# Quantitative functional renormalization for three-dimensional quantum Heisenberg models

### Submission summary

 As Contributors: Nils Niggemann Arxiv Link: https://arxiv.org/abs/2112.08104v2 (pdf) Date accepted: 2022-04-20 Date submitted: 2022-03-22 07:07 Submitted by: Niggemann, Nils Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Condensed Matter Physics - Computational Approaches: Theoretical, Computational

### Abstract

We employ a recently developed variant of the functional renormalization group method for spin systems, the so-called pseudo Majorana functional renormalization group, to investigate three-dimensional spin-1/2 Heisenberg models at finite temperatures. We study unfrustrated and frustrated Heisenberg systems on the simple cubic and pyrochlore lattices. Comparing our results with other quantum many-body techniques, we demonstrate a high quantitative accuracy of our method. Particularly, for the unfrustrated simple cubic lattice antiferromagnet ordering temperatures obtained from finite-size scaling of one-loop data deviate from error controlled quantum Monte Carlo results by $\sim5\%$ and we further confirm the established values for the critical exponent $\nu$ and the anomalous dimension $\eta$. As the PMFRG yields results in good agreement with QMC, but remains applicable when the system is frustrated, we next treat the pyrochlore Heisenberg antiferromagnet as a paradigmatic magnetically disordered system and find nearly perfect agreement of our two-loop static homogeneous susceptibility with other methods. We further investigate the broadening of pinch points in the spin structure factor as a result of quantum and thermal fluctuations and confirm a finite width in the extrapolated limit $T\rightarrow0$. While extensions towards higher loop orders $\ell$ seem to systematically improve our approach for magnetically disordered systems we also discuss subtleties when increasing $\ell$ in the presence of magnetic order. Overall, the pseudo Majorana functional renormalization group is established as a powerful many-body technique in quantum magnetism with a wealth of possible future applications.

Published as SciPost Phys. 12, 156 (2022)

### Author comments upon resubmission

______Response to the first referee:______
We thank the first referee for their helpful comments. Please refer to the comment posted by us recently before this resubmission. In this comment, we adress all points in detail.

______Response to the second referee:______
We thank the second referee for the comments provided. However, we feel that there are a number of important conceptual misinterpretations by the referee that lead to criticism that we deem unjustified. We explain these confusions in the following:

Confusion 1: In our manuscript, the FRG treatment of spin-systems relies on a pseudo-MAJORANA representation of spin (-> PMFRG), which is conceptually different from the established (complex) pseudo-FERMION representation (-> PFFRG) which exists for a decade. As we explain at length in Sec. 1+2, the faithfulness of the MAJORANA representation allows the PMFRG to be applied at finite temperatures, whereas the PFFRG is limited to zero temperature.

Confusion 2: References 16+17 [Kiese et al, Thoenniss et al] have implemented the multiloop FRG formalism only for the pseudo-FERMION representation, i.e. for PFFRG. The only existing PMFRG paper (our work in Ref. 13, [Niggemann et al]) was limited to a one-loop formalism. In so far, two-loop PMFRG is a novelty and there is no overlap with existing literature.

Confusion 3: Two-loop (or multiloop) FRG is not the main point in our manuscript. In fact, all the results for finite-temperature phase transition in the simple cubic lattice model in Sec. 3 are obtained in a one-loop scheme. These results show good agreement with Quantum-Monte Carlo results and we are thus convinced that the "quantitative" in the title is justified.
We only turn to the two-loop formalism for the study of paramagnetic models like the pyrochlore lattice in Sec. 4 where we indeed observe quantitative improvements over one-loop results, justifying our claim about "systematic improvement with loop order" in the abstract. Our two-loop susceptbilitity data in Fig. 4 is then again in QUANTITATIVE agreement with diagrammatic Monte Carlo results.
We agree with the referee that it is of course desirable to eventually reach full convergence in the loop order, as achieved for the PFFRG and the Hubbard model. Research efforts in this direction for the PMFRG are ongoing.

We thank the referee for pointing out two references concerning multiloop calculations for the Hubbard model, [SciPost Phys. 6, 009 (2019)] and [Phys. Rev. Research 2, 033372]. We have cited these accordingly.

Yours sincerely,

Nils Niggemann, Johannes Reuther and Björn Sbierski

### List of changes

____List of changes____

-pg. 5: Added panel (d) in Fig. 1 testing scaling behaviour equivalently to (c) but with mean-field exponents instead.
-pg. 6: Added references to other methods for the extraction of mean-field exponents
-pg. 6: Added discussion of new figure
-pg. 8: Clarified difference between PM- and PFFRG for finite temperature applications
-pg. 11: Motivated inclusion of two-loop corrections.
-pg. 14: Clarified statement on the detection of first-order transitions with PMFRG
-pg. 12: Added references to recent multiloop works in Hubbard models

### Submission & Refereeing History

#### Published as SciPost Phys. 12, 156 (2022)

Resubmission 2112.08104v2 on 22 March 2022
Submission scipost_202201_00005v1 on 11 January 2022

## Reports on this Submission

### Anonymous Report 1 on 2022-4-6 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2112.08104v2, delivered 2022-04-06, doi: 10.21468/SciPost.Report.4875

### Report

The authors have considered all the comments and questions that I formulated in my first report and they have revised their manuscript accordingly. In particular, they have added a discussion on the extraction of the critical exponents as well as the validity of the two-loop approximation, which I believe helps to clarify and contextualize some of their statements. The included changes are appropriate and satisfactory within the scope set in the paper.

As I have stated before, I believe that the manuscript introduces and applies a very promising new formulation within the FRG framework and it presents a series of very interesting and non-trivial results. Therefore, I now recommend publication.

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