Masters Defense: Emil Boesen

The (2,N) quantum-dot cellular automaton

Summary: In this master thesis, a special kind of quantum-dot cellular automaton, know as the (2,N)-QDCA, are described and mapped to an anti ferromagnetic, nearest neighbour, quantum Ising model with open boundary conditions. Ignoring the quantum dynamics, the system acts as a classical Ising model in a longitudinal field. The equilibrium properties of the classical Ising model are therefore explored. From statistical average calculations it is shown that for small temperatures and small field, the system localizes around a zero-field AFM ground state. Increasing the field over a certain threshold is predicted to cause the system to localize around a zero-field FM ground state with a magnetization opposite to the field.
To describe the non-equilibrium dynamics of the classical model, the kinetic Ising model is introduced. This allows one to simulate how a given initial configuration will act when coupled to a Glauber heat bath. To simulate this, the Gillespie algorithm is implemented using Cython code and can be used in Python. From numerical results, it is shown that for zero longitudinal fields and zero temperature, the system randomly picks a zero-field AFM ground state, which it decays into. The average decay time is measured to be independent of the initial state in general and depends on the system size via a power law with an exponent of 2.05188 ± 0.00006. When the field is non-zero but less than a given threshold, any given initial state decays into a disordered state with domain walls in the bulk. At the threshold, domain wall pairs can be created and annihilated spontaneously which introduces an enormous amount of disorder. For fields over the threshold, the system decays into the zero-field FM ground with magnetizations opposite to the field. At the end of the thesis, the eigenvalues of the quantum Hamiltonian is evaluated semi-analytical.