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Topological two-body interaction obstructing trivial ground states: an indicator of fractional Chern insulators

Nobuyuki Okuma, Tomonari Mizoguchi

TL;DR

The work investigates how the topology of two-body interactions, as encoded by the common matrix $\\hat{S}_{\\bm{q}}$, can influence the stabilization of fractional Chern insulators (FCIs). By formulating both the effective interaction projected onto a Chern band and the two-particle problem with the same $\\hat{S}_{\\bm{q}}$ and examining the small Hamiltonian $h_{\\bm{q}} = \\hat{S}_{\\bm{q}}\\hat{S}^{\\dagger}_{\\bm{q}}$, the authors connect the two-particle Chern-band structure to FCI stability, showing that bound-state bands with unit Chern number correlate with robust FCIs at $\\nu = 1/3$ in a checkerboard lattice. Numerical results reveal that the top two two-particle bands carry unit Chern number and that increasing band hybridization or altering interaction ranges degrades the many-body gap, while limiting the two-particle orbital set can preserve a topological bound-state. The paper argues that topology intrinsic to the interaction can obstruct trivial ground states and provides various basis pictures—cylindrical, radially localized, and biorthogonal—that illuminate how topological two-body interactions act as a diagnostic and potential engine for exotic states beyond FCIs.

Abstract

The search for candidate materials for fractional Chern insulators (FCIs) has mainly focused on the topological and geometrical structures of single-particle Chern bands. However, there are inherent limitations in approaches that neglect interaction effects, highlighting the need for complementary methods. In this work, we discuss how the Chern number defined for the effective interaction projected onto a Chern band is related to the stabilization of FCIs. Specifically, by formulating both the effective interaction and the two-particle problem using a common matrix, we establish a connection between the two-particle band structure and the effective interaction. This formulation allows us to characterize the effective interaction through the topology of the two-particle band. To investigate the relationship between topological effective interactions and FCIs, we perform numerical calculations primarily based on exact diagonalization. We find a notable correlation between the fact that the dominant two-particle bands carry a unit Chern number and the realization of a robust FCI at the filling fraction $ν= 1/3$. This result is consistent with the presumed correspondence between pseudopotentials in the fractional quantum Hall effect and the two-particle band structure. From another perspective, our findings suggest that the topology inherent in the interaction itself can obstruct trivial ground states. We also discuss this in the context of scattering channels. Extending such topological two-body interactions could pave the way for realizing exotic states beyond FCIs.

Topological two-body interaction obstructing trivial ground states: an indicator of fractional Chern insulators

TL;DR

The work investigates how the topology of two-body interactions, as encoded by the common matrix , can influence the stabilization of fractional Chern insulators (FCIs). By formulating both the effective interaction projected onto a Chern band and the two-particle problem with the same and examining the small Hamiltonian , the authors connect the two-particle Chern-band structure to FCI stability, showing that bound-state bands with unit Chern number correlate with robust FCIs at in a checkerboard lattice. Numerical results reveal that the top two two-particle bands carry unit Chern number and that increasing band hybridization or altering interaction ranges degrades the many-body gap, while limiting the two-particle orbital set can preserve a topological bound-state. The paper argues that topology intrinsic to the interaction can obstruct trivial ground states and provides various basis pictures—cylindrical, radially localized, and biorthogonal—that illuminate how topological two-body interactions act as a diagnostic and potential engine for exotic states beyond FCIs.

Abstract

The search for candidate materials for fractional Chern insulators (FCIs) has mainly focused on the topological and geometrical structures of single-particle Chern bands. However, there are inherent limitations in approaches that neglect interaction effects, highlighting the need for complementary methods. In this work, we discuss how the Chern number defined for the effective interaction projected onto a Chern band is related to the stabilization of FCIs. Specifically, by formulating both the effective interaction and the two-particle problem using a common matrix, we establish a connection between the two-particle band structure and the effective interaction. This formulation allows us to characterize the effective interaction through the topology of the two-particle band. To investigate the relationship between topological effective interactions and FCIs, we perform numerical calculations primarily based on exact diagonalization. We find a notable correlation between the fact that the dominant two-particle bands carry a unit Chern number and the realization of a robust FCI at the filling fraction . This result is consistent with the presumed correspondence between pseudopotentials in the fractional quantum Hall effect and the two-particle band structure. From another perspective, our findings suggest that the topology inherent in the interaction itself can obstruct trivial ground states. We also discuss this in the context of scattering channels. Extending such topological two-body interactions could pave the way for realizing exotic states beyond FCIs.

Paper Structure

This paper contains 18 sections, 62 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Convention and notation
  3. Non-interacting part
  4. Two-body interaction under projection
  5. Two-particle orbitals and symmetrized Hamiltonian
  6. Two-particle problem
  7. Large and small Hamiltonian
  8. Spectral and singular value decompositions
  9. Real-space picture
  10. Pseudopotential versus Two-particle topological bands
  11. Numerical calculations
  12. Two-body topological obstruction to trivial ground state
  13. Cylindrical basis
  14. Radially-localized basis
  15. Biorthogonal basis
  16. ...and 3 more sections

Figures (4)

  • Figure 1: Real-space pictures of (a) two-particle hopping and (b) fermionic exchange process.
  • Figure 2: Two-particle problem in the checkerboard lattice model. (a) Unit-cell structure. Red arrows represent two-particle orbitals. (b) Band structure of the bound states. (c) Berry curvature of the top two bands.
  • Figure 3: Exact Diagonalizations of the checkerboard lattice model under parameter variation. The system size is $L_1\times L_2=6\times4$. Total momentum sectors are labeled by $(n_1,n_2)$. (a) Hopping dependence of two- and eight-particle spectra. (b) Interaction dependence of two- and eight-particle spectra.
  • Figure 4: Schematic pictures of scattering basis in effective interactions. Red and blue arrows represent incoming and outgoing particles, respectively. (a) Wannier basis. Under the topological obstruction, the scattering channels become widely distributed. (b) Cylindrical basis. On this basis, the scattering can be interpreted as occurring in a one-dimensional manner. Under the flux insertion, the interaction itself is topologically pumped. (c) Radially-localized basis. This basis provides an analogy to the analytical structure found in Landau levels. (d) Biorthogonal basis. On this basis, two particles scatter into localized channels and are emitted from delocalized channels.