Topological materials have become the focus of intense research in recent years, since they
exhibit fundamentally new physical phenomena with potential applications for novel devices …

The discovery of the quantum spin Hall effect and topological insulators more than a decade
ago has revolutionized modern condensed matter physics. Today, the field of topological …

Long-range entanglement—the backbone of topologically ordered states—cannot be
created in finite time using local unitary circuits, or, equivalently, adiabatic state preparation …

A fundamental distinction between many-body quantum states are those with short-and long-
range entanglement (SRE and LRE). The latter cannot be created by finite-depth circuits …

The theory of electric polarization in crystals defines the dipole moment of an insulator in
terms of a Berry phase (geometric phase) associated with its electronic ground state,. This …

Symmetry-protected topological (SPT) phases are many-body quantum states that are
topologically nontrivial as long as the relevant symmetries are unbroken. In this work we …

Classifying phases of matter is key to our understanding of many problems in physics. For
quantum-mechanical systems in particular, the task can be daunting due to the …

We develop a theory of symmetry in open quantum systems. Using the operator-state
map**, we characterize symmetry of Liouvillian superoperators for the open quantum …

This review focuses on the field of quantum entanglement applied to condensed matter
physics systems with strong correlations, a domain which has rapidly grown over the last …

We examine the interplay of symmetry and topological order in 2+ 1-dimensional topological
quantum phases of matter. We present a precise definition of the topological symmetry …