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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Prove that Majorana mass term is Lorentz invariance

I have a homework to prove that using one kind of chirality, let's say left-handed, we can construct a mass term. The argument is to show that this term $\psi^TCP_L\psi$ is satisfy the dimensionality ...
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Amplitude of fermion decay with Majorana mass insertion

Consider the following Lagrangian, $$ L \supset \frac{1}{2} m \overline{\psi}\psi^C + g \phi\, \overline{\chi}\,P_R\,\psi + \rm{h.c.} $$ where $\psi$ is a heavy fermion with a Majorana mass $m$, $\phi$...
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Charge conjugation of symplectic Majorana spinors in 4+1 dimensions

In the book "Supergravity" written by Freedman & van Proeyen, a symplectic Majorana spinor is defined in eq. (3.86) $$ \chi^i = \varepsilon^{ij} (\chi^j)^C, \tag{3.86}$$ where the upper ...
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What is a p-wave superconductor in 1-dimension often referred in the Kitaev model? What term in the hamiltonian makes it P wave?

I am currently reading papers related to Majorana zero modes observation in 1D nanowire systems. I am very new to the field and I read everywhere that the presence of spin-orbit interaction, magnetic ...
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Solving the quantum XX model using drone fermions

Suppose we wish to solve the XX model in 1D, which describes spin-$\frac{1}{2}$ particles interacting with their nearest neighbor. Assuming open boundary conditions for simplicity, the Hamiltonian ...
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Unitary Transformation Taking a 4$\pi$ Periodic Wave Function to 2$\pi$ Periodic Wave Function

I am reading the following paper, which discusses Majorana fermions in Josephson junction arrays. Initially, the paper starts with a model such that the wavefunctions are $4\pi$ periodic. These ...
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How could the neutrino be a Dirac particle

I'm reading up about neutrinos now and I understand the possibility of neutrinos being Majorana particles and further theories can be thought of from that (like the seesaw mechanism). I'm still very ...
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Do we have any guarantee that the Noether current of continuous symmetry is non-zero? (Inspired by calculating $SO(2)$ charge of Majorana Fermion)

Let me first describe how I got to that problem. We know that Majorana Lagrangian (here I choose left-handed but for right-handed problem is analogue) $${\cal L}=\psi_{L}^{\dagger}i\bar{\sigma}^{\mu}\...
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Is the real spinor representation of the Lorentz group irreducible?

Specifically the $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ representation. Given that we label representations by the corresponding representations of the complexified Lie group, the direct sum can be ...
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$Z$-boson decay into two neutrinos depends on the Dirac or Majorana nature of the neutrinos?

The decay rate of the $Z$-boson into two active neutrinos $Z \rightarrow \nu \overline{\nu}$ can be calculated straightforwardly and I obtained the same as in the literature. However, I was wondering, ...
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Finding the Bogoliubov- De Gennes Hamiltonian of Pauli Matrices

I'm studying a book about Majorana Fermions and condensed matter and I'm stuck with an exercise about finding the BdG Hamiltonian of some Hamiltonians in $p$ dependence and Pauli Matrices. I know how ...
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Real-space correlations of massless Majorana fermions

Consider an action of free, massless Majorana fermions in real time and 1+1 dimensions of the form $$ S[\psi] = \frac{1}{2} \int d^2x \ \psi^{T}\gamma^0 (i \gamma^{\mu} \partial_{\mu}) \psi $$ Here, $\...
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Majorana Dirac equation in 1+1 dimensions

While working out the continuum limit of the transverse field Ising model in the Majorana representation (see, for example, either Fradkin's Field Theories of Condensed Matter Physics or Shankar's ...
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Is it possible to express any quadratic fermionic system in terms of a quadratic majorana system and viceversa?

Can one always write, for some suitable matrix $M$ $$ H= \sum^N_{jk}(A_{jk}c^\dagger_jc_k+B_{jk}c_jc_k+h.c.)=i\sum^{2N}_{jk} M_{jk} \gamma_j\gamma_k, $$ for any $A,B$? And viceversa, can one always ...
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If neutrinos are majorana fermions, will it be possible for a beta minus decay to emit a neutrino instead of an antineutrino?

If neutrinos are majorana fermions, neutrinos and antineutrinos should be equivalent. As a result, reactions like $n\rightarrow p+e^{-}+\nu$, $\nu+p\rightarrow n+e^{+}$ should be kinematically ...
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Proof of correlation revival in Kitaev chain

I am searching for detailed notes on the Kitaev chain. In particular, I am looking for a proof of the correlation revival in the single-particle correlator from one edge to the other. Do you know how ...
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Reference for Kitaev chain correlation functions

I am searching for detailed notes on the Kitaev chain to get familiar with the computations that one should perform on it. In particular, I am looking for a proof of the correlation revival in the ...
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In a hypothetical neutrinoless beta decay, how the energy spectrum will look like?

If we consider a hypothetical situation in which $\beta^-$ deacay is happening without emitting any anti-neutrino, to be precise, a neutrinoless beta decay is happening. So, in this case, how the ...
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What causes the degrees of freedom to be halved for Majorana fermions?

In many textbooks and on this site (nanophys answer here) it is stated that 'Majorana Spinors have half the degrees of freedom of a typical Dirac spinor'. A generic spinor in 3+1D has 8 degrees of ...
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The $\rm SO(8)$ invariant interaction piece in Fidkowski and Kitaev's model

In this paper (arXiv link), the authors demonstrate the existence of a quartic interaction $W$ involving the 8 majorana operators $c_1 \ldots c_8$ (eq. 8) which is invariant under an $\rm SO(7)$ ...
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Where do I see the difference in Majorana vs. Dirac pair annihilation?

If I have a Lagrangian density $\cal{L}=\bar{\chi}\gamma^{\mu}{\rm A}_{\mu}\chi+\bar{f}\gamma^{\mu}{\rm A} _{\mu}f$ and I calculate the diagram $\chi \bar{\chi}\to f \bar{f}$ where $f$ is (let's say) ...
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Antimatter Majorana and DIrac

i have a question , how did Dirac derive the existence of antimmater from his equation ?? $$i \hbar \gamma^\mu \partial_\mu \psi - m c \psi = 0$$ and why for the Majorana equation predicts the ...
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CFT with central charge different from 1 and 1/2?

In $1+1$ dimensions, I know the $c=1$ $\rm CFT$ which describes a massless bosonic mode, and the $c=\frac{1}{2}$ $\rm CFT$, which describes a Majorana mode. Is it possible to write a Lagrangian ...
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2d CFT: Majorana = Weyl spinors?

I am studying the free fermion CFT using the book by Di Francesco et al. I am very confused with the notion of Majorana fermions, sparked by the following introduction to the chapter: In two ...
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Symplectic Majorana spinors in 5D

According to the book "Supergravity" written by Freedman & van Proeyen in 5D for the existence of Majorana spinors it is necessary to introduce so called sympletic ones which requires ...
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Zero energy eigenstates of the Kitaev chain Hamiltonian

I am thinking about a two-site Kitaev Hamiltonian. Namely, $$H = -\mu c^\dagger_1 c_1 -\mu c^\dagger_2 c_2-t c^\dagger_{2} c_1-t c^\dagger_{1} c_2 +\Delta c_1 c_{2}+\Delta c_2^\dagger c_{1}^\dagger.$$ ...
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Error in Majorana Operators squaring to the identity

Consider the Majorana fermions expressed mathematically in terms of the creation and annihilation operators of second quantization, the ordinary fermionic annihilation and creation operators $\alpha$ ...
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Underlying Hilbert space of Kitaev's exactly solvable models

In Kitaevs's paper (Anyons in an exactly solved model and beyond) section 2.1-2.2, he seems to be extending the Hilbert space of a multi-spin system using Majorana operators. More specifically, if ...
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Can photons turn into neutrinos, similar to 'pair production' of electrons and positrons?

Since neutrinos ate much lighter, and are their own antiparticles (Majorana), it should be much 'easier' than turning a photon into an electron-positron pair, correct?
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Partition function in SYK Model

In SYK model, we have the partition function for $N$-interacting fermions as \begin{equation} z=\int d^{N} \psi \exp \left(\imath^{q / 2} \sum J_{a_{1} a_{2} \ldots a_{q}} \psi_{a_{1} a_{2} \ldots a_{...
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Majorana Fermions and the Divergence of Currents

I am working on exercise 3.4d in Peskin and Schroeder's Introduction to Quantum Field Theory. In part c I found the Dirac Lagrangian density to be $$L = i\chi_1^\dagger \bar{\sigma}^\mu \partial_\mu \...
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How to get the generators of $\mathfrak{so}(3)$ in the paper by Fidkowski and Kitaev?

In the paper by Fidkowski and Kitaev, they aim to study the interaction of 8 parallel Majorana wires, and they work on $\mathfrak{so(8)}$ Lie Algebra. They first start with just 4 parallel Majorana ...
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Deriving the Majorana equations of motion from a Lagrangian density

This is regarding exercise 3.4b in Peskin and Schroeder's Introduction to Quantum Field Theory. This problem asks us to find the equations of motion for the Majorana field (the equations of motion ...
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Basis state of non-interacting fermions

I am trying to calculate the periodic dynamics of many-body systems (spin-$1/2$ $XY$) Hamiltonian, where, \begin{equation} H_1 = \sum_{i=1}^{N-1}(\sigma^{x}_{i}\sigma^{x}_{i+1}+\sigma^{y}_{i}\sigma^{...
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A Complex Field Meaning

Here are some conceptual questions I have been thinking about but I either am not sure about the answer or do not know if my thoughts are correct. Some might seem silly, but I ask nevertheless. Any ...
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Eigenvalues of Majorana fermions

I have a somewhat naive question about Majorana fermions. Typically, two Majorana fermion modes $\gamma_{i,1}$ and $\gamma_{i,2}$ are defined by writing a single ordinary fermion $c_i$ in terms of &...
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If neutrinos were Majorana fermions, then what would be the lepton number of the neutrino?

Majorana particles are their own antiparticles and the lepton number of antimatter is -1 whereas for matter is +1, so if neutrinos were Majorana fermions, then what would be the lepton number of the ...
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Are Dirac/Weyl/Majorana fermions exclusive?

I think we can be pretty sure that fermions exist. We have several ways to describe them (Dirac, Weyl, Majorana, maybe someone I'm missing?), with different equations and number of components. My ...
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Has neutrinoless double-beta decay been observed?

After searching on this question in the archives here, it would appear that as of 2018, neutrinoless double beta decay had still not been observed. Has the situation changed since then? Can anybody ...
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What is the definition of a Majorana fermion in conformal field theory?

Majorana spinors background According to Eq. (4.84) and (4.85) of these notes, charge conjugation of the spinor $\Psi$ is defined as $$ \Psi^{(c)} = C \Psi^*,$$ where $C$ is the unitary charge ...
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Majorana fermions in Euclidean and Minkowski signatures - contradiction with Wikipedia Table

In this wonderful lecture note on Clifford Algebra and Spin(N) Representations, http://hitoshi.berkeley.edu/230A/clifford.pdf Somehow I find some inconsistency with his Tables of Euclidean and ...
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Dirac equation for neutrinos

Neutrinos are described by spinors. But they don't have a defined mass (they are in a superposition). How can I write the Dirac (Majorana) equation for neutrinos? Maybe $$i \gamma^{\mu} \partial_{\mu} ...
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Is it possible to diagonalize a Hamiltonian with both quadratic and linear terms in the fermi operators?

A quadratic Hamiltonian in the fermi operators is exactly diagonalizable. The most convenient way of describing these Hamiltonians is of the form: $$\mathcal{H}=\displaystyle \sum_{j,k}(\alpha_{jk}a_{...
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Dirac operator: Hermitian or antiHermitian parts

We know that $i \partial_t$ and $-i \partial_x$ are both Hermitian operators. If I understand correctly the Dirac operator $$ i \gamma^\mu \partial_\mu $$ contains the $\gamma^\mu$ such that in the ...
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Number and masses of right-handed neutrinos in the seesaw mechanism

In the Wikipedia article about the seesaw mechanism, it is argued that the seesaw mechanism "extends the Standard Model by assuming two or more additional right-handed neutrino fields", with ...
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Majorana fermions algebra confusion

This is a quiet embarrasing question. Consider the Majorana fermion fields $\psi_i(x)$ and $\psi_j(x)$, where $i$ and $j$ denote lattice sites and $x$ is a spatial coordinate, which satisfy the ...
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Majorana fermions - Scattering amplitude/Feynman rules

I am looking at 4 field scattering processes ($\phi \phi \rightarrow \psi \psi$) given by the Lagrangian is \begin{equation} \mathcal{L}_i \supset \frac{\lambda_s}{\Lambda} \overline{ \psi} \psi |\...
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Connection between Majorana and boson operators

I recently was thinking about the mathematical similarity between two very different physical systems. For bosons, we can write down the usual canonical relations \begin{align} b &= \frac{1}{\...
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Classifying elementary fermions

Familiar elementary (non-composite) relativistic fermions are of the Dirac, Weyl, and Majorana kinds. Are there other kinds allowed in principle by relativistic quantum physics? If not, why not? Are ...
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Are neutrinos and anti-neutrinos the same thing? [duplicate]

Neutrinos are subatomic particles with no electrical charge. Anti-matter has the same mass, spin, etc. as that of regular matter except the charge of anti-matter is opposite to that of the normal ...
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