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arXiv:1409.1877

Density of states at disorder-induced phase transitions in a multichannel Majorana wireMaria-Theresa Rieder, Piet W. Brouwer

Maria-Theresa Rieder, Piet W. Brouwer

Published Sep 5, 2014• 0 citations
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Abstract

An $N$-channel spinless p-wave superconducting wire is known to go through a series of $N$ topological phase transitions upon increasing the disorder strength. Here, we show that at each of those transitions the density of states shows a Dyson singularity $\nu(\varepsilon) \propto \varepsilon^{-1}|\ln\varepsilon|^{-3} $, whereas $\nu(\varepsilon) \propto \varepsilon^{|\alpha|-1}$ has a power-law singularity for small energies $\varepsilon$ away from the critical points. Using the concept of superuniversality [Gruzberg, Read, and Vishveshwara, Phys. Rev. B 71, 245124 (2005)], we are able to relate the exponent $\alpha$ to the wire's transport properties at zero energy and, hence, to the mean free path $l$ and the superconducting coherence length $\xi$.

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An NNN-channel spinless p-wave superconducting wire is known to go through a series of NNN topological phase transitions upon increasing the disorder strength. Here, we show that at each of those transitions the density of states shows a Dyson singularity ν(ε)∝ε−1∣ln⁡ε∣−3\nu(\varepsilon) \propto \varepsilon^{-1}|\ln\varepsilon|^{-3} ν(ε)∝ε−1∣lnε∣−3, whereas ν(ε)∝ε∣α∣−1\nu(\varepsilon) \propto \varepsilon^{|\alpha|-1}ν(ε)∝ε∣α∣−1 has a power-law singularity for small energies ε\varepsilonε away from the critical points. Using the concept of superuniversality [Gruzberg, Read, and Vishveshwara, Phys. Rev. B 71, 245124 (2005)], we are able to relate the exponent α\alphaα to the wire's transport properties at zero energy and, hence, to the mean free path lll and the superconducting coherence length ξ\xiξ.
Superconducting wires with broken time-reversal and spin-rotational symmetries can exhibit two distinct topological gapped phases and host bound Majorana states at the phase boundaries. When the wire is tuned to the transition between these two phases and the gap is closed, Majorana states become delocalized leading to a peculiar critical state of the system. We study transport properties of this critical state as a function of the length LLL of a disordered multichannel wire. Applying a nonlinear supersymmetric sigma model of symmetry class D with two replicas, we identify the average conductance, its variance, and the third cumulant in the whole range of LLL from the Ohmic limit of short wires to the regime of a broad conductance distribution when LLL exceeds the correlation length of the system. In addition, we calculate the average thermal shot noise power and variance of the topological index for arbitrary LLL. The general approach developed in the paper can also be applied to study combined effects of disorder and topology in wires of other symmetries.
We study the interplay between disorder and interaction in one-dimensional topological superconductors which carry localized Majorana zero-energy states. Using Abelian bosonization and the perturbative renormalization group approach, we obtain the renormalization group flow and the associated scaling dimensions of the parameters and identify the critical points of the low-energy theory. We predict a quantum phase transition from a topological superconducting phase to a nontopological localized phase, and obtain the phase boundary between these two phases as a function of the electron-electron interaction and the disorder strength in the nanowire. Based on an instanton analysis which incorporates the effect of disorder, we also identify a large regime of stability of the Majorana-carrying topological phase in the parameter space of the model.
We present a numerical study of the quasi-particle density of states (DoS) of two-dimensional d-wave superconductors in the presence of smooth disorder. We find power law scaling of the DoS with an exponent depending on the strength of the disorder and the superconducting order parameter in quantitative agreement with the theory of Nersesyan et al. (Phys.Rev.Lett. 72, 2628 (1994)). For strong disorder a transition to a constant DoS occurs. Our results are in contrast to the case of short-ranged disorder.
We present a comprehensive theoretical study of interacting and disordered topological phases of coupled Kitaev wires, which may support further realistic applications of Majorana fermions. We develop a variety of analytical, mathematical, and numerical methods for one- and two-coupled wires, associated with a topological marker accessible from real-space correlation functions on the wire(s). We verify the stability of the topological superconducting phase and quantify disorder effects close to the quantum phase transitions, e.g., through two-point correlation functions or using a renormalization group analysis of disorder. We show that the double critical Ising (DCI) phase---a fractional Majorana liquid characterized by a pair of half central charges and topological numbers---is stabilized by strong interactions against disorder, which respects the inversion symmetry between the wires (i.e., parity conservation on each wire). In the presence of an interwire hopping term, the DCI phase turns into a protected topological phase with a bulk gap. We study the localization physics developing along the critical line for weaker interactions.
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Majorana zero modes (MZMs) appear as edge states in topological superconductors and are envisioned as promising candidates for application towards fault-tolerant quantum computation. Quantum information may be stored in a degenerate ground state subspace constructed from MZMs while quantum gates may be performed by braiding MZMs and exploiting their non-Abelian properties. While the topological protection of MZMs ensures that they are robust against decoherence, their dynamical manipulation can nonetheless induce undesirable transitions to excited states, manifesting as a “diabatic error.”This thesis studies the diabatic error accumulated as MZMs are transported across a superconducting wire, which is an essential building block for physical braiding schemes. The transport protocol considered relies on appropriately tuning sections (“piano keys”) of a wire between distinct topological phases via electric gates to control local chemical potentials. First, the diabatic error is examined for transport which uses multiple piano keys. The error is calculated through numerical simulation of the transport protocol and is complemented by approximate analytical expressions calculated using an effective, low-energy model. It is found that an optimal number of keys can be used to reduce the error. Furthermore, this thesis explores transport protocols performed on wires which feature disorder on the chemical potential and noise in the piano key tuning. The rise in the diabatic error due to these additional effects with respect to parameters-of-interest including the disorder strength, disorder correlation length, noise amplitude, and noise frequency cut-offs is characterized using numerical methods. For disorder, it is found that the error’s behaviour is principally controlled by the statistics of the minimum bulk energy gap. Spatially correlated disorder is also studied which presents non-monotonic behaviour for the error. Finally, for noise, it is found that optical transitions can cause a substantial increase in the error as the frequency modes within the noise approach the minimum energy gap.