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In this study, the Majorana equation for particles with arbitrary spin is solved for a half-integer spin free particle. The solution for the fundamental state, corresponding to the reference frame in which the particle is at rest, is compared with that obtained using the Dirac equation, especially as regards the approximation in the relativistic limit, in which the speed of the particle is close to that of light. Furthermore, the solutions that Majorana defines unphysical, proving that their occupation probability increases with the particle velocity, are taken into consideration. The anomalous behavior exhibited by these states also shows that for high-energy particles with small mass, transitions from a bradyonic state to a tachyonic state become possible., Comment: 17 pages, 2 figures

A Majorana fermion is the single fermionic particle that is its own antiparticle. Its dynamics is determined by the Majorana equation, where the spinor field is by definition equal to its charge-conjugate field. In this paper, we investigated Shannon's entropy of linear Majorana fermions to understand how this quantity is modified due to an external potential of the linear type linear. Subsequently, we turn our attention to the construction of an ensemble of these Majorana particles to study the thermodynamic properties of the model. Finally, we show how Shannon's entropy and thermodynamic properties are modified under the linear potential action., 26 pages, 7 figures

In general, the relativistic wave equation considered to mathematically describe the so-called Majorana particle is the Dirac equation with a real Lorentz scalar potential plus the so-called Majorana condition. Certainly, depending on the representation that one uses, the resulting differential equation changes. It could be a real or a complex system of coupled equations, or it could even be a single complex equation for a single component of the entire wave function. Any of these equations or systems of equations could be referred to as a Majorana equation or Majorana system of equations because it can be used to describe the Majorana particle. For example, in the Weyl representation, in (3+1) dimensions, we can have two non-equivalent explicitly covariant complex first-order equations; in contrast, in (1+1) dimensions, we have a complex system of coupled equations. In any case, whichever equation or system of equations is used, the wave function that describes the Majorana particle in (3+1) or (1+1) dimensions is determined by four or two real quantities. The aim of this paper is to study and discuss all these issues from an algebraic point of view, highlighting the similarities and differences that arise between these equations in the cases of (3+1) and (1+1) dimensions in the Dirac, Weyl, and Majorana representations. Additionally, to reinforce this task, we rederive and use results that come from a procedure already introduced by Case to obtain a two-component Majorana equation in (3+1) dimensions. Likewise, we introduce for the first time a somewhat analogous procedure in (1+1) dimensions and then use the results we obtain., Comment: 38 pages

We present a pedagogical review of topological superconductivity and its consequences in spin-orbit-coupled semiconductor/superconductor heterostructures. We start by reviewing the historical origins of the notions of Dirac and Majorana fermions in particle physics and discuss how lower dimensional versions of these emerge in one-dimensional superconductors. Ultimately, we focus on Majorana zero modes, which emerge at defects in the Majorana equation. We then review the definition of the topological invariant, and how it allows the prediction of such Majorana modes from the bulk bandstructure of realistic superconductors, which do not have Lorentz invariance. Finally, we end with a discussion of protocols for how to detect such Majorana modes and use them for topological quantum computation.

A biquaternionic version of the Dirac Equation is introduced, with a procedure for converting four-component spinors to elements of the Pauli algebra. In this version, mass appears as a coefficient between the 4-gradient of a spinor and its image under an outer automorphism of the Pauli algebra. The charge conjugation operator takes a particulary simple form in this formulation and switches the sign of the mass coefficient, so that for a solution invariant under charge conjugation the mass has to equal zero. The multiple of the charge conjugation operator by the imaginary unit turns out to be a complex Lorentz transformation. It commutes with the outer automorphism, while the charge conjugation operator itself anticommutes with it, providing a second more algebraic proof of the main theorem. Considering the Majorana equation, it is shown that non-zero mass of its solution is imaginary.

In A symmetric theory of electrons and positrons [1] of 1937 Majorana published his theoretical discovery of the completely neutral spinors. Among the known elementary particles, neutrinos may be such, namely equal to their antiparticles. Other cases are hypothetically foreseen by Supersymmetry, such as the neutralino and the gluino. The Majorana solution of the Dirac equation, his representation of the Dirac γ matrices and the properties of his spinor are described in the first chapter of this book by A. Zichichi, to which I refer the reader (AZ in the following).

A Majorana fermion is the single fermionic particle that is its own antiparticle. Its dynamics is determined by the Majorana equation where the spinor field $(\psi)$ is by definition equal to its charge-conjugate field $(\psi_c)$. Here, we study the dynamics of Majorana fermions in the presence of the most general static external field in $1+1$ dimensions, which is just a scalar potential, by implementing for the first time the methods of supersymmetric quantum mechanics. In particular, for potentials for which shape invariance holds, we show how to obtain analytical solutions. We find that although this equation does have bound states, it does not have stationary states. The approach is illustrated with a linear potential., Comment: 11 pages, 1 figure

Topological superconductor is described by a full pairing gap in the bulk with nonzero topological number and gapless surface states consisting of Majorana fermion that is a hypothetical particle of its own. Here, being its own means that a Majorana fermion should be an equal superposition of an electron and a hole state. The emergence of Majorana fermions is the most prominent characteristic of topological superconductors. In particle physics, it is still unclear if there are some elementary particles that are Majorana fermions, but importantly, they are likely to exist as quasiparticle excitations in certain condensed matter systems. It has drawn much attention in the content of Majorana fermions recently in particular the condensed matter physics area, since these Majorana states are ideal platform for non-Abelian statistics studies and can be used to fabricate as topological qubit, thus having great potential application in fault-tolerant topological quantum computation. A general introduction to the remarkable properties of Majorana fermions in condensed matter systems will be introduced following the story from Dirac equation to Majorana equation. Then this review elaborates a variety of routes to topological superconductivity in the realm of condensed mater physics, from Majorana modes in zero dimension to one dimension, that is, Majorana bound state and Majorana edge state. These Majorana states are localized states and propagating states respectively, but still obey non-Abelian statistics and remain their topological properties. Specifically, one-dimensional tight-binding model of a p-wave superconductor and a magnetic Fe chain with superconducting pairing are representative platforms for hosting Majorana bound states at the wires ends. Some emergent topological material systems, such as topological insulator, quantum spin Hall insulator, and quantum anomalous Hall insulator are two-dimensional material systems that can not only accommodate Majorana bound states but also Majorana edge states. Different material systems, from one-dimension quantum wire to two-dimension material system, from hybridized system to intrinsic system, will also be discussed specifically. Relevant theoretical studies and experimental results that show possible signatures of topological superconductivity and Majorana states are summarized as well. Particularly, experimental signatures of these Majorana states, such as zero-bias conductance peaks in tunneling spectra due to Majorana bound states, quantization of conductance in magneto-electric transport measurements due to chiral Majorana edge states, and some unconventional superconductivities in intrinsic topological superconductors will be briefly reviewed. Insights about methods to perspectives to realize topological quantum computation is provided at the end, emphasizing the braiding using different Majorana states. The goal of this review is to provide a general introduction to the subject for either experimentalists or theorists who are new to this field, focusing on the aspects and current progresses which are most important for understanding these basic physics.

This study reconsiders the decay of an ordinary particle in bradyons, tachyons and luxons in the field of the relativistic quantum mechanics. Lemke already investigated this from the perspective of covariant kinematics. Since the decay involves both spacelike and timelike particles, the study uses the Majorana equation for particles with an arbitrary spin. The equation describes the tachyonic and bradyonic realms of massive particles, and approaches the problem of how spacelike particles might develop. This method confirms the kinematic constraints that Lemke theory provided and proves that some possible decays are more favourable than others are., 9 pages

This paper focuses on Hermann Weyl’s two-component theory and frames it within the early development of different theories of spinors and the history of the discovery of parity violation in weak interactions. In order to show the implications of Weyl’s theory, the paper discusses the case study of Ettore Majorana’s symmetric theory of electron and positron (1937), as well as its role in inspiring Case’s formulation of parity violation for massive neutrinos in 1957. In doing so, this paper clarifies the relevance of Weyl’s and Majorana’s theories for the foundations of neutrino physics and emphasizes which conceptual aspects of Weyl’s approach led to Lee’s and Yang’s works on neutrino physics and to the solution of the theta-tau puzzle in 1957. This contribution thus sheds a light on the alleged “re-discovery” of Weyl’s and Majorana’s theories in 1957, by showing that this did not happen all of a sudden. On the contrary, the scientific community was well versed in applying these theories in the 1950s on the ground of previous studies that involved important actors in both Europe and United States.

We suggest that the Majorana neutrino should be regarded as a Bogoliubov quasiparticle that is consistently understood only by use of a relativistic analogue of the Bogoliubov transformation. The unitary charge conjugation condition ${\cal C}\psi{\cal C}^{\dagger}=\psi$ is not maintained in the definition of a quantum Majorana fermion from a Weyl fermion. This is remedied by the Bogoliubov transformation accompanying a redefinition of the charge conjugation properties of vacuum, such that a C-noninvariant fermion number violating term (condensate) is converted to a Dirac mass. We also comment on the chiral symmetry of a Majorana fermion; a massless Majorana fermion is invariant under a global chiral transformation $\psi\rightarrow \exp[i\alpha\gamma_{5}]\psi$ and different Majorana fermions are distinguished by different chiral $U(1)$ charge assignments. The reversed process, namely, the definition of a Weyl fermion from a well-defined massless Majorana fermion is also briefly discussed., Comment: 15 pages; correction of the formula after eq. (6), in the version published in Phys. Lett. B

We study Majorana states near Abrikosov vortices in a 2D topological superconductor in the applied magnetic field B. The Majorana fermions always arise in pairs. In the considered case, the first Majorana fermion localizes in the vortex core while the second, exterior, Majorana fermion localizes at the distance r ∝ 1/B away from the core. We calculate the hybridization between the vortex-core and the exterior Majorana fermions in the cases of a single vortex, two vortices, and the Abrikosov vortex lattice. We show that the hybridization can be effectively governed by the applied magnetic field if the chemical potential is tuned near the Dirac point. We also show that in the case of the vortex lattice, the hybridization between the vortex-core and external Majorana fermions affects significantly low-energy spectrum giving rise to the gap between two lowest Majorana energy bands.

Majorana spinors are constructed in terms of the multivectors of relativistic Cl1,3 algebra. Such spinors are found to be multiplied by primitive idempotents which drastically change spinor properties. Running electronic waves are used to solve the real Dirac–Majorana equation transformed to Cl1,3 algebra. From the analysis of the solution it is concluded that free Majorana particles do not exist, because relativistic Cl1,3 algebra requires the massive Majorana particle to move with light velocity.

In this study, a novel theory to investigate the mass oscillation of particles is proposed. It has been proven that, at high-energy conditions, the fermion field described by Dirac Lagrangian interacts with the half-integer spin tachyon field with negative energy, causing the formation of composite particles whose mass depends on the total angular momentum. The proposed theory is based on a new interpretation of the Majorana equation for particles with arbitrary spin and shows that mass oscillation is a phenomenon in which the component of particle decay prevails over that of mixing mass states. Using the kinematic of Lemke for spacelike particle decay, we propose a mechanism able to explain the neutrino flavour change. The proposed mechanism is also investigated concerning the shape of its spectrum. Finally, the Lagrangian field of composite particles is formulated., Comment: 18 pages, 1 figure

We present a basic formalism for using the Weyl spinor notation in Feynman rules. We focus on Weyl spinors with mixed Dirac and Majorana mass terms. To clarify the definitions we derive the Feynman rules from the path integral and present two examples: loop corrections for a fermion propagator and a tree level analysis of a seesaw toy model.

We consider different configurations of ac driven quantum dots coupled to superconductor leads where Majorana fermions can exist as collective quasiparticles. The main goal is to tune the existence, localization and properties of these zero energy quasiparticles by means of periodically driven external gates. In particular, we analyze the relevance of the system and driving symmetry. We predict the existence of different sweet spots with Floquet Majorana fermions in configurations where they are not present in the undriven system., We acknowledge the Spanish Ministry of Economy and Competitiveness through Project no. MAT2014-58241-P and the associated FPI scholarship (M.B.).

We study the full counting statistics of interferometers for chiral Majorana fermions with two incoming and two outgoing Dirac fermion channels. In the absence of interactions, the FCS can be obtained from the 4×4 scattering matrix S that relates the outgoing Dirac fermions to the incoming Dirac fermions. After presenting explicit expressions for the higher-order current correlations for a modified Hanbury Brown–Twiss interferometer, we note that the cumulant-generating function can be interpreted such that unit-charge transfer processes correspond to two independent half-charge transfer processes, or alternatively, to two independent electron-hole conversion processes. By a combination of analytical and numerical approaches, we verify that this factorization property holds for a general SO(4) scattering matrix, i.e. for a general interferometer geometry.

We evaluate the Majorana phases for a general $3\times3$ complex symmetric neutrino mass matrix on the basis of Mohapatra-Rodejohann's phase convention using the three rephasing invariant quantities $I_{12}$, $I_{13}$ and $I_{23}$ proposed by Sarkar and Singh. We find them interesting as they allow us to evaluate each Majorana phase in a model independent way even if one eigenvalue is zero. Utilizing the solution of a general complex symmetric mass matrix for eigenvalues and mixing angles we determine the Majorana phases for both the hierarchies, normal and inverted, taking into account the constraints from neutrino oscillation global fit data as well as bound on the sum of the three light neutrino masses ($\Sigma_im_i$) and the neutrinoless double beta decay ($\beta\beta_{0\nu}$) parameter $|m_{11}|$. This methodology of finding the Majorana phases is applied thereafter in some predictive models for both the hierarchical cases (normal and inverted) to evaluate the corresponding Majorana phases and it is shown that all the sub cases presented in inverted hierarchy section can be realized in a model with texture zeros and scaling ansatz within the framework of inverse seesaw although one of the sub case following the normal hierarchy is yet to be established. Except the case of quasi degenerate neutrinos, the methodology obtained in this work is able to evaluate the corresponding Majorana phases, given any model of neutrino masses., Comment: 27 pages, 16 figures, 4 tables, to appear in NPB

We describe a scheme for universal quantum computation with Majorana fermions. We investigate two possible dissipative couplings of Majorana fermions to external systems, including metallic leads and local phonons. While the dissipation when coupling to metallic leads to uninteresting states for the Majorana fermions, we show that coupling the Majorana fermions to local phonons allows to generate arbitrary dissipations and therefore universal quantum operations on a single QuBit that can be enhanced by additional two-QuBit operations.

The idea that the Majorana neutrino should be identified as a Bogoliubov quasiparticle is applied to the seesaw mechanism for the three generations of neutrinos in the Standard Model. A relativistic analogue of the Bogoliubov transformation in the present context is a CP-preserving canonical transformation but modifies charge conjugation properties in such a way that the C-noninvariant fermion number violating term (condensate) is converted to a Dirac mass term. Puzzling aspects associated with the charge conjugation of chiral Weyl fermions are clarified. By invoking the Coleman--Weinberg mechanism in the framework of dimensional regularization, it is also noted that seesaw models become unnatural in some parameter regions which induce the hierarchy problems in the bosonic sector., Comment: 20 pages

We analize the non-cyclic geometric phase for neutrinos. We find that the geometric phase and the total phase associated to the mixing phenomenon provide a tool to distinguish between Dirac and Majorana neutrinos. Our results hold for neutrinos propagating in vacuum and through the matter. Future experiments, based on interferometry, could reveal the nature of neutrinos., 5 pages, 2 figures

We investigate the generation of small neutrino masses in a clockwork framework which includes Dirac mass terms as well as Majorana mass terms for the new fermions. We derive analytic formulas for the masses of the new particles and for their Yukawa couplings to the lepton doublets, in the scenario where the clockwork parameters are universal. When the Majorana masses all vanish, the zero mode of the clockwork sector forms a Dirac pair with the active neutrino, with a mass which is in agreement with oscillations experiments for a sufficiently large number of clockwork gears. On the other hand, when the Majorana masses do not vanish, neutrino masses are generated via the seesaw mechanism. In this case, and due to the fact that the effective Yukawa couplings of the higher modes can be sizable, neutrino masses can only be suppressed by postulating a large Majorana mass for all the gears. Finally, we discuss the constraints on the mass scale of the clockwork fermions from the non-observation of the rare leptonic decay $\mu\rightarrow e\gamma$., Comment: 11 pages, 7 figures

In this paper, we use the exact master equation approach to investigate the decoherence dynamics of Majorana zero modes in the Kitaev model, a 1D $p\phantom{\rule{0.16em}{0ex}}$-wave spinless topological superconducting chain (TSC) that is disturbed by gate-induced charge fluctuations. The exact master equation is derived by extending Feynman-Vernon influence functional technique to fermionic open systems involving pairing excitations. We obtain the exact master equation for the zero-energy Bogoliubov quasiparticle (bogoliubon) in the TSC, and then transfer it into the master equation for the Majorana zero modes. Within this exact master equation formalism, we can describe in detail the non-Markovian decoherence dynamics of the zero-energy bogoliubon as well as Majorana zero modes under local perturbations. We find that at zero temperature, local charge fluctuations induce level broadening to one of the Majorana zero modes but there is an isolated peak (localized bound state) located at zero energy that partially protects the Majorana zero mode from decoherence. At finite temperatures, the zero-energy localized bound state does not precisely exist, but the coherence of the Majorana zero mode can still be partially but weakly protected, due to the sharp dip of the spectral density near the zero frequency. The decoherence will be enhanced as one increases the charge fluctuations and/or the temperature of the gate.

In this paper we analyze the formal and conceptual steps made by Ettore Majorana in a wide set of unpublished (handwritten) manuscripts (Quaderni, Fascicoli, Volumetti) written in later 20's and earlier 30's where, starting from the Dirac equation for spin-1/2 particles, he developed quantum relativistic wave equations for different (integer and half-integer) spins. In such a way Majorana obtained a Dirac-like equation for the photon and an infinite component quantum field theory for particles of any spin, thus anticipating the modern supersymmetry and string theories.

We revisit the charge-conjugation operation for the Dirac equation in its chiral representation. A new decomposition of the Dirac spinor field is suggested and achieved by means of projection operators based on charge conjugation, which is discussed here in a non-standard way. Thus, two separate two-component Majorana-type field equations for the eigenfields of the charge-conjugation operator are obtained. The corresponding free fields are entirely separated without a gauge field, but remain mixed and coupled together through an electromagnetic field term. For fermions that are charged and, thus, subjected to the gauge field of electrodynamics, these two Majorana fields can be reassembled into a doublet, which is equivalent to a standard four-component Dirac spinor field. In this way, the Dirac equation is retained in a new guise, which is fully equivalent to that equation in its chiral form.

Ettore Majorana (1906-1938) disappeared while traveling by ship from Palermo to Naples in 1938. His fate has never been fully resolved and several articles have been written that explore the mystery itself. His demise intrigues us still today because of his seminal work, published the previous year, that established symmetric solutions to the Dirac equation that describe a fermionic particle that is its own anti-particle. This work has long had a significant impact in neutrino physics, where this fundamental question regarding the particle remains unanswered. But the formalism he developed has found many uses as there are now a number of candidate spin-1/2 neutral particles that may be truly neutral with no quantum number to distinguish them from their anti-particles. If such particles exist, they will influence many areas of nuclear and particle physics. Most notably the process of neutrinoless double beta decay can only exist if neutrinos are massive Majorana particles. Hence, many efforts to search for this process are underway. Majorana's influence doesn't stop with particle physics, however, even though that was his original consideration. The equations he derived also arise in solid state physics where they describe electronic states in materials with superconducting order. Of special interest here is the class of solutions of the Majorana equation in one and two spatial dimensions at exactly zero energy. These Majorana zero modes are endowed with some remarkable physical properties that may lead to advances in quantum computing and, in fact, there is evidence that they have been experimentally observed. This review first summarizes the basics of Majorana's theory and its implications. It then provides an overview of the rich experimental programs trying to find a fermion that is its own anti-particle in nuclear, particle, and solid state physics., Comment: Invited review for Reviews of Modern Physics; v2: shortened and reorganized version to appear in RMP, 28 pages 12 figures. This version emphasizes the distinction between the generic Majorana fermions in particle physics and Majorana zero modes discussed in condensed matter physics

A real version of the Dirac equation is derived and its coupling to the electromagnetic field considered. First the four-component real Majorana equation is briefly discussed. Then the complex Dirac equation including an electromagnetic field will be written as an eight-component real spinor equation by separating it into its real and imaginary parts. Through this decomposition, what becomes obvious is the way in which the electromagnetic field couples to charged fermions (electron and positron) when being described by real spinor fields. Thus, contrary to common expectation, charged fermions can also be described by a real Dirac equation if they are considered as a doublet related to the SO(2) symmetry group, which enables a matrix coupling to the electromagnetic field and corresponds to the usual U(1) gauge symmetry of the standard Dirac equation.

Majorana stars are visual representation for a quantum pure state. For some states, the corresponding majorana stars are located on one curve on the Block sphere. However, it is lack of exact curve equations for them. To find the exact equations, we consider a superposition of two bosonic coherent states with an arbitrary relative phase. We analytically give the curve equation and find that the curve always goes through the North pole on the Block sphere. Furthermore, for the superpositions of SU(1,1) coherent states, we find the same curve equation.

We review the properties of Majorana fermions in particle physics and point out that Majorana modes in solid state systems are significantly different. The key reason is the concept of anti-particle in solid state systems is different from its counterpart in particle physics. We define Majorana modes as the eigenstates of Majorana operators and find that they can exist both at edges and in the bulk. According to our definition, only one single Majorana mode can exist in a system no matter at edges or in the bulk. Kitaev’s spinless p-wave superconductor is used to illustrate our results and the dynamical behavior of the Majorana modes.

Majorana zero modes, which behave like Majorana fermions, are quasiparticle excitations in condensed matter systems. They obey non-Abelian statistics, and have been proposed as building blocks of topological quantum computers. They are predicted to exist in the vortex of topological superconductors. In 2012, such a topological superconductor was engineered by depositing topological insulator thin films on top of an s-wave superconductor. Thereafter, several evidences have been reported to prove the Majorana zero modes’ existence in the vortex. In this review, by putting all experimental and theoretical results together, we show that these experimental evidences are consistent and they are also strongly supported by the theories, so the existence of Majorana zero mode is firmly established. Moreover, the adjacent Majorana zero modes annihilate when two vortices are close enough, which demonstrate that they have the nature of Majorana fermions. Finally, their potential application in topological quantum computing is discussed.

We construct a dual-bosonized description of a massless Majorana fermion in (2+1)$d$. In contrast to Dirac fermions, for which a bosonized description can be constructed using a flux attachment procedure, neutral Majorana fermions call for a different approach.We argue that the dual theory is an $\mathrm{SO}{(N)}_{1}$ Chern-Simons gauge theory with a critical $\mathrm{SO}(N)$ vector bosonic matter field ($N\ensuremath{\ge}3$). The monopole of the $\mathrm{SO}(N)$ gauge field is identified with the Majorana fermion. We provide evidence for the duality by establishing the correspondence of adjacent gapped phases and by a parton construction. We also propose a generalization of the duality to ${N}_{f}$ flavors of Majorana fermions, and discuss possible resolutions of a caveat associated with an emergent global ${Z}_{2}$ symmetry. Finally, we conjecture a dual description of an $\mathcal{N}=1$ supersymmetric fixed point in (2+1)$d$, which is realized by tuning a single flavor of Majorana fermions to an interacting (Gross-Neveu) critical point.

We show how angular momentum conservation can stabilize a symmetry-protected quasitopological phase of matter supporting Majorana quasiparticles as edge modes in one-dimensional cold atom gases. We investigate a number-conserving four-species Hubbard model in the presence of spin-orbit coupling. The latter reduces the global spin symmetry to an angular momentum parity symmetry, which provides an extremely robust protection mechanism that does not rely on any coupling to additional reservoirs. The emergence of Majorana edge modes is elucidated using field theory techniques, and corroborated by density-matrix-renormalization-group simulations. Our results pave the way toward the observation of Majorana edge modes with alkaline-earth-like fermions in optical lattices, where all basic ingredients for our recipe---spin-orbit coupling and strong interorbital interactions---have been experimentally realized over the last two years.

Majorana fermion (MF), an exotic particle that is identical to its own antiparticle, was recently found in solid matter as a quasiparticle excitation, the Majorana zero mode (MZM), in the vortex of an artificial topological superconductor (TSC). This artificial TSC, first proposed by Fu and Kane in 2008, is a heterostructure made of a topological insulator Bi2Te3 and an s-wave superconductor NbSe2. This paper will briefly review the experimental progresses based on the Bi2Te3/NbSe2 heterostructure. All evidences are self-consistent and reveal that the MZM exists in the center of vortex. Those experimental results are also supported by theory. This finding is a milestone in the research of Majorana fermions in solid state physics and a starting point of MZM’s application in topological quantum computation.

We study interacting Majorana fermions in two dimensions as a low-energy effective model of a vortex lattice in two-dimensional time-reversal-invariant topological superconductors. For that purpose, we implement ab-initio quantum Monte Carlo simulation to the Majorana fermion system in which the path-integral measure is given by a semi-positive Pfaffian. We discuss spontaneous breaking of time-reversal symmetry at finite temperature.

No experiment could directly test the particle/antiparticle duality of Majorana fermions, so far. However, this property represents a necessary ingredient towards the realization of topological quantum computing schemes. Here, we show how to complete this task by using microwave techniques. The direct coupling between a pair of overlapping Majorana bound states and the electric field from a microwave cavity is extremely difficult to detect due to the self-adjoint character of Majorana fermions which forbids direct energy exchanges with the cavity. We show theoretically how this problem can be circumvented by using photo-assisted tunneling to fermionic reservoirs. The absence of direct microwave transition inside the Majorana pair in spite of the light-Majorana coupling would represent a smoking gun for the Majorana self-adjoint character., 6 pages, 4 figures

Because Majorana fermions are their own antiparticles, their electric and magnetic dipole moments must vanish, leaving the anapole moment as their only static electromagnetic property. But the existence of induced dipole moments is not necessarily prohibited. Through a study real Compton scattering, we explore the constraints that the Majorana fermion's self-conjugate nature has on induced moments. In terms of the Compton amplitude, we find no constraints if the interactions are separately invariant under charge conjugation, parity, and time reversal. However, if the interactions are odd under parity and even under time reversal, then these contributions to the Compton amplitude must vanish. We employ a simple model to confirm these general findings via explicit calculation of the Majorana fermion's polarizabilities. We then use these polarizabilities to estimate the cross-section for $s$-wave annihilation of two Majorana fermions into photons. The cross-section is larger than a na\"ive estimate might suggest.

We study the formation of Majorana states in superconductors using the Majorana polarization, which can locally evaluate the Majorana character of a given state. We introduce the definition of the Majorana polarization vector and the corresponding criterion to identify a Majorana state, and we apply it to some simple cases such as a one-dimensional wire with spin-orbit coupling, subject to a Zeeman magnetic field, and proximitized by a superconductor, as well as to an NS junction made with such a wire. We also apply this criterion to two-dimensional finite-size strips and squares subject to the same physical conditions. Our analysis demonstrates the necessity of using the Majorana polarization local order parameter to characterize the Majorana states, particularly in finite-size systems., Comment: 16 pages, 11 figures, review article published in Comptes Rendus Physique

B meson semi-leptonic rare decays are sensitive to new physics beyond standard model. We study the $B^{-}\to \pi^{-}\mu^{+}\mu^{-}$ process and investigate the Majorana neutrino contribution to its decay width. The constraints on the Majorana neutrino mass and mixing parameter are obtained from this decay channel with the latest LHCb data. Utilizing the best fit for the parameters, we study the lepton number violating decay $B^{-}\to \pi^{+}\mu^{-}\mu^{-}$, and find its branching ratio is about $6.4\times10^{-10}$, which is consistent with the LHCb data reported recently., Comment: 10 pages, 3 figures

We propose a scheme to simulate the 1D Majorana equation with two Cooper pair boxes coupled to a 1D superconducting transmission line resonator, where strong coupling limit can be achieved. With proper chosen of systematic parameters, we are able to engineer different kinds of interaction, which is indispensable in simulating the Majorana equation in an enlarged real Hilbert space. Measurement of the conserved observable of pseudo-helicity via transmission spectrum of the cavity field can verify the simulated Majorana wave function. The measurement results are experimentally resolvable based on our estimation with conservative parameters., first submitted on March 2012; v2 published version

We investigate transport properties of an Aharonov-Bohm interferometer with an embedded quantum dot with Majorana bound states at the end of the topological superconductor. The differential conductance is calculated by the Keldysh Green function formalism. The Fano resonance of the symmetric shape with the maximum value of 2e 2 /h emerges as a function of the bias voltage. We find that when the energy of the quantum dot is fixed to the energy of the Majorana bound state, the conductance shows π periodicity as a function of the Aharonov-Bohm phase induced by the magnetic flux penetrating the interferometer which differs from the case of the superconducting lead.

It is generally believed that Majorana’s paper [4] describes particles which have no antiparticles. A thorough analysis have shown that this paper gives rise to the questions the answers to which have not been realized up to now. By the Majorana equation we mean the equation that describes stable, massive, and neutral leptons. Fulfillment of these three requirements leads strictly and unambiguously to the conclusion that the Majorana equation describes a pair of stable, massive, and neutral leptons, a particle and an antiparticle. This is similar to the Dirac equation, but the spin properties of the leptons of this type differ from the spin properties of the electron or positron.

This work clarifies the relation between Maxwell, Dirac and Majorana neutrino equations presenting an original way to derive the Dirac and neutrino equation from the chiral electrodynamics leading, perhaps, to novel conception in the mass generation by electromagnetic fields. In the present article, it is shown that Maxwell equations can be written in the same form as the two components Dirac and neutrino equations, that is the vector representation of electromagnetic theory can be factorized into a pair of two-component spinor field equations. We propose a simple approach with the electric field E parallel to the magnetic field H. Our analysis is based on the chiral or Weyl form of the Maxwell equations in a chiral vacuum. This theory is a new quantum mechanics (QM) interpretation for Dirac and neutrino equation. The below research proves that the QM of particles represents the electrodynamics of the curvilinear closed chiral waves. Electromagnetic properties of neutrinos are discussed.