# Quantum phase transitions driven by rhombic-type single-ion anisotropy in the S=1 Haldane chain

## TL;DR

This paper maps the phase diagram of the spin-1 Haldane chain with rhombic and uniaxial single-ion anisotropies, revealing topological phase transitions and critical points with high precision, and suggests experimental realization in specific compounds.

## Contribution

It provides the first precise ground-state phase diagram including the effects of rhombic anisotropy and introduces a new parity technique for level spectroscopy in DMRG.

## Key findings

- Topological quantum phase transitions identified between Haldane and large anisotropy phases.
- Critical point for Haldane-Large-D transition determined with unprecedented accuracy.
- Small rhombic anisotropy can induce a transition to the y-Néel phase.

## Abstract

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, $D(S^z)^2$, and the rhombic-type, $E[(S^x)^2-(S^y)^2]$, single-ion anisotropies. Here, we provide a precise ground-state phase diagram for spin-1 Haldane chain with these single-ion anisotropies. Using quantum numbers, we find that the $\mathbb{Z}_2$ 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-$E_x$, Large-$E_y$, 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 unprecedentedly precision, $(D/J)_c=0.9684713(1)$. Close to this critical point, a small rhombic single-ion anisotropy $|E|/J\ll1$ can destroy the Haldane phase and bring the system into a $y$-N\'eel phase. We propose that the compound [Ni(HF$_2$)(3-Clpy)$_4$]BF$_4$ is a candidate system to search for the $y$-N\'eel phase.