Finite Size Geometry Research Articles

Two-dimensional Dirac fermions in a mass superlattice

We study two-dimensional (2D) Dirac fermions in the presence of a periodic mass term alternating between positive and negative values along one direction. This scenario could be realized for a graphene monolayer or for the surface states of topological insulators. The low-energy physics is governed by chiral Jackiw-Rebbi modes propagating along zero-mass lines, with the energy dispersion of the Bloch states given by an anisotropic Dirac cone. By means of the transfer matrix approach, we obtain exact results for a piecewise constant mass superlattice. On top of Bloch states, two different classes of boundary and/or interface modes can exist in a finite-size geometry or in a nonuniform electrostatic potential, respectively. We compute the dispersion relation for both types of boundary and interface modes, which originate either from states close to the superlattice Brillouin zone (BZ) center or, via a Lifshitz transition, from states near the BZ boundary. In the presence of a potential step, we predict that the interface modes, the Bloch wave functions, and the electrical conductance will sensitively depend on the step position relative to the mass superlattice.

CFD study of drag and lift coefficients of non-spherical particles

The prediction of correct drag and lift forces acting on immersed bodies is vital for optimized efficiency and industrial economics. Particulate suspensions involving non-spherical particles are extensively used in the chemical and medical industry. In recent years, the hydrodynamics of non-spherical particles have attracted much attention. The hydrodynamic forces acting on these particles are the function of particle size, shape, orientation, and the physical properties of fluid. In the current research, the drag and lift coefficient of non-spherical particles of disc and prolate shapes at different inclination angles and Reynolds number (Re) is calculated using the simplified Eulerian approach of computational fluid dynamics (CFD). The method adopted for simulation is first validated for a simple case of sphere. The effect of orientation and Re on drag and lift coefficient (CD and CL, respectively) is studied and compared with the available data. It is found that CD and CL increase as the angle of inclination increases, but decrease with Re. CD and CL are smaller for prolate at given Re and orientation than for the case of disc. The effects of finite size geometry cause some deviation in results from the literature. The results obtained with this simple and less computationally intensive method agree well with highly resolved numerical simulations with minor deviations.

Quasicrystalline electronic states in 30∘ rotated twisted bilayer graphene

The recently realized bilayer graphene system with a twist angle of $30^\circ$ offers a new type of quasicrystal which unites the dodecagonal quasicrystalline nature and graphene's relativistic properties. Here, we introduce a concise theoretical framework that fully respects both the dodecagonal rotational symmetry and the massless Dirac nature, to describe the electronic states of the system. We find that the electronic spectrum consists of resonant states labeled by 12-fold quantized angular momentum, together with the extended relativistic states. The resulting quasi-band structure is composed of the nearly flat bands with spiky peaks in the density of states, where the wave functions exhibit characteristic patterns which fit to the fractal inflations of the quasicrystal tiling. We also demonstrate that the 12-fold resonant states appear as spatially-localized states in a finite-size geometry, which is another hallmark of quasicrystal. The theoretical method introduced here is applicable to a broad class of "extrinsic quasicrystals" composed of a pair of two-dimensional crystals overlaid on top of the other with incommensurate configurations.

Entanglement entropy of composite Fermi liquid states on the lattice: In support of the Widom formula

Quantum phases characterized by surfaces of gapless excitations are known to violate the otherwise ubiquitous boundary law of entanglement entropy in the form of a multiplicative log correction: $S\sim L^{d-1} \log L$. Using variational Monte Carlo, we calculate the second R\'enyi entropy for a model wavefunction of the $\nu=1/2$ composite Fermi liquid (CFL) state defined on the two-dimensional triangular lattice. By carefully studying the scaling of the total R\'enyi entropy and, crucially, its contributions from the modulus and sign of the wavefunction on various finite-size geometries, we argue that the prefactor of the leading $L \log L$ term is equivalent to that in the analogous free fermion wavefunction. In contrast to the recent results of Shao et al. [PRL 114, 206402 (2015)], we thus conclude that the "Widom formula" holds even in this non-Fermi liquid CFL state. More generally, our results further elucidate---and place on a more quantitative footing---the relationship between nontrivial wavefunction sign structure and $S\sim L \log L$ entanglement scaling in such highly entangled gapless phases.

Efimov-Like Behaviour in Low-Dimensional Polymer Models

In the quantum Efimov effect, identical bosons form infinitely many bound trimer states at the bound dimer dissociation threshold, with their energy spectrum obeying a universal geometrical scaling law. Inspired by the formal correspondence between the possible trajectories of a quantum particle and the possible conformations of a polymer chain, the existence of a triple-stranded DNA bound state when a double-stranded DNA is not stable was recently predicted by modelling three directed polymer chains in low-dimensional lattices, both fractal ( $$d<1$$ ) and euclidean ( $$d=1$$ ). A finite melting temperature for double-stranded DNA requires in $$d\le 2$$ the introduction of a weighting factor penalizing the formation of denaturation bubbles, that is non-base paired portions of DNA. The details of how bubble weighting is defined for a three-chain system were shown to crucially affect the presence of Efimov-like behaviour on a fractal lattice. Here we assess the same dependence on the euclidean $$1+1$$ lattice, by setting up the transfer matrix method for three infinitely long chains confined in a finite size geometry. This allows us to discriminate unambiguously between the absence of Efimov-like behaviour and its presence in a very narrow temperature range, in close correspondence with what was already found on the fractal lattice. When present, however, no evidence is found for triple-stranded bound states other than the ground state at the two-chain melting temperature.

Significance of accurate diffraction corrections for the second harmonic wave in determining the acoustic nonlinearity parameter

The accurate measurement of acoustic nonlinearity parameter β for fluids or solids generally requires making corrections for diffraction effects due to finite size geometry of transmitter and receiver. These effects are well known in linear acoustics, while those for second harmonic waves have not been well addressed and therefore not properly considered in previous studies. In this work, we explicitly define the attenuation and diffraction corrections using the multi-Gaussian beam (MGB) equations which were developed from the quasilinear solutions of the KZK equation. The effects of making these corrections are examined through the simulation of β determination in water. Diffraction corrections are found to have more significant effects than attenuation corrections, and the β values of water can be estimated experimentally with less than 5% errors when the exact second harmonic diffraction corrections are used together with the negligible attenuation correction effects on the basis of linear frequency dependence between attenuation coefficients, α2 ≃ 2α1.

Population of resonance and metastable atoms in a cylindrical volume of finite size

A solution method of the Holstein-Biberman equation in the case of two-dimensional finite-size geometry by means of transformation of the integral operator to a four-dimensional matrix is presented. Using this matrix the array of two-dimensional eigenvalues and eigenfunctions of the radiation transport operator in the case of finite cylinder is determined. The exact two-dimensional characteristics have been compared with approximate functions determined as a combination of corresponding eigenvalues and eigenfunctions for the one-dimensional problems (cylinder of infinite length and slab). The spatiotemporal evolution of excited atom densities for two typical forms of the excitation source in a nonequilibrium plasma has been analyzed. The reasons for the distinct difference in the formation of spatiotemporal distributions of resonance and metastable atoms in the case when the spatial distribution of the excitation source does not coincide with the fundamental mode are discussed. Resonance atoms follow the excitation source while the diffusion effectively takes metastable atoms out from the excitation source. Rearrangement of metastable atoms to the fundamental mode during their decay lasts about one effective diffusion lifetime while the corresponding process for the resonance atoms takes much longer (several effective lifetimes). The differences are caused by the effective suppression of higher diffusion modes compared with radiation modes. The developed solution method treats the radiation transport processes at the same accuracy level as diffusion transport of other plasma components and it is suitable for a self-consistent modeling of nonequilibrium plasmas.

Light propagation in atomic Mott Insulators

We study radiation-matter interaction in a system of ultracold atoms trapped in an optical lattice in a Mott insulator phase. We develop a fully general quantum model, and we perform calculations for a one-dimensional geometry at normal incidence. Both two- and three-level L atomic configurations are studied. The polariton dispersion and the reflectivity spectra are characterized in the different regimes, for both semi-infinite and finite-size geometries. We apply this model to propose a photon energy lifter experiment: a device which is able to shift the carrier frequency of a slowly travelling wavepacket without affecting the pulse shape nor its coherence.

Universal correlations of trapped one-dimensional impenetrable bosons

We calculate the asymptotic behaviour of the one body density matrix of one-dimensional impenetrable bosons in finite size geometries. Our approach is based on a modification of the Replica Method from the theory of disordered systems. We obtain explicit expressions for oscillating terms, similar to fermionic Friedel oscillations. These terms are universal and originate from the strong short-range correlations between bosons in one dimension.

Spin flop transition in a finite antiferromagnetic superlattice: evolution of the magnetic structure.

An antiferromagnetic (AF) superlattice of Fe/Cr(211) is used as a model system to study magnetic transitions in a finite-size geometry. With polarization neutron reflectometry the magnetic structure at the surface spin-flop transition and its evolution with field is determined. A domain wall created near the surface penetrates the superlattice with increasing field, splitting it into two antiphase, AF domains. After reaching the center the spin-flopped phase spreads throughout the superlattice. The experimental results are in substantial agreement with theoretical and numerical predictions.