Geometry of Free Fermion Commutants

Quantum Physics [Submitted on 6 Apr 2026] Geometry of Free Fermion Commutants Marco Lastres, Sanjay Moudgalya Understanding the structure of operators that commute with k identical replicas of unitary ensembles, also known as their k-commutants, is an important problem in quantum many-body physics with deep implications for the late-time behavior of physical quantities such as correlation functions and entanglement entropies under unitary evolution. In this work, we study the k-commutants of free-fermion unitary systems, which are heuristically known to contain SO(k) and SU(k) groups without and with particle number conservation respectively, with formal derivations of projectors onto these commutants appearing only very recently. We establish a complementary perspective by highlighting a larger O(2k) replica symmetry (or SU(2k) respectively) that the k-commutant transforms irreducibly under, which leads to a simple geometric understanding of the commutant in terms of coherent states parametrized by a Grassmannian manifold. We derive this structure by mapping the k-commutant to the ground state of effective ferromagnetic Heisenberg models, analogous to the ones that appear in the noisy circuit literature, which we solve exactly using standard representation theory methods. Further, we show that the Grassmannian manifold of the k-commutant is exactly the manifold of fermionic Gaussian states on 2k sites, which reveals a duality between real space and replica space in free-fermion systems. This geometric understanding also provides a compact projection formula onto the k-commutant, based on the resolution of identity for coherent states, which can prove advantageous in analytical calculations of averaged non-linear functionals of Gaussian states, as we demonstrate using some examples for the entanglement entropies. In all, this work provides a geometric perspective on the k-commutant of free-fermions that naturally connects to problems in quantum many-body physics. Comments: 13+13 pages Subjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph) Cite as: arXiv:2604.05031 [quant-ph]   (or arXiv:2604.05031v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.05031 Focus to learn more Submission history From: Marco Lastres [view email] [v1] Mon, 6 Apr 2026 18:00:03 UTC (85 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: cond-mat cond-mat.stat-mech cond-mat.str-el hep-th math math-ph math.MP References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)