Geometry of Free Fermion Commutants

Abstract

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.