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This is a \cut-down" version of the paper, ontaining all of the text but only some of the (cid:12)gures. The full set of (cid:12)gures is available via anonymous ftp from suna.amtp.liv.ac.ukas /pub/pss/fullpaper.ps.Z . Login as `ftp'and send your email address as password. The (cid:12)gures are available as compressed PostScript (cid:12)les. The full LaT X source is also there. Don't forget to set the transfer mode to binary! E Paddy Spencer. 4 9 9 1 n a J 2 1 1 1 0 1 0 4 9 / t a l - p e h Liverpool Preprint: LTH 328 December 22, 1993 Topography of the cooled O(3) vacuum `\and what is the use of a book," thought Alice, \without pictures:::?"' [1]. P.S. Spencer and C. Michael DAMTP, University of Liverpool, Liverpool L69 3BX, UK Abstract Weinvestigatethetopography oftheO(3)vacuumaftervariousdegreesofunder-relaxed cooling,usingthe na(cid:127)(cid:16)veactiondensity andgeometricand(cid:12)eld theoretical de(cid:12)nitionsof the topologicalcharge density. The results are presented graphically. 1 Introduction The O(3) model has been widely studied [2] in 2 dimensions because of its similarities to non-Abelian gauge theories in 4 dimensions: it is asymptotically free, becomes non- perturbative in the infra-red regime and has a non-trivial topology due to the homotopy class (cid:25)2(S2) = Zof windings from S2 ! S2 . The O(3) model is well understood in some respects|for instance anexactexpression forthe massgapis known[3]. Itis thusanatural candidate for a theory which one can try and understand in terms of the vacuum structure. In this paper we explore the nature of the vacuum via numerical simulation in Euclidean time. In order to focus on important features in the vacuum, we use a procedure to smooth out local (cid:13)uctuations. A cooling algorithm has been used previously to achieve this. The basic idea is that a local smoothing of the (cid:12)elds should preserve long range features such as instantons. We explore this by varying the details of the cooling procedure and checking whether the resulting topological charge distribution is universal. The main result we present in this paper is a series of (cid:12)gures showing the action and topological charge densities after cooling. In another paper [4], we analyse these distributions in terms of models|for instance the distributions of sizes of topological objects (instantons and anti-instantons) were calculated and compared with a distribution relation derived from a modi(cid:12)cation of Coleman's argument for the SU(2) case [5]. The paper is organised as follows: in section 2 we outline the formalism of the O(3) model, givetwode(cid:12)nitions ofthelatticetopologicalcharge,anddiscusstheirsuitability. We also give formulae relevant to single-instanton solutions. In section 3 we detail the cooling 1 algorithm we used and discuss the di(cid:11)erent e(cid:11)ects that di(cid:11)erent degrees of cooling have upon an initial, uncooled con(cid:12)guration. We then discuss apossible criterion forquantifying the degree of cooling required to remove the short-range (cid:13)uctuations. Finally, in section 5 we show examples of the vacuum obtained from di(cid:11)erent initial con(cid:12)gurations, using the method given in section 3. 2 The Model. The continuum 2d-O(3) Euclidean action SE is de(cid:12)ned as: Z 1 2 2 SE = 2 d x(@(cid:22)(cid:30)) (1) 2g 2 with (cid:30) a 3-component vector and the constraint (cid:30) = 1. On the lattice the simplest discretization gives: 1 X X 2 SL = 2 ((cid:30)x+(cid:22)(cid:0)(cid:30)x) (2) 2g x (cid:22) 1 X = 2 (2(cid:0)(cid:30)F(x)(cid:1)(cid:30)x) g x with (cid:30)F de(cid:12)ned by: X (cid:30)F(x)= (cid:30)x+(cid:22): (3) (cid:22) T The continuum topological charge Q is given by: Z T 1 2 Qf = "(cid:22)(cid:23)"ijk(cid:30)i@(cid:22)(cid:30)j@(cid:23)(cid:30)kd x (4) 8(cid:25) and is an integer. However, if this `(cid:12)eld theoretical' de(cid:12)nition is na(cid:127)(cid:16)vely discretised, it returns non-integer values and acquires a renormalisation factor [6]. T There is also a`geometric' lattice de(cid:12)nition ofQ based on the mapping S2 ! S2 of the O(3)(cid:12)elds tothelattice trianglesformedbythe(cid:12)elds aroundaplaquette. Thecontribution (cid:3) to the total charge from a site x on the dual lattice is given by [7{9]: T (cid:3) 1 Qg(x ) = (((cid:27)A)((cid:30)1;(cid:30)2;(cid:30)3)+((cid:27)A)((cid:30)3;(cid:30)4;(cid:30)1)) (5) 4(cid:25) where (cid:27)((cid:30)1;(cid:30)2;(cid:30)3)= sign((cid:30)1(cid:1)(cid:30)2(cid:2)(cid:30)3) ; (6) A((cid:30)1;(cid:30)2;(cid:30)3) is the area of the spherical triangle on the unit sphere mapped out by the (cid:30) (cid:12)elds, and the subscripts 1;2;3;4refer to the corners of a plaquette labelled anti-clockwise, as in (cid:12)g. 1. Obviously, there is a choice of triangulation, and for con(cid:12)gurations with small action the twoequivalent possibilities give equal contributions tothe topological charge but for other con(cid:12)gurations the di(cid:11)erence in the contribution can be (cid:6)1. There are arguments against the use of such geometric de(cid:12)nitions, particularly in cal- T N(cid:0)1 culations of the topological susceptibility, (cid:31) , in CP theories with small values of N. These argumentsarise because these lattice models are plagued by dislocations: unphysical T con(cid:12)gurations that contribute to (cid:31) . Campostrini et al have shown [10] that for values of 2 T N = 10 and larger, the geometric formulation gives a sensible de(cid:12)nition of (cid:31) , but fails to do so for lower values of N. When (cid:30), which is purely real, is written in terms of the complex projective (cid:12)elds !, !: !+! !(cid:0)! !!(cid:0)1 (cid:30)1 = ; (cid:30)2 = ; (cid:30)1 = (7) !!+1 i(!!+1) !!+1 (cid:30)1+i(cid:30)2 ! = 1(cid:0)(cid:30)3 the continuum action and topological charge densities are seen to be closely related to one another [11]. Speci(cid:12)cally, in the case of (anti-)instanton solutions, a (cid:12)eld con(cid:12)guration representing a single instanton of size (cid:26) at position r can be constructed via (cid:26) ! = ; with z = x+it: (8) z(cid:0)r An instanton so constructed is circular in the continuum, but, being an extended object, is only circular on the lattice if (cid:26) is small compared to L, the dimension of the lattice. With (cid:30) and ! as in eqs. 7 and 8 above, the action and topological charge densities become: 2 (cid:26) S(z); Q(z)/ : (9) 2 2 2 ((cid:26) + j z(cid:0)r j ) A similar relation holds for multi-instanton con(cid:12)gurations. It should be noted that, for a general (cid:12)eld con(cid:12)guration, the action is positive de(cid:12)nite, but the topological charge is not, having contributions from both anti- and instantons for which the charge has opposite signs. A single (anti-)instanton solution has continuum action 4(cid:25) SI = 2 (10) g In general a multi instanton{anti-instanton solution has action 4(cid:25) T S (cid:21) jQ j (11) 2 g 3 Cooling. Our aim is to study the topography|the distribution of topological charge in a con(cid:12)guration. As can be seen from (cid:12)g. 2, little if anything can be deduced about a con(cid:12)guration's 2 topography from an uncooled con(cid:12)guration|a thermalised, i.e. simulated at non-zero g , con(cid:12)guration contains too many short-range (cid:13)uctuations, and the longer scale structure is obscured. Cooling is a local update procedure, and as such is used with the intention of removing the short scale (cid:13)uctuations while leaving intact the larger scale structure of interest. We employed an under-relaxed cooling: (cid:30)x ! (cid:11)(cid:30)x+(cid:30)F(x) (12) 2 (normalised to (cid:30)x = 1 and with (cid:30)F(x) as in eq. 3 above). The cooling usually performed 2 (as in e.g. [6]) is a Metropolis-style accept-reject cooling at g = 0, equivalent to only 3 accepting new (cid:12)eld values that lower the action. Cooling using (cid:11) = 0 in eq.12 is equivalent to a Heatbath algorithm, or a Metropolis update where each site is updated very many times before moving on tothe next site. Taking non-zero (cid:11) has, forour purposes, a number of advantages: (cid:12)rstly, our implementation is deterministic which is computationally less demanding; secondly, the parameter (cid:11), together with the number of cooling sweeps used, allows the nature ofthe cooling performed to be controlled. Provided the cooling algorithm is implemented sensibly, a higher (cid:11)-value requires a larger number of cooling sweeps to reach the same N-sector as can be reached by fewer sweeps at a lower (cid:11)-value. Ideally a checkerboard approach should be adopted, as this updates only non-interacting sites on eachsweep andis thusmaximally insensitive tothe orderin which (cid:12)eld variables arecooled. 2 2 We work on a 64 lattice at g = 0:8 using the na(cid:127)(cid:16)ve discretisation of the action given above in eq. 2. Con(cid:12)gurations are separated by 1000 Monte Carlo sweeps, each consisting 2 of 1 heatbath and 9 over-relaxation updates. The correlation length at this value of g , we measure to be (cid:24) = 3:82(3). Studies at other correlation lengths will be presented elsewhere [4]. Some examples of the cooled vacuum are shown in section 5. It is convenient to de(cid:12)ne X Nf;g = jQf;g(x)j (13) x Generally, di(cid:11)erent degrees of cooling will result in di(cid:11)erent stabilities versus number of cooling sweeps for the di(cid:11)erent N-sectors, as can be seen from (cid:12)g. 3. The more gentle cooling, thatwithalarger(cid:11), will tendtoisolatethevarious N-sectorsforlonger. Plotssuch as (cid:12)g. 3 have appeared before in the literature and the plateaux were described by multi- instanton con(cid:12)gurations; the jumps between the plateaux were ascribed to the annihilation ofan(anti-)instanton bythecooling process. This canbe observedby following thehistory ofthecon(cid:12)gurationin computertimeasitiscooled. Whathappensisthetopological object is shrunk by the cooling process from its uncooled size, and eventually, under prolonged cooling, it is reduced to a point and then destroyed, at which stage in the cooling a jump occurs in N. Wenowinvestigatewhethervariations in thecooling algorithmgivedi(cid:11)erenttopological distributions. It is necessary to look beyond plots such as (cid:12)g. 3 to study this. Indeed, each N-sector is not constructed uniquely from instanton{anti-instanton solutions|a con(cid:12)guration with two instantons and one anti-instanton will have the same value of N as one with one instanton and two anti-instantons, but surely lies in a di(cid:11)erent topological sector. Similarly, two di(cid:11)erent cooling processes need not lead to the same topological sector. We have observed con(cid:12)gurations which, when cooled under di(cid:11)ering degrees of cooling, lead to (cid:12)nal con(cid:12)gurations where the topological objects removed by the cooling process are not the same for the di(cid:11)erent coolings, and consequently, although N is identical for these resultant (cid:12)eld con(cid:12)gurations, they may lie in di(cid:11)erent topological sectors. There are further consequences to this. Prolonged cooling of a con(cid:12)guration should be avoided, as the structure remaining after a large amount of cooling is more an artefact of the cooling process than any large-scale physics that was present before the cooling. As a criterion for how much cooling should be performed to remove the short-range (cid:13)uctuations without adversely a(cid:11)ecting the physics present in a con(cid:12)guration, we investigated the amount of cooling required to annihilate single-instanton con(cid:12)gurations generated via eq. 8. Some results are shown in (cid:12)g. 4. As can be seen there is a linear relation between the value of (cid:11) and the number of sweeps needed to remove an instanton of a given size. This is equivalent to cooling until a particular value of S=(VSI) is attained, where V is 4 the volume of the lattice and the bar denotes average taken over con(cid:12)gurations. It appears to be the case that, whilst matching the number of cooling sweeps to the value of (cid:11) in this way leads to broadly the same (cid:12)nal con(cid:12)guration, the con(cid:12)gurations become (visually) much more alike if more sweeps of a gentler cooling is used. For example, (cid:12)gs. 5a to 5e are the (cid:12)nal con(cid:12)gurations for the same initial con(cid:12)guration where the amount and degree of cooling have been chosen from relationships similar to (cid:12)g. 4. In this case the criterion was 2 that an instanton of size (cid:26) = 2 on a 64 lattice be annihilated by the cooling|equivalent 2 to S=(VSI) = 0:0017(3) calculated on 145 con(cid:12)gurations. For this value of g , the various degrees of cooling employed appear to produce very nearly identical end con(cid:12)gurations, for non-zero(cid:11). Asouraimistodisturbthelarge-scale structuremuchlessthantheshort-range (cid:13)uctuations, we therefore chose to cool subsequent con(cid:12)gurations at (cid:11) = 2 for 90 sweeps, as the di(cid:11)erences between (cid:12)gs. 5a and 5b are much more signi(cid:12)cant than those between (cid:12)g. 5b and the rest; clearly fewer sweeps are desirable in order to minimise the extent to which the long-range objects are a(cid:11)ected. It has been suggested in [12] that, particularly in gauge theories, lattice instantons are unstableundercoolingbecause thelatticeinstantonactionissmallerthaninthecontinuum. They propose a so-called \over-improved" action that stabilises the instantons. Certainly L cont our O(3) lattice instantons have S (cid:20) S , and are shrunk by the cooling process. We I I concentrateonrelatively largetopological objectsunderrelatively mild cooling soweexpect little bias. It would be interesting to explore the use of over-improved actions to improve the stability of instantons under cooling. Another approach, the \perfect" action, has been proposed [13] and this would allow instanton e(cid:11)ects to be studied closer to the continuum limit. 4 Conclusions Fig. 6 shows the absolute topological charge, calculated using both the geometric and (cid:12)eld theoretic de(cid:12)nitions and the normalised action, as one con(cid:12)guration is cooled at (cid:11) = 2, and (cid:12)g. 7 shows S=SI and Qf;g for the same con(cid:12)guration. It is interesting to note, in the light of the discussion in section 2, that after a certain amount of cooling, Ng and S=SI track T each other much more closely than Nf. This relation between S and Q will be useful in T calculations in SU(N) gauge theories, where the de(cid:12)nition of Q is non-trivial and any L calculation of the topological charge is much more computationally intensive than a similar calculation of SL. Asademonstrationofpoints madeabove, wepresent(cid:12)gs.8a{8l. Theseshowaction and geometric topological charge densities for di(cid:11)erent vacuum con(cid:12)gurations, each generated by cooling using (cid:11)= 2, for 90 sweeps. This combination was chosen to match the criterion de(cid:12)ned in section 3above. Thecriterion herewasagaintoannihilate a (cid:26)= 2instantonona 64(cid:2)64 lattice|equivalent to S=(VSI) = 0:0017(3)as stated earlier. The close relationship between S(x;t) and Qg(x;t) is readily apparent, and is shown more clearly in (cid:12)g. 9 which shows contour plots of the last vacuum example. In this paper we show that cooling con(cid:12)gurations can give reliable information about the O(3) vacuum topography. Moreover, after such cooling the topological and action densities trackeach other closely sothatthere is su(cid:14)cient information contained within the action density for topological calculations that do not depend on the sign of the topological objects. In particular, a study of the topological susceptibility is plagued by the di(cid:14)culty 5 of determining the contribution of small-size topological objects on a lattice. By looking at the full distribution itself, it is possible to focus on those properties which should be well determined by traditional lattice methods|namely the distribution of instantons above a cetain size. We return to this in [4]. References [1] Alice's Adventures in Wonderland, L. Carrol. [2] For a review see V.A.Novikov et al, Phys. Rep. 116, 103, (1984). [3] P. Hasenfratz, P. Maggiore, F. Niedermayer, Phys. Lett. B265, 522 (1990) [4] O(3) Instanton Size Distribution C. Michael & P.S. Spencer, in preparation. [5] The Uses of Instantons,S. Coleman, lectures given at Ettore Majorana, 1977. [6] Topological Susceptibility on the Lattice: the 2d O(3) Sigma Model, A. Di Giacomo et al, CERN preprint TH-6382/92. [7] M. Lu(cid:127)scher, Nuc. Phys. B200, 61, (1982). [8] B. Berg, M. Lu(cid:127)scher, Nucl. Phys. B190, 365, (1981). [9] B. Berg, Phys. Lett. 104B, 475, (1981). N(cid:0)1 [10] Topological susceptibility and string tension in the lattice CP models, M. Cam- postrini et al, Pisa preprint IFUP-TH 34/92. [11] E. Mottola & A. Wipf, Phys. Rev. D 39, 588, (1989). [12] Instantons from over-improved cooling, Margarita Garcia Perez, Antonio Gonzalez- Arroyo, Jeroen Snippe and Pierre van Baal, preprint INLO-PUB-11/93, FTUAM-93/31. [13] Perfect Lattice Action For Asymptotically Free Theories P. Hasenfratz and F. Nieder- mayer, BUTP-93/17. 6 5 (cid:11) = 0 (cid:11) = 2 4 (cid:11) = 4 3 S=SI 2 1 0 0 500 1000 1500 2000 2500 3000 3500 4000 # cooling sweeps Figure 3: S=SI, for a con(cid:12)guration as it is cooled with (cid:11) = 0; 2; 4. 8 c 13000 c 12000 (cid:26)= 6 3 (cid:26)= 10 2 (cid:26)= 16 4 4 11000 (cid:26)= 28 c c 4 10000 c 4 2 9000 c 2 4 8000 c 2 4 # 7000 2 c 4 3 2 6000 3 c 4 2 3 5000 4 2 3 c 3 4000 2 4 3 c 2 3 3000 4 3 c 2 3 2000 4 2 3 1000 3 0 1 2 3 4 5 6 7 8 9 10 (cid:11) Figure 4: The number of cooling sweeps required to annihilate a single, size (cid:26) (as de(cid:12)ned in eq. 8) instanton con(cid:12)guration at a given value of (cid:11). Data calculated on a 64(cid:2)64 lattice with the instanton positioned at the centre of the lattice. 9 5 S=SI Ng 4 Nf 3 S=SI; Nf;g 2 1 0 0 250 500 750 1000 1250 1500 1750 2000 2250 # cooling sweeps Figure 6: Nf;g as de(cid:12)ned in eq. 13, and the normalised action S=SI. These results were obtained by cooling with (cid:11) = 2. 5 S=SI 4 Qg Qf 3 2 S=SI; Qf;g 1 0 -1 -2 0 250 500 750 1000 1250 1500 1750 2000 2250 # cooling sweeps Figure 7: Qf;g and the normalised action S=SI. These results were obtained by the same process that generated (cid:12)g. 6. 13