Lattice Operators Research Articles

One-loop operator matching in the static heavy and domain-wall light quark system with O(a) improvement

We discuss perturbative O(g^2a) matching with static heavy quarks and domain-wall light quarks for lattice operators relevant to B-meson decays and $B^0$-$\bar{B}^0$ mixing. The chiral symmetry of the light domain-wall quarks does not prohibit operator mixing at O(a) for these operators. The O(a) corrections to physical quantities are non-negligible and must be included to obtain high-precision simulation results for CKM physics. We provide results using plaquette, Symanzik, Iwasaki and DBW2 gluon actions and applying APE, HYP1 and HYP2 link-smearing for the static quark action.

Lattice results for low moments of light meson distribution amplitudes

As part of the UKQCD and RBC collaborations' N_f=2+1 domain-wall fermion phenomenology programme, we calculate the first two moments of the light-cone distribution amplitudes of the pseudoscalar mesons pion and kaon and the (longitudinally-polarised) vector mesons rho, K-star and phi. We obtain the desired quantities with good precision and are able to discern the expected quark-mass dependence of SU(3)-flavour breaking effects. An important ingredient of the calculation is the nonperturbative renormalisation of lattice operators using the RI'/MOM technique.

Perturbative renormalization factors andO(a2)corrections for lattice four-fermion operators with improved fermion/gluon actions

In this work we calculate the corrections to the amputated Green's functions of 4-fermion operators, in 1-loop Lattice Perturbation theory. One of the novel aspects of our calculations is that they are carried out to O(a^2) (a: lattice spacing). We employ the Wilson/clover action for massless fermions (also applicable for the twisted mass action in the chiral limit) and a family of Symanzik improved actions for gluons. Our calculations have been carried out in a general covariant gauge. Results have been obtained for several popular choices of values for the Symanzik coefficients. While our Green's function calculations regard any pointlike 4-fermion operators which do not mix with lower dimension ones, we pay particular attention to DF=2 operators, both Parity Conserving and Parity Violating (F: flavour). We compute the perturbative renormalization constants for a complete basis of 4-fermion operators and we study their mixing pattern. For some of the actions considered here, even O(a^0) results did not exist in the literature to date. The correction terms which we calculate are essential ingredients for minimizing the lattice artifacts which are present in non-perturbative evaluations of renormalization constants with the RI'-MOM method. Our perturbative results, for the matrix elements of DF=2 operators and for the corresponding renormalization matrices, depend on a large number of parameters: coupling constant, number of colors, lattice spacing, external momentum, clover parameter, Symanzik coefficients, gauge parameter. To make these results most easily accessible, we have included them in the distribution package of this paper, as an ASCII file named: 4-fermi.m; the file is best perused as Mathematica input. The main results of this work have been applied to improve non-perturbative estimates of the B_K-parameter in N_F=2 twisted mass lattice QCD.

EE<sub>8</sub>-Lattices and Dihedral Groups

We classify integral rootless lattices which are sums of pairs of EE8-lattices (lattices isometric to √ 2 times the E8-lattice) and which define dihedral groups of orders less than or equal to 12. Most of these may be seen in the Leech lattice. Our classification may help understand Miyamoto involutions on lattice type vertex operator algebras and give a context for the dihedral groups which occur in the Glauberman-Norton moonshine theory.

Single flavor staggered fermions

Based on recent work by Adams, I construct a lattice fermion operator that fully lifts the staggered flavor degeneracy. The resulting operator is of Wilson type but smaller by a factor of 4, better conditioned and contains 3 instead of 15 doublers. It is further suggested that this operator may be used as a candidate kernel operator to an overlap construction. Prospects for practical applications and potential problems of the new discretizations are briefly discussed.

Magnetic-Field-Induced Insulator-Conductor Transition inSU(2)Quenched Lattice Gauge Theory

We study the correlator of two vector currents in quenched SU(2) lattice gauge theory with a chirally invariant lattice Dirac operator with a constant external magnetic field. It is found that in the confinement phase the correlator of the components of the current parallel to the magnetic field decays much slower than in the absence of a magnetic field, while for other components the correlation length slightly decreases. We apply the maximal entropy method to extract the corresponding spectral function. In the limit of zero frequency this spectral function yields the electric conductivity of quenched theory. We find that in the confinement phase the external magnetic field induces nonzero electric conductivity along the direction of the field, transforming the system from an insulator into an anisotropic conductor. In the deconfinement phase the conductivity does not exhibit any sizable dependence on the magnetic field.

Distance preconditioning for lattice Dirac operators

We propose a preconditioning of the Dirac operator based on the factorisation of a predefined function related to the decay of the propagator with the distance. We show that it can improve the accuracy of correlators involving heavy quarks at large distances and accelerate the computation of light quark propagators.

Formulation of supersymmetry on a lattice as a representation of a deformed superalgebra

The lattice superalgebra of the link approach is shown to satisfy a Hopf algebraic supersymmetry where the difference operator is introduced as a momentum operator. The breakdown of the Leibniz rule for the lattice difference operator is accommodated as a coproduct operation of (quasi)triangular Hopf algebra and the associated field theory is consistently defined as a braided quantum field theory. Algebraic formulation of path integral is perturbatively defined and Ward-Takahashi identity can be derived on the lattice. The claimed inconsistency of the link approach leading to the ordering ambiguity for a product of fields is solved by introducing an almost trivial braiding structure corresponding to the triangular structure of the Hopf algebraic superalgebra. This could be seen as a generalization of spin and statistics relation on the lattice. From the consistency of this braiding structure of fields a grading nature for the momentum operator is required.

Essential spectra and exponential estimates of eigenfunctions of lattice operators of quantum mechanics

This paper is devoted to estimates of the exponential decay of eigenfunctions of difference operators on the lattice which are discrete analogs of the Schrödinger, Dirac and square-root Klein–Gordon operators. Our investigation of the essential spectra and the exponential decay of eigenfunctions of the discrete spectra is based on the calculus of pseudodifference operators (i.e., pseudodifferential operators on the group with analytic symbols), and the limit operators method. We obtain a description of the location of the essential spectra and estimates of the eigenfunctions of the discrete spectra of the main lattice operators of quantum mechanics, namely: matrix Schrödinger operators on , Dirac operators on and square root Klein–Gordon operators on .

Constants of motion in an interacting electron-lattice system

The electron-lattice interaction term of the Bloch-FrOhlich Hamiltonian, written in the occupation number representation, is bilinear in the electron creation and annihilation operators but linear in the phonon raising and lowering operators. This asymmetry leads to the non-commutability of the phonon number operator and the interaction term of the Hamiltonian. It is shown that by introducing a bilinear form for each phonon operator the commutability of these two terms can be achieved. Formally, the new lattice operators arise from the quantum field-theoretical treatment of the normal modes. Specifically, in the harmonic approximation and coordinate representation of the lattice Hamiltonian in phonon space, this is achieved by the replacement of wave functions by a coordinate-dependent operator field. One obtains then such a field for every normal mode of the lattice. The operator field expansion coefficients are the lattice vibrational mode creation and annihilation operators which appear only in a bilinear combination throughout the theory. To obtain an equivalence relation between the field-theoretical and wave-mechanical formulations, one requires that only one lattice quasiparticle per lattice vibrational mode be allowed. For this case, the relation between the lattice field operators and the conventional phonon operators assumes a simple form. Finally, the entire Bloch-FrOhlich Hamiltonian is rewritten in terms of the new set of lattice operators and the conservation of the new lattice quasiparticles is demonstrated. Thus, both the number of electrons and the number of lattice quasiparticles are shown to be constants of motion of an interacting electron-lattice system.

A spectral function for an interacting electron-lattice system

The lattice quasiparticle conserving Hamiltonian for an interacting electron-lattice system is employed in calculating the electromagnetic spectral response function in absorption for an electron located in a deep trap such that the electronic transition energies are much higher than the cutoff of the lattice vibrational spectrum. The zero-order Hamiltonian for the system is then diagonal in electronic quantum numbers and linear in the lattice vibrational mode coordinates. The spectral response is calculated for a dipole mediated transition by employing the temperature-dependent double-time Green's function technique. It is shown that because of the properties of the lattice field operators, the hierarchy of Green's function equations decouples and no higher-order Green's functions are required for obtaining an explicit solution to a given order. An exact solution for the absorption of spectrum in the lowest order in electromagnetic field is given in terms of delta-function lines. Broadening effects, numerical calculation restrictions and applications to molecular vibronic spectra are briefly discussed.

Frame Stabilizers for Framed Vertex Operator Algebras Associated to Lattices Having 4-Frames

In this paper, we study certain Virasoro frames for lattice vertex operator algebras (VOAs) and their -orbifolds using linear codes over . We also compute the corresponding frame stabilizer from the viewpoint of binary codes and -codes. As an application, we determine the frame stabilizers of several Virasoro frames of the VOA and the moonshine vertex operator algebra V ♮ .

The lattice structure and refinement operators for the hypothesis space bounded by a bottom clause

Searching the hypothesis space bounded below by a bottom clause is the basis of several state-of-the-art ILP systems (e.g. Progol, Aleph). These systems use refinement operators together with search heuristics to explore a bounded hypothesis space. It is known that the search space of these systems is limited to a sub-graph of the general subsumption lattice. However, the structure and properties of this sub-graph have not been properly characterised. In this paper firstly, we characterise the hypothesis space considered by the ILP systems which use a bottom clause to constrain the search. In particular, we discuss refinement in Progol as a representative of these ILP systems. Secondly, we study the lattice structure of this bounded hypothesis space. Thirdly, we give a new analysis of refinement operators, least generalisation and greatest specialisation in the subsumption order relative to a bottom clause. The results of this study are important for better understanding of the constrained refinement space of ILP systems such as Progol and Aleph, which proved to be successful for solving real-world problems (despite being incomplete with respect to the general subsumption order). Moreover, characterising this refinement sub-lattice can lead to more efficient ILP algorithms and operators for searching this particular sub-lattice. For example, it is shown that, unlike for the general subsumption order, efficient least generalisation operators can be designed for the subsumption order relative to a bottom clause.

Generalizing the Ginsparg-Wilson relation: Lattice supersymmetry from blocking transformations

The Ginsparg-Wilson relation is extended to interacting field theories with general linear symmetries. Our relation encodes the remnant of the original symmetry in terms of the blocked fields and guides the construction of invariant lattice actions. We apply this approach in the case of lattice supersymmetry. An additional constraint has to be satisfied because of the appearance of a derivative operator in the symmetry transformations. The solution of this constraint leads to nonlocal SLAC-type derivatives. We investigate the corresponding kinetic operators on the lattice within an exact solution of supersymmetric quantum mechanics. These solutions---analogs of the overlap operator for supersymmetry---can be made local through a specific choice of the blocking kernel. We show that the symmetry relation allows for local lattice symmetry operators as well as local lattice actions. We argue that for interacting theories the lattice action is polynomial in the fields only under special circumstances, which is exemplified within an exact solution.

Transfinite Semantics in the Form of Greatest Fixpoint

Transfinite semantics is a semantics according to which program executions can continue working after an infinite number of steps. Such a view of programs can be useful in the theory of program transformations. So far, transfinite semantics have been succesfully defined for iterative loops. This paper provides an exhaustive definition for semantics that enable also infinitely deep recursion. The definition is actually a parametric schema that defines a family of different transfinite semantics. As standard semantics also match the same schema, our framework describes both standard and transfinite semantics in a uniform way. All semantics are expressed as greatest fixpoints of monotone operators on some complete lattices. It turns out that, for transfinite semantics, the corresponding lattice operators are cocontinuous. According to Kleene’s theorem, this shows that transfinite semantics can be expressed as a limit of iteration which is not transfinite.

Lattice Operators and Topologies

Working within a complete (not necessarily atomic) Boolean algebra, we use a sublattice to define a topology on that algebra. Our operators generalizecomplementon a lattice which in turn abstracts the set theoretic operator. Less restricted than those of Banaschewski and Samuel, the operators exhibit some surprising behaviors. We consider properties of such lattices and their interrelations. Many of these properties are abstractions and generalizations of topological spaces. The approach is similar to that of Bachman and Cohen. It is in the spirit of Alexandroff, Frolík, and Nöbeling, although the setting is more general. Proceeding in this manner, we can handle diverse topological theorems systematically before specializing to get as corollaries as the topological results of Alexandroff, Alo and Shapiro, Dykes, Frolík, and Ramsay.

Parking functions and vertex operators

We introduce several associative algebras and families of vector spaces associated to these algebras. Using lattice vertex operators, we obtain dimension and character formulae for these spaces. In particular, we define a family of representations of symmetric groups which turn out to be isomorphic to parking function modules. We also construct families of vector spaces whose dimensions are Catalan numbers and Fuss–Catalan numbers respectively. Conjecturally, these spaces are related to spaces of global sections of vector bundles on (zero fibres of) Hilbert schemes and representations of rational Cherednik algebras.

Non-compact QED3at finite temperature: the confinement-deconfinement transition

The confinement-deconfinement phase transition is explored by lattice numerical simulations in non-compact (2+1)-dimensional quantum electrodynamics with massive fermions at finite temperature. The existence of two phases, one with and the other without confinement of fractional charges, is related to the realization of the symmetry. The order parameter of this transition can be clearly identified. We show that it is possible to detect the critical temperature for a given value of the fermion mass, by exploiting suitable lattice operators as probes of the symmetry. Moreover, the large-distance behavior of the correlation of these operators permits to distinguish the phase with Coulomb-confinement from the Debye-screened phase. The resulting scenario is compatible with the existence of a Berezinsky-Kosterlitz-Thouless transition. Some investigations are presented on the possible relation between chiral and deconfinement transitions and on the role of ``monopoles''.

Representing and Manipulating Situation Hierarchies using Situation Lattices

Situations, the semantic interpretations of context, provide a better basis for selecting adaptive behaviours than context itself. The definition of situations typically rests on the ability to define logical expressions and inference methods to identify particular situations. In this paper we extend this approach to provide for efficient organisation and selection in systems with large numbers of situations having structured relationships to each other. We apply lattice theory to define a specialisation relationship across situations, and show how this can be used to improve the identification of situations using lattice operators and uncertain reasoning. We demonstrate the technique against a real-world dataset.

Nonperturbative renormalization of quark bilinear operators andBKusing domain wall fermions

We present a calculation of the renormalization coefficients of the quark bilinear operators and the $K\ensuremath{-}\overline{K}$ mixing parameter ${B}_{K}$. The coefficients relating the bare lattice operators to those in the RI/MOM scheme are computed nonperturbatively and then matched perturbatively to the $\overline{\mathrm{MS}}$ scheme. The coefficients are calculated on the RBC/UKQCD $2+1$ flavor dynamical lattice configurations. Specifically we use a ${16}^{3}\ifmmode\times\else\texttimes\fi{}32$ lattice volume, the Iwasaki gauge action at $\ensuremath{\beta}=2.13$ and domain wall fermions with ${L}_{s}=16$.