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# Block Spin Effective Action for 4d SU(2) Finite Temperature Lattice Gauge Theory

MS¨CTPI¨C97¨C13

Block Spin E?ective Action

arXiv:hep-lat/9710076v1 22 Oct 1997

for 4d SU(2) Finite Temperature Lattice Gauge Theory
Klaus Pinn
?

and Stefano Vinti

?

Institut f¡§ ur Theoretische Physik I Universit¡§ at M¡§ unster Wilhelm¨CKlemm¨CStr. 9 D¨C48149 M¡§ unster, Germany

Abstract The Svetitsky¨CYa?e conjecture for ?nite temperature 4d SU (2) lattice gauge theory is con?rmed by observing matching of block spin e?ective actions of the gauge model with those of the 3d Ising model. The e?ective action for the gauge model is de?ned by blocking the signs of the Polyakov loops with the majority rule. To compute it numerically, we apply a variant of the IMCRG method of Gupta and Cordery.

? ?

e¨Cmail: pinn@uni¨Cmuenster.de e¨Cmail: vinti@uni¨Cmuenster.de

1

Introduction

Block spin renormalization group [1] has become an important tool in the qualitative and quantitative understanding of critical phenomena in classical statistical mechanics and Euclidean quantum ?eld theory. As a basic ingredient, it introduces e?ective Hamiltonians (actions in ?eld theoretic language) which govern block spin degrees of freedom. The block spins are determined from the original degrees of freedom by averaging them over blocks. In principle, renormalization group (RG) solves the problems posed by critical or nearly critical statistical systems. Under iterated application of the block transformation, either the correlation length in the system becomes small, or the ?ow of e?ective Hamiltonians eventually reaches a ?xed point which determines the universal properties of the system. A major drawback of the RG approach stems from the fact that e?ective Hamiltonians in general contain an in?nite number of couplings, in contrast to the original Hamiltonians one starts from, which usually contain only a small number of interaction terms. The proliferation of couplings has a number of consequences. It is, e.g., not always clear how a certain truncation to a ?nite number of couplings a?ects the physical results. Furthermore, even if one relies on a certain truncation scheme, it might be tedious to explicitly compute the e?ective couplings. This might be an explanation why the task of explicit computation of block spin e?ective actions has not received very much attention in the literature. See, however, e.g., Refs. [2] and [3]. Svetitsky and Ya?e [4] have conjectured that a (continuous) decon?nement transition of a (d + 1)¨Cdimensional ?nite temperature lattice gauge theory should be in the same universality class as the phase transition of a corresponding d¨Cdimensional spin system. This spin system should have the center of the gauge group as a global symmetry group. The Svetitsky¨CYa?e conjecture for SU (2) gauge theory at ?nite temperature o?ers the possibility of an interesting application of the block spin renormalization group. First, it has never been rigorously proved that this model belongs to the Ising universality class. On the other hand, the conjecture has been checked several times by comparison of Monte Carlo (MC) estimates for the critical indices (which were found in good agreement [5]), as well as with a mean ?eld like analytical approach (which gives also predictions for SU (N ) decon?nement temperatures [6]). However, so far there have been no numerical attempts to explicitly compute the e?ective action 1

for the Polyakov loops and compare it with that of the Ising model. With this article, we intend to ?ll this gap. We will demonstrate that actions for the degrees of freedom relevant for the decon?nement transition can well be computed by Monte Carlo. Comparing them with the corresponding actions for the Ising model we are able to con?rm the validity of the Svetitsky¨CYa?e conjecture in a very fundamental way. The article is organized as follows: In Section 2 we introduce the notations for ?nite temperature lattice gauge theory and recall the Svetitsky¨CYa?e conjecture. In Section 3 we introduce the block spin renormalization group. Section 4 explains the idea of ?ow matching, and the application of Improved Monte Carlo Renormalization Group (IMCRG) [7] to SU (2) lattice gauge theory is described in some detail. In Section 5 we discuss some details of our Monte Carlo methods and present the results. Conclusions follow.

2

Finite Temperature Lattice Gauge Theory

Let us brie?y review the formulation of ?nite temperature gauge theory on a lattice (see for instance Ref. [8]). Consider an SU (N ) gauge system on a (d + 1)¨Cdimensional hypercubic lattice of size Ld ¡¤NT , where L and NT are the spatial and temporal extensions, respectively, in units of the lattice spacing a. A Euclidean quantum ?eld theory at ?nite temperature is obtained if one compacti?es the (imaginary) temporal direction, keeping in?nite the spatial directions. In a ?nite lattice formulation one therefore assumes L ? NT . The compacti?cation length is proportional to the inverse physical temperature T 1 NT ¡¤ a = . (1) T Denote with U? (n) the SU (N ) group element belonging to the link with origin in the site n ¡Ô (x, t) and pointing in the ?¨Cdirection. The usual Wilson action reads Sg ( U ) = ¦Â
P

(N ? Re TrUP ) ,

(2) (3)

¦Â =

2N d?1 a , g2

where UP is the product of the group elements around the plaquette P . The 2

partition function is given by Z=
n,?

dU? (n) exp [?Sg (U )] .

(4)

Because of the periodicity in the temporal direction, the system is also invariant under a global ZN symmetry, i.e. the center of the gauge group: its spontaneous symmetry breaking at a ?nite temperature Tc is the signal of the decon?nement transition. The Polyakov loop is an order parameter for the ?nite temperature decon?nement transition. It is the trace of the ordered product of all timelike links with the same space coordinate, wrapping in the time direction
NT

L(x) = Tr
t=1

U0 (x, t) .

(5)

It is a non¨Ctrivial observable from a topological point of view: its vacuum expectation value is not invariant under ZN transformations. It is zero in the con?ning phase, while it acquires a ?nite value in the decon?ned phase. According to the 15 years old Svetitsky¨CYa?e conjecture [4], integrating out the space¨Clike degrees of freedom one should obtain an e?ective action for the Polyakov loops which is short ranged and has the center of SU (N ) as a global symmetry group. Thus, given a d¨Cdimensional classical spin system with the same symmetry properties, undergoing a continuous phase transition, the (d + 1)¨C dimensional quantum gauge model is expected to be in its universality class if the decon?nement transition is a continuous one and the e?ective Hamiltonian has good locality properties. This applies in particular to the 4¨Cdimensional SU (2) gauge model which should belong to the 3d Ising universality class.

3

Block Spin Renormalization Group

To de?ne the block spin transformation, consider a magnetic system consisting of spins ¦Ò on the sites of a d¨Cdimensional lattice, de?ned by a Hamiltonian H and a set of couplings {K }, H (¦Ò ) = ?
¦Á

K¦Á S ¦Á ( ¦Ò ) . 3

(6)

{¦Ò}

exp [?H (¦Ò )] .

(7)

The ¡°operators¡± S¦Á (¦Ò ) are in general all possible products of spins compatible with the symmetry of the Hamiltonian. Explicit examples will be given below. A block spin transformation maps the ?ne Ld lattice onto the block lattice of size L¡äd , where L = LB L¡ä . This is achieved by averaging the original spins over cubical blocks of side length LB according to a certain rule. The Hamiltonian H ¡ä of the block spins {?} assigned to the sites of the block lattice is de?ned by exp [?H ¡ä (?)] =
{¦Ò}

P (?, ¦Ò ) exp [?H (¦Ò )] ,

(8)

where P encodes the mapping from the ?ne to the coarse lattice. It obeys P (?, ¦Ò ) ¡Ý 0 and
{?}

P (?, ¦Ò ) = 1 .

(9)

The latter property ensures that the partition function remains unchanged, Z=
{?}

exp [?H ¡ä (?)] .

(10)

In this work we use the majority rule prescription (i.e. the ? spins take values plus or minus one)
(blocks)

P (?, ¦Ò ) =
x¡ä

1? ¦Òx ? 1 + ?x¡ä sign 2 ¡ä x¡Êx

?

?

.

(11)

The sign function sign(x) in Eq. (11) is de?ned such that it vanishes for x = 0. This ensures that in case of a zero sum of spins inside a block a positive (negative) ?x¡ä is selected with probability one half. ¡ä The block Hamiltonian H ¡ä can be expressed in terms of operators S¦Á (?), de?ned on the block lattice, H ¡ä (? ) = ?
¦Á ¡ä ¡ä K¦Á S¦Á ( ? ) .

(12)

4

In the case of the 4¨Cdimensional SU (2) gauge model, the 3d e?ective action for the signs of the Polyakov loops shares (by de?nition) the Z2 symmetry with the 3d Ising model. To de?ne this action we assign to each Polyakov loop its sign ¦Òx (U ) = sign L(x) . (13) Then we block the ¦Ò ¨Cspins with the majority rule to obtain Ising type block ?¨Cspins. It follows that, similarly to Eq. (8), the e?ective Hamiltonian H¡ä for the ?nite temperature gauge system is given by exp[?H¡ä (?)] = with P (?, U ) =
x¡ä (blocks)

DU P (?, U ) exp[?Sg (U )] , 1? 1 + ?x¡ä sign ¦Òx (U )? . 2 x¡Êx¡ä
? ?

(14)

(15)

Other procedures of blocking, like ?rst averaging the Polyakov loops inside the blocks and then take as Ising type spin its sign, could also be employed. We close this section by de?ning a renormalization group ?ow. A natural way to do it would be to ?x a block length, e.g., LB = 2, and then iterate the transformation (15). We do not stick to this de?nition here. Instead we de?ne the ?ow by just increasing the block size LB . This allows us to compute Hamiltonians not only for scales 2n , but also for arbitrary scales LB , with LB integer.

4

Monte Carlo Renormalization Group for Polyakov Loops

In this section we ?rst recall the RG matching idea. Then we show how to apply the Improved Monte Carlo Renormalization Group (IMCRG) method by Gupta and Cordery [7] to 4d SU (2) gauge theory at ?nite temperature.

4.1

Matching of RG Trajectories

In the in?nite¨Cdimensional space of couplings {K }, a renormalization group transformation R can be looked at as a mapping of the original bare Hamiltonian H onto a new Hamiltonian H ¡ä = R(H ), de?ned by the couplings {K ¡ä }. 5

model A

fixed point

model B

Figure 1: Matching of the RG trajectories of two critical models A
and B in the neighbourhood of an RG ?xed point.

The RG ?ow obtained under iterated RG transformations will eventually end in a ?xed point {K ? }, de?ned through H ? = R(H ? ). The critical surface is identi?ed by all ?ows connected to the ?xed point in this way. The RG matching method is based on the assumption that di?erent physical systems, belonging to the same universality class, will follow RG ?ows which originate from di?erent ¡°bare couplings¡± on the critical surface and eventually match in a neighbourhood of the common ?xed point. Of course, a matching close to a non¨Ctrivial ?xed point will only take place if both models under consideration are at criticality. A matching thus con?rms both universality and allows to check for criticality. This matching method has been successfully applied in the context of spin models, see e.g. [9]. The feasibility of matching di?erent critical ?ows by means of MC methods mainly relies on the assumption that the di?erent trajectories come close to each other, i.e. approximately match, before the ?xed point is actually reached, c.f. Figure 1. As we shall see, this condition is met for our models.

4.2

E?ective Couplings from IMCRG

The Improved Monte Carlo Renormalization Group method [7] allows to compute e?ective actions for Ising type block spins.1
1

A generalization to systems with continuous block variables is not straightforward.

6

The main idea is to avoid simulations of the original partition function. Instead, consider a modi?ed system de?ned through Zc =
{?}

? (? ) exp ?H ¡ä (?) + H ? (? ) , P (?, ¦Ò ) exp ?H (¦Ò ) + H
{?} {¦Ò}

= where

(16)

? (? ) = ? H
¦Á

¡ä ¡ä ?¦Á K S¦Á ( ? )

(17)

is a guess for H ¡ä (?). ? are given. Note the plus This system can be simulated once H and H ? in Eq. (16). sign in front of H The system with partition function Zc is non¨Ccritical, even in the case of a critical Hamiltonian H (¦Ò ). Consider the expectation values
¡ä < S¦Á >c

= =

1 Zc 1 Zc

¡ä S¦Á e {?}

? (? ) ?H ¡ä (?) + H ? (? ) ?H (¦Ò ) + H

(18) . (19)

¡ä S¦Á P (?, ¦Ò ) e {¦Ò} {?}

If the guess is exact, i.e.,

? (? ) = H ¡ä (? ) , H

(20)

the block spins ?x¡ä completely decouple and ?uctuate independently. In other words, the system is non¨Ccritical and the correlations in Zc are bounded by the block size LB . The correlations functions are then known exactly,
¡ä < S¦Á >o = 0 , ¡ä ¡ä < S¦Á S¦Â >o = n¦Á ¦Ä¦Á¦Â ,

(21)

where n¦Á are trivial multiplicity factors. ? (?) is close to H ¡ä (?). Then a ?rst order expansion Let us assume that H gives ¡ä ¡ä ¡ä ¡ä ¡ä 2 ?¦Á ?¦Á (22) < S¦Á >c = n¦Á (K¦Á ?K ) + O ( K¦Á ?K ) .
¡ä ? ¡ä . Usually a few Solving this equation for K¦Á allows to improve the guess K ¦Á iterations ?¡ä ¡ú K ? ¡ä + n?1 < S ¡ä >c , K (23) ¦Á ¦Á ¦Á ¦Á

7

where the expectation values are determined by simulation of the system (16), are su?cient to determine H ¡ä to a good precision. To apply the IMCRG procedure to the SU (2) gauge system, one has to simulate the partition function Zc =
{?}

? (? ) exp ?H¡ä (?) + H ? (? ) . DU P (?, U ) exp ?Sg (U ) + H
{?}

=

(24)

Remember that the ?¨Cvariables are de?ned as the majority rule block spins of the signs of the Polyakov loops. It is straightforward to design a MC procedure for the updating of this system. Details will be given in the next section.

5

Monte Carlo Simulations

We applied the IMCRG method to three di?erent systems: the 3d standard Ising model (with nearest¨Cneighbour coupling), the 3d ¡°I3 ¡± model, which includes also a third, cube¨Cdiagonal neighbour coupling [10], and the 4d SU (2) pure gauge model. We simulated the system de?ned by Eq. (16) for the spin models and by Eq. (24) for the SU (2) gauge model. For the updating of the Ising model we used a Metropolis algorithm: A single spin ¦Òx is proposed to be ?ipped. It is checked whether this update leads to a ?ip of the block spin ?x¡ä , with x ¡Ê x¡ä . The total change of ? (?) is then computed and used in the usual Metropolis energy ?H (¦Ò ) ? ?H acceptance/rejectance step. In case of the SU (2) model, only the temporal links couple to the compensating block Hamiltonian. The space¨Clike links are updated using the incomplete Kennedy¨CPendleton heat bath sweep [11] supplied with a number of overrelaxation sweeps. For temporal links one employs again a Metropolis procedure: A proposed change of a link matrix leads to a change of the Polyakov loop L(x) of which it is member. If the sign ¦Òx changes, the block ? (?). The relevant spin ?x¡ä in turn might ?ip and give rise to a change of H ? (?). energy change for the Metropolis step is ?Sg (U ) ? ?H In practical calculations one has to truncate the interactions in H ¡ä and ? . We chose to include in the ansatz eight 2¨Cpoint couplings and six 4¨Cpoint H couplings. The 2¨Cpoint couplings can be labelled by specifying the relative 8

K9

K10

K11

K12
Figure 2:

K13

K14

Graphical de?nition of 4¨Cspin couplings included in the e?ective Hamiltonian.

position of the interacting spins (up to obvious symmetries): Our couplings K1 . . . K8 then correspond to 001, 011, 111, 002, 012, 112, 022, 122. The 4¨Cpoint couplings K9 . . . K14 are de?ned in an obvious way through Figure 2. The corresponding interaction terms in the e?ective Hamiltonian are denoted ¡ä by S¦Á , ¦Á = 1 . . . 14.

5.1

Reduction of Critical Slowing Down

A merit of the IMCRG method is that block spin observables are (nearly) decorrelated and the critical slowing down problem is less severe than in standard simulations. In Figure 3 we show scatterplots (MC time history) of measurements of the nearest neighbour block spin 2¨Cpoint function. The comparison is between a simulation of the pure gauge system (without IMCRG compensation on the block level) and the system de?ned through Eq. (24). The plot clearly shows that the IMCRG type simulation su?ers much less from critical slowing down. Analogous observations were made in case of the Ising model simulations with the compensating block Hamiltonian switched on. It is the reduction 9

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

block size 2, L=12, standard simulation

0.15 0.1 0.05 0 -0.05 -0.1 -0.15

block size 2, L=12, IMCRG simulation

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

1000 1500 2000 sweep block size 4, L=24, standard simulation

500

0

0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2

1000 1500 2000 sweep block size 4, L=24, IMCRG simulation

500

500

1000 sweep

1500

2000

0

500

1000 sweep

1500

2000

Figure 3:

Comparing scatter plots of the nearest neighbour block correlation function observable in SU (2) simulations with NT = 2. The standard simulations (no compensation on block level) are shown on the left, the IMCRG simulations on the right.

10

model standard Ising I3 Ising

couplings 0.2216544 (0.128003, 0.051201)

LB 3,5,7,9,13 3,5,7,9,13,17

Table 1:

Block sizes LB and couplings used for the IMCRG simulations of the standard and I3 Ising models. For all runs we used L¡ä = 8.

of critical slowing down obtained from the compensation even on moderate lattice sizes which enabled us to obtain reasonable results with moderate CPU expense.

5.2

Matching of the Two Ising Models

We started by comparing the RG ?ow of the two Ising models. In Table 1 we summarize some parameters of the MC simulations. We made simulations on lattices consisting of 83 blocks of size LB at the in?nite volume critical couplings ¦Â = 0.2216544 [12] for the ordinary Ising and (¦Â1 , ¦Â3 ) = (0.128003, 0.051201) [10] for the I3 model. At each RG step (?xed LB value) we made usually two, three or four IMCRG iterations in order to have the guesses of the e?ective couplings converge to reasonable precision. The number of sweeps in each run ranged from ? 105 for the largest lattice sizes to a ? 6 ¡¤ 106 for the smallest ones. As the ?nal error of the estimate for an e?ective coupling, we took the maximum of the statistical error and the last change of guess in the IMCRG procedure. In Figure 4 we show the results for the ?ow of the two leading ¡ä ¡ä , with increasing block size LB . To achieve matching, , and K2 couplings K1 the block sizes LB of the I3 model have been rescaled by a factor of ¦Ë = 0.59. A rescaling is always needed to obtain matching. The reason for this is that the two models have a di?erent distance to ¡°travel¡± before they meet on a common trajectory, c.f. Figure 1. The ?gure shows that the two ?ows collapse nicely on a single trajectory, indicating that they are approaching a common ?xed point. This happens also with the other 12 couplings not shown in the plot. It turns out that the approach to the ?xed point can be well ?tted by a power law,
¦Ñ ? K¦Á (LB ) = K¦Á + a¦Á ¡¤ L? B .

(25)

11

0.2 0.195 0.19 0.185 0.18 0.175 0.17 0.165 0.16 0.155 0.15 0.145 2 4

flow of K_1

6

0.036 0.034 0.032 0.03 0.028 0.026 0.024 0.022

8 10 L_B flow of K_2

12

14

2

4

6

8 L_B

10

12

14

Figure 4: Flows of nearest and second nearest neighbour couplings in
the standard (diamonds) and the I3 (bars) Ising model with increasing block size LB . In order to obtain matching, the block sizes of the I3 model were rescaled by a factor ¦Ë = 0.59. The dotted lines are ?ts of the ?ows with a power law.

12

¦Á, model 001, 001, 001, 001, 001, Ising Ising I3 I3 combined

LB ¡Ý 3 5 3 5 7 7 5 5 5 5 5 5 5 5

? K¦Á

a¦Á ?0.077(4) ?0.15(7) ?0.240(8) ?0.33(9) ?0.10(3) ?0.27(9) 0.023(2) 0.055(6) 0.015(2) 0.029(2) 0.047(2) 0.059(8) ?0.006(1) ?0.005(3)

¦Ñ 1.37(7) 1.9(3) 1.60(4) 1.80(14) 1.67(20) 1.67 ?x 1.67 ?x 1.67 ?x 1.67 ?x 1.67 ?x 1.67 ?x 1.67 ?x 1.67 ?x

¦Ö2 /dof 0.99 0.01 0.50 0.09 0.17 0.94 0.27 0.15 1.12 1.53 0.38 0.10 0.17

0.1990(5) 0.1977(7) 0.1981(4) 0.1975(5) 0.1979(6) 0.0224(1) 0.0225(2) 0.0013(1) 0.0013(1) ?0.0202(1) ?0.0201(3) 0.00210(4) 0.00215(10)

011, Ising 011, I3 111, Ising 111, I3 002, Ising 002, I3 9, Ising 9, I3

Table 2: Fit results for the Ising model ?ows for a number of 2¨Cpoint
couplings and for the largest 4¨Cpoint coupling K9 . The ?ts were done with Eq. (25). The second column gives the minimum block sizes that ? are the estimates for the ?xed point values. were used in the ?t. K¦Á A ¡°?x¡± after a parameter means that the value was kept ?xed during the ?tting procedure.

? If the ?ows of two models obey such a law, with the same ?xed point K¦Á and exponent ¦Ñ, but di?erent ¡°amplitudes¡± a and a¡ä , the rescale factor ¦Ë to obtain matching is 1 a¡ä ¦Ñ . (26) ¦Ë= a

We always used as a reference trajectory the ?ow of the standard Ising model and rescaled the block sizes of the other models by an appropriate factor. The results of our various power law ?ts of the Ising model are summarized in Table 2. We ?tted the models separately, checking also the e?ect of discarding the e?ective couplings for the smallest block sizes. That the ?xed point value and the exponent of the two models coincide is con?rmed by a 13

NT 1 2 2 2 2

¦Â 0.8730 1.871 1.874 1.877 1.880

LB 2,3,4,5,6,7 3,4,5,6 3,4,5,6 2,3,4,5,6 2,3,4,5,6

L 3 ¡¤ NT 423 ¡¤ 1 363 ¡¤ 2 363 ¡¤ 2 363 ¡¤ 2 363 ¡¤ 2

Table 3: Lattice sizes and values of ¦Â used for the 4d SU (2) model
at ?nite temperature. In the last column the maximum lattice size is shown. For all runs we used L¡ä = 6.

common ?t of the two Ising ?ows, where in the ?t function only the amplitudes a¦Á were allowed to depend on the model. This yields the results quoted in the last two lines of the ?rst block of the table. We then ?xed ¦Ñ = 1.67 and ?tted the ?ows of the non¨Cleading couplings with two parameters (?xed point value and amplitude of power law correction). We found a very nice agreement of the resulting ?xed point values for all couplings. The value of the exponent ¦Ñ turns out to be too big to be identi?ed with the ?rst correction to scaling exponent ¦Ø ¡Ö 0.8. We expected that ¦Ø should be the leading exponent. A possible explanation of the present observation is the following: The amplitude of a power term with ¦Ø as exponent is too small to be detected within our precision. The exponent ¦Ñ with its relatively large amplitudes is due to the presence of a redundant operator of the particular blocking scheme we used.

5.3

Matching of the 4d SU(2) with the Ising Models

We then turned to the 4d SU (2) gauge model at ?nite temperature. Informations on MC simulations made in this case are given in Table 3. We made MC simulations for the NT = 1 and NT = 2 cases on lattices consisting of 63 blocks of size LB . As the critical decon?nement transition value we used the gauge couplings ¦Âc = 0.8730(2) [13] for NT = 1. For NT = 2 we studied a neighbourhood of the critical value ¦Âc = 1.880(3) [14] (see Table 3). For the SU (2) model statistic has necessarily been reduced compared to the Ising models. For NT = 2 measurements ranged from ? 104 for the 14

0.195 0.19 0.185 0.18 0.175 0.17 0.165 0.16 0.155 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 L_B
Figure 5:
Flow of the nearest neighbour coupling in the e?ective action for NT = 1 lattice gauge theory with gauge coupling ¦Â = 0.8730 (squares). Also shown is the standard Ising (diamonds) and and I3 model (bars). The rescaling factor of the gauge block size with respect to the standard Ising scale is ¦Ë = 0.61.

largest sizes up to ? 5 ¡¤ 105 for the smaller ones. ¡ä The K1 leading coupling result for NT = 1 is shown in Figure 5. Also shown is the Ising ?ow. Notice that here and in the ?gures which follow the Ising ?ows are the same as in Figure 4, which also means that the ?t lines plotted are those obtained from the Ising data. Fits were made again according to Eq. (25). Omitting the LB = 2 value, they gave results consistent with the ?ts of the Ising data, both for the exponent and the asymptotic values, however with bigger errors due the lower statistics. The rescaling of the gauge block sizes with respect to the standard Ising scale used in this case is ¦Ë = 0.61. 15

0.2 0.195 0.19 0.185 0.18 0.175 0.17 0.165 0.16 0.155 0.15 0.145 2 4 6

K_1

0.036 0.034 0.032 0.03 0.028 0.026 0.024 0.022

K_2

0

8 L_B K_4

10

12

14

2

4

6

0.003 0.0025 0.002

8 L_B K_9

10

12

14

-0.005

-0.01 0.0015 -0.015 0.001 -0.02 0.0005 0

-0.025

2

4

6

8 L_B

10

12

14

2

4

6

8 L_B

10

12

14

Figure 6:

Flows of four di?erent e?ective couplings for NT = 2 SU (2) lattice gauge theory at ¦Â = 1.877, matching with the two Ising models (bars, diamonds, and ?t lines). The block sizes of the gauge model are rescaled by a factor ¦Ë = 0.65 with respect to the standard Ising scale.

16

¦Á 001 011 111 002 012 112 022 122 9 10 11 12 13 14 stat

LB = 3 0.15962(94) 0.02918(57) 0.00668(73) ?0.00728(94) ?0.00277(46) ?0.00115(42) 0.00042(63) ?0.00030(52) 0.00085(34) 0.00005(20) 0.00026(55) ?0.00013(18) ?0.00020(36) 0.00012(59) 12 ¡¤ 103 [50] [118] [103] [177] [70] [62] [131] [33] [31] [13] [32] [6] [9] [4]

LB = 4 0.17442(52) 0.02800(32) 0.00432(35) ?0.01282(41) ?0.00444(21) ?0.00177(21) ?0.00041(29) 0.00012(21) 0.00146(17) 0.00014(10) ?0.00096(28) ?0.00013(10) ?0.00020(20) 0.00012(33) 27 ¡¤ 103 [34] [4] [24] [62] [12] [5] [59] [42] [20] [20] [70] [9] [15] [5]

LB = 5 0.18333(70) 0.02689(42) 0.00357(45) ?0.01514(49) ?0.00500(25) ?0.00179(26) ?0.00041(39) ?0.00043(30) 0.00182(21) 0.00014(12) 0.00006(33) ?0.00013(12) ?0.00029(23) 0.00012(39) 25 ¡¤ 103 [27] [25] [68] [77] [56] [59] [14] [55] [48] [11] [23] [11] [41] [11]

LB = 6 0.18800(92) 0.02500(57) 0.00300(61) ?0.01691(69) ?0.00505(38) ?0.00164(35) ?0.00077(49) 0.00046(36) 0.00212(28) 0.00020(16) ?0.00040(43) ?0.00030(16) 0.00004(31) 0.00033(53) 20 ¡¤ 103 [48] [58] [80] [139] [35] [53] [77] [110] [45] [3] [8] [10] [66] [95]

Table 4: Values of the e?ective couplings for the 4d SU (2) model at NT = 2, ¦Â = 1.877 for di?erent block size LB . In the bottom row the statistics of the last IMCRG iteration (?xed LB ) is given. Statistical errors are given in parenthesis. Square brackets contain the change of ¡ä in the last IMCRG iteration. the coupling ?K¦Á

The ?ows of the e?ective couplings for NT = 2 at ¦Â = 1.877 are given in Figure 6. One can see a clear matching with the two Ising models (bars and diamonds) and the ?t lines for the ?rst two couplings. The LB ¡¯s of the SU (2) gauge model were rescaled in this case by a factor of ¦Ë = 0.65. We found that, among the di?erent gauge couplings used for NT = 2, the supposed critical coupling ¦Â = 1.880 is actually ruled out, no matter which value of the rescaling parameter ¦Ë is chosen. This can be clearly seen in Figure 7 where the ?ows of the NN coupling in the e?ective action are shown for the four di?erent gauge couplings ¦Â = 1.880, 1.877, 1.874, 1871 and compared with the ?tted curve of the Ising model. For ¦Â = 1.880 the system is de?nitely in the decon?ned phase, whereas the ?ow for ¦Â = 1.871 moves away towards the high temperature ?xed point (con?nement phase). The e?ective coupling values for the best matching 17

0.2 0.19 0.18 0.17 0.16 0.15

K_1

0.14 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 L_B
Figure 7: Flows of the NN coupling in the e?ective action for NT = 2
4d SU (2) at four di?erent gauge couplings ¦Â = 1.880, 1.877, 1.874, 1871 (triangles, diagonal crosses, squares and stars respectively). Also shown is the Ising ?ow (diamonds, bars and ?tted curve). The rescaling of the gauge block sizes with respect to the standard Ising scale is ¦Ë = 0.65.

18

¦Á 001 011 111 002 012 112 022 122 ¡ä K9 ¡ä K10 ¡ä K11 ¡ä K12 ¡ä K13 ¡ä K14 stat

L¡ä = 8 0.19532(20) 0.02307(6) 0.00172(14) ?0.01899(7) ?0.00551(5) ?0.00168(3) ?0.00002(19) 0.00010(3) 0.00195(2) 0.00024(2) ?0.00050(3) ?0.00010(1) ?0.00014(8) ?0.00003(7) 1.3 ¡¤ 106

L¡ä = 6 0.19529(60) 0.02315(13) 0.00192(51) ?0.01946(16) ?0.00558(11) ?0.00168(11) 0.00002(13) 0.00024(29) 0.00191(9) 0.00024(4) ?0.00049(11) ?0.00007(9) ?0.00010(8) ?0.00025(29) 3 ¡¤ 105

Table 5: Comparison of e?ective critical couplings for di?erent sizes of the coarse lattice. The example shown is the ordinary Ising, with LB = 9. In the bottom line the statistics is given.

trajectory of ¦Â = 1.877 are given in Table 4. The less signi?cant values, at LB = 2, have been omitted. We explicitly reported the statistical errors ¡ä and the ?K¦Á variations in the last IMCRG iteration (the latter in square brackets). Let us conclude this analysis with two remarks. First, it is worthwhile to stress that weak ?nite size e?ects are present within this approach: Comparing the 63 block lattice of the gauge model with the 83 of the Ising model should not give sizable systematic errors within our precision. As a check, in Table 5 the e?ective coupling values of the ordinary Ising model are reported for two di?erent block lattice sizes, L¡ä = 8 and L¡ä = 6. The result con?rms that all couplings are consistent within errors. Finally, let us notice that within our statistic the ¦Â = 1.874 ?ow can also be made compatible with the Ising trajectory: A better resolution to 19

discriminate between the two beta values would have required to extend the MC analysis to bigger block sizes LB , of course with much more CPU time consuming. Even though, using the block sizes at our disposal the corresponding ?t is not as good as that of the ¦Â = 1.877 value. Therefore we assume the latter as the critical coupling value for NT = 2, consistently (within one standard deviation) with Ref. [14].

6

Conclusion and Outlook

The discussion of MC results shows that the Svetitsky¨CYa?e conjecture is con?rmed in a very fundamental way by observing matching of the SU (2) RG trajectory with that of the Ising model. At the same time, we showed that IMCRG works well as a method to compute the e?ective action of Ising type degrees of freedom in a genuine non¨CIsing model like 4d ?nite temperature SU (2) gauge theory. Notice also that this kind of calculations could be done on workstations, with relatively small computer resources. An extension to NT greater than two would be interesting but more expensive. The reason is that with increasing temporal size the small LB actions move farther away from the ?xed points, i.e. they need to be blocked more in order to come close to the reference Ising ?ows. This observation is in agreement with the fact that also in more standard approaches, e.g. via the Binder cumulant, the spatial size of the lattice has to be increased very much with increasing NT . Finally, it would be of interest to check this approach with di?erent blocking prescriptions. The rate of approaching the RG ?xed point is in fact very sensitive to the blocking rule used and a faster convergence can in principle be obtained using a more sophisticated blocking scheme than the majority rule.

Acknowledgements
We would like to thank M. Caselle for interesting exchange of opinions. Access to CPU resources of the Torino I.N.F.N. is gratefully acknowledged.

20

References
[1] K. G. Wilson and J. Kogut, Phys. Rep. 12 (1974) 76. [2] W. Bock and J. Kuti, Phys. Lett. B 367 (1996) 242. [3] A.P. Gottlob, M. Hasenbusch, and K. Pinn, Phys. Rev. D 54 (1996) 1736. [4] B. Svetitsky and L. G. Ya?e, Nucl. Phys. B 210[FS6] (1982) 423. [5] J. Engels, J. Fingberg, and M. Weber, Nucl. Phys. B 332 (1990) 737; J. Engels, J. Fingberg, and D. E. Miller, Nucl. Phys. B 387 (1992) 501. [6] M. Bill? o, M. Caselle, A. D¡¯Adda, and S. Panzeri, Int. J. Mod. Phys. A 12 (1997) 1783. [7] R. Gupta and R. Cordery, Phys. Lett. A 105 (1984) 415. [8] B. Svetitsky, Phys. Rep. 132 (1986) 1. [9] M. Hasenbusch and K. Pinn, J. Phys. A 30 (1997) 63. [10] H.W.J. Bl¡§ ote, E. Luijten and J.R. Heringa, J. Phys. A 28 (1995) 6289. [11] A. Kennedy and B. Pendleton, Phys. Lett. B 156 (1985) 393. [12] A.L. Talapov and H.W.J. Bl¡§ ote, J. Phys. A 29 (1996) 5727. [13] R. Ben-Av, H.G. Evertz, M. Marcu, and S. Solomon, Phys. Rev. D 44 (1991) 2953. [14] J. Fingberg, U. Heller, and F. Karsch, Nucl. Phys. B 392 (1993) 493.

21

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