Topological interpretation of Dirac equation and geometrization of physical interactions

The Generalization of Dirac’s Theoremfor HypergraphsEndre Szemer´e di1,Andrzej Ruci´n ski2, ,and Vojtˇe ch R¨o dl3,1Rutgers University,New Brunswickszemered@2A.Mickiewicz University,Pozna´n,Polandrucinski@.pl3Emory University,Atlanta,GArodl@1Introduction and Main ResultA substantial amount of research in graph theory continues to concentrate on the existence of hamiltonian cycles and perfect matchings.A classic theorem of Dirac states that a sufficient condition for an n-vertex graph to be hamiltonian, and thus,for n even,to have a perfect matching,is that the minimum degree is at least n/2.Moreover,there are obvious counterexamples showing that this is best possible.The study of hamiltonian cycles in hypergraphs was initiated in[1]where, however,a different definition than the one considered here was introduced. Given an integer k≥2,a k-uniform hypergraph is a hypergraph(a set system) where every edge(set)is of size k.By a cycle we mean a k-uniform hypergraph whose vertices can be or-dered cyclically v1,...,v l in such a way that for each i=1,...,l,the set {v i,v i+1,...,v i+k−1}is an edge,where for h>l we set v h=v h−l.A hamil-tonian cycle in a k-uniform hypergraph H is a spanning cycle in H,that is,a sub-hypergraph of H which is a cycle and contains all vertices of H.A k-uniform hypergraph containing a hamiltonian cycle is called hamiltonian.This notion and its generalizations have a potential to be applicable in many contexts which still need to be explored.An application in the relational database theory can be found in[2].As observed in[5],the square of a(graph)hamiltonian cycle naturally coincides with a hamiltonian cycle in a hypergraph built on top of the triangles of the graph.More precisely,given a graph G,let T r(G)be the set of triangles in G.Define a hypergraph H T r(G)=(V(G),T r(G)).Then there is a one-to-one correspondence between hamiltonian cycles in H T r(G)and the squares of hamiltonian cycles in G.For results about the existence of squares of hamiltonian cycles see,e.g.,[6].As another potential application consider a seriously ill patient taking24 different pills on a daily basis,one at a time every hour.Certain combinations Research supported by KBN grant2P03A01523.Part of research performed at Emory University,Atlanta.Research supported by NSF grant DMS-0300529.J.J edrzejowicz and A.Szepietowski(Eds.):MFCS2004,LNCS3618,pp.52–56,2005.c Springer-Verlag Berlin Heidelberg2005The Generalization of Dirac’s Theorem for Hypergraphs 53of three pills can be deadly if taken within 2.5hour.Let D be the set of deadly triplets of pills.Then any safe schedule corresponds to a hamiltonian cycle in the hypergraph which is precisely the complement of D .A natural extension of Dirac’s theorem to k -graphs,k ≥2,has been con-jectured in [5],where as a sufficient condition one demands that every (k −1)-element set of vertices is contained in at least n/2 edges.The following con-struction of a k -uniform hypergraph H 0,also from [5],shows that the above conjecture,if true,is nearly best possible (best possible for k =3).Let V =V ∪{v },|V |=n .Split V =X ∪Y ,where,|X |= n −12 and |Y |= n −12 .The edges of H 0are all k -element subsets S of V such that |X ∩S |= k 2 or v ∈S .It is shown in [5]that H 0is not hamiltonian,while every (k −1)-element set of vertices belongs to at least n −k +12 edges.In [9]we proved an approximate version of the conjecture from [5]for k =3,and in [11]we give a generalization of that result to k -uniform hypergraphs for arbitrary k .Theorem 1([11]).Let k ≥3and γ>0.Then,for sufficiently large n ,every k -uniform hypergraph on n -vertices such that each (k −1)-element set of vertices is contained in at least (1/2+γ)n edges is hamiltonian.2The Idea of ProofThe idea of the proof is as follows.As a preliminary step,we find in H a powerful path A ,called absorbing which has the property that every not too large subset of vertices can be “absorbed”by that path.We also put aside a small subset of vertices R which preserves the degree properties of the entire hypergraph.On the sub-hypergraph H =H −(A ∪R )we find a collection of long,disjoint paths which cover almost all vertices of H .Then,using R we “glue”them and the absorbing path A together to form a long cycle in H .In the final step,the vertices which are not yet on the cycle are absorbed by A to form a hamiltonian cycle in H .The main tool allowing to cover almost all vertices by disjoint paths is a generalization of the regularity lemma from [12].Given a k -uniform hypergraph H and k non-empty,disjoint subsets A i ⊂V (H ),i =1,...,k ,we define e H (A 1,...,A k )to be the number of edges in H with one vertex in each A i ,and the density of H with respect to (A 1,...,A k )asd H (A 1,...,A k )=e H (A 1,...,A k )|A 1|···|A k |.A k -uniform hypergraph H is k -partite if there is a partition V (H )=V 1∪···∪V k such that every edge of H intersects each set V i in precisely one vertex.For a k -uniform,k -partite hypergraph H ,we will write d H for d H (V 1,...,V k )and call it the density of H .We say that a k -uniform,k -partite hypergraph H is -regular if for all A i ⊆V i with |A i |≥ |V i |,i =1,...,k ,we have|d H (A 1,...,A k )−d H |≤ .54 E.Szemer´e di,A.Ruci´n ski,and V.R ¨o dlThe following result,called weak regularity lemma as opposed to the stronger result in [4],is a straightforward generalization of the graph regularity lemma from [12].Lemma 1(Weak regularity lemma for hypergraphs).For all k ≥2,every >0and every integer t 0there exist T 0and n 0such that the following holds.For every k -uniform hypergraph H on n >n 0vertices there is,for some t 0≤t ≤T 0,a partition V (H )=V 1∪···∪V t such that |V 1|≤|V 2|≤···≤|V t |≤|V 1|+1and for all but at most t k sets of partition classes {V i 1,...,V i k },the induced k -uniform,k -partite sub-hypergraph H [V i 1,...,V i k ]of H is -regular.The above regularity lemma,combined with the fact that every dense -regular hypergraph contains an almost perfect path-cover,yields an almost per-fect path-cover of the entire hypergraph H .3Results for MatchingsA perfect matching in a k -uniform hypergraph on n vertices,n divisible by k ,is a set of n/k disjoint edges.Clearly,every hamiltonian,k -uniform hypergraph with the number of vertices n divisible by k contains a perfect matching.Given a k -uniform hypergraph H and a (k −1)-tuple of vertices v 1,...,v k −1,we denote by N H (v 1,...,v k −1)the set of vertices v ∈V (H )such that {v 1,...,v k −1,v }∈H .Let δk −1(H )=δk −1be the minimum of |N H (v 1,...,v k −1)|over all (k −1)-tuples of vertices in H .For all integer k ≥2and n divisible by k ,denote by t k (n )the smallest integer t such that every k -uniform hypergraph on n vertices and with δk −1≥t contains a perfect matching.For k =2,that is,in the case of graphs,we have t 2(n )=n/2.Indeed,the lower bound is delivered by the complete bipartite graph K n/2−1,n/2+1,while the upper bound is a trivial corollary of Dirac’s condition [3]for the existence of Hamilton cycles.In [10]we study t k for k ≥3.As a by-product of our result about hamiltonian cycles in [11](see Theorem 2above),it follows that t k (n )=n/2+o (n ).K ¨u hn and Osthus proved in [7]thatn 2−k +1≤t k (n )≤n 2+3k 2 n log n.The lower bound follows by a simple construction,which,in fact,for k odd yields t k (n )≥n/2−k +2.For instance,when k =3and n/2is an odd integer,split the vertex set into sets A and B of size n/2each,and take as edges all triples of vertices which are either disjoint from A or intersect A in precisely two elements.In [10]we improve the upper bound from [7].Theorem 2.For every integer k ≥3there exists a constant C >0such that for sufficiently large n ,t k (n )≤n 2+C log n.The Generalization of Dirac’s Theorem for Hypergraphs55 It is very likely that the true value of t k(n)is yet closer to n/2.Indeed,in[5]it is conjectured thatδk−1≥n/2is sufficient for the existence of a Hamilton cycle, and thus,when n is divisible by k,the existence of a perfect matching.Based on this conjecture and on the above mentioned construction from[7],we believe that t k(n)=n/2−O(1).In fact,for k=3,we conjecture that t3(n)= n/2 −1.Our belief that t k(n)=n/2−O(1)is supported by some partial results.For example,we are able to show that the threshold function t k(n)has a stability property,in the sense that hypergraphs that are“away”from the“extreme case”H0,described in Section1,contain a perfect matching even whenδk−1is smaller than but not too far from n/2.Interestingly,if we were satisfied with only a partial matching,covering all but a constant number of vertices,then this is guaranteed already with n/2+o(n) replaced by n/k,that is,whenδk−1≥n/k.We have also another related result,about the existence of a fractional perfect matching,which is a simple consequence of Farkas’Lemma(see,e.g.,[8]).A fractional perfect matching in a k-uniform hypergraph H=(V,E)is a function w:E→[0,1]such that for each v∈V we havew(e)=1.e vIn particular,it follows from our result that ifδk−1(H)≥n/k then H has a fractional perfect matching,so,again,the threshold is much lower than that for perfect matchings.References1.J.C.Bermond et al.,Hypergraphes hamiltoniens,b.Theorie GraphOrsay260(1976)39-43.2.J.Demetrovics,G.O.H.Katona and A.Sali,Design type problems motivated bydatabase theory,Journal of Statistical Planning and Inference72(1998)149-164.3.G.A.Dirac,Some theorems for abstract graphs,Proc.London Math.Soc.(3)2(1952)69-81.4.P.Frankl and V.R¨o dl,Extremal problems on set systems,Random Struct.Algo-rithms20,no.2,(2002)131-164.5.Gyula Y.Katona and H.A.Kierstead,Hamiltonian chains in hypergraphs,J.Graph Theory30(1999)205-212.6.J.Koml´o s,G.N.S´a rk¨o zy and E.Szemer´e di,On the P´o sa-Seymour conjecture,J.Graph Theory29(1998)167-176.7.D.Kuhn and D.Osthus,Matchings in hypergraphs of large minimum degree,sub-mited.8.L.Lov´a sz&M.D.Plummer,Matching theory.North-Holland Mathematics Studies121,Annals of Discrete Mathematics29,North-Holland Publishing Co.,Amster-dam;Akad´e miai Kiad´o,Budapest,19869.V.R¨o dl,A.Ruci´n ski and E.Szemer´e di,A Dirac-type theorem for3-uniform hy-pergraphs,Combinatorics,Probability and Computing,to appear.10.V.R¨o dl,A.Ruci´n ski and E.Szemer´e di,Perfect matchings in uniform hypergraphswith large minimum degree,submitted.56 E.Szemer´e di,A.Ruci´n ski,and V.R¨o dl11.V.R¨o dl,A.Ruci´n ski and E.Szemer´e di,An approximative Dirac-type theorem fork-uniform hypergraphs,submitted.12.E.Szemer´e di,Regular partitions of graphs.Problemes combinatoires et theorie desgraphes(RS,Univ.Orsay,Orsay,1976),pp.399–401,Colloq.RS,260,CNRS,Paris,1978.。

拓扑陈数概念解析一、引言拓扑陈数作为拓扑学中的一个重要概念，被广泛应用于凝聚态物理学和拓扑量子计算等领域。

它以独特的方式描述了物质体系中的拓扑性质和量子态之间的关系。

本文将对拓扑陈数进行深入解析，从概念的起源到数学表达形式及其意义等多个方面进行探讨，旨在帮助读者更好地理解和应用该概念。

二、概念起源与发展1. 定义与历史背景拓扑陈数最早由物理学家斯蒂芬·亨利·沃兹尼亚克于1983年提出，他在研究量子霍尔效应时发现了这一概念。

拓扑陈数在描述物质的拓扑性质时能够很好地解释量子霍尔效应和拓扑绝缘体等物理现象。

2. 拓扑数学的基础拓扑陈数的概念来源于拓扑数学领域，它涉及拓扑空间和连续映射的性质。

拓扑数学研究的对象是具有连续性质的空间，而不考虑其具体的度量和形状。

三、拓扑陈数的数学表达1. 拓扑不变量的计算拓扑陈数通常被定义为相应拓扑不变量的积分。

对于二维拓扑陈数，可以使用Berry联络和曲率来计算；对于三维拓扑陈数，则需要使用Berry联络和Berry曲面等工具进行计算。

2. 拓扑不变量的几何表示拓扑陈数可以通过拓扑不变量的几何表示来理解。

二维拓扑陈数可以通过将系统的波函数视为在动量空间中运动的粒子，并计算路径在波函数空间的闭合路径，从而得到表示系统拓扑性质的导数值。

四、拓扑陈数的物理意义1. 拓扑保护性质拓扑陈数描述了物质体系在拓扑变化下的保护性质。

它能够有效地区分拓扑态和非拓扑态，并且对外界扰动表现出一定的抗干扰能力。

2. 拓扑量子计算拓扑陈数在拓扑量子计算中有重要应用。

通过控制拓扑陈数的变化，可以实现量子比特的存储、控制和相互作用等操作，从而提高量子计算的可靠性和稳定性。

3. 实验观测与应用拓扑陈数的物理意义可以通过实验观测进行验证，在凝聚态物理学和拓扑材料研究中有广泛应用。

量子霍尔效应的实验观测和利用拓扑绝缘体进行信息传输等。

五、总结与展望拓扑陈数作为拓扑学中一个重要的概念，具有深远的理论和实际意义。

什么是拓扑绝缘体？[教材]什么是拓扑绝缘体,拓扑绝缘体(topological insulator，简称TI)是这两年凝聚态理论里面很热的一个方向，最早提出这一概念的应该是UPenn的Kane，然后就是Stanford的张守晟组，主要是在Quantum Spin Hall体系中的TI。

按照电子态结构的不同，传统意义上的材料被分为“金属”和“绝缘体”两大类。

而拓扑绝缘体是一种新的量子物质态，完成不同于传统意义上的“金属”和“绝缘体”。

这种物质态的体电子态是有能隙的绝缘体，而其表面则是无能隙的金属态。

这种无能隙的表面金属态也完全不同于一般意义上的由于表面未饱和键或者是表面重构导致的表面态，拓扑绝缘体的表面金属态完全是由材料的体电子态的拓扑结构所决定，是由对称性所决定的，与表面的具体结构无关。

也正是因为该表面金属态的出现是有对称性所决定的，他的存在非常稳定，基本不受到杂质与无序的影响。

除此之外，拓扑绝缘体的基本性质是由“量子力学”和“相对论”共同作用的结果，由于自旋轨道耦合耦合作用，在表面上会产生由时间反演对称性保护的无能隙的自旋分辨的表面电子态。

这种表面态形成一种无有效质量的二维电子气(与有效质量近似下的二维电子气完全不同:例如广泛使用的场效应晶体管中的二维电子气)，它需要用狄拉克方程描述，而不能用薛定谔方程。

正是由于这些迷人的重要特征保证了拓扑绝缘体将有可能在未来的电子技术发展中获得重要的应用，有着巨大的应用潜在。

寻找具有足够大的体能隙并且具有化学稳定性的强拓扑绝缘体材料成为了人们目前关注的重要焦点和难点。

拓扑绝缘体的表面金属态完全是由材料的体电子态的拓扑结构所决定，是由对称性所决定的，与表面的具体结构无关。

这句话的意思是拓扑绝缘体的“拓扑”，不是实空间的拓扑结构，而是动量空间的拓扑结构。

说起拓扑，大家也许会联想到Möbius带，或者Klein瓶的东西，但实际上拓扑绝缘体与实空间的这些几何结构都没有关系，它的表面形貌和其它材料没有什么差别。