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11/14 - Seminar (theoretical, informal): Raffaele Resta, University of Trieste, Italy

Raffaele Resta - speaker's webpage
Department of Physics
University of Trieste, Italy

Topological Order in Electronic Wavefunctions

Topology is defined as a branch of mathematics that describes properties which remain unchanged under continuous deformations; such properties are labelled by integer numbers, named topological invariants. If a measurable physical property of a macroscopic system is an integer (in appropriate units), then it can be measured in principle with infinite precision.

To this day, the most spectacular manifestation of topology in nonrelativistic quantum mechanics is the quantum Hall effect; the "integerness" of the relevant invariant is experimentally verified to 10−9. The last five years witnessed a booming interest in topological insulators (TIs), defined as materials whose ground state shows topological order in absence of an applied magnetic field.

So far, topological order has been invariably addressed in k space, where different values of the invariant characterize ground-state wavefunctions which are differently "knotted", and cannot be transformed into each other by a continuous deformation. But topological order must also be observable in the bulk of the material, even for inhomogeneous samples, and independently of the boundary conditions. We have shown that a "topological marker" -- sampling the Chern number locally in r space -- can indeed be defined. Simulations on finite samples, performed within open boundary conditions, prove the effectiveness of our marker for several cases: crystalline, disordered, and inhomogeneous (heterojunctions) [1].

[1] R. Bianco and R. Resta, Mapping topological order in coordinate space, Phys. Rev. B 84, 241106(R) (2011).