The geometry of the Hermitian matrix space and the Schrieffer–Wolff transformation
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AbstractIn quantum mechanics, the Schrieffer–Wolff (SW) transformation (also called quasi-degenerate perturbation theory) is known as an approximative method to reduce the dimension of the Hamiltonian. We present a geometric interpretation of the SW transformation: We prove that it induces a local coordinate chart in the space of Hermitian matrices near a $k$-fold degeneracy submanifold. Inspired by this result, we establish a `distance theorem': we show that the standard deviation of $k$ neighboring eigenvalues of a Hamiltonian equals the distance of this Hamiltonian from the corresponding $k$-fold degeneracy submanifold, divided by $\sqrt{k}$. Furthermore, we investigate one-parameter perturbations of a degenerate Hamiltonian, and prove that the standard deviation and the pairwise differences of the eigenvalues lead to the same order of splitting of the energy eigenvalues, which in turn is the same as the order of distancing from the degeneracy submanifold. As applications, we prove the `protection' of Weyl points using the transversality theorem, and infer geometrical properties of certain degeneracy submanifolds based on results from quantum error correction and topological order.Featured image: A local coordinate system (chart) defined by the Schrieffer-Wolff transformation in the space of Hermitian matrices near a submanifold of matrices with degenerate spectra (orange).Popular summaryMany physical systems—from topological materials to quantum error-correcting codes—share a common feature: their behaviour is shaped by degeneracies, parameter points where two or more energy levels become equal. Understanding how systems behave near such degeneracies is important, yet often technically challenging. In this work, we revisit a standard tool of quantum theory, the Schrieffer–Wolff transformation, and uncover a new geometric perspective on it. Rather than viewing it only as a perturbative method to simplify Hamiltonians, we show that it naturally defines a local coordinate system in the space of Hermitian matrices near a manifold of degenerate spectra. Based on this geometric viewpoint, we find that the spread of nearby energy levels directly measures how far the system is from an exact degeneracy. This relation provides an intuitive bridge between spectral properties and geometry. It unifies phenomena that appear across very different areas of physics—such as protected band crossings in topological materials and the structure of logical subspaces in quantum error correction—by identifying a common underlying structure: proximity to degeneracy manifolds. Beyond its conceptual appeal, this framework offers a new way to quantify and analyze robustness in quantum systems. By translating spectral features into geometric distances, it provides a tool that may prove useful wherever degeneracies play a central role, from condensed matter physics to quantum information.► BibTeX data@article{Pinter2026geometryofhermitian, doi = {10.22331/q-2026-03-27-2047}, url = {https://doi.org/10.22331/q-2026-03-27-2047}, title = {The geometry of the {H}ermitian matrix space and the {S}chrieffer–{W}olff transformation}, author = {Pinter, Gergo and Frank, Gyorgy and Varjas, Daniel and Palyi, Andras}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2047}, month = mar, year = {2026} }► References [1] J. R. Schriefferand P. A. Wolff ``Relation between the Anderson and Kondo Hamiltonians'' Phys. Rev. 149, 491-492 (1966). https://doi.org/10.1103/PhysRev.149.491 [2] J. M. Luttingerand W. Kohn ``Motion of Electrons and Holes in Perturbed Periodic Fields'' Phys. Rev. 97, 869–883 (1955). https://doi.org/10.1103/PhysRev.97.869 [3] Sergey Bravyi, David P. DiVincenzo, and Daniel Loss, ``Schrieffer–Wolff transformation for quantum many-body systems'' Annals of Physics 326, 2793–2826 (2011). https://doi.org/10.1016/j.aop.2011.06.004 [4] Roland Winkler ``Spin-orbit Coupling Effects in Two-Dimensional Electron and Hole Systems'' Springer (2003). https://doi.org/10.1007/978-3-540-36616-4_12 [5] Isidora Araya Day, Sebastian Miles, Hugo K. Kerstens, Daniel Varjas, and Anton R. Akhmerov, ``Pymablock: An algorithm and a package for quasi-degenerate perturbation theory'' SciPost Phys. Codebases 50 (2025). https://doi.org/10.21468/SciPostPhysCodeb.50 [6] J. H. Van Vleck ``On ${\sigma}$-Type Doubling and Electron Spin in the Spectra of Diatomic Molecules'' Phys. Rev. 33, 467–506 (1929). https://doi.org/10.1103/PhysRev.33.467 [7] Per‐Olov Löwdin ``Studies in Perturbation Theory. IV. Solution of Eigenvalue Problem by Projection Operator Formalism'' Journal of Mathematical Physics 3, 969–982 (1962). https://doi.org/10.1063/1.1724312 [8] N. P. Armitage, E. J. Mele, and Ashvin Vishwanath, ``Weyl and Dirac semimetals in three-dimensional solids'' Rev. Mod. Phys. 90, 015001 (2018). https://doi.org/10.1103/RevModPhys.90.015001 [9] Ferenc Iglóiand Heiko Rieger ``Random transverse Ising spin chain and random walks'' Phys. Rev. B 57, 11404–11420 (1998). https://doi.org/10.1103/PhysRevB.57.11404 [10] Bogdan Damskiand Marek M Rams ``Exact results for fidelity susceptibility of the quantum Ising model: the interplay between parity, system size, and magnetic field'' Journal of Physics A: Mathematical and Theoretical 47, 025303 (2013). https://doi.org/10.1088/1751-8113/47/2/025303 [11] Róbert Juhász ``Exact bounds on the energy gap of transverse-field Ising chains by mapping to random walks'' Phys. Rev. B 106, 064204 (2022). https://doi.org/10.1103/PhysRevB.106.064204 [12] W. P. Su, J. R. Schrieffer, and A. J. Heeger, ``Solitons in Polyacetylene'' Phys. Rev. Lett. 42, 1698–1701 (1979). https://doi.org/10.1103/PhysRevLett.42.1698 [13] J. K. Asbóth, L. Oroszlány, and A. Pályi, ``A Short Course on Topological Insulators'' Springer (2016). https://doi.org/10.1007/978-3-319-25607-8 [14] A Yu Kitaev ``Unpaired Majorana fermions in quantum wires'' Physics-Uspekhi 44, 131 (2001). https://doi.org/10.1070/1063-7869/44/10S/S29 [15] A.Yu. Kitaev ``Fault-tolerant quantum computation by anyons'' Annals of Physics 303, 2–30 (2003). https://doi.org/10.1016/S0003-4916(02)00018-0 https://www.sciencedirect.com/science/article/pii/S0003491602000180 [16] Daniel Gottesman ``Stabilizer Codes and Quantum Error Correction'' thesis (1997). https://doi.org/10.48550/arXiv.quant-ph/9705052 [17] Hermann Weyl ``Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)'' Mathematische Annalen 71, 441–479 (1912). https://doi.org/10.1007/BF01456804 [18] J. von Neumannand E. P. Wigner ``Über das Verhalten von Eigenwerten bei adiabatischen Prozessen'' Physikalische Zeitschrift 30, 467 (1929). https://doi.org/10.1007/978-3-662-02781-3_20 [19] V. I. Arnold ``Remarks on Eigenvalues and Eigenvectors of Hermitian Matrices, Berry Phase, Adiabatic Connections and Quantum Hall Effect'' Selecta Mathematica 1, 1 (1995). https://doi.org/10.1007/BF01614072 [20] Victor A Vassiliev ``Spaces of Hermitian operators with simple spectra and their finite-order cohomology'' arXiv preprint arXiv:1407.7238 (2014). https://doi.org/10.48550/arXiv.1407.7238 https://arxiv.org/abs/1407.7238 [21] Carl Eckartand Gale Young ``The Approximation of One Matrix by Another of Lower Rank'' Psychometrika 1, 211–218 (1936). https://doi.org/10.1007/BF02288367 [22] L. Mirsky ``SYMMETRIC GAUGE FUNCTIONS AND UNITARILY INVARIANT NORMS'' The Quarterly Journal of Mathematics 11, 50–59 (1960). https://doi.org/10.1093/qmath/11.1.50 [23] Paul Breiding, Khazhgali Kozhasov, and Antonio Lerario, ``On the Geometry of the Set of Symmetric Matrices with Repeated Eigenvalues'' Arnold Mathematical Journal 4, 423–443 (2018). https://doi.org/10.1007/s40598-018-0095-0 [24] Hiroshi Teramoto, Kenji Kondo, Shyūichi Izumiya, Mikito Toda, and Tamiki Komatsuzaki, ``Classification of Hamiltonians in neighborhoods of band crossings in terms of the theory of singularities'' J. Math. Phys. 58, 073502 (2017). https://doi.org/10.1063/1.4991662 [25] Hiroshi Teramoto, Asahi Tsuchida, Kenji Kondo, Shyuichi Izumiya, Mikito Toda, and Tamiki Komatsuzaki, ``Application of singularity theory to bifurcation of band structures in crystals'' Journal of Singularities 21, 289–302 (2020). https://doi.org/10.5427/jsing.2020.21p [26] Gergo Pintér, György Frank, Dániel Varjas, and András Pályi, ``Upper bound on the number of Weyl points born from a nongeneric degeneracy point'' Phys. Rev. B 110, 245124 (2024). https://doi.org/10.1103/PhysRevB.110.245124 [27] Gabriele Naselli, György Frank, Dániel Varjas, Ion Cosma Fulga, Gergo Pintér, András Pályi, and Viktor Könye, ``Stability of Weyl Node Merging Processes under Symmetry Constraints'' Phys. Rev. Lett. 133, 196602 (2024). https://doi.org/10.1103/PhysRevLett.133.196602 [28] Victor Guilleminand Alan Pollack ``Differential topology'' American Mathematical Soc. (2010). https://doi.org/10.1090/chel/370 [29] Martin Golubitskyand Victor Guillemin ``Stable mappings and their singularities'' Springer New York, NY (2012). https://doi.org/10.1007/978-1-4615-7904-5 [30] Zoltán Scherübl, András Pályi, György Frank, István Endre Lukács, Gergő Fülöp, Bálint Fülöp, Jesper Nygård, Kenji Watanabe, Takashi Taniguchi, Gergely Zaránd, and Szabolcs Csonka, ``Observation of spin–orbit coupling induced Weyl points in a two-electron double quantum dot'' Communications Physics 2, 108 (2019). https://doi.org/10.1038/s42005-019-0200-2 https://www.nature.com/articles/s42005-019-0200-2 [31] György Frank, Zoltán Scherübl, Szabolcs Csonka, Gergely Zaránd, and András Pályi, ``Magnetic degeneracy points in interacting two-spin systems: Geometrical patterns, topological charge distributions, and their stability'' Physical Review B 101, 245409 (2020). https://doi.org/10.1103/PhysRevB.101.245409 [32] Roman-Pascal Riwar, Manuel Houzet, Julia S. Meyer, and Yuli V. Nazarov, ``Multi-terminal Josephson junctions as topological matter'' Nature Communications 7, 11167 (2016). https://doi.org/10.1038/ncomms11167 [33] Valla Fatemi, Anton R. Akhmerov, and Landry Bretheau, ``Weyl Josephson circuits'' Phys. Rev. Research 3, 013288 (2021). https://doi.org/10.1103/PhysRevResearch.3.013288 [34] György Frank, Dániel Varjas, Péter Vrana, Gergő Pintér, and András Pályi, ``Topological charge distributions of an interacting two-spin system'' Physical Review B 105, 035414 (2022). https://doi.org/10.1103/PhysRevB.105.035414 [35] Israel Klich ``On the stability of topological phases on a lattice'' Annals of Physics 325, 2120–2131 (2010). https://doi.org/10.1016/j.aop.2010.05.002 [36] Sergey Bravyi, Matthew B. Hastings, and Spyridon Michalakis, ``Topological quantum order: Stability under local perturbations'' Journal of Mathematical Physics 51, 093512 (2010). https://doi.org/10.1063/1.3490195 [37] Sergey Bravyiand Matthew B. Hastings ``A Short Proof of Stability of Topological Order under Local Perturbations'' Communications in Mathematical Physics 307, 609–627 (2011). https://doi.org/10.1007/s00220-011-1346-2 [38] Ali Lavasani, Michael J. Gullans, Victor V. Albert, and Maissam Barkeshli, ``On stability of k-local quantum phases of matter'' arXiv:2405.19412 (2024). https://doi.org/10.48550/arXiv.2405.19412 https://arxiv.org/abs/2405.19412 [39] Yaodong Li, Nicholas O'Dea, and Vedika Khemani, ``Perturbative Stability and Error-Correction Thresholds of Quantum Codes'' PRX Quantum 6, 010327 (2025). https://doi.org/10.1103/PRXQuantum.6.010327 [40] Nikolas P.
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Dupont ``Fibre Bundles and Chern-Weil Theory'' University of Aarhus, Department of Mathematics (2003). https://books.google.hu/books?id=6zUZAQAAIAAJ [49] D. Husemöller ``Fibre Bundles'' Springer (1994). https://doi.org/10.1007/978-1-4757-2261-1 https://books.google.hu/books?id=DPr_BSH89cACCited byCould not fetch Crossref cited-by data during last attempt 2026-03-27 11:44:26: Could not fetch cited-by data for 10.22331/q-2026-03-27-2047 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-27 11:44:27: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractIn quantum mechanics, the Schrieffer–Wolff (SW) transformation (also called quasi-degenerate perturbation theory) is known as an approximative method to reduce the dimension of the Hamiltonian. We present a geometric interpretation of the SW transformation: We prove that it induces a local coordinate chart in the space of Hermitian matrices near a $k$-fold degeneracy submanifold. Inspired by this result, we establish a `distance theorem': we show that the standard deviation of $k$ neighboring eigenvalues of a Hamiltonian equals the distance of this Hamiltonian from the corresponding $k$-fold degeneracy submanifold, divided by $\sqrt{k}$. Furthermore, we investigate one-parameter perturbations of a degenerate Hamiltonian, and prove that the standard deviation and the pairwise differences of the eigenvalues lead to the same order of splitting of the energy eigenvalues, which in turn is the same as the order of distancing from the degeneracy submanifold. As applications, we prove the `protection' of Weyl points using the transversality theorem, and infer geometrical properties of certain degeneracy submanifolds based on results from quantum error correction and topological order.Featured image: A local coordinate system (chart) defined by the Schrieffer-Wolff transformation in the space of Hermitian matrices near a submanifold of matrices with degenerate spectra (orange).Popular summaryMany physical systems—from topological materials to quantum error-correcting codes—share a common feature: their behaviour is shaped by degeneracies, parameter points where two or more energy levels become equal. Understanding how systems behave near such degeneracies is important, yet often technically challenging. In this work, we revisit a standard tool of quantum theory, the Schrieffer–Wolff transformation, and uncover a new geometric perspective on it. Rather than viewing it only as a perturbative method to simplify Hamiltonians, we show that it naturally defines a local coordinate system in the space of Hermitian matrices near a manifold of degenerate spectra. Based on this geometric viewpoint, we find that the spread of nearby energy levels directly measures how far the system is from an exact degeneracy. This relation provides an intuitive bridge between spectral properties and geometry. It unifies phenomena that appear across very different areas of physics—such as protected band crossings in topological materials and the structure of logical subspaces in quantum error correction—by identifying a common underlying structure: proximity to degeneracy manifolds. Beyond its conceptual appeal, this framework offers a new way to quantify and analyze robustness in quantum systems. By translating spectral features into geometric distances, it provides a tool that may prove useful wherever degeneracies play a central role, from condensed matter physics to quantum information.► BibTeX data@article{Pinter2026geometryofhermitian, doi = {10.22331/q-2026-03-27-2047}, url = {https://doi.org/10.22331/q-2026-03-27-2047}, title = {The geometry of the {H}ermitian matrix space and the {S}chrieffer–{W}olff transformation}, author = {Pinter, Gergo and Frank, Gyorgy and Varjas, Daniel and Palyi, Andras}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2047}, month = mar, year = {2026} }► References [1] J. R. Schriefferand P. A. Wolff ``Relation between the Anderson and Kondo Hamiltonians'' Phys. Rev. 149, 491-492 (1966). https://doi.org/10.1103/PhysRev.149.491 [2] J. M. Luttingerand W. Kohn ``Motion of Electrons and Holes in Perturbed Periodic Fields'' Phys. Rev. 97, 869–883 (1955). https://doi.org/10.1103/PhysRev.97.869 [3] Sergey Bravyi, David P. DiVincenzo, and Daniel Loss, ``Schrieffer–Wolff transformation for quantum many-body systems'' Annals of Physics 326, 2793–2826 (2011). https://doi.org/10.1016/j.aop.2011.06.004 [4] Roland Winkler ``Spin-orbit Coupling Effects in Two-Dimensional Electron and Hole Systems'' Springer (2003). https://doi.org/10.1007/978-3-540-36616-4_12 [5] Isidora Araya Day, Sebastian Miles, Hugo K. Kerstens, Daniel Varjas, and Anton R. Akhmerov, ``Pymablock: An algorithm and a package for quasi-degenerate perturbation theory'' SciPost Phys. Codebases 50 (2025). https://doi.org/10.21468/SciPostPhysCodeb.50 [6] J. H. Van Vleck ``On ${\sigma}$-Type Doubling and Electron Spin in the Spectra of Diatomic Molecules'' Phys. Rev. 33, 467–506 (1929). https://doi.org/10.1103/PhysRev.33.467 [7] Per‐Olov Löwdin ``Studies in Perturbation Theory. IV. Solution of Eigenvalue Problem by Projection Operator Formalism'' Journal of Mathematical Physics 3, 969–982 (1962). https://doi.org/10.1063/1.1724312 [8] N. P. Armitage, E. J. Mele, and Ashvin Vishwanath, ``Weyl and Dirac semimetals in three-dimensional solids'' Rev. Mod. Phys. 90, 015001 (2018). https://doi.org/10.1103/RevModPhys.90.015001 [9] Ferenc Iglóiand Heiko Rieger ``Random transverse Ising spin chain and random walks'' Phys. Rev. B 57, 11404–11420 (1998). https://doi.org/10.1103/PhysRevB.57.11404 [10] Bogdan Damskiand Marek M Rams ``Exact results for fidelity susceptibility of the quantum Ising model: the interplay between parity, system size, and magnetic field'' Journal of Physics A: Mathematical and Theoretical 47, 025303 (2013). https://doi.org/10.1088/1751-8113/47/2/025303 [11] Róbert Juhász ``Exact bounds on the energy gap of transverse-field Ising chains by mapping to random walks'' Phys. Rev. B 106, 064204 (2022). https://doi.org/10.1103/PhysRevB.106.064204 [12] W. P. Su, J. R. Schrieffer, and A. J. Heeger, ``Solitons in Polyacetylene'' Phys. Rev. Lett. 42, 1698–1701 (1979). https://doi.org/10.1103/PhysRevLett.42.1698 [13] J. K. Asbóth, L. Oroszlány, and A. Pályi, ``A Short Course on Topological Insulators'' Springer (2016). https://doi.org/10.1007/978-3-319-25607-8 [14] A Yu Kitaev ``Unpaired Majorana fermions in quantum wires'' Physics-Uspekhi 44, 131 (2001). https://doi.org/10.1070/1063-7869/44/10S/S29 [15] A.Yu. Kitaev ``Fault-tolerant quantum computation by anyons'' Annals of Physics 303, 2–30 (2003). https://doi.org/10.1016/S0003-4916(02)00018-0 https://www.sciencedirect.com/science/article/pii/S0003491602000180 [16] Daniel Gottesman ``Stabilizer Codes and Quantum Error Correction'' thesis (1997). https://doi.org/10.48550/arXiv.quant-ph/9705052 [17] Hermann Weyl ``Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)'' Mathematische Annalen 71, 441–479 (1912). https://doi.org/10.1007/BF01456804 [18] J. von Neumannand E. P. Wigner ``Über das Verhalten von Eigenwerten bei adiabatischen Prozessen'' Physikalische Zeitschrift 30, 467 (1929). https://doi.org/10.1007/978-3-662-02781-3_20 [19] V. I. Arnold ``Remarks on Eigenvalues and Eigenvectors of Hermitian Matrices, Berry Phase, Adiabatic Connections and Quantum Hall Effect'' Selecta Mathematica 1, 1 (1995). https://doi.org/10.1007/BF01614072 [20] Victor A Vassiliev ``Spaces of Hermitian operators with simple spectra and their finite-order cohomology'' arXiv preprint arXiv:1407.7238 (2014). https://doi.org/10.48550/arXiv.1407.7238 https://arxiv.org/abs/1407.7238 [21] Carl Eckartand Gale Young ``The Approximation of One Matrix by Another of Lower Rank'' Psychometrika 1, 211–218 (1936). https://doi.org/10.1007/BF02288367 [22] L. Mirsky ``SYMMETRIC GAUGE FUNCTIONS AND UNITARILY INVARIANT NORMS'' The Quarterly Journal of Mathematics 11, 50–59 (1960). https://doi.org/10.1093/qmath/11.1.50 [23] Paul Breiding, Khazhgali Kozhasov, and Antonio Lerario, ``On the Geometry of the Set of Symmetric Matrices with Repeated Eigenvalues'' Arnold Mathematical Journal 4, 423–443 (2018). https://doi.org/10.1007/s40598-018-0095-0 [24] Hiroshi Teramoto, Kenji Kondo, Shyūichi Izumiya, Mikito Toda, and Tamiki Komatsuzaki, ``Classification of Hamiltonians in neighborhoods of band crossings in terms of the theory of singularities'' J. Math. Phys. 58, 073502 (2017). https://doi.org/10.1063/1.4991662 [25] Hiroshi Teramoto, Asahi Tsuchida, Kenji Kondo, Shyuichi Izumiya, Mikito Toda, and Tamiki Komatsuzaki, ``Application of singularity theory to bifurcation of band structures in crystals'' Journal of Singularities 21, 289–302 (2020). https://doi.org/10.5427/jsing.2020.21p [26] Gergo Pintér, György Frank, Dániel Varjas, and András Pályi, ``Upper bound on the number of Weyl points born from a nongeneric degeneracy point'' Phys. Rev. B 110, 245124 (2024). https://doi.org/10.1103/PhysRevB.110.245124 [27] Gabriele Naselli, György Frank, Dániel Varjas, Ion Cosma Fulga, Gergo Pintér, András Pályi, and Viktor Könye, ``Stability of Weyl Node Merging Processes under Symmetry Constraints'' Phys. Rev. Lett. 133, 196602 (2024). https://doi.org/10.1103/PhysRevLett.133.196602 [28] Victor Guilleminand Alan Pollack ``Differential topology'' American Mathematical Soc. (2010). https://doi.org/10.1090/chel/370 [29] Martin Golubitskyand Victor Guillemin ``Stable mappings and their singularities'' Springer New York, NY (2012). https://doi.org/10.1007/978-1-4615-7904-5 [30] Zoltán Scherübl, András Pályi, György Frank, István Endre Lukács, Gergő Fülöp, Bálint Fülöp, Jesper Nygård, Kenji Watanabe, Takashi Taniguchi, Gergely Zaránd, and Szabolcs Csonka, ``Observation of spin–orbit coupling induced Weyl points in a two-electron double quantum dot'' Communications Physics 2, 108 (2019). https://doi.org/10.1038/s42005-019-0200-2 https://www.nature.com/articles/s42005-019-0200-2 [31] György Frank, Zoltán Scherübl, Szabolcs Csonka, Gergely Zaránd, and András Pályi, ``Magnetic degeneracy points in interacting two-spin systems: Geometrical patterns, topological charge distributions, and their stability'' Physical Review B 101, 245409 (2020). https://doi.org/10.1103/PhysRevB.101.245409 [32] Roman-Pascal Riwar, Manuel Houzet, Julia S. Meyer, and Yuli V. Nazarov, ``Multi-terminal Josephson junctions as topological matter'' Nature Communications 7, 11167 (2016). https://doi.org/10.1038/ncomms11167 [33] Valla Fatemi, Anton R. Akhmerov, and Landry Bretheau, ``Weyl Josephson circuits'' Phys. Rev. Research 3, 013288 (2021). https://doi.org/10.1103/PhysRevResearch.3.013288 [34] György Frank, Dániel Varjas, Péter Vrana, Gergő Pintér, and András Pályi, ``Topological charge distributions of an interacting two-spin system'' Physical Review B 105, 035414 (2022). https://doi.org/10.1103/PhysRevB.105.035414 [35] Israel Klich ``On the stability of topological phases on a lattice'' Annals of Physics 325, 2120–2131 (2010). https://doi.org/10.1016/j.aop.2010.05.002 [36] Sergey Bravyi, Matthew B. Hastings, and Spyridon Michalakis, ``Topological quantum order: Stability under local perturbations'' Journal of Mathematical Physics 51, 093512 (2010). https://doi.org/10.1063/1.3490195 [37] Sergey Bravyiand Matthew B. Hastings ``A Short Proof of Stability of Topological Order under Local Perturbations'' Communications in Mathematical Physics 307, 609–627 (2011). https://doi.org/10.1007/s00220-011-1346-2 [38] Ali Lavasani, Michael J. Gullans, Victor V. Albert, and Maissam Barkeshli, ``On stability of k-local quantum phases of matter'' arXiv:2405.19412 (2024). https://doi.org/10.48550/arXiv.2405.19412 https://arxiv.org/abs/2405.19412 [39] Yaodong Li, Nicholas O'Dea, and Vedika Khemani, ``Perturbative Stability and Error-Correction Thresholds of Quantum Codes'' PRX Quantum 6, 010327 (2025). https://doi.org/10.1103/PRXQuantum.6.010327 [40] Nikolas P.
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