Matter can exist in many phases e.g. ice, water and water vapor. The transition between different
states (a phase transition) is a remarkable example of emergence and cooperative behavior. When
the temperature of the substance decreases, quantum fluctuations take the upper hand and new
and unexpected phenomena arise. Such quantum phases were well described before the 1980's by
the breaking of certain symmetries, but since then new exotic phases have been discovered that
cannot be understood in this way. These phases are called topological phases and their bulk
properties are very different from their properties at the edge of the sample. Symmetry plays an
important role in their classification and a symmetry that is violated at the quantum level is called
an anomaly. Anomalies were only recently discovered to be present in these new topological
phases and their investigation provides new ways of classifying topological phases.
Quantum entanglement is key in understanding these phases and in the last decade tensor
networks, which naturally encode entanglement, have been very successful in understanding
these phases both theoretically and numerically. This proposal aims at a better understanding of
these phases from the new perspective of anomalies using tensor networks and the development
of new and efficient numerical algorithms to probe this highly fascinating new quantum order for
which the 2016 Nobel Prize of physics was awarded.