A Relation Between Topological Quantum Field Theory and the Kodama State

电流诱导的斯格明子电流诱导的斯格明子1. 引言电流诱导的斯格明子是一个令人着迷的物理现象，它在凝聚态物理学领域引起了广泛的关注和研究。

斯格明子是一种拥有非阿贝尔任意子统计特性的量子激发态，在拓扑量子计算和量子信息传输等领域具有重要的应用潜力。

本文将深入探讨电流诱导的斯格明子，包括其形成机制、特性和应用前景。

2. 什么是斯格明子斯格明子最早由诺贝尔物理学奖得主斯图尔特·斯格明（StewartSutherland）在1977年提出。

斯格明子是一种拓扑孤立激发态，与费米子和玻色子不同，具有非阿贝尔任意子统计特性。

斯格明子的存在和性质可以通过拓扑相变的理论描述。

在二维拓扑绝缘体中，斯格明子可以通过局域的扰动导致量子霍尔效应的转变，从而产生电流诱导的斯格明子。

3. 电流诱导的斯格明子的形成机制电流诱导的斯格明子形成机制可以通过量子霍尔效应的理论来解释。

在二维系统中，当外加磁场强度达到一定值时，系统会发生量子霍尔效应。

在特定的边界条件下，电流通过二维系统时会导致斯格明边界模式的出现。

这些斯格明边界模式具有非阿贝尔任意子统计特性，形成了电流诱导的斯格明子。

4. 电流诱导的斯格明子的特性电流诱导的斯格明子具有许多特殊的性质，使其在量子计算和量子信息传输等领域具有重要的应用潜力。

（1）非阿贝尔任意子统计特性：斯格明子具有非阿贝尔任意子统计特性，可以实现量子纠缠和量子计算的高效率。

（2）鲁棒性：电流诱导的斯格明子形成后，对外界的微弱扰动具有鲁棒性，能够保持其拓扑性质和非阿贝尔任意子统计。

（3）操控性：通过调节外界磁场和电流的强度，可以操控电流诱导的斯格明子的位置和数量，实现对量子信息的操控。

5. 电流诱导的斯格明子的应用前景电流诱导的斯格明子在量子计算和量子信息传输等领域具有广泛的应用前景。

（1）拓扑量子计算：电流诱导的斯格明子可以作为量子比特来实现拓扑量子计算，具有较强的抗误差性能和可扩展性。

（2）量子信息传输：斯格明子的非阿贝尔任意子统计特性使其能够实现量子纠缠和量子通信的高效率，有望应用于量子通信网络的构建。

有效质量英语Effective MassThe concept of effective mass is a fundamental principle in physics that has far-reaching implications in various fields, from semiconductor technology to particle physics. Effective mass is a crucial parameter that describes the behavior of particles, particularly in the context of their interaction with external forces or fields.At its core, effective mass is a measure of the "apparent" or "effective" mass of a particle, which can differ from its actual or rest mass. This difference arises due to the complex interactions between the particle and its surrounding environment, such as the potential energy fields or the crystal structure of a material.In the case of a free particle, such as an electron in a vacuum, its effective mass is equal to its rest mass. However, when a particle is subjected to a potential energy field or is embedded within a material, its effective mass can deviate significantly from its rest mass. This phenomenon is particularly evident in semiconductor materials, where the effective mass of charge carriers (electrons and holes) plays a crucial role in determining the material's electronic andoptical properties.In semiconductor materials, the periodic potential created by the crystal lattice can significantly modify the effective mass of charge carriers. This modification is a result of the complex interactions between the charge carriers and the periodic potential, which can be described using the principles of quantum mechanics.The effective mass of charge carriers in semiconductors is a key parameter that determines the mobility, conductivity, and other important characteristics of the material. For example, in the design of electronic devices such as transistors and integrated circuits, the effective mass of charge carriers is a critical factor in optimizing device performance and efficiency.Moreover, the concept of effective mass extends beyond the realm of semiconductor physics. In particle physics, the effective mass of particles, such as subatomic particles or quasiparticles, is crucial for understanding their behavior and interactions within complex systems. For instance, the effective mass of particles in high-energy physics experiments can provide insights into the fundamental nature of matter and the forces that govern the universe.One of the most fascinating aspects of effective mass is its ability to exhibit exotic and counterintuitive behavior. In certain materials, suchas graphene or topological insulators, the charge carriers can exhibit an effective mass that is negative or even diverges to infinity. These unusual effective mass properties can lead to the emergence of novel physical phenomena, such as the quantum Hall effect or the formation of Dirac or Weyl fermions.The study of effective mass has also been instrumental in the development of cutting-edge technologies, such as quantum computing and spintronics. In these fields, the manipulation and control of the effective mass of charge carriers or spin-polarized particles are crucial for the realization of advanced devices and the exploration of quantum mechanical effects.In conclusion, the concept of effective mass is a fundamental principle in physics that has far-reaching implications across various disciplines. From semiconductor technology to particle physics, the effective mass of particles plays a vital role in understanding and predicting the behavior of complex systems. As scientific research continues to push the boundaries of our understanding, the study of effective mass will undoubtedly remain a crucial area of investigation, with the potential to unlock new frontiers in our quest to unravel the mysteries of the physical world.。

本科毕业论文（本科毕业设计题目：新型拓扑绝缘材料的研究摘要拓扑绝缘体是一种新的量子物态，为近几年来凝聚态物理学的重要科学前沿之一，已经引起的巨大的研究热潮。

拓扑绝缘体具有新奇的性质，虽然与普通绝缘体一样具有能隙，但拓扑性质不同，在自旋一轨道耦合作用下，在其表面或与普通绝缘体的界面上会出现无能隙、自旋劈裂且具有线性色散关系的表面/界面态。

这些态受时间反演对称性保护，不会受到杂质和无序的影响，由无质量的狄拉克(Dirac)方程所描述。

从广义上来说，拓扑绝缘体可以分为两大类：一类是破坏时间反演的量子霍尔体系，另一类是新近发现的时间反演不变的拓扑绝缘体，这些材料的奇特物理性质存在着很好的应用前景。

理论上预言，拓扑绝缘体和磁性材料或超导材料的界面，还可能发现新的物质相和预言的Majorana费米子，它们在未来的自旋电子学和量子计算中将会有重要应用。

拓扑绝缘体还与近年的研究热点如量子霍尔效应、量子自旋霍尔效应等领域紧密相连，其基本特征都是利用物质中电子能带的拓扑性质来实现各种新奇的物理性质。

关键词：拓扑绝缘体，量子霍尔效应，量子自旋霍尔效应，Majorana费米子AbstractIn recent years, one of the important frontiers in condensed matter physics, topological insulators are a new quantum state, which has attract many researchers attention. Topological insulators show some novel properties, although normal insulator has the same energy gap, but topological properties are different. Under the action of spin-orbit coupling interaction, on the surface or or with normal insulator interface will appear gapless, spin-splitting and with the linear dispersion relation of surface or interface states. These states are conserved by the time reversal symmetry and are not affected by the effect of the impurities and disorder, which is described by the massless Dirac equation. Broadly defined, topological insulators can be separated into two categories: a class is destroy time reversal of the quantum Hall system, another kind is the newly discovered time reversal invariant topological insulators, peculiar physical properties of these materials exist very good application prospect. Theoretically predicted, the interface of topological insulators and magnetic or superconducting material, may also find new material phase and the prophecy of Majorana fermion, they will have important applications in the future spintronics and quantum computing . Topological insulators also are closely linked with the research hotspot in recent years, such as the quantum Hall effect, quantum spin Hall effect and other fields. Its basic characteristics are to achieve a variety of novel physical properties by using the topological property of the material of the electronic band.Keywords:Topological insulator;quantum hall effect;quantum spin-Hall effect;Majorana fermion目录引言 (1)第一章拓扑绝缘体简介 (2)1.1 绝缘体、导体和拓扑绝缘 (2)1.2 二维拓扑绝缘体 (3)1.3三维拓扑绝缘体 (3)第二章拓扑绝缘体的研究进展与现状 (5)2.1拓扑绝缘体研究进展 (5)2.2拓扑绝缘体的研究现状 (5)第三章拓扑绝缘体材料的制备方法与特性 (7)3.1 拓扑绝缘体Bi Se的结构 (7)233.2 拓扑绝缘体的制备Bi Se的制备 (8)233.3 SnTe拓扑晶态绝缘体制备 (8)3.4拓扑绝缘体的特性 (9)结论 (10)参考文献 (11)谢辞 (13)引言拓扑绝缘体是一种新的量子物态，为近几年来凝聚态物理学的重要科学前沿之一，已经引起的巨大的研究热潮。

新的125个科学问题2005年，在庆祝创刊125周年之际，Science公布了125个最具挑战性的科学问题，对指引近十几年的科学发展产生积极影响。

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SJTU & Science 125个科学问题Mathematical Sciences数学1. What makes prime numbers so special?1.什么使素数如此特别？2. Will the Navier–Stokes problem ever be solved?2.纳维尔-斯托克斯问题会得到解决吗？3. Is the Riemann hypothesis true?3.黎曼猜想是真的吗？Chemistry1. Are there more color pigments to discover?1.还有更多色彩元素可发现吗？2. Will the periodic table ever be complete?2.元素周期表会完整吗？3. How can we measure interface phenomena on the microscopic level?3.如何在微观层面测量界面现象？4. What is the future for energy storage？4.能量存储的未来是怎样的？5. Why does life require chirality?5.为什么生命需要手性？6. How can we better manage the world's plastic waste?6.我们如何更好地管理世界上的塑料废物？7. Will AI redefine the future of chemistry?7.AI会重新定义化学的未来吗？8. How can matter be programmed into living materials?8.物质如何被编码而成为生命材料？9. What drives reproduction in living systems?9.是什么驱动生命系统的复制？Medicine & Health医学与健康1. Can we predict the next pandemic?1.我们可以预测下一次流行病吗？2. Will we ever find a cure for the common cold?2.我们会找到治疗感冒的方法吗？3. Can we design and manufacture medicines customized for individual people?3.我们可以设计和制造出为个人定制的药物吗？4. Can a human tissue or organ be fully regenerated?4.人体组织或器官可以完全再生吗？5. How is immune homeostasis maintained and regulated?5.如何维持和调节免疫稳态？6. Is there a scientific basis to the Meridian System in traditional Chinese medicine?6.中医的经络系统有科学依据吗？7. How will the next generation of vaccines be made?7.下一代疫苗将如何生产？8. Can we ever overcome antibiotic resistance?8.我们能否克服抗生素耐药性？9. What is the etiology of autism?9.自闭症的病因是什么？10. What role does our microbiome play in health and disease?10.我们的微生物组在健康和疾病中扮演什么角色？11. Can xenotransplantation solve the shortage of donor organs?11.异种移植能否解决供体器官的短缺问题？Biology生命科学1. What could help conservation of the oceans?1.什么可以帮助保护海洋？2. Can we stop ourselves from aging?2.我们可以阻止自己衰老吗？3. Why can only some cells become other cells?3.为什么只有一些细胞会变成其他细胞？4. Why are some genomes so big and others very small?4.为什么有些基因组非常大而另一些却很小？5. Will it be possible to cure all cancers?5.有可能治愈所有癌症吗？6. What genes make us uniquely human?6.哪些基因使我们人类与众不同？7. How do migratory animals know where they're going?7.迁徙动物如何知道它们要去哪里？8. How many species are there on Earth?8.地球上有多少物种？9. How do organisms evolve?9.有机体是如何进化的？10. Why did dinosaurs grow to be so big?10.为什么恐龙长得如此之大？11. Did ancient humans interbreed with other human-like ancestors?11.远古人类是否曾与其他类人祖先杂交？12. Why do humans get so attached to dogs and cats?12.人类为什么会对猫狗如此着迷？13. Will the world's population keep growing indefinitely?13.世界人口会无限增长吗？14. Why do we stop growing?14.我们为什么会停止生长？15. Is de-extinction possible?15.能否复活灭绝生物？16. Can humans hibernate?16.人类可以冬眠吗？17. Where do human emotions originate?17.人类的情感源于何处？18. Will humans look physically different in the future?18.未来人类的外貌会有所不同吗？19. Why were there species explosions and mass extinction?19.为什么会发生物种大爆发和大灭绝？20. How might genome editing be used to cure disease?20.基因组编辑将如何用于治疗疾病？21. Can a cell be artificially synthesized?21.可以人工合成细胞吗？22. How are biomolecules organized in cells to function orderly and effectively?22.细胞内的生物分子是如何组织从而有序有效发挥作用的？Astronomy天文学1. How many dimensions are there in space?1.空间中有多少个维度？2. What is the shape of the universe?2.宇宙的形状是怎样的？3. Where did the big bang start?3.大爆炸从何处开始？4. Why don't the orbits of planets decay and cause them to crash into each other?4.为什么行星的轨道不衰减并导致它们相互碰撞？5. When will the universe die? Will it continue to expand?5.宇宙何时消亡？它会继续膨胀吗？6. Is it possible to live permanently on another planet?6.我们有可能在另一个星球上长期居住吗？7. Why do black holes exist?7.为什么存在黑洞？8. What is the universe made of?8.宇宙是由什么构成的？9. Are we alone in the universe?9.我们是宇宙中唯一的生命体吗？10. What is the origin of cosmic rays?10.宇宙射线的起源是什么？11. What is the origin of mass?11.物质的起源是什么？12. What is the smallest scale of space-time?12.时空的最小尺度是是多少？13. Is water necessary for all life in the universe, or just on Earth?13.水是宇宙中所有生命所必需的么，还是仅对地球生命？14. What is preventing humans from carrying out deep-space exploration?14.是什么阻止了人类进行深空探测？15. Is Einstein's general theory of relativity correct?15.爱因斯坦的广义相对论是正确的吗？16. How are pulsars formed?16.脉冲星是如何形成的？17. Is our Milky Way Galaxy special?17.我们的银河系特别吗？18. What is the volume, composition, and significance of the deep biosphere?18.深层生物圈的规模、组成和意义是什么？19. Will humans one day have to leave the planet (or die trying)?19.人类有一天会不得不离开地球吗（还是会在尝试中死去）？20. Where do the heavy elements in the universe come from?20.宇宙中的重元素来自何处？21. Is it possible to understand the structure of compact stars and matter?21.有可能了解致密恒星和物质的结构吗？22. What is the origin of the high-energy cosmic neutrinos?22.高能宇宙中微子的起源是什么？23. What is gravity?23.什么是重力？Physics物理学1. Is there a diffraction limit?1.有衍射极限吗？2. What is the microscopic mechanism for high-temperature superconductivity?2.高温超导的微观机理是什么？3. What are the limits of heat transfer in matter?3.物质传热的极限是什么？4. What are the fundamental principles of collective motion?4.集体运动的基本原理是什么？5. What are the smallest building blocks of matter?5.什么是物质的最小组成部分？6. Will we ever travel at the speed of light?6.我们会以光速行驶吗？7. What is quantum uncertainty and why is it important?7.什么是量子不确定性，为什么它很重要？8. Will there ever be a "theory of everything"?8.会有“万有理论”吗？9. Why does time seem to flow in only one direction?9.为什么时间似乎只朝一个方向流动？10. What is dark matter?10.什么是暗物质？11. Can we make a real, human-size invisibility cloak?11.我们可以制作出真人大小的隐形斗篷吗？12. Are there any particles that behave oppositely to the properties or states of photons?12.是否存在与光子性质或状态相反的粒子？13. Will the Bose-Einstein condensate be widely used in the future?13.玻色-爱因斯坦冷凝体未来会被广泛使用吗？14. Can humans make intense lasers with incoherence comparable to sunlight?14.人类能制造出与太阳光相似的非相干强激光吗？15. What is the maximum speed to which we can acceleratea particle?15.我们最多可以将粒子加速到多快？16. Is quantum many-body entanglement more fundamental than quantum fields?16.量子多体纠缠比量子场更基本吗？17. What is the optimum hardware for quantum computers?17.量子计算机的最佳硬件是什么？18. Can we accurately simulate the macro- and microworld?18.我们可以精确模拟宏观和微观世界吗？Information Science信息科学1. Is there an upper limit to computer processing speed?1.计算机处理速度是否有上限？2. Can AI replace a doctor?2.AI可以代替医生吗？3. Can topological quantum computing be realized?3.拓扑量子计算可以实现吗？4. Can DNA act as an information storage medium?4.DNA可以用作信息存储介质吗？Engineering & Material Science工程与材料科学1. What is the ultimate statistical invariances of turbulence?1.湍流的最终统计不变性是什么？2. How can we break the current limit of energy conversion efficiencies?2.我们如何突破当前的能量转换效率极限？3. How can we develop manufacturing systems on Mars?3.我们如何在火星上开发制造系统？4. Is a future of only self-driving cars realistic?4.纯无人驾驶汽车的未来是否现实？Neuroscience神经科学1. What are the coding principles embedded in neuronal spike trains?1.神经元放电序列的编码准则是什么？2. Where does consciousness lie?2.意识存在于何处？3. Can human memory be stored, manipulated, and transplanted digitally?3.能否数字化地存储、操控和移植人类记忆？4. Why do we need sleep?4.为什么我们需要睡眠？5. What is addiction and how does it work?5.什么是成瘾？6. Why do we fall in love?6.为什么我们会坠入爱河？7. How did speech evolve and what parts of the brain control it?7.言语如何演变形成，大脑的哪些部分对其进行控制？8. How smart are nonhuman animals?8.除人类以外的其他动物有多聪明？9. Why are most people right-handed?9.为什么大多数人都是右撇子？10. Can we cure neurodegenerative diseases?10.我们可以治愈神经退行性疾病吗？11. Is it possible to predict the future?11.有可能预知未来吗？12. Can we more effectively diagnose and treat complex mental disorders?12.精神障碍能否有效诊断和治疗？Ecology生态学1. Can we stop global climate change?1.我们可以阻止全球气候变化吗？2. Where do we put all the excess carbon dioxide?2.我们能把过量的二氧化碳存到何处？3. What creates the Earth's magnetic field (and why does it move)?3.是什么创造了地球的磁场（为什么它会移动）？4. Will we be able to predict catastrophic weather events (tsunami, hurricanes, earthquakes) more accurately?4.我们是否能够更准确地预测灾害性事件（海啸、飓风、地震）？5. What happens if all the ice on the planet melts?5.如果地球上所有的冰融化会怎样？6. Can we create an environmentally friendly replacement for plastics?6.我们可以创造一种环保的塑料替代品吗？7. Can we achieve a situation where essentially every material can be recycled and reused?7.几乎所有材料都可以回收再利用是否可以实现？8. Will we soon see the end of monocultures like wheat, maize, rice, and soy?8.我们会很快看到小麦、玉米、大米和大豆等单一作物的终结吗？Energy Science能源科学1. Could we live in a fossil-fuel-free world?1.我们可以生活在一个去化石燃料的世界中吗？2. What is the future of hydrogen energy?2.氢能的未来是怎样的？3. Will cold fusion ever be possible?3.冷聚变有可能实现吗？Artificial Intelligence人工智能1. Will injectable, disease-fighting nanobots ever be a reality?1.可注射的抗病纳米机器人会成为现实吗？2. Will it be possible to create sentient robots?2.是否有可能创建有感知力的机器人？3. Is there a limit to human intelligence?3.人类智力是否有极限？4. Will artificial intelligence replace humans?4.人工智能会取代人类吗？5. How does group intelligence emerge?5.群体智能是如何出现的？6. Can robots or AIs have human creativity?6.机器人或 AI 可以具有人类创造力吗？7. Can quantum artificial intelligence imitate the human brain?7.量子人工智能可以模仿人脑吗？8. Could we integrate with computers to form a human-machine hybrid species?8.我们可以和计算机结合以形成人机混合物种吗？。

a r X i v :q u a n t -p h /0102108v 2 9 O c t 2001Quantum Kolmogorov Complexity Based onClassical DescriptionsPaul M.B.Vit´a nyiAbstract —We develop a theory of the algorithmic informa-tion in bits contained in an individual pure quantum state.This extends classical Kolmogorov complexity to the quan-tum domain retaining classical descriptions.Quantum Kol-mogorov complexity coincides with the classical Kolmogorov complexity on the classical domain.Quantum Kolmogorov complexity is upper bounded and can be effectively approx-imated from above under certain conditions.With high probability a quantum object is incompressible.Upper-and lower bounds of the quantum complexity of multiple copies of individual pure quantum states are derived and may shed some light on the no-cloning properties of quantum states.In the quantum situation complexity is not sub-additive.We discuss some relations with “no-cloning”and “approximate cloning”properties.Keywords —Algorithmic information theory,quantum;classical descriptions of quantum states;information the-ory,quantum;Kolmogorov complexity,quantum;quantum cloning.I.IntroductionQUANTUM information theory,the quantum mechan-ical analogue of classical information theory [6],is ex-periencing a renaissance [2]due to the rising interest in the notion of quantum computation and the possibility of re-alizing a quantum computer [16].While Kolmogorov com-plexity [12]is the accepted absolute measure of information content in a individual classical finite object,a similar ab-solute notion is needed for the information content of an individual pure quantum state.One motivation is to extend probabilistic quantum information theory to Kolmogorov’s absolute individual notion.Another reason is to try and duplicate the success of classical Kolmogorov complexity as a general proof method in applications ranging from com-binatorics to the analysis of algorithms,and from pattern recognition to learning theory [13].We propose a theory of quantum Kolmogorov complexity based on classical de-scriptions and derive the results given in the abstract.A preliminary partial version appeared as [19].What are the problems and choices to be made develop-ing a theory of quantum Kolmogorov complexity?Quan-tum theory assumes that every complex vector of unit length represents a realizable pure quantum state [17].There arises the question of how to design the equipment that prepares such a pure state.While there are contin-uously many pure states in a finite-dimensional complexPartially supported by the EU fifth framework project QAIP,IST–1999–11234,the NoE QUIPROCONE IST–1999–29064,the ESF QiT Programmme,and the EU Fourth Framework BRA NeuroCOLT II Working Group EP 27150.Part of this work was done during the author’s 1998stay at Tokyo Institute of Technology,Tokyo,Japan,as Gaikoku-Jin Kenkyuin at INCOCSAT,and appeared in a preliminary version [19]archived as quant-ph/9907035.Address:CWI,Kruislaan 413,1098SJ Amsterdam,The Netherlands.Email:paulv@cwi.nlvector space—corresponding to all vectors of unit length—we can finitely describe only a countable subset.Imposing effectiveness on such descriptions leads to constructive pro-cedures.The most general such procedures satisfying uni-versally agreed-upon logical principles of effectiveness are quantum Turing machines,[3].To define quantum Kol-mogorov complexity by way of quantum Turing machines leaves essentially two options:1.We want to describe every quantum superposition ex-actly;or2.we want to take into account the number of bits/qubits in the specification as well the accuracy of the quantum state produced.We have to deal with three problems:•There are continuously many quantum Turing machines;•There are continuously many pure quantum states;•There are continuously many qubit descriptions.There are uncountably many quantum Turing machines only if we allow arbitrary real rotations in the definition of machines.Then,a quantum Turing machine can only be universal in the sense that it can approximate the compu-tation of an arbitrary machine,[3].In descriptions using universal quantum Turing machines we would have to ac-count for the closeness of approximation,the number of steps required to get this precision,and the like.In con-trast,if we fix the rotation of all contemplated machines to a single primitive rotation θwith cos θ=35,then there are only countably many Turing machines and the universal machine simulates the others exactly [1].Ev-ery quantum Turing machine computation,using arbitrary real rotations to obtain a target pure quantum state,can be approximated to every precision by machines with fixed rotation θbut in general cannot be simulated exactly—just like in the case of the simulation of arbitrary quantum Turing machines by a universal quantum Turing machine.Since exact simulation is impossible by a fixed universal quantum Turing machine anyhow,but arbitrarily close ap-proximations are possible by Turing machines using a fixed rotation like θ,we are motivated to fix Q 1,Q 2,...as a stan-dard enumeration of quantum Turing machines using only rotation θ.Our next question is whether we want programs (descrip-tions)to be in classical bits or in qubits?The intuitive no-tion of computability requires the programs to be ly,to prepare a quantum state requires a physical ap-paratus that “computes”this quantum state from classical specifications.Since such specifications have effective de-scriptions,every quantum state that can be prepared can be described effectively in descriptions consisting of classi-cal bits.Descriptions consisting of arbitrary pure quantumstates allows noncomputable(or hard to compute)informa-tion to be hidden in the bits of the amplitudes.In Defini-tion4we call a pure quantum state directly computable if there is a(classical)program such that the universal quan-tum Turing machine computes that state from the program and then halts in an appropriate fashion.In a computa-tional setting we naturally require that directly computable pure quantum states can be prepared.By repeating the preparation we can obtain arbitrarily many copies of the pure quantum state.If descriptions are not effective then we are not going to use them in our algorithms except possibly on inputs from an“unprepared”origin.Every quantum state used in a quantum computation arises from some classically prepa-ration or is possibly captured from some unknown origin. If the latter,then we can consume it as conditional side-information or an oracle.Restricting ourselves to an effective enumeration of quan-tum Turing machines and classical descriptions to describe by approximation continuously many pure quantum states is reminiscent of the construction of continuously many real numbers from Cauchy sequences of rational numbers,the rationals being effectively enumerable.Kolmogorov complexity:We summarize some basic definitions in Appendix A(see also this journal[20])in order to establish notations and recall the notion of short-est effective descriptions.More details can be found in the textbook[13].Shortest effective descriptions are“effective”in the sense that they are programs:we can compute the described objects from them.Unfortunately,[12],there is no algorithm that computes the shortest program and then halts,that is,there is no general method to compute the length of a shortest description(the Kolmogorov com-plexity)from the object being described.This obviously impedes actual use.Instead,one needs to consider com-putable approximations to shortest descriptions,for exam-ple by restricting the allowable approximation time.Apart from computability and approximability,there is another property of descriptions that is important to us.A set of descriptions is prefix-free if no description is a proper prefix of another description.Such a set is called a prefix code. Since a code message consists of concatenated code words, we have to parse it into its constituent code words to re-trieve the encoded source message.If the code is uniquely decodable,then every code message can be decoded in only one way.The importance of prefix-codes stems from the fact that(i)they are uniquely decodable from left to right without backing up,and(ii)for every uniquely decodable code there is a prefix code with the same length code words. Therefore,we can restrict ourselves to prefix codes.In our setting we require the set of programs to be prefix-free and hence to be a prefix-code for the objects being described.It is well-known that with every prefix-code there corresponds a probability distribution P(·)such that the prefix-code is a Shannon-Fano code1that assigns prefix code length l x=−log P(x)to x—irrespective of the regularities in x. 1In what follows,“log”denotes the binary logarithm.For example,with the uniform distribution P(x)=2−n on the set of n-bit source words,the Shannon-Fano code word length of an all-zero source word equals the code word length of a truly irregular source word.The Shannon-Fano code gives an expected code word length close to the en-tropy,and,by Shannon’s Noiseless Coding Theorem,it possesses the optimal expected code word length.But the Shannon-Fano code is not optimal for individual elements: it does not take advantage of the regularity in some ele-ments to encode those shorter.In contrast,one can view the Kolmogorov complexity K(x)as the code word length of the shortest program x∗for x,the set of shortest pro-grams consitituting the Shannon-Fano code of the so-called “universal distribution”m(x)=2−K(x).The code consist-ing of the shortest programs has the remarkable property that it achieves(i)an expected code length that is about optimal since it is close to the entropy,and simultaneously, (ii)every individual object is coded as short as is effectively possible,that is,squeezing out all regularity.In this sense the set of shortest programs constitutes the optimal effec-tive Shannon-Fano code,induced by the optimal effective distribution(the universal distribution).Quantum Computing:We summarize some basic def-initions in Appendix B in order to establish notations and briefly review the notion of a quantum Turing machine computation.See also this journal’s survey[2]on quan-tum information theory.More details can be found in the textbook[16].Loosely speaking,like randomized compu-tation is a generalization of deterministic computation,so is quantum computation a generalization of randomized computation.Realizing a mathematical random source to drive a random computation is,in its ideal form,presum-ably impossible(or impossible to certify)in practice.Thus, in applications an algorithmic random number generator is used.Strictly speaking this invalidates the analysis based on mathematical randomized computation.As John von Neumann[15]put it:“Any one who considers arithmetical methods of producing random digits is,of course,in a state of sin.For,as has been pointed out several times,there is no such thing as a random number—there are only meth-ods to produce random numbers,and a strict arithmetical procedure is of course not such a method.”In practice ran-domized computations reasonably satisfy theoretical anal-ysis.In the quantum computation setting,the practical problem is that the ideal coherent superposition cannot re-ally be maintained during computation but deteriorates—it decoheres.In our analysis we abstract from that problem and one hopes that in practice anti-decoherence techniques will suffice to approximate the idealized performance suffi-ciently.We view a quantum Turing machine as a generalization of the classic probabilistic(that is,randomized)Turing machine.The probabilistic Turing machine computation follows multiple computation paths in parallel,each path with a certain associated probability.The quantum Turing machine computation follows multiple computation paths in parallel,but now every path has an associated complex probability amplitude.If it is possible to reach the sameVIT´ANYI:QUANTUM KOLMOGOROV COMPLEXITY BASED ON CLASSICAL DESCRIPTIONS3state via different paths,then in the probabilistic case the probability of observing that state is simply the sum of the path probabilities.In the quantum case it is the squared norm of the summed path probability amplitudes.Since the probability amplitudes can be of opposite sign,the ob-servation probability can vanish;if the path probability amplitudes are of equal sign then the observation probabil-ity can get boosted since it is the square of the sum norm. While this generalizes the probabilistic aspect,and boosts the computation power through the phenomenon of inter-ference between parallel computation paths,there are extra restrictions vis-a-vis probabilistic computation in that the quantum evolution must be unitary.Quantum Kolmogorov Complexity:We define the Kolmogorov complexity of a pure quantum state as the length of the shortest two-part code consisting of a classical program to compute an approximate pure quantum state and the negative log-fidelity of the approximation to the target quantum state.We show that the resulting quantum Kolmogorov complexity coincides with the classical self-delimiting complexity on the domain of classical objects; and that certain properties that we love and cherish in the classical Kolmogorov complexity are shared by the new quantum Kolmogorov complexity:quantum Kolmogorov complexity of an n-qubit object is upper bounded by about 2n;it is not computable but can under certain conditions be approximated from above by a computable process;and with high probability a quantum object is incompressible. We may call this quantum Kolmogorov complexity the bit complexity of a pure quantum state|φ (using Dirac’s“ket”notation)and denote it by K(|φ ).From now on,we will denote by+<an inequality to within an additive constant, and by+=the situation when both+<and+>hold.For exam-ple,we will show that,for n-qubit states|φ ,the complexity satisfies K(|φ |n)+<2n.For certain restricted pure quan-tum states,quantum kolmogorov complexity satisfies the sub-additive property:K(|φ,ψ )+<K(|φ )+K(|ψ ||φ ). But,in general,quantum Kolmogorov complexity is not sub-additive.Although“cloning”of non-orthogonal states is forbidden in the quantum setting[21],[7],m copies of the same quantum state have combined complexity that can be considerable lower than m times the complexity of a single copy.In fact,quantum Kolmogorov complex-ity appears to enable us to express and partially quantify “non-clonability”and“approximate clonability”of individ-ual pure quantum states.Related Work:In the classical situation there are sev-eral variants of Kolmogorov complexity that are very mean-ingful in their respective settings:plain Kolmogorov com-plexity,prefix complexity,monotone complexity,uniform complexity,negative logarithm of universal measure,and so on[13].It is therefore not surprising that in the more com-plicated situation of quantum information several different choices of complexity can be meaningful and unavoidable in different settings.Following the preliminary version[19] of this work there have been alternative proposals:Qubit Descriptions:The most straightforward way to define a notion of quantum Kolmogorov complexity is to consider the shortest effective qubit description of a pure quantum state which is studied in[4].(This qubit com-plexity can also be formulated in terms of the conditional version of bit complexity as in[19].)An advantage of qubit complexity is that the upper bound on the complexity of a pure quantum state is immediately given by the number of qubits involved in the literal description of that pure quan-tum state.Let us denote the resulting qubit complexity of a pure quantum state|φ by KQ(|φ ).While it is clear that(just as with the previous aproach) the qubit complexity is not computable,it is unlikely that one can approximate the qubit complexity from above by a computable process in some meaningful sense.In particu-lar,the dovetailing approach we used in our approach now doesn’t seem applicable due to the non-countability of the potentential qubit program candidates.The quantitative incompressibility properties are much like the classical case (this is important for future applications).There are some interesting exceptions in case of objects consisting of multi-ple copies related to the“no-cloning”property of quantum objects,[21],[7].Qubit complexity does not satisfy the sub-additive property,and a certain version of it(bounded fidelity)is bounded above by the von Neumann entropy. Density Matrices:In classical algorithmic informa-tion theory it turns out that the negative logarithm of the “largest”probability distribution effectively approximable from below—the universal distribution—coincides with the self-delimiting Kolmogorov complexity.In[8]G´a cs defines two notions of complexities based on the negative loga-rithm of the“largest”density matrixµeffectively approx-imable from below.There arise two different complexi-ties of|φ based on whether we take the logarithm inside as KG(|φ )=− φ|logµ|φ or outside as Kg(|φ )=−log φ|µ|φ .It turns out that Kg(|φ )+<KG(|φ ). This approach serves to compare the two approaches above: It was shown that Kg(|φ )is within a factor four of K(|φ ); that KG(|φ )essentially is a lower bound on KQ(|φ )and an oracle version of KG is essentially an upper bound on qubit complexity KQ.Since qubit complexity is trivially+<n and it was shown that bit complexity is typically close to2n,atfirst glance this leaves the possibility that the two complexities are within a factor two of each other.This turns out to be not the case since it was shown that the Kg complexity can for some arguments be much smaller than the KG complexity,so that the bit complexity is in these cases also much smaller than the qubit complexity.As[8] states:this is due to the permissive way the bit complexity deals with approximation.The von Neumann entropy of a computable density matrix is within an additive constant (the complexity of the program computing the density ma-trix)of a notion of average complexity.The drawback of density matrix based complexity is that we seem to have lost the direct relation with a meaningful interpretation in terms of description length:a crucial aspect of classical Kolmogorov complexity in most applications[13].Real Descriptions:A version of quantum Kolmogorov4IEEE TRANSACTIONS ON INFORMATION THEORYcomplexity briefly considered in[19]uses computable real parameters to describe the pure quantum state with com-plex probability amplitudes.This requires two reals per complex probability amplitude,that is,for n qubits one requires2n+1real numbers in the worst case.A real num-ber is computable if there is afixed program that outputs consecutive bits of the binary expansion of the number for-ever.Since every computable real number may require a separate program,a computable n-qubit pure state may re-quire2n+1finite programs.Most n-qubit pure states have parameters that are noncomputable and increased preci-sion will require increasingly long programs.For exam-ple,if the parameters are recursively enumerable(the po-sitions of the“1”s in the binary expansion is a recursively enumerable set),then a log k length program per parame-ter,to achieve k bits precision per recursively enumerable real,is sufficient and for some recursively enumerable re-als also necessary.In certain contexts where the approx-imation of the real parameters is a central concern,such considerations may be useful.While this approach does not allow the development of a clean theory in the sense of the previous approaches,it can be directly developed in terms of algorithmic thermodynamics—an extension of Kolmogorov complexity to randomness of infinite sequences (such as binary expansions of real numbers)in terms of coarse-graining and sequential Martin-L¨o f tests,analogous to the classical case in[9],[13].But this is outside the scope of the present paper.II.Quantum Turing Machine ModelWe assume the notation and definitions in Appendices A, B.Our model of computation is a quantum Turing ma-chine equipped with a input tape that is one-way infinite with the classical input(the program)in binary left ad-justed from the beginning.We require that the input tape is read-only from left-to-right without backing up.This automatically yields a property we require in the sequel: The set of halting programs is prefix-free.Additionaly,the machine contains a one-way infinite work tape containing qubits,a one-way infinite auxiliary tape containing qubits, and a one-way infinite output tape containing qubits.Ini-tially,the input tape contains a classical binary program p, and all(qu)bits of the work tape,auxiliary tape,and out-put tape qubits are set to|0 .In case the Turing machine has an auxiliary input(classical or quantum)then initially the leftmost qubits of the auxiliary tape contain this in-put.A quantum Turing machine Q with classical program p and auxiliary input y computes until it halts with output Q(p,y)on its output tape or it computes forever.Halt-ing is a more complicated matter here than in the classical case since quantum Turing machines are reversible,which means that there must be an ongoing evolution with non-repeating configurations.There are various ways to resolve this problem[3]and we do not discuss this matter further. We only consider quantum Turing machine that do not modify the output tape after halting.Another—related—problem is that after halting the quantum state on the out-put tape may be“entangled”with the quantum state of the remainder of the machine,that is,the input tape,thefinite control,the work tape,and the auxilliary tape.This hasthe effect that the output state viewed in isolation may notbe a pure quantum state but a mixture of pure quantumstates.This problem does not arise if the output and the remainder of the machine form a tensor product so that theoutput is un-entangled with the remainder.The results inthis paper are invariant under these different assumptions,but considering output entangled with the remainder ofthe machine complicates formulas and calculations.Corre-spondingly,we restrict consideration to outputs that forma tensor product with the remainder of the machine,withthe understanding that the same results hold with aboutthe same proofs if we choose the other option—except inthe case of Theorem4item(ii),see the pertinent caveat there.Note that the Kolmogorov complexity based on en-tangled output tapes is at most(and conceivably less than)the Kolmogorov complexity based on un-entangled outputtapes.Definition1:Define the output Q(p,y)of a quantumTuring machine Q with classical program p and auxil-iary input y as the pure quantum state|ψ resulting of Q computing until it halts with output|ψ on its ouputtape.Moreover,|ψ doesn’t change after halting,andit is un-entangled with the remainder of Q’s configura-tion.We write Q(p,y)<∞.If there is no such|ψthen Q(p,y)is undefined and we write Q(p,y)=∞.By definition the input tape is read-only from left-to-rightwithout backing up:therefore the set of halting programsP y={p:Q(p,y)<∞}is prefix-free:no program in P y is a proper prefix of another program in P y.Put differ-ently,the Turing machine scans all of a halting program p but never scans the bit following the last bit of p:it isself-delimiting.Wefix the rotation of all contemplated machines to a sin-gle primitive rotationθwith cosθ=35.Thereare only countably many such Turing ing astandard ordering,wefix Q1,Q2,...as a standard enumer-ation of quantum Turing machines using only rotationθ. By[1],there is a universal machine U in this enumeration that simulates the others exactly:U(1i0p,y)=Q i(p,y), for all i,p,y.(Instead of the many-bit encoding1i0for i we can use a shorter self-delimiting code like i′in Ap-pendix A.)As noted in the Introduction,every quantum Turing machine computation using arbitrary real rotations can be approximated to arbitrary precision by machines withfixed rotationθbut in general cannot be simulated exactly.Remark1:There are two possible interpretations for the computation relation Q(p,y)=|x .In the narrow interpre-tation we require that Q with p on the input tape and y on the conditional tape halts with|x on the output tape.In the wide interpretation we can define pure quantum states by requiring that for every precision parameter k>0the computation of Q with p on the input tape and y on the conditional tape,with k on a special new tape where the precision is to be supplied,halts with|x′ on the output tape and|| x|x′ ||2≥1−1/2k.Such a notion of“com-VIT ´ANYI:QUANTUM KOLMOGOROV COMPLEXITY BASED ON CLASSICAL DESCRIPTIONS5putable”or “recursive”pure quantum states is similar to Turing’s notion of “computable numbers.”In the remain-der of this section we use the narrow interpretation.Remark 2:As remarked in [8],the notion of a quan-tum computer is not essential to the theory here or in [4],[8].Since the computation time of the machine is not limited in the theory of description complexity as de-veloped here,a quantum computer can be simulated by a classical computer to every desired degree of precision.We can rephrase everything in terms of the standard enu-meration of T 1,T 2,...of classical Turing machines.Let |x = N −1i =0αi |e i (N =2n )be an n -qubit state.We can write T (p )=|x if T either outputs(i)algebraic definitions of the coefficients of |x (in case these are algebraic),or(ii)a sequence of approximations (α0,k ,...,αN −1,k )for k =1,2,...where αi,k is an algebraic approximation of αi to within 2−k .III.Classical Descriptions of Pure QuantumStates The complex quantity x |z is the inner product of vec-tors x |and |z .Since pure quantum states |x ,|z have unit length,|| x |z ||=|cos θ|where θis the angle between vectors |x and |z .The quantity || x |z ||2,the fidelity between |x and |z ,is a measure of how “close”or “con-fusable”the vectors |x and |z are.It is the probability of outcome |x being measured from state |z .Essentially,we project |z on outcome |x using projection |x x |resulting in x |z |x .Definition 2:The (self-delimiting)complexity of |x with respect to quantum Turing machine Q with y as conditional input given for free isK Q (|x |y )=min p{l (p )+⌈−log || z |x ||2⌉:Q (p,y )=|z }(1)where l (p )is the number of bits in the program p ,auxiliary y is an input (possibly quantum)state,and |x is the target state that one is trying to describe.Note that |z is the quantum state produced by the com-putation Q (p,y ),and therefore,given Q and y ,completely determined by p .Therefore,we obtain the minimum of the right-hand side of the equality by minimizing over p only.We call the |z that minimizes the right-hand sidethe directly computed part of |x while ⌈−log || z |x ||2⌉is the approximation part .Quantum Kolmogorov complexity is the sum of two terms:the first term is the integral length of a binary pro-gram,and the second term,the minlog probability term,corresponds to the length of the corresponding code word in the Shannon-Fano code associated with that probabil-ity distribution,see for example [6],and is thus also ex-pressed in an integral number of bits.Let us consider this relation more closely:For a quantum system |z the quantity P (x )=|| z |x ||2is the probability that the system passes a test for |x ,and vice versa.The term ⌈−log || z |x ||2⌉can be viewed as the code word lengthto redescribe |x ,given |z and an orthonormal basis with |x as one of the basis vectors,using the Shannon-Fano pre-fix code.This works as follows:Write N =2n .For every state |z in (2n )-dimensional Hilbert space with basis vec-tors B ={|e 0 ,...,|e N −1 }we have N −1i =0|| e i |z ||2=1.If the basis has |x as one of the basis vectors,then we can consider |z as a random variable that assumes value |x with probability || x |z ||2.The Shannon-Fano code word for |x in the probabilistic ensemble B ,(|| e i |z ||2)iisbased on the probability || x |z ||2of |x ,given |z ,and haslength ⌈−log || x |z ||2⌉.Considering a canonical method of constructing an orthonormal basis B =|e 0 ,...,|e N −1 from a given basis vector,we can choose B such thatK (B )+=min i {K (|e i )}.The Shannon-Fano code is ap-propriate for our purpose since it is optimal in that it achieves the least expected code word length—the expec-tation taken over the probability of the source words—up to 1bit by Shannon’s Noiseless Coding Theorem.As in the classical case the quantum Kolmogorov complexity is an integral number.The main property required to be able to develop a meaningful theory is that our definition satisfies a so-called Invariance Theorem (see also Appendix A).Below we use “U ”to denote a special type of universal (quantum)Turing machine rather than a unitary matrix.Theorem 1(Invariance)There is a universal machine U ,such that for all machines Q ,there is a constant c Q (the length of the description of the index of Q in the enumera-tion),such that for all quantum states |x and all auxiliary inputs y we have:K U (|x |y )≤K Q (|x |y )+c Q .Proof:Assume that the program p that minimizes the right-hand side of (1)is p 0and the computed |z is |z 0 :K Q (|x |y )=l (p 0)+⌈−log || z 0|x ||2⌉.There is a universal quantum Turing machine U in the standard enumeration Q 1,Q 2,...such that for every quan-tum Turing machine Q in the enumeration there is a self-delimiting program i Q (the index of Q )and U (i Q p,y )=Q (p,y )for all p,y :if Q (p,y )=|z then U (i Q p,y )=|z .In particular,this holds for p 0such that Q with auxiliary input y halts with output |z 0 .But U with auxiliary input y halts on input i Q p 0also with output |z 0 .Consequently,the program q that minimizes the right-hand side of (1)with U substituted for Q ,and computes U (q,y )=|u for some state |u possibly different from |z ,satisfiesK U (|x |y )=l (q )+⌈−log || u |x ||2⌉≤l (i Q p 0)+⌈−log || z 0|x ||2⌉.Combining the two displayed inequalities,and setting c Q =l (i Q ),proves the theorem.。

八卦一下量子机器学习的历史人工智能和量子信息在讲量子机器学习之前我们先来八卦一下人工智能和量子信息。

1956，达特茅斯，十位大牛聚集于此，麦卡锡（John McCarthy）给这个活动起了个别出心裁的名字：“人工智能夏季研讨会”（Summer Research Project on Artificial Intelligence），现在被普遍认为是人工智能的起点。

AI的历史是非常曲折的，从符号派到联结派，从逻辑推理到统计学习，从经历70年代和80年代两次大规模的政府经费削减，到90年代开始提出神经网络，默默无闻直到2006年Hinton提出深层神经网络的层级预训练方法，从专注于算法到李飞飞引入ImageNet，大家开始注意到数据的重要性，大数据的土壤加上计算力的摩尔定律迎来了现在深度学习的火热。

量子信息的历史则更为悠久和艰难。

这一切都可以归结到1935年，爱因斯坦，波多尔斯基和罗森在“Can Quantum-Mechanical Description of Physical Reality be Considered Complete?”一文中提出了EPR悖论，从而引出了量子纠缠这个概念。

回溯到更早一点，1927年第五次索尔维会议，世界上最主要的物理学家聚在一起讨论新近表述的量子理论。

会议上爱因斯坦和波尔起了争执，爱因斯坦用“上帝不会掷骰子”的观点来反对海森堡的不确定性原理，而玻尔反驳道，“爱因斯坦，不要告诉上帝怎么做”。

这一论战持续了很多年，伴随着量子力学的发展，直到爱因斯坦在1955年去世。

爱因斯坦直到去世也还一直坚持这个世界没有随机性这种东西，所有的物理规律都是确定性的，给定初态和演化规律，物理学家就能推算出任意时刻系统的状态。

而量子力学生来就伴随了不确定性，一只猫在没测量前可以同时“生”和'死'，不具备一个确定的状态，只有测量后这只猫才具备“生”和'死'其中的一种状态，至于具体是哪一种状态量子力学只能告诉我们每一种态的概率，给不出一个确定的结果。

弱hopf群余代数的maschke型定理(英文) Maschke's theorem for weak Hopf algebras is an important result that describes how certain representations of weak Hopf algebras behave. This theorem provides a description of all representations of a weak Hopf algebras in terms of a single irreducible representation.1. IntroductionThe weak Hopf algebra is a kind of algebraic structure and can be used to describe a lot of physical and mathematical information. It was first proposed by Jeffery Hoffnung in 1994 and was intensively studied afterwards. It can be considered as an analogue of group algebras and has many properties in common with them. Maschke’s theorem is an important result inspired by group algebras which describes the behavior of representations of weak Hopf algebras.2. The Definition of Weak Hopf AlgebraIn general, an algebra is a vector space with multiplication defined on it. A weak Hopf algebra is a kind of algebra that is, besides linear over a field, equipped with a multiplication and two different kinds of maps: comultiplication and antipode. Both maps are bijective and can be defined in terms of each other.3. The Statement of Maschke's TheoremMaschke's theorem for weak Hopf algebras is an important result that describes how representations of weak Hopf algebras behave. It states that any representation of a finite-dimensional weak Hopf algebra can bedecomposed into direct sums of irreducible representations. Furthermore, the decomposition is unique, up to isomorphism. In other words, the Maschke's theorem states that all representations of a weak Hopf algebra can be described using a single irreducible representation.4. Proof of Maschke's TheoremIn order to prove Maschke's theorem, it is necessary to first show that the decomposition of a repeated irreducible representation is unique. This can be achieved by showing that any two irreducible representations are either equivalent or orthogonal, and so cannot both be included in a single decomposition. To prove this, a set of linear independent vectors is defined which can be used to construct an isomorphism that shows the equivalence or orthogonality of any two irreducible representations.Once the uniqueness of decomposition has been established, the proof of Maschke's theorem follows easily. It can be demonstrated that any representation of a weak Hopf algebra can be decomposed into direct sums of irreducible representations. Since the decomposition is unique, the theorem is proved.5. ApplicationsMaschke's theorem for weak Hopf algebras has many important applications in Physics, ranging from the study of quasiparticles in condensed matter to the description of topological insulators. It is also used in the study of quantum entanglement and quantum computers. In Mathematics, the theorem is used in the study of Graph Theory and Knot Theory.6. ConclusionIn summary, Maschke's theorem for weak Hopf algebras is an important result that describes how certain representations of weak Hopf algebras behave. This theorem provides a description of all representations of a weak Hopf algebras in terms of a single irreducible representation. It has many important applications in the fields of Physics, Mathematics and Computer Science, and is a very useful tool in the study of such topics.。

2013-12-3 08:55 |个人分类:系列科普|系统分类:科普集锦|关键词:量子自旋霍尔效应时间反演 拓扑31.拓扑绝缘体（续）（系列完结篇）上节中介绍的石墨烯，由于它独特的物理性质而引起了人们的兴趣。

它的无质量的相对论性准粒子，被观察到的整数及分数量子霍尔效应，为基础物理研究的许多方面，提供了理论模型和实验依据。

它优异的电子输运性质，又使其在自旋电子学等工程领域可能得到广泛的实际应用。

图31.1列出了石墨烯及量子霍尔态等几种物态在费米能级附近的能带图。

从图31.1中的（a）和（b），我们可以看到双层和单层碳原子结构能带形状的不同。

前者是抛物线型接触，而后者是线性的。

（必须提醒注意的是，我们所说的这两种石墨烯能带图都是指在二维空间中能无限延伸的理想晶体之能带图。

）那么，量子霍尔态的能带形状又如何呢？图31.1：两种石墨烯及量子霍尔态等能带图之比较图31.1c是量子霍尔态的能带示意图。

它的导带及价带在费米能级附近的形状，接近抛物线，类似于普通绝缘体。

但是，我们在上一节中也说过，量子霍尔态体内虽然是绝缘体，但它们由于边缘态的存在而导电。

在图中，量子霍尔态的边缘态是一条连接导带和价带的直线。

因此，量子霍尔态在低能态附近的行为，和石墨烯相仿，能量和动量的关系也是线性的，也存在无质量的相对论性准粒子。

因为量子霍尔态的实现需要强大的外磁场，由此人们将兴趣转向不需要磁场的量子自旋霍尔效应，并且在实验室里已经多次观察到了此种现象。

对量子自旋霍尔态而言，不同的自旋有不同的边界态，因此，拓扑绝缘体简介在图31.1d所示的自旋霍尔态能带图中，有两条直线连接导带和价带，它们分别对应于自旋上和自旋下的边缘电流。

这种情形下的能带图，看起来与理想石墨烯的能带图更为类似了。

普通的绝缘体，也可能产生边缘态而形成边缘导电，但却和前面两种情形下的边缘态有本质的区别。

图31.1e画出了普通绝缘体的能带。

图中的边缘态曲线与费米能级相交，意味着在此绝缘体中可以存在边缘电流。

凝聚态物理英语Condensed matter physics is a branch of physics that deals with the macroscopic and microscopic properties of matter, specifically the behavior of solids and liquids. It is a highly interdisciplinary field that explores the quantum mechanical properties of matter in bulk. This field of study has led to numerous technological advancements and has greatly impacted our understanding of the physical world.One of the key concepts in condensed matter physics is the study of phase transitions. Phase transitions occur when a material changes from one state to another, such as from a solid to a liquid or from a liquid to a gas. These transitions are characterized by changes in the material's physical properties, such as density, conductivity, and magnetization. Understanding the mechanisms behind phase transitions is essential for developing new materials with novel properties and applications.Another important area of research in condensed matter physics is the study of quantum phenomena in solids. Quantum mechanics describes the behavior of particles on the atomic and subatomic levels, and in condensed matter physics, these principles are applied to understand the interactions between atoms in solids. Quantum phenomena in solids can lead tounique properties such as superconductivity, where materials exhibit zero resistance to electrical current, and magnetism, where materials exhibit magnetic properties due to the alignment of electron spins.Condensed matter physics also plays a crucial role in the development of new materials for various technologies. Researchers in this field work to understand the properties of materials at the atomic level and how they can be manipulated to create materials with specific properties. This knowledge has led to the development of new materials for applications in electronics, energy storage, and medical devices.In recent years, there has been a growing interest in the study of topological materials in condensed matter physics. Topological materials are materials that exhibit unique electronic properties due to their topological structure, which is determined by the arrangement of atoms in the material. These materials have the potential to revolutionize electronics and computing by enabling the development of new types of electronic devices with improved performance and efficiency.Overall, condensed matter physics is a diverse and dynamic field of study that continues to push the boundaries of our understanding of the physical world. Through the study of phasetransitions, quantum phenomena, and novel materials, researchers in this field are continually expanding our knowledge of matter and developing new technologies that have the potential to transform society.In conclusion, condensed matter physics is a fascinating and important field of study that has a profound impact on our understanding of the physical world and on technological advancements. Researchers in this field continue to make significant contributions to science and technology, and the future of condensed matter physics holds great promise for exciting new discoveries and innovations.。

a r X i v :h e p -t h /9609063v 1 8 S e p 1996Calogero-Sutherland Particles as QuasisemionsGiovanni AMELINO-CAMELIACenter for Theoretical Physics,MIT,Cambridge,Massachusetts 02139,USAandTheoretical Physics,University of Oxford,1Keble Rd.,Oxford OX13NP,UK 1ABSTRACTThe ultraviolet structure of the Calogero-Sutherland models is examined,and,in par-ticular,semions result to have special properties.An analogy with ultraviolet structures known in anyon quantum mechanics is drawn,and it is used to suggest possible physical consequences of the observed semionic properties.MIT-CTP-2432/OUTP-96-12P hep-th/9609063Modern Physics Letters A (1996)in pressRecently,there has been renewed interest[1-8]in the Calogero-Sutherland models[9-11], especially in connection with the study of fractional exclusion statistics[4,7,12-16]in1+1di-mensions.These quantum mechanical models describe particles whose dynamics is governed by a Hamiltonian of the form− i d2L2sin2(π(x i−x j)/L)+V({x i}),(1)where i,j=1,2,...,N,x i denotes the i-th particle position on a circle of radius L,βis a non-negative real parameter,and V is a regular(i.e.finite for every{x i})potential.The parameterβhas been found to characterize the exclusion statistics of the particles[6,7]; in particular,(once appropriate boundary conditions are imposed[6])β=0corresponds to bosons andβ=1corresponds to fermions2.The cases V=0(“free Calogero-Sutherland particles”)and V= i<j(x i−x j)2(“Calogero-Sutherland particles with an harmonic potential”)have been completely solved[10,11]both forfinite L and in the limit L→∞.A very important open problem[5,6,8]is the one offinding a formulation of the Calogero-Sutherland models in the formalism of non-relativistic quantumfield theory.In the case of anyons[17],particles in2+1dimensions that have fractional exchange statistics[17],such a formulation is given by a Chern-Simonsfield theory,and has been very useful[18-20]in the understanding of the statistics.In this Letter I propose a technique of investigation of the Calogero-Sutherland models which should allow to uncover some of the features of their yet-to-be-foundfield theoretical formulation.My analysis is indeed motivated by an analogy with the case of the anyon models.In that context it has been recently realized[20,21]that the ultraviolet structure of the perturbative expansions in the statistical parameter is closely related to the structure of the Chern-Simonsfield theoretical formulation,which,for example,leads to Feynman diagrams affected by ultraviolet divergences reproducing the ones encountered in the quan-tum mechanical perturbative framework[20,21].I am therefore interested in an analogous perturbative expansion for the Calogero-Sutherland models.I start by analyzing the ultraviolet problems of such an expansion.For simplicity,I limit the discussion to the case of two Calogero-Sutherland particles with0≤β≤1,an harmonic oscillator potential,and L∼∞;the relative motion is therefore described by the HamiltonianHβ=−d2x2,(2)where x is the relative coordinate,and,since we are dealing with two identical particles on the line,the configuration space is x≥0.The harmonic potential is introduced[9]in order to discretize the spectrum,so that the dependence onβcan be examined more easily.The eigenfunctions of Hβthat are regular at the point x=0,where the particle positions coincide,are[9](the Lµn are Laguerre polynomials and the Nβn are normalization constants)|Ψn,β>=Nβn xβe−x22Note that the potential sin−2(πx/L),which is singular at the points of coincidence of particle positions and causes the fractionality of the exclusion statistics,has vanishing coefficient forβ=0,1.describe the fractional exclusion statistics of particles with anyβin terms of the one of particles withβ=β0.Important building blocks of such a perturbative expansion are the matrix elements<Ψn,β0|1x2−2β0,(5)but these are(ultraviolet)divergent for everyβ0≤1/2.An ultraviolet problem somewhat analogous to this one is encountered in the study of anyons.In that case one is interested in perturbative expansions depending on the statistical parameterν[17,23],which also has bosonic limitν=0and fermionic limitν=1,and one encounters logarithmic ultravio-let divergences when expanding around the special(bosonic)valueν=0.This divergent bosonic end perturbation theory of anyons can be handled[23,24]by using the formalism of renormalization for quantum mechanics[25],and a direct relation between the structure of the renormalized perturbative approach and some features of the Chern-Simonsfield theo-retical formulation of anyons has been found[19-21].The hope that such a program might be completed also for the Calogero-Sutherland models is confronted by the realization that the ultraviolet problems illustrated by Eq.(5)are much worse than the ones of the anyon case.Rather than being specific of a certain choice of the center of the expansionβ0,these ultraviolet problems are encountered for any of a continuous of choices ofβ0,and in general the divergences are worse-than-logarithmic.However,from Eq.(5)one can see that the ex-pansion aroundβ0=1/2is only affected by logarithmic divergences,and therefore this type of expansion is the best candidate for a generalization of the results obtained for anyons with the bosonic end perturbation theory.Motivated by this observation,I now consider more carefully the possibility of study-ing Calogero-Sutherland particles with anyβas perturbations of“Calogero-Sutherland semions”,i.e.Calogero-Sutherland particles withβ=1/2.Following the usual path of renor-malization theory in quantum mechanics(see,for example,Refs.[20,23-25]),I add(only for the perturbation theory)the counterterm3(β−1/2) i<jδ(x i−x j)/(x i−x j)to the original Calogero-Sutherland Hamiltonian.A very general verification of the validity of this proce-dure will be given in detail elsewhere[26],but here I intend to briefly describe(to second order in the eigenenergies andfirst order in the eigenfunctions)how the exact two-body solutions(3)and(4)are correctly reproduced in this way.Let me start by setting up the renormalized quasisemionic description of the Hβeigenproblem.The zero-th order Hamil-tonian,wave functions,and energies are obviously the ones for semions,i.e.H1/2,|Ψn,1/2>, and E n,1/2[see Eqs.(2),(3),and(4)].The renormalized perturbative Hamiltonian isH Rpert1/2≡Hβ−H1/2+(β−1/2)δ(x)x+(β−1/2)2x2f(x)≡ ∞1/Λdx13As I shall show in a longer paper[26],the formδ(x)/x of the counterterm can be derived using symmetries and the appropriate power-counting rules for renormalization in quantum mechanics,and the value of the overall coefficient(β−1/2)can be obtained by looking for afixed point of the renormalization groupflow.∞0dxδ(x )|x |δ(x )|Ψn,1/2>=2(β−1/2),(9)and from Eq.(4)one sees that E (1)n,β=E n,β−E n,1/2,i.e.the first order result (9)already reproduces the exact result,as expected since the latter is linear in (β−1/2).The first order eigenfunctions are given by|Ψ(1)n,β>=m (=n )<Ψm,1/2|(β−1/2)E n,1/2−E m,1/2|Ψm,1/2>=−(β−1/2)π m (=n )L 0m (x 2)2,(10)which,as it can be seen using properties of the Laguerre polynomials,is in agreement with the expansion of Eq.(3)to first order in (β−1/2).Concerning the second order energies,I have verified that<Ψn,1/2|(β−1/2)2|x |δ(x )|Ψ(1)n,β>+O (1x 2–type interaction),and they obviously requirea corresponding number of copies of the same two-body δ(x )would be the ideal starting point for a bosonizedfield theoretical formulation.Research in this direction is certainly encouraged by my results for the quasisemionic perturbative ap-proach,but,as I showed,the structure of the divergences in the bosonic limit is substantially different from the one of the divergences I regularized/renormalized here.The special role played by Calogero-Sutherland semions in my analysis should have deeper physical roots(probably related to the special properties of particles withβ=1/2pointed out in Refs.[15,29])than the rather formal ones I noticed here.In particular,the fact that the renormalizability of the semionic perturbation theory that I considered arises in complete analogy with the renormalizability of the bosonic perturbation theory used for anyons,might suggest that semions play a special role in(1+1-dimensional)exclusion statistics,just like bosons have a special role4in(2+1-dimensional)exchange statistics.From the point of view of mathematical physics it is noteworthy that one more appli-cation of renormalization in quantum mechanics has been here found.There are not many such applications and this one should be particularly easy to examine because the problem is1+1-dimensional and all exact solutions are known.In particular,certain comparisons between the exact solutions and the renormalization-requiring perturbative results might lead to insight in the physics behind the general regularization/renormalization procedure; for example,since the exact solutions(3)-(4)are well-defined at every scale,my analysis is consistent with the idea[20,24]that the necessity of a cut-offis simply an artifact of the perturbative methods used,and not a relict of some unknown ultraviolet physics.Finally,I want to emphasize that I chose to consider only the regular Calogero-Sutherland eigenfunctions because they have a clearer physical interpretation[9]and allow a scale-invariant5analysis[26],but,based on the experience with anyons[20,27,28],I expect that additional insight into the nature of1+1-dimensional fractional exclusion statistics might be gained by looking at the renormalized perturbative expansion of the Calogero-Sutherland eigenfunctions that are singular at the points of coincidence of particle positions.I want to thank D.Sen for a conversation on recent results for the Calogero-Sutherland models,which contributed to my increasing interest in thisfield.I also happily acknowledge conversations with D.Bak,M.Bergeron,R.Jackiw,V.Pasquier,and 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