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Wegner's Ising gauge spins versus Kitaev's Majorana partons: Mapping and application to anisotropic confinement in spin-orbital liquids
by Urban F. P. Seifert, Sergej Moroz
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Submission summary
| Authors (as registered SciPost users): | Sergej Moroz · Urban Seifert |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2306.09405v1 (pdf) |
| Date submitted: | 2023-09-29 18:35 |
| Submitted by: | Seifert, Urban |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Emergent gauge theories take a prominent role in the description of quantum matter, supporting deconfined phases with topological order and fractionalized excitations. A common construction of $\mathbb{Z}_2$ lattice gauge theories, first introduced by Wegner, involves Ising gauge spins placed on links and subject to a discrete $\mathbb{Z}_2$ Gauss law constraint. As shown by Kitaev, $\mathbb{Z}_2$ lattice gauge theories also emerge in the exact solution of certain spin systems with bond-dependent interactions. In this context, the $\mathbb{Z}_2$ gauge field is constructed from Majorana fermions, with gauge constraints given by the parity of Majorana fermions on each site. In this work, we provide an explicit Jordan-Wigner transformation that maps between these two formulations on the square lattice, where the Kitaev-type gauge theory emerges as the exact solution of a spin-orbital (Kugel-Khomskii) Hamiltonian. We then apply our mapping to study local perturbations to the spin-orbital Hamiltonian, which correspond to anisotropic interactions between electric-field variables in the $\mathbb{Z}_2$ gauge theory. These are shown to induce anisotropic confinement that is characterized by emergence of weakly-coupled one-dimensional spin chains. We study the nature of these phases and corresponding confinement transitions in both absence and presence of itinerant fermionic matter degrees of freedom. Finally, we discuss how our mapping can be applied to the Kitaev spin-1/2 model on the honeycomb lattice.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 3) on 2023-11-10 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2306.09405v1, delivered 2023-11-10, doi: 10.21468/SciPost.Report.8093
Strengths
1. Clearly written.
2. Of interest to community of lattice gauge theories and quantum magnetism.
Weaknesses
1. Despite the methods not particularly novel, there is very limited acknowledgement and citations to precedents in the literature.
Report
The authors discuss a map between Wegner Z2 gauge theory and the Majorana parton representation of a spin orbital liquid model. The map is a Jordan-Wigner transformation between the gauge degrees of freedom, which are Ising spins in the Wegner Z2 gauge theory and the gauge Majorana bilinears. The authors apply the map to study an anisotropic confinement transition. The results are interesting and relevant for the to community of lattice gauge theories and quantum magnetism. Therefore I recommend publication.
However the paper has a very limited acknowledgement and missing citations to a series of previous works in the literature dealing with closely related constructions where dual representations of gauge theories are constructed with Jordan Wigner transformations. Below are some suggestions to improve the references.
- Their Ref. 51 has related ideas (but it is only cited as dealing with something "more involved").
The following references (not cited) discuss or use related constructions/methods:
-S. B. Bravyi and A. Y. Kitaev, Annals of Physics 298, 210 (2002).
- F. Verstraete and J. I. Cirac, Journal of Statistical Mechanics: Theory and Experiment 2005, P09012 (2005).
- M. Levin and X.-G. Wen, Physical Review B 73, 035122 (2006).
- R. C. Ball, Phys. Rev. Lett. 95, 176407 (2005).
- Y.-A. Chen and A. Kapustin, Physical Review B 100, 245127 (2019).
- D. Radicevic, arXiv preprint arXiv:1809.07757 (2018).
- Y.-A. Chen Phys. Rev. Research 2, 033527 (2020).
- H. C- Po, arXiv:2107.10842 (2021).
- P. Rao and I. Sodemann, Phys. Rev. Research 3, 023120 (2021).
- K. Li and H. Chun Po, Phys. Rev. B 106, 115109 (2022).
-W. Cao, M.o Yamazaki, and Y. Zheng, Phys. Rev. B 106, 075150 (2022).
- C. Chen, P. Rao, and I. Sodemann, Phys. Rev. Research 4, 043003 (2022).
- L Goller, I. S. Villadiego, arXiv:2309.13116 (2023).
- Y.-A. Chen and Y. Xu, PRX Quantum 4, 010326 (2023).
Requested changes
Improving the acknowledgement and citations to precedents in the literature.
Report #1 by Anonymous (Referee 4) on 2023-10-30 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2306.09405v1, delivered 2023-10-30, doi: 10.21468/SciPost.Report.8023
Report
The authors bridges the gap between the Wegner’s Z2 lattice gauge theory with bosonic matter, and the Kitaev-like spin-orbital model with fermionic matter. Even though the spin-orbital model and the fermionization scheme has been uncovered in some of the author’s previous work, a direct comparison with the Wegner’s lattice gauge model remains elusive in the literature. In particular, the electric field is a fundamental degree of freedom in the lattice gauge theory, which fluctuates the gauge flux and drives a versatile of gauge phenomena such as confinement. However, it is less discussed in the Kitaev-like spin-orbital model, where people either consider a solvable model with static gauge flux, or deal with the Kitaev model beyond solvable limit only in the spin basis. In this report, the authors show a two-step mapping: (spin-orbital basis) —> (fractionalized Majorana basis) —> (bosonic lattice gauge basis). As an application, the authors discuss the anisotropic confinement consequence from a natural spin-orbital interaction, treating that from the pure gauge limit to the matter-gauge coupling scenario.
Based on my estimate, their work could appeal to readers from both the frustrated magnetism and lattice gauge theory community. Despite the limited discussion of gauge phenomena, they have opened an avenue for more discussions of the gauge theories in spin-orbital liquids. Sufficient details are provided in the writing. Overall, I recommend this manuscript to be published by SciPost Physics.
In addition, I leave the following questions for the authors:
1. Eq.18 expresses the electric field sigma^{x(y)} in terms of the fractionalized Majorana fermion b. But since the interests originate from the spin-orbital model, how is the electric field expressed in terms of the original spin-orbital degrees of freedom (s, tau), which is by definition automatically a gauge invariant basis?
2. Relatedly, for the natural perturbation or solvability breaking fields of (s, tau) in different angles, how are they expressed in the gauge language?
3. While the authors have discussed the confinement transition, equally interesting is the Higgs transition from the gauge theory perspective: the Higgs transition from deconfined Z2 to trivial phase; and the un-Higgs transition from Z2 to deconfined U(1) state. The latter used to be observed in the lattice gauge theory, the toric code or the Kitaev honeycomb model beyond solvable limit. I wonder whether the authors can comment or relate their formalism to these transitions, charting out a guiding landscape for the gauge phases of matter in the spin-orbital liquids.
Author: Urban Seifert on 2024-05-03 [id 4468]
(in reply to Report 1 on 2023-10-30)
We are grateful for the Referee's reading of our manuscript and are happy that they recommend our manuscript to be published in SciPost Physics. We have made several edits in response to their questions:
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and 2. We have provided a detailed explanation for expressing the electric-field operators in the $\mathbb{Z}_2$ gauge theory in terms of spin-orbital in Sec. 3.1: we find that $\sigma^x$ and $\sigma^y$ do not admit simple closed-form expressions in terms of the spin-orbital degrees of freedom, however pairs of such electric field operators along appropriate bonds map onto spin-orbital degrees of freedom (see, e.g., Eq. (24)).
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The Referee raises highly interesting open questions: Indeed, the Higgs transition from U(1) to $\mathbb{Z}_2$ lattice gauge theory could be studied by developing a U(1) lattice gauge theory with appropriate deformations. Its continuum limit could lead to new insights into the appropriate critical phenomena. By studying the deformed U(1) LGT that mimics the effect of finite couplings $g_1$ and $g_2$ (see Eq. (41)) might also allow novel insights into the critical theory of the deconfinement transition driven by these couplings. We further comment that our work can also be extended the $\mathbb{Z}_N$ lattice gauge theories, where we expect that one might construct a mapping from a "Wegner-type" theory (with clock and shift matrices placed on bonds) to a (para-)fermionic theory by means of generalized Jordan-Wigner transformations, and a subsequent rewriting in terms of a generalized spin-orbital theory. As these require significant technical efforts, we leave these topic for future study.
Author: Urban Seifert on 2024-05-03 [id 4467]
(in reply to Report 2 on 2023-11-10)We thank the Referee for their feedback and overall positive assessment of our work. We regret having neglected referencing previous constructions that may appear to be of similar nature. In the resubmitted manuscript, we have added several key references. However, we stress that our work is distinct from many previous contributions in the fact that we do not construct two-dimensional bosonization (i.e., a mapping between an ungauged fermionic problem and a $\mathbb{Z}_2$ gauged spin-1/2 model), but rather our key achievement consists in relating to common constructions of $\mathbb{Z}_2$ gauge theories (of "Kitaev"- and "Wegner"-type) to each other.