Questions tagged [majorana-fermions]

Majorana fermions are the particles associated to the real representation (hence the real solution) of the Dirac equation, as established by the Italian physicist Ettore Majorana in the late 30's. By extension, a Majorana fermion will be any particle (emergent or fundamental) having the property to be its own anti-particle.

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Expanding a density matrix in terms of operators

In Lukasz's paper: https://arxiv.org/pdf/0909.2654.pdf He writes "consider a density matrix ρ, written as a polynomial of the 2N Majoranas cj in such a way that each cj occurs to the power 0 or 1 in ...
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Topological superconductors

I have been reading a lot about quantum computation by using topological materials and I could see that the standard approach is to engineer a p-wave superconductor by using a 1-D semiconductor wire ...
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How to plot Majorana energies for Kitaev chain as function of mu?

My first attempt was, using the Python library Kwant, getting the lowest eigenvalues for the 1D p-wave superconductor as a function of the chemical potential mu (In the image below I show what I got). ...
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Connection between $2n$ real fermions and $SO(2n)$

In section 11.4 of "Basic Concepts of String Theory" by Blumenhagen et al, they say: Consider a system of $2n$ two-dimensional real fermion (...) transforming as a vector of $SO(2n)$. I guess they ...
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What are the fermions in the SYK model doing?

The Hamiltonian of the SYK model is \begin{equation} H = \mathcal{N}\sum_{ijkl}^N J^{ijkl} \chi_i \chi_j \chi _k \chi _l \end{equation} where $\mathcal{N}$ is some normalization to make the energy ...
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What mechanism will generate Dirac neutrinos?

Giving mass to neutrinos is not a problem. The problem is to explain its smallness. Once right-handed neutrinos $\nu_R$'s are included in the Standard Model (SM), the Majorana mass for $\nu_R$ must ...
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Spin 1/2 Fermion With Both Majorana and Dirac Masses?

I will use Van der Waerden notation in the following: It is easy to construct both Majorana and Dirac mass terms in the lagrangian of a Dirac spinor with left-handed component $\phi_{\alpha}$ and ...
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Kitaev's reason for looking at p-wave superconductors

I would really like to understand the thought process that lead Kitaev to proposing the Kitaev chain but most expositions really just start from the Hamiltonian and derive the Majoranas without any ...
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The logic behind a specific step of SUSY variation

I'm following Neil Lambert's Supersymmetry notes, and there's a step in equation 5.67 which has me stumped. He says he uses the fact that "$C\gamma_\mu$ is symmetric". I don't see how that helps, and ...
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What's the difference between cylindrical geometry and disk with a hole for topological chiral $p_x\!+\!i p_y$ superconductor?

We know that for a topological chiral p-wave superconductor with a cylindrical geometry, i.e. one conserved momentum $k$ and one open boundary direction, there exists edge modes with opposite ...
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Fermionizing the Gell-Mann Algebra

In condensed matter physics one often solves a spin Hamiltonian by transcribing the Pauli matrices into fermionic operators. For instance, in the Kitaev model you can introduce four Majorana modes for ...
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How many dimensions has a Majorana Fermion state space?

For an exercise question, I'm wondering on how I can determine the dimensionality of a state space spanned by Majorana fermions. A (Dirac) fermionic excitation ($f , f^\dagger$) has a state space of ...
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A Naive Question about SUSY Variation

I am following BUSSTEPP Lectures on Supersymmetry to learn supersymmetry. My simple question is the following. My Lagrangian for the Wess-Zumino model in $4D$ is $$\mathcal{L}=-\frac{1}{2}(\...
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Transpose of fermion bilinears

TL;DR When we take the transpose of two Grassmann-valued spinors (fermions), should we add a minus sign because we end up anticommutating the two spinors? More details. I'm studying the behavior of ...
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The Majorana condition and C violation

Is the Majorana condition $$ \psi = \psi^c = C \overline{\psi}^T, $$ general? The point is often made that Majorana particles should be defined by CPT symmetry and not C as generally theories do not ...
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Do I need Gamma matrices in Majorana representation in the Lagrangian of a Majorana fermion?

I understand that the Majorana representation of the Gamma matrices are the real representations of the associated Clifford algebra. A Majorana fermion is defined as a fermion that equals to its ...
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Majorana Neutrinos in the KamLAND-Zen

I'm reading the following paper concerning the KamLAND-Zen experiment "Search for Majorana Neutrinos near the Inverted Mass Hierarchy Region with KamLAND-Zen" In the second paragraph of the first ...
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References to learn about Majorana Zero Modes

I'm a masters student currently trying to learn about Majorana zero modes in condensed matter physics. But so far the references I have checked have been not really useful for learning. I even read ...
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Equivalence between Dirac and Majorana action in CFT

In Mussardo's Statistical field theory Chapter 12, section 12.3 about the conformal field theory of a free fermion field he talks about the complex fermion field (Dirac field) $$ \Psi(z,\bar{z}) = \...
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Majorana fermions

If you write the Majorana spinors as $$\chi = \begin{pmatrix}\psi_L\\ i\sigma_2\psi_L^* \end{pmatrix} \tag1$$ It satisfies the Dirac equation that leads you to the Majorana equation $$i\bar{\sigma}^\...
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Why is the notion of neutrinos as possiblly Majorana fermions still promoted given the observation of neutrino oscillations?

A Majorana particle is a fermion whose antiparticle is itself. In other words, a Majorana fermion has only two components, rather than 4 components for a Dirac fermion. Neutrinos are alleged as ...
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Plugging Majorana Spinor into Dirac Lagrangian does not give Majorana Lagrangian?

This seems like it should be simple but somehow I do not see how. The Majorana Lagrangian can be written in terms of a left handed Weyl spinor $\psi_L$ as $$ \mathcal{L}_M= i \psi_L^\dagger \bar{\...
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Majorana Fermion Coherent States

I was wondering if there are coherent states for Majorana operators, so, states that fulfill the relation \begin{align} \hat{\gamma}_A |a,b\rangle &= a |a,b\rangle \\ \hat{\gamma}_B |a,b\...
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Why two different spinors are Grassmann quantities?

In Rydberg Quantum Field Theory page 441 (this edition, unfortunately page 441 is not in the link) it says If $\xi$ and $\eta$ are Majorana spinors [...] and since $\xi$ and $\eta$ are Grassmann ...
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Real Majorana wavefunction / field: What is the big deal?

It is known that there is a set of gamma matrices that can be purely imaginary (called Majorana basis), thus one can solve the 1st quantized Majorana wave function in terms of real wave function. ...
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When considering a general mass term with Majorana masses, how do you exclude the existence of the left-handed neutrino mass?

Considering a general mass term describing the neutrino masses: $$ -\frac{1}{2} \begin{pmatrix} \bar\nu_{R} & \bar\nu_L \end{pmatrix} \begin{pmatrix} M_R & m \\ m & M_L \end{pmatrix} \...
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What's the reason experiments look for the half-life of neutrinoless double beta decay?

I've been reading some papers about looking for neutrinoless double beta decay. A couple of them talk about finding a lower limit for neutrinoless double beta decay. From what I understand double beta ...
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What if Dark Matter is Majorana Fermion?

Few reasons to believe Dark Matter is Majorana fermion: https://arxiv.org/abs/1711.03877: second contra argument by Jordan and Pauli. "A non vanishing zero-point energy of elementary fields should ...
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Computing Feynman Diagrams with Majorana and Dirac Fermions

there is some literature explaining systematic algorithms for computing Feynman diagrams for scattering processes, but I cannot see why the calculations for such processes require a choice of ...
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Can experiments measure the Majorana phases?

Unlike the Dirac CP-phase $\delta$, the Majorana CP-phases $\alpha,\beta$ do not appear in the formula of neutrino flavour oscillation probability. Hence, they cannot be measured from oscillation ...
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Neutrino-Antineutrino oscillation

I read somewhere about $\nu$-$\bar{\nu}$ oscillations. For this purpose Majorana mass of neutrinos was considered. But I could not understand through the mathematics (involving Lagrangian) how it is ...
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Why have we not observed neutrinoless double beta decay yet? What obstacles are there?

Neutrinoless Double Beta Decay has been theorized by Majorana many, many years ago. Yet still, we have not detected this form of decay yet. The Theory makes sense. What is preventing us from ...
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Majorana Flip Relations

In the Supergravity book of Freedman et.al, which uses the signature $(+,-,\dots,-)$, we have defined the charge conjugation matrix for general Clifford Algebra as $(C\Gamma^{(r)})^T = -t_rC \Gamma^{(...
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Is a quadratic Majorana Hamiltonian exactly solvable?

Given $2N$ Majorana operators $\{a_i\}$ where $i=1,2,3,4,\cdots,2N$ The system Hamiltonian is the most general quadratic form: $H=\sum A_{ij}a_i a_j$ where $ \{a_i,a_j\}=2\delta_{ij} \quad a^\...
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Express a product of Majorana Fermions as an exponential of number operators

I am a computational chemist who is moving into, what I understand is, condensed matter physics. I have started reading Blaizot and Ripka "Quantum Theory of Finite Systems" and I am having trouble ...
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Physical states of a non-relativistic Majorana fermion

Would it be correct to say that the space of all possible physical states of a non-relativistic Majorana fermion is a circle? (analogous to the Bloch sphere in the case of a regular two-level system) ...
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Majorana mass term

I see somewhere that we can add a Majorana mass term, $m\psi^Ti\sigma^2\psi+h.c.$, to a Weyl fermion Lagrangian, where $\psi$ is a two-component spinor. However, it seems that this term simply ...
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Unfamiliar with some notations of BdG Hamiltonian

In a paper Majorana Fermions in Equilibrium and in Driven Cold-Atom Quantum Wires. The authors come to a Hamiltonian for BdG, eqn(2). $$ H = \sum_{p} a_{p}^{\dagger}(\epsilon'_{p}-\mu+u p \sigma_{z}+B\...
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Difference between $c=1$ Dirac CFT and two copies of $c=\frac{1}{2}$ Majorana CFT?

The $c = \frac{1}{2}$ (non-chiral) Majorana CFT has six primary fields: the vacuum 1, the two Majorana fields $\eta, \bar \eta$ $\left( \Delta_\eta = \Delta_{\bar \eta} = \frac{1}{2}\right)$, the ...
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Can neutrino oscillates into antineutrino?

Unlike antineutron, an antineutrino aren't made up of antiquarks so how do we tell an antineutrino from ordinary neutrino? Can majorana particle oscillates between matter particle and antiparticle?
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How to get the Green's function of free modes in the Sachdev-Ye-Kitaev (SYK) model?

In studying this paper "Remarks on the Sachdev-Ye-Kitaev (SYK) model" (http://arXiv.org/abs/1604.07818), I have no idea how to get the free Majorana fermion Green's function (Eq. (2.5) in the above ...
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What is a Higgs Portal Scale?

I am reading about the prospective decay of Higgs into Majorana fermions (possibly sterile neutrinos). In the image the branching ratio of Higgs to a pair of Majorana fermions is given for different ...
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What is the order of magnitude of the energy released in Majorana fermions collision/annihilation?

Majorana fermion experiment It has been observed a quantum state in a one atom thick wire which in a certain energy range behaves like a Majorana fermion. It is a quasiparticle that arises out of the ...
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About Kitaev p-wave superconductor model and Majorana Zero mode

The Kitaev $p$-wave spinless superconductor model has Hamiltonian as $$H = \sum_{j=1}^{N-1} tc_j^\dagger c_{j+1} + \Delta c_jc_{j+1} + h.c. + \sum_{j=1}^N \mu c_j^\dagger c_j $$ which has topological ...
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Expansion of a Majorana field operator in creation and annihilation operators

A Dirac spinor $\Psi=\left(\begin{array}{c}\chi_\alpha\\\psi^\dot{\alpha}\end{array} \right)$ can be expanded in the following way:$\Psi=\int \frac{d^3p}{(2\pi)^3}\sqrt{\frac{1}{2E_p}} \sum_s\left( ...
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Ising anyon topological order and its edge $c=1/2$ CFT

We know that conformal field theories are closely related to two-dimensional topological orders via edge-boundary correspondence. An Ising topological order can be obtained by gauging the fermion ...
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Is this a topological $\mathbb Z_2$ (Majorana-)invariant in *any* dimension?

Consider a non-interacting superconducting Hamiltonian in an arbitrary dimension. This is most conveniently expressed in terms of Majorana modes, which are defined as $$\gamma_{2n-1} = c_n + c_n^\...
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Zero Energy Wavefunctions in 1D P-Wave Kitaev Model

$\bf{Setup}$ Hi! I am trying to derive the wavefunctions of the zero energy solutions of the Schrodinger equation in a 1D p-wave superconductor (Kitaev model). I am starting with the Hamiltonian $ \...
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Is $N_R$ a Majorana field in the Seesaw Lagrangian?

Consider the Lagrangian for the type-I seesaw given by $$-\mathcal{L}=\bar{\nu}_{L}m_DN_{R}+\frac{1}{2}\overline{(N_{R})^c}M_R N_{R}+\text{h.c.}.$$ $\bullet$ In this Lagrangian, what is the nature ...
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Can a Majorana field $\psi$ be charged under some $U(1)$ with a charge other than zero?

I know Majorana particles have to be electrically neutral because electric charged is conserved. My question, however, is whether at all a Majorana field $\psi$ be charged under any $U(1)$ (other ...