Using an action of pure Wilson form (trace of plaque in the fundamental representation), we obtain results with high statistics. Using actions that contain higher representations of the group, we look for actions that are closer to the continuum limit. We will take advantage of this freedom later when choosing the scheduling action.
Strong Coupling Expansions
The Wilson loop measures the interaction of the measuring field with an external current loop. Let's expand the e's in powers of \. we find that Mr. the lowest extension classes give zero. To see this, take the integral involving one of the contour angles, C.
Renormalization on the Lattice
The effective theory is the renormalization of the original; the links are "running" links. We would be working in the canonical ensemble rather than the grand canonical ensemble of the theory. 2 Because the i-th configuration arises from the i-l-th, these configurations will be highly correlated.
Fermions
What is needed is an efficient way to find Q-1[U], the fermion propagator, in an external gauge field, which depends on all U-variables of the lattice. It is the bottom envelope of the data that determines the string tension; reliability should be judged by how well the envelope follows the result of the continuum renormalization group (the sets of 3 solid lines). The outer curve is the contour C, the inner squares are the plaques of the extension of the action that cancel out the exposed phase factors.
The Glueball Mass on an Array of Computers
Parallel Processing and Lattice Gauge Theories
Labeled processors (0-3) are cluster nodes which have bidirectional communication paths shown with solid lines. This processor acts as the cluster controller and also as a data buffer between the VAX and the cluster. We will now discuss some of the details of pure 4-dimensional measurement algorithms on a 4-node concurrent processor.
The network is partitioned between computers so that neighboring network variables are either on the same node or on neighboring nodes. For the 2x2 square, the 4-dimensional grid is split into 2 of its dimensions, the other 2 dimensions are "pushed" to the processors: if the total grid is 4x4x4x4, each node stores a 2x2X4X4 subcell of the grid. After the matrix is placed on the mailbox, the mailbox ftag is set, indicating to processor 1 that the mailbox is ready to be read.
Once all this is accomplished, the processors update all their ~ and then proceed to the next link, staying in step due to the nature of the communications software. Since we are working with a finite subset, only integers that label the members of the subset need to be communicated between the processors; actual matrices are not passed. As a partial check of the correctness of the algorithm on the 4 node machine, we have verified that the usual mean plaquette results were obtained - see fig.
Since the speed of each node is ~h VAX, you would naively expect the performance of the four-node machine to be that of ~ of a VAX.
The Glueball Mass
Approximately two-thirds of the time was spent updating and one-third measuring. Rather than giving the details here, an explanation of the algorithm used is given in Appendix 1. We will assume that the mass of spin 4 is above that of spin 0 and interpret our lowest state as spin 0.
The glue ball mass (times the grid spacing) as a function of the coupling is plotted in the figure. The lines drawn in the figure correspond to the prediction of the continuum renormalization group in the two-loop perturbation theory. Our belief that this is the case is strengthened by research into the specific heat in the same coupling region.
To reliably find the mass of the o+ glueballL using the fundamental representation action of Eq. Berg and Billoire [16] also note that the mass values appear too low, again this occurs at the location of the specific heat peak. The parity transformation (i.e. inversion through the origin) of this loop cannot be achieved by a rotation of the loop.
The points on the line AB are the locations of the first-order phase transitions.
DITJ
Migdal-Kadanotl Renormalization for Gauge Theories
We will explain the technique in a similar way to the example of the 2-dimensional Ising model from Chapter 1. To construct this integral, it is convenient to expand the signs of the group (the traces of the various irreducible representation matrices). This can be continued until the integration of links in the other direction (horizontal links in Figure 3.1).
The entire term is raised to the r.ct-2 power because, in d-dimensions, integrating a link requires shifting d-2 planes of plaques. Starting with any action, using this transformation, traces of plaque matrices will appear in all irreducible representations of the group. The systematic improvement of the Migdal-Kadanoff program of Martinelli and Parisi [5] can in principle also find the non-local terms (no one has tried this yet for gauge theories because it is technically difficult).
The Results of Bitar, Gottlieb and Zachos
III.2.3)
III.2 .5)
Results
Although our choice of couplings is motivated by the Migdal–Kadanoff renormalization technique, our results do not depend on the accuracy of the technique. In contrast to the situation on the fJA=O axis, the lower envelope of the masses appears to fall smoothly and monotonically. These lines depend on the value of the mean plaquette, for which Grossman and Samuel use a weak coupling approximation.
Our result that the masses, when mapped onto the Wilson axis, follow the scale curve can now be reinterpreted as evidence that in the region of couplings in which we have been working we can find the ratio of the glue ball mass to the square root of the string tension must be constant. We have shown that actions that incorporate a negative component of the adjunct representation are closer to the continuum limit than actions along the Wilson axis. Working on the stable line of the approximate Migdal-Kadanoff renormalization currents, we have shown that the mass of the o+ glue ball divided by the square root of the string tension is a constant, something which nQl is true on the Wilson axis for the torque range which is studied.
The dotted line is the extrapolation of the critical line to the Wilson axis and intersects it at the location of the peak in the specific heat (S). Including dynamic fermions in Monte Carlo calculations is one of the most important problems in the field of lattice gauge theories. In Chapter 2 the pseudo-fermion method is introduced and the results of its application to 1 + 1 QED are given.
The results of the pseudo-fermion method applied to the Schwinger model have previously been described in [3].
Exact Algorithm, Stochastic Method
- KOO 32.00 40.00 48.00 56.00 64.0
Suppose we sweep through all the gauge links of the grid, recalculating (QtQ) -1 for each update (as one strictly must). Our original intention was to study the severity of the systematic error by updating subsets of all the gauge links from the same pseudo Monte Carlo. The matrix Q in the fermion part of the action is identical to that in Eq.
The Z(200) subset was used to approximate U(1) in the pure action meter part and the Metropolis algorithm was used to update the mesh. Such observables can be expressed in terms of the fermion propagator in the external gauge field: The pseudofermion Monte Carlo was run only once for each gauge sweep through the mesh, i.e., updating the entire gauge field configuration as in Eq.
Even if the error in the determinant (or more generally, in the action) due to the finite statistics is often large (Fig. 4.2), the validity of the approximation must ultimately be judged by the error on the mass gap (or other) observables). This implies that, in the time evolution of the U-field for k steps, the contribution of this term is,. We have done some preliminary studies on the application of the pseudo-fermion method to the real theory: SU(3) in 4 dimensions.
The Observable to be compared is the mean value of the tile or 1x 1 Wilson loop. The pure measuring part of the action was taken as the usual Wilson form (Eq. We see that throughout most of the sweep the ratios are very close to 1 - the "old" determinant is the same as the true determinant.
Wilson Loops on the Homogeneous Machine
One minor complication is the fact that the 2-point correlation, which is a global observation, cannot be easily calculated within a homogeneous machine. First, all Wi.lson loops on the mesh are computed at the machine nodes. The zero-quantity operators are then found by adding loops in the spatial directions - this is done by adding the results and passing the results towards processor 0.
The zero momentum operators are now in processor 0 for each time slice and all that remains to be done is to calculate the two-point correlations between the different slices. Instead of doing this in processor 0, we pass the numbers to the Intermediate Host (IH) and let it calculate the correlations. Because the amount of work the IH has to do is small compared to the amount of work required to move through a subcell, the IH has no problem keeping up with the array.
Gauss-Seidel Inversion
Langevin equations are equivalent to the Path Integral
This proves that the ensemble obtained by solving the Langevin equations has its members distributed according to the Boltzmann factor, e, and is therefore equivalent to the path integral. The distance the rp field traverses through configuration space, which is proportional to YN in Monte Carlo, goes as :* in the Langevin approach since this multiplies the width of the distribution in Fig. We see that the size of the Metropolis hit, 6 U , is proportional to T* since this is, again, the width of the distribution.
