Dirac Mode Research Articles

Geometry-independent tight-binding method for massless Dirac fermions in two dimensions

The Nielsen-Ninomiya theorem, dubbed `fermion-doubling', poses a problem for the naive discretization of a single (massless) Dirac cone on a two-dimensional surface. The inevitable appearance of an additional, unphysical fermionic mode can, for example, be circumvented by introducing an extra dimension to spatially separate Dirac cones. In this work, we propose a geometry-independent protocol based on a tight-binding model for a three-dimensional topological insulator on a cubic lattice. The low-energy theory, below the bulk gap, corresponds to a Dirac cone on its two-dimensional surface which can have an arbitrary geometry. We introduce a method where only a thin shell of the topological insulator needs to be simulated. Depending on the setup, we propose to gap out the states on the undesired surfaces either by breaking the time-reversal symmetry or by introducing a superconducting pairing. We show that it is enough to have a thickness of the topological-insulator shell of three to nine lattice constants. This leads to an effectively two-dimensional scaling with minimal and fixed shell thickness. We test the idea by comparing the spectrum and probability distribution to analytical results for both a proximitized Dirac mode and a Dirac mode on a sphere which exhibits a nontrivial spin-connection. The protocol yields a tight-binding model on a cubic lattice simulating Dirac cones on arbitrary surfaces with only a small overhead due to the finite thickness of the shell.

Fermionic Bell violation in the presence of background electromagnetic fields in the cosmological de Sitter spacetime

The violation of the Bell inequality for Dirac fermions is investigated in the cosmological de Sitter spacetime, in the presence of background electromagnetic fields of constant strengths. The orthonormal Dirac mode functions are obtained and the relevant in-out squeezed state expansion in terms of the Bogoliubov coefficients are found. We focus on two scenarios here: strong electric field and heavy mass limits (with respect to the Hubble constant). Using the squeezed state expansion, we then demonstrate the Bell violations for the vacuum and some maximally entangled initial states. Even though a background magnetic field alone cannot create particles, in the presence of background electric field and or spacetime curvature, it can affect the particle creation rate. Thus, our chief aim here is to investigate the role of the background magnetic field strength in the Bell violation. Qualitative differences in this regard for different maximally entangled initial states are shown. Further extension of these results to the so-called $\ensuremath{\alpha}$ vacua are also discussed.

Chirality inversion of Majorana edge modes in a Fu–Kane heterostructure

Fu and Kane have discovered that a topological insulator with induced s-wave superconductivity (gap Δ0, Fermi velocity v F, Fermi energy μ) supports chiral Majorana modes propagating on the surface along the edge with a magnetic insulator. We show that the direction of motion of the Majorana fermions can be inverted by the counterflow of supercurrent, when the Cooper pair momentum along the boundary exceeds . The chirality inversion is signaled by a doubling of the thermal conductance of a channel parallel to the supercurrent. Moreover, the inverted edge can transport a nonzero electrical current, carried by a Dirac mode that appears when the Majorana mode switches chirality. The chirality inversion is a unique signature of Majorana fermions in a spinful topological superconductor: it does not exist for spinless chiral p-wave pairing.

Observation of the Dirac mode guidance in Kagome lattice of photonic crystals

Conventionally, the behavior of light transmission in optical devices is based on total internal reflection or photonic bandgaps. The Dirac mode is regarded as another guiding mechanism that relies on the zero-density of radiation states around the Dirac point to avoid power leakage and thus achieves field confinement. Unlike conventional light guiding mechanisms, the Dirac mode features a unique algebraic decay profile around the Dirac frequency. In this study, the Dirac mode in the Kagome lattice of photonic crystals (PCs) at a Dirac frequency that is beyond any complete photonic bandgaps is observed, and its features are verified using finite difference time domain analyses. The results indicate that the Dirac mode in the triangular, honeycomb, and two-compound lattices of PCs also occurs in the Kagome lattice and has an extremely high quality factor of 1.03×106. Furthermore, analyses of the Dirac mode serve as a convenient method to control the all-range Dirac frequency by modifying the structural parameters accordingly, thus extending the applications of the Dirac mode in modern integrated optical devices.

Reconfinement, localization, and thermal monopoles in SU(3) trace-deformed Yang-Mills theory

We study, by means of numerical lattice simulations, the properties of the reconfinement phase transition taking place in trace deformed $SU(3)$ Yang-Mills theory defined on $\mathbb{R}^3\times S^1$, in which center symmetry is recovered even for small compactification radii. We show, by means of a finite size scaling analysis, that the reconfinement phase transition is first-order, like the usual $SU(3)$ thermal phase transition. We then investigate two different physical phenomena, which are known to characterize the standard confinement/deconfinement phase transition, namely the condensation of thermal magnetic monopoles and the change in the localization properties of the eigenmodes of the Dirac operator. Regarding the latter, we show that the mobility edge signalling the Anderson-like transition in the Dirac spectrum vanishes as one enters the reconfined phase, as it happens in the standard confined phase. Thermal monopoles, instead, show a peculiar behavior: their density decreases going through reconfinement, at odds with the standard thermal theory; nonetheless, they condense at reconfinement, like at the usual confinement transition. The coincidence of monopole condensation and Dirac mode delocalization, even in a framework different from that of the standard confinement transition, suggests the existence of a strict link between them.

Jackiw-Rebbi zero modes in non-uniform topological insulator nanowire

We theoretically investigate the emergence of Jackiw-Rebbi zero modes and their conductance signature in non-uniform topological insulator nano-wires. We modelled the non-uniform nano-wires as junction between two cylindrical nano-wires with different radius. In the limit of wire length being much larger than its radius, the surface state of the nanowire splits into one dimensional Dirac modes propagating along the axis of the cylinder owing to radial confinement. The sign of the mass gap in each of these Dirac mode is decided by angular momentum quantum number corresponding to the rotational motion of the electron about the axis of the cylindrical. Application of an external magnetic flux through the cylindrical nanowires enables us to tune the mass gap from positive to negative value across the junction. Due to this flux tunable band inversion, controlled by the external magnetic filed, Jackiw-Rebbi zero modes can be made to appear or disappear at the junction. We compute differential conductance of our topological insulator nanowire junction and show that a quantized conductance peak appears at zero-energy (zero-bias) in the presence of the Jackiw-Rebbi mode.

Observation of Dirac mode in modified honeycomb hollow core photonic crystal fiber

The Dirac mode in the hollow core photonic crystal fibers (HC-PCFs) traditionally rely on high refractive index glasses with a high air-filling factor. In this paper, the Dirac mode was observed in the modified honeycomb lattice HC-PCFs, which breaks the constraint of refractive index, and can be realized in pure silica glasses for the first time. The band structure of the modified honeycomb lattice was calculated to elaborate the formation process of the Dirac point. The Dirac mode in the HC-PCFs was established at the Dirac frequency outside the bandgaps. In addition, the relationship between the Dirac frequency and the air-filling fraction and the refractive index were also analyzed. This design provides a modified honeycomb lattice HC-PCFs with substantially reduced fabrication effort and expands the application of Dirac mode in the conventional silica-based optical fiber devices.

Light trapping and circularly polarization at a Dirac point in 2D plasma photonic crystals

Light trapping at the Dirac point in 2D plasma photonic crystal has been obtained. The new localized mode, Dirac mode, is attributable to neither photonic bandgap nor total internal reflection. It exhibits a unique algebraic profile and possesses a high-Q factor resonator of about 105. The Dirac point could be modulated by tuning the filling factor, plasma frequency and plasma cyclotron frequency, respectively. When a magnetic field parallel to the wave vector is applied, Dirac modes for right circularly polarized and left circularly polarized waves could be obtained at different frequencies, and the Q factor could be tuned. This property will add more controllability and flexibility to the design and modulation of novel photonic devices. It is also valuable for the possibilities of Dirac modes in photonic crystal containing other kinds of metamaterials.

Light confinement at a Dirac point in honeycomb-like lattice photonic crystal

Optical waveguides and cavities are important components among modern optical devices. Traditional optical cavities rely on total internal reflection or photonic bandgaps to achieve field confinement. Recently, a new type of trapped mode, the Dirac mode, has been reported to occur in triangular or honeycomb lattice photonic crystal. This novel localized mode is attribute to neither of the traditional light-guiding mechanisms and owns different characteristics. Here we report the discovery of the Dirac mode in honeycomb-like photonic crystal lattice. The Dirac mode occurs at a Dirac frequency, which is beyond the complete photonic bandgaps. It has a different algebraic-decay feature. Our discovery extends applicability of the Dirac mode in designing for optical devices.

Light trapping at Dirac point in 2D triangular Archimedean-like lattice photonic crystal.

Optical cavities and waveguides are critical parts of modern optical devices. Traditionally, optical cavities and waveguides rely on photonic bandgaps, or total internal reflection, to achieve light trapping. It has been reported that a novel light trapping, which exists in triangular and honeycomb lattices, is attributed to the so-called Dirac point. Our analysis reveals that 2D triangular Archimedean-like lattice photonic crystals also can support this Dirac mode with similar characteristics. This is a new type of localized mode with a different algebraic field profile at a different specified Dirac frequency, which is also beyond any complete photonic bandgap. The new wave localization has different features and can be applied to the design of new optical devices.

Infrared regime of SU(2) with one adjoint Dirac flavor

SU(2) gauge theory with one Dirac flavour in the adjoint representation is investigated on a lattice. Initial results for the gluonic and mesonic spectrum, static potential from Wilson and Polyakov loops, and the anomalous dimension of the fermionic condensate from the Dirac mode number are presented. The results found are not consistent with conventional confining behaviour, instead tentatively pointing towards a theory lying within or very near the onset of the conformal window, with the anomalous dimension of the fermionic condensate in the range $0.9 \lesssim \gamma_* \lesssim 0.95$. The implications of our work for building a viable theory of strongly interacting dynamics beyond the standard model are discussed.

Relation between confinement and chiral symmetry breaking in temporally odd-number lattice QCD

In the lattice QCD formalism, we investigate the relation between confinement and chiral symmetry breaking. A gauge-invariant analytical relation connecting the Polyakov loop and the Dirac modes is derived on a temporally odd-number lattice, where the temporal lattice size is odd, with the normal (nontwisted) periodic boundary condition for link variables. This analytical relation indicates that low-lying Dirac modes have little contribution to the Polyakov loop, and it is numerically confirmed at the quenched level in both confinement and deconfinement phases. This fact indicates no direct one-to-one correspondence between confinement and chiral symmetry breaking in QCD. Using the relation, we also investigate the contribution from each Dirac mode to the Polyakov loop. In the confinement phase, we find a new ``positive/negative symmetry'' of the Dirac-mode matrix element of the link-variable operator, and this symmetry leads to the zero value of the Polyakov loop. In the deconfinement phase, there is no such symmetry and the Polyakov loop is nonzero. Also, we develop a new method for spin-diagonalizing the Dirac operator on the temporally odd-number lattice modifying the Kogut-Susskind formalism.

Analytical relation between quark confinement and chiral symmetry breaking in odd-number lattice QCD

To clarify the relation between confinement and chiral symmetry breaking in QCD, we consider a temporally odd-number lattice, with the temporal lattice size $N_t$ being odd. We here use an ordinary square lattice with the normal (nontwisted) periodic boundary condition for link-variables in the temporal direction. By considering ${\rm Tr} (\hat{U}_4\hat{\not D}^{N_t-1})$, we analytically derive a gauge-invariant relation between the Polyakov loop $\langle L_P \rangle$ and the Dirac eigenvalues $\lambda_n$ in QCD, i.e., $\langle L_P \rangle \propto \sum_n \lambda_n^{N_t -1} \langle n|\hat U_4|n \rangle$, which is a Dirac spectral representation of the Polyakov loop in terms of Dirac eigenmodes $|n\rangle$. Owing to the factor $\lambda_n^{N_t -1}$ in the Dirac spectral sum, this relation generally indicates fairly small contribution of low-lying Dirac modes to the Polyakov loop, while the low-lying Dirac modes are essential for chiral symmetry breaking. Also in lattice QCD calculations in both confined and deconfined phases, we numerically confirm the analytical relation, non-zero finiteness of $\langle n|\hat U_4|n \rangle$ for each Dirac mode, and negligibly small contribution from low-lying Dirac modes to the Polyakov loop, i.e., the Polyakov loop is almost unchanged even by removing low-lying Dirac-mode contribution from the QCD vacuum generated by lattice QCD simulations. We thus conclude that low-lying Dirac modes are not essential modes for confinement, which indicates no direct one-to-one correspondence between confinement and chiral symmetry breaking in QCD.

Theory of the random potential and conductivity at the surface of a topological insulator

We study the disorder potential induced by random Coulomb impurities at the surface of a topological insulator (TI). We use a simple model in which positive and negative impurities are distributed uniformly throughout the bulk of the TI, and we derive the magnitude of the disorder potential at the TI surface using a self-consistent theory based on the Thomas-Fermi approximation for screening by the Dirac mode. Simple formulas are presented for the mean squared potential both at the Dirac point and far from it, as well as for the characteristic size of electron/hole puddles at the Dirac point and the total concentration of electrons/holes that they contain. We also derive an expression for the autocorrelation function for the potential at the surface and show that it has an unusually slow decay, which can be used to verify the bulk origin of disorder. The implications of our model for the electron conductivity of the surface are also presented.

Lattice fermions with Majorana couplings

We analyse stability of almost massless Dirac mode in gauge models with boundary (domain wall) fermoions, and consider the possibility of decoupling one of its chiral component by giving it a Majorana mass of the order of the inverse lattice spacing. We argue that the chiral spectrum in such models is always uncharged, so they can be implemented for defining the Weyl fermions only in the real representation of the gauge group, for instance, in SUSY models.

Perturbation theory at finite extent of fifth dimension for vacuum overlap formula of chiral determinant

We develop a perturbation theory for the vacuum overlap formula of chiral determinant at finite extent of fifth dimension in the lattice regularization. We adopt a fermion boundary condition which is derived from the lattice action of Wilson fermion. As an application, we calculate the vacuum polarization in four-dimensional theory. Each contribution to the vacuum polarization is vector-like. It is the existence of light (Dirac) mode in the spectrum of fermion with positive homogeneous mass that leads to the chiral normalization.