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Interaction, dynamics and information on quantum networks
Thesis subject
In recent years we were interested in topological properties of quantum condensed matter systems:

Micromagnetic textures in ferromagnets and chiral metals exhibit a rich phenomenology due to the competition between exchange, anisotropy and spinorbit interactions. We studied the dynamics of skyrmions driven by spin polarized currents using a semiclassical approach to model the spin transfer torque together with methods issued from nonlinear physics (Verga, 2014)[1].

Topological insulators possess an inverted electronic band structure, which results in the existence of robust edge states. We analyzed the quantization of the conductance in the presence of quenched magnetic impurities. We demonstrated the existence of a quantum phase transition induced by strong disorder, between a quantum spin Hall phase and a quantum anomalous Hall phase (Raymond et al. 2015)[2].

More recently, we investigated the topologically protected edge states using discrete quantum walks of the same topological class than quantum Hall systems, in a two dimensional lattice (Verga, 2016)[3]. This last work makes a first connexion, in our research, between materials systems and quantum information: a quantum walk is a succession of unitary gates applied to a quantum state, followed by a measurement.
The present thesis project aims at developing this relationship between the physics of topological nontrivial systems and quantum information concepts.
We are in particular interested in the link between disorder, interactions and entanglement. Indeed, entanglement emerges as a central concept in the understanding of manybody interacting systems such as fractional Hall states, topological phases and nonthermal states in manybody localization. It is also central in quantum computing: correlations created by entanglement are an informational resource without classical equivalent; the speedup of quantum algorithms depends essentially on this resource. However, the physical mechanism producing entanglement is not so obvious: while entanglement of quantum states is nonlocal, interactions are strictly local, at least at the microscopic level.
In fact, interaction creates longrange correlations through the generation of entanglement. These correlations characterize a kind of order, not directly related with symmetry breaking, beyond the usual Landau description in terms of order parameters. One example is given by the topological phases in spin liquids, satisfying the area law of entanglement. In this project we want to investigate the emergence of complexity in physical systems. To pursue this objective, we will exploit the observation that these special states of matter correspond to some sort of quantum graph state. Now, graph states are naturally associated with quantum networks.
Quantum states are defined at each node: a superposition of product states gives a graph state.
The candidate will start with the study of interacting quantum walks on graphs. This system is suitable to reveal the influence of interactions on entanglement, by changing the graph topology and the type of interaction; informationtheoretic quantities, such as entanglement entropy and negativity, are appropriated to characterize the information content and the structure of the quantum state. A natural further development of this simple model, is to investigate dynamical quantum networks. We will introduce a local rule at the network nodes (the equivalent of coin operators of the quantum walk) analog to an effective hamiltonian describing the particle internal degrees of freedom and neighboring interactions, together with a dynamical local rule to rewire the network edges. The question to explore is: can this system, combining energy and evolving topology, selforganize into some hierarchical structure from which macroscopic correlations (phases, geometry) can emerge?
References
[1] Alberto D. Verga, Skyrmion to ferromagnetic state transition: A description of the topological change as a finitetime singularity in the skyrmion dynamics, Phys. Rev. B, 90, 174428, 2014.
[2] Laurent Raymond, Alberto D. Verga, and Arnaud Demion, Anomalous quantum {H}all effect induced by disorder in topological insulators, Phys. Rev. B, 92, 075101, 2015.
[3] Alberto D. Verga, Edge states in a twodimensional quantum walk with disorder, ArXiv eprints: 1606.00613, June 2016 (to be published in EPJB, 2017).