Anyonic Haldane Insulator in One Dimension

We demonstrate numerically the existence of a nontrivial topological Haldane phase for the one-dimensional extended ($U\text{−}V$) Hubbard model with a mean density of one particle per site, not only for bosons but also for anyons, despite a broken reflection parity symmetry. The Haldane insulator, surrounded by superfluid, Mott insulator, and density-wave phases in the $V\text{−}U$ parameter plane, is protected by combined (modified) spatial-inversion and time-reversal symmetries, which is verified within our matrix-product-state based infinite density-matrix renormalization group scheme by analyzing generalized transfer matrices. With regard to an experimental verification of the anyonic Haldane insulator state the calculated asymmetry of the dynamical density structure factor should be of particular importance.

Figure 1

Order parameter $O$, defined by Eq. (15), selecting the topological state in the EAHM at fixed $U/t=5$ and $\theta =\pi /4$ for different $n_p$ (a), and at fixed $\theta =\pi /2$ and $\pi$ for $n_p=2$ (b). Data obtained by iDMRG calculations with a (relatively small) bond dimension $\chi =100$.

Figure 2

Ground-state phase diagram of the extended anyon-Hubbard model in one dimension, where the particle density $\rho =1$, $n_p=2$, and $\theta =\pi /4$. Most notably the Haldane insulator (HI), located between Mott insulator (MI) and density wave (DW) insulating phases in the EBHM, survives for any $\theta >0$, i.e., in the anyonic case. Likewise, the superfluid (SF) appears in the very weak-coupling regime. The MI-HI (squares) and HI-DW (circles) transition points can be determined by a divergent correlation length ${\xi }_{\chi }$ as $\chi$ increases; i.e., the model becomes critical with the central charge $c=1$ and $c=1/2$, respectively (see Ref. [27]). For comparison, the dotted (dashed) line marks the MI-HI (HI-DW) transition in the EBHM ($\theta =0$) [34]. The dash-dotted line with triangles up denotes the first-order MI-DW phase transition for $\theta =\pi$.

Figure 3

Intensity plots of the dynamical structure factor $S(k,\omega )$ in the EBHM ($\theta =0$, upper panels) and in the EAHM ($\theta =\pi /4$, lower panels) for characteristic values of $V/t$ at fixed $U/t=5$. Again, the maximum number of particles per site is limited to $n_p=2$. Dashed lines in panels (b)–(d) and (g)–(i) mark the highest intensity of $S(k,\omega )$ in the $k\text{−}\omega$ plane.