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The algebra of the spinor differential operator in Berry geometry

Miftachul Hadi

Published Aug 6, 2026• 0 citations
Publisher

Abstract

We study the algebraic structure of a spinor differential operator associated with the Berry connection, showing that it is a rank-one operator with a simple multiplication law. We establish its relation to the Berry projector through its anti-Hermitian part. These results provide an operator framework for Berry geometry and a basis for future extensions.

DOI
10.5281/zenodo.21826533

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We study the algebraic structure of a spinor differential operator associated with the Berry connection, showing that it is a rank-one operator with a simple multiplication law. We establish its relation to the Berry projector through its anti-Hermitian part. These results provide an operator framework for Berry geometry and a basis for future extensions.
An isolated group of kkk bands in an NNN-level quantum system defines a rank-kkk spectral projector and hence a map into the complex Grassmannian Gr(k,N)\mathrm{Gr}(k,N)Gr(k,N). For a smooth gapped Hamiltonian, this projector is globally smooth and periodic even when band topology obstructs a globally smooth periodic, or symmetry-compatible, Bloch frame. We take the globally defined differential $\dd P$ as the central object: it is the tangent field of the Grassmannian map, removes unphysical rotations within the selected subspace, and retains the physical interband transition. The interband block of this tangent data simultaneously determines the quantum metric and Berry curvature; the associated horizontal generator produces finite Grassmannian motion, while wedge products of $\dd P$ enter topological forms. We construct a shortest path between two projectors in the ambient Grassmannian and show that the singular values of its horizontal generator block are the principal angles between the endpoint subspaces. This construction provides a piecewise-geodesic interpretation of discrete geometric phases. In a separate development, we derive a basis-independent expression for the determinant of a multiband Wilson loop from traces of powers of an ordered projector product, without decomposing a degenerate band multiplet into individual bands. Finally, the same tangent-vector calculus organizes Chern characters, chiral winding numbers, and the time-reversal Z2\mathbb Z_2Z2​ index. The resulting framework unifies local quantum geometry, finite subspace distance, holonomy, and topology while avoiding global gauge fixing.
We introduce a spinor differential operator whose expectation value reproduces the Berry connection exactly. We derive its transformation under local U(1)U(1)U(1) gauge transformations and show that the conventional Berry gauge transformation follows directly through projection onto the normalized spinor. The proposed formulation provides an operator-level formulation of Berry geometry whose projection reproduces the conventional Berry connection and its gauge transformation.
In this paper we discuss a simplified approach to the symplectic Clifford algebra, the symplectic Clifford group and the symplectic spinor by first extending the Heisenberg algebra. We do this by adding a new idempotent element to the algebra. It turns out that this element is the projection operator onto the Dirac standard ket. When this algebra is transformed into the Fock algebra, the corresponding idempotent is the projector onto the vacuum. These additional elements give a very simple way to write down expressions for symplectic spinors as elements of the symplectic Clifford algebra. When this algebra structure is applied to an optical system we find a new way to understand the Gouy effect. It is seen to arise from the covering group, which in this case is the symplectic Clifford group, also known as the metaplectic group. We relate our approach to that of Simon and Mukunda who have shown the relation of the Gouy phase to the Berry phase.
The Hamiltonian of an so(5) algebraic structure is constructed by extending an so(3) system to an so(5) dynamical system. We exactly solve the Hamiltonian by using the coherent state operator of the so(5) algebra. It is shown that the eigenstates corresponding to the same eigenvalue are multi-fold degenerate. The Berry connection associated with the so(5) coherent states is calculated and is found to be decomposed into Abelian and non-Abelian Berry connections. Based on the above theory, the Berry connections of the p-wave superconductivity model are determined explicitly.