Scaling and asymptotic scaling in
2차원 CPN−1 모델은 네 차원의 SU(N) 게이지 이론과 동적 특성이 공유하는 동시에 수치적으로 탐구하기 쉬운 모델이다. 따라서 라군 QCD에 사용할 알고리즘 및 방법을 테스트하고 정확한 이론적 예측과 비교하기 위한 좋은 테스트베드는 될 수 있다.
본 연구에서는 CPN−1 모델의 스케일링, 비아스키밍, 그리고 유한 크기 스케일링 특성을 조사했다.
먼저, Wall-wall 상호관계 함수를 통해 스케일링을 확인했다. 두 개의 상호관계 길이(ξw와 ξdw)를 정의하고, 그들의 비(ξdw/ξw)를 플롯하여 회전 불변성을 검증했다. 결과는 회전 불변성을 보인다.
또한, 새로운 상호관계 함수(ξ2G)로 스케일링을 확인했다. 스케일링이 만족되는 값을 찾은 후, 그 다음 값으로 스케일링이 얼마나 잘되느냐를 비교하였다. 결과는 모든 값에서 스케일링이 잘 보였다.
비아스키밍을 연구하기 위해, 두 개의 비아스키밍을 정의하고(β와 βE), 비아스키밍에 대한 분석을 수행했다. 결과는 기존의 β를 사용한 경우 비아스키밍 불량과 다른 라군 액션 간의 불일치를 보인 반면, 새로운 βE를 사용한 경우가 훨씬 더 좋은 비아스키밍 결과를 보였다.
유한 크기 스케일링을 연구하기 위해, 유한 크기 스케일링 함수(fO(L/ξ))를 정의하고, 그에 대한 분석을 수행했다. CP1 모델에서는 기존의 예측과 일치하였다. 대조적으로, N이 커지는 경우에는 유한 크기 효과가 달라질 것이라는 기대와 일치한다.
한글 요약 끝.
Scaling and asymptotic scaling in
arXiv:hep-lat/9210010v1 8 Oct 1992IFUP-TH 46/92Scaling and asymptotic scaling intwo-dimensional CPN−1 models ∗Massimo Campostrini, Paolo Rossi, and Ettore VicariIstituto Nazionale di Fisica Nucleare, Sezione di Pisa,Dipartimento di Fisica dell’Universit`a, I-56126 Pisa, Italy.October 6th, 1992AbstractTwo-dimensional CPN−1 models are investigated by Monte Carlo meth-ods on the lattice, for values of N ranging from 2 to 21.Scaling androtation invariance are studied by comparing different definitions of cor-relation length ξ. Several lattice formulations are compared and shown toenjoy scaling for ξ as small as 2.5.Asymptotic scaling is investigated using as bare coupling constant boththe usual β and βE (related to the internal energy); the latter is shown toimprove asymptotic scaling properties.
Studies of finite size effects showtheir N-dependence to be highly non-trivial, due to the increasing radiusof the ¯zz bound states at large N.1IntroductionTwo-dimensional CPN−1 models share many dynamical features with four-dimensional SU(N) gauge theories, but are much easier to explore numericallyand are amenable to analytical analysis by means of the 1/N expansion. CPN−1models can therefore be useful as a test-bed for algorithms and numerical meth-ods to be applied to lattice QCD, since systematic errors can be explored indetails and numerical results can be compared with precise theoretical predic-tions.
Indeed several studies of CPN−1 and related models have been presentedin this Conference.In this paper we summarize studies of the scaling, asymptotic scaling, andfinite-size scaling properties of different lattice formulations of CPN−1 models.∗Talk presented at the Lattice ’92 Conference, Amsterdam.1
The full details of the simulations and analyses of different aspects of the modelscan be found in Refs. [1, 2].Many recent lattice studies of CPN−1 models used the “quartic” actionS1 = −NβXn,µ|¯zn+µzn|2 .
(1)We performed most of our simulations using the “gauge” actionSg = −NβXn,µ¯zn+µλn,µzn + ¯zn¯λn,µzn+µ. (2)zn is an N-component complex scalar field constrained by the condition ¯znzn =1, and λn,µ is a U(1) gauge field.
Sg enjoys several advantages over S1, themost important being a better scaling behavior. We also considered the tree-level Symanzik-improved versions of both actions, which will be denoted bySSym1and SSymg.We performed Monte Carlo simulations of the CP1, CP3, CP9, and CP20models, for values of the correlation length ξ ranging from 2.5 to 30.2Correlation Length and ScalingThe most interesting correlation function of the model selects the SU(N)-adjoint,gauge-neutral channel:GP (x) = ⟨Tr{P(x)P(0)}⟩conn ,Pij(x) = ¯zi(x)zj(x).
(3)We defined the usual wall-wall correlation length ξw from the exponential de-cay of the zero-momentum projection of GP . In order to check for rotationinvariance, we also defined the diagonal wall-wall correlation length ξdw in theobvious way.
Both ξw and ξdw tend to the inverse mass gap in the scaling limit.The ratio ξdw/ξw is plotted in Figs. 1a, 2a, 3, and 4 for different values of Nand different actions, showing rotation invariance even for the smallest valuesof ξ.We also chose a different definition of correlation length:ξ2G =14 sin2 π/L" eGP (0, 0)eGP (0, 1)−1#,(4)where eGP (p1, p2) is the Fourier transform of GP ; eGP (0, 0) is the magnetic sus-ceptibility χ.
In the scaling limit, ξG tends to the second moment of the cor-relation function. The physical quantity ξG/ξw must be independent of β andon the choice of the action in the scaling region.
ξG/ξw is plotted in Figs. 1b,2b, 3, and 4 for different values of N and different actions, showing scaling forall the values of ξ.
In the following we will choose ξG as length scale, since it isless noisy then ξw or ξdw.2
3Asymptotic ScalingWe compared the β-dependence of the lattice mass scale 1/ξG with the two-loop perturbative formula f(β) = (2πβ)2/N exp(−2πβ), by considering the ratioMG/Λlatt ≡[ξGf(β)]−1. In the asymptotic regime, MG/Λlatt must be constantplus corrections O(1/β); the β →∞limit may therefore be approached veryslowly.
In order to compare different lattice formulations, we reexpressed thespecific Λlatt of each formulation in terms Λg.We used two different schemes for the analysis: in the “standard” schemethe inverse bare coupling is simply β, the action parameter. In the “energy”scheme, suggested by G. Parisi [3], the inverse bare coupling is βE ≡1/(2E);E is the expectation value of the link action, normalized to E(β = ∞) = 0.
Atthe perturbative level, β and βE are simply two different definitions of couplingconstant. We reexpressed Λlatt in the “energy” scheme in terms of Λg in the“standard” scheme.The results for CP1, CP3, and CP9, are presented in Figs.
5–7. Resultsfor the “standard” scheme show asymptotic scaling violations and disagreementbetween different lattice actions, but both effects are compatible with the ex-pected 1/β behavior.
On the other hand, results for the “energy” scheme areremarkably β- and action-independent. In the case of CP1, the “energy” schemeresult is in agreement with the exact result of Ref.
[4]:MGΛg= 8e√32 expπ4∼= 36.51 . (5)4Finite Size ScalingWe examined finite size scaling effects by studying the finite size scaling functionof an observable O: fO(L/ξ) ≡OL(β)/O∞(β), where OL is the expectationvalue of the observable O measured on a L×L lattice.
In the scaling region, fOmust be a universal function of L/ξ, independent of β and of the lattice action.We present here results for the magnetic susceptibility χ; fξ has a behavior verysimilar to fχ.We expect a qualitative difference in the finite size behavior of CPN−1 modelsat small and large N. Finite size effects for CP1 are expected to be dominatedby the inverse mass gap, therefore following the predictions of Ref. [5]: fO =1+O(exp{−L/ξ}).
fχ is plotted in Fig. 8, and it is in agreement with the aboveprediction.No clear theoretical predictions are available for the CP3 model.
fχ is plottedin Fig. 9.The large-N expansion predicts a radius of the lowest-lying state propor-tional to ξN 1/3 [6].
Therefore we expect, for N large enough, finite size effectsto be dominated by the bound state radius rather then by the inverse mass gap.Quantities like ξ and χ are expected to increase when L is decreased, in contrast3
with the CP1 case. Moreover the function fOL/(ξN 1/3)should not dependon N (at least for N<∼100; for huge values of N, the bound state decouples fromχ and ξG).
Fig. 10 shows fχ plotted as a function of L/(ξN 1/3), for both CP9and CP20, and it is in agreement with the large-N expansion.References[1] M. Campostrini, P. Rossi, and E. Vicari, Phys.
Rev. D46 (1992) 2647.
[2] M. Campostrini, P. Rossi, and E. Vicari, Pisa preprint IFUP-TH 34/92,Phys. Rev.
D, in press. [3] G. Parisi, in High Energy Physics—1980, Proceedings of the XXth Confer-ence, Madison, Wisconsin, 1980, AIP Conf.
Proc. No.
68 (AIP, New York,1981). [4] P. Hasenfratz, M. Maggiore, and F. Niedermayer, Phys.
Lett. B245 (1990)522.
[5] M. L¨uscher, Phys. Lett.
118B (1982) 391. [6] E. Witten, Nucl.
Phys. B149 (1979) 285.4
Figure captionsFig. 1: Scaling tests for CP1.Fig.
2: Scaling tests for CP3.Fig. 3: Scaling tests for CP9.Fig.
4: Scaling tests for CP20.Fig. 5: Asymptotic scaling tests for CP1.Fig.
6: Asymptotic scaling tests for CP3.Fig. 7: Asymptotic scaling tests for CP9.Fig.
8: Finite size scaling for CP1.Fig. 9: Finite size scaling for CP3.Fig.
10: Finite size scaling for CP9 and CP20.5
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