Characterization of a Topological Mott Insulator in One Dimension

We investigate properties of a topological Mott insulator in one dimension by examining the bulk topological invariant and the entanglement spectrum of a correlated electron model. We clarify how gapless edge states in a noninteracting topological band insulator evolve into spinon edge states in a topological Mott insulator. Furthermore, we propose a topological Mott transition, which is a new type of topological phase transition which has never been observed in free fermion systems. This unconventional transition occurs in spin liquid phases in the Mott insulator and is accompanied by zeros of the single-electron Green’s function and a gap closing in the spin excitation spectrum.

Figure 1

Entanglement spectrum for $(J,V)=(0,-0.4)$ and $L=30$. (a) Lowest five entanglement energy levels $E_{\alpha \beta }=-2\log ({\lambda }_{\alpha }{\lambda }_{\beta })$ as a function of interaction strength $U$. (b) Highest 19 Schmidt eigenvalues for $U=0$ and $U=0.5$, which connect entanglement energy with the relation $E_{\alpha \beta }=-2\log ({\lambda }_{\alpha }{\lambda }_{\beta })$.

Figure 2

Results obtained for $(J,V)=(0,-0.4)$. (a) Winding number for $L=11$, where $L$ denotes chain length, (b) double occupancy of orbital $a$ at the edge site for several values of chain length, and (c) [(d)] single-electron [spin] excitation gap under the open boundary condition.

Figure 3

Winding number for $(J,V)=(-1.5,-1.6)$ and $L=11$. Left: Winding number as a function of interaction strength. Right: Locus of $g_{ab}(k)=G_{ab\sigma }(i\omega =0,k)$ for several values of interaction strength. In the nontrivial phase, the Green’s function $g_{ab}$ is positive and real at $k=k_{\min }$. As the momentum $k$ increased, the $g_{ab}(k)$ draws its locus clockwise. In the trivial phase, at $k=k_{\min }$, $g_{ab}$ is positive and real and draws its locus clockwise.

Figure 4

Left: Lowest five ES for each value of interaction strength $U$ under periodic boundary conditions at $(J,V)=(-1.5,-1.6)$. Right: Spectral gaps as functions of interaction strength $U$ for the same parameter set; single-electron excitation (spin excitation) is denoted as ${\mathrm{\Delta }}_c$ (${\mathrm{\Delta }}_s$), respectively. These values are extrapolated to the thermodynamic limit with scaling $L^{-1}$.