Multi-particle content of Majorana zero-modes in the interacting p-wave wire

In the topological phase of p-wave superconductors, zero-energy Majorana quasi-particle excitations can be well-defined in the presence of local density-density interactions. Here we examine this phenomenon from the perspective of matrix representations of the commutator $\mathcal{H} =[H,\bullet]$ ,with the aim of characterising the multi-particle content of the many-body Majorana mode. To do this we show that, for quadratic fermionic systems, $\mathcal{H}$ can always be decomposed into sub-blocks that act as multi-particle generalisations of the BdG/Majorana forms that encode single-particle excitations. In this picture, density-density like interactions will break this exact excitation-number symmetry, coupling different sub-blocks and lifting degeneracies so that the eigen-operators of the commutator $\mathcal{H}$ take the form of individual eigenstate transitions $|n\rangle \langle m|$. However, the Majorana mode is special in that zero-energy transitions are not destroyed by local interactions and it becomes possible to define many-body Majoranas as the odd-parity zero-energy solutions of $\mathcal{H}$ that minimise their excitation number. This idea forms the basis for an algorithm which is used to characterise the multi-particle excitation content of the Majorana zero modes of the one-dimensional p-wave lattice model. We find that the multi-particle content of the Majorana zero-mode operators is significant even at modest interaction strengths. This has important consequences for the stability of Majorana based qubits when they are coupled to a heat bath. We will also discuss how these findings differ from previous work regarding the structure of the many-body-Majorana operators and point out that this should affect how certain experimental features are interpreted.Comment: 16 pages , 11 figure

Many-body Majorana operators and the equivalence of parity sectors

The one-dimensional p-wave topological superconductor model with open-boundary conditions is examined in its topological phase. Using the eigenbasis of the non-interacting system I show that, provided the interactions are local and do not result in a closing of the gap, then even and odd parity sectors are unitarily equivalent. Following on from this, it is possible to define two many-body operators that connect each state in one sector with a degenerate counterpart in the sector with opposite parity. This result applies to all states in the system and therefore establishes, for a long enough wire, that all even-odd eigenpairs remain essentially degenerate in the presence of local interactions. Building on this observation I then set out a full definition of the related many-body Majorana operators and point out that their structure cannot be fully revealed using cross-correlation data obtained from the ground state manifold alone. Although all results are formulated in the context of the 1-dimensional p-wave model, I argue why they should also apply to more realistic realisations (e.g. the multi-channel p-wave wire and proximity coupled models) of topological superconductivity.Comment: 8 pages, 1 figur

Near-zero-energy end states in topologically trivial spin-orbit coupled superconducting nanowires with a smooth confinement

A one-dimensional spin-orbit coupled nanowire with proximity-induced pairing from a nearby s-wave superconductor may be in a topological nontrivial state, in which it has a zero energy Majorana bound state at each end. We find that the topological trivial phase may have fermionic end states with an exponentially small energy, if the confinement potential at the wire's ends is smooth. The possible existence of such near-zero energy levels implies that the mere observation of a zero-bias peak in the tunneling conductance is not an exclusive signature of a topological superconducting phase even in the ideal clean single channel limit.Comment: 4 pages, 4 figure

Extraction of energy from gravitational waves by laser interferometer detectors

In this paper we discuss the energy interaction between gravitational waves and laser interferom- eter gravitational wave detectors. We show that the widely held view that the laser interferometer gravitational wave detector absorbs no energy from gravitational waves is only valid under the approximation of a frequency-independent optomechanical coupling strength and a pump laser without detuning with respect to the resonance of the interferometer. For a strongly detuned interferometer, the optical-damping dynamics dissipates gravitational wave energy through the interaction between the test masses and the optical field. For a non-detuned interferometer, the frequency-dependence of the optomechanical coupling strength causes a tiny energy dissipation, which is proved to be equivalent to the Doppler friction raised by Braginsky et.al.Comment: 20 pages, 7 figure

Topological Blocking in Quantum Quench Dynamics

We study the non-equilibrium 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 non-local maps of each of these systems, we effectively study spinless fermionic $p$-wave paired 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.Comment: 18 pages, 11 figure

Zero energy and chiral edge modes in a p-wave magnetic spin model

In this work we discuss the formation of zero energy vortex and chiral edge modes in a fermionic representation of the Kitaev honeycomb model. We introduce the representation and show how the associated Jordan-Wigner procedure naturally defines the so-called branch cuts that connect the topological vortex excitations. Using this notion of the branch cuts we show how to, in the non-Abelian phase of the model, describe the Majorana zero mode structure associated with vortex excitations. Furthermore we show how, by intersecting the edges between Abelian and non-Abelian domains, the branch cuts dictate the character of the chiral edge modes. In particular we will see in what situations the exact zero energy Majorana edge modes exist. On a cylinder, and for the particular instances where the Abelian phase of the model is the full vacuum, we have been able to exactly solve for the systems edge energy eigensolutions and derive a recursive formula that exactly describes the edge mode structure. Penetration depth is also calculated and shown to be dependent on the momentum of the edge mode. These solutions also describe the overall character of the fully open non- Abelian domain and are excellent approximations at moderate distances from the corners

Exact results for the star lattice chiral spin liquid

We examine the star lattice Kitaev model whose ground state is a a chiral spin liquid. We fermionize the model such that the fermionic vacua are toric code states on an effective Kagome lattice. This implies that the Abelian phase of the system is inherited from the fermionic vacua and that time reversal symmetry is spontaneously broken at the level of the vacuum. In terms of these fermions we derive the Bloch-matrix Hamiltonians for the vortex free sector and its time reversed counterpart and illuminate the relationships between the sectors. The phase diagram for the model is shown to be a sphere in the space of coupling parameters around the triangles of the lattices. The abelian phase lies inside the sphere and the critical boundary between topologically distinct Abelian and non-Abelian phases lies on the surface. Outside the sphere the system is generically gapped except in the planes where the coupling parameters are zero. These cases correspond to bipartite lattice structures and the dispersion relations are similar to that of the original Kitaev honeycomb model. In a further analysis we demonstrate the three-fold non-Abelian groundstate degeneracy on a torus by explicit calculation.Comment: 7 pages, 8 figure

Model of Thermal Wavefront Distortion in Interferometric Gravitational-Wave Detectors I: Thermal Focusing

We develop a steady-state analytical and numerical model of the optical response of power-recycled Fabry-Perot Michelson laser gravitational-wave detectors to thermal focusing in optical substrates. We assume that the thermal distortions are small enough that we can represent the unperturbed intracavity field anywhere in the detector as a linear combination of basis functions related to the eigenmodes of one of the Fabry-Perot arm cavities, and we take great care to preserve numerically the nearly ideal longitudinal phase resonance conditions that would otherwise be provided by an external servo-locking control system. We have included the effects of nonlinear thermal focusing due to power absorption in both the substrates and coatings of the mirrors and beamsplitter, the effects of a finite mismatch between the curvatures of the laser wavefront and the mirror surface, and the diffraction by the mirror aperture at each instance of reflection and transmission. We demonstrate a detailed numerical example of this model using the MATLAB program Melody for the initial LIGO detector in the Hermite-Gauss basis, and compare the resulting computations of intracavity fields in two special cases with those of a fast Fourier transform field propagation model. Additional systematic perturbations (e.g., mirror tilt, thermoelastic surface deformations, and other optical imperfections) can be included easily by incorporating the appropriate operators into the transfer matrices describing reflection and transmission for the mirrors and beamsplitter.Comment: 24 pages, 22 figures. Submitted to JOSA