We propose a family of renormalization group transformations characterized by
free parameters that may be tuned in order to reduce the truncation effects. As
a check we test them in the three dimensional XY model. The Schwinger--Dyson
equations are used to study the renormalization group flow.Comment: Contribution to Lattice'94. uuencoded postscript fil

A. Muñoz Sudupe

A. Tarancón

Falcioni

Fernández

J.J. Ruiz-Lorenzo

L.A. Fernández

Wolff

English

arXiv

© 1995 Elsevier Science B.V. International Symposium on Lattice Field Theory (LATTICE 94) (12. 1994. Bielefeld, Germany). We acknowledge CICyT for partial financial support with the projects AEN93-0604, AEN93-0776 and AEN94-218. One of us (JJRL) is granted by MEC (Spain).We propose a family of renormalization group transformations characterized by free parameters that may be tuned in order to reduce the truncation effects. As a check we test them in the three dimensional XY model. The Schwinger-Dyson equations are used to study the renormalization group flow.CICyT, SpainMEC, SpainDepto. de Física TeóricaFac. de Ciencias FísicasTRUEpu

Fernández Pérez, Luis Antonio

Muñoz Sudupe, Antonio

Ruiz Lorenzo, J. J.

Tarancón, A.

Docta Complutense

! LI[OI l ~,'.I i l l  .,,,i .- k-+",] [ab",l PROCEEDINGS SUPPLEMENTS ELSEVIER Nuclear Physics B (Proc. Suppl.) 42 (1995) 802-804 A proposal  of a Monte Carlo renormalizat ion group t ransformat ion  L. A. Ferngndez a, A. Mufioz Sudupe a, J.J.  Ruiz-Lorenzo b and A. Tarancfn  ¢ aDepar tamento  de Ffsica Tedrica, Universidad Complutense de Madrid, 28040 Madrid, Spain, b Dipar t imento di Fisica, Universit~ di Roma I, P. A. Moro 2, 00185 Roma,  I taly CDepartamento de Ffsica Tedrica, Universidad de Zaragoza, 50009 Zaragoza, Spain. We propose a family of renormalization group transformations characterized by free parameters that may be tuned in order to reduce the truncation effects. As a check we test them in the three dimensional XY model. The Schwinger-Dyson equations are used to study the renormalization group flow. 1. I N T R O D U C T I O N  One of the key problems of the application of the renormalization group (RG) ideas in the con- text  of Monte Carlo Simulations is the growth, under the RG transformations,  of significant cou- plings not considered in the original action. This effect being a consequence of a RG fixed point that  is si tuated relatively far away from the sim- ulation point. Our proposal will be to improve one of the ingredients of the RG transformation,  namely the field renormalization definition, in such a way that  it will bring the RG fixed point closer to the simulation point so as to reduce the truncation effects. At the same t ime we want to keep the RG transformation as simple as possible because we are mainly interested in gauge theo- ries. In order to test the reliability of the method we have chosen the three dimensional XY model, with well known critical properties. Our procedure will be to generate configura- tions at the critical point •1,c -~" 0.45420(2), where the appropriate  observables will be mea- sured. We will then perform iteratively the RG transformations for different parameters  and measure the observables on the blocked lattices. As we will see it is possible, through the lattice Schwinger-Dyson equations [1], to obtain from these measures the (blocked) renormalized cou- plings at each blocking level. We are thus able to follow the RG flow in the two dimensional param-  eter space of couplings /~1, f12, nearest neighbors and next to nearest neighbors, respectively. We 0920-5632/95/$09.50© 1995 Elsevier Sc ience  B . V .  All rights reserved.  SSDI 0920-5632(95)00386-X compute also the thermal  exponent v from the RG flow and from a Finite Size Scaling Analysis. 2. R G  T R A N S F O R M A T I O N  A RG transformation consists on renormaliz- ing the fields and blocking the lattice from L to L/2.  For gauge theories one is forced, in order to preserve gauge invariance, to take the mean over ordered products of links along fixed~ends trajectories. This procedure can become hard to compute specially for parallel machines. Our approach would be to perform a smearing (re- laxation) step previous to the actual field renor- malization (decimation). These relaxational tech- niques are well known in other problems like spec- troscopy or topological studies [2]. There are different ways to define the smearing step, a plausible guess would be to consider the relaxation driven by the heat equation a~(x, r) a~/ However, at finite lattice spacing it is not obvious which actual t ransformation performs better.  We will choose a smearing transformation that  a pos- teriori will show to have better  properties. The criterion that  we impose to deduce it is that  it should preserve locally the form of the Hamilto-  nian. For the XY model for instance, the contri- bution of the spin at site n to the energy can be written as Re{e i°~ y~.+~ e -ie"+~ } = Repe i'~, p > L.A. Ferntindez et al./Nuclear Pt~vsics B (Proc. Suppl.) 42 (1995) 802-804 803 0. With the transformation 0n,8+1 = On,8 + eArg Z ei(°"+"-°") it is easily seen that,  locally, the energy changes to pcos[(1 - e)a]. Other options have been tested and we have found that  they are less efficient. The smearing parameter  ~ can be combined with the iterative number of smearing steps ns in order to select the RG transformation that  has its fixed point as close as possible to the simula- tion (critical) point. We use the S-D equations to follow the RG flow in the coupling space. They give us a linear systems of equations at every blocking level. For the XY model, assuming that  the interaction is exclusively nearest neighbor f l  ~ 0 at all blocking levels the S-D may be written as Should we consider also next to nearest neighbors interaction, the S-D equations would provide a 2 × 2 system of linear equations (see [3] for de- tails). We have mainly used the Wolff's Single Clus- ter (WSC) algorithm [4] with lattice sizes L -- 8, 16,32 and 64, where we have generated 100, 50, 40 and l0 thousands of configurations respec- tively. Successive configurations are separated by a mean of 200 single cluster spin updates and one sweep of Metropolis. We have the interesting op- portunity to study the performance of this updat- ing algorithm at different scales and to compare it with others such as the Metropolis algorithm see Fig.1. In it we have plotted the integrated au- tocorrelation time for the pure WSC, Metropolis and the one used in this paper respectively. There it can be seen that,  as expected, the WSC per- forms better at large scales while Metropolis do the opposite. Notice also that by adding just one Metropolis step to every 200 WSC updates all the scales are similarly treated. The results presented amount for a total CPU time of only one month of Workstation. We re- mark that our interest is mainly on the RG trans- ! ' " l ' " ' l ' " ' l ' ' " l " "  3 1 ' ' " 1 " " 1 " " 1 ' ' " 1 ' " 4  ~ , , , ,  i ,  , , ,  i ,  • , ,  I , , ,  ,.L,.r.IJ. ....f. 0 lO 20 30 40 10 20 30 40 20 40 60 80 t Figure 1. Comparation of the integrated autoeor- relation times for from left to rigth: 1 Metropolis for each 200 Wolff's Single Cluster update, only Wolff's Single cluster and Metropolis. The solid line corresponds to L = 16, the dashed to L = 8, the dashed-dotted to L = 4 and the dotted to L = 2. We plot the integral of the normalized autocorrelation between 0 and t. formation itself and that  we apply it to the 3d XY model only as a test. As a first test of the efficiency we try to tune the smearing parameters in such a way that the fixed point of the so defined RG transformation would be as close as possible to f* = (tic, 0, 0,...). To do that we compute the blocked couplings f l  b, calculated with (1) in a 163 lattice for different smearing parameters choices. We obtained that the matching of the blocked couplings is optimal for the RG defined by (n~, e) = (2, 0.285). As a second test it is possible, using a two di- mensional generalization of (1), to compute the nearest and next to nearest neighbors couplings f ] ,  f2 at blocking level b. Thus we can draw the RG flow in the two dimensional parameter space (fix,fl2) (see Fig. 2). There it is clearly seen the importance of the appropriate tuning of the RG parameters (n,,  e). For the choice (2, 0.285) it can be seen that  in the first two RG steps the flow approaches the fixed point ~ (0.43, 0.02) while in the third step and, more obviously, in the fourth step it leaves the fixed point in the relevant direc- tion, following the renormalized trajectory. The difference between the coupling ~ computed in 804 L.A. Ferndndez et al./Nuclear Physics B (Proc. Suppl.) 42 (1995) 802-804 I . . . .  I . . . .  I . . . .  I . . . .  1 ' ' - ~  0.06 \ \  0.oz "-..xx,.. 00 " %  1 I , , , , I , , , , I . . . .  I , , , , " q k ~ t ~ l  ).25 0.30 0.35 0.40 0.45 131 Figure 2. Flow of the Renormalization Group transformations in the two dimensional coupling space for a 323 lattice. The solid lines correspond to the selection (n~ = 2, e = 0.285), the dashed ones to (ns -- 2, ~ - 0.2). ous formula for the plaquette operator as well as the fll coupling obtained from the S-D equations. 0.72 0.70 0.68 0.66 0.64 (D T Figure 3. Critical exponent v obtained from Fi- nite Size Scaling analysis of the plaquette opera- tor (filled circles) and fll coupling respectively. the one coupling (nearest neighbors) approxima- tion and when two couplings are considered , in a 643 , is small (5%) for the first four levels of cou- pling giving us an estimation of the order of the systematic errors introduced. Strong finite size effects manifest themselves in a higher difference between the two calculations of the coupling in the last blocking level. We have also computed the critical exponent v from derivatives of the couplings, obtaining values with systematic errors lower than 3% just after one RG step. For the L = 64 lattice we obtain from a linear fit v = 0.0649(20). Another somewhat independent measure can be obtained from Finite Size Scaling arguments. Starting from different lattice sizes L1, L2 we ar- rive, after several RG steps, to the same given size L (usually L - 2). By comparing the /3- derivatives of some observable on the L = 2 lat- tice coming from L1, L2, we obtain directly the critical exponent through ! : log (O)L,,I~ d~ (O)L2,fl B=fl~ (2) v log In Fig. 3 we plot the estimation using the previ- In summary we have proposed a family of RG transformations characterized by the parameters (n,,  e) whose main advantages are: it is easily im- plemented, has small truncation effects (at least for the tested model) and it may be ported with small modifications to parallel machines. We acknowledge CICyT for partial financial support with the projects AEN93-0604, AEN93- 0776 and AEN94-218. One of us (JJRL) is granted by MEC (Spain). R E F E R E N C E S  1. M. Falcioni et. al. Nucl. Phys. B265  (1986) 187. 2. Ape Collaboration Phys. Lett. 192B (1987) 163. 3. L.A. Fernandez et. al. Phys. Rev. D50 (1994) 5935. 4. U. Wolff, Phys. Rev. Lett 62 (1989) 361. 

A proposal of a Renormalization Group transformation

Abstract

Similar works

Full text

Available Versions

Docta Complutense

Crossref