Instability of the Luttinger liquids towards an exotic quantum state of matter with highly degenerate ground states: an anisotropic extension of the ferromagnetic spin-1 biquadratic model

TL;DR

This paper explores an anisotropic extension of the ferromagnetic spin-1 biquadratic model, revealing a rich phase diagram including trivial, fractal, and Luttinger liquid phases, and discovering a new universality class with highly degenerate ground states.

Contribution

It introduces an anisotropic model exhibiting a novel universality class arising from Luttinger liquid instability towards an exotic quantum state with degenerate ground states.

Findings

Identification of three trivial phases

Discovery of three fractal phases

Observation of six Luttinger liquid phases

Abstract

An extensive investigation, both numerical and analytical, is performed for an anisotropic extension of the ferromagnetic spin-1 biquadratic model. The ground state phase diagram accommodates three symmetry-protected trivial phases, three coexisting fractal phases and six Luttinger liquid phases. A novel universality class arises from an instability of a Luttinger liquid towards an exotic quantum state of matter with infinitely degenerate ground states. The latter in turn is a scale-invariant quantum state of matter, which may be attributed to the coexistence of ${\rm SU}(2)$ spontaneous symmetry breaking with one type-B Goldstone mode on the characteristic line: $J_y=J_z$, and ${\rm U}(1)$ spontaneous symmetry breaking without any gapless Goldstone mode on the characteristic line $J_x/J_z=0$, together with their cyclic permutations with respect to $x$, $y$ and $z$.