Monte Carlo calculation with unquenched Wilson fermions

Abstract

A Monte Carlo updating procedure taking into account the virtual quark loops is described. It is based on high-order hopping parameter expansion of the quark determinant for Wilson fermions. In a first test run Wilson loop expectation values are measured on a 64 lattice at 13 = 5.70 using 16th order hopping parameter expansion for the quark determinant.The inclusion of the effects of virtual fermion loops in Monte Carlo simulations of lattice gauge theories is an important and challenging problem. At present it is to a large extent unknown how much the Monte Carlo resuks obtained in the pure colour gauge sector or in the "quenched approximation" will be changed once light dynamical quarks are properly included in the computations of lattice quantum chromodynamics. There is a class of problems, like the screening of (fundamental) colour charge or the fragmentation of fast quarks into hadrons etc., which cannot even be formulated without virtual quark loops.From the computational point of view the inclusion of dynamical fermions is, however, rather difficult because of the long range interaction induced by the virtual light fermion loops. Several methods proposed recently, like the pseudo-fermion method [1][2][3], stochastic method [4,5] or the microcanonical method [6] are promising, but it is usually difficult to control their accuracy due to some approximations which affect the results only rather indirectly. In a recently proposed method based on high order numerical hopping parameter expansion [7,8] the committed errors are easier to control. The fermion determinant is, however, not included in the updating but it is treated essentially as a part of the expectation value. In the case of light fermions this is the source of large fluctuations prohibiting the collection of enough statistics in a reasonable amount of computer time.i Supported by Bundesministerium f~ir Forschung und Technologie, Bonn, Germany. 70In the present letter I describe an updating procedure with the fermion determinant evaluated to high order in the hopping parameter expansion at every link and included in the Monte Carlo updating. The use of the hopping parameter expansion of fermion determinant in the updating was already attempted previously in a different way by Lang and Nicolai [9]. They, however, sorted out the contributions of the fermion determinant to the effective action according to closed curves passing through a given link. The number of such curves increases very rapidly with the length (i.e. with the order of expansion): at 12th order there are more than 4 × 106 such curves and e.g. at 16th order already more than 6 × 109 [10]. (Actually, these numbers refer to an infinite lattice. On a finite periodic lattice the number is still increased by the curves wrapping around the lattice.) It is clearly impossible to handle such large numbers of curves with any reasonable approximation (in ref.[9] typically 20 curves with length not more than 12 were taken). The solution I have chosen is to use the fast iterative method [11] for ...

Unquenching and Unitarising Mesons in Quark Models and on the Lattice

“…The term "unquenched" first appeared in the title of a lattice paper due to I. Montvay [15] back in 1984, describing a Monte Carlo calculation with virtual quark loops through a quark determinant for Wilson fermions. Progressively, many other lattice groups started to carry out QCD simulations with dynamical quarks.…”

An algorithm for gluinos on the lattice

“…The optimized hopping parameter expansion of the previous subsection can also be used in the old fermion algorithm based on the direct evaluation of the change of the fermion determinant [21,20]. Let us remember that according to (9) the change of the fermion matrix is…”

“…Let us now turn to the question, how to implement the fermion determinant factor in eq. ( 23) in the path integrals appearing, for instance, in the expectation values ( 18)- (21). Following ref.…”