Fermion Matrix Research Articles

Comment on "Skeleton boson realizations of collective subalgebras"

The skeleton boson realizations of fermion collective subalgebras proposed by Kim and Vincent are discussed. The Dyson boson mapping is found to play an essential role in the possibility of such realizations. A simple but instructive example is given to show that a straightforward application of the ``skeletonization procedure'' to the Holstein-Primakoff boson mapping may lead to serious errors in reproducing the fermion matrix elements. A possible way of constructing the modified skeleton Holstein-Primakoff realization of the collective subalgebra is suggested.

Block-conjugate-gradient method.

It is shown that by using the block-conjugate-gradient method several, say s, columns of the inverse Kogut-Susskind fermion matrix can be found simultaneously, in less time than it would take to run the standard conjugate-gradient algorithm s times. The method improves in efficiency relative to the standard conjugate-gradient algorithm as the fermion mass is decreased and as the value of the coupling is pushed to its limit before the finite-size effects become important. Thus it is potentially useful for measuring propagators in large lattice-gauge-theory calculations of the particle spectrum.

Preconditioning the Kogut-Susskind fermion matrix.

Preconditioning has been successfully used to accelerate the conjugate-gradient algorithm for inverting the lattice Dirac operator in the Wilson formulation. However, technical problems have arisen when trying to apply it in the Kogut-Susskind formulation. We discuss attempts to circumvent the technical problems and to find a successful way to precondition the Kogut-Susskind fermion matrix.

Wilson fermions on the lattice—A study of the eigenvalue spectrum

We have studied the eigenvalues of the Wilson fermion matrix on various lattice sizes and for a range of coupling constants, β, in the quenched approximation. The implications for the phase structure of lattice QCD in the presence of Wilson fermions are discussed.

Identity of the conjugate gradient and Lanczos algorithms for matrix inversion in lattice fermion calculations

Two of the methods that are widely used in lattice gauge theory calculations requiring inversion of the fermion matrix are the Lanczos and the conjugate gradient algorithms. Those algorithms are already known to be closely related. In fact for matrix inversion, in exact arithmetic, they give identical results at each iteration and are just alternative formulations of a single algorithm. This equivalence survives rounding errors. We give the identities between the coefficients of the two formulations, enabling many of the best features of them to be combined.

Fourier acceleration in lattice gauge theories. I. Landau gauge fixing.

Fourier acceleration is a useful technique which can be applied to many different numerical algorithms in order to alleviate the problem of critical slowing down. Here we describe its application to an optimization problem in the simulation of lattice gauge theories, that of gauge fixing a configuration of link fields to the Landau gauge (${\ensuremath{\partial}}_{\ensuremath{\mu}}$${A}^{\ensuremath{\mu}}$=0). We find that a steepest-descents method of gauge fixing link fields at \ensuremath{\beta}=5.8 on an ${8}^{4}$ lattice can be made 5 times faster using Fourier acceleration. This factor will grow as the volume of the lattice is increased. We also discuss other gauges that are useful to lattice-gauge-theory simulations, among them one that is a combination of the axial and Landau gauges. This seems to be the optimal gauge to impose for the Fourier acceleration of two other important algorithms, the inversion of the fermion matrix and the updating of gauge field configurations.

Spurious components in the ideal boson states.

The nature of the appearance of the spurious states in the ideal boson basis states is investigated. It is shown that the occurrence of the spurious states can be mainly attributed to the orthogonality of the boson basis states. It is further observed that the matrix element of the pairing interaction between the ideal boson basis is identical to the corresponding fermion matrix element.

Algorithms for calculating quark propagators on large lattices

We describe methods for solving large sparse systems of linear equations on computers with limited fast memory, or high ratios of processor speed to bandwidth between main and fast memory. Our algorithms are designed for calculating quark propagators, columns of the inverse of the fermion matrix in Lattice Quantum Chromodynamics, but are more generally applicable. We compare their rates of convergence and the balance between CPU time and I/O overhead. We present a block-iterative algorithm which when implemented on the DAP is 5 times as efficient as the Conjugate Gradient Method for 16 3 × 24 lattices (complex linear systems of size approximately 3 × 10 5).

Hadron mass calculations using susskind fermions at β = 5.7 and 6.0

Hadron masses are calculated in quenched quantum chromodynamics (QCD) using pure gauge configurations on 16 4 lattices, periodically extended in time where necessary. We use the Susskind formulation of lattice fermions and construct the propagators for hadrons created by local lattice operators. The inversion of the fermion matrix is accomplished using either the even/odd partitioned conjugate gradient algorithm, or block successive over-relaxation, depending on lattice size. Low statistics results at β = 5.7 confirm the high value of the nucleon-to-rho mass ratio found on smaller lattices. Together with the absence of flavour symmetry in the meson spectrum, this leads us to conclude that continuum behaviour is not observed at this coupling. Higher statistics results at β = 6.0 give a nucleon-to-rho mass ratio of approximately 1.5 and satisfy flavour symmetry in the meson sector to within 10%. We observe two baryon timeslice propagators which become increasingly different as the quark mass is decreased. This effect appears to be exaggerated by our choice of antiperiodic boundary conditions in space and is interpreted as a finite size effect.

Mesons in the Eguchi Kawai model

Fermions in the fundamental representation of theSU(N) gauge group are incorporated into the Eguchi Kawai reduction scheme. The proof for the equivalence of reduced and extended theories is given. This equivalence can be used, to calculate chiral condensate and meson propagators from the fermion matrix of a partially reduced TEK model.

The fermion propagator matrix in lattice QCD

The fermion propagator matrix is introduced and analyzed in lattice QCD. It can be related directly to the inverse of the fermion matrix and is similar in some ways to the transfer matrix. It is shown how the Lanczos algorithm can be used to diagonalise it and this is illustrated by calculating the eigenvalues on 4 4 QCD configurations and relating the results to finite-density calculations of the chiral condensate and quark currents. This sets the groundwork for calculations on larger lattices which should, in principle, give the hadron spectrum in a way which has some advantages over usual methods.

A classical instanton on a four-dimensional periodic lattice

On a four-dimensional periodic lattice, we construct an SU (2) gauge field configuration which is analogous to the classical instanton. This is shown to have topological charge equal to one. We calculate the eigenvalue spectrum of the fermion matrix and demonstrate the existence of an approximate zero mode.

Large distance propagators for hadrons on a 56 × 16 3 lattice

We have computed hadron propagators in the quenched approximation to Wilson fermions at β = 6 on 24 SU(3) background fields of size 56 × 16 3. The inversion of the fermion matrix was achieved by the approximate block diagonalization method. We find a clean exponential decay of the ϱ, N and Δ propagators for time separations 8 ≲ t ⩽ 54 at quark masses am q = 0.04 and beyond.

Lattice QCD with light quark masses: Does chiral symmetry get broken spontaneously?

We present a first direct calculation of the properties of QCD for the small quark masses of phenomenological interest without extrapolations. We describe methods specially adapted to invert the fermion matrix at small quark masses. We use these methods to calculate 〈 ψψ〉 directly on presently used lattice sizes with different boundary conditions. As is to be expected for a finite system, we do not observe spontaneous chiral symmetry breaking. By comparing the results obtained on lattices of different size we see, however, indications that are eventual spontaneous chiral symmetry breaking in the infinite volume limit. Our calculations underline the importance of using antiperiodic boundary conditions for fermions.

Property of Many-Phonon Norm Matrix

It is shown that the many-phonon states built up from collective quasiparticle pairs satisfy the orthogonality condition in a good approximation, if the approximation scheme recently proposed by Holzwarth, Janssen and Jolos is applied to the intrinsic space in the quasiparticle state space, which is orthogonal to the pairing rotational and vibrational exci­ tations. Recently, Holzwarth, Janssen and Jolos 1l have proposed a new approximation method to evaluate anharmonicity effects associated with low-frequency quadrupole modes in transitional nuclei_ The essence of this method consists in truncating the quasiparticle state space by building up many-phonon states from collective two-quasiparticle operators with J = 2+ and evaluating the fermion matrix elements within this collective phonon space. According to a more rigorous formulation given by Iwasaki, Sakata and Takada,) this method is regarded as the first-order approximation in the expansion with respect to the order of commutators involving phonon operators_ In principle, the norm matrix of many-phonon states have to be diagonalized in order to exactly take account of the Pauli principle. However, if the many-phonon states satisfy the orthogonality property in a good approximation, our task is reduced to calculating only normalization constants and, accordingly, physical interpretation of the resulting expressions may be greatly simplified_ The orthogonality implies that we can classify the many-phonon states defined in fermion space in terms of the quantum numbers which characterize the five-dimensional harmonic oscillators (i.e., the quadrupole boson states). In this paper, we show that this property holds in a good approximation, if the collective phonon sPace is defined in the intrinsic sPace of the quasiparticle state space_ The concept of intrinsic space has been introduced in Ref. 3), which is defined to be orthogonal to the pairing degrees of freedom (pairing rotation and pairing vibration). In § 2, the method of Holzwarth et al. D is applied to the intrinsic space so that the collective phonon space never involves the spurious components associated with the nucleon-number non-conservation in the quasiparticle representation. The condition under which the many-phonon states satisfy the orthogonality property is given in

Energy-momentum tensors and z-diagrams

The discussion of Z-diagrams is extended to equal-time commutators of the charges Q D α (0) = ∫ d 3 x x m T 0 m α ( x) with space and time components of currents J μ and their divergences. (We have denoted by T μν α any symmetric energy-momentum tensor within the Callan-Coleman-Jackiw class parametrized by α.) The requirement of absence of Z-diagrams in the infinite momentum limit for boson matrix elements of either of the equal time commutators [ Q D α (0), ∂ μ J μ (0)], [ T μ αμ (0)] or [ Q D α (0), J k (0) restricts the energy momentum tensor to the new and improved one. This tensor T μν satisfies the preceeding requirement and, as a consequence, Z-diagrams do not contribute in the infinite momentum limit to boson matrix elements of the commutators [ Q D(0), ∂ μ J μ (0)], [ T μ μ (0), Q(0)] and [ Q D(0), J k (0)] (with Q D the dilatation charge). For fermion matrix elements of [ Q D(0), ∂ μ J μ (0)] Z-diagrams do give a contribution in the infinite momentum limit. Furthermore, in this limit no Z-diagrams contribute to fermion matrix elements of [ Q D(0), J μ (0)] and to boson matrix elements of [ Q D α (0), J o (0)] (for any α). Moreover, assuming one particle saturation in the infinite momentum limit of both meson and fermion matrix elements of [ Q D(0), ∂ μ J μ (0)] and [ T μ μ (0), Q(0)] one consistently derives that ∂ μ J μ has dimension two if it has a dimension at all.