Fig: a) Conduction electrons coupled to Kitaev's toric code constitute a solvable toy model of gapless systems with topological order. b) The corresponding phase diagram contains a reconstructed Fermi surface without symmetry breaking.
A central puzzle of present day research on strongly correlated electron materials is the experimental evidence for changes in the Fermi surface volume without symmetry breaking, as reported for example in cuprate and heavy fermion superconductors. It has been suggested that this phenomenon may be explained by topological order, e.g. in Z2 gauge theories. Independently, this form of quantum order is also employed in topological quantum error correction codes.
In this ongoing research direction  I harness the synergies between materials science and quantum information theory and study a model for gapless conduction electrons coupled to a topological toy model (Kitaev's toric code). On the basis of this model, it is possible to design an anologue quantum computer for electrons in a Z2 gauge theory [1,3].
I complement this research direction with the study of tractable toy models of deconfining gauge theories, i.e. frustrated impurity models , which display signatures of emergent non-Abelian anyons.
 EJ König, P Coleman, AM Tsvelik, Physical Review B 102 (15), 155143 (2020)
 EJ König, P Coleman, Y Komijani, Phys. Rev. B 104, 115103 (2021)
 EJ König, in preparation.