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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\n \n k<\/mml:mi>\n <\/mml:math>\n -fold degeneracy submanifold. Inspired by this result, we establish a `distance theorem': we show that the standard deviation of\n \n k<\/mml:mi>\n <\/mml:math>\n neighboring eigenvalues of a Hamiltonian equals the distance of this Hamiltonian from the corresponding\n \n k<\/mml:mi>\n <\/mml:math>\n -fold degeneracy submanifold, divided by\n \n \n k<\/mml:mi>\n <\/mml:msqrt>\n <\/mml:math>\n . 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.\n <\/jats:p>","DOI":"10.22331\/q-2026-03-27-2047","type":"journal-article","created":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T11:52:44Z","timestamp":1774612364000},"page":"2047","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["The geometry of the Hermitian matrix space and the Schrieffer\u2013Wolff transformation"],"prefix":"10.22331","volume":"10","author":[{"given":"Gergo","family":"Pinter","sequence":"first","affiliation":[{"name":"Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, Muegyetem rkp. 3., H-1111 Budapest, Hungary"},{"name":"HUN-REN-BME-BCE Quantum Technology Research Group, M\u0171egyetem rkp. 3., H-1111 Budapest, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gyorgy","family":"Frank","sequence":"additional","affiliation":[{"name":"Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, Muegyetem rkp. 3., H-1111 Budapest, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniel","family":"Varjas","sequence":"additional","affiliation":[{"name":"Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, Muegyetem rkp. 3., H-1111 Budapest, Hungary"},{"name":"IFW Dresden and W\u00fcrzburg-Dresden Cluster of Excellence ct.qmat, Helmholtzstrasse 20, 01069 Dresden, Germany"},{"name":"Max Planck Institute for the Physics of Complex Systems, N\u00f6thnitzer Strasse 38, 01187 Dresden, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andras","family":"Palyi","sequence":"additional","affiliation":[{"name":"Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, Muegyetem rkp. 3., H-1111 Budapest, Hungary"},{"name":"HUN-REN-BME-BCE Quantum Technology Research Group, M\u0171egyetem rkp. 3., H-1111 Budapest, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2026,3,27]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"J. 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