All spin-1 topological phases in a single spin-2 chain

TL;DR

This paper demonstrates that a spin-2 quantum chain can host multiple distinct spin-1 symmetry-protected topological phases, including the Haldane phase and an $SO(5)$ phase, with continuous transitions described by conformal field theory.

Contribution

It reveals the coexistence of various spin-1 SPT phases within a single spin-2 chain and characterizes their continuous phase transitions using advanced numerical and theoretical methods.

Findings

Identification of four different spin-1 SPT phases in a spin-2 chain.

Observation of continuous phase transitions between these SPT phases.

Evidence for a $c=2$ conformal field theory at the transition points.

Abstract

Here we study the emergence of different Symmetry-Protected Topological (SPT) phases in a spin-2 quantum chain. We consider a Heisenberg-like model with bilinear, biquadratic, bicubic, and biquartic nearest-neighbor interactions, as well as uniaxial anisotropy. We show that this model contains four different effective spin-1 SPT phases, corresponding to different representations of the $(\mathbb{Z}_2 \times \mathbb{Z}_2) + T$ symmetry group, where $\mathbb{Z}_2$ is some $\pi$-rotation in the spin internal space and $T$ is time-reversal. One of these phases is equivalent to the usual spin-1 Haldane phase, while the other three are different but also typical of spin-1 systems. The model also exhibits an $SO(5)$-Haldane phase. Moreover, we also find that the transitions between the different effective spin-1 SPT phases are continuous, and can be described by a $c=2$ conformal field theory. At such transitions, indirect evidence suggests a possible effective field theory of four massless Majorana fermions. The results are obtained by approximating the ground state of the system in the thermodynamic limit using Matrix Product States via the infinite Time Evolving Block Decimation method, as well as by effective field theory considerations. Our findings show, for the first time, that different large effective spin-1 SPT phases separated by continuous quantum phase transitions can be stabilized in a simple quantum spin chain.