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arXiv:1612.00605 · Physical Review Letters

Anyonic Haldane Insulator in One Dimension

Florian Lange, Satoshi Ejima, Holger Fehske

Published Mar 23, 2017• 26 citations
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Abstract

We demonstrate numerically the existence of a nontrivial topological Haldane phase for the one-dimensional extended (U-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-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.

DOI
10.1103/physrevlett.118.120401

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We demonstrate numerically the existence of a nontrivial topological Haldane phase for the one-dimensional extended (U-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-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.
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We study one-dimensional lattice anyons with extended Hubbard interactions at unit filling using bosonization and numerical simulations. The behavior can be continuously tuned from bosonic to fermionic statistics by adjusting the topological exchange angle θ, which leads to a competition of different instabilities. We present the bosonization theory in the presence of dynamic gauge fields, which predicts a phase diagram of four different gapped phases with distinct dominant correlations. Advanced numerical simulations determine and analyze the exact phase transitions between Mott insulator, charge density wave, dimerized state, and Haldane insulator, all of which meet at a multicritical line in the parameter space of anyonic angle θ, on-site interaction U, and nearest neighbor repulsion V. Superfluid and pair-superfluid phases are stable in a region of small V.
We investigate the entanglement properties of a nontrivial topological phase in the one-dimensional (1D) Bose-Hubbard model with additional nearest-neighbor repulsion. Employing the large-scale density-matrix renormalization group technique we show that a gapped insulating phase protected by lattice inversion symmetry, the so-called Haldane insulator, appears between the Mott and density wave phases in the intermediate-coupling regime. The phase boundaries were determined from the central charge via the von Neumann entropy. The Haldane insulator reveals a characteristic degeneracy in the entanglement spectra. Breaking the lattice inversion symmetry strongly affects the distinctive gapped dispersion of the dynamical charge response of the bosonic Haldane insulator.
By performing large-scale density-matrix renormalization group simulations, we investigate a one-dimensional correlated bosonic lattice model with a synthetic spin-orbit coupling realized in recent experiments. In the insulating regime, this model exhibits a symmetry-protected topological phase. This symmetry-protected topological phase is stabilized by time-reversal symmetry and it is identified as a Haldane phase. We confirm our conclusions further by analyzing the entanglement spectrum. In addition, we find four conventional phases: a Mott insulating phase with no long range order, a ferromagnetic superfluid phase, a ferromagnetic insulating phase, and a density-wave phase.
We carry out an extensive study of the phase diagram of the extended Bose Hubbard model, with a mean filling of one boson per site, in one dimension by using the density matrix renormalization group and show that it can have Superfluid (SF), Mott-insulator (MI), density-wave (DW) and Haldane-insulator (HI) phases depending on the precise value of filling and how edge states are handled. We show that the critical exponents and central charge for the HI-DW, MI-HI and SF-MI transitions are consistent with those for models in the two-dimensional Ising, Gaussian, and Berezinskii-Kosterlitz-Thouless (BKT) universality classes, respectively; and we suggest that the SF-HI transition may be more exotic than a simple BKT transition.
We investigate the ground-state phase diagram of a modified spinless Haldane-Hubbard model, with broken threefold rotational symmetry, employing exact diagonalization calculations. The interplay of asymmetry, interactions, and topology gives rise to a rich phase diagram. The noninteracting limit of the Hamiltonian exhibits a higher-order topological insulator characterized by the existence of corner modes, in contrast to known chiral edge metallic states of the standard Haldane model. Our investigation demonstrates that these symmetry-protected states are robust to the presence of finite interactions. Furthermore, in certain regimes of parameters, we show that a topological Mott insulator exists in this model, where a nontrivial topological bulk coexists with an interaction-driven charge-density wave, whose emergence is characterized by a Z2{Z}_{2}Z2​-symmetry breaking within the 3d-Ising3d\text{-Ising}3d-Ising universality class.