Classification of trivial spin-1 tensor network states on a square lattice

TL;DR

This paper develops a tensor network classification scheme for trivial spin-1 quantum spin liquid states on a square lattice, identifying 32 distinct classes and exploring their physical properties through explicit wave function constructions.

Contribution

It introduces a general method for classifying trivial spin-1 quantum spin liquids on a square lattice, expanding understanding beyond spin-1/2 systems and providing explicit wave functions and phase diagrams.

Findings

Identified 32 classes of spin-1 QSL states respecting all symmetries.

Found both gapped and gapless states with trivial topological entanglement entropy.

Discovered a rich phase diagram including plaquette-ordered, RVB, and critical phases.

Abstract

Classification of possible quantum spin liquid (QSL) states of interacting spin-1/2's in two dimensions has been a fascinating topic of condensed matter for decades, resulting in enormous progress in our understanding of low-dimensional quantum matter. By contrast, relatively little work exists on the identification, let alone classification, of QSL phases for spin-1 systems in dimensions higher than one. Employing the powerful ideas of tensor network theory and its classification, we develop general methods for writing QSL wave functions of spin-1 respecting all the lattice symmetries, spin rotation, and time reversal with trivial gauge structure on the square lattice. We find $2^5$ distinct classes characterized by five binary quantum numbers. Several explicit constructions of such wave functions are given for bond dimensions $D$ ranging from two to four, along with thorough numerical analyses to identify their physical characters. Both gapless and gapped states are found. The topological entanglement entropy of the gapped states are close to zero, indicative of topologically trivial states. In $D=4$, several different tensors can be linearly combined to produce a family of states within the same symmetry class. A rich "phase diagram" can be worked out among the phases of these tensors, as well as the phase transitions among them. Among the states we identified in this putative phase diagram is the plaquette-ordered phase, gapped resonating valence bond phase, and a critical phase. A continuous transition separates the plaquette-ordered phase from the resonating valence bond phase.