Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Weak symmetry breaking and topological order in a 3D compressible
  quantum liquid   We introduce a new type of 3D compressible quantum phase, in which the U(1)
charge conservation symmetry is weakly broken by a rigid string-like order
parameter, and no local order parameter exists. We show that this gapless phase
is completely stable and described at low energy by an infinite-component
Chern-Simons-Maxwell theory. We determine the emergent symmetry group, which
contains U(1) 0-form planar symmetries and an unusual subgroup of the dual U(1)
1-form symmetry supported on cylindrical surfaces. Through the associated 't
Hooft anomaly, we examine how the filling condition is fulfilled in the
low-energy theory. We also demonstrate that the phase exhibits a kind of
fractonic topological order, signified by extensively many different types of
topologically nontrivial quasiparticles formed out of vortices of the weak
superfluid. A microscopic model realizing the weak superfluid phase is
constructed using an array of strongly coupled Luttinger liquid wires, and the
connection to the field theory is established through boson-vortex duality.

3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
2
The categories of this paper are: math
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
30
The categories of this paper are: q-alg
31
The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
18
The categories of this paper are: econ
28
The categories of this paper are: alg-geom
34
The categories of this paper are: chem-ph