Session E29: Majorana Fermions

Spinon Majorana fermion

A new realization of Majorana fermions is proposed in the frustrated magnets via the topological proximity effect. Specifically, we consider the interface between a topological insulator and a frustrated magnetic material. Using the renormalization group-based mean-field theory, and calculating the self-energy correction due to the topological insulator, we find that the spin texture and the spin-momentum locking of the Dirac cone will be inherited by the spinons in the nearby frustrated magnets. This leads to a particular topological state of matter that supports the Majorana excitation. Unlike the conventional ones, these Majorana fermions are the composite states of spinons and anti-spinons, rather than electrons and holes. They can also participate in the transport of spinons, resulting in nontrivial spin current, while the charge current is completely frozen.   

Chiral Majorana Interference in Quantum Anomalous Hall-Superconductor Junctions

We study the interference of the edge chiral Majorana fermions in junctions of quantum anomalous Hall (QAH) insulators and superconductors (SCs). We show the two chiral Majorana fermions on a QAH edge in contact with an SC generically have a momentum difference $\Delta k$ that depends on the chemical potentials of both the QAH insulator and the SC. Due to the spacial interference induced by $\Delta k$, the longitudinal conductance of normal SC/QAH junctions exhibits an oscillation with respect to the edge lengths and the chemical potentials, which can be measured in transport experiments. Further, we show for QAH/chiral topological SC junctions where there is only one edge chiral Majorana fermion, the dynamical fluctuation of the SC phase yields a geometrical correction to the longitudinal conductance usually derived.   

Majorana Fermions at the End of Topological Insulator Nanoribbon

Majorana zero modes can exist at the end of a 1D p-wave SC. 1D semiconductor nanowire approximated a s-wave superconductor is a famous one of those proposals. In which, strong Zeeman field is required to have a large topological region, but unfortunately suppresses superconducting pairing and makes the system more sensitive to disorder. Here we propose a Nanoribbon system made of 2D topological insulator where finite size effect due to the narrow width between two edges plays an important role. A ferromagnetic insulator and an s-wave superconductor are attached at each edge, respectively. We introduce a low energy effective model to investigate the superconducting phase diagram. And, the disorder effect is studied analytically by using the self-consistent Born approximation(SCBA). Furthermore, realistic numerically calculation is carried out with a tight-binding model. We demonstrate that, strong Zeeman field generates a large topological region, and at the same time enhances superconducting pairing and makes the system more immune to disorder.   

Molecular Andreev bound states and Majorana modes in a double dot system

Nanostructured systems such as quantum dots (QD) connected to superconductors has attracted a lot of attention in the recent years. One of the well known phenomena in such a system is the formation of a pair of bound called Andreev bound states (ABS)\footnote{T. Meng, S. Florens, and P. Simon, Phys. Rev. B {\bf 79}, 224521 (2009).}. Recently, it have been shown that when a QD is coupled to a topological superconductor wire, a Majorana bound state (MBS) leaks from the end of the wire into the dot\footnote{E. Vernek, P. H. Penteado, A. C. Seridonio, and J. C. Egues, Phys. Rev. B {\bf 89}, 165314 (2014).}. The character of these bound states is much reacher in structures like molecules and is far from being completely understood. In this work we study a system composed by a two inter-connected QDs in which one of then is coupled to a normal superconductor and to a normal lead while the other is coupled to a topological superconductor and to a distinct normal metallic lead. We show that in the atomic limit (for small interdot coupling), one of the dot has a pair of ABS whereas the other has a single a MBS. More interestingly, in the molecular regime (large inter-dot coupling) we observe a localized Majorana mode coexisting with a delocalized molecular ABS.   

Coupled wire model of symmetric Majorana surfaces of topological superconductors I: 4-fermion gapping interactions

Time reversal symmetric topological superconductors in three spatial dimensions carry gapless surface Majorana fermions. They are robust against any time reversal symmetric single-body perturbation weaker than the bulk energy gap. We mimic the massless surface Majorana's by coupled wire models in two spatial dimensions. We introduce explicit many-body interwire interactions that preserve time reversal symmetry and give energy gaps to all low energy degrees of freedom. The gapping 4-fermion interactions are constructed by interwire Kac-Moody current backscattering and rely on the fractionalization or conformal embedding of the Majorana wires.   

Coupled wire model of symmetric Majorana surfaces of topological superconductors II: 32-fold periodic topological orders

We mimic the massless surface Majorana's of topological superconductors by coupled wire models in two spatial dimensions, and introduce many-body gapping interactions that preserve time reversal symmetry. Coupling with a $\mathbb{Z}_2$ gauge theory, the symmetric gapped surface generically carries a non-trivial $G_N$ topological order, where $N$ is the number of Majorana species and $G_N$ is some $SO(r)_1$ or $SO(3)_3$-like topological state. These form a 32-fold periodic class $G_N\cong G_{N+32}$, and a $\mathbb{Z}_{32}$ {\em relative} tensor product structure $G_{N_1}\otimes_bG_{N_2}\cong G_{N_1+N_2}$ by anyon condensation. We present the anyon structures of these topological states, and understand the topological orders through bulk-boundary correspondence and the Wilson structures on a torus geometry.   

Topological Massive Dirac Edge Modes and Long-Range Superconducting Hamiltonians

We discover novel topological effects in the one-dimensional Kitaev chain modified by long-range Hamiltonian deformations in the hopping and pairing terms. This class of models display symmetry-protected topological order measured by the Berry phase of the ground state and the winding number of the Hamiltonians. For exponentially-decaying hopping amplitudes, the topological sector can be significantly augmented as the penetration length increases, something experimentally achievable. For power-law decaying superconducting pairings, the massless Majorana modes at the edges get paired together into a massive non-local Dirac fermion localised at both edges of the chain: a new topological quasiparticle that we call topological massive Dirac fermion. This topological phase has fractional topological numbers as a consequence of the long-range couplings. Possible applications to current experimental setups and topological quantum computation are also discussed.   

Local and non-local conductance of one-dimensional topological superconductor with Majorana bound states

A one-dimensional topological superconductor has a full pairing gap in the bulk and Majorana bound states on the boundary. It is well known that the existence of Majorana bound states enables a quantized resonant local Andreev reflection. Here we find that non-local topological signatures can also be induced by the Majorana bound states in a simple setup, even though the bulk is fully gapped. The non-local conductance, which can even be quantized, depends crucially on the symmetries, the topological index, and the mesoscopic properties of the bulk. Coulomb interactions can further drive the hybrid system into a novel phase, which can be understood by a set of renormalization-group flow equations.   

Phase diagrams and transport signatures of Luttinger liquid - Majorana Kramers pair junction.

Majorana Kramers pairs appearing at the ends of a time reversal invariant topological superconductor lead to a quantized conductance of {\$}4e\textasciicircum 2/h{\$} due to perfect Andreev reflection. We study the stability of Andreev reflection fixed point with respect to electron-electron interactions present in the nanowire and calculate the phase diagram for the system. We find that the low energy properties are determined by local or crossed Andreev reflection fixed points. We analyze transport properties of the junction at these fixed points.   

Topological Phases of Inhomogeneous Superconductivity

We theoretically consider the effect of a spatially periodic modulation of the superconducting order parameter on the formation of Majorana fermions induced by a one-dimensional system with magnetic impurities brought into close proximity to an $s$-wave superconductor. When the magnetic exchange energy is larger than the inter-impurity electron hopping we model the effective system as a chain of coupled Shiba states. While in the opposite regime, the effective system is accurately described by a quantum wire model. Upon including a spatially modulated superconducting pairing, we find, for sufficiently large magnetic exchange energy, the system is able to support a single pair of Majorana fermions with one Majorana fermion on the left end of the system and one on the right end. When the modulation of superconductivity is large compared to the magnetic exchange energy, the Shiba chain returns to a trivially gapped regime while the quantum wire enters a new topological phase capable of supporting two pairs of Majorana fermions.   

Kramers pair of Majoranas in contact with a Luttinger Liquid

I will discuss the signatures of the Kramers pair of Majorana coupled to a quantum wire. I will start from the non-interacting case, showing perfect Andreev process at zero energy -- either intrachannel, or interchannel. I will show that turning on the interactions will make one of the Andreev process stable, and another unstable. I will discuss the stability diagram of the fixed points as a function of interactions strength and show that strong enough interactions lead to a non-trivial fixed point.   

Superconductor disorder and strong proximity coupling effects in Majorana nanowires

Topological superconductivity induced by proximity to a conventional superconductor is only robust against moderate disorder in the parent superconductor, and only when the energy scale of the interface coupling is much smaller than the parent gap. I present detailed calculations of proximity-induced superconductivity in one-dimensional, spin-orbit coupled, semiconductor nanowires when the parent superconductor disorder and interface coupling exceed this limit. This parameter regime is characterized by unique spectroscopic signatures on both sides of the external field tuned topological phase transition.   

Disorder-induced topological transitions in multichannel Majorana wires

In this work, we investigate the effect of disorder on the topological properties of multichannel superconductor nanowires. While the standard expectation is that the spectral gap is closed and opened at transitions changing the topological property of the ground state, we show that the closing and opening of a \textit{transport} gap can also cause topological transitions, even in the presence of (localized) states at both sides of the transition. Such transport gaps, induced by disorder, can thus change the topological index, driving a topologically trivial wire into a nontrivial state. We focus on nanowires exhibiting \textit{p}-wave superconductivity as well as Rashba semiconductor nanowires in proximity to a conventional superconductor, and obtain analytical formulas for topological transitions in these wires, valid for generic realizations of disorder, generalizing earlier results. Full tight-binding simulations show excellent agreement with our analytical results without any fitting parameters.