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量子多体系统的理论模型引言量子力学是描述微观物质行为的基本理论。
在量子力学中，描述一个系统的基本单位是量子态，而量子多体系统则是由多个量子态组成的系统。
由于量子多体系统的复杂性，需要借助一些理论模型来描述和研究。
本文将介绍一些常见的量子多体系统的理论模型，包括自旋链模型、玻色-爱因斯坦凝聚模型和费米气体模型等。
通过对这些模型的研究，我们可以深入了解量子多体系统的行为和性质。
自旋链模型自旋链模型是描述自旋之间相互作用的量子多体系统的模型。
在自旋链模型中，每个粒子可以处于自旋向上或向下的两种状态。
粒子之间通过自旋-自旋相互作用产生相互作用。
常见的自旋链模型包括Ising模型和Heisenberg模型。
Ising模型Ising模型是最简单的自旋链模型之一。
在一维Ising模型中，每个自旋可以取向上（+1）或向下（-1）。
自旋之间通过简单的相邻自旋相互作用来影响彼此的取向。
可以使用以下哈密顿量来描述一维Ising模型：$$H = -J\\sum_{i=1}^{N}s_is_{i+1}$$其中，J为相邻自旋之间的交换耦合常数，s i为第i个自旋的取向。
Heisenberg模型Heisenberg模型是描述自旋间相互作用的模型，与Ising模型不同的是，Heisenberg模型中的自旋可以沿任意方向取向。
常见的一维Heisenberg模型可以使用以下哈密顿量来描述：$$H = \\sum_{i=1}^{N} J\\mathbf{S}_i \\cdot \\mathbf{S}_{i+1}$$其中，$\\mathbf{S}_i$为第i个自旋的自旋算符，J为自旋间的交换耦合常数。
玻色-爱因斯坦凝聚模型玻色-爱因斯坦凝聚是一种量子多体系统的现象，它描述了玻色子统计的粒子在低温下向基态排列的行为。
玻色-爱因斯坦凝聚模型可以使用用薛定谔方程来描述：$$i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r},t) = -\\frac{\\hbar^2}{2m}\ abla^2\\Psi(\\mathbf{r},t) +V(\\mathbf{r})\\Psi(\\mathbf{r},t) +g|\\Psi(\\mathbf{r},t)|^2\\Psi(\\mathbf{r},t)$$其中，$\\Psi(\\mathbf{r},t)$是波函数，m是粒子的质量，$V(\\mathbf{r})$是外势场，g是粒子之间的相互作用常数。
WAYS TO UNDERSTAND T-X PHASE DIAGRAMS浙江大学材料科学与工程系张昶Summary：Phase diagrams, especially T-x phase diagrams, is useful in showing the physical changes that mixtures undergo when the temperature or their composition is changed. But to understand a phase diagram is not easy, especially when several kinds of substances exist and they are not stable. In this article we summarize the points, lines and regions that may appear in a phase diagram, and then come up with rules to understand T-x phase diagrams. This article is not a research paper, but a learning paper. It is helpful in the process of learning and reviewing the chapter of phase diagram.Keywords: points, lines, regions, laws, rules, examplesChapter One: Geometry ElementsIn this chapter we discuss geometry elements that may appear in a phase diagram. We should remember their characteristics and the substances, phases in the regions so that we can find and understand them easily in a real phase diagram.1-1Points1-1-1 AzeotropesDeviations from ideal solutions sometimesresult in azeotropes, which can be divided intohigh-boiling azeotropes and low-boilingazeotropes according to whether there is ahighest boiling point of the mixture.It’s easy to recognize the two lines in theleft pictures with regard to the temperature.The one above is gas-phase line and the otherone is liquid-phase line. Hence the regionabove both of them is gas-phase region (P=1)and the one below them is liquid-phase region(P=1) and the one between them is the regionwhere the both phases exist (P=2).The characteristic of an azeotrope is that itis the place where two slim ellipses meet.Notice thatthe two ellipses always turn towards the sameside.*If we replace ‘g ’ by ‘l ’, and ‘l ’ by ‘s ’ in the phase diagrams above, we can also get a solid-solid phase diagram whose solids are completely miscible. There will be a highest constant melting point or a lowest constant melting point as well.1-1-2 EutecticsEutectic is the lowest melting point of a eutectic composition. Now we discuss the congruent melting. Look at the picture below. Line JKL is the bond line of different regions. It means a limit, but not the condition of the whole system. The horizontal line through F is a three-phase line. So the region above line JKL is liquid- phase region (P=1).The region between the two lines is the twophase region which can be labeled ‘liquid+Bi ’or ‘liquid+Cd ’ (P=2). The region below thethree-phase line is solid-phase region. We canalso divide the region into two parts due to theformation of the eutectic mixture Cd BiE . Let ’s draw a vertical line through K. That ’s the puresubstance line of Cd BiE . On the left of it lies the Cd Bi E +Bi region and Cd BiE +Cd region on the right. Thesymbol of the a eutecticis1-1-3 Incongruent Melting PointsThe melting of an instable solid solution at a certaintemperature results in the formation of an incongruent meltingpoint. The left part below the line refers to the condition thatboth solid solution I and liquid exist (P=2). The right partrefers to the condition that both solid solution I and solidsolution II exist (P=2). The middle part refers to the conditionthat only solid solution I exist (P=1).1-2 L ines1-2-1 Pure Substance LinePure substance line is a vertical line. It means only one composition exists on this line. According to the phase law: 1*+=Φ+C f . Because only the temperature ischangeable, *f=1. Because the line exists in one region, Φ=1. So we come to the conclusion that C=1. We can only find this line in solid phase.There are two kinds of pure substance lines, now we’ll introduce them below..1-2-1-1 Stable Pure Substance LinesThe first one is stable pure substance line which has got acurved line on it and they look like an umbrella. The freezingpoint is depressed because other substances are added in.1-2-1-2 Instable Pure Substance LinesThe second one is instable pure substance line which has got ahorizontal line on it and they look like a letter ‘T’.Because the substance is instable, it is decomposed into liquidand another kind of solid above a certain temperature.1-2-3 Three-Phase LinesAny horizontal line is a three-phase line. According to thephase law:1*+f. Because the temperature is fixed,+C=Φ*f=0. Because there are two compositions, C=2. So we come to the conclusion that Φ=3.Eutectic line, incongruent melting line are both three-phase lines.*We don’t discuss eutectic lines and incongruent melting lines in this section because they have been discussed in 1-1 as points.1-3 Regions1-3-1 Partially Miscible LiquidsThere are three kinds of regions in the phase diagrams ofpartially miscible liquids: regions with an upper criticaltemperature, regions with a lower critical temperature, regionswith both an upper critical temperature and a lower criticaltemperature. But only the first one is common.The characteristic of the regions is that they are in a closed curve like cap. What’s more, the number of phases in the cap-shaped region is two and the number of phases in the outside region is one.1-3-2 Partially Miscible SolidsThere are two kinds of regions in the phase diagram of partially miscible solids: regions with eutectics, regions with incongruent melting points. The substances and phases in different regions have been discussed in 1-1-2 and 1-1-3, so we don’t write them again. We should remember the basic shapes of the two phase diagram by heart.Chapter Two: Laws and RulesIn this chapter we introduce rules and laws to help understand phase diagrams. It’s OK to understand phase diagrams only with the geometry elements in chapter 1, but with these rules we can solve the problems more easily.2-1 A region of only one kind solid solution is usually not surrounded by a three-phase line.2-2 A region of miscible solutions is usually not surrounded by a three-phase line.2-3 Neighboring phase region law: If we minus the number of phases of a region by the number of phases of a neighboring region, the result will be either 1 or -1 (We regard the pure-substance line as a one-phase region and the three-phase line as a three-phase region).2-4 The region between two one-phase regions is a two-phase region where both the substances in the two one-phase regions exist.Interpretation:(1)*2-1, 2-2 and 2-3 are just special cases of Palatnik and Landau’s rule, you can see the full expression of it in the reference essay [6].(2)2-4 is a deduction of 2-3.(3)With these two rules we can easily find one-phase regions in phase diagrams. After we have determined the one-phase regions with 2-1, 2-2 and three-phase lines (regions) with 1-1-3, we can determine the left two-phase regions due to 2-3.(4)Substances in two-phase regions sometimes can be determined by 2-4.Chapter Three: Steps and Examples3-1 STEPS: Now we introduce the basic steps to understand a phase diagram based on the geometry elements and laws discussed above.(1)Recognize the obvious regions such as liquid or gas regions.(2)Find regions that are not surrounded by three-phase lines, they are probably one-phase regions according to 2-1 and 2-2. Then find vertical lines that refer to pure substance lines according to 1-2-1. Then label the two-phase region between two one-phase regions according to 2-3 and 2-4.(3)Find geometry elements in the phase diagram. If the two components are the similar to those that are discussed in chapter one, then imitate them to label the regions.*Step 2 and 3 can be swapped, but do as the steps introduced above will be more convenient.3-2-1 Example 1: Label the regions of the phase diagram below. State what substances exist in each region. Label each substance in each region as solid, liquid or gas.Analysis:In this example we don’t use the rules in chapter 2.First, it ’s easy to know that region 1 is liquid.Second, we can find that AB is an instable pure substance line according to 1-2-1-2. Then we know the instable substance is Z, which is made from A and B. According to 2-4, we know that the substances in region 2 to 4 can be determined by the bonds. So, region 2 is X(s) +l, region 3 is Z(s) +l, region 4 is X(s) +l.Third, we can find those region 6 to 10 are the regions of solid solutions with incongruent melting points according to 1-3-2. So we can just copy the substances and phases of the second picture in 1-3-2. Region 6 is )(s β, region 7 is )(s β+l, region 8 is )(s α+l, region 9 is )(s α, region 10 is )()(s s βα+.3-3-2 Example 2: Label the regions of the phase diagram below. State what substances exist in each region. Label each substance in each region as solid, liquid, or gas. Region 7 and region 11 is already known.Analysis ：There are some unfamiliar regions in the phase diagram so we should make use of the rules and laws in chapter 2.First, it ’s easy to understand that region 1 is liquid.Second, according to 2-4, we can find that region 5 is )(s α+l, region 6 is )(s β+l, region 8 is )()(s s βα+, region 9 and 10 are )(s β+ l. (We may also determine region 9 to 11 by H, a lowest constant melting point.)Third, according to 1-2-1-1 and 1-1-2, we find BC is the stable pure substance line of C and A is a eutectic of pure X and pure Z (compound of X and Y). So we copy the substances and phases in 1-1-2. Region 2 is X(s) + l, region 3 is Z(s) + l, region 4 is X(s) + Z(s).*Now we can find that it’s easier to understand the phase diagrams with the laws and rules introduced in chapter 2.Reference Books and Essays:[1]《物理化学学习指导》南大第五版 高等教育出版社 p235-p243[2]《Atkins ’ Physical Chemistry 影印版》, 高等教育出版社[3]《材料科学基础》 二元相图分析 上海交通大学出版社[4]工程材料课程课件 2.2合金的结晶 某石油大学[5]《固体物理》 黄昆著 第八章 相图 高等教育出版社[6]恒压相图中紧邻相区及其边界的关系 吉林大学化学系 赵慕愚。
第6章光源和放大器在光纤系统，光纤光源产生的光束携带的信息。
激光二极管和发光二极管是两种最常见的来源。
他们的微小尺寸与小直径的光纤兼容，其坚固的结构和低功耗要求与现代的固态电子兼容。
在以下几个GHz的工作系统，大部分（或数Gb /秒），信息贴到光束通过调节输入电流源。
外部调制（在第4、10章讨论）被认为是当这些率超标。
我们二极管LED和激光研究，包括操作方法，转移特性和调制。
我们计划以获得其他好的或理念的差异的两个来源，什么情况下调用。
当纤维损失导致信号功率低于要求的水平，光放大器都需要增强信号到有效的水平。
通过他们的使用，光纤链路可以延长。
因为光源和光放大器，如此多的共同点，他们都是在这一章处理。
1.发光二极管一个发光二极管[1，2]是一个PN结的半导体发光时正向偏置。
图6.1显示的连接器件、电路符号，能量块和二极管关联。
能带理论提供了对一个）简单的解释半导体发射器（和探测器）。
允许能带通过的是工作组，其显示的宽度能在图中，相隔一禁止区域（带隙）。
在上层能带称为导带，电子不一定要到移动单个原子都是免费的。
洞中有一个正电荷。
它们存在于原子电子的地点已经从一个中立带走，留下的电荷原子与净正。
自由电子与空穴重新结合可以，返回的中性原子状态。
能量被释放时，发生这种情况。
一个n -型半导体拥有自由电子数，如图图英寸6.1。
p型半导体有孔数自由。
当一种P型和一种N型材料费米能级（WF）的P和N的材料一致，并外加电压上作用时，产生的能垒如显示的数字所示。
重参杂材料，这种情况提供许多电子传到和过程中需要排放的孔。
在图中，电子能量增加垂直向上，能增加洞垂直向下。
因此，在N地区的自由电子没有足够的能量去穿越阻碍而移动到P区。
同样，空穴缺乏足够的能量克服障碍而移动进入n区。
当没有外加电压时，由于两种材料不同的费米能级产生的的能量阻碍，就不能自由移动。
外加电压通过升高的N端势能，降低一侧的P端势能，从而是阻碍减小。
如果供电电压（电子伏特）与能级（工作组）相同，自由电子和自由空穴就有足够的能量移动到交界区，如底部的数字显示，当一个自由电子在交界区遇到了一个空穴，电子可以下降到价带，并与空穴重组。
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Modeling of morphology evolution in the injection moldingprocess of thermoplastic polymersR.Pantani,I.Coccorullo,V.Speranza,G.Titomanlio* Department of Chemical and Food Engineering,University of Salerno,via Ponte don Melillo,I-84084Fisciano(Salerno),Italy Received13May2005;received in revised form30August2005;accepted12September2005AbstractA thorough analysis of the effect of operative conditions of injection molding process on the morphology distribution inside the obtained moldings is performed,with particular reference to semi-crystalline polymers.The paper is divided into two parts:in the first part,the state of the art on the subject is outlined and discussed;in the second part,an example of the characterization required for a satisfactorily understanding and description of the phenomena is presented,starting from material characterization,passing through the monitoring of the process cycle and arriving to a deep analysis of morphology distribution inside the moldings.In particular,fully characterized injection molding tests are presented using an isotactic polypropylene,previously carefully characterized as far as most of properties of interest.The effects of both injectionflow rate and mold temperature are analyzed.The resulting moldings morphology(in terms of distribution of crystallinity degree,molecular orientation and crystals structure and dimensions)are analyzed by adopting different experimental techniques(optical,electronic and atomic force microscopy,IR and WAXS analysis).Final morphological characteristics of the samples are compared with the predictions of a simulation code developed at University of Salerno for the simulation of the injection molding process.q2005Elsevier Ltd.All rights reserved.Keywords:Injection molding;Crystallization kinetics;Morphology;Modeling;Isotactic polypropyleneContents1.Introduction (1186)1.1.Morphology distribution in injection molded iPP parts:state of the art (1189)1.1.1.Modeling of the injection molding process (1190)1.1.2.Modeling of the crystallization kinetics (1190)1.1.3.Modeling of the morphology evolution (1191)1.1.4.Modeling of the effect of crystallinity on rheology (1192)1.1.5.Modeling of the molecular orientation (1193)1.1.6.Modeling of theflow-induced crystallization (1195)ments on the state of the art (1197)2.Material and characterization (1198)2.1.PVT description (1198)*Corresponding author.Tel.:C39089964152;fax:C39089964057.E-mail address:gtitomanlio@unisa.it(G.Titomanlio).2.2.Quiescent crystallization kinetics (1198)2.3.Viscosity (1199)2.4.Viscoelastic behavior (1200)3.Injection molding tests and analysis of the moldings (1200)3.1.Injection molding tests and sample preparation (1200)3.2.Microscopy (1202)3.2.1.Optical microscopy (1202)3.2.2.SEM and AFM analysis (1202)3.3.Distribution of crystallinity (1202)3.3.1.IR analysis (1202)3.3.2.X-ray analysis (1203)3.4.Distribution of molecular orientation (1203)4.Analysis of experimental results (1203)4.1.Injection molding tests (1203)4.2.Morphology distribution along thickness direction (1204)4.2.1.Optical microscopy (1204)4.2.2.SEM and AFM analysis (1204)4.3.Morphology distribution alongflow direction (1208)4.4.Distribution of crystallinity (1210)4.4.1.Distribution of crystallinity along thickness direction (1210)4.4.2.Crystallinity distribution alongflow direction (1212)4.5.Distribution of molecular orientation (1212)4.5.1.Orientation along thickness direction (1212)4.5.2.Orientation alongflow direction (1213)4.5.3.Direction of orientation (1214)5.Simulation (1214)5.1.Pressure curves (1215)5.2.Morphology distribution (1215)5.3.Molecular orientation (1216)5.3.1.Molecular orientation distribution along thickness direction (1216)5.3.2.Molecular orientation distribution alongflow direction (1216)5.3.3.Direction of orientation (1217)5.4.Crystallinity distribution (1217)6.Conclusions (1217)References (1219)1.IntroductionInjection molding is one of the most widely employed methods for manufacturing polymeric products.Three main steps are recognized in the molding:filling,packing/holding and cooling.During thefilling stage,a hot polymer melt rapidlyfills a cold mold reproducing a cavity of the desired product shape. During the packing/holding stage,the pressure is raised and extra material is forced into the mold to compensate for the effects that both temperature decrease and crystallinity development determine on density during solidification.The cooling stage starts at the solidification of a thin section at cavity entrance (gate),starting from that instant no more material can enter or exit from the mold impression and holding pressure can be released.When the solid layer on the mold surface reaches a thickness sufficient to assure required rigidity,the product is ejected from the mold.Due to the thermomechanical history experienced by the polymer during processing,macromolecules in injection-molded objects present a local order.This order is referred to as‘morphology’which literally means‘the study of the form’where form stands for the shape and arrangement of parts of the object.When referred to polymers,the word morphology is adopted to indicate:–crystallinity,which is the relative volume occupied by each of the crystalline phases,including mesophases;–dimensions,shape,distribution and orientation of the crystallites;–orientation of amorphous phase.R.Pantani et al./Prog.Polym.Sci.30(2005)1185–1222 1186R.Pantani et al./Prog.Polym.Sci.30(2005)1185–12221187Apart from the scientific interest in understandingthe mechanisms leading to different order levels inside a polymer,the great technological importance of morphology relies on the fact that polymer character-istics (above all mechanical,but also optical,electrical,transport and chemical)are to a great extent affected by morphology.For instance,crystallinity has a pro-nounced effect on the mechanical properties of the bulk material since crystals are generally stiffer than amorphous material,and also orientation induces anisotropy and other changes in mechanical properties.In this work,a thorough analysis of the effect of injection molding operative conditions on morphology distribution in moldings with particular reference to crystalline materials is performed.The aim of the paper is twofold:first,to outline the state of the art on the subject;second,to present an example of the characterization required for asatisfactorilyR.Pantani et al./Prog.Polym.Sci.30(2005)1185–12221188understanding and description of the phenomena, starting from material description,passing through the monitoring of the process cycle and arriving to a deep analysis of morphology distribution inside the mold-ings.To these purposes,fully characterized injection molding tests were performed using an isotactic polypropylene,previously carefully characterized as far as most of properties of interest,in particular quiescent nucleation density,spherulitic growth rate and rheological properties(viscosity and relaxation time)were determined.The resulting moldings mor-phology(in terms of distribution of crystallinity degree, molecular orientation and crystals structure and dimensions)was analyzed by adopting different experimental techniques(optical,electronic and atomic force microscopy,IR and WAXS analysis).Final morphological characteristics of the samples were compared with the predictions of a simulation code developed at University of Salerno for the simulation of the injection molding process.The effects of both injectionflow rate and mold temperature were analyzed.1.1.Morphology distribution in injection molded iPP parts:state of the artFrom many experimental observations,it is shown that a highly oriented lamellar crystallite microstructure, usually referred to as‘skin layer’forms close to the surface of injection molded articles of semi-crystalline polymers.Far from the wall,the melt is allowed to crystallize three dimensionally to form spherulitic structures.Relative dimensions and morphology of both skin and core layers are dependent on local thermo-mechanical history,which is characterized on the surface by high stress levels,decreasing to very small values toward the core region.As a result,the skin and the core reveal distinct characteristics across the thickness and also along theflow path[1].Structural and morphological characterization of the injection molded polypropylene has attracted the interest of researchers in the past three decades.In the early seventies,Kantz et al.[2]studied the morphology of injection molded iPP tensile bars by using optical microscopy and X-ray diffraction.The microscopic results revealed the presence of three distinct crystalline zones on the cross-section:a highly oriented non-spherulitic skin;a shear zone with molecular chains oriented essentially parallel to the injection direction;a spherulitic core with essentially no preferred orientation.The X-ray diffraction studies indicated that the skin layer contains biaxially oriented crystallites due to the biaxial extensionalflow at theflow front.A similar multilayered morphology was also reported by Menges et al.[3].Later on,Fujiyama et al.[4] investigated the skin–core morphology of injection molded iPP samples using X-ray Small and Wide Angle Scattering techniques,and suggested that the shear region contains shish–kebab structures.The same shish–kebab structure was observed by Wenig and Herzog in the shear region of their molded samples[5].A similar investigation was conducted by Titomanlio and co-workers[6],who analyzed the morphology distribution in injection moldings of iPP. They observed a skin–core morphology distribution with an isotropic spherulitic core,a skin layer characterized by afine crystalline structure and an intermediate layer appearing as a dark band in crossed polarized light,this layer being characterized by high crystallinity.Kalay and Bevis[7]pointed out that,although iPP crystallizes essentially in the a-form,a small amount of b-form can be found in the skin layer and in the shear region.The amount of b-form was found to increase by effect of high shear rates[8].A wide analysis on the effect of processing conditions on the morphology of injection molded iPP was conducted by Viana et al.[9]and,more recently, by Mendoza et al.[10].In particular,Mendoza et al. report that the highest level of crystallinity orientation is found inside the shear zone and that a high level of orientation was also found in the skin layer,with an orientation angle tilted toward the core.It is rather difficult to theoretically establish the relationship between the observed microstructure and processing conditions.Indeed,a model of the injection molding process able to predict morphology distribution in thefinal samples is not yet available,even if it would be of enormous strategic importance.This is mainly because a complete understanding of crystallization kinetics in processing conditions(high cooling rates and pressures,strong and complexflowfields)has not yet been reached.In this section,the most relevant aspects for process modeling and morphology development are identified. In particular,a successful path leading to a reliable description of morphology evolution during polymer processing should necessarily pass through:–a good description of morphology evolution under quiescent conditions(accounting all competing crystallization processes),including the range of cooling rates characteristic of processing operations (from1to10008C/s);R.Pantani et al./Prog.Polym.Sci.30(2005)1185–12221189–a description capturing the main features of melt morphology(orientation and stretch)evolution under processing conditions;–a good coupling of the two(quiescent crystallization and orientation)in order to capture the effect of crystallinity on viscosity and the effect offlow on crystallization kinetics.The points listed above outline the strategy to be followed in order to achieve the basic understanding for a satisfactory description of morphology evolution during all polymer processing operations.In the following,the state of art for each of those points will be analyzed in a dedicated section.1.1.1.Modeling of the injection molding processThefirst step in the prediction of the morphology distribution within injection moldings is obviously the thermo-mechanical simulation of the process.Much of the efforts in the past were focused on the prediction of pressure and temperature evolution during the process and on the prediction of the melt front advancement [11–15].The simulation of injection molding involves the simultaneous solution of the mass,energy and momentum balance equations.Thefluid is non-New-tonian(and viscoelastic)with all parameters dependent upon temperature,pressure,crystallinity,which are all function of pressibility cannot be neglected as theflow during the packing/holding step is determined by density changes due to temperature, pressure and crystallinity evolution.Indeed,apart from some attempts to introduce a full 3D approach[16–19],the analysis is currently still often restricted to the Hele–Shaw(or thinfilm) approximation,which is warranted by the fact that most injection molded parts have the characteristic of being thin.Furthermore,it is recognized that the viscoelastic behavior of the polymer only marginally influences theflow kinematics[20–22]thus the melt is normally considered as a non-Newtonian viscousfluid for the description of pressure and velocity gradients evolution.Some examples of adopting a viscoelastic constitutive equation in the momentum balance equations are found in the literature[23],but the improvements in accuracy do not justify a considerable extension of computational effort.It has to be mentioned that the analysis of some features of kinematics and temperature gradients affecting the description of morphology need a more accurate description with respect to the analysis of pressure distributions.Some aspects of the process which were often neglected and may have a critical importance are the description of the heat transfer at polymer–mold interface[24–26]and of the effect of mold deformation[24,27,28].Another aspect of particular interest to the develop-ment of morphology is the fountainflow[29–32], which is often neglected being restricted to a rather small region at theflow front and close to the mold walls.1.1.2.Modeling of the crystallization kineticsIt is obvious that the description of crystallization kinetics is necessary if thefinal morphology of the molded object wants to be described.Also,the development of a crystalline degree during the process influences the evolution of all material properties like density and,above all,viscosity(see below).Further-more,crystallization kinetics enters explicitly in the generation term of the energy balance,through the latent heat of crystallization[26,33].It is therefore clear that the crystallinity degree is not only a result of simulation but also(and above all)a phenomenon to be kept into account in each step of process modeling.In spite of its dramatic influence on the process,the efforts to simulate the injection molding of semi-crystalline polymers are crude in most of the commercial software for processing simulation and rather scarce in the fleur and Kamal[34],Papatanasiu[35], Titomanlio et al.[15],Han and Wang[36],Ito et al.[37],Manzione[38],Guo and Isayev[26],and Hieber [25]adopted the following equation(Kolmogoroff–Avrami–Evans,KAE)to predict the development of crystallinityd xd tZð1K xÞd d cd t(1)where x is the relative degree of crystallization;d c is the undisturbed volume fraction of the crystals(if no impingement would occur).A significant improvement in the prediction of crystallinity development was introduced by Titoman-lio and co-workers[39]who kept into account the possibility of the formation of different crystalline phases.This was done by assuming a parallel of several non-interacting kinetic processes competing for the available amorphous volume.The evolution of each phase can thus be described byd x id tZð1K xÞd d c id t(2)where the subscript i stands for a particular phase,x i is the relative degree of crystallization,x ZPix i and d c iR.Pantani et al./Prog.Polym.Sci.30(2005)1185–1222 1190is the expectancy of volume fraction of each phase if no impingement would occur.Eq.(2)assumes that,for each phase,the probability of the fraction increase of a single crystalline phase is simply the product of the rate of growth of the corresponding undisturbed volume fraction and of the amount of available amorphous fraction.By summing up the phase evolution equations of all phases(Eq.(2))over the index i,and solving the resulting differential equation,one simply obtainsxðtÞZ1K exp½K d cðtÞ (3)where d c Z Pid c i and Eq.(1)is recovered.It was shown by Coccorullo et al.[40]with reference to an iPP,that the description of the kinetic competition between phases is crucial to a reliable prediction of solidified structures:indeed,it is not possible to describe iPP crystallization kinetics in the range of cooling rates of interest for processing(i.e.up to several hundreds of8C/s)if the mesomorphic phase is neglected:in the cooling rate range10–1008C/s, spherulite crystals in the a-phase are overcome by the formation of the mesophase.Furthermore,it has been found that in some conditions(mainly at pressures higher than100MPa,and low cooling rates),the g-phase can also form[41].In spite of this,the presence of different crystalline phases is usually neglected in the literature,essentially because the range of cooling rates investigated for characterization falls in the DSC range (well lower than typical cooling rates of interest for the process)and only one crystalline phase is formed for iPP at low cooling rates.It has to be noticed that for iPP,which presents a T g well lower than ambient temperature,high values of crystallinity degree are always found in solids which passed through ambient temperature,and the cooling rate can only determine which crystalline phase forms, roughly a-phase at low cooling rates(below about 508C/s)and mesomorphic phase at higher cooling rates.The most widespread approach to the description of kinetic constant is the isokinetic approach introduced by Nakamura et al.According to this model,d c in Eq.(1)is calculated asd cðtÞZ ln2ðt0KðTðsÞÞd s2 435n(4)where K is the kinetic constant and n is the so-called Avrami index.When introduced as in Eq.(4),the reciprocal of the kinetic constant is a characteristic time for crystallization,namely the crystallization half-time, t05.If a polymer is cooled through the crystallization temperature,crystallization takes place at the tempera-ture at which crystallization half-time is of the order of characteristic cooling time t q defined ast q Z D T=q(5) where q is the cooling rate and D T is a temperature interval over which the crystallization kinetic constant changes of at least one order of magnitude.The temperature dependence of the kinetic constant is modeled using some analytical function which,in the simplest approach,is described by a Gaussian shaped curve:KðTÞZ K0exp K4ln2ðT K T maxÞ2D2(6)The following Hoffman–Lauritzen expression[42] is also commonly adopted:K½TðtÞ Z K0exp KUÃR$ðTðtÞK T NÞ!exp KKÃ$ðTðtÞC T mÞ2TðtÞ2$ðT m K TðtÞÞð7ÞBoth equations describe a bell shaped curve with a maximum which for Eq.(6)is located at T Z T max and for Eq.(7)lies at a temperature between T m(the melting temperature)and T N(which is classically assumed to be 308C below the glass transition temperature).Accord-ing to Eq.(7),the kinetic constant is exactly zero at T Z T m and at T Z T N,whereas Eq.(6)describes a reduction of several orders of magnitude when the temperature departs from T max of a value higher than2D.It is worth mentioning that only three parameters are needed for Eq.(6),whereas Eq.(7)needs the definition offive parameters.Some authors[43,44]couple the above equations with the so-called‘induction time’,which can be defined as the time the crystallization process starts, when the temperature is below the equilibrium melting temperature.It is normally described as[45]Dt indDtZðT0m K TÞat m(8)where t m,T0m and a are material constants.It should be mentioned that it has been found[46,47]that there is no need to explicitly incorporate an induction time when the modeling is based upon the KAE equation(Eq.(1)).1.1.3.Modeling of the morphology evolutionDespite of the fact that the approaches based on Eq.(4)do represent a significant step toward the descriptionR.Pantani et al./Prog.Polym.Sci.30(2005)1185–12221191of morphology,it has often been pointed out in the literature that the isokinetic approach on which Nakamura’s equation (Eq.(4))is based does not describe details of structure formation [48].For instance,the well-known experience that,with many polymers,the number of spherulites in the final solid sample increases strongly with increasing cooling rate,is indeed not taken into account by this approach.Furthermore,Eq.(4)describes an increase of crystal-linity (at constant temperature)depending only on the current value of crystallinity degree itself,whereas it is expected that the crystallization rate should depend also on the number of crystalline entities present in the material.These limits are overcome by considering the crystallization phenomenon as the consequence of nucleation and growth.Kolmogoroff’s model [49],which describes crystallinity evolution accounting of the number of nuclei per unit volume and spherulitic growth rate can then be applied.In this case,d c in Eq.(1)is described asd ðt ÞZ C m ðt 0d N ðs Þd s$ðt sG ðu Þd u 2435nd s (9)where C m is a shape factor (C 3Z 4/3p ,for spherical growth),G (T (t ))is the linear growth rate,and N (T (t ))is the nucleation density.The following Hoffman–Lauritzen expression is normally adopted for the growth rateG ½T ðt Þ Z G 0exp KUR $ðT ðt ÞK T N Þ!exp K K g $ðT ðt ÞC T m Þ2T ðt Þ2$ðT m K T ðt ÞÞð10ÞEqs.(7)and (10)have the same form,however the values of the constants are different.The nucleation mechanism can be either homo-geneous or heterogeneous.In the case of heterogeneous nucleation,two equations are reported in the literature,both describing the nucleation density as a function of temperature [37,50]:N ðT ðt ÞÞZ N 0exp ½j $ðT m K T ðt ÞÞ (11)N ðT ðt ÞÞZ N 0exp K 3$T mT ðt ÞðT m K T ðt ÞÞ(12)In the case of homogeneous nucleation,the nucleation rate rather than the nucleation density is function of temperature,and a Hoffman–Lauritzen expression isadoptedd N ðT ðt ÞÞd t Z N 0exp K C 1ðT ðt ÞK T N Þ!exp KC 2$ðT ðt ÞC T m ÞT ðt Þ$ðT m K T ðt ÞÞð13ÞConcentration of nucleating particles is usually quite significant in commercial polymers,and thus hetero-geneous nucleation becomes the dominant mechanism.When Kolmogoroff’s approach is followed,the number N a of active nuclei at the end of the crystal-lization process can be calculated as [48]N a ;final Zðt final 0d N ½T ðs Þd sð1K x ðs ÞÞd s (14)and the average dimension of crystalline structures can be attained by geometrical considerations.Pantani et al.[51]and Zuidema et al.[22]exploited this method to describe the distribution of crystallinity and the final average radius of the spherulites in injection moldings of polypropylene;in particular,they adopted the following equationR Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3x a ;final 4p N a ;final 3s (15)A different approach is also present in the literature,somehow halfway between Nakamura’s and Kolmo-goroff’s models:the growth rate (G )and the kinetic constant (K )are described independently,and the number of active nuclei (and consequently the average dimensions of crystalline entities)can be obtained by coupling Eqs.(4)and (9)asN a ðT ÞZ 3ln 24p K ðT ÞG ðT Þ 3(16)where heterogeneous nucleation and spherical growth is assumed (Avrami’s index Z 3).Guo et al.[43]adopted this approach to describe the dimensions of spherulites in injection moldings of polypropylene.1.1.4.Modeling of the effect of crystallinity on rheology As mentioned above,crystallization has a dramatic influence on material viscosity.This phenomenon must obviously be taken into account and,indeed,the solidification of a semi-crystalline material is essen-tially caused by crystallization rather than by tempera-ture in normal processing conditions.Despite of the importance of the subject,the relevant literature on the effect of crystallinity on viscosity isR.Pantani et al./Prog.Polym.Sci.30(2005)1185–12221192rather scarce.This might be due to the difficulties in measuring simultaneously rheological properties and crystallinity evolution during the same tests.Apart from some attempts to obtain simultaneous measure-ments of crystallinity and viscosity by special setups [52,53],more often viscosity and crystallinity are measured during separate tests having the same thermal history,thus greatly simplifying the experimental approach.Nevertheless,very few works can be retrieved in the literature in which(shear or complex) viscosity can be somehow linked to a crystallinity development.This is the case of Winter and co-workers [54],Vleeshouwers and Meijer[55](crystallinity evolution can be drawn from Swartjes[56]),Boutahar et al.[57],Titomanlio et al.[15],Han and Wang[36], Floudas et al.[58],Wassner and Maier[59],Pantani et al.[60],Pogodina et al.[61],Acierno and Grizzuti[62].All the authors essentially agree that melt viscosity experiences an abrupt increase when crystallinity degree reaches a certain‘critical’value,x c[15]. However,little agreement is found in the literature on the value of this critical crystallinity degree:assuming that x c is reached when the viscosity increases of one order of magnitude with respect to the molten state,it is found in the literature that,for iPP,x c ranges from a value of a few percent[15,62,60,58]up to values of20–30%[58,61]or even higher than40%[59,54,57].Some studies are also reported on the secondary effects of relevant variables such as temperature or shear rate(or frequency)on the dependence of crystallinity on viscosity.As for the effect of temperature,Titomanlio[15]found for an iPP that the increase of viscosity for the same crystallinity degree was higher at lower temperatures,whereas Winter[63] reports the opposite trend for a thermoplastic elasto-meric polypropylene.As for the effect of shear rate,a general agreement is found in the literature that the increase of viscosity for the same crystallinity degree is lower at higher deformation rates[62,61,57].Essentially,the equations adopted to describe the effect of crystallinity on viscosity of polymers can be grouped into two main categories:–equations based on suspensions theories(for a review,see[64]or[65]);–empirical equations.Some of the equations adopted in the literature with regard to polymer processing are summarized in Table1.Apart from Eq.(17)adopted by Katayama and Yoon [66],all equations predict a sharp increase of viscosity on increasing crystallinity,sometimes reaching infinite (Eqs.(18)and(21)).All authors consider that the relevant variable is the volume occupied by crystalline entities(i.e.x),even if the dimensions of the crystals should reasonably have an effect.1.1.5.Modeling of the molecular orientationOne of the most challenging problems to present day polymer science regards the reliable prediction of molecular orientation during transformation processes. Indeed,although pressure and velocity distribution during injection molding can be satisfactorily described by viscous models,details of the viscoelastic nature of the polymer need to be accounted for in the descriptionTable1List of the most used equations to describe the effect of crystallinity on viscosityEquation Author Derivation Parameters h=h0Z1C a0x(17)Katayama[66]Suspensions a Z99h=h0Z1=ðx K x cÞa0(18)Ziabicki[67]Empirical x c Z0.1h=h0Z1C a1expðK a2=x a3Þ(19)Titomanlio[15],also adopted byGuo[68]and Hieber[25]Empiricalh=h0Z expða1x a2Þ(20)Shimizu[69],also adopted byZuidema[22]and Hieber[25]Empiricalh=h0Z1Cðx=a1Þa2=ð1Kðx=a1Þa2Þ(21)Tanner[70]Empirical,basedon suspensionsa1Z0.44for compact crystallitesa1Z0.68for spherical crystallitesh=h0Z expða1x C a2x2Þ(22)Han[36]Empiricalh=h0Z1C a1x C a2x2(23)Tanner[71]Empirical a1Z0.54,a2Z4,x!0.4h=h0Zð1K x=a0ÞK2(24)Metzner[65],also adopted byTanner[70]Suspensions a Z0.68for smooth spheresR.Pantani et al./Prog.Polym.Sci.30(2005)1185–12221193。
Phase diagramHello everybody, welcome to my class. Today, we will talk about phase diagram and Gibbs phase rule, as well as how to calculate the corresponding proportion of liquid phase and solid phase.译文：大家好，欢迎来到我的课程。
今天，我们将讨论相图，吉布斯相律，以及如何计算液相和固相的相对含量。
First of all, let’s introduce the definition of phase. Phase is defined as a homogeneous part or aggregation of material. This homogenous part is distinguished from another part due to difference in structure, composition, or both. The different structures form an interface to difference in structure and composition. （这里要注意相的概念，相是指在结构和组成方面与其它部分不同的均匀体。
）译文：我们首先学习相的定义。
相是指在一种材料中，结构、组成，或两者同时不同于其他部分的均匀体或聚集体部分。
不同部分间形成界面，也就是相与相之间的分界面。
Some solid materials have the capability of changing their crystal structure under the varying conditions of pressure and temperature, causing an ability of phase-change.译文：一些固体材料随着压力和温度条件的改变而发生结晶结构变化，具有相变的能力。
生物信息学复习题名词解释1. Homology (同源):来源于共同祖先的序列相似的序列及同源序列。
序列相似序列并不一定是同源序列。
2.Orthologs（直系同源）：指由于物种形成的特殊事件来自一个共同祖先的不同物种中的同源序列，它们具有相似的功能。
3.Paralogs（旁系（并系）同源）：指同一个物种中具有共同祖先，通过基因复制产生的一组基因，这些基因在功能上的可能发生了改变。
基因复制事件是促进新基因进化的重要推动力。
4.Xenologs (异同源)：通过横向转移，来源于共生或病毒侵染而产生的相似的序列，为异同源。
5.Identity Score：The sum of the number of identical matches and conservative (high scoring) substitutions in a sequence alignment divided by the total number of aligned sequence characters. Gap总是不计入总数中。
6.点矩阵（dot matrix）：构建一个二维矩阵，其X轴是一条序列，Y轴是另一个序列，然后在2个序列相同碱基的对应位置（x，y）加点，如果两条序列完全相同则会形成一条主对角线，如果两条序列相似则会出现一条或者几条直线；如果完全没有相似性则不能连成直线。
7. E值：得分大于等于某个分值S的不同的比对的数目在随机的数据库搜索中发生的可能性。
衡量序列之间相似性是否显著的期望值。
E值大小说明了可以找到与查询序列（query）相匹配的随机或无关序列的概率，E值越小意味着序列的相似性偶然发生的机会越小，也即相似性越能反映真实的生物学意义，E值越接近零，越不可能找到其他匹配序列。
8.P值：得分为所要求的分值比对或更好的比对随机发生的概率。
它是将观测得到的比对得分S，与同样长度和组成的随机序列作为查询序列进行数据库搜索进行比较得到的HSP（高分片段对）得分的期望分布联系起来计算的。
分支历史表法（Phylogenetic Tree），也称为系统发生树或进化树，是一种用于描述生物物种之间演化关系的图形化表示方法。
它通过树状结构展示了不同物种的分支演化关系，帮助我们理解生物界的进化历程和亲缘关系。
以下是一些分支历史表法中常用的术语解释：
节点（Node）：树状图中的分支交叉点，表示共同的祖先物种。
节点表示物种的分化和分裂事件。
分支（Branch）：树状图中连接节点的线段，表示物种的演化过程和亲缘关系。
进化距离（Evolutionary Distance）：表示不同物种之间的差异程度，可以用来衡量物种之间的相似性或差异性。
分类单元（Taxonomic Unit）：分支历史表法中的末梢节点，代表具体的生物物种。
内部节点（Internal Node）：位于分支历史表法中两个或多个分支之间的节点，代表共同的祖先物种。
进化支持度（Bootstrap Support）：表示树状图上每个节点的可靠性程度，常用百分比表示。
高支持度表示该节点的演化关系是可靠的。
距离矩阵（Distance Matrix）：用于计算不同物种之间的进化距离的矩阵，通过比较基因序列、蛋白质序列或形态特征等来测量差异。
分支历史表法在生物学、系统学和进化生物学等领域得到广泛应用。
它可以帮助我们理解物种的进化过程、亲缘关系和分类系统，以及探索物种间的共同祖先和演化途径。
这种图形化表示方法为研究生物多样性、种群遗传和物种起源等提供了重要的工具和参考。
a r X i v :c o n d -m a t /9903213v 1 12 M a r 1999Phase Diagrams from Topological Transitions:The Hubbard Chain with Correlated HoppingA.A.Aligia a ,K.Hallberg a ,C.D.Batista a ,and G.Ortiz baComisi´o n Nacional de Energ´ıa At´o mica,Centro At´o mico Bariloche and Instituto Balseiro,8400S.C.de Bariloche,ArgentinabTheoretical Division,Los Alamos National Laboratory,Los Alamos,NM 87545(Received February 1,2008)The quantum phase diagram of the Hubbard chain with correlated hopping is accurately determined through jumps in πin the charge and spin Berry phases.The nature of each thermodynamic phase,and the existence of charge and spin gaps,is confirmed by calculating correlation functions and other fundamental quantities using numerical methods,and symmetry arguments.Remarkably we find striking similarities between the stable phases for moderate on-site Coulomb repulsion:spin Peierls,spin-density-wave and triplet superconductor,and those measured in (TMTSF)2X.The search for electronic mechanisms of superconduc-tivity and the study of superconducting and Mott phase transitions are among the most interesting subjects of the physics of strongly correlated systems.In few cases,exact results have helped to elucidate the nature of these transitions [1–3].In general one has to rely on numerical calculations of finite systems for which quantities like the Drude weight D c (which should vanish for an insulator in the thermodynamic limit [4]),or any other correlation function,vary smoothly at the transition.Consequently,for instance,the boundaries between a charge-density-wave (CDW)or spin-density-wave (SDW)insulators and metallic phases in half-filled generalized Hubbard models were difficult to establish [5,6].The Berry phase is a general geometrical concept which finds realizations in various physical problems [7].It is the anholonomy associated to the parallel transport of a vector state in a certain parameter space.In condensed matter,the charge Berry phase γc is a measure of the macroscopic electric polarization in band or Mott insula-tors [8]while the spin Berry phase γs represents its spin polarization [9,10].In systems with inversion symmetry γc and γs can attain only two values:0or π(mod(2π)).Thus,if two thermodynamic phases differ in the topolog-ical vector γ=(γc ,γs )this sharp difference allows us to unambiguously identify the transition point even in finite systems.This “order parameter”was recently used to de-tect metallic,insulator and metal-insulator transitions in one-dimensional lattice fermion models [9,10].In this Letter we determine the quantum phase dia-gram of the Hubbard chain with correlated hopping at half-filling using topological transitions.The phase di-agram is very rich showing two metallic and two insu-lating thermodynamic phases each characterized by one of the four possible values of the topological vector γ.One of the metallic phases corresponds to a Tomonaga-Luttinger liquid with dominant triplet superconducting correlations at large distances (TS).This is interesting since there is experimental evidence indicating that the Bechgaard salts (TMTSF)2ClO 4and (TMTSF)PF 6un-der pressure are TS [11–13].Furthermore,the insulat-ing SDW and spin gapped spin-Peierls phase observed in (TMTSF)PF 6as the pressure is lowered [14]are also present in the model phase diagram.The effective model Hamiltonian is:[3]H = i,j σ(c †iσc jσ+h.c.){t AA (1−n i ¯σ)(1−n j ¯σ)+t BB n i ¯σn j ¯σ+t AB [n i ¯σ(1−n j ¯σ)+n j ¯σ(1−n i ¯σ)]}+U i (n i ↑−12).(1)H contains the most general form of hopping term de-scribing the low energy physics of a broad class of sys-tem Hamiltonians in which four states per effective siteare retained.In particular,the Hamiltonian H in Eq.(1)has been derived and studied for transition met-als,organic molecules and compounds [2],intermedi-ate valence systems,cuprates and other superconduc-tors [15].In the continuum limit,the only relevant in-teractions at half-filling are U and t AA +t BB −2t AB [16].Therefore,we restrict the present study to the electron-hole symmetric case (t AA =t BB =1)which has spin and pseudospin SU(2)symmetries,the latterwith generators η+= i (−1)i c †i ↑c †i ↓,η−=(η+)†,and ηz =1changes in the Berry phase by:∆P↑±∆P↓=e∆γc,s/2π(mod(e))[8–10].Thus,a phase transition will be de-tected by a jump inπofγc(γs)if and only if both ther-modynamic phases differ in P↑+P↓(P↑−P↓)by e/2 (mod(e)).For example,if one of the phases is a CDW with maximum order parameter(CDWM)and the other a N´e el state(N),one is transformed into the other trans-porting half of the charges(those with a given spin)one lattice parameter.In addition,as explained below,in the present model topological transitions inγc andγs indicate the opening of the charge and spin gap∆c,∆s. Wefind that the minimum of the GS energy as a func-tion offluxes,E g(φ↑,φ↓),corresponds to the so-called closed shell conditions(CSC):if the number of sites(as-sumed even)is L=2(mod(4)),then K=φσ=0,while for L multiple of four K=φσ=π(which is equivalent to taking antiperiodic boundary conditions and¯K=0inthecor-atspintwo topological transitions occur in the model,correspond-ing to a jump in eitherγc orγs.We have determined those transitions in rings of length L=6,8,10,12using the Lanczos method.The results extrapolated with a cu-bic polynomial in1/L are represented in Fig.1.In con-trast to other physical quantities which show largefinite-size effects,particularly near t AB=0[6],the topolog-ical transitions converge rapidly to the thermodynamic limit(for example,for t AB=0.05,γc jumps at U=3.451, 3.681,3.788,and3.846for L=6−12,and the ex-trapolated value is U=3.932).The numerical conver-gence becomes problematic for smaller t AB values.At t AB=0the transition points are determined from the exact solution[3]as those values of U where∆c and∆s open.Those critical values are U c,s=±4and match smoothly with the rest of the curves in Fig.1.It is easy to see that under CT the geometrical phases transform asγc←→γs+π[9].Thus,as seen in Fig.1,a jump in γc at U c(full line)implies a jump inγs at−U c(dashed line),and vice versa.In the case where all particles are localized one can easily determine the value of γasγc,s=Im ln z c,sL,where z c,s L= g|e i2πofdospin excitations form a subset of all charge excitations. In our case the charge velocity v c and∆c(computed as usual infinite systems L≤12[19])coincide with their pseudospin counterparts vηand∆η(computed from v s and∆s for opposite U)[21].This is consistent with the exact solution for t AB=0where the charge excitations of lowest energy are pseudospin ones[3].In addition, the charge-charge and spin-spin correlation functions are interchanged as U changes sign(see below).The open-ing of∆c whereγc jumps from0toπis also consistent with calculations of z c L,D c,superconducting correlation functions,Kρand central charge c.For t AB>1,∆c opens more slowly and the detection of the transition becomes more difficult.However,as shown in Fig.3,D c and z c L display a similar behavior near the jump inγc (U c=-1.702)as in the quarter-filled infinite U extended Hubbard model as a function of the nearest-neighbor re-pulsion V[18],where a metal-insulator transition takes place at V=2t.At large|U|the only relevant energy scale is4t2AB/|U|and therefore,D c increases with U for large negative U.Near U c there is a drastic change of behavior of D c and z L vs U,and for U>U c,the extrap-olated values suggest a tendency to reach the insulating values(D c=0,z c L=1),in the thermodynamic limit.z c L always decreases(increases)with L at the left(right)of U c.This is thefirst accurate(DMRG)calculation of z c L in a system of more than16sites.In order to further characterize each thermodynamic we use symmetry arguments and the following cor-functions(CF)(see Fig.4):χt(s)(d)=12(n0↑+n0↓−1)(n d↑+n d↓−1)χbsdw(d)=1All CF decay exponen-tially.To understand the nature of this insulating GS let us analyze the limit t AB→∞,where effectively there is a sequence of spin and pseudospin dimers.Each dimer is the ground state of the model for two sites and two par-ticles.Including the dimer-dimer interaction in second-order perturbation theory the resulting GS energy per site is e≈−1.0625t AB(e DMRG=−1.0808t AB).The energy difference between the two lowest singlet states (with¯K=0and¯K=π)decays nearly exponentially with L,indicating a breaking of the translational sym-metry in the thermodynamic limit.b) γ=(π,π),∆c=0,∆s=0.The system forms a Luther-Emery liquid withχs andχc CF decaying as1/d (neglecting logarithmic corrections.)d) γ=(0,π),∆c=0,∆s=0.metry arguments,for U =0,all CF except χt =χbsdw show the same 1/d 2decay (apparently without logarith-mic corrections).Renormalization group arguments [22]show that when ∆s =0,χt should decay more slowly than χs ,and therefore dominate for small |U |.As U in-creases,χt decays more rapidly,while the opposite hap-pens with χsz .Near the opening of ∆c (at U c ≈2.05for tdistance d as a function of d for t AB =0.6:nearest-neighbor triplet |χt (d )|=|χbsdw (d )|(squares)and singlet χs (d )(cir-cles)pair CF,on-site pair and charge CF |χos (d )|=|χc (d )|(downward triangles),and spin CF χsz (d )(upward triangles).See Eq.(4).In conclusion,we have constructed the quantum phase diagram of Eq.(1)from topological considerations.Each thermodynamic phase is associated to a topological vec-tor,and changes in that quantity signal the transition point.A key finding is the identification of a triplet superconducting phase degenerate with a bond-located SDW for t AB <1and small |U |.Taking into account the slight dimerization in (TMTSF)2X compounds,an effective Hamiltonian similar to Eq.(1)can be real-ized,where only four low-energy states per unit cell are kept.Assuming that additional interactions stabilize the TS with respect to the BSDW it is remarkable that,forU >0,as t AB is decreased a similar sequence of quantum phase transitions takes place as pressure is applied [14].This general approach can be extended to spatial dimen-sions higher than one [8],and applied to more general models which do not necessarily have SU(2)pseudospin symmetry [9].One of us (A.A.A.)thanks J.Voit and L.Arrachea for discussions.We acknowledge computer time at Max-Planck Institute f¨u r Physik Komplexer Systeme.K.H.and C.D.B.are supported by CONICET,while G.O.is supported by the US Depart-of Energy.。
