Massless Dirac Fermions Research Articles

Luttinger liquid tensor network: Sine versus tangent dispersion of massless Dirac fermions

To apply the powerful many-body techniques of tensor networks to massless Dirac fermions, one wants to discretize the p·σ Hamiltonian and construct a matrix-product-operator (MPO) representation. We compare two alternative discretization schemes, one with a sine dispersion, the other with a tangent dispersion, applied to a one-dimensional Luttinger liquid with Hubbard interaction. Both types of lattice fermions allow for an exact MPO representation of low bond dimension, so they are efficiently computable, but only the tangent dispersion gives a power law decay of the propagator in agreement with the continuum limit: The sine dispersion is gapped by the interactions, evidenced by an exponentially decaying propagator. Our construction of a tensor network with an unpaired Dirac cone works around the fermion-doubling obstruction by exploiting the fact that the Hamiltonian of tangent fermions permits a generalized eigenproblem. Published by the American Physical Society 2024

Topological edge states of massless fermions with nonquantized and zero Berry phases

We investigate the bulk-boundary correspondence for massless Dirac fermions in α−T3 lattice where the Berry phase can be continuously tuned from π (graphene) to 0 (T3 or dice lattice) without modifying the energy dispersion. The topological origin of edge states is revealed by the unitary transformation of the Hamiltonian, which maps an α−T3 zigzag ribbon (ZR) onto a stub Su-Schrieffer-Heeger (SSH) chain. In the Majorana representation of the eigenstates, the topological invariant is manifested by the azimuthal winding number on the Bloch sphere. Particularly, T3 ZR exemplifies a topologically nontrivial system with zero Berry phase. Contrary to conventional topological insulators, the gap closing in the corresponding stub SSH chain is not accompanied by a topological phase transition. Published by the American Physical Society 2024

Robust massless Dirac fermions in hydro-/halogenated trigonal borophene

The striking electronic characteristics of graphene trigger immense interests and continual explora-tions for new two-dimensional (2D) Dirac materials. By first-principles electronic structure calculations, we here identify a new set of 2D semimetals in hydro-/halogen embedding trigonalδ6borophene, namelyδ6-B3X (X = H, F, Cl), that possess the graphene-like massless Dirac fermions. Owing to the central hollow B atoms strongly hybridized to the hydro-/halogen adatoms, adequate charge transfer is induced from the hollow B to the basal honeycomb B sublattice, which electronically stabilizes the 2D sheet and decisively endows a robust (intrinsic and stable-against-strains) graphene-like Dirac cone state. The predicted high energetic, dynamic and thermal stabilities, combined with pretty geometrical match to the commonly utilized Ag/Au(111) substrates, support their experimental viabilities. Our prediction provides a new branch for exploring the intriguing 2D Dirac fermionic states in versatile boron element and its derivatives.

Total and symmetry resolved entanglement spectra in some fermionic CFTs from the BCFT approach

In this work, we study the universal total and symmetry-resolved entanglement spectra for a single interval of some 2d Fermionic CFTs using the Boundary Conformal Field theory (BCFT) approach. In this approach, the partition of Hilbert space is achieved by cutting out discs around the entangling boundary points and imposing boundary conditions preserving the extended symmetry under scrutiny. The reduced density moments are then related to the BCFT partition functions and are also found to be diagonal in the symmetry charge sectors. In particular, we first study the entanglement spectra of massless Dirac fermion and modular invariant Z2-gauged Dirac fermion by considering the boundary conditions preserving either the axial or the vector U(1) symmetry. The total entanglement spectra of the modular invariant Z2-gauged Dirac fermion are shown to match with the compact boson result at the compactification radius where the Bose-Fermi duality holds, while for the massless Dirac fermion, it is found that the boundary entropy term doesn’t match with the self-dual compact boson. The symmetry-resolved entanglement is found to be the same in all cases, except for the charge spectrum which is dependent on both the symmetry and the theory. We also study the entanglement spectra of N massless Dirac fermions by considering boundary conditions preserving different chiral U(1)N symmetries. Entanglement spectra are studied for U(1)M subgroups, where M ≤ N, by imposing boundary conditions preserving different chiral symmetries. The total entanglement spectra are found to be sensitive to the representations of the U(1)M symmetry in the boundary theory among other behaviours at O(1). Similar results are also found for the Symmetry resolved entanglement entropies. The characteristic log log (ℓ/ϵ) term of the U(1) symmetry is found to be proportional to M in the symmetry-resolved entanglement spectra.

Two-dimensional bilayer blue phosphorus Dirac-like material: a multi-orbital tight-binding investigation.

This study presents a theoretical examination of the electronic band structure of AA (AB) stacked bilayer blue phosphorus system within the fifth intralayer (5NN) and second interlayer nearest-neighbor (2NN) multi-orbital tight-binding (MOTB) approach. The variation of energy levels has been investigated through the symmetrical tensile strain of the low-buckled honeycomb lattice. Here, the primary objective is to examine the existence of Dirac electronic features in hexagonal stacked bilayer geometry. Our theoretical calculations predict that the AA bilayer is a new hexagonal two-dimensional material with px,y-orbital Dirac-like states at the high-symmetry K point. Consequently, these systems can host massless (massive) Dirac fermions. In particular, the AA bilayer exhibits zero-gap Dirac-like properties and manifests distinguishable Dirac-like cones in the presence of weak spin-orbit coupling when a modest stretch of 2.30% is achieved with a remarkably high Fermi velocity of approximately vf ≈ 0.12 × 105 m s-1. The behavior of the dispersion bands aligns reasonably well with recent experimental observations. Moreover, a stretch of 7.17% breaks some of the sublattice equivalence and enhances the spin-orbit interaction, resulting in the emergence of an electronic band gap of approximately ≈ 0.27 eV in the proximity of the high-symmetry K point. Furthermore, the tiny gap induced by the spin-orbit interaction implies topological nontriviality in the electronic state (quantum anomalous Hall state) of the honeycomb lattice. These findings categorize the AA bilayer as a rare two-dimensional Dirac-like material. This work provides, to the extent of our knowledge, a pioneering investigation into the existence of Dirac electronic properties in bilayer blue phosphorus. In addition, we present the first derivation of the MOTB model. However, the identified electronic characteristics designate this two-dimensional system as an ideal candidate for high-performance nanoelectronic devices.

Pseudospin-filter tunneling of massless Dirac fermions

Abstract The tunneling of the massless Dirac fermions through a vector potential barrier are theoretically investigated, where the vector potential can be introduced by the very high and very thin (δ-function) magnetic potential barriers. We show that, distinct from the previously studied electric barrier tunneling, the vector potential barriers are more transparent for pseudospin-1/2 Dirac fermions but more obstructive for pseudospin-1 Dirac fermions. By tuning the height of the vector potential barrier, the pseudospin-1/2 Dirac fermions remain transmitted, whereas the transmission of the pseudospin-1 Dirac fermions is forbidden, leading to a pseudospin filtering effect for massless Dirac fermions.

Helical Luttinger Liquid on a Space-Time Lattice.

The Luttinger model is a paradigm for the breakdown due to interactions of the Fermi liquid description of one-dimensional massless Dirac fermions. Attempts to discretize the model on a one-dimensional lattice have failed to reproduce the established bosonization results because of the fermion-doubling obstruction: a local and symmetry-preserving discretization of the Hamiltonian introduces a spurious second species of low-energy excitations, while a nonlocal discretization opens a single-particle gap at the Dirac point. Here, we show how to work around this obstruction by discretizing both space and time to obtain a local Lagrangian for a helical Luttinger liquid with Hubbard interaction. The approach enables quantum MonteCarlo simulations that preserve the topological protection of an unpaired Dirac cone.

Electronic analogue of Fourier optics with massless Dirac fermions scattered by quantum dot lattice

Abstract The field of electron optics exploits the analogy between the movement of electrons or charged quasiparticles, primarily in two-dimensional materials subjected to electric and magnetic (EM) fields and the propagation of electromagnetic waves in a dielectric medium with varied refractive index. We significantly extend this analogy by introducing an electronic analogue of Fourier optics dubbed as Fourier electron optics (FEO) with massless Dirac fermions (MDF), namely the charge carriers of single-layer graphene under ambient conditions, by considering their scattering from a two-dimensional quantum dot lattice (TDQDL) treated within Lippmann–Schwinger formalism. By considering the scattering of MDF from TDQDL with a defect region, as well as the moiré pattern of twisted TDQDLs, we establish an electronic analogue of Babinet’s principle in optics. Exploiting the similarity of the resulting differential scattering cross-section with the Fraunhofer diffraction pattern, we construct a dictionary for such FEO. Subsequently, we evaluate the resistivity of such scattered MDF using the Boltzmann approach as a function of the angle made between the direction of propagation of these charge-carriers and the symmetry axis of the dot-lattice, and Fourier analyze them to show that the spatial frequency associated with the angle-resolved resistivity gets filtered according to the structural changes in the dot lattice, indicating wider applicability of FEO of MDF.

A note on degeneracy of excited energy levels in massless Dirac fermions

We propose a mechanism to construct the eigenvalues and eigenfunctions of the massless Dirac-Weyl equation in the presences of magnetic flux Φ localized in a restricted region of the plane. Using this mechanism we analyze the degeneracy of the existed energy levels. We find that the zero and first energy level has the same N+1 degeneracy, where N is the integer part of Φ2π. In addition, and contrary to what is described in the literature regarding graphene, we show that higher energy levels are N+m degenerate, being m the level of energy. In other words, this implies an indefinite growth of degenerate states as the energy level grows.

New Manifestations of the Strong Nuclear Interactions in Solids

The discovery of global macroscopic, including optical, characteristics with the addition of one neutron made crystals LiH, LiD, Diamond, Graphene an excellent model for studying the strong nuclear interaction. New manifestations of strong interaction in solids are measurements of its dependence on the distance between nucleons, finding a neutron - electron binding energy (106 meV), which is in excellent agreement with the prediction of the theory (106,7 meV). Moreover, the creation of mass in massless Dirac fermions in graphene by strong nuclear interaction has been demonstrated for the first time.

Topologically protected Casimir effect for lattice fermions

The electromagnetic Casimir effect has a fermionic counterpart in topological insulators: Zero-point fluctuations of a massless Dirac fermion field mediate a force between magnetic scatterers. The Casimir force is insensitive to disorder that preserves the topological protection of an unpaired Dirac cone. The protection may be broken if the Dirac equation is discretized, and an exponential suppression of the Casimir effect will result if a gap opens at the Dirac point. Here, we show how this lattice artifact may be avoided, by applying a recently developed local discretization of the Euclidean action that does not suffer from the fermion-doubling obstruction of local discretizations of the Hamiltonian. Published by the American Physical Society 2024

Quantum transport of massless Dirac fermions through wormhole-shaped curved graphene in presence of constant axial magnetic flux

In this work, we have studied the spin-dependent quantum transport of charged fermion on (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-dimensional spacetime, whose spatial part is described by a wormhole-type geometry in the presence of constant axial magnetic flux. Choosing the solutions of the Dirac equation associated with real energy and momentum, we explored the spin-dependent transmission probabilities and giant magnetoresistance (GMR) through a single layer of wormhole graphene with an external magnetic field, using the transition matrix (T-Matrix) approach. The spin-up and spin-down components within the A and B sublattices of graphene in the matrix of 4×1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4\ imes 1$$\\end{document} wave function are coupled to each other due to the wormhole structure and the magnetic field. We have found that transport properties strongly depend on the magnetic field, incident energy, and geometric parameters of the system. We observed that the transmission probability increases as the radius of the wormhole increases, and the length of the wormhole decreases. The higher energies lead to a decrease in the transmission probabilities of particles. Furthermore, we observed that the probability of the spin-flip effect is almost larger than that of the non-spin-flip effect, illustrating that electrons lose their spins during transmission. These findings highlight the complex and interesting behavior of wormhole graphene in the presence of external magnetic fields and suggest that these nano structures can have potential applications in electronic and spintronic devices.

Spectral Asymmetry Induces a Re-Entrant Quantum Hall Effect in a Topological Insulator.

The band inversion of topological materials in three spatial dimensions is intimately connected to the parity anomaly of 2D massless Dirac fermions, known from quantum field theory. At finite magnetic fields, the parity anomaly reveals itself as a non-zero spectral asymmetry, i.e., an imbalance between the number of conduction and valence band Landau levels, due to the unpaired zero Landau level. This work reports the realization of this 2D Dirac physics at a single surface of the 3D topological insulator (Hg,Mn)Te. An unconventional re-entrant sequence of quantized Hall plateaus in the measured Hall resistance can be directly related to the occurrence of spectral asymmetry in a single topological surface state. The effect should be observable in any topological insulator where the transport is dominated by a single Dirac surfacestate.

Possibility of massless Dirac fermions in an Aubry–André–Harper potential

In this study, we present a one-dimensional tight-binding model designed to explore the impact of electric fields on an incommensurate quantum system. We specifically focus on the Aubry–André–Harper model, a quasiperiodic model known to exhibit a metal–insulator transition at a critical potential value of λc = 2. This model combines Anderson and Aubry–André–Harper localization phenomena in a quantum system, leading to intriguing effects on the lattice band structure upon the application of an electric field F to the Aubry–André–Harper potential. Our investigation reveals that by choosing a specific value for the applied electric field, it becomes feasible to generate effective massless Dirac fermions within our Aubry–André–Harper system. Furthermore, we note that the extension or localization of the massless particle wave function is contingent upon the potential strength value λ within our incommensurate model. Importantly, our findings highlight the potential for detecting this intriguing phenomenon through experimental means.

Nonlinear Valley Hall Effect.

The valley Hall effect arises from valley-contrasting Berry curvature and requires inversion symmetry breaking. Here, we propose a nonlinear mechanism to generate a valley Hall current in systems with both inversion and time-reversal symmetry, where the linear and second-order charge Hall currents vanish along with the linear valley Hall current. We show that a second-order valley Hall signal emerges from the electric field correction to the Berry curvature, provided a valley-contrasting anisotropic dispersion is engineered. We demonstrate the nonlinear valley Hall effect in tilted massless Dirac fermions in strained graphene and organic semiconductors. Our Letter opens up the possibility of controlling the valley degree of freedom in inversion symmetric systems via nonlinear valleytronics.

Scaling dimension of the 4π -flux monopole operator in four-flavor three-dimensional QED using lattice simulation

We numerically address the issue of which monopole operators are relevant under renormalization group flow in three-dimensional parity-invariant noncompact QED with four flavors of massless two-component Dirac fermion. Using lattice simulation and finite-size scaling analysis of the free energy to introduce monopole-antimonopole pairs in N=4 and N=12 flavor noncompact QED3, we estimate the infrared scaling dimensions of monopole operators that introduce 2π and 4π fluxes around them. We first show that the estimates for the monopole scaling dimensions are consistent with the large-N expectations for N=12 QED3. Applying the same procedure in N=4 QED3, we estimate the scaling dimension of 4π flux monopole operator to be 3.7(3), which allows the possibility of the operator being irrelevant. This finding offers support to the scenario in which higher-flux monopoles are irrelevant deformations to the Dirac spin liquid phase that could be realized on certain non-bipartite lattices by forbidding 2π-flux monopoles. Published by the American Physical Society 2024

Neutral current neutrino-electron scattering and generalized uncertainty principle

In this article, we study the effect of quantum gravitational corrections on neutrino electron scattering in electroweak theory SU(2) L × U(1) Y . We consider a neutrino that scatters off an electron by an exchange of a massive neutral gauge boson. In the minimal length formalism, starting from the equation of motion for a massive gauge field we obtain the propagator. And currents are obtained from the action for a massless Dirac fermion. Scattering amplitudes and differential cross sections as function of inelasticity variable y are obtained using expressions of the propagator and leptonic current. All usual expressions are recovered when the deformation parameter α tends to zero. We find that quantum gravitational effects become important only for a highly energetic process with a momentum transfer being much greater than the neutral gauge boson mass.

Symmetry resolution of the computable cross-norm negativity of two disjoint intervals in the massless Dirac field theory

We investigate how entanglement in the mixed state of a quantum field theory can be described using the cross-computable norm or realignment (CCNR) criterion, employing a recently introduced negativity. We study its symmetry resolution for two disjoint intervals in the ground state of the massless Dirac fermion field theory, extending previous results for the case of adjacent intervals. By applying the replica trick, this problem boils down to computing the charged moments of the realignment matrix. We show that, for two disjoint intervals, they correspond to the partition function of the theory on a torus with a non-contractible charged loop. This confers a great advantage compared to the negativity based on the partial transposition, for which the Riemann surfaces generated by the replica trick have higher genus. This result empowers us to carry out the replica limit, yielding analytic expressions for the symmetry-resolved CCNR negativity. Furthermore, these expressions provide also the symmetry decomposition of other related quantities such as the operator entanglement of the reduced density matrix or the reflected entropy.

Multi-charged moments and symmetry-resolved Rényi entropy of free compact boson for multiple disjoint intervals

We study multi-charged moments and symmetry-resolved Rényi entropy of free compact boson for multiple disjoint intervals. The Rényi entropy evaluation involves computing the partition function of the theory on Riemann surfaces with genus g > 1. This makes Rényi entropy sensitive to the local conformal algebra of the theory. The free compact boson possesses a global U(1) symmetry with respect to which we resolve Rényi entropy. The multi-charged moments are obtained by studying the correlation function of flux-generating vertex operators on the associated Riemann surface. Symmetry-resolved Rényi entropy is then obtained from the Fourier transforms of the charged moments. Rényi entropy is shown to have the familiar equipartition into local charge sectors upto the leading order. The multi-charged moments are also essential in studying the symmetry resolution of mutual information. The multi-charged moments of the self-dual compact boson and massless Dirac fermion are also shown to match for the cases when the associated reduced density moments are known to be the same. Finally, we numerically check our results against the tight-binding model.

Influcence of external electric field and B/N doping on the band gap of stanene

Stanene possesses excellent properties, including an extremely high charge carrier density, massless Dirac fermions, and high thermal conductivity. Moreover, it exhibits band inversion phenomena, being made a candidate for a topological insulator. Topological insulators can generate dissipationless electric currents under certain conditions, showing great application potentials. However, the presence of a Dirac cone in the band structure of stanene at the high-symmetry point <i>K</i> in the Brillouin zone, resulting in a zero band gap, significantly limits its applications in the semiconductor field. This study adopts the method of doping B/N elements in stanene and applying an electric field perpendicular to the stanene to open the band gap at the <i>K</i> point. The effects of doping and the intensity of the applied electric field on the structural and electronic properties of stanene are investigated. The results reveal that both doping B elements and applying a vertical electric field can open the band gap at the <i>K</i> point while preserving the topological properties of stanene. Additionally, there is a positive correlation between the applied vertical electric field intensity and the band gap at the <i>K</i> point. Simultaneously doping B elements and applying a vertical electric field can increase the band gap at the <i>K</i> point, reaching 0.092 eV when the electric field intensity is 0.5 V/Å. After doping N elements, stanene is transformed into an indirect band gap semiconductor with a band gap of 0.183 eV. Applying a vertical electric field cannot change the structure of N-doped stanene, and the intensity of the applied vertical electric field is negatively correlated with the band gap at the <i>K</i> point. When the electric field intensity is 0.5 V/Å, the band gap at the <i>K</i> point decreases to 0.153 eV.