From trivial to topological paramagnets: The case of $\mathbb{Z}_2$ and $\mathbb{Z}_2^3$ symmetries in two dimensions

TL;DR

This study uses quantum Monte Carlo simulations to explore the phase diagram of two-dimensional bosonic systems with $Z_2$ and $Z_2^3$ symmetries, revealing intermediate symmetry-breaking phases between trivial and topological states.

Contribution

It provides the first comprehensive phase diagram for these symmetry-protected topological phases in two dimensions, identifying intermediate symmetry-breaking phases and their magnetic orders.

Findings

Intermediate phases break protecting symmetries.

Various magnetic orders identified, including ferromagnetism and stripe orders.

Finite-size scaling used to analyze critical properties.

Abstract

Using quantum Monte Carlo simulations, we map out the phase diagram of Hamiltonians interpolating between trivial and non-trivial bosonic symmetry-protected topological phases, protected by $\mathbb{Z}_2$ and $\mathbb{Z}_2^3$ symmetries, in two dimensions. In all cases, we find that the trivial and the topological phases are separated by an intermediate phase in which the protecting symmetry is spontaneously broken. Depending on the model, we identify a variety of magnetic orders on the triangular lattice, including ferromagnetism, $\sqrt{3}\times\sqrt{3}$ order, and stripe orders (both commensurate and incommensurate). Critical properties are determined through a finite-size scaling analysis. Possible scenarios regarding the nature of the phase transitions are discussed.