Emerging Majorana Modes in Junctions of One-Dimensional Spin Systems
Domenico Giuliano, Andrea Trombettoni, Pasquale Sodano
备注：
29 pages, 2 .eps figures, Article to appear in "Roman Jackiw: 80th Birthday Festschrift", edited by A. Niemi, T. Tomboulis, and K. K. Phua (World Scientific, 2020)
The non-local effects induced by Majorana fermions in field theories for condensed matter systems are deeply related to the fermion charge fractionalization discovered by Roman Jackiw in relativistic field theories. We show how the presence of Majorana fermions may be mimicked in pertinent networks of spin chains inducing a spin analogue of the multi-channel Kondo effect. The relevance of this spin analogue of the Kondo effect for networks of Josephson arrays and Tonks-Girardeau gases is highlighted.
Electron transport through [normal metal]-[quantum dot]-[topological superconductor] junction is studied and reveals interlacing physics of Kondo correlations with two Majorana fermions bound state residing on the opposite edges of the topological superconductor. When the strength of the Majorana fermion coupling exceeds the temperature $T$, this combination of Kono-Majorana fermion physics can be observed: The usual peak of the temperature dependent zero biased conductance $σ(V=0,T)$ splits and the conductance has a {\it dip} at T=0. The height of the conductance side-peaks decreases with mag…
Exact zero modes in interacting Majorana X- and Y-junctions
Kavli Institute of Nanoscience, Delft University of Technology（代尔夫特理工大学卡弗里纳米科学研究所）；Rudolf Peierls Centre for Theoretical Physics, University of Oxford（牛津大学鲁道夫·皮尔斯理论物理中心）
We demonstrate that the current-voltage (I-V) characteristics of resistively and capacitively shunted Josephson junctions (RCSJs) hosting localized subgap Majorana states provides a phase sensitive method for their detection. The I-V characteristics of such RCSJs, in contrast to their resistively shunted counterparts, exhibit subharmonic odd Shapiro steps; such steps occur even in the absence of any $2π$ periodic terms in the current-phase relation of these junctions. These steps, owing to their subharmonic nature, exhibit qualitatively different properties compared to harmonic odd steps of co…
Planar Josephson junctions provide a versatile platform, alternative to the nanowire-based geometry, for the generation of the Majorana bound states, due to the additional phase tunability of the topological superconductivity. The proximity induction of chiral magnetism and superconductivity in a two-dimensional electron gas showed remarkable promises to manipulate topological superconductivity. Here, we consider a Josephson junction involving a skyrmion crystal and show that the chiral magnetism of the skyrmions can create and control the Majorana bound states without the requirement of an in…
We present a class of exactly solvable tri-junctions of one- and two-dimensional spin systems. Based on the geometric criterion for solvability, we clarify the sufficient condition for the junctions so that the spin Hamiltonian becomes equivalent to Majorana quadratic forms. Then we examine spin tri-junctions using the obtained solvable models. We consider the transverse magnetic field Ising spin chains and reveal how Majorana zero modes appear at the tri-junctions of the chains. Local terms of the tri-junction crucially affect the appearance of Majorna zero modes, and the tri-junction may sup…
One of the most promising proposals for engineering topological superconductivity and Majorana fermions employs a spin-orbit coupled nanowire subjected to a magnetic field and proximate to an s-wave superconductor. When only part of the wire's length contacts to the superconductor, the remaining conducting portion serves as a natural lead that can be used to probe these Majorana modes via tunneling. The enhanced role of interactions in one dimension dictates that this configuration should be viewed as a superconductor-Luttinger liquid junction. We investigate such junctions between both helica…
We found that QED in the lowest eigenstate of the microwave-cavity, capacitively coupled with Majorana populated Josephson junctions array, is described with an effective $ϕ^{4}$ model emerging in the finite domain of the system parameters. Strong anharmonicity of the cavity mode arises due to electric-dipole phase transition facilitated by the single-electron zero mode and Cooper pair tunneling between superconducting islands. An instantonic meandering of the electromagnetic field potential at finite temperatures is predicted.
We present a powerful and general approach to describe the coupling of Majorana fermions to external leads, of interacting or non-interacting electrons. Our picture has the Klein factors of bosonization appearing as extra Majoranas hybridizing with the physical ones. We demonstrate the power of this approach by solving a highly nontrivial SO(M) Kondo problem arising in topological superconductors with M Majorana-lead couplings, allowing for arbitrary M and for conduction electron interactions. We find that these topological Kondo problems give rise to robust non-Fermi liquid behavior, even for…
The search for Majorana bound states in solid-state physics has been limited to materials which display a gap in their bulk spectrum. We show that such unpaired states appear in certain quasi-one-dimensional Josephson junctions arrays with gapless bulk excitations. The bulk modes mediate a coupling between Majorana bound states via the Ruderman-Kittel-Yosida-Kasuya mechanism. As a consequence, the lowest energy doublet acquires a finite energy difference. For realistic set of parameters this energy splitting remains much smaller than the energy of the bulk eigenstates even for short chains of …
An array of superconducting islands with semiconducting nanowires in the right regime provides a macroscopic implementation of Kitaev's toy model for Majorana wires. We show that a capacitive coupling between adjacent islands leads to an effective interaction between the Majorana modes. We demonstrate that even though strong repulsive interaction eventually drive the system into a Mott insulating state the competition between the (trivial) band-insulator and the (trivial) Mott insulator leads to an interjacent topological insulating state for arbitrary strong interactions.
Time-reversal invariant topological superconductors are characterized by the presence of Majorana Kramers pairs localized at defects. One of the transport signatures of Majorana Kramers pairs is the quantized differential conductance of $4e^2/h$ when such a one-dimensional superconductor is coupled to a normal-metal lead. The resonant Andreev reflection, responsible for this phenomenon, can be understood as the boundary condition change for lead electrons at low energies. In this paper, we study the stability of the Andreev reflection fixed point with respect to electron-electron interactions …
The point contact tunnel junctions between a one-dimensional topological superconductor and single-channel quantum Hall (QH) liquids are investigated theoretically with bosonization technology and renormalization group methods. For the $ν=1$ integer QH liquid, the universal low-energy tunneling transport is governed by the perfect Andreev reflection fixed point with quantized zero-bias conductance $G(0)=2e^{2}/h$, which can serve as a definitive fingerprint of the existence of a Majorana fermion. For the $ν=1/m$ Laughlin fractional QH liquids, its transport is governed by the perfect normal re…
Experimental evidence was recently obtained for topological superconductivity in spin-orbit coupled nano wires in a magnetic field, proximate to an s-wave superconductor. When only part of the wire contacts the superconductor, a localized Majorana mode exists at the junction between superconducting and normal parts of the nanowire. We consider here the case of a T-junction between the superconductor and two normal nanowires and also the case of a single wire with two (or more) partially filled bands in the normal part. We find that coupling this 2-channel Luttinger liquid to the single Majoran…
We study a Majorana island coupled to a bulk superconductor via a Josephson junction and to multiple external normal leads. In the absence of the Josephson coupling, the system displays a topological Kondo state, which had been largely studied recently. However, we find that this state is unstable even to small Josephson coupling, which instead leads at low temperature $T$ to a new fixed point. Most interesting is the case of three external leads, forming a minimal electronic realization of the long sought two-channel Kondo effect. While the $T=0$ conductance corresponds to simple resonant And…
We study theoretically charge transport properties of normal metal (N) / ferromagnet insulator (FI) / superconductor (S) junction and S/FI/S junction formed on the surface of three-dimensional topological insulator (TI), where chiral Majorana mode (CMM) exists at FI/S interface. We find that CMM generated in N/FI/S and S/FI/S junctions are very sensitively controlled by the direction of the magnetization ${\bm m}$ in FI region. Especially, the current-phase relation of Josephson current in S/FI/S junctions has a phase shift neither 0 nor $π$, which can be tuned continuously by the component of…
We consider a Josephson contact mediated by 1D chiral modes on a surface of a 3D topological insulator with superimposed superconducting and magnetic layers. The system represents an interferometer in which 1D chiral Majorana modes on the boundaries of superconducting electrodes are linked by ballistic chiral Dirac channels. We investigate the noise of the Josephson current as a function of the dc phase bias and the Aharonov-Bohm flux. Starting from the scattering formalism, a Majorana representation of the Keldysh generating action for cumulants of the transmitted charge is found. At temperat…
We study Majorana modes and transport in one-dimensional systems with a $p$-wave superconductor (SC) and normal metal leads. For a system with a SC lying between two leads, it is known that there is a Majorana mode at the junction between the SC and each lead. If the $p$-wave pairing $Δ$ changes sign or if a strong impurity is present at some point inside the SC, two additional Majorana modes appear near that point. We study the effect of all these modes on the sub-gap conductance between the leads and the SC. We derive an analytical expression as a function of $Δ$ and the length $L$ of the SC…
We study the proximity effect between an s-wave superconductor and the surface states of a strong topological insulator. The resulting two dimensional state resembles a spinless p_x+ip_y superconductor, but does not break time reversal symmetry. This state supports Majorana bound states at vortices. We show that linear junctions between superconductors mediated by the topological insulator form a non chiral 1 dimensional wire for Majorana fermions, and that circuits formed from these junctions provide a method for creating, manipulating and fusing Majorana bound states.
Majorana fermions are predicted to arise at the ends of nanowire devices which combine superconductivity, strong spin-orbit coupling and an external magnetic field. By manipulating networks of these devices with suitable gating, it has been suggested that braiding operations may be performed which act as logic operations, suitable for quantum computation. However, the unavoidable misalignment of the magnetic field in any realistic device geometry has raised questions about the feasibility of such braiding. In this paper, we numerically simulate braiding operations in devices with Y-junction an…
Topological superconductivity supports exotic Majorana bound states (MBS) which are chargeless zero-energy emergent quasiparticles. With their non-Abelian exchange statistics and fractionalization of a single electron stored nonlocally as a spatially separated MBS, they are particularly suitable for implementing fault-tolerant topological quantum computing. While the main efforts to realize MBS have focused on one-dimensional systems, the onset of topological superconductivity requires delicate parameter tuning and geometric constraints pose significant challenges for their control and demonst…
We propose and analyze theoretically an experimental setup for detecting the elusive Majorana particle in semiconductor-superconductor heterostructures. The experimental system consists of one-dimensional semiconductor wire with strong spin-orbit Rashba interaction embedded into a superconducting quantum interference device. We show that the energy spectra of the Andreev bound states at the junction are qualitatively different in topologically trivial (i.e., not containing any Majorana) and nontrivial phases having an even and odd number of crossings at zero energy, respectively. The measureme…
In this paper we study a novel superconductor-ferromagnet-superconductor (SC-FM-SC) Josephson junction array deposited on top of a two-dimensional quantum spin Hall (QSH) insulator. The existence of Majorana bound states at the interface between SC and FM gives rise to charge-e tunneling, in addition to the usual charge-2e Cooper pair tunneling, between neighboring superconductor islands. Moreover, because Majorana fermions encode the information of charge number parity, an exact Z_2 gauge structure naturally emerges and leads to many new insulating phases, including a deconfined phase where e…
While the application of out-of-plane magnetic fields was, so far, believed to be detrimental for the formation of Majorana phases in artificially engineered hybrid superconducting-semiconducting junctions, several recent theoretical studies have found it indeed useful in establishing such topological phases 1-5. Majorana phases emerge as quantized plateaus in the magnetoconductance of the hybrid junctions based on two-dimensional electron gases (2DEG) under fully out-of-plane magnetic fields. The large transverse Rashba spin-orbit interaction in 2DEG, together with a strong magneto-orbital ef…
We study the effects of Majorana modes and interactions between electrons on transport in a one-dimensional system with a junction of three p-wave superconductors (SCs) which are connected to normal metal leads. For sufficiently long SCs, there are zero energy Majorana modes at the junctions between the SCs and the leads,and, depending on the signs of the p-wave pairings in the three SCs, there can also be one or three Majorana modes at the junction of the three SCs. We show that the various sub-gap conductances have peaks occurring at the energies of all these modes; we therefore get a rich p…
