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 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.
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.
Topological superconductors are believed to host exotic quasiparticle excitations known as Majorana zero modes (MZMs), with much of the evidence based on BCS mean-field theory. The direct application of mean-field arguments is tenuous in finite, isolated systems relevant in some experiments. Here, we develop a new correlation-based method for identifying MZMs in interacting, number-conserving systems. Using the density matrix renormalization group, we study fermion number-conserving models with long-range interactions, which under periodic boundary conditions exhibit robust topological and nontopological superconductivity, tuned by the strength of interaction [Ortiz et al., Phys. Rev. Lett. 113, 267002 (2014)PRLTAO0031-900710.1103/PhysRevLett.113.267002]. We find evidence that, on the topological side, Majorana edge modes appear in open chains, manifesting as the vanishing of the energy splitting between odd- and even-parity ground states with increasing system size. Additionally, off-diagonal two-point correlation functions show nonlocal, parity-dependent edge effects. These correlations reveal the spatial structure of Majorana modes in the many-body wave function. We show that the correlation diagnostic applies broadly, including to short-range interacting models, where topological superconductivity is more fragile due to the absence of a bulk excitation gap.
The topological properties of the Kitaev chain model with extended-range interactions are investigated, focusing on cases where the topological winding number is preserved. We assume that the pairing and hopping terms decay algebraically in space with exponents α and β, respectively. We show that in the truncated-range scenario, there are as many distinct topological phases as the number of coupled neighboring sites. In addition, an explicit analytical formulation is provided to evaluate the topological invariant and the phase transitions that emerge in these systems. Besides the analytical description, we introduce a new physical insight into the topological excitations of the ground-state by measuring the spatial distribution of the edge modes with the Majorana average position. Taking the next-nearest neighbor Kitaev chains as a probe, various numerical calculations of Majorana edge states were performed in finite-size clusters to determine the sensitivity of the topological zero energy modes to the parameters of interest. The occupation of edge-to-edge non-local fermion states is computed and defined as an effective parity. Such an effective parity exhibits new interesting features beyond the energetic exchange from the ground-state fermion parity switches, which are related to the distribution of the respective edge modes. Our calculations show a direct correlation between the ground-state fermion parity and the edge occupation numbers, which are translated into localization and delocalization of the Majorana average position.
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.
Majorana zero modes provide a route to fault-tolerant qubits by encoding information non-locally in fermion parity. Their sensitivity to noise is expected to decrease exponentially with increasing separation between the Majoranas, a suppression known as topological protection. Kitaev chains engineered in quantum dot-superconductor arrays provide a tunable platform in which separated Majorana zero modes can emerge at the ends of the chain, even in two-site chains. These minimal-chain modes are known as poor man's Majoranas and retain characteristic Majorana properties, including near-zero energy and equal electron-hole character, but have only limited protection. A key outstanding challenge is to move beyond identifying such modes in electrical transport measurements and achieve coherent qubit control in the time domain. Here, we demonstrate a Majorana parity qubit by realizing coherent coupling between two-site Kitaev chains. Since total fermion parity is conserved, the system separates into global even and odd parity manifolds. We observe coherent parity oscillations in both manifolds with equal oscillation frequencies at the Majorana sweet spot, as predicted for isolated Majorana zero modes. We further show that the oscillation frequency and coherence depend systematically on inter-chain coupling and quantum-dot detunings, in close agreement with our model for short, partially protected chains. Our results establish the first coherent control of a Majorana qubit, encoded in the fermion parity of Majorana zero modes in minimal Kitaev chains.