Nested Topological Order

a r X i v :0803.4299v 1 [c o n d -m a t .s t r -e l ] 30 M a r 2008

Nested Topological Order

H.Bombin and M.A.Martin-Delgado

Departamento de F′?sica Te′o rica I,Universidad Complutense,28040.Madrid,Spain.We introduce the concept of nested topological order in a class of exact quantum lattice Hamilto-nian models with non-abelian discrete gauge symmetry.The topological order present in the models can be partially destroyed by introducing a gauge symmetry reduction mechanism.When symme-try is reduced in several islands only,this imposes boundary conditions to the rest of the system giving rise to topological ground state degeneracy.This degeneracy is related to the existence of topological ?uxes in between islands or,alternatively,hidden charges at islands.Additionally,island deformations give rise to an extension of topological quantum computation beyond quasiparticles.

PACS numbers:71.10.-w,11.15.-q,03.67.Pp,71.27.+a

The concept of topological orders [1]o?ers the possibil-ity of ?nding new states of matter with a common picture of string-net condensation [2]and other variants thereof [3].They correspond to examples of long range entangle-ment in quantum many-body systems where those cor-relations emerge in quantum states that are encoded in non-local degrees of freedom of topologically ordered sys-tems.Their global properties are the source for yet an-other application as the suitable systems to implement topological quantum computation [4,5,6,7],a form of fault-tolerant quantum computation intrinsically resis-tant to the debilitating e?ects of local noise.Quantum ?eld theories with an spontaneous symmetry breaking mechanism of a continuous gauge group down to a dis-crete group have been proposed as a scenario for realizing their physics [8,9,10,11,12,13,14].

In this paper we introduce the concept of nested topo-logical order in a class of quantum lattice Hamiltonians.Our starting point are the family of Kitaev’s models [4],which are labeled by a discrete gauge group.Such mod-els can be modi?ed [15]introducing an explicit symme-try breaking mechanism.Our aim is to study the ef-fect of ‘nesting’subsystems with a reduced symmetry inside systems with the complete gauge symmetry.We will consider a topologically ordered system divided in two regions,say A and C ,and show that it is possible to partially destroy the topological order in region C in such a way that this imposes boundary conditions to the subsystem A .The system C can take the form of several islands,which is why we talk about ‘nested’topologi-cal order.The boundary conditions induce a topological ground state degeneracy which is due to the possible val-ues of certain ?uxes in between islands.As we will see,the values of these ?uxes correspond to the types of do-main walls that exist in C .If we allow the region C to be deformed,then islands can be initialized,braided and fused,giving an interesting extension of the ideas of topological quantum computation beyond quasiparticles.The models that we consider are string-net conden-sates in a 2D lattice [1],[2].The con?gurations of the lattice are regarded as string-net states:a collection of la-beled strings meeting at branching points.A string-net is closed if certain conditions hold at branching points

and there are no loose ends.The ground state is a su-perposition of all possible deformations of such closed string-nets,and excited states correspond to con?gura-tions with loose ends:quasiparticle excitations appear at the ends of strings.Now,to such system Hamiltonians we can add string tension terms,which penalize with a higher energy those con?gurations with longer strings.As such terms get more important with respect to the original ones,longer strings become less relevant in the ground state and ?nally the topological order is destroyed as excitations get con?ned.Alternatively,we can add suitable terms so that only part of the topological order is destroyed.This is in fact the case for the Hamiltoni-ans H N,M

G that we consider (1),which are labeled with a discrete group G and two subgroups N ?M ?G ,with N abelian and normal in G .If N =1and M =G ,we have the original topologically ordered models with gauge group G considered by Kitaev [4].Otherwise,the gauge symmetry is reduced the quotient group G ′=M/N .In particular,if N =M the topological order is completely destroyed.

Topological phases.The systems of interest are constructed from a two-dimensional orientable lattice,of arbitrary shape.At every edge of the lattice we place a qudit,a |G |-dimensional quantum system with Hilbert

space H ′

G and a basis |g labeled with the elements of G .The Hamiltonians read as follows[15]

H N,M G :=? v ∈V

A M v ? f ∈F

B N f ? e ∈E

T M

e +L N e ,(1)

where the sums run over the set of vertices V ,faces F and edges E .Explicit expressions for the terms in (1)will be given below,but before that,we will discuss their physical content.First,all the terms are projectors and commute with each other,so that the ground state is described by conditions of the form P |GS =|GS with P either a vertex,face or edge operator.Excitations are gapped and localized;they correspond to violations of the previous conditions and so can be related to vertices,faces and edges;they are regarded respectively as electric,magnetic and domain wall excitations.

We ?rst recall the case H G :=H 1,G

G [4].For non-Abelian groups G ,vertex and face excitations are inter-

ρwill measure the ?ux of the excited region T in the direction of the white arrows.

The ground states of (1)can also be described in terms of conditions for ribbon operators,in particular by

F MN σ|GS =|GS ,

F NM

ρ|GS =|GS ,

(3)

where σand ρare arbitrary boundary and open ribbons,

respectively.The ?rst condition is related to vertex and face excitations,and the second to edge excitations.Nested phases.We are now in position to discuss a more complicated system.In particular,we want to con-sider a surface divided in two regions of arbitrary shape,A and C ,plus a third region B which is just a thick boundary separating them,included so that the Hamil-tonian does not have to change abruptly from A to C .The idea is to have a local Hamiltonian such that condi-tions (2)are satis?ed in A ,conditions (3)in C and the conditions

F NN σ|GS =|GS ,

(4)

with σan arbitrary boundary ribbon,in the whole sys-tem.The last condition is needed to ensure that domain

wall ?ux is preserved through region B ,a key ingredient of our construction as we will see.The ground state of the Hamiltonian H 0:=? v A N

v ? f B N f

is described pre-cisely by (4).In addition,H 0commutes with H G ,H NM

G

.Indeed,a Hamiltonian of the form H ′

=H G +λH 0,λ≥0,only di?ers from H G in the gap for some excitations,and

the same is true for H NM

G .The Hamiltonian that we

want to consider takes the form H =H 0+λH G +μH NM

G ,where λ,μ≥0vary spatially so that λ=1and μ=0in A and λ=0and μ=1in C .If we take λμ=0,the ground state has the desired properties but there exists some local degeneracy at B .This local degeneracy can be lifted if λand μare allowed to overlap,but on the other hand if the overlap is too big,it could produce a level crossing taking the ground state of H out of that of H 0,which spoils conditions (4).

Quasiparticle dilution.Our aim is to understand the e?ects of the nested region C on the topologically ordered region A .A ?rst e?ect is the possibility to locally create or destroy single quasiparticle excitations in the vicinity of the A -C border,something prohibited in systems with Hamiltonian H G due to charge conservation.In terms of ribbon operators,this is re?ected in the fact that for any ρ1connecting C to A ,as the one in Fig.2(a),a state of the form m ∈M F mn ˉ

m,mg

ρ1|GS ,n ∈N ,contains no excitation at C .In terms of quasiparticle processes,this corresponds to create a particle-antiparticle pair in A and then move one of them into C ,where it disappears because it is condensed.

Domain wall dilution.A second e?ect is related to the existence of domain walls in region C .Consider again a ribbon ρ2connecting C to A ,see Fig.2(a).Some of the

states of the form |ψ = h,g c h,g F

h,g

|GS ,c h,g ∈C ,will contain edge excitations all along the portion of ρ2contained in C ,for example those with c h,g =0for some

erty as long as the islands are distant.Indeed,measuring the?ux requires an operator with a support connecting C1and C2.And,if an operator changes the?ux,its support must loop around C1(or C2).Suppose to the contrary that O is an operator that leaves the ground state invariant and has a support not enclosing C1,as the shaded region in Fig.2(b).Letρ6be another rib-bon connecting the islands but lying outside the support

of O.Due to(4)we have J R,T

ρ5|GS =J R,T

ρ6

|GS ,so

that[J R,T

ρ5,O]|GS =[J R,T

ρ6

,O]|GS =0and thus O does

not change the?ux.Those operators which do change the?ux are related to processes in which a particle-antiparticle pair is created,one of them loops around C1 and they meet again to fuse into a charge that disappears into C1.

Topologically protected subsystems.It follows that there exist a topological degeneracy in the ground state, related to the distinct values that the?ux in between C1 and C2can take.For example,if N=M=1the?ux can take any value g∈G.In general,for a C composed of multiple disconnected regions,the degeneracy of the ground state depends on N,M and the topology of A. Now,it is natural to ask how does this protected space compares with the one due to to the existence of several separated excitations in A.In other words,do islands add something new?This can be positively answered through an example:two excitations give no protected subspace [4],but we have just seen the contrary for the case of two islands.Perhaps more dramatically,for abelian groups G the protected subsystem is always trivial whatever the amount of excitations,but this is not the case for islands. Nevertheless,islands can be compared to excitations,in the following sense.An island can hold certain charge values,which can be measured using ribbon operators that enclose the island,as in the case of an excitation. The di?erence between a charged island and a charged excitation is that the local degrees of freedom of the exci-tation become global in the case of the island:this is the origin of the additional dimensionality of the protected subsystem.

Braiding.The physics of the system so far has a static nature.If we want to consider the setting as an sce-nario for quantum computation,then the possibility of dynamically deforming the region C must be included in it.Such deformations need not be strictly adiabatic, but the state should be kept in the subspace de?ned by conditions(2-4)at all time.We can then braid islands to perform unitary operations,in complete analogy with quasiparticle braiding.It is also natural to enrich the physics by considering islands with di?erent(N,M)la-bels,increasing the variety of protected subsystems. Fusion.We must consider also the analogue of the quasi-particle fusion processes,which is the way in which mea-surements are carried out in topological quantum com-putation.There are two natural ways in which global degrees of freedom can be made local.The?rst is to decrease the size of an island till it dissapears leaving a small charged region.The outcome of such a process is the charge,which can be measured but not changed locally.The second way is closer to the idea of fusion. Indeed,it is also a fusion,but of islands instead of quasi-particles.The idea is depicted in Fig.2(c).As two is-lands of the same(N,M)type get closer,some of the ribbon operators connecting them become small and thus the?ux between the islands is exposed to local measure-ments.If we continue the approach till the islands meet, the?ux will take the form of a domain wall excitation at the meeting place,as in Fig.2(c).Due to con?nement the domain wall can decay to several smaller walls,but there is something that will not disappear,the two bor-der charges in its ends on B.As explained in the caption of Fig.2(c),the appearance of this border charges can be seen directly in terms of ribbon operators.Regarding the initialization of the system,reverse processes can be used.That is,if an island is divided in two,the topolog-ical?ux in between them will be trivial,and if an island is created from the vacuum,it will have trivial charge. In both cases the reason is that topological properties cannot be changed by local processes. Acknowledgements We acknowledge?nancial support from a PFI grant of the EJ-GV(H.B.),DGS grants un-der contracts BFM2003-05316-C02-01,FIS2006-04885 (H.B.,M.A.M.D,),and the ESF Science Programme IN-STANS2005-2010(M.A.M.D.).

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采购订单PURCHASE ORDER

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PurchaseOrder模板

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输入必要的信息，save , 系统提示创建了计划协议5500000001。 使用me35l批准计划协议（否则不能做库存收货）， 点击Release+save批准计划协议。 2. 维护交货计划deliveryschedule （tcode:me38 ） path:logistics->materialmanagement->purchasing->out lineagreement->schedulingagreement ->deliveryschedule->maintain enter , 选中行，点击deliveryschedule 按钮（或者F2）, 假设企业对物料b-16mnR-12的需求是均衡的，每个月 计划交货100kg , save。 说明一下，这里秀才犯了一个错误，前面计划协议的targetquantity=7000 ,但是这里的交行计划合计数只有700 （本意是每个月1000）,这属于低级失误。不过不影响后续工作。 3. 收货（tcode:migo_gR ） path:logistics->materialmanagement->inventorymanage ment->goodsmovement->goods

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最小重量MiximumWeight 总计容量TotalCapacity 出库单DeliveryList 拣货PickingGoods 拣货区PickingArea 转储单TransferOrder 检验报告单InspectionDocument 物料清单BillofMaterial 料号PartNumber 电子单据ElectronicsNote 码盘Pallet-Sorting 分拣Picking/Sorting 采购订单PurchaseOrder（PO）手写单HandNote 到货通知ArrivalNotice 报关到货ImportingGoods 检验单InspectionNote 入库扫描EntryScanning 扫描Scan 存货InStock 库存Inventory/Stock 库存清单StockList 发货区ShippingArea 发货单Deliverysheet

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世界奢侈品大全及品牌介绍 前言 本书以华美的图文(20万字、546多幅彩图)详尽介绍了所选品牌的品牌特色、精神内涵、传奇故事及性能特点。是目前国内第一本收录类别最全、文字内容最详尽、图片资料最经典的大型的奢华品牌图鉴。 《名品盛宴》莅选了西方百年经典品牌、全球最有价值的高级消费品和奢侈品集团的付表性品牌，荟萃了时装、珠宝、钟表、饰品、男士用品、女士用品、饮品、美食、汽车、影音、酒店、度假胜地等27类235个奢华品牌。

男装物语：风华无限 优雅的风度，大气的线条，内敛的奢华，国际顶级品牌完美地诠释了男装的独特魅力，展现了高贵典雅的绅士气质。让你在举手投足间透射出自由、俊逸、无畏的鲜活气息和清洌神秘的叛逆韵味，微妙地触动新锐们那最易感动的心弦。这让你明白，男人力与美的魅力并不完全来自肉体，而来自精神与气质，服饰亲顺理成章地成为这种力量忠实的外化与延伸。 如何感悟高雅时尚的色彩与陎料，让轻盈，挺括取付传统的深沉、厚重？男装的气韵与奢华，在于每一处的精雕细琢。 起风的街头，一个男人，身着柏帛丽风衣、登喜路风镜、路易威登黑色旅行箱向你走来时，你的心沉了又起，你会想巴黎最后一班地铁的男主角是他，那是怎样的一种气韵...... 1.(意)乔治·阿玛尼：不着痕迹的优雅 (意)乔治·阿玛尼Giorgio Armani--不着痕迹的优雅 >>品牌精神

在两性性别愈趋混淆的年付，乔治·阿玛尼打破阴柔与阳刚的界限，引领女装走向中性化，崇尚自我风格的流露，成为世界高层主管和好莱坞影星们的最爱。好莱坞那句流行语：?当你不知道秔什么时，秔Armani就没错了?成为阿玛尼产品品质的最佳诠释。理智的分析态度及世界均衡观念是他设计服装的准则。他的禅学的理念和中性风格对时装界影响至深。 >>品牌故事 阿玛尼说：?我只知风格是一种方法。?他的设计理念归纳为：删除不必要装饰，强调舒适性和表现不繁复的优雅。乔治·阿玛尼于1934年出生在意大利的皮亚琴察。他在校内修科学课程，大学念医科，服兵役时担任助理医官。70年付中期他与情人西雅治·盖利奥提开始了男女装的生产。他重构传统男装，强调了倒三角形，使一种无结构亲装显得浪漫洒脱。 经过了20世纪六七十年付?嬉皮士?、?朋克?的纷杂混乱和变幻莫测，人们对那种光怪陆离的打扮方式也心存倦意，这时候阿玛尼那种高雅简洁，庄重洒脱的服装风格，十足的意大利大家风范，恰好满足了人们新的时装需求，使人耳目一新。然后他以?美国的乔治?装扮成理查德介入主流男性的潜意识，从而提供给美国男人意大利的优雅选择。《时付》杂志在1982年以封陎人物的方式授予了阿玛尼极高的荣誉，这是继迪奥以来第二位获得此殊荣的设计师。并被誉为?夹克衫之王?、?80年付的夏奈尔?。乔治·阿玛尼终于凭着天才与努力在米兰这个世界时装之都赢得了教皇般的显赫地位。 与整天戴太阳镜摇着纸扇的拉格菲尔德（Karl Lagerfield）和华丽放纵、作风另类的范思哲（Gianni Versace）相比，阿玛尼更像一位苦行僧——风格既不新潮亦非传统。追溯阿玛尼的经营历史，很少有可笑的或非常过时的设计。他能够在市场需求和优雅时尚之间创造一种近乎完美、仙人惊司的帄衡：中性色系，优雅的裁剪仙人无需刻意炫耀，同时删去设计中无关的细节，这也是对男装裁剪的一大贡献。 虽然男人来自火星，女人来自地球，但顶级男装却为每个季节增添了情调、风韵、姿态、气度。把你黑、灰色彩衣橱的革命进行到底吧。把时装的品位?品?成一种味道。 >>品牌鉴赏

奢侈品-LOGO大全

Bentley 宾利主要产品：车 Estée Lauder 雅诗兰黛主要产品：化妆品 Chanel 香奈尔主要产品：时装、香水 Rolex 劳力士主要产品：腕表 Tiffany 蒂芬尼主要产品：珠宝 梅塞德斯(奔驰) 主要产品：轿车 Gianni versace 范思哲主要产品：时装

Louis Vuitton 路易威登主要产品：皮具、箱包《LV》 Hennessy 轩尼诗主要产品：高级干邑 American Express 美国运通主要产品：旅行信用卡 Prada 普拉达主要产品：时装、眼镜 Lancome 兰蔻主要产品：化妆品 Chivas 芝华士主要产品：威士忌

FERRARI 法拉利主要产品：车 BMW 宝马主要产品：轿车、摩托 Armani 阿玛尼主要产品：时装 Calvin Klein 卡尔文·克莱恩主要产品：内衣、香水 Rolls-Royce 劳斯莱斯主要产品：汽车 Harley·Davidson 哈雷戴维森主要产品：摩托 Ralph Lauren 拉尔夫劳伦主要产品：时装

Gucci 古姿主要产品：时装、香水 Remy Martin 人头马主要产品：名酒 Jaguar 捷豹主要产品：轿车 Starbucks 星巴克主要产品：咖啡店 Oakley 奥克利主要产品：眼镜 porsche 保时捷主要产品：名车 Dior 迪奥主要产品：时装、化妆品

CADILLAC 凯迪拉克主要产品：名车 Lamborghini 兰博基尼主要产品：名车 PAGANI 帕格尼主要产品：名车 Bugatti 布加迪主要产品：名车 Patek Philippe 百达翡丽主要产品：名表Bvgari 宝嘉丽主要产品：珠宝 Parker 派克主要产品：名笔

世界三大奢侈品集团及旗下品牌

世界三大奢侈品集团及旗下品牌 第一大奢侈品集团：法国的LVMH路易威登集团 公司类型：私有公司 成立时间：1987年 总部地点：巴黎 重要人物贝尔纳·阿尔诺 口号：创新突破，永不言息；超卓产品，完美无暇；品牌形象，集团灵魂；业务分散，员工进取；追求卓越，永无休止。 旗下拥有50多个世界奢侈顶级品牌，最主要的当然是Lovis Vuitton，LV在LVMH公司的营业总额中，占四分之一强，而在公司经营利润中，LV更是占到了60%左右。 法国酩悦·轩尼诗－路易·威登集团（Mo?t Hennessy - Louis Vuitton, LVMH Group）由贝尔纳·阿尔诺（Bernard Arnault）将全球著名的皮件公司Louis Vuitton与酒业家族Mo?t Hennessy於1987年合并而成， 员工约五万六千人，旗下拥有50多个品牌，是当今世界最大的精品集团。 集团主要业务包括以下五个领域： 葡萄酒及烈酒（Wines & Spirits）、 时装及皮革制品（Fashion & Leather Goods）、 香水及化妆品（Perfumes & Cosmetics）、 钟表及珠宝（Watches & Jewelry）、 精品零售（Selective Retailing）。

旗下品牌 葡萄酒及烈酒 酩悦香槟（Mo?t & Chandon）：成立於1743年，主营香槟。 香槟王（Dom Périgno n）：成立於18世纪，主营香槟。 凯歌香槟（Veuve Clicquot）：成立於1772年，主营香槟。 库克香槟（Krug）：成立於1843年，主营香槟。 梅西耶香槟（Mercier）：成立於1858年，主营香槟。 修纳尔香槟（Ruinart）：成立於1729年，主营香槟。 伊更堡（Chateau d'Yquem）：成立於1593年，主营高级葡萄酒。 轩尼诗（Hennessy）：成立於1765年，主营干邑。 格兰摩兰吉（Glenmorangie）：成立於1893年，主营苏格兰威士忌。 Domaine Chandon(California)：成立於1973年，主营香槟及加州葡萄酒。Bodegas Chandon(Argentina)：成立於1959年，主营香槟及阿根廷葡萄酒。Domaine Chandon(Australia) Green Point：成立於1986年，主营香槟及澳洲葡萄酒。云湾（Cloudy Bay）：成立於1985年，主营纽西兰葡萄酒。 曼达岬（Cape Mentelle）：成立於1976年，主营澳洲葡萄酒。 纽顿（Newton）：成立於1984年，主营加州葡萄酒。 安地斯之阶（Terrazas de los Andes）：成立於1999年，主营葡萄酒。 时装及皮革制品 路易威登（Louis Vuitton）：成立於1854年，主营皮革制品、成衣、鞋履、腕表、珠宝、纺织品及书写用品。 罗威（Loewe）：成立於1846年，主营皮革制品、成衣、丝绸饰品及香水。