Topological Signatures for the Kitaev Chain in and out of Equilibrium
This thesis investigates aspects of the topological properties and non-equilibrium dynamics of one-dimensional quadratic fermionic systems, with a primary focus on the Kitaev chain and related lattice models. A unified theoretical framework is developed based on quadratic Hamiltonians, their diagonalisation, and the use of correlation matrices to efficiently characterise both static and time-dependent many-body states. Within this framework, several paradigmatic models – including the Kitaev chain, the Creutz ladder, and the Su-Schrieffer-Heeger model – are analysed to determine their phase structure, excitation spectra, and the presence of topologically protected edge modes. Bulk topological invariants, such as the Zak phase and the winding number, are examined and shown to signal transitions between distinct gapped phases. A complementary perspective is also developed, based on non-local order parameters calculated using Toeplitz determinant techniques, providing an analytic expression for a proposed nonlocal topological order parameter. We also investigate the dynamical behaviour of these systems under slow parameter ramps across phase transitions, drawing connections to the Landau-Zener framework and the Kibble-Zurek mechanism where applicable. Particular emphasis is placed on the excitation density, the evolution of geometric phases, and the role of symmetries in constraining the dynamics. These ideas are explored in both the Kitaev chain and the Creutz ladder, revealing how topology and symmetry influence non-adiabatic dynamics. Specifically, no dynamics was observed in the Zak phase due to the inversion symmetry present in each of the three models. The breaking of this symmetry led to non-trivial dynamics. For parameter ramps avoiding the critical point, exponential Landau-Zener scaling was observed for the non-adiabatic corrections to the Zak phase as well as the excitation density. For faster ramps, the scaling of the excitation density was observed to change from exponential Landau-Zener scaling to power-law Kibble-Zurek scaling, resulting in two scaling regimes.
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