Composite Boson Theory of Fractional Chern Insulators

Abstract

The understanding of fractional Chern insulators (FCIs) has been deeply guided by band topology and quantum geometry. Here, we introduce a real-space theoretical framework in which FCIs are understood in terms of composite bosons, local objects consisting of electrons bound to their energetically excluded surrounding orbitals. The central element of our framework is the construction of a radially ordered set of maximally localized basis for Chern bands without requiring continuous rotational symmetry. Within this basis, the complex many-body problem simplifies to a real-space organizing principle: a stable FCI occurs if the orbitals excluded around central electrons are those maximizing the two-body interaction energy. We validate this with direct numerical evidence for composite boson formation in the Haldane model, demonstrating that our criterion reliably characterizes FCIs. Importantly, our analysis illustrates that the composite boson framework bridges the fractional quantum Hall effect in continuum and lattice paradigms, providing a unified and intuitive real-space interpretation for distinct correlated phases. It thus establishes a foundation for diagnosing and guiding the design of both Abelian and non-Abelian topologically ordered phases across distinct platforms.

Figures (2)

• Figure 1: Conceptual workflow of the composite boson theory for fractional Chern insulators (FCIs). The diagram outlines the key steps of the theoretical framework: (i) Construction of a radially ordered set of maximally localized orbital basis within a topological flat band. (ii) Calculation and comparison of the two-body interaction energy between an electron in the central orbital ($m=0$) and a test electron in the $m$-th orbital. A stable composite boson is formed if the orbitals it excludes (e.g., $m=1, 2$ for Laughlin state) are those maximizing the two-body interaction energy. (iii) Condensation of these stable composite bosons gives rise to the FCI phase. This workflow establishes a universal real-space organizing principle for understanding and predicting FCIs.
• Figure 2: Numerical implementation and evidence for the real-space composite-boson framework. (a) Lattice geometry and parameters of the Haldane model. (b) The entanglement spectrum of the many-body ground state at filling $\nu=1/3$, exhibiting the characteristic low-lying level counting $1, 1, 2, 3, 5, ...$, a hallmark of the underlying topological order and strong evidence for the FCI phase. (c) Schematic of the matrix product state (MPS) compression scheme used to measure the non-local occupation correlation $\langle \Psi|P_n P_0|\Psi\rangle$. (d) Measured occupation correlation $\langle \Psi|P_n P_0|\Psi\rangle$, showing a pronounced suppression of electron occupation for the first two orbitals ($n=1,2$) adjacent to an occupied central orbital. This pattern provides direct numerical evidence for real-space composite-boson formation, analogous to flux attachment in the fractional quantum Hall effect. (e) The first few radially ordered, maximally localized orbitals, labeled by the orbital index $m = 0, 1, 2, \ldots$, near the chosen origin.