We analyze the lattice fermion kinetic term using PT symmetry, R-hermiticity, and $γ_{5}$-hermiticity. R-hermiticity is a condition for Hermite action and it is related to $γ_{5}$-hermiticity and PT symmetry. Assuming that a translation-invariant kinetic term with continuum and periodic function does not have PT symmetry, it can have R-hermiticity or $γ_{5}$-hermiticity. We prove that a kinetic term with continuum and periodic function that is PT symmetric does not reduce doublers. As a simple example, we analyze the two-dimensional two-flavor Gross-Neveu model with minimal doubling fermions. The minimal doubling fermions break PT symmetry and R-hermiticity, hence complex or non-Hermite coupling constants are caused by quantum correction.
We formulate new two dimensional fermions breaking $γ_{5}$hermiticity, based on the minimal doubling fermion. We investigate their properties: (I) Symmetries, (II) eigenvalue distributions, and (III)the number of poles. As a simple application of the fermions, the Gross-Neveu model in two dimensions is studied using the fermion. We obtain the parity broken phase diagram called Aoki phase and the chiral broken phase diagram of the model with an imaginary chemical potential.
We construct 2D non-$γ_{5}$hermiticity fermions based on the minimal doubling fermion. We investigate symmetries, reflection positivity, eigenvalue distribution and the number of poles for our fermions. As simple tests for application to the fermion, the Gross-Neveu model in two dimensions is studied using the non-$γ_{5}$hermiticity fermion. We draw the parity broken phase diagram, called Aoki phase and the chiral broken phase diagram for the model with an imaginary chemical potential.
We propose a new method for Hybrid Monte Carlo (HMC) simulations with odd numbers of dynamical fermions on the lattice. It employs a different approach from polynomial or rational HMC. In this method, gamma-five hermiticity of the lattice Dirac operators is crucial and it can be applied to Wilson, domain-wall, and overlap fermions. We compare HMC simulations with two degenerate flavors and (1 + 1) degenerate flavors using optimal domain-wall fermions. The ratio of the efficiency, (number of accepted trajectories) / (simulation time), is about 3:2. The relation between pseudofermion action of c…
We discuss the naive lattice fermion without the issue of doublers. A local lattice massless fermion action with chiral symmetry and hermiticity cannot avoid the doubling problem from the Nielsen-Ninomiya theorem. Here we adopt the forward finite-difference deforming the $γ_5$-hermiticity but preserving the continuum chiral-symmetry. The lattice momentum is not hermitian without the continuum limit now. We demonstrate that there is no doubling issue from an exact solution. The propagator only has one pole in the first-order accuracy. Therefore, it is hard to know the avoiding due to the non-he…
Species doubling is a problem that infects most numerical methods that use a spatial lattice. An understanding of species doubling can be found in the Nielsen-Ninomiya theorem which gives a set of conditions that require species doubling. The transverse lattice approach to solving field theories, which has at least one spatial lattice, fails one of the conditions of the Nielsen-Ninomiya theorem nevertheless one still finds species doubling for the standard Lagrangian formulation of the transverse lattice. We will show that the Supersymmetric Discrete Light Cone Quantization (SDLCQ) formulation…
We discuss the vacuum structure of QCD with two flavors of Wilson fermions. We derive two possible scenarios: (i) If the spectral density $ρ_U(λ,κ)$ of the overlap hamiltonian in a fixed background gauge field is not symmetric in $λ$, Hermiticity is violated and Hermiticity violation effects could influence numerical determinations of the $η$ meson mass if we are not near enough to the continuum limit, where Hermiticity should be recovered; (ii) otherwise we argue that, under certain assumptions, new phases appear beside the Aoki phase, which can be characterized by a nonvanishing vacuum expec…
It is shown that the Wilson fermion doubling phenomenon on irregular lattices (simplicial complexes) does exist. This means that the irregular (not smooth) zero or soft modes exist. The statement is proved on 4 Dimensional lattice by means of the Atiyah-Singer index theorem, then it is extended easily into the cases $D<4$. But there is a fundamental difference between doubled quanta on regular and irregular lattices: in the latter case the propagator decreases exponentially. This means that the doubled quanta on irregular lattice are "bad" quasiparticles.
In this paper I propose the use of a lattice derivative operator that is equivalent to the ideal SLAC derivative operator in all lattice calculations, but without the prohibitively expensive computational cost of the latter. A pedagogical motivation and derivation of the closed form of the SLAC derivative in position space is presented, and the proposed method for its cost-effective implementation is presented in detail.
In this paper we generalize to higher dimensions several types of fermion actions on the hyperdiamond lattice including a two-parameter class of minimal-doubling fermions "Creutz fermion" and a simple fermion with sufficient discrete symmetry "BBTW fermion". Then it is shown that they possess some properties in common with the four-dimensional case: BBTW fermions in higher even dimensions inevitably yield unphysical degrees of freedom. Creutz fermions are defined on the distorted lattices, and they lose the high discrete symmetry of the original lattices. We also find properties specific to th…
We propose a method to control the number of species of lattice fermions, which yields new classes of minimally doubled lattice fermions with one exact chiral symmetry and exact locality. We classify all the known minimally doubled fermions into two types based on the locations of the propagator poles in the Brillouin zone. We also study higher-dimensional extension of them and show it tends to be more difficult to realize minimal-doubling in higher dimensions.
In a previous paper I showed how the ideal SLAC derivative and second-derivative operators for an infinite lattice can be obtained in simple closed form in position space, and implemented very efficiently in a stochastic fashion for practical calculations on finite lattices. In this second paper I show how the small (order 1/N) errors introduced by truncating the operators to a finite lattice may be removed by a small adjustment of coefficients, without incurring any additional computational cost. The derivation of these results is again presented in a simple, pedagogical fashion.
Minimally-doubled chiral fermions have the unusual property of a single local field creating two fermionic species. Spreading the field over hypercubes allows construction of combinations that isolate specific modes. Combining these fields into bilinears produces meson fields of specific quantum numbers.
We consider a nonlocal lattice action for fermions fermion doubling in lattice theories. It is shown, that it is possible to avoid the fermionic doubling in the case of free fermions, but this approach does not reproduce results for the effective action for gauge fields in the continuum theory, because the high frequency fermion modes have a strong dependence on the gauge field.
We show that the string bit model suffers from doubling in the fermionic sector. The doubling leads to strong violation of supersymmetry in the limit $N\to\infty$. Since there is an exact correspondence between string bits and the algebra of BMN operators even at finite $N$, doubling is expected also on the side of super-Yang--Mills theory. We discuss the origin of the doubling in the BMN sector.
We study minimal-doubling fermion actions on hyperdiamond and deformed hyperdiamond lattices, with emphasis on the real-space construction of them and Lorentz covariance of excitations from fermion poles. We propose the improved spatial construction of Creutz fermion action on a deformed hyperdiamond lattice, and discuss conditions for a hyperdiamond-lattice action to produce Lorentz-covariant excitations from poles of fermion propagators. It is pointed out that the non-nearest-site hoppings are essential for the correct excitations. We propose a class of minimal-doubling actions defined on a …
The graphene-inspired fermion actions recently proposed by Creutz and Borici have sparked interest in the use of non-orthogonal lattices in lattice QCD. These fermion actions have the desired chiral symmetry and have the minimal doubling required by the Nielsen-Ninomiya no-go theorem. However, due to the lack of discrete symmetries, radiative corrections in the gauged lattice theory will lead to the generation of unwanted relevant and marginal operators. Other similarly motivated non-orthogonal fermion actions avoid these unwanted operators, but introduce incorrect continuum behavior or excess…
Random-lattice fermions have been shown to be free of the doubling problem if there are no interactions or interactions of a non-gauge nature. On the other hand, gauge interactions impose stringent constraints as expressed by the Ward-Takahashi identities which could revive the free-field suppressed doubler modes in loop diagrams. Comparing random lattice, naive and Wilson fermions in two dimensional abelian background gauge theory, we show that indeed the doublers are revived for random lattices in the continuum limit. Some implications of the persistent doubling phenomenon on random lattices…
We present some arguments showing spectrum doubling of matrix models in the limit $N\to\infty$ which is connected with fermionic determinant behaviour. The problems are similar to ones encountered in the lattice gauge theories with chiral fermions. One may discuss the ``physical meaning'' of the doubling states or ways to eliminate them. We briefly consider both situations.
In this paper, we show that the Hamiltonian approach to loop quantum gravity has a fermion doubling problem. To obtain this result, we couple loop quantum gravity to a free massless scalar and a chiral fermion field, gauge fixing the many fingered time gauge invariance by interpreting the scalar field as a physical clock. We expand around a quantum gravity state based on a regular lattice and consider the limit where the bare cosmological constant is large but the fermonic excitations have energies low in Planck units. We then make the case for identifying the energy spectrum in this approxima…
Based on the Pauli principle and Heisenberg uncertainty relationship, we show that the phenomenon of vectorlike fermion-doubling in quantum field theories on a lattice is related to the phenomenon of pair-creation of massless fermions, due to quantum-field fluctuations upon the zero-energy level of the ground state of regularized quantum field theories on the lattice.
In our previous paper (hep-th/9911087), we proposed a resolution for the fermion doubling problem in discrete field theories based on the fuzzy sphere and its cartesian products. In this paper after a review of that work, we bring out its relationship to the Ginsparg-Wilson approach.
We develop an approach to classical and quantum mechanics where continuous time is extended by an infinitesimal parameter $T$ and equations of motion converted into difference equations. These equations are solved and the physical limit $T \rightarrow 0$ then taken. In principle this strategy should recover all standard solutions to the original continuous time differential equations. We find this is valid for bosonic variables whereas with fermions, additional solutions occur. For both bosons and fermions, the difference equations of motion can be related to Möbius transformations in projecti…
The fermion doubling problem has an important impact on quantum gravity, by revealing the tension between fermion and the fundamental discreteness of quantum spacetime. In this work, we discover that in Loop Quantum Gravity, the quantum geometry involving superposition of states associated with lattice refinements provides a resolution to the fermion doubling problem. We construct and analyze the fermion propagator on the quantum geometry, and we show that all fermion doubler modes are suppressed in the propagator. Our result suggests that the superposition nature of quantum geometry should re…
The Ginsparg-Wilson algebra is the algebra underlying the Ginsparg-Wilson solution of the fermion doubling problem in lattice gauge theory. The Dirac operator of the fuzzy sphere is not afflicted with this problem. Previously we have indicated that there is a Ginsparg-Wilson operator underlying it as well in the absence of gauge fields and instantons. Here we develop this observation systematically and establish a Dirac operator theory for the fuzzy sphere with or without gauge fields, and always with the Ginsparg-Wilson algebra. There is no fermion doubling in this theory. The association of …
A few years ago some attention has been given to a fermionic action on the lattice, with a Wilson-like term which is chirally invariant but breaks the hypercubic space-time lattice symmetry. This action describes two Dirac fields in the continuum limit, provided the coefficient $λ$ of the Wilson-like term satisfies $λ> \frac{1}{2}$. In this letter it is shown that for $\frac{1}{2} < λ\leq 1$ the theory is link-reflection positive. The propagator has the expected real energy poles. Modulo a phase shift on the fermions, the only relevant terms which can be added to the action respecting its symm…
Motivated by recent examples of three-dimensional lattice Hamiltonians with massless Dirac fermions in their (bulk) spectrum, I revisit the problem of fermion doubling on bipartite lattices. The number of components of the Dirac fermion in a time-reversal and parity invariant d-dimensional lattice system is determined by the minimal representation of the Clifford algebra of $d+1$ Hermitian Dirac matrices that allows a construction of the time-reversal operator with the square of unity, and it equals $2^d$ for $d=2,3$. Possible mass-terms for (spinless) Dirac fermions are listed and discussed. …
In loop quantum gravity the discrete nature of quantum geometry acts as a natural regulator for matter theories. Studies of quantum field theory in quantum space-times in spherical symmetry in the canonical approach have shown that the main effect of the quantum geometry is to discretize the equations of matter fields. This raises the possibility that in the case of fermion fields one could confront the usual fermion doubling problem that arises in lattice gauge theories. We suggest, again based on recent results on spherical symmetry, that since the background space-times will generically inv…
A free fermion without doubler is formulated on 1+D dimensional discrete Minkowski space-time. The action is not hermitian but causes no harm. In 1+3 dimensional massless case the equation describes a single species of Dirac particle in the continuous space-time limit. In 1+1 dimensional massless case the equation is the same as the automaton equation by 't Hooft and describes a chiral fermion. The time evolution operator is unitary and the norm is conserved. For interacting fermions with gauge fields the evolution operator is not unitary. If it is considered as an approximation for the theory…
We describe a method for evaluating chiral gauge theories that is not plagued by the doubling problem. To demonstrate the efficiency of the approach, we apply our ideas to the chiral Schwinger model.
The QCD partition function for the Wilson Dirac operator, $D_W$, at nonzero lattice spacing $a$ can be expressed in terms of a chiral Lagrangian as a systematic expansion in the quark mass, the momentum and $a^2$. Starting from this chiral Lagrangian we obtain an analytical expression for the spectral density of $γ_5 (D_W+m)$ in the microscopic domain. It is shown that the $γ_5$-Hermiticity of the Dirac operator necessarily leads to a coefficient of the $a^2$ term that is consistent with the existence of an Aoki phase. The transition to the Aoki phase is explained, and the interplay of the ind…
We propose a discretisation scheme based on the Dirac-Kahler formalism (DK) in which the algebraic relations between continuum operators ${\wedge, d, \star}$ are captured by their discrete analogues, allowing the construction of the relevant projection operators necessary to prevent species doubling. We thus avoid the traditional form of species doubling as well as spectral doubling, which does not occur in the DK setting. Chirality is also captured, since we have $\star$ from geometric discretisation.
We only require generalized chiral symmetry and $γ_5$-hermiticity, which leads to a large class of Dirac operators describing massless fermions on the lattice, and use this framework to give an overview of developments in this field. Spectral representations turn out to be a powerful tool for obtaining detailed properties of the operators and a general construction of them. A basic unitary operator is seen to play a central rôle in this context. We discuss a number of special cases of the operators and elaborate on various aspects of index relations. We also show that our weaker conditions lea…
Chiral gauge groups acting on a lattice fermion field are constructed such that all fermion modes (doublers) have the same charge. Details are given for an abelian axial gauge group within a perturbative framework. An action based on this group correctly reproduces the continuum gauge-current anomaly, while preserving global chiral symmetry, locality, rotational symmetry and hermiticity. A Wess-Zumino-like scalar field enters naturally to enforce exact chiral gauge invariance. The degeneracy amongst the doublers can be lifted as in a vector model.
In this letter it will be demonstrated explicitly that the finite-element formulation of quantum electrodynamics is free from fermion doubling. We do this by (1) examining the lattice fermion propagator and using it to compute the one-loop vacuum polarization on the lattice, and (2) by an explict computation of vector and axial-vector current anomalies for an arbitrary rectangular lattice in the Schwinger model. There it is shown that requiring that the vector current be conserved necessitates the use of a square lattice, in which case the axial-vector current is anomalous.
A new approach to the problem of doubling is presented with the Dirac-Kahler (DK) theory as a starting point and using Geometric Discretisation providing us with a new way of extracting the Dirac field in the discrete setting of a hyper-cubic complex.
Recently the Lattice Strong Dynamics Collaboration reported a reduction of the electroweak S parameter for SU(3) gauge theory with Nf=6 fermions in the fundamental representation, compared to scaled-up QCD (arXiv:1009.5967). Here I provide additional details of our calculation. I discuss our use of conserved lattice currents; the relation to vector--axial parity doubling; finite-volume effects; and the sensitivity of our results to the number of fermion doublets with chiral electroweak couplings. Results presented here include additional data, and do not affect our previously-published conclus…
Based upon the lattice Dirac operator satisfying the Ginsparg-Wilson relation, we investigate canonical formulation of massless fermion on the spatial lattice. For free fermion system exact chiral symmetry can be implemented without species doubling. In the presence of gauge couplings the chiral symmetry is violated. We show that the divergence of the axial vector current is related to the chiral anomaly in the classical continuum limit.
Fermions moving in a two-dimensional honeycomb lattice (graphene) have, at low energies, chiral symmetry. Generalizing this construction to four dimensions potentially provides fermions with chiral symmetry and only the minimal fermion doubling demanded by the Nielsen-Ninomiya no-go theorem. The practical usefulness of such fermions hinges on whether the action has a necessary set of discrete symmetries of the lattice. If this is the case, one avoids the generation of dimension three and four operators which require fine tuning. We construct hyperdiamond lattice actions with enough symmetries …
We formulate Dirac fermions on a (1+1)-dimensional lattice based on a Hamiltonian formalism. The species doubling problem of the lattice fermion is resolved by introducing hopping interactions that mix left- and right-handed fermions around the momentum boundary. Approximate chiral symmetry is realized on the lattice. The deviation of the fermion propagator from the continuum one is small.
The species doubling problem of the lattice fermion is resolved by introducing hopping interactions that mix left- and right-handed fermions around the momentum boundary. Approximate chiral symmetry is realized on the lattice. The deviation of the fermion propagator from the continuum one is small.
Exact chiral symmetry at finite lattice spacing would preclude the axial anomaly. In order to describe a continuum quantum field theory of Dirac fermions, lattice actions with purported exact chiral symmetry must break the flavor-singlet axial symmetry. We demonstrate that this is indeed the case by using a minimally doubled fermion action. For simplicity we consider the Abelian axial anomaly in two dimensions. At finite lattice spacing and with gauge interactions, the axial anomaly arises from non-conservation of the flavor-singlet current. Similar non-conservation also leads to the axial ano…
We propose a formulation of lattice fermions with one-sided differences that is hermitian, chirally symmetric (barring a bare mass term) and completely free of doubling. To obtain the axial anomaly in perturbation theory it was necessary to break chiral symmetry on the lattice only through a bare mass term for the physical fermion. The chiral limit may be taken once the continuum limit is reached. We comment on the role of the mass term with examples elsewhere in field theory.
Random-lattice fermions have been shown to be free of the doubling problem if there are no interactions or interactions of a non-gauge nature. However, gauge interactions impose stringent constraints as expressed by the Ward-Takahashi identities which could revive the free-field suppressed doubler modes in loop diagrams. After introducing a formulation for fermions on a new kind of random lattice, we compare random, naive and Wilson fermions in two dimensional Abelian background gauge theory. We show that the doublers are revived for random lattices in the continuum limit, while demonstrating …
The fermionic part of the Schrödinger functional of QCD is formulated in the lattice regularization with the staggered fermion. The boundary condition imposed on the staggered fermion field are examined in terms of the four-component Dirac spinor. The boundary terms are different from those of the Symanzik's theory in the flavor structure due to the species doubling. It is argued that, in the case of the homogeneous Dirichlet boundary condition, surface divergence does not occur if the link variables of gauge field are introduced on the original lattice, not on the blocked one. Its application…
In this paper we propose a new approach to formulate the field theory on a lattice. This approach can eliminate the Fermion doubling problem, preserve the chiral symmetry and get the same dispersion relation for both Fermion and Boson fields. This gives us the possibility to write down the chiral model (such as the Weinberg-Salam model) on a lattice.
Novel chirally symmetric fermion actions containing the minimum amount of fermion doubling have been recently proposed in the literature. We study the symmetries and renormalization of these actions and find that in each case, discrete symmetries, such as parity and time-reversal, are explicitly broken. Consequently, when the gauge interactions are included, these theories radiatively generate relevant and marginal operators. Thus the restoration of these symmetries and the approach to the continuum limit require the fine-tuning of several parameters. With some assumptions, we show that this b…
We study the mass spectrum of mesons at high temperature in $SU(2)$ gauge theory with two flavors of Dirac fundamental fermions. Numerical simulations are carried out on anisotropic lattices using Wilson fermions, with lattice parameters tuned so that Euclidean symmetry is restored at low energy. We determine the pseudo-critical temperature $T_c$ using renormalized Polyakov loops. We calculate temporal and spatial meson correlation functions across $T_c$, and observe a clear sign of parity doublings above $T_c$ in both vector and scalar channels. The degeneracy between parity partners in the s…
We investigate numerically the effect of regulating fermions in the presence of singular background fields in three dimensions. For this, we couple free lattice fermions to a background compact U(1) gauge field consisting of a monopole-anti-monopole pair of magnetic charge $\pm Q$ separated by a distance $s$ in a periodic $L^3$ lattice, and study the low-lying eigenvalues of different lattice Dirac operators under a continuum limit defined by taking $L\to\infty$ at fixed $s/L$. As the background gauge field is parity even, we look for a two-fold degeneracy of the Dirac spectrum that is expecte…
We address the problems of fermions in light front QCD on a transverse lattice. We propose and numerically investigate different approaches of formulating fermions on the light front transverse lattice. In one approach we use forward and backward derivatives. There is no fermion doubling and the helicity flip term proportional to the fermion mass in the full light front QCD becomes an irrelevant term in the free field limit. In the second approach with symmetric derivative (which has been employed previously in the literature), doublers appear and their occurrence is due to the decoupling of e…
According to the Nielsen-Ninomiya No-Go theorem, the doubling of fermions on the lattice cannot be suppressed in a chiral theory. Whereas Wilson and staggered fermions suppress doublers with explicit breaking of chiral symmetry, the random lattice does so by spontaneous chiral symmetry breaking even in the free theory. I present results for meson masses, the chiral condensate and fermionic eigenvalues from simulations of quenched QCD on random lattices in four dimensions, focusing on chiral symmetry breaking.
We discuss the chiral phase diagram in the parameter space of lattice QCD with minimal-doubling fermions, which can be seen as lattice fermions with flavored chemical potential terms. We study strong-coupling lattice QCD with the Karsten-Wilczek formulation, which has one relevant parameter $μ_{3}$ as well as gauge coupling and a mass parameter. We find a nontrivial chiral phase structure with a second-order phase transition between chiral symmetric and broken phases. To capture the whole structure of the phase diagram, we study the related lattice Gross-Neveu model. The result indicates that …
The $SU(N_f)_L \otimes SU(N_f)_R$ chiral symmetry of QCD is of central importance for the nonperturbative low-energy dynamics of light quarks and gluons. Lattice field theory provides a theoretical framework in which these dynamics can be studied from first principles. The implementation of chiral symmetry on the lattice is a nontrivial issue. In particular, local lattice fermion actions with the chiral symmetry of the continuum theory suffer from the fermion doubling problem. The Ginsparg-Wilson relation implies Lüscher's lattice variant of chiral symmetry which agrees with the usual one in t…
