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Majorana Diagrammatics for Quantum Spin-1/2 Models
by Thibault Noblet, Laura Messio, Riccardo Rossi
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Laura Messio |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2508.19734v1 (pdf) |
| Date submitted: | Aug. 29, 2025, 4:06 p.m. |
| Submitted by: | Laura Messio |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
A diagrammatic formalism for lattices of 1/2 is developed. It is based on an unconstrained mapping between spin and Majorana operators. This allows the use of standard tools of diagrammatic quantum many-body theory without requiring projections. We derive, in particular, the Feynman rules for the expansion around a color-preserving mean-field theory. We then present the numerical results obtained by computing the corrections up to second order for the Heisenberg model in one and two dimensions, showing that perturbative corrections are not only numerically important, but also qualitatively improve the results of mean-field theory. These results pave the way for the use of Majorana diagrammatic tools in theoretical and numerical studies of quantum spin systems.
Current status:
Reports on this Submission
Strengths
1) This manuscript provides a very nice and rather complete introduction into the Majorana representation of spin-1/2 operators and its possible application for the analysis of the Heisenberg model.
2) The redundancy, i.e., the presence of multiple copies of each physical state is explained very clearly. The closely related Z2 gauge freedom in the mean field description is very thoroughly analysed.
3) The analysis of the exactly solvable 2-site and 4-site (numerically exact) cases is very useful to develop intuition.
4) The perturbative results in 1D and 2D obtained using the diagrammatic approach indeed improve the mean-field results.
Weaknesses
1) Some extra details about the diagrammatic calculations would definitely improve the clarity of this part of the paper (see requested changes).
2) The perturbative approach seems to be useful only for short range physics. There should be a clear physical explanation of this.
Report
The first part of the paper provides a very complete introduction into the Majorana representation, the issue of multiple copies and the Z2 gauge freedom.
The manuscript makes an important contribution into the development of the perturbative techniques based on the Majorana spin representation.
Requested changes
1) The fact that the mean-field Hamiltonian is a part of the perturbation seems on the first sight to be ignored. Only after some reading is becomes clear that diagrams of Fig. 1d are omitted exactly to take this into account. This is not explained and also the caption of Fig. 1d does not really help. I would recommend to clarify this.
2) It is never mentioned that even the mean-field treatment requires a numerical diagonalisation of a large (not exponentially large) matrix. I would recommend to mention this for clarity and explain how the Green's function of the mean-field Hamiltonian are calculated and how the self-consistency is achieved.
3) It is not clear that the perturbation parameter $\xi$ is taken to be equal one in both 1D and 2D numerical calculations. I would state this explicitly.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report #1 by Alexei Tsvelik (Referee 1) on 2025-9-6 (Invited Report)
Strengths
- It has been a long time since the Majorana field representation of spin-1/2 operators was discovered. Its advantage is the absence of the Hilbert space constraint which must be imposed leading to all sorts of complications. The paper represents an important step towards a practical utilization of this remarkable representation.
- The authors have formulated the diagrammatic expansion for the spin-1/2 Heisenberg model in the Majorana representation and demonstrated its convergence to known results for several lattices.
- It is especially remarkable that the second order perturbation theory result for the spin susceptibility in 1D well reproduces the Bethe ansatz results. Likewise, the perturbation theory for the square lattice shows a clear tendency to the antiferromagnetic order.
Report
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)