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Interparticle interactions, presence of impurities or disorder, and topology and
geometry of wavefunctions determine the low temperature behaviors of quantum
materials, which crucially depend on their dimensionality. However, crystals are
naturally bound to live within three spatial dimensions, which thereby provides a
natural constraint on the possible quantum phases that can be observed in nature.
In this talk, first I will promote a general principle of constructing effective
Hamiltonian of a lower-dimensional brane or subsystem (quasi-crystalline and
crystalline), embedded within the higher dimensional parent crystals via the Schur
complement. Then I will show a plethora of examples to establish that the Schur
complemented Hamiltonian of the brane provide a window into the landscape of the
topological phases in higher dimensions in terms of the bulk-boundary
correspondence, probing through the lattice dislocations, and the quantum transport,
e.g. the chiral anomaly, when projecting it from parent two- or three-dimensional
topological crystals. Furthermore, this construction successfully harnesses discrete
crystalline symmetry protected topological phases on the projected branes that are,
however, only present on their parent higher-dimensional crystals as well as
topological superconductors.
Finally, I will show that such projected two-dimensional branes feature the quantum
phase diagrams and quantum criticality of three-dimensional disordered Anderson
model and dirty Weyl semimetals. Some of the possible experimental platforms to
test these theoretical predictions and future directions of this general theme of
pursuit will be highlighted.
[1] A. Panigrahi, V.Juricic, and B. Roy, Communications Physics 5, 230 (2023).
[2] A. Panigrahi and B. Roy, Phys. Rev. B 113, 035301 (2025).
[3] A. C. Tyner, V. Juricic, and B. Roy, arXiv: 2507.23780 |