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Scientia Sinica Mathematica

Operator geometry and an index formula for Floquet topological phases

Chen Zeqian, Sun Mu, Zhi Yin

Published Dec 21, 2022• 0 citations
Publisher

Abstract

For giving a geometric interpretation of the observable-geometric phase with application to quantum computation, operator geometry on a finite-dimensional complex Hilbert space was developed by Chen (2020), i.e., geometry over the operator-principal G-bundle of the unitary operator group based on the observable space. In this paper, we further develop operator geometry on an infinite-dimensional separable complex Hilbert space. To this end, we use the theory of unbounded operators for describing the differential structure of infinite-dimensional observable spaces and defining quantum connections and quantum parallel transportation. Then, we introduce the notion of the observable-geometric phase for infinite-dimensional quantum systems and give its geometric interpretation in terms of operator geometry. Finally, we present a topological index for Floquet topological phases and prove a local formula for it in terms of the observable-geometric phases, which are measurable in the physical sense.

DOI
10.1360/ssm-2022-0167

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For giving a geometric interpretation of the observable-geometric phase with application to quantum computation, operator geometry on a finite-dimensional complex Hilbert space was developed by Chen (2020), i.e., geometry over the operator-principal G-bundle of the unitary operator group based on the observable space. In this paper, we further develop operator geometry on an infinite-dimensional separable complex Hilbert space. To this end, we use the theory of unbounded operators for describing the differential structure of infinite-dimensional observable spaces and defining quantum connections and quantum parallel transportation. Then, we introduce the notion of the observable-geometric phase for infinite-dimensional quantum systems and give its geometric interpretation in terms of operator geometry. Finally, we present a topological index for Floquet topological phases and prove a local formula for it in terms of the observable-geometric phases, which are measurable in the physical sense.
This paper examines the geometric structure underlying quantum state space and operator representations. Building on the interpretation of quantum phase in terms of holonomy and parallel transport, Hilbert space is analyzed as a representation of transport structure in complex line and vector bundles. Quantum states are interpreted as equivalence classes under phase transformations, and operators arise as generators of infinitesimal transformations associated with the connection. The construction provides a geometric perspective on state space, inner products, and operator structure within standard quantum mechanics.
Quantum mechanics is traditionally formulated on complex Hilbert spaces, whereas gauge theories are formulated on fiber bundles equipped with connections and curvature. Despite the central role of geometry in modern physics, the Hilbert-space structure of quantum theory is usually postulated rather than derived. In this work, we investigate a differential-geometric formulation in which quantum states are represented as points on a coherence manifold characterized by a radial coherence coordinate and a holonomy-induced phase coordinate. The resulting geometry naturally admits the structure of a fiber bundle whose base space is associated with a coherence lattice and whose fiber corresponds to a phase circle. Within this framework, Hilbert spaces emerge as function spaces defined on the coherence manifold, while Schrödinger dynamics arise from the Laplace–Beltrami operator associated with the underlying geometry. The construction unifies concepts from geometric quantum mechanics, fiber-bundle gauge theory, holonomy, and coherence geometry within a common differential-geometric framework.
Quantum mechanics is traditionally formulated in terms of complex Hilbert spaces, wavefunctions, and linear evolution equations. While extraordinarily successful, this framework leaves open several conceptual questions concerning the geometric origin of the wavefunction, the meaning of quantum phase, and the status of Hilbert space as a fundamental structure. In this work, we investigate a coherence-geometric reformulation in which the fundamental object is a coherence coordinate , where represents radial coherence and represents phase coherence associated with geometric transport. We show that the standard structures of quantum mechanics admit a natural interpretation within a differential-geometric framework. The wavefunction emerges from coherence coordinates, probability density is generated by radial coherence, probability flow is generated by phase coherence, and Hilbert spaces appear as spaces of square-integrable functions defined on a coherence manifold. The resulting formulation provides a unified geometric language connecting quantum mechanics, geometric phases, fiber bundles, holonomies, gauge theory, and differential geometry
The Hilbert-Pólya conjecture asserts that the imaginary parts of the nontrivial zeros of the Riemann zeta function (the Riemann zeros) are the eigenvalues of a self-adjoint operator (a quantum mechanical Hamiltonian, in the physical sense), as a promising approach to prove the Riemann hypothesis (cf.\cite{SH2011}). Instead of the eigenvalues, in this paper we consider observable-geometric phases as the realization of the Riemann zeros in a periodically driven quantum system, which were introduced in \cite{Chen2020} for the study of geometric quantum computation. To this end, we further introduce the notion of non-Abelian observable-geometric phases, involving which we give an approach to finding a physical system to study the Riemann zeros. Since the observable-geometric phases are connected with the geometry of the observable space according to the evolution of the Heisenberg equation, this sheds some light on the investigation of the Riemann hypothesis.
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.
Abstract We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional, i.e. we do not restrict considerations to finite-dimensional Hilbert spaces, contrary to many other works on the geometry of quantum mechanics, and include a Lagrangian formalism in which self-adjoint (Schrödinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we also obtain results concerning coadjoint orbits of the unitary group in infinite dimensions, embedding of pure states in the unitary group, and self-adjoint extensions of symmetric relations.