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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What is the “BCS” ground state of a p-wave superconductor?

$\newcommand{\Ket}[1]{\left|#1\right>}$ In BCS theory (I always understood this as referring only to s-wave pairing), the ground state wavefunction takes the form $\Ket{BCS} = \prod_k(u_k + v_k c_{...
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Is superposition of charge possible?

Maybe majorana fermions could exist but is very different from both quasi particle pair and particle hole pair, it could have both positive and negative charge in superposition until it is being ...
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How are Majorana zero-modes protected against fermionic operators?

I am learning about Majorana fermions in topological quantum computation, and more particularly about the Kitaev chain, described by $$ H = -\mu \sum_{i=1}^N c_i^\dagger c_i - \sum_{i=1}^{N-1} \left(t ...
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Majorana Mass term for $SU(2)_L$ doublets

Assume we add a colorless and hyperchargeless $SU(2)_L$ fermion doublet to the Standard Model. $$L=\left(\begin{matrix}L_u\\L_d\end{matrix}\right)$$ Then, considering gauge invariance, the ...
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Is the symmetry factor of a Majorana fermion doubled compared to Dirac fermions?

I think the title says it all. If I have to contract e.g. an expression of the following form: $$\left\langle (...)\ \bar{\psi}(x_1)\ \psi(x_1)\ (...)\ \bar{\psi}(x_2)\ \psi(x_2)\ (...) \right\rangle,...
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Kinetic term of the Majorana Lagrangian

I want to expand the kinetic term of the Majorana Lagrangian (considering only a left handed Weyl spinor) from $$\mathcal{L}= \frac{1}{2} \bar{\psi}_M i \gamma^{\mu}\partial_{\mu} \psi$$ So I wrote ...
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Neutrinos, quasiparticles and Majorana fermions

In connection with topological quantum computing I encountered term Majorana fermions. According to Wikipedia these are: A Majorana fermion, also referred to as a Majorana particle, is a fermion ...
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Internal flavor symmetry of the $N$ left-handed complex Weyl spinors v.s. $N$ real Majorana spinors: ${\rm U}(N)$ vs. ${\rm O}(2N)$ or ${\rm O}(N)$

Consider 4d spacetime, it seems that for massless particles, we can easily change the left-handed complex Weyl spinor basis (2 component in complex $\mathbb{C}$ for Euclidean spacetime Spin(4)) to ...
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Constructing Lagrangian from Hamiltonian for Majorana fermions

The text gives the Hamiltonian density as \begin{equation}{\cal H}=\frac{v}{2}\Big(\psi^\dagger\frac{\partial\psi^\dagger}{\partial x}-\psi\frac{\partial\psi}{\partial x}\Big)+\Delta\Psi^\dagger\Psi \...
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How to check conformal invariance of a Lagrangian?

The Lagrangian is $$\mathcal{L}=\frac{-\dot\iota}{2}\Big(\Phi^\dagger\frac{\partial\Phi}{\partial x^0}+\Phi\frac{\partial\Phi^\dagger}{\partial x^0}+\Phi^\dagger\frac{\partial\Phi^\dagger}{\partial x^...
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Partition function for quantum Ising model

I have hamiltonian for fermionic field as $${\cal H}_F=E_0+\int dx[\frac{v}{2}(\Psi^\dagger\frac{\partial \Psi^\dagger}{\partial x}-\Psi\frac{\partial \Psi}{\partial x})+\Delta\Psi^\dagger\Psi]\tag{1}$...
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Why do people say that neutrinos are either Dirac or Majorana fermions?

The question of whether a given particle "is" a Dirac or Majorana fermion is more subtle than is sometimes presented. For example, if we just consider the "old" Standard Model with massless neutrinos, ...
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Describing Majorana fermions with operations

I'm reading a book on topological quantum theory and one of the exercises says that Majorana fermions $\gamma_j$ are such that $\{\gamma_j,\gamma_i\}=\delta_{ij}$ and that $\gamma_j=\gamma_j^\dagger$, ...
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Show that $\nu^c _L = (\nu ^c)_R $

Considering Majorana neutrinos, how can I show that $\nu^c _L = (\nu ^c)_R $? I don't know how to answer this question. And what is the difference between $\nu ^c _R $ and $(\nu ^c)_R $? I know it ...
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For two majorana field $\psi$ and $\chi$, is it true that $\bar{\psi} \chi = - \bar{\chi} \psi$?

For two majorana field $\psi$ and $\chi$, which satisfy $\psi_{c}=\psi$ and $\chi_{c} = \chi$, where the charge-conjugation operation we define as $$ \Psi_{c} = C \Psi^{\ast} $$ where we work in the ...
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Possible to have Dirac AND Majorana mass?

Supposing you have a lagrangian consisting of $(1/2,0)\oplus (0,1/2)$ representation. Writing in terms of Weyl fermions, the following terms are possible: $$-\frac{m_1}{2} (\psi_R^T \epsilon \psi_R -...
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Would Majorana antinetrinos have opposite weak isospin?

There was a question about the difference between a neutrino and antineutrino, and answers suggested that weak isospin is different, as well as handedness. This would make sense if you expect pair ...
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Is there a vacuum for Majorana fermions?

I know this question sounds somewhat odd. But if $\psi = \psi^\dagger$ is the Majorana operator, what would $\psi |0\rangle$ be? Is it the zero state or a Majorana fermion? This is confusing, because $...
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Help me in my attempt of finding the Lagrangian for the Majorana field

The Majorana field $\psi$ can be thought of as a reality condition $\psi=\psi^c$ (and $\overline{\psi}=\overline{\psi^c}$) on the Dirac field. So how does one write the Lagrangian for the Majorana ...
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Dirac sea for Majorana equation

To resolve the problem of infinite energies, Dirac has proposed the conception of the Dirac sea. Is the same concept present for the case of Majorana spinors? How it is affected by the absence of ...
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How are topological qubits encoded in the Majorana-fermion-based platform for quantum computing?

Where is “the two level system” of a topological qubit encoded in the Majorana-fermion-based platform of quantum computing? If the Hamiltonian in a topological quantum field theory is absent (H=0), ...
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Why we use majorana fermions the SYK model?

Does anyone know why do we use majorana fermions in the SYK model. why we can't use Dirac spinors? Is there any specific reason why we use majorana fermions in this model?
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Question about the Dyson-Schwinger equation for 4-point function of the SYK model

In studying the SYK model, I have no idea how to get the kernel $K(t_a,t_b,t_3,t_4)$ and $\Gamma_0(t_1,t_2,t_3,t_4)$ and also ladder diagrams in the Dyson-Schwinger equation for 4 point function of ...
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What is the Hilbert space of the SYK model?

In reading a recent preprint [1] contrasting bosonic models with local (tensor product) Hilbert spaces with SYK-like models of fermions, I realized I was confused about something. While I have a vague ...
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Variation of Majorana action

In the Supergravity book by Freedman, Van Proeyen on page 59 it states that the Majorana action is $$S[\Psi]=-\frac{1}{2}\int d^Dx\bar{\Psi}(\gamma^{\mu}\partial_{\mu}-m)\Psi.\tag{3.88}$$ After ...
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Non-Abelian Statistic of Majorana Fermions and Quantum Computing

I read in many sources that braiding operations of Majorana excitations in (topological) superconductors do not commute, and that this fact is useful in quantum computing. However, about this last ...
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What's the difference between Majorana fermion and Majorana zero mode?

Why does Majorana zero mode obey non-Abelian statistics while Majorana fermion obey Fermi-Dirac statistics?
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Are the two definitions of a Majorana fermion equivalent?

There seems to be two definitions of a Majorana fermion and I can't work out how to show they are equivalent. The first definition, which is on Wikipedia, is that a Majorana fermion is described by ...
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Why can't a Weyl Fermion have mass?

My understanding was that a particle may have mass if there is a quadratic term in the fields without derivatives. For a single left-handed Weyl fermion, the following expression is lorentz invariant, ...
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Stuck at Peskin Problem 3.4(e), how to proceed?

Here is my solution for problem (e), but I don't know what to do with the two $\sqrt{p\cdot\sigma}$. Majorana equation: $$ i\bar\sigma\cdot\partial\chi-im\sigma^2\chi^*=0 $$ Dirac equation: $$ \left(\...
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Question about Majorana fermion current conservation of problem 3.4(d) in Peskin's QFT book?

I have some trouble proving the current conservation mentioned in problem 3.4(d) of Peskin&Schroeder's QFT book. Relevant sections of the problem: (b). Does the Majorana equation $$ i\bar\...
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Representation of spin-$1/2$ operators in terms of Majorana fermions

I am reading Quantum Field Theory in Condensed Matter Physics by A.M. Tsvelik. In Chapter 20, it is claimed that introducing three Majorana fermions $\gamma^\mu_i$ on each site $i$ of the lattice (...
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Why does the majorana equation preserve handedness?

In the "QFT Nutshell" by A. Zee, it is stated that The Majorana equation is $$i\not\partial\psi=m\psi_c$$ where $\psi_c$ is the charge conjugated spinor $\psi_c = \left(C\gamma^0\right)\psi^*$....
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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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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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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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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}^\...