Fermion Doubling Research Articles

Floquet Insulators and Lattice Fermions.

Floquet insulators are periodically driven quantum systems that can host novel topological phases as a function of the drive parameters. These new phases exhibit features reminiscent of fermion doubling in discrete-time lattice fermion theories. We make this suggestion concrete by mapping the spectrum of a noninteracting (1+1)D Floquet insulator for certain drive parameters onto that of a discrete-time lattice fermion theory with a time-independent Hamiltonian. The resulting Hamiltonian is distinct from the Floquet Hamiltonian that generates stroboscopic dynamics. It can take the form of a discrete-time Su-Schrieffer-Heeger model with half the number of spatial sites of the original model, or of a (1+1)D Wilson-Dirac theory with one quarter of the spatial sites.

Remarks on discrete Dirac operators and their continuum limits

We discuss possible definitions of discrete Dirac operators, and discuss their continuum limits. It is well known in the lattice field theory that the straightforward discretization of the Dirac operator introduces unwanted spectral subspaces, and it is known as the fermion doubling . In oder to overcome this difficulty, two methods were proposed. The first one is to introduce a new term, called the Wilson term , and the second one is the KS-fermion model or the staggered fermion model . We discuss mathematical formulations of these, and study their continuum limits.

Floquet insulators and lattice fermions beyond naive time discretization

Periodically driven quantum systems known as Floquet insulators can host topologically protected bound states known as “π modes” that exhibit response at half the frequency of the drive. Such states can also appear in undriven lattice field theories when time is discretized as a result of fermion doubling, raising the question of whether these two phenomena could be connected. Recently we demonstrated such a connection at the level of an explicit mapping between the spectra of a continuous-time Floquet model and a discrete-time undriven lattice fermion model. However, this mapping relied on a symmetry of the single-particle spectrum that is not present for generic drive parameters. Inspired by the example of the temporal Wilson term in lattice field theory, in this paper we extend this mapping to the full drive parameter space by allowing the parameters of the discrete-time model to be frequency-dependent. The spectra of the resulting lattice fermion models exactly match the quasienergy spectrum of the Floquet model in the thermodynamic limit. Our results demonstrate that spectral features characteristic of beyond-equilibrium physics in Floquet systems can be replicated in static systems with appropriate time discretization. Published by the American Physical Society 2024

Tangent Fermions: Dirac or Majorana Fermions on a Lattice Without Fermion Doubling

AbstractMethods to discretize the Hamiltonian of a topological insulator or topological superconductor, without giving up on the topological protection of the massless excitations (respectively, Dirac fermions or Majorana fermions) are reviewed. The method of tangent fermions, pioneered by Richard Stacey, is singled out as being uniquely suited for this purpose. Tangent fermions propagate on a dimensional space‐time lattice with a tangent dispersion: in dimensionless units. They avoid the fermion doubling lattice artefact that will spoil the topological protection, while preserving the fundamental symmetries of the Dirac Hamiltonian. Although the discretized Hamiltonian is nonlocal, as required by the fermion‐doubling no‐go theorem, it is possible to transform the wave equation into a generalized eigenproblem that is local in space and time. Applications that are discussed include Klein tunneling of Dirac fermions through a potential barrier, the absence of localization by disorder, the anomalous quantum Hall effect in a magnetic field, and the thermal metal of Majorana fermions.

Twisted quantum walks, generalised Dirac equation and Fermion doubling

Quantum discrete-time walkers have, since their introduction, demonstrated applications in algorithmic and in modeling and simulating a wide range of transport phenomena. They have long been considered the discrete-time and discrete space analogue of the Dirac equation and have been used as a primitive to simulate quantum field theories precisely because of some of their internal symmetries. In this paper we introduce a new family of quantum walks, said twisted, which admits, as continuous limit, a generalized Dirac operator equipped with a dispersion term. Moreover, this quadratic term in the energy spectrum acts as an effective mass, leading to a regularization of the well known Fermion doubling problem.

CT or P problem and symmetric gapped fermion solution

An analogous ``strong $CP$ problem'' is identified in a toy model in two-dimensional spacetime: a general $1+1\mathrm{d}$ Abelian U(1) anomaly free chiral fermion and chiral gauge theory with a generic theta instanton term $\frac{\ensuremath{\theta}}{2\ensuremath{\pi}}\ensuremath{\int}F$. The theta term alone violates the charge-conjugation-time-reversal $CT$ and the parity $P$ discrete symmetries. The analogous puzzle here is the $CT$ or $P$ problem in $1+1\mathrm{d}$: Why can the $\overline{\ensuremath{\theta}}$ angle (including the effect of $\ensuremath{\theta}$ and the complex phase of a mass matrix) be zero or small for a natural reason? We show that this $CT$ or $P$ problem can be solved by a symmetric mass generation mechanism (SMG, namely generating a mass or energy gap while preserving an anomaly free symmetry). This $1+1\mathrm{d}$ toy model mimics several features of the $3+1\mathrm{d}$ Standard Model: chiral matter content, confinement, and Anderson-Higgs mass by Yukawa-Higgs term. One solution replaces some chiral fermion's Higgs-induced mean-field mass with SMG-induced non-mean-field mass. Another solution enriches this toy model by introducing several new physics beyond the Standard Model: a parity-reflection $PR$ discrete symmetry maps between the chiral and mirror fermions as fermion doubling localized on two domain walls at high energy, and SMG dynamically generates mass to the mirror fermion while still preserving the anomaly free chiral symmetry at an intermediate energy scale, much before the Higgs mechanism generates mass to the chiral fermion at lower energy. Without loss of generality, an arguably simplest $1+1\mathrm{d}$ U(1) symmetric anomaly free chiral fermion/gauge theory (e.g., Weyl fermions with ${3}_{L}\text{\ensuremath{-}}{4}_{L}\text{\ensuremath{-}}{5}_{R}\text{\ensuremath{-}}{0}_{R}$ U(1) charges) is demonstrated in detail. As an analogy to the superfluid-insulator or order-disorder quantum phase transition, in contrast to the conventional Peccei-Quinn solution sitting in the (quasi-long-range-order)-superfluid ordered phase, our solution is in the SMG insulator disordered phase.

Resolving the spurious-state problem in the Dirac equation with the finite-difference method

To solve the Dirac equation with the finite difference method, one has to face up to the spurious-state problem due to the fermion doubling problem when using the conventional central difference formula to calculate the first-order derivative on the equal interval lattices. This problem is resolved by replacing the central difference formula with the asymmetric difference formula, i.e., the backward or forward difference formula. To guarantee the hermitian of the Hamiltonian matrix, the backward and forward difference formula should be used alternatively according to the parity of the wavefunction. This provides a simple and efficient numerical prescription to solve various relativistic problems in the microscopic world.

Massless Dirac Fermions on a Space‐Time Lattice with a Topologically Protected Dirac Cone

AbstractThe symmetries that protect massless Dirac fermions from a gap opening may become ineffective if the Dirac equation is discretized in space and time, either because of scattering between multiple Dirac cones in the Brillouin zone (fermion doubling) or because of singularities at zone boundaries. Here an implementation of Dirac fermions on a space‐time lattice that removes both obstructions is introduced. The quasi‐energy band structure has a tangent dispersion with a single Dirac cone that cannot be gapped without breaking both time‐reversal and chiral symmetries. It is shown that this topological protection is absent in the familiar single‐cone discretization with a linear sawtooth dispersion, as a consequence of the fact that there the time‐evolution operator is discontinuous at Brillouin zone boundaries.

Quantum anomalous semimetals

The topological states of matter and topological materials have been attracting extensive interests as one of the frontier topics in condensed matter physics and materials science since the discovery of quantum Hall effect in 1980s. So far all the topological phases such as integer quantum Hall effect and topological insulators are characterized by integer topological invariants. None is a half integer or fractional. Here we propose a type of semimetals which hosts a single cone of Wilson fermions. The Wilson fermions possess linear dispersion near the Dirac point, but break the chiral or parity symmetry such that an unpaired Dirac cone can be realized on a lattice. In order to avoid the fermion doubling problem, the chiral symmetry or parity symmetry must be broken explicitly if the hermiticity, locality and translational invariance all hold. We find that the system can be classified by the relative homotopy group, and a half-integer topological invariant. We term the nontrivial quantum phase as quantum anomalous semimetal. The work opens the door towards exploring novel states of matter with fractional topological charge.

Generalized fermion doubling theorems: Classification of two-dimensional nodal systems in terms of wallpaper groups

The Nielsen-Ninomiya Theorem has set up a ground rule for the minimal number of the topological points in a Brillouin zone. Notably, in the 2D Brillouin zone, chiral symmetry and space-time inversion symmetry can properly define topological invariants as charges characterizing the stability of the nodal points so that the non-zero charges protect these points. Due to the charge neutralization, the Nielsen-Ninomiya Theorem requires at least two stable topological points in the entire Brillouin zone. However, additional crystalline symmetries might duplicate the points. In this regard, for the wallpaper groups with crystalline symmetries, the minimal number of the nodal points in the Brillouin zone might be more than two. In this work, we determine the minimal numbers of the nodal points for the wallpaper groups in chiral-symmetric and space-time-inversion-symmetric systems separately and provide examples for new topological materials, such as topological nodal time-reversal-symmetric superconductors and Dirac semimetals. This generalized Nielsen-Ninomiya Theorem serves as a guide to search for 2D topological nodal materials and new platforms for twistronics. Furthermore, we show the Nielsen-Ninomiya Theorem can be extended to 2D non-Hermitian systems hosting topologically protected exceptional points and Fermi points for the 17 wallpaper groups and use the violation of the theorem on the surface to classify 3D Hermitian and non-Hermitian topological bulks.

Reflectionless Klein tunneling of Dirac fermions: comparison of split-operator and staggered-lattice discretization of the Dirac equation

Massless Dirac fermions in an electric field propagate along the field lines without backscattering, due to the combination of spin-momentum locking and spin conservation. This phenomenon, known as ‘Klein tunneling’, may be lost if the Dirac equation is discretized in space and time, because of scattering between multiple Dirac cones in the Brillouin zone. To avoid this, a staggered space-time lattice discretization has been developed in the literature, with one single Dirac cone in the Brillouin zone of the original square lattice. Here we show that the staggering doubles the size of the Brillouin zone, which actually contains two Dirac cones. We find that this fermion doubling causes a spurious breakdown of Klein tunneling, which can be avoided by an alternative single-cone discretization scheme based on a split-operator approach.

Network of Topological Nodal Planes, Multifold Degeneracies, and Weyl Points in CoSi.

We showcase the importance of global band topology in a study of the Weyl semimetal CoSi as a representative of chiral space group (SG) 198. We identify a network of band crossings comprising topological nodal planes, multifold degeneracies, and Weyl points consistent with the fermion doubling theorem. To confirm these findings, we combined the general analysis of the band topology of SG 198 with Shubnikov-de Haas oscillations and material-specific calculations of the electronic structure and Berry curvature. The observation of two nearly dispersionless Shubnikov-de Haas frequency branches provides unambiguous evidence of four Fermi surface sheets at the R point that reflect the symmetry-enforced orthogonality of the underlying wave functions at the intersections with the nodal planes. Hence, irrespective of the spin-orbit coupling strength, SG 198 features always six- and fourfold degenerate crossings at R and Γ that are intimately connected to the topological charges distributed across the network.

Topological properties of minimally doubled fermions in two spacetime dimensions

The two-dimensional Schwinger model is used to explore how lattice fermion operators perceive the global topological charge $q \in \mathbb{Z}$ of a given background gauge field. We focus on Karsten-Wilczek and Borici-Creutz fermions, which are minimally doubled, and compare them to Wilson, Brillouin, naive, staggered and Adams fermions. For each operator the eigenvalue spectrum in a background with $q \neq 0$ is determined along with the chiralities of the eigenmodes, and the spectral flow of the pertinent hermitean operator is worked out. We find that Karsten-Wilczek and Borici-Creutz fermions perceive the global topological charge $q$ in the same way as staggered and naive fermions do.

Non-Hermitian Spatial Symmetries and Their Stabilized Normal and Exceptional Topological Semimetals.

We study non-Hermitian spatial symmetries-a class of symmetries that have no counterparts in Hermitian systems-and study how normal and exceptional semimetals can be stabilized by these symmetries. Different from internal ones, spatial symmetries act nonlocally in momentum space and enforce global constraints on both band degeneracies and topological quantities at different locations. Inderiving general constraints on band degeneracies and topological invariants, we demonstrate that non-Hermitian spatial symmetries are on an equal footing with, but are essentially different from Hermitian ones. First, we discover the nonlocal Hermitian conjugate pair of exceptional or normal band degeneracies that are enforced by non-Hermitian spatial symmetries. Remarkably, we find that these pairs lead to the symmetry-enforced violation of the Fermion doubling theorem in the long-time limit. Second, with the topological constraints, we unravel that a certain exceptional manifold is only compatible with and stabilized by non-Hermitian spatial symmetries but is intrinsically incompatible with Hermitian spatial symmetries. We illustrate these findings using two three-dimensional models of a non-Hermitian Weyl semimetal and an exceptional unconventional Weyl semimetal. Experimental cold-atom realizations of both models are also proposed.

Moiré Landau levels of a C4 -symmetric twisted bilayer system in the absence of a magnetic field

It is widely known that the twisted bilayer graphene (TBG) shows flat bands at magic angles, which can be well described by the effective continuum model derived by Bistritzer and MacDonald (BM). We propose in this paper a similar twisted bilayer system but defined on the square lattice with $\pi$-flux per plaquette, and study its spectrum using the BM Hamiltonian with a mass term which is originated from the staggered potential. The basic difference between the TBG and the present model is simply rotational symmetry, $C_3$ vs $C_4$, as well as a mass term. Nevertheless, the feature of the flat bands is quite different: Those of the TBG appear at magic angles only, while the present model shows many flat bands, which are reminiscent of Landau levels, quite stably at any angles even in the absence of a magnetic field other than $\pi$-flux which keeps time reversal (TR) symmetry. Moreover, flat bands emerge in the mass gap of the Dirac spectrum, and each state composing these flat bands is well-localized at the position forming the moir\'e lattice. It turns out that the moir\'e potential serves as a periodic magnetic field, which can give energies smaller that the gap around moir\'e lattice positions. We derive a local Hamiltonian valid around the moir\'e lattice sites and show that it indeed reproduces the energies of the flat bands within the mass gap. Since these mid-gap states are localized at the moir\'e lattice, they form degenerate levels, which may be referred to as moir\'e Landau levels, although the mechanism of degeneracies are different from the conventional Landau levels. Interestingly, doubled fermions of the BH Hamiltonian associated with two layers have opposite charges when they couple with the effective moir\'e magnetic filed, which concern TR symmetry.

One generation of standard model Weyl representations as a single copy of [formula omitted

Peering in from the outside, A:=R⊗C⊗H⊗O looks to be an ideal mathematical structure for particle physics. It is 32 C-dimensional: exactly the size of one full generation of fermions. Furthermore, as alluded to earlier in [1], it supplies a richer algebraic structure, which can be used, for example, to replace SU(5) with the SU(3)×SU(2)×U(1)/Z6 symmetry of the standard model.However, this line of research has been weighted down by a difficulty known as the fermion doubling problem. That is, a satisfactory description of SL(2,C) symmetries has so far only been achieved by taking two copies of the algebra, instead of one. Arguably, this doubling of states betrays much of A's original appeal. In this article, we solve the fermion doubling problem in the context of A.Furthermore, we give an explicit description of the standard model symmetries, ▪, its gauge bosons, Higgs, and a generation of fermions, each in the compact language of this 32 C-dimensional algebra. Most importantly, we seek out the subalgebra of ▪ which is invariant under the complex conjugate - and find that it is given by su(3)C⊕u(1)EM. Could this new result provide a clue as to why the standard model symmetries break in the way that they do?

Spectral action and the electroweak \u03b8-terms for the Standard Model without fermion doubling

We compute the leading terms of the spectral action for a noncommutative geometry model that has no fermion doubling. The spectral triple describing it, which is chiral and allows for CP-symmetry breaking, has the Dirac operator that is not of the product type. Using Wick rotation we derive explicitly the Lagrangian of the model from the spectral action for a flat metric, demonstrating the appearance of the topological θ-terms for the electroweak gauge fields.

Quantum Hall effect of the topological insulator state of cadmium arsenide in Corbino geometry

The surfaces of a three-dimensional topological insulator each host a single Dirac fermion, which, in a strong magnetic field, contribute to the transverse conductance in integer-and-a-half multiples of the conductance quantum. The direct observation of this extra half integer, the hallmark of the two-dimensional Dirac state, is usually thwarted by the fermion doubling theorem—top and bottom surfaces are not measured independently. Here, we employ a Corbino measurement geometry in which a current, induced by an ac magnetic field, is driven around the ring, as a complementary measurement to the conventional Hall bar geometry. As the device enters the quantum Hall regime, the transverse voltage reaches a series of plateaus when the current is carried by the incompressible bulk states. We compare the results with the corresponding Hall bar measurements.

Discovery of {hat{boldsymbol{C}}}_2 rotation anomaly in topological crystalline insulator SrPb

Topological crystalline insulators (TCIs) are insulating electronic states with nontrivial topology protected by crystalline symmetries. Recently, theory has proposed new classes of TCIs protected by rotation symmetries hat C_n, which have surface rotation anomaly evading the fermion doubling theorem, i.e., n instead of 2n Dirac cones on the surface preserving the rotation symmetry. Here, we report the first realization of the hat C_2 rotation anomaly in a binary compound SrPb. Our first-principles calculations reveal two massless Dirac fermions protected by the combination of time-reversal symmetry hat T and hat C_{2y} on the (010) surface. Using angle-resolved photoemission spectroscopy, we identify two Dirac surface states inside the bulk band gap of SrPb, confirming the hat C_2 rotation anomaly in the new classes of TCIs. The findings enrich the classification of topological phases, which pave the way for exploring exotic behavior of the new classes of TCIs.

Fermion Doubling Theorems in Two-Dimensional Non-Hermitian Systems for Fermi Points and Exceptional Points.

The fermion doubling theorem plays a pivotal role in Hermitian topological materials. It states, for example, that Weyl points must come in pairs in three-dimensional semimetals. Here, we present an extension of the doubling theorem to non-Hermitian lattice Hamiltonians. We focus on two-dimensional non-Hermitian systems without any symmetry constraints, which can host two different types of topological point nodes, namely, (i)Fermi points and (ii)exceptional points. We show that these two types of protected point nodes obey doubling theorems, which require that the point nodes come in pairs. To prove the doubling theorem for exceptional points, we introduce a generalized winding number invariant, which we call the "discriminant number." Importantly, this invariant is applicable to any two-dimensional non-Hermitian Hamiltonian with exceptional points of arbitrary order and, moreover, can also be used to characterize nondefective degeneracy points. Furthermore, we show that a surface of a three-dimensional system can violate the non-Hermitian doubling theorems, which implies unusual bulk physics.