Spectral degeneracy of the lattice dirac equation as a function of lattice shape (original) (raw)
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Microscopic Universality in the Spectrum of the Lattice Dirac Operator
Physical Review Letters, 1998
Large ensembles of complete spectra of the Euclidean Dirac operator for staggered fermions are calculated for SU(2) lattice gauge theory. The accumulation of eigenvalues near zero is analyzed as a signal of chiral symmetry breaking and compared with parameter-free predictions from chiral random matrix theory. Excellent agreement for the distribution of the smallest eigenvalue and the microscopic spectral density is found. This provides direct evidence for the conjecture that these quantities are universal functions.
From the Dirac operator to Wess–Zumino models on spatial lattices
Annals of Physics, 2005
We investigate two-dimensional Wess-Zumino models in the continuum and on spatial lattices in detail. We show that a non-antisymmetric lattice derivative not only excludes chiral fermions but in addition introduces supersymmetry breaking lattice artifacts. We study the nonlocal and antisymmetric SLAC derivative which allows for chiral fermions without doublers and minimizes those artifacts. The supercharges of the lattice Wess-Zumino models are obtained by dimensional reduction of Dirac operators in high-dimensional spaces. The normalizable zero modes of the models with N = 1 and N = 2 supersymmetry are counted and constructed in the weak-and strong-coupling limits. Together with known methods from operator theory this gives us complete control of the zero mode sector of these theories for arbitrary coupling.
Dirac Fermions on a Two-Dimensional Lattice and the Intermediate Metallic Phase
Consequences of different discretizations of the two-dimensional Dirac operator on low energy properties (e.g., the number of nodes) and their relations to gauge properties are discussed. Breaking of the gauge invariance was suggested in a recent work by M. Bocquet, D. Serban, and M.R. Zirnbauer [cond-mat/9910480] in order to destroy an intermediate metallic phase of lattice Dirac fermions with random mass. It is explained that such a procedure is inconsistent with the underlying lattice physics. Previous results point out that the logarithmic growth of the slope of the average density of states with the system size, obtained in the field-theoretical calculation of M. Bocquet et al., could be a precursor for the appearence of an intermediate metallic phase.
Symmetry-controlled edge states in the type-II phase of Dirac photonic lattices
Nature Communications
The exceptional properties exhibited by two-dimensional materials, such as graphene, are rooted in the underlying physics of the relativistic Dirac equation that describes the low energy excitations of such molecular systems. In this study, we explore a periodic lattice that provides access to the full solution spectrum of the extended Dirac Hamiltonian. Employing its photonic implementation of evanescently coupled waveguides, we indicate its ability to independently perturb the symmetries of the discrete model (breaking, also, the barrier towards the type-II phase) and arbitrarily define the location, anisotropy, and tilt of Dirac cones in the bulk. This unique aspect of topological control gives rise to highly versatile edge states, including an unusual class that emerges from the type-II degeneracies residing in the complex space of k. By probing these states, we investigate the topological nature of tilt and shed light on novel transport dynamics supported by Dirac configuration...
Lattice Dirac fermions on a simplicial Riemannian manifold
Physical Review D
The lattice Dirac equation is formulated on a simplicial complex which approximates a smooth Riemann manifold by introducing a lattice vierbein on each site and a lattice spin connection on each link. Care is taken so the construction applies to any smooth D-dimensional Riemannian manifold that permits a spin connection. It is tested numerically in 2D for the projective sphere S 2 in the limit of an increasingly refined sequence of triangles. The eigenspectrum and eigenvectors are shown to converge rapidly to the exact result in the continuum limit. In addition comparison is made with the continuum Ising conformal field theory on S 2. Convergence is tested for the two point, (x1) (x2) , and the four point, σ(x1) (x2) (x3)σ(x4) , correlators for the energy, (x) = iψ(x)ψ(x), and twist operators, σ(x), respectively.
Dirac equation as a quantum walk over the honeycomb and triangular lattices
Physical Review A, 2018
A discrete-time Quantum Walk (QW) is essentially an operator driving the evolution of a single particle on the lattice, through local unitaries. Some QWs admit a continuum limit, leading to well-known physics partial differential equations, such as the Dirac equation. We show that these simulation results need not rely on the grid: the Dirac equation in (2+1)-dimensions can also be simulated, through local unitaries, on the honeycomb or the triangular lattice, both of interest in the study of quantum propagation on the non-rectangular grids, as in graphene-like materials. The latter, in particular, we argue, opens the door for a generalization of the Dirac equation to arbitrary discrete surfaces.
The most fundamental characteristic of a physical system can often be deduced from its behaviour under discrete symmetry transformations, such as time reversal, parity and chirality. Here, we review some of the basic symmetry properties of the relativistic quantum theories for free electrons in (2+1)- and (1+1)-dimensional spacetime. Additional flavour degrees of freedom are necessary to properly define symmetry operations in (2+1) dimensions, and are generally present in physical realizations of such systems, for example in single sheets of graphite. We find that there exist two possibilities for defining any flavour-coupling discrete symmetry operation of the two-flavour (2+1)-dimensional Dirac theory. Some physical implications of this previously unnoticed duplicity are discussed.
The construction of generalized Dirac operators on the lattice 1
2000
We discuss the steps to construct Dirac operators which have arbitrary fermion offsets, gauge paths, a general structure in Dirac space and satisfy the basic symmetries (gauge symmetry, hermiticity condition, charge conjugation, hypercubic rotations and reflections) on the lattice. We give an extensive set of examples and offer help to add further structures. Dirac operators which satisfy the Ginsparg-Wilson (GW) relation γ 5 D −1 + D −1 γ 5 = γ 5 2R, (1) where R is a local operator trivial in Dirac space, define lattice regularized theories whose chiral properties are the same as those of the corresponding
Emergence of Type-II Dirac Points in Graphynelike Photonic Lattices
Physical Review Letters
We theoretically demonstrate that a type-II class of tilted Dirac cones can emerge in generalized two-dimensional anisotropic lattice arrangements. This is achieved by introducing a special set of graphyne-like exchange bonds by means of which the complete spectrum of the underlying Weyl Hamiltonian can be realized. Our ab-initio calculations demonstrate a unique class of eigensolutions corresponding to a type-II class of Dirac fermionic excitations. Based on our approach, one can systematically synthesize a wide range of strongly anisotropic band diagrams having tilted Dirac cones with variable location and orientation. Moreover, we show that asymmetric conical diffraction as well as edge states, can arise in these configurations. Our results can provide a versatile platform to observe, for the first time, optical transport around type-II Dirac points in two-dimensional optical settings under linear, nonlinear, and non-Hermitian conditions.
Dirac lattices, zero-range potentials, and self-adjoint extension
Physical Review D, 2015
We consider the electromagnetic field in the presence of polarizable point dipoles. In the corresponding effective Maxwell equation these dipoles are described by three dimensional delta function potentials. We review the approaches handling these: the selfadjoint extension, regularization/renormalisation and the zero range potential methods. Their close interrelations are discussed in detail and compared with the electrostatic approach which drops the contributions from the self fields. For a homogeneous two dimensional lattice of dipoles we write down the complete solutions, which allow, for example, for an easy numerical treatment of the scattering of the electromagnetic field on the lattice or for investigating plasmons. Using these formulas, we consider the limiting case of vanishing lattice spacing, i.e., the transition to a continuous sheet. For a scalar field and for the TE polarization of the electromagnetic field this transition is smooth and results in the results known from the continuous sheet. Especially for the TE polarization, we reproduce the results known from the hydrodynamic model describing a two dimensional electron gas. For the TM polarization, for polarizability parallel and perpendicular to the lattice, in both cases, the transition is singular. For the parallel polarizability this is surprising and different from the hydrodynamic model. For perpendicular polarizability this is what was known in literature. We also investigate the case when the transition is done with dipoles described by smeared delta function, i.e., keeping a regularization. Here, for TM polarization for parallel polarizability, when subsequently doing the limit of vanishing lattice spacing, we reproduce the result known from the hydrodynamic model. In case of perpendicular polarizability we need an additional renormalization to reproduce the result obtained previously by stepping back from the dipole approximation.