Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: From trivial to topological paramagnets: The case of $\mathbb{Z}_2$ and
  $\mathbb{Z}_2^3$ symmetries in two dimensions   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.

16
The categories of this paper are: stat
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
33
The categories of this paper are: plasm-ph
2
The categories of this paper are: math
17
The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
8
The categories of this paper are: hep-ex
15
The categories of this paper are: nucl-ex
21
The categories of this paper are: adap-org
34
The categories of this paper are: chem-ph
20
The categories of this paper are: acc-phys