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arXiv:2104.05863 · International Journal of Quantum Chemistry

Statistical properties of linear Majorana fermions

Francisco Cleiton E. Lima, Allan R. P. Moreira, Laura E. S. Machado, Carlos Alberto S. Almeida

Published Jun 5, 2021• 20 citations
arXivHTMLPDF

Abstract

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. Keywords: Majorana Fermions; Thermodynamic properties; Shannon’s Entropy.

DOI
10.1002/qua.26749

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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. Keywords: Majorana Fermions; Thermodynamic properties; Shannon’s Entropy.
Abstract We propose a closed form for the statistical distribution of non-interacting Majorana fermions at low temperature. Majorana particles often appear in the contemporary many-body literature in the Kitaev, Fu–Kane, or Sachdev–Ye–Kitaev models, where the Majorana condition of self-conjugacy immediately results in non-conserved particle number, non-trivial braiding statistics, and the absence of a non-interacting limit. We deviate from this description and instead consider a gas of non-interacting, spin-1/2 Majorana fermions that obey the spin-statistics theorem via imposing a condensed matter analog of momentum conservation. This allows us to build a quantum statistical theory of the Majorana system in the low temperature, low density limit without the need to account for strong fluctuations in the particle number. A combinatorial analysis leads to a configurational entropy which deviates from the fermionic result with an increasing number of available microstates. A number-conserving Majorana distribution function is derived which shows signatures of a sharply-defined Fermi surface at finite temperatures. Such a distribution is then re-derived from a microscopic model in the form of a modified Kitaev chain with a bosonic pair interaction. The thermodynamics of this free Majorana system is found to be nearly identical to that of a free Fermi gas, except now distinguished by a two-fold ground state degeneracy and, subsequently, a residual entropy at zero temperature. Despite clear differences with the anyonic or Sachdev–Ye–Kitaev models, we nevertheless find surprising agreement between our theory and experimental signatures of Majorana excitations in several materials. Experimental realization of our exactly solvable model is also discussed in the realm of astrophysical and high-energy phenomena.
Abstract In the context of non‐relativistic quantum mechanics, we investigated Shannon's entropy of a non‐Hermitian system to understand how this quantity is modified with the cyclotron frequency. Subsequently, we turn our attention to the construction of an ensemble of these spinless particles in the presence of a uniform magnetic field. Then, we study the thermodynamic properties of the model. Finally, we show how Shannon's entropy and thermodynamic properties are modified with the action of the magnetic field.
The majorana fermion represents a kind of particle which is its own antiparticle. This paper aims to analyze the majorana fermions from theoretical aspect and application aspect, including the derivation of Dirac equation and the practicality of topological quantum computer. This paper explores the idea of Majorana fermions from Dirac equation and the anyons which is an important quasi-particle to build a topological quantum computer. After that, the potentials of quantum computer are emphasized as well. Several current major difficulties faced by building a quantum computer have also been discussed, including the decoherence of qubit and the errors during the operation of qubit. The advantages of topological computer are mentioned as well, especially the high resistance to local perturbation. The important property of two-dimensional non-abelian anyons has been discussed as well. Finally, the important relation of majorana fermions with anyons are introduced and the reason why majorana fermions is important has also been revealed.
Majorana phase-space representations for fermions map fermionic many-body physics into a distribution over one of the Cartan symmetric spaces of Lie group theory. The representation is in terms of $2M\ifmmode\times\else\texttimes\fi{}2M$ complex antisymmetric matrices, which generate the Gaussian Majorana operators. Here we show how this expansion can be utilized to calculate quantities arising in quantum thermodynamics and quantum information. Purity and the linear entropy are calculated, as well as the quantum fidelity between two general fermionic states, with numerical examples for pure states. We describe the geometrical properties of the phase space, and show that the overlap between two Gaussian Majorana states depends on the product of their antisymmetric matrices. Fermionic phase space is divided up into two orthogonal subspaces of different number parity, whose matrix representations differ by an orthogonal reflection in the phase space.
A Majorana fermion is a fermionic particle that is its own antiparticle. Since the theoretical discovery in 1937, the exotic particle has long been searched in particle physics. In the last few decades, however, it has attracted renewed interest in condensed matter physics, where it can be realized as an elementary excitation (quasiparticle) in quantum states of matter. In this review, we discuss another platform for Majorana fermions, the quantum spin liquid, in which interacting magnetic moments remain disordered down to the lowest temperature under strong quantum fluctuations. They are characterized by topological entanglement and fractional excitations, whose possible application to topological quantum computation is recently discussed intensively. As a prime candidate for such exotic states, we here focus on the Kitaev magnets, a subgroup of the spin-orbit Mott insulators. After a brief overview of the Kitaev model and the fractionalization of spins in the exact ground state, we review recent explosive development in this rapidly growing field, with a focus on numerical solutions of the Kitaev model at finite temperatures and the comparison with experiments. The key concept is thermal fractionalization --- two types of fractional excitations manifest themselves at largely different temperatures. This leads to distinct thermodynamics and spin dynamics in a variety of experimentally measurable quantities. We discuss such peculiar behaviors as the signatures of fractional quasiparticles, in careful comparison with the available experimental data for the candidate materials of the Kitaev magnets. Our review gives an overview of the current status of the identification of Majorana fermions in the Kitaev magnets, which would serve as a basis for further experimental and theoretical studies toward the manipulation of the exotic particles for topological quantum computation.
We propose the concept of birefringent Majorana fermions as collective excitations of quantum interacting many-body states motivated by the already known quasiparticles including Dirac fermions, birefringent Dirac fermions and Majorana fermions. Low energy effective models are constructed to understand the basic mathematical structures of equations of motion and physical properties. We find that the equation of motion, with two different "speed of light", describing birefringent Majorana fermions in condensed matter physics is the birefringence version of the original Majorana equation of real relativistic fermions.