Density Matrix Renormalization Group: Probing the Topology of Quantum States
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Matter occurs in various phases with different properties. Usually these phases
are characterized in terms of symmetry breaking. A major discovery in the 1980s
was the quantum Hall effect which forms a new kind of “topological” order. This
order represents exotic phases with unusual properties and cannot be understood
in terms of symmetry breaking. Since then, a growing number of instances of
topological phases has accumulated, and important applications – not least
topological quantum computers – have been proposed, but a characterization and
classification of these new phenomena has been slow to emerge. In parallel, DMRG
has arrived as a powerful numerical method with ex- tensions to two dimensional
systems and time-dependent phenomena. I will show how to use DMRG to develop new
frameworks that help to understand topologically ordered systems. For example,
it is now possible to extract characterizing properties of the anyonic
excitations di- rectly from the ground state of fractional quantum Hall systems.
This approach further makes contact with “measurable” quantities (Hall
viscosity) and field theories (central charge at critical points). Other
remarkable examples are symmetry protected topological phases in one-dimensional
systems for which DMRG provides a complete characterization.