We theoretically explore quench dynamics in a finite-sized topological fermionic p-wave superconducting wire with the goal of demonstrating that topological order can have marked effects on such non-equilibrium dynamics. In the case studied here, topological order is reflected in the presence of two (nearly) isolated Majorana fermionic end bound modes together forming an electronic state that can be occupied or not, leading to two (nearly) degenerate ground states characterized by fermion parity. Our study begins with a characterization of the static properties of the finite-sized wire, including the behavior of the Majorana end modes and the form of the tunnel coupling between them; a transfer matrix approach to analytically determine the locations of the zero energy contours where this coupling vanishes; and a Pfaffian approach to map the ground state parity in the associated phase diagram. We next study the quench dynamics resulting from initializing the system in a topological ground state and then dynamically tuning one of the parameters of the Hamiltonian. For this, we develop a dynamic quantum many-body technique that invokes a Wick's theorem for Majorana fermions, vastly reducing the numerical effort given the exponentially large Hilbert space. We investigate the salient and detailed features of two dynamic quantities-the overlap between the time-evolved state and the instantaneous ground state (adiabatic fidelity) and the residual energy. When the parity of the instantaneous ground state flips successively with time, we find that the time-evolved state can dramatically switch back and forth between this state and an excited state even when the quenching is very slow, a phenomenon that we term `parity blocking'. This parity blocking becomes prominently manifest as non-analytic jumps as a function of time in both dynamic quantities.
We theoretically explore quench dynamics in a finite-sized topological fermionic p-wave superconducting wire with the goal of demonstrating that topological order can have marked effects on such non-equilibrium dynamics. In the case studied here, topological order is reflected in the presence of two (nearly) isolated Majorana fermionic end bound modes together forming an electronic state that can be occupied or not, leading to two (nearly) degenerate ground states characterized by fermion parity. Our study begins with a characterization of the static properties of the finite-sized wire, including the behavior of the Majorana end modes and the form of the tunnel coupling between them; a transfer matrix approach to analytically determine the locations of the zero energy contours where this coupling vanishes; and a Pfaffian approach to map the ground state parity in the associated phase diagram. We next study the quench dynamics resulting from initializing the system in a topological ground state and then dynamically tuning one of the parameters of the Hamiltonian. For this, we develop a dynamic quantum many-body technique that invokes a Wick's theorem for Majorana fermions, vastly reducing the numerical effort given the exponentially large Hilbert space. We investigate the salient and detailed features of two dynamic quantities-the overlap between the time-evolved state and the instantaneous ground state (adiabatic fidelity) and the residual energy. When the parity of the instantaneous ground state flips successively with time, we find that the time-evolved state can dramatically switch back and forth between this state and an excited state even when the quenching is very slow, a phenomenon that we term `parity blocking'. This parity blocking becomes prominently manifest as non-analytic jumps as a function of time in both dynamic quantities.
We theoretically explore quench dynamics in a finite-sized topological fermionic p -wave superconducting wire with the goal of demonstrating that topological order can have marked effects on such non-equilibrium dynamics. In the case studied here, topological order is reflected in the presence of two (nearly) isolated Majorana fermionic end bound modes together forming an electronic state that can be occupied or not, leading to two (nearly) degenerate ground states characterized by fermion parity. Our study begins with a characterization of the static properties of the finite-sized wire, including the behavior of the Majorana end modes and the form of the tunnel coupling between them; a transfer matrix approach to analytically determine the locations of the zero energy contours where this coupling vanishes; and a Pfaffian approach to map the ground state parity in the associated phase diagram. We next study the quench dynamics resulting from initializing the system in a topological ground state and then dynamically tuning one of the parameters of the Hamiltonian. For this, we develop a dynamic quantum many-body technique that invokes a Wick's theorem for Majorana fermions, vastly reducing the numerical effort given the exponentially large Hilbert space. We investigate the salient and detailed features of two dynamic quantities—the overlap between the time-evolved state and the instantaneous ground state (adiabatic fidelity) and the residual energy. When the parity of the instantaneous ground state flips successively with time, we find that the time-evolved state can dramatically switch back and forth between this state and an excited state even when the quenching is very slow, a phenomenon that we term 'parity blocking'. This parity blocking becomes prominently manifest as non-analytic jumps as a function of time in both dynamic quantities.
This dissertation deals with two broad topics - Majorana modes in Kitaev chain and parallels of black hole phenomena in the quantum Hall effect. Majorana modes in topological superconductors are of fundamental importance as realizations of real solutions to the Dirac equation and for their anyonic exchange statistics. They are realised as zero energy edge modes in one-dimensional topological superconductors, modeled by the Kitaev chain Hamiltonian. Here an extensive study is made on the wavefunction features of these Majorana modes. It is shown that the Majorana wavefunction has two distinct features- a decaying envelope and underlying oscillations. The latter becomes important when one considers the coupling between the Majorana modes in a finite-sized chain. The coupled Majorana modes form a non-local Dirac fermionic state which determines the ground state fermion parity. The dependance of the fermion parity on the parameters of the system is purely determined by the oscillatory part of the Majorana wavefunctions. Using transfer matrix method, one can uncover a new boundary in the phase diagram, termed as `circle of oscillations', across which the oscillations in the wavefunction and the ground-state fermionic parity cease to exist. This is closely related to the circle that appears in the context of transverse field XY spin chain, within which the spin-spin correlations have oscillations. For a finite sized system, the circle is further split into mutliple ellipses called `parity sectors'. The parity oscillations have a scaling behaviour i.e oscillations for different superconducting gaps can be scaled to collapse to a single plot. Making use of results from random matrix theory for class D systems, one can also predict the robustness of certain features of fermion parity switches in the presence of disorder and comment on the critical properties of the MBS wavefunctions and level crossings near zero energy. These results could provide directions for making measurements on zero-bias conductance oscillations and the parameter range of operations for robust parity switches in realistic disordered system. On the front of non-equilibrium dynamics, the effect of Majorana modes on the dynamical evolution of the ground state under time variation of a Hamiltonian parameter is studied. The key result is the failure of the ground state to evolve into opposite parity sectors under the dynamical tuning of the system within the topological phase. This dramatic lack of adiabaticity is termed as ‘parity blocking’. A real-space time-dependent formalism is also developed using Pfaffian correlations, where simple momentum space methods fail. This formalism can be used for calculating the non-equilibrium quantities, such as adiabatic fidelity and the residual energy in a system with open boundaries. The consideration of Majorana modes in non-equilibrium dynamics lead to deviation from Kibble-Zurek physics and non-analyticities in adiabatic fidelity even within the topological phase. The second part of the thesis deals with uncovering structural parallels of black hole phenomena such as the Hawking-Unruh effect and quasinormal modes in quantum Hall systems. The Hawking-Unruh effect is the emergence of a thermal state when a vacuum of a quantum field theory on a given spacetime is restricted to a submanifold bounded by an event horizon. The thermal state manifests as Hawking radiation in the context of a black hole spacetime with an event horizon. The Unruh effect is a simpler example where a family of accelerating observers in Minkowski spacetime are confined by the lightcone structure and the Minkowski vacuum looks as a thermal state to them. The key element in understanding the Hawking-Unruh effect is the Rindler Hamiltonian or the boost. The boost acts as the generator of time translation for the quantum states in the Rindler wedge giving rise to thermality. In this thesis it is shown that due to an exact isomorphism between the Lorentz algebra in Minkowksi spacetime and the algebra of area preserving transformations in the lowest Landau level of quantum hall effect, an applied saddle potential acts as an equivalent to the Rindler Hamiltonian giving rise to a parallel of Hawking-Unurh effect. In the lowest Landau level, the saddle potential is reduced to the problem of scattering off an inverted harmonic oscillator(IHO) and the tunneling probability assumes the form of a thermal distribution. The IHO also has scattering resonances which are poles of the scattering matrix in the complex energy plane. The scattering resonances are states with time-decaying behavior and have purely incoming/outgoing probability current. These states are identified as quasinormal modes analogous to those occurring the scattering off an effective potential in black hole spacetimes. The quasinormal decay is an unexplored effect in quantum Hall systems and provides a new class of time-dependent probe of quantum Hall physics. The parallels between the relativistic symmetry generators and the potentials applied in the lowest Landau level also open up an avenue for studying Lorentz Kinematics and symplectic phase space dynamics in the lowest Landau level. These parallels open up new avenues of exploration in the quantum Hall effect.
We study the decay and oscillations of Majorana fermion wave functions and ground-state (GS) fermion parity in one-dimensional topological superconducting lattice systems. Using a Majorana transfer matrix method, we find that Majorana wave-function properties are encoded in the associated Lyapunov exponent, which in turn is the sum of two independent components: a ``superconducting component,'' which characterizes the gap induced decay, and the ``normal component,'' which determines the oscillations and response to chemical potential configurations. The topological phase transition separating phases with and without Majorana end modes is seen to be a cancellation of these two components. We show that Majorana wave-function oscillations are completely determined by an underlying nonsuperconducting tight-binding model and are solely responsible for GS fermion parity switches in finite-sized systems. These observations enable us to analytically chart out wave-function oscillations, the resultant GS parity configuration as a function of parameter space in uniform wires, and special parity switch points where degenerate zero energy Majorana modes are restored in spite of finite size effects. For disordered wires, we find that band oscillations are completely washed out leading to a second localization length for the Majorana mode and the remnant oscillations are randomized as per Anderson localization physics in normal systems. Our transfer matrix method further allows us to (i) reproduce known results on the scaling of midgap Majorana states and demonstrate the origin of its log-normal distribution, (ii) identify contrasting behavior of disorder-dependent GS parity switches for the cases of even versus odd number of lattice sites, and (iii) chart out the GS parity configuration and associated parity switch points as a function of disorder strength.
We study the nonequilibrium dynamics of quenching through a quantum critical point in topological systems, focusing on one of their defining features: ground-state degeneracies and associated topological sectors. We present the notion of ``topological blocking,'' experienced by the dynamics due to a mismatch in degeneracies between two phases, and we argue that the dynamic evolution of the quench depends strongly on the topological sector being probed. We demonstrate this interplay between quench and topology in models stemming from two extensively studied systems, the transverse Ising chain and the Kitaev honeycomb model. Through nonlocal maps of each of these systems, we effectively study spinless fermionic p-wave paired topological superconductors. Confining the systems to ring and toroidal geometries, respectively, enables us to cleanly address degeneracies, subtle issues of fermion occupation and parity, and mismatches between topological sectors. We show that various features of the quench, which are related to Kibble-Zurek physics, are sensitive to the topological sector being probed, in particular, the overlap between the time-evolved initial ground state and an appropriate low-energy state of the final Hamiltonian. While most of our study is confined to translationally invariant systems, where momentum is a convenient quantum number, we briefly consider the effect of disorder and illustrate how this can influence the quench in a qualitatively different way depending on the topological sector considered.
We explore the salient features of the `Kitaev ladder', a two-legged ladder version of the spin-1/2 Kitaev model on a honeycomb lattice, by mapping it to a one-dimensional fermionic p-wave superconducting system. We examine the connections between spin phases and topologically non-trivial phases of non-interacting fermionic systems, demonstrating the equivalence between the spontaneous breaking of global Z(2) symmetry in spin systems and the existence of isolated Majorana modes. In the Kitaev ladder, we investigate topological properties of the system in different sectors characterized by the presence or absence of a vortex in each plaquette of the ladder. We show that vortex patterns can yield a rich parameter space for tuning into topologically non-trivial phases. We introduce and employ a new topological invariant for explicitly determining the presence of zero energy Majorana modes at the boundaries of such phases. Finally, we discuss dynamic quenching between topologically non-trivial phases in the Kitaev ladder and, in particular, the post-quench dynamics governed by tuning through a quantum critical point.
We explore the salient features of the 'Kitaev ladder', a two-legged ladder version of the spin-1/2 Kitaev model on a honeycomb lattice, by mapping it to a one-dimensional fermionic p-wave superconducting system. We examine the connections between spin phases and topologically non-trivial phases of non-interacting fermionic systems, demonstrating the equivalence between the spontaneous breaking of global Z 2 symmetry in spin systems and the existence of isolated Majorana modes. In the Kitaev ladder, we investigate topological properties of the system in different sectors characterized by the presence or absence of a vortex in each plaquette of the ladder. We show that vortex patterns can yield a rich parameter space for tuning into topologically non-trivial phases. We introduce and employ a new topological invariant for explicitly determining the presence of zero energy Majorana modes at the boundaries of such phases. Finally, we discuss dynamic quenching between topologically non-trivial phases in the Kitaev ladder and, in particular, the post-quench dynamics governed by tuning through a quantum critical point.