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dvi ?le made on February 1,2008Angular Momentum Distribution Function of the Laughlin Droplet Sami Mitra and A.H.MacDonald Department of Physics,Indiana University,Bloomington IN 47405Abstract We have evaluated the angular-momentum distribution functions for ?nite numbers of electrons in Laughlin states.For very small numbers of elec-trons the angular-momentum state occupation numbers have been evaluated exactly while for larger numbers of electrons they have been obtained from Monte-Carlo estimates of the one-particle density matrix.An exact relation-ship,valid for any number of electrons,has been derived for the ratio of the occupation numbers of the two outermost orbitals of the Laughlin droplet and is used to test the accuracy of the Monte-Carlo calculations.We compare the occupation numbers near the outer edges of the droplets with predictions based on the chiral Luttinger liquid picture of Laughlin state edges and dis-
cuss the surprisingly large oscillations in occupation numbers which occur for
angular momenta far from the edge.
PACS numbers:73.20Dx,73.d0.Mf
Typeset using REVT E X
In the strong magnetic?eld limit all electrons in a two dimensional electron gas occupy only the single-particle orbitals which have the minimum quantized kinetic energy,those of the lowest Landau level.In the symmetric gauge,appropriate for a?nite droplet of electrons with circular symmetry,the lowest Landau level orbitals[1]are labeled by angular momentum:φm(z)=(2πm!2m)?1/2z m exp(?|z|2/4),m=0,1,2,...where z i=x i+i y i. (We use the magnetic length,?≡(ˉh c/eB)1/2,as the unit of length.)The orbital with
√
angular momentum m has its radial coordinate localized within～?of R m=
is the ground state when the droplet occupies[6–8]an area～MA MDD
N
.The M=1Laughlin state is the maximum density droplet.We restrict our attention here to these many-body states.
We evaluate the one-body density matrix,n(z,z′)de?ned by
n(z,z′)=N d2z2... d2z NψM(z,z2,...z N)×
ψ?M(z′,z2,...z N)/Q N(2) where Q N is the normalization integral for the N-electron Laughlin wave function.The second quantized form of Eq.(2)relates n(z,z′)to the AMDF:
n(z,z′)=M(N?1)
m=0 n m φ?m(z′)φm(z),(3)
To obtain Eq.(3)we have noted that no orbital angular momentum larger than M(N?1)is ever occupied in the N-electron Laughlin droplet and that the Laughlin state is an eigenstate of total angular momentum.(M T OT=MN(N?1)/2).Eq.(3)can be inverted to express n m in terms of n(z,z′):
n m = d2z d2z′φ?m(z)n(z,z′)φm(z′).(4) For su?ciently small N and M the AMDF for a Laughlin droplet can be evaluated analyt-ically.Some results[9]for very small droplets are listed in Table I.
For larger droplets we have been unable to evaluate the AMDF analytically[10]but some properties of the distribution function are known.One such property follows from the expansion ofψM[z]in decreasing powers of z1:
ψ(N)
M
(z1,···,z N)=
[z1M(N?1)?M(N?1)z M(N?1)?1
1
ˉZ+···]×
exp(?|z1|2/4)ψ(N?1)
M
(z2,···,z N).(5)
In Eq.(5),ψ(N?1)
M
(z2,···,z N)is the N?1electron droplet,ˉZ is its center of mass coordinate and only the two highest powers of z1have been https://www.doczj.com/doc/f216473067.html,ing Eq.(5)in Eq.(4)we obtain
n m o =N Q N?1
2m o?1(m o?1)!M2(N?1)2 ˉZ2 N?1.(7)
Q N
It is easy to show a Laughlin droplet has de?nite center-of-mass angular momentum equal to zero[8]from which it follows that ˉZ2 N?1=2?2/(N?1)and hence that for any N
n m o?1 =M n m o .(8) This result can be extended to orbitals farther from the outer edge using an argument due to Wen[2].The density,n(z)≡n(z,z)far outside the droplet may be determined up to a constant from Eq.(2)and Eq.(3)by using a plasma analogy[5].The result for r=|z|?R m
is[2]
o
/r2)?m.(9) n(r)∝exp(?r2/2)(r2/2)m o(1?R2m
o
Expanding the right hand side of Eq.(9)and comparing with Eq.(4)gives
n m o?k = k+M?1
M?1 n m o (10) which reduces to Eq.(8)for k=1.We remark that while Eq.(8)is exact for any number of particles Eq.(10)(for k>1)becomes exact only in the limit N1/2?k.(For example,we see that the occupation numbers listed in Table I do not satisfy Eq.(10).Eq.(10)implies that for N1/2?k?M, n m o?k ∝k M?1.The behavior of the AMDF in this limit is thus consistent with Luttinger liquid[12]behavior with a critical exponent related to the quantized Hall conductance as argued on more general grounds by Wen[2].This property of the AMDF presumably holds as long as the fractional quantum Hall e?ect occurs and is not unique to the Laughlin state.
To examine the AMDF in the interior of the droplet and to test how well Eq.(10)is satis?ed for?nite size droplets it is su?cient to evaluate n(z,z′)for the case of|z|=|z′|=r .Using the explicit form of the one-particle orbitals Eq.(3)simpli?es for this case to
n(r,r;θ)=1
m!
r2m
f m(r)
M(N?1)
j=0e iθj m n(r,r;θj)
2πm!
r2m
M(N?1)+1
.(14)
We evaluate the AMDF numerically by combining a Monte-Carlo evaluation of n(r,r;θj) with Eq.(12).
It is interesting to note that the full AMDF can be determined,and therefore that the full one-particle density matrix can be reconstructed,from the angular dependence of n(z,z′)at any common radius,|z|=|z′|=r.This property is unique[11]to two-dimensional systems in the strong magnetic?eld limit where all electrons are restricted to a single Landau level.However,the Monte-Carlo values of n(r,r;θj)will inevitably have some statistical uncertainty.From Eq.(12)we see that the resulting uncertainty in the occupation number will be a minimum when r is near the maximum in f m(r)which occurs at at r=
√
W(z2,...,z N)=
N
1
With this weighting factor
n(r,r;θ)=N
Q N?1
(2πM)?1,from which it follows that n m approaches M?1.Our Monte-Carlo calculations show that this limit is approached slowly as N increases.The largest occupation numbers occur near the edge of the droplet and the relative excess in this region is larger for larger M.For the droplets illustrated the largest occupation numbers are n50 =0.52±0.02for M=3, n72 =0.47±0.04for M=5and n75 =0.43±0.03for M=7.The oscillations in occupation number are related to the oscillations in charge density expected near the edge of a?nite2D plasma[15]but are much more pronounced since the density averages occupation numbers of orbitals with R m near r.
As we see from the plots of n m ,the statistical uncertainties,obtained by averaging over independent runs,are smaller closer to the edge of the droplet.We believe that this is because fewer orbitals contribute importantly to the one-particle density matrix.In fact the occupation numbers for orbitals very close to the edge are actually more reliably calculated by evaluating the angular dependence of the density-matrix at r substantially larger than R m
.In closing we remark that it is in principle possible to determine the occupation num-o
bers uniquely if the radial dependence of the particle-density is known precisely.However, this procedure is extremely ill-conditioned and we believe that it is practical only where the particle-density is known analytically.We have found it to be impossible to determine the AMDF accurately from Monte-Carlo particle densities even for quite small droplets.
This work was supported by the National Science Foundation under grant DMR-9113911. The authors are grateful to Mats Wallin for generous and helpful advice on the Monte Carlo calculations.AHM acknowledges informative conversations with X.-G.Wen,and E.H.Rezayi.
REFERENCES
[1]See for example,A.H.MacDonald,in Quantum Coherence in Mesoscopic Systems edited
by B.Kramer(Plenum,New York,1991).
[2]X.-G.Wen,Phys.Rev.B41,12838(1990);X.G.Wen,Phys.Rev.B43,11025(1991);
X.-G.Wen,Phys.Rev.B44,5708(1991);X.-G.Wen,Inter.J.Mod.Phys.B6,1711 (1992)and work cited therein.
[3]A.H.MacDonald,Phys.Rev.Lett.64,222(1990).
[4]Rezayi and Haldane have made a numerical exact diagonalization study of the mo-
mentum distribution function for Laughlin states on tori.E.H.Rezayi and F.D.M.
Haldane,“Laughlin state on stretched and squeezed cylinders”,preprint(1992).
[5]https://www.doczj.com/doc/f216473067.html,ughlin,Phys.Rev.Lett.50,1395(1983).
[6]The area occupied by a droplet depends on the magnetic?eld strength and on the nature
of the potential which con?nes the electrons to a?nite area.For typical con?nement potential and a?nite number of electrons there will be a?nite range of magnetic?elds over which each of the Laughlin droplet states becomes the ground state.
[7]A.H.MacDonald,and M.D.Johnson,submitted to Phys.Rev.Lett.(1993).
[8]A.H.MacDonald,S.-R.E.Yang,and M.D.Johnson,Australian Journal of Physics,to
appear(1993).
[9]Angular momentum distribution functions for Laughlin droplets with N up to7for
M=3,for N up to6for M=5,for N up to5for M=7and for N up to4for M=9 are available from the authors.These were obtained using a special purpose symbolic manipulation program described in an earlier publication:A.H.MacDonald and D.B.
Murray,Phys.Rev.B32,2707(1985).
[10]Techniques which allow the corresponding integrals to be evaluated analytically for
closely related one dimensional many-body wave functions fail in this case.Jean-Louis Pichard and A.H.MacDonald,unpublished(1991).See also B.Sutherland,J.Math.
Phys.12,246(1971);12,251(1971);Phys.Rev.A4,2019(1971);5,1372(1972).
[11]See for example,A.H.MacDonald and S.M.Girvin,Phys.Rev.B38,6295(1988).
[12]F.D.M.Haldane,J.Phys.C14,2585(1981).
[13]N.Metropolis,A.W.Rosenbluth,M.N.Rosenbluth,A.H.Teller and E.Teller,J.
Chem.Phys.21,1087(1953).
[14]Some physical consequences of Luttinger liquid behavior are discussed in Ref.[2]and in
K.Moon,H.Yi,C.L.Kane,S.M.Girvin,and M.P.A.Fisher,submitted to Phys.Rev.
Lett.(1993).See also C.L.Kane and M.P.A.Fisher,Phys.Rev.B46,7268(1992)and work cited therein.
[15]These oscillations have been studied numerically by R.Ferrari and N.Datta,Bull.Am.
Phys.Soc.37,164(1992).
FIG.1.Occupation numbers for M=3,M=5,and M=7,Laughlin droplets with N=25, N=20,and N=15respectively plotted vs.m=R2m/2.The solid line is the particle density in (2π?2)?1units plotted vs.r2/2calculated from the occupation numbers using Eq.(3).The dashed line shows the particle density calculated directly;the di?erence in these two quantities is a measure of the error introduced in extracting the occupation numbers from the density matrix.
https://www.doczj.com/doc/f216473067.html,parison of Monte Carlo occupation numbers near the edge with the formula derived by Wen,Eq.(10).
TABLE I.Angular momentum state occupation numbers for some representative small N Laughlin droplets
M,N m=0m=1m=2m=3m=4m=5m=6
摘要 自意大利学者M. Dorigo于1991年提出蚁群算法后，该算法引起了学者们的极大关注，在短短十多年的时间里，已在组合优化、网络路由、函数优化、数据挖掘、机器人路径规划等领域获得了广泛应用，并取得了较好的效果。本文首先讨论了该算法的基本原理，接着介绍了旅行商问题，然后对蚁群算法及其二种改进算法进行了分析，并通过计算机仿真来说明蚁群算法基本原理，然后分析了聚类算法原理和蚁群聚类算法的数学模型，通过调整传统的蚁群算法构建了求解聚类问题的蚁群聚类算法。最后，本文还研究了一种依赖信息素解决聚类问题的蚁群聚类算法，并把此蚁群聚类算法应用到对人工数据进行分类，还利用该算法对2005年中国24所高校综合实力进行分类，得到的分类结果与实际情况相符，说明了蚁群算法在聚类分析中能够收到较为理想的结果。 【关键词】蚁群算法；计算机仿真；聚类；蚁群聚类
Study on Ant Colony Algorithm and its Application in Clustering Abstract： As the ant colony algorithm was proposed by M. Dorigo in 1991,it bringed a extremely large attention of scholars, in past short more than ten years, optimized, the network route, the function in the combination optimizes, domains and so on data mining, robot way plan has obtained the widespread application, and has obtained the good effect.This acticle discussed the basic principle of it at first, then introduced the TSP,this acticle also analysed the ant colony algorithm and its improved algorithm, and explanated it by the computer simulates, then it analysed the clustering algorithm and the ant clustering algorithm, builded the ant clustering algorith to solution the clustering by the traditioned ant algorithm. At last, this article also proposed the ant clustering algorith to soluted the clustering dependent on pheromon. Carry on the classification to the artificial data using this ant clustering algorithm; Use this algorithm to carry on the classification of the synthesize strength of the 2005 Chinese 24 universities; we can obtain the classified result which matches to the actual situation case. In the next work, we also should do the different cluster algorithm respective good and bad points as well as the classified performance aspect the comparison research; distinguish the different performance of different algorithm in the analysis when the dates are different. Key words： Ant colony algorithm; Computer simulation; clustering; Ant clustering 目录
蚁群优化算法
目录 [隐藏]
? ?
比较
1 2
蚁群算法的提出： 人工蚂蚁与真实蚂蚁的异同
o o ? ? ? ? ?
3 4 5 6 7
2.1 2.2
相同点比较 不同点比较
蚁群算法的流程图 基本蚁群算法的实现步骤 蚁群算法的 matlab 源程序 蚁群算法仿真结果 版权声明
[编辑]蚁群算法的提出：
人类认识事物的能力来源于与自然界的相互作用，自然界一直是人类创造力 的源泉。 自然界有许多自适应的优化现象不断地给人以启示，生物和自然中的生 态系 统可以利用自身的演化来让许多在人类看来高度复杂的优化问题得到几乎完美 的解决。近些年来，一些与经典的数学问题思想不同的，试图通过模拟自然生态 系统 来求解复杂优化问题的仿生学算法相继出现，如蚁群算法、遗传算法、粒子群算 法等。 这些算法大大丰富了现在优化技术，也为那些传统最优化技术难以处理的 组 合优化问题提供了切实可行的解决方案。 生物学家通过对蚂蚁的长期的观察发现，每只蚂蚁的智能并不高，看起来没 有集中的指挥，但它们却能协同工作，集中事物，建起坚固漂亮的蚁穴并抚养后 代， 依靠群体能力发挥出超出个体的智能。 蚁群算法是最新发展的一种模拟昆虫王国 中蚂蚁群体智能行为的仿生优化算法，它具有较强的鲁棒性、优良的分布式计算 机 制、易于与其他方法相结合等优点。尽管蚁群算法的严格理论基础尚未奠定，国 内外的相关研究还处于实验阶段， 但是目前人们对蚁群算法的研究已经由当初单 一 的旅行商问题(TSP)领域渗透到了多个应用领域，由解决一维静态优化问题发展 到解决多维动态组合优化问题， 由离散域范围内的研究逐渐扩展到了连续域范围 内的
蚁群优化算法的JAVA实现 一、蚁群算法简介 蚁群算法是群智能算法的一种，所谓的群智能是一种由无智能或简单智能的个体通过任何形式的聚集协同而表现出智能行为，它为在没有集中控制且不提供全局模型的前提下寻找复杂的分布式问题求解方案提供了基础，比如常见的蚂蚁觅食，大雁南飞等行为。蚁群算法是模拟自然界中蚂蚁觅食的一种随机搜索算法，由Dorigo等人于1991年在第一届欧洲人工生命会议上提出[1] 。 二、蚁群算法的生物原理 通过观察发现，蚁群在觅食的时候，总能找到一条从蚁巢到食物之间的一条最短的路径。这个现象引起了生物学家的注意，根据研究，原来是蚂蚁在行进的过程中，会分泌一种化学物质——信息素，而蚂蚁在行进时，总是倾向于选择信息素浓度比较高的路线。这样，在蚁巢和食物之间假如有多条路径，初始的时候，每条路径上都会有蚂蚁爬过，但是随着时间的推迟，单位时间内最短的那条路上爬过的蚂蚁数量会比较多，释放的信息素就相对来说比较多，那么以后蚂蚁选择的时候会大部分都选择信息素比较多的路径，从而会把最短路径找出来。 蚁群算法正是模拟这种蚁群觅食的原理，构造人工蚂蚁，用来求解许多组合优化问题。 有关蚁群算法的详细信息，可参考[2]——[5]。 三、蚁群算法的JAVA实现 我个人认为利用JAVA编写一些计算密集型的算法不是一个好的选择。本身一些算法是要要求高效率的，但是我感觉JAVA语言的性能不够，所以编写算法最好用c，其次也可以用c++。当然，这仅是一家之言，欢迎拍砖。 四、算法说明 算法以求解TSP问题为例，用来演示ACO的框架。 算法设定了两个类，一个是ACO，用来处理文件信息的读入，信息素的更新，路径的计算等；另一个是ant，即蚂蚁的信息。 算法中用到的数据，可以从TSPLib 上面下载，使用的是对称TSP数据，为了简化算法的代码，下载下来的数据需要做一个简单处理，即把TSP文件中除去城市节点信息部分之外的内容都删除掉，然后在文件首插入一行，写入此文件包含的城市的数目，以att48.tsp为例，下载下来的文件内容如下：
蚁群优化算法ACO 一、蚁群算法的背景信息 蚁群优化算法(ACO)是一种模拟蚂蚁觅食行为的模拟优化算法，它是由意大利学者Dorigo M等人于1991年首先提出，之后，又系统研究了蚁群算法的基本原理和数学模型，并结合TSP优化问题与遗传算法、禁忌搜索算法、模拟退火算法、爬山法等进行了仿真实验比较，为蚁群算法的发展奠定了基础，并引起了全世界学者的关注与研究 蚁群算法是一种基于种群的启发式仿生进化系统。蚁群算法最早成功应用于解决著名的旅行商问题(TSP)，该算法采用了分布式正反馈并行计算机制，易于与其他方法结合，而且具有较强的鲁棒性。 二、蚁群算法的原理[1] 蚁群算法是对自然界蚂蚁的寻径方式进行模似而得出的一种仿生算法。蚂蚁在运动过程中，能够在它所经过的路径上留下一种称之为外激素(pheromone)的物质进行信息传递，而且蚂蚁在运动过程中能够感知这种物质，并以此指导自己的运动方向，因此由大量蚂蚁组成的蚁群集体行为便表现出一种信息正反馈现象：某一路径上走过的蚂蚁越多，则后来者选择该路径的概率就越大。 基本的ACO模型由下面三个公式描述： 式（2-1）、式(2-2)和式(2-3)中：m为蚂蚁个数；n为迭代次数；i为蚂蚁所在位置；j为蚂蚁可以到达的置；为蚂蚁可以到达位置的集合；为启发性信息
，这里为由i到j的路径的能见度，即；为目标函数，这里为两点间欧氏(Euclidean)距离；为由i到j的路径的信息素强度；为蚂蚁k由i到j的路径上留下的信息素数量；为路径权；为启发性信息的权；为路径上信息素数量的蒸发系数；Q为信息素质量系数；为蚂蚁k从位置i移动到位置j的转移概率。 三、改进的蚁群算法[3] 蚁群算法具有如下一些优点：①通用性较强，能够解决很多可以转换为连通图结构的路径优化问题；②同时具有正负反馈的特点，通过正反馈特点利用局部解构造全局解，通过负反馈特点也就是信息素的挥发来避免算法陷入局部最优； ③有间接通讯和自组织的特点，蚂蚁之间并没有直接联系，而是通过路径上的信息素来进行间接的信息传递，自组织性使得群体的力量能够解决问题。 但是，基本蚁群算法也存在一些缺点：①从蚁群算法的复杂度来看，该算法与其他算法相比，所需要的搜索时间较长；②该算法在搜索进行到一定程度以后，容易出现所有蚂蚁所发现的解完全一致这种“停滞现象”，使得搜索空间受到限制 3.1基于遗传学的改进蚁群算法研究 该文献[2]提出的算法弥补了基本蚁群算法中“容易陷入停滞状态”和“盲目随机搜索”的不足。文献中提出的解决办法是在每一次迭代搜索后，都把当前解和最优解进行交叉变异,这样既能搜索更大的解空间，又能使系统陷入局部最优后跳出停滞状态。 这种基于遗传学的蚁群算法（G-蚁群算法）的基本思想是在以蚁群算法为主体的基础上引入遗传算法的思想，目的是让蚁群算法经过迭代产生遗传算法所需的初始种群数据，提高种群数据的多样性。然后，遗传算法经过选择、交叉和变异操作,将处理后的数据交由蚁群算法模块进行判断处理，若满足结束条件则退出系统，否则重新进行迭代操作。 该文献中的交叉操作采用了Davis提出的OX交叉算子，即按照交叉概率Pc进行交叉操作，通过一个亲体中挑选出的子序列路径作为后代相对位置的子序列，变异操作以变异概率Pm执行变异操作，在子代序列路径中随机选择两点进行变异操作，在交叉变异操作结束后，判断当前解是否满足收敛条件，若满足收敛条件则更新当前最优解，返回蚁群算法程序,执行下一次的迭代操作。若不满足收敛条件则继续进行遗传算法的交叉变异操作。