The Eigenvalue Spectrum of the Fermion Matrix in Lattice Higgs Systems

[摘要] In the first part of this thesis we consider the performance of various block algorithms for the inversion of large sparse matrices. By computing the eigenvalue spectra of the matrices under consideration we are able to directly relate the performance of the algorithms to the difficulty of the calculation. We find that the block Lanczos algorithm is superior to all others considered for the inversion of the Kogut Susskind fermion matrix. Furthermore we investigate the performance of the block Lanczos algorithm on matrices constructed to have specific eigenvalue spectra. From this study we are able to make quantitative predictive statements about the number of iterations that the algorithm will take to converge given the form of the eigenvalue spectrum of the matrix whose inversion is attempted. The rest of this thesis is concerned with lattice Higgs systems. Specifically we study a model where staggered fermions are coupled to Ising spins via an on-site Yukawa term with coupling constant y. This is a very simple model that seems to embody most of the relevant phenomena observed in more complicated systems. Most importantly there are two symmetric regions PM1 and PM2 where the renormalised fermion mass my is non-zero for large y in the PM2 region despite the scalar field having zero expectation value. We study the model in the quenched approximation and by examining the distribution of the eigenvalues of the fermion matrix M in the complex plane we qualitatively explain the features of the model as being due to the transition of eigenvalues from the imaginary to the real axis via the origin as y is increased. An approximate method for calculating mf from the value of a fermion condensate is developed and we reproduce the values for mf obtained by other authors who calculate it using the standard method involving the fermion propagator. However, our method has the advantage that it is applicable on very small volumes where the propagator definition breaks down. We investigate the behaviour in the quenched infinite volume limit by evaluating the low lying eigenvalues of the matrix A+M. We show that the small eigenvalues observed in the spectrum of M at intermediate y on finite lattices imply that there is a finite density of zero modes in the infinite volume limit. By performing dynamical simulations on a small lattice we determine the phase diagram of the model and demonstrate the validity of mean field calculations of the phase boundaries. From calculations of mf we identify the PM1 and PM2 phases. It is shown that the inclusion of fermion dynamics eliminates the small eigenvalues of M present in the quenched model and as y is increased the eigenvalues now transfer from the real to imaginary axis via a path avoiding the origin. It is only by using the block Lanczos algorithm that simulations in certain regions of the phase plane are feasible, and only by our method of considering a fermion condensate can we calculate mf on such a small volume.

[发布机构] University:University of Glasgow

[关键词] Theoretical physics