Critical exponents from seven-loop strong-coupling φ 4 theory in three dimensions
Critical exponents from seven-loop strong-coupling φ 4 theory in three dimensions
复制标题
三维七环强耦合 φ 4 理论的临界指数
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发表时间：
1998
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通讯作者：
H. Kleinert
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作者：
H. Kleinert
场论临界指数的精确计算提出了一个理论上的挑战，因为相关的信息只能从不同的幂级数展开。这些结果也具有实际意义，因为它们预测了许多二阶相变的许多可能的未来实验的结果。在最近的工作@1#，我们已经开发了一种新的方法，通过强耦合理论的标量场与f 4相互作用，从这样的扩展提取这些指数。假设这些域有n个分量，其动作是O（n）对称的。作为一个应用，我们利用三维空间中重整化常数的六圈微扰展开式@2 - 4 #计算了所有O（n）普适类的临界指数，并获得了高精度.强耦合理论也适用于42 e维@6，并且能够在非线性s模型的42 e和21 e维之间进行插值。本文的目的是利用临界指数n和h在8 #下的新的七圈展开系数，以及最重要的是，应用比以前更强的无穷阶外推方法，显著地提高我们以前在3维1 #下的结果的精度.后者使我们的结果一样准确的Guida和Zinn-Justin @ 9 #通过一个更复杂的remation技术的基础上解析映射和Borel变换，其中还考虑到信息的大规模增长的展开系数。我们达到这个准确性，而不使用的信息，我们将证明在结束时，在第二节。实际上对结果没有影响，除了稍微降低v小于0.2%）。在我们的方法中的大顺序信息的重要性不大的原因是，临界指数是从无限裸耦合的膨胀的评价。另一方面，关于高阶行为的信息指定了从复耦合常数平面@10#的原点开始的左侧切割的尖端处的不连续性。这与无限耦合极限相差太远，因此没有相关性。在裸耦合常数的幂次膨胀的再膨胀方案中，接近标度v的临界指数起着重要作用，其通过相同方案的精确计算对于获得所有其他临界指数的高精度是至关重要的。它是由重整化耦合强度g在强耦合极限中与常数g* 相反的条件决定的。知识的v是更产生比大的顺序信息在以前的remation计划，其中的临界指数被确定为一个函数的重整化耦合常数g附近的g*，这是为了单位，从而躺在一个有限的距离，从左手切在复杂的g平面。虽然这些测定对切割顶部的不连续性很敏感，但必须认识到，由于领先瞬子的逸度很小，切割的影响非常小，它带有玻尔兹曼因子e 2const/ g。我们简单地回顾一下重整化耦合g[g/m]在裸耦合g <$0 [g 0/m对于所有O（n）的可用展开@4#，
The accurate calculation of critical exponents from field theory presents a theoretical challenge, since the relevant information is available only from divergent power series expansions. The results are also of practical relevance, since they predict the outcome of many possible future experiments on many second-order phase transitions. In recent work @1# we have developed a novel method for extracting these exponents from such expansions via a strong-coupling theory of scalar fields with a f 4 interaction. The fields are assumed to have n components with an action which is O(n) symmetric. As an application, we have used available sixloop perturbation expansions of the renormalization constants in three dimensions @2‐4# to calculate the critical exponents for all O( n) universality classes with high precision. Strong-coupling theory works also in 42e dimensions @6#, and is capable of interpolating between the expansions in 42e with those in 21e dimensions of the nonlinear s model @7#. The purpose of this note is to improve significantly the accuracy of our earlier results in three dimensions @1# by making use of new seven-loop expansion coefficients for the critical exponents n and h @8# and, most importantly, by applying a more powerful extrapolation method to infinite order than before. The latter makes our results as accurate as those obtained by Guida and Zinn-Justin @9# via a more sophisticated resummation technique based on analytic mapping and Borel transformations, which in addition takes into account information on the large-order growth of the expansion coefficients. We reach this accuracy without using that information which, as we shall demonstrate at the end in Sec. V, has practically no influence on the results, except for lowering v slightly ~by less than ;0.2%). The reason for the little importance of the large-order information in our approach is that the critical exponents are obtained from evaluations of expansions at infinite bare couplings. The information on the large-order behavior, on the other hand, specifies the discontinuity at the tip of the left-hand cut which starts at the origin of the complex-coupling constant plane @10#. This is too far from the infinite-coupling limit to be of relevance. In our resummation scheme for expansion in powers of the bare coupling constant, an important role is played by the critical exponent of approach to scaling v, whose precise calculation by the same scheme is crucial for obtaining high accuracies in all other critical exponents. It is determined by the condition that the renormalized coupling strength g goes against a constant g* in the strong-coupling limit. The knowledge of v is more yielding than the large-order information in previous resummation schemes in which the critical exponents are determined as a function of the renormalized coupling constant g near g* which is of order unity, thus lying a finite distance away from the left-hand cut in the complex g plane. Although these determinations are sensitive to the discontinuity at the top of the cut, it must be realized that the influence of the cut is very small due to the smallness of the fugacity of the leading instanton, which carries a Boltzmann factor e 2const/ g . We briefly recall the available expansions @4# of the renormalized coupling g[g/m in terms of the bare coupling g ¯ 0[g0 /m for all O( n),