Abstract:  
The fundamental understanding of phases and their transitions has been a central theme in condensed matter
physics. Until recently, it was largely believed that the Landau symmetry breaking principle was effective in
distinguishing different phases of matter, with broken symmetries signaling the phase transition. But with the
discovery of topological phases which are beyond the Landau symmetry breaking principle, the identification and
classification of quantum phases at absolute zero has opened up new unexplored avenues thus leading to
exciting theoretical discoveries further propelling technological advancement. Topological phases of matter are
characterized by the notion of topological order and in this work we aim to explore and understand topological
phases by introducing novel signatures which characterize topological order. The robustness of these phases
to external perturbation makes them an ideal candidate to store and manipulate quantum information thus
making them an unique and interesting prospect for realizing quantum computers.
There have been several signatures to characterize intrinsic topological order, for instance the invariance of the
topologically ordered state under local operators, the dependence of ground state degeneracy on the underlying
manifold and its robustness to external perturbation, topological entanglement entropy related to the quantum
dimension of the supersselection sectors, the inability to construct the topologically ordered state from a product
state via constant depth unitary transformations. With toric code as the toy model, we analyze the robustness of
topological order on a manifold supporting open boundaries by computing some of the above signatures which
effectively detect a topological to trivial phase transition. We then probe the existence of a quantum criticality
between distinct topological phases obtained by varying the underlying manifold. In these scenarios, most of the
above signatures turn out to be ineffective in detecting the distinct phases leading to the introduction of an non
local order parameter whose construction is facilitated by the phenomenon of anyon condensation.
The signatures for quantitatively and qualitatively characterizing intrinsic topological order being highly
scenario dependent and also with its definition for mixed states being elusive we introduce an operational
definition based on concepts of topological error correction. We define a state to be topologically ordered
if the errors in the state can be corrected by an error correction circuit of finite depth. To concretize
the notion of topological to trivial phase transition in an open setting we turn to nonequilibrium
phenomenon, for example: Directed Percolation, with the change in percolation rate driving a dynamical phase
transition between absorbing and active states with the former being topologically ordered while the latter
being topologically trivial. Additionally, we explore the notion of topological phase transitions between
distinct topological phases obtained by varying underlying topology in an open setting, analogous to the
closed setting discussed earlier. To summarize, we have introduced various mixed states which exhibit
topological order and also an operational definition to quantify topological order applicable across
multitude of scenarios.
We extend the above operational definition to quantify and detect quantum phase transitions in the case of
Symmetry Protected Topological (SPT) phases. To further validate the above notion, we consider the perturbed
variants of the SuSchrieffer–Heeger (SSH) models and detect quantum phase transitions to a high
accuracy by employing the techniques from the framework of tensor networks. It is significant to note the
distinction of the error correction algorithms applied earlier in the case of intrinsic topological order
were independent of symmetry constraints while in the current scenario we impose additional symmetry
constraints to accurately detect the phase transition. In addition, we also devise error correction
strategies with respect to topologically trivial states to detect quantum phase transitions which do
not involve topological phases. This gives rise to a very fundamental question on whether error correction
statistics with a well defined error correction algorithm, not necessarily optimal, are capable of detecting
a equivalence classes of phases and thereby acting as a reliable probe to effectively detect topological/quantum
phase transitions?
From theoretical and numerical end of the spectrum we shift gears to explore possible experimental platforms
with an aim to realize some of the quantum manybody phenomenon discussed earlier. While there have been
several innovative experimental avenues to realize the above, one such promising candidate has been ultracold
polar molecules setups that offer additional degrees of freedom due to the rovibrational degrees of freedom.
Based on the chemical reaction between atoms and molecules which results in a quantum Zenobased
blockade, we devise several optimal strategies to efficiently detect molecules using atom as probe, we further
extend the above technique to entangle the internal states of molecules and atoms. In addition, we also present
optimal strategies for dissipative state engineering using the atommolecule interactions.


License of this version:  CC BY 3.0 DE  http://creativecommons.org/licenses/by/3.0/de/ 
Publication type:  doctoralThesis 
Publishing status:  publishedVersion 
Publication date:  2021 
Keywords german:  Offene Quantensysteme, Topologische Quantenphasenübergänge, Quantenfehlerkorrektur, Ultrakalte polare Moleküle 
Keywords english:  Topological order, Open quantum systems, topological quantum phase transitions, quantum error correction, ultracold polar molecules, engineered dissipation 
DDC:  530  Physik 