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.
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 …
We present a new staggered discretization of the Dirac operator. In comparison with standard staggered fermions, real and imaginary parts are located in different nodes. Doubling gives only a doublet of Dirac fermions which we propose to interpret as a physical (lepton or quark) doublet. Contrary to usual staggered fermions, we have no exact chiral symmetry but obtain a generalization of the Ginsparg-Wilson relation.
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.
We analyse the pure Chern-Simons theory on an Euclidean infinite lattice. We point out that, as a consequence of its symmetries, the Chern-Simons theory does not have an integrable kernel. Due to the linearity of the action in the derivatives, the situation is very similar to the one arising in the lattice formulation of fermionic theories. Doubling of bosonic degrees of freedom is removed by adding a Maxwell term with a mechanism similar to the one proposed by Wilson for fermionic models.
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…
The Szymanzik improvement program for gauge theories is most commonly implemented using forward finite difference corrections to the Wilson action. Central symmetric schemes naively applied, suffer from a doubling of degrees of freedom, identical to the well known fermion doubling phenomenon. And while adding a complex Wilson term remedies the problem for fermions, it does not easily transfer to real-valued gauge fields. In this talk I report on recent progress in formulating symmetric discretization schemes for classical actions of simple one-dimensional problems. They avoid doubling by explo…
Iterating renormalization group transformations for lattice fermions the Wilson action is driven to fixed points of the renormalization group. A line of fixed points is found and the fixed point actions are computed analytically. They are local and may be used to improve scaling in lattice QCD. The action at the line's endpoint is chirally invariant and still has no fermion doubling. The Nielsen-Ninomiya theorem is evaded because in this case the fixed point action is nonlocal. The use of this action for a construction of lattice chiral fermions is discussed.
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.
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 give a perturbative proof that U(1) lattice gauge theories generate the axial anomaly in the continuum limit under very general conditions on the lattice Dirac operator. These conditions are locality, gauge covariance and the absense of species doubling. They hold for Wilson fermions as well as for realizations of the Dirac operator that satisfy the Ginsparg-Wilson relation. The proof is based on the lattice power counting theorem.
It is shown that the Ginsparg-Wilson relation implies an exact symmetry of the fermion action, which may be regarded as a lattice form of an infinitesimal chiral rotation. Using this result it is straightforward to construct lattice Yukawa models with unbroken flavour and chiral symmetries and no doubling of the fermion spectrum. A contradiction with the Nielsen-Ninomiya theorem is avoided, because the chiral symmetry is realized in a different way than has been assumed when proving the theorem.
We give a perturbative proof that U(1) lattice gauge theories generate the axial anomaly in the continuum limit under very general conditions on the lattice Dirac operator. These conditions are locality, gauge covariance and the absense of species doubling. They hold for Wilson fermions as well as for realizations of the Dirac operator that satisfy the Ginsparg-Wilson relation. The proof is based on the lattice power counting theorem. The results generalize to non-abelian gauge theories.
Important recent discoveries suggest that Ginsparg-Wilson-Luscher (GWL) symmetry has analogous dynamical consequences for the theory on the lattice as chiral symmetry does in the continuum. While it is well known that inherent property of lattice chiral symmetry is fermion doubling, we show here that inherent property of GWL symmetry is that the infinitesimal symmetry transformation couples fermionic degrees of freedom at arbitrarily large lattice distances (non-ultralocality). The consequences of this result for ultralocality of symmetric actions are discussed.
Baryonic correlation functions provide an ideal tool to study parity doubling and chiral symmetry using lattice simulations. We present a study using $2+1$ flavors of anisotropic Wilson clover fermions on the FASTSUM ensembles and find clear evidence that parity doubling emerges in the quark-gluon plasma. This result is confirmed on the level of spectral functions, which are obtained using a MEM reconstruction. We further highlight the importance of Gaussian smearing in this study.
First principle study of QCD at finite temperature $T$ and chemical potential $μ$ is essential for understanding a wide range of phenomena from heavy-ion collisions to cosmology and neutron stars. However, in the presence of finite density, the critical behavior lattice gauge theory without species doubling, is unknown. At strong coupling, we examine the phase structure on the $(μ,T)$ plane, using Hamiltonian lattice QCD with Wilson fermions. A tricritical point is found, separating the first and second order chiral phase transitions. Such a tricritical point at finite $T$ has not been found i…
We study properties of the thermal transition in QCD, using anisotropic, fixed-scale lattice simulations with $N_f = 2+1$ flavours of Wilson fermion. Observables are compared for two values of the pion mass, focusing on chiral properties. Results are presented for the Polyakov loop, various susceptibilities, the chiral condensate and its susceptibility, and the onset of parity doubling in the light and strange baryonic sector.
We apply the $δ$-expansion to the Gross-Neveu model in the large $N$ limit with Wilson fermion and investigate dynamical mass generation from inverse-mass expansion. The dimensionless mass $M$ defined via the effective potential is employed as the expansion parameter of the bare coupling constant $β$ which is partially renormalized by the subtraction of linear divergence. We show that $δ$-expansion of the $1/M$ series of $β$ is compatible with the mass renormalization. After the confirmation of the continuum scaling of the bare coupling without fermion doubling, we attempt to estimate dynamica…
It is shown that the conflict between lattice chiral symmetry and the Majorana condition in the presence of Yukawa couplings, which was noted in our previous paper, is related in an essential way to the basic properties of Ginsparg-Wilson operators, namely, locality and species doubling.
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…
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 evaluate the axial anomaly for the general Ginsparg-Wilson fermion operator $D = D_c (\Id + R D_c)^{-1}$ with $R = r \Id$. For any chirally symmetric $D_c$ which in the free fermion limit, is free of species doubling and behaves like $i γ_μp_μ$ as $p \to 0$, the axial anomaly $\tr[γ_5 (R D) (x,x)]$ for U(1) lattice gauge theory with single fermion flavor is equal to $e^2/(32 π^2) ε_{μνλσ} F_{μν}(x) F_{λσ}(x+\hatμ+\hatν)$ plus terms which are higher orders and/or non-perturbative contribtuons. The $F \tilde{F}$ term is r-invariant and has the correct continuum limit.
It is suggested that the fermion determinant for a vector-like gauge theory with strictly massless quarks can be represented on the lattice as $\det{{1+V}\over 2}$, where $V=X(X^\dagger X)^{-1/2}$ and $X$ is the Wilson-Dirac lattice operator with a negative mass term. There is no undesired doubling and no need for any fine tuning. Several other appealing features of the formula are pointed out.
We prove that lattice QCD generates the axial anomaly in the continuum limit under very general conditions on the lattice action, which includes the case of Ginsparg-Wilson fermions. The ingredients going into the proof are gauge invariance, locality of the Dirac operator, absence of fermion doubling, the general form of the lattice Ward identity, and the power counting theorem of Reisz. The results generalize in an obvious way to SU(N) lattice gauge theories.
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.
The formulation of massless relativistic fermions in lattice gauge theories is hampered by the fundamental problem of species doubling, namely, the rise of spurious fermions modifying the underlying physics. A suitable tailoring of the fermion masses prevents such abundance of species, and leads to the so-called Wilson fermions. Here we show that ultracold atoms provide us with the first controllable realization of these paradigmatic fermions, thus generating a quantum simulator of fermionic lattice gauge theories. We describe a novel scheme that exploits laser-assisted tunneling in a cubic op…
The lattice fermion determinants, in a given background gauge field, are evaluated for two different kinds of random lattices and compared to those of naive and wilson fermions in the continuum limit. While the fermion doubling is confirmed on one kind of lattices, there is positive evidence that it may be absent for the other, at least for vector interactions in two dimensions. Combined with previous studies, arbitrary randomness by itself is shown to be not a sufficient condition to remove the fermion doublers.
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 …
We investigate the structure of center vortices in maximal center gauge of SU(2) lattice gauge theory at zero and finite temperature. In center projection the vortices (called P-vortices) form connected two dimensional surfaces on the dual four-dimensional lattice. At zero temperature we find, in agreement with the area law behaviour of Wilson loops, that most of the P-vortex plaquettes are parts of a single huge vortex. Small P-vortices, and short-range fluctuations of the large vortex surface, do not contribute to the string tension. All of the huge vortices detected in several thousand fiel…
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…
The space-time reduced model of large N QCD with two adjoint Wilson fermions is constructed by applying the symmetric twist boundary conditions with non-vanishing flux $k$. For large but finite $N=L^2$, the model should behave as the large N version of the ordinary lattice gauge model on a $V=L^4$ space-time volume. We perform a comparison of the $N$ dependence of several quantities in this model and in the $k$=0 model (corresponding to periodic boundary conditions). Although the Z$^4$(N) symmetry seems unbroken in all cases, the N-dependence analysis favours the use of the same values of $k$ …
We address the problem of the fermion species doubling on the Lattice. Our strategy is to factorize the fermion doubles from the action. The mass term of the Dirac-Wilson action is changed. In this case the extra roots which appear in the action of free fermions in the moment representation are independent of the mass and can be factorized from the fermion propagator. However the gauge couplings suffer from the pathological ghost poles which are common to non-local actions. This action can be used to find a solution of the Ginsparg Wilson relation, which is cured from the non-local pathology. …
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…
A representation of the continuum fermionic propagator as a sum of directed random walks on a lattice is presented. Also a random walk representation for the lattice fermionic propagators is developed for the case of the naive, the Wilson, and the Kogut-Susskind fermions. For the naive fermions the phenomenon of fermion doubling appears as having $2^{D}$ distinct spin factors being associated with a single path in $D$-dimensions. In the case of the Wilson and the Kogut-Susskind fermions, in the naive continuum limit, the path integral representation coincides with the path integral representat…
We revisit 2D $\mathcal{N}=(2,2)$ super Yang-Mills lattice formulation (Sugino model) to investigate its fermion action with two (Majorana) fermion flavors and exact chiral-$U(1)_{R}$ symmetry. We show that the reconcilement of chiral symmetry and absence of further species-doubling originates in the 4D clifford algebra structure of the action, where 2D two flavors are spuriously treated as a single 4D four-spinor with four 4D gamma matrices introduced into kinetic and Wilson terms. This fermion construction based on the higher-dimensional clifford algebra is extended to four dimensions in two…
A simplified test of universality in Lattice QCD is performed by analytically evaluating the continuous Euclidean time limits of various lattice fermion determinants, both with and without a Wilson term to lift the fermion doubling on the Euclidean time axis, and comparing them with each other and with the zeta-regularised fermion determinant in the continuous time--lattice space setting. The determinant relations expected from universality considerations are found to be violated by a certain gauge field-dependent factor, i.e. we uncover a "universality anomaly". The physical significance, or …
The domain wall fermion formalism in lattice gauge theory is much investigated recently. This is set up by reducing 4+1 dimensional theory to low energy effective 4 dimensional one. In order to look around other possibilities of realizing chiral fermion on the lattice, we construct vortex fermion by reducing 4+2 dimensional theory to low energy effective 4 dimensional one on the lattice. In extra 2 dimensions we propose a new lattice regularization which has a discrete rotational invariance but not a translational one. In order to eliminate doubling species in the naive construction we introdu…
A non-perturbative lattice regularization of chiral fermions and bosons with anomaly-free symmetry $G$ in 1+1D spacetime is proposed. More precisely, we ask "whether there is a local short-range quantum Hamiltonian with a finite Hilbert space for a finite system realizing onsite symmetry $G$ defined on a 1D spatial lattice, such that its low energy physics produces a 1+1D anomaly-free chiral matter theory of symmetry $G$?" In particular, we propose that the 3$_L$-5$_R$-4$_L$-0$_R$ U(1) chiral fermion theory, with two left-moving fermions of charge-3 and 4, and two right-moving fermions of char…
