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.
We discuss the existence of a nontrivial topological phase in one-dimensional interacting systems described by the extended Bose-Hubbard model with a mean filling of one boson per site. Performing large-scale density-matrix renormalization group calculations we show that the presence of nearest-neighbor repulsion enriches the ground-state phase diagram of the paradigmatic Bose-Hubbard model by stabilizing a novel gapped insulating state, the so-called Haldane insulator, which, embedded into superfluid, Mott insulator, and density wave phases, is protected by the lattice inversion symmetry. The quantum phase transitions between the different insulating phases were determined from the central charge via the von Neumann entropy. The Haldane phase reveals a characteristic fourfold degeneracy of the entanglement spectrum. We finally demonstrate that the intensity maximum of the dynamical charge structure factor, accessible by Bragg spectroscopy, features the gapped dispersion known from the spin-1 Heisenberg chain.
Exploiting the entanglement concept within a matrix-product-state based infinite density-matrix renormalization group approach, we show that the spin-density-wave and bond-order-wave ground states of the one-dimensional half-filled extended Hubbard model give way to a symmetry-protected topological Haldane state in case an additional alternating ferromagnetic spin interaction is added. In the Haldane insulator, the lowest entanglement level features a characteristic twofold degeneracy. Increasing the ratio between the nearest-neighbor and local Coulomb interaction
V/U, the enhancement of the entanglement entropy, the variation of the charge, spin, and neutral gaps, and the dynamical spin and density response signal a quantum phase transition to a charge-ordered state. Below a critical point, which belongs to the universality class of the tricritical Ising model with central charge 7/10, the model is critical with
c=1/2 along the transition line. Above this point, the transition between the Haldane insulator and charge-density-wave phases becomes first order.
According to the classification using projective representations of the SO(3) group, there exist two topologically distinct gapped phases in spin-1 chains. The symmetry-protected topological (SPT) phase possesses half-integer projective representations of the SO(3) group, while the trivial phase possesses integer linear representations. In the present work, we implement non-Abelian symmetries in the density matrix renormalization group (DMRG) method, allowing us to keep track of (and also control) the virtual bond representations and to readily distinguish the SPT phase from the trivial one by evaluating the multiplet entanglement spectrum. In particular, using the entropies
SI (
SH) of integer (half-integer) representations, we can define an entanglement gap $G={S}^{I}\ensuremath{-}{S}^{H}$, which equals 1 in the SPT phase, and $\ensuremath{-}1$ in the trivial phase. As an application of our proposal, we study the spin-1 models on various one-dimensional (1D) and quasi-1D lattices, including the bilinear-biquadratic model on a single chain, and the Heisenberg model on a two-leg ladder and a three-leg tube. Among others, we confirm the existence of a SPT phase in the spin-1 tube model and reveal that the phase transition between the SPT and the trivial phase is a continuous one. The transition point is found to be critical, with conformal central charge
c=3 determined by fits to the block entanglement entropy.
We study the bond-alternating Heisenberg model using the finite-size density-matrix renormalization-group (DMRG) technique and analytical arguments based on the matrix product state, where we pay particular attention to the boundary-condition dependence on the entanglement spectrum of the system. We show that, in the antiperiodic boundary condition (APBC), the parity quantum numbers are equivalent to the topological invariants characterizing the topological phases protected by the bond-centered inversion and $\ensuremath{\pi}$ rotation about the
z axis. We also show that the odd parity in the APBC, which characterizes topologically nontrivial phases, can be extracted as a twofold degeneracy in the entanglement spectrum even with finite system size. We then determine the phase diagram of the model with the uniaxial single-ion anisotropy using the level spectroscopy method in the DMRG technique. These results not only suggest the detectability of the symmetry protected topological (SPT) phases via general twisted boundary conditions but also provide a useful and precise numerical tool for discussing the SPT phases in the exact diagonalization and DMRG techniques.
The spin-1 Haldane chain is an example of the symmetry-protected-topological (SPT) phase in one dimension. Experimental realization of the spin chain materials usually involves both the uniaxial-type,
D(Sz)2, and the rhombic-type, $E[{({S}^{x})}^{2}\ensuremath{-}{({S}^{y})}^{2}]$, single-ion anisotropies. Here, we provide a precise ground-state phase diagram for a spin-1 Haldane chain with these single-ion anisotropies. Using quantum numbers, we find that the
Z2 symmetry breaking phase can be characterized by double degeneracy in the entanglement spectrum. Topological quantum phase transitions take place on particular paths in the phase diagram, from the Haldane phase to the large-
Ex, large-
Ey, or large-
D phases. The topological critical points are determined by the level spectroscopy method with a newly developed parity technique in the density matrix renormalization group [Phys. Rev. B 86, 024403 (2012)], and the Haldane-large-
D critical point is obtained with an unprecedented precision,
(D/J)c=0.9684713(1). Close to this critical point, a small rhombic single-ion anisotropy $|E|/J\ensuremath{\ll}1$ can destroy the Haldane phase and bring the system into a
y-N\'eel phase. We propose that the compound $[\mathrm{Ni}({\mathrm{HF}}_{2}){(3\text{\ensuremath{-}}\mathrm{Clpy})}_{4}]{\mathrm{BF}}_{4}$ is a candidate system to search for the
y-N\'eel phase.