ" in the title of Ref. [42] that is central to the present work is misprinted and names such as "Néel" in the title of Ref. [28] should not be lowercased either.

Without further discussion of the quantitative accuracy of DMMFT, I believe that the present work would be more suitable for SciPost Physics Core, see

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

Anonymous Report 1 on 2024-3-3 (Invited Report)


The manuscript under consideration proposes a novel formulation of the mean-field theory by accounting for short-range entanglement via the reduced density matrix of the subsystem. The idea makes sense and can be useful to high-dimensional ordered systems (higher than one-dimensional) with significant quantum fluctuations.


The approach is not intuitive and a simple illustration of it on some simple toy model would help to understand it better.


I can not comment on the numerical side of the proposal, but the two applications of the approach - the Haldane chain and the triangular XXZ spin-1/2 model in an external magnetic field - look quite promising. The authors find that (in the second example), the 1/3 magnetization plateau (the UUD phase) extends relative to the conventional cluster mean-field theory. This signals that DMMFT captures more quantum fluctuations than standard MFT.

Requested changes

Fig.2b also shows that M(h) behavior of MFT and DMMFT calculations is different below h_{c1}, the red DMMFT curve is quite nonlinear. This leads to the question: which of these is closer to the correct M(h) curve? It would be good for authors to comment on this and/or to compare it with the “exact” DMRG result if that exists.

  • validity: good
  • significance: good
  • originality: high
  • clarity: good
  • formatting: good
  • grammar: good

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