Diagnosing Topological Quantum Matter via Entanglement Patterns

Quantum matter involves the study of entanglement patterns in the ground states of many-body
systems. Of significant interest have recently been topological states of matter,
which exhibit characteristics only described globally. As such they are robust to local deformations.
In this thesis, we study inter-correlations of many-body states through the entanglement spectrum,
obtained by a bipartition of both topological and non-topological systems.

In particular, we introduce two novel diagnostics which operate on entanglement spectra.
For topological phases supporting edge states on open boundaries
we take a quantum-information inspired approach by invoking
the monogamy relations obeyed by multi-partite systems.
Within a strictly single-particle framework,
we establish a correspondence between highly entangled mode and the existence of edge states.
In the many-body context, we introduce the interaction distance
of a mixed state.
Exclusively via the entanglement spectrum it
determines how close a free-fermion state lies and what the emergent free quasiparticles are.

We apply these two measures to diagnose the properties of a variety of free and interacting fermionic topological systems and reinterpret their properties from a fresh point of view.
Our case studies revolve around Kitaev's honeycomb model,
which supports both short-range and long-range topological order,
constituting it thus relevant to both the monogamy qualifier and the interaction distance.
The possibility to diagnose whether a model has zero interacting distance or if it supports maximally entangled states provides central and compact information about the behaviour of complex quantum systems.