Comparative density-matrix renormalization group study of symmetry-protected topological phases in spin-1 chain and Bose-Hubbard models

TL;DR

This study uses advanced DMRG techniques to map out the phase diagram of a spin-1 chain, revealing the nature of the topological Haldane phase and its relation to the Bose-Hubbard model, with implications for experimental detection.

Contribution

It provides a detailed density-matrix renormalization group analysis of symmetry-protected topological phases in spin-1 and Bose-Hubbard models, highlighting their similarities and differences.

Findings

Identification of the Haldane phase between trivial phases.

Degeneracy of the lowest entanglement level in the Haldane phase.

Distinct dynamical spin structure factors in different phases.

Abstract

We reexamine the one-dimensional spin-1 $XXZ$ model with on-site uniaxial single-ion anisotropy as to the appearance and characterization of the symmetry-protected topological Haldane phase. By means of large-scale density-matrix renormalization group (DMRG) calculations the central charge can be determined numerically via the von Neumann entropy, from which the ground-sate phase diagram of the model can be derived with high precision. The nontrivial gapped Haldane phase shows up in between the trivial gapped even Haldane and N{\'e}el phases, appearing at large single-ion and spin--exchange interaction anisotropies, respectively. We furthermore carve out a characteristic degeneracy of the lowest entanglement level in the topological Haldane phase, which is determined using a conventional finite-system DMRG technique with both periodic and open boundary conditions. Defining the spin and neutral gaps in analogy to the single-particle and neutral gaps in the intimately connected extended Bose-Hubbard model, we show that the excitation gaps in the spin model qualitatively behave just as for the bosonic system. We finally compute the dynamical spin structure factor in the three different gapped phases and find significant differences in the intensity maximum which might be used to distinguish these phases experimentally.