We study symmetry-enriched topological order in two-dimensional tensor network states by using graded matrix product operator algebras to represent symmetry induced domain walls. A close connection to the theory of graded unitary fusion categories is established. Tensor network representations of the topological defect superselection sectors are constructed for all domain walls. The emergent symmetry-enriched topological order is extracted from these representations, including the symmetry action on the underlying anyons. Dual phase transitions, induced by gauging a global symmetry, and condensation of a bosonic subtheory, are analyzed and the relationship between topological orders on either side of the transition is derived. Several examples are worked through explicitly.

We show that the mutual, through‐space compression of atomic volume experienced by approaching topological atoms causes an exponential increase in the intra‐atomic energy of those atoms, regardless of approach orientation. This insight was obtained using the modern energy partitioning method called interacting quantum atoms (IQA). This behaviour is consistent for all atoms except hydrogen, which can behave differently depending on its environment. Whilst all atoms experience charge transfer when they interact, the intra‐atomic energy of the hydrogen atom is more vulnerable to these changes than larger atoms. The difference in behaviour is found to be due to hydrogen's lack of a core of electrons, which, in heavier atoms, consistently provide repulsion when compressed. As such, hydrogen atoms do not always provide steric hindrance. In accounting for hydrogen's unusual behaviour and demonstrating the exponential character of the intra‐atomic energy in all other atoms, we provide evidence for IQA's intra‐atomic energy as a quantitative description of steric energy.

Subsystem symmetry has emerged as a powerful organizing principle for unconventional quantum phases of matter, most prominently fracton topological orders. Here, we focus on a special subclass of such symmetries, known as higher-form subsystem symmetries, which allow us to adapt tools from the study of conventional topological phases to the fracton setting. We demonstrate that certain transitions out of familiar fracton phases, including the X-cube model, can be understood in terms of the spontaneous breaking of higher-form subsystem symmetries. We find simple pictures for these seemingly complicated fracton topological phase transitions by relating them in an exact manner, via gauging, to spontaneous higher-form subsystem symmetry breaking phase transitions of decoupled stacks of lower-dimensional models. We harness this perspective to construct a sequence of unconventional subdimensional critical points in two and three spatial dimensions based on the stacking and gauging of canonical models with higher-form symmetry. Through numerous examples, we illustrate the ubiquity of coupled layer constructions in theories with higher-form subsystem symmetries. Contents2 This symmetry motivates our choice of notation 2 H (0,1) Z 2 for the Hamiltonian of the plaquette Ising model: the (0, 1) in the superscript indicates that the model has a Z (0,1) 2 (F x , F y ) ∼ symmetry, and the 2 denotes its spatial dimension. We define an infinite family d H (q,k) Z 2 of similar models in §2.1.

Invited for this month's cover picture is the group of Paul L. A. Popelier from Manchester Institute of Biotechnology (UK). The cover picture shows the quantum topological atoms in a configuration of the complex HF⋅⋅⋅OH2, where F is green and O is red. Read the full text of their Full Paper at 10.1002/open.201800275.