Many-body Euler topology

Abstract

Integer and fractional Chern insulators exhibit a nonzero quantized anomalous Hall conductivity due to a spontaneous breaking of time reversal symmetry. To identify nontrivial topology in their time-reversal symmetric many-body spectra, we introduce many-body Euler numbers as a counterpart to many-body Chern numbers. Exemplarily, we perform calculations in a topological Hubbard model that can realize Chern and fractional Chern insulating phases. Furthermore, we lay out a classification scheme to realize different topological phases in interacting systems using symmetry indicators in analogy to topological band theory.

Figures (2)

• Figure 1: Left column: Low-energy spectrum of Eq. \ref{['Eq:projected_Hamiltonian']} with $t = 1, t' = 1/\sqrt{2}, \gamma = 0.1, U = 1$ (top) and for $t = 1, t' = 1/\sqrt{2}, \gamma = 0, U = 1$ with an onsite potential $\Delta = 2.5$ (bottom, see Eq. \ref{['Eq:onsite']}). For clarity, energies have been shifted so that the two (quasi)degenerate ground states (shown in blue) are centered around 0. In the plot with $\gamma = 0.1$, finite-size splittings between the ground states are smaller than the resolution of the figure. An energy gap protects the ground states from the lowest excitations (shown in red) for all $\boldsymbol{\theta}$. Darker shades indicate lower energy values. Right column: Wilson loop flows of the two ground states, computed using the same parameters as in the corresponding energy spectra shown in the left column.
• Figure 2: Left image: Low-energy spectrum of Eq. \ref{['Eq:fractional_projected_hamiltonian']} exhibiting spectral flow with $t = 1, t' = 1/\sqrt{2}, \gamma = 0.1, U = 1, V = 0.1$. The finite-size splitting of the six (quasi)degenerate ground states (shown in blue) is smaller than the linewidth. The excitation spectrum (shown in red) is gapped from the ground state spectrum for all $\boldsymbol{\theta}$. Right image: Wilson loop flow for the ground states in the fractional Chern insulating phase.