Effective Field Theory of Triangular-Lattice Three-Spin Interaction Model

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金兹堡朗道理论

Ginzburg–Landau theoryFrom Wikipedia, the free encyclopediaIn physics, Ginzburg–Landau theory, named after Vitaly Lazarevich Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. Later, a version of Ginzburg–Landau theory was derived from the Bardeen-Cooper-Schrieffer microscopic theory by Lev Gor'kov, thus showing that it also appears in some limit of microscopic theory and giving microscopic interpretation of all its parameters.Contents•1Introduction•2Simple interpretation•3Coherence length and penetration depth•4Fluctuations in the Ginzburg–Landau model•5Classification of superconductors based on Ginzburg–Landau theory•6Landau–Ginzburg theories in string theory•7See also•8References•8.1PapersIntroduction[edit]Based on Landau's previously-established theory of second-order phase transitions, Ginzburg and Landau argued that the free energy, F, of a superconductor near the superconducting transition can be expressed in terms ofa complex order parameter field, ψ, which is nonzero below a phase transition into a superconducting state and isrelated to the density of the superconducting component, although no direct interpretation of this parameter was given in the original paper. Assuming smallness of |ψ| and smallness of its gradients, the free energy has the form ofa field theory.where F n is the free energy in the normal phase, α and β in the initial argument were treated as phenomenologicalparameters, m is an effective mass, e is the charge of an electron, A is the magnetic vector potential, and is the magnetic field. By minimizing the free energy with respect to variations in the order parameter and the vector potential, one arrives at the Ginzburg–Landau equationswhere j denotes the dissipation-less electric current density and Re the real part. The first equation — which bears some similarities to the time-independent Schrödinger equation, but is principally different due to a nonlinear term —determines the order parameter, ψ. The second equation then provides the superconducting current.Simple interpretation[edit]Consider a homogeneous superconductor where there is no superconducting current and the equation for ψ simplifies to:This equation has a trivial solution: ψ = 0. This corresponds to the normal state of the superconductor, that is for temperatures above the superconducting transition temperature, T>T c.Below the superconducting transition temperature, the above equation is expected to have a non-trivial solution (that is ψ ≠ 0). Under this assumption the equation above can be rearranged into:When the right hand side of this equation is positive, there is a nonzero solution for ψ (remember that the magnitude of a complex number can be positive or zero). This can be achieved by assuming the following temperature dependence of α: α(T) = α0 (T - T c) with α0/ β > 0:•Above the superconducting transition temperature, T > T c, the expression α(T) / β is positive and the right hand side of the equation above is negative. The magnitude of a complex number must be a non-negative number, so only ψ = 0 solves the Ginzburg–Landau equation.•Below the superconducting transition temperature, T < T c, the right hand side of the equation above is positive and there is a non-trivial solution for ψ. Furthermorethat is ψ approaches zero as T gets closer to T c from below. Such a behaviour is typical for a second order phase transition.In Ginzburg–Landau theory the electrons that contribute to superconductivity were proposed to forma superfluid.[1] In this interpretation, |ψ|2 indicates the fraction of electrons that have condensed into a superfluid.[1] Coherence length and penetration depth[edit]The Ginzburg–Landau equations predicted two new characteristic lengths in a superconductor which wastermed coherence length, ξ. For T > T c (normal phase), it is given bywhile for T < T c (superconducting phase), where it is more relevant, it is given byIt sets the exponential law according to which small perturbations of density of superconducting electrons recover their equilibrium value ψ0. Thus this theory characterized all superconductors by two length scales. The second one is the penetration depth, λ. It was previously introduced by the London brothers in their London theory. Expressed in terms of the parameters of Ginzburg-Landau model it iswhere ψ0 is the equilibrium value of the order parameter in the absence of an electromagnetic field. The penetration depth sets the exponential law according to which an external magnetic field decays inside the superconductor. The original idea on the parameter "k" belongs to Landau. The ratio κ = λ/ξ is presently known asthe Ginzburg–Landau parameter. It has been proposed by Landau that Type I superconductors are those with 0 < κ< 1/√2, and Type II superconductors those with κ> 1/√2.The exponential decay of the magnetic field is equivalent with the Higgs mechanism in high-energy physics. Fluctuations in the Ginzburg–Landau model[edit]Taking into account fluctuations. For Type II superconductors, the phase transition from the normal state is of second order, as demonstrated by Dasgupta and Halperin. While for Type I superconductors it is of first order as demonstrated by Halperin, Lubensky and Ma.Classification of superconductors based on Ginzburg–Landau theory[edit]In the original paper Ginzburg and Landau observed the existence of two types of superconductors depending on the energy of the interface between the normal and superconducting states.The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value H c. Depending on the geometry of the sample, one may obtain an intermediate state[2] consisting of a baroque pattern[3] of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductors, raising the applied field past a critical value H c1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength H c2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called fluxons because the flux carried by these vortices is quantized. Most pure elemental superconductors, except niobium and carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II.The most important finding from Ginzburg–Landau theory was made by Alexei Abrikosov in 1957. He used Ginzburg–Landau theory to explain experiments on superconducting alloys and thin films. He found that in a type-II superconductor in a high magnetic field, the field penetrates in a triangular lattice of quantized tubes offlux vortices.[citation needed]Landau–Ginzburg theories in string theory[edit]In particle physics, any quantum field theory with a unique classical vacuum state and a potential energy witha degenerate critical point is called a Landau–Ginzburg theory. The generalization to N=(2,2) supersymmetric theories in 2 spacetime dimensions was proposed by Cumrun Vafa and Nicholas Warner in the November 1988 article Catastrophes and the Classification of Conformal Theories, in this generalization one imposes thatthe superpotential possess a degenerate critical point. The same month, together with Brian Greene they argued that these theories are related by a renormalization group flow to sigma models on Calabi–Yau manifolds in thepaper Calabi–Yau Manifolds and Renormalization Group Flows. In his 1993 paper Phases of N=2 theories intwo-dimensions, Edward Witten argued that Landau–Ginzburg theories and sigma models on Calabi–Yau manifolds are different phases of the same theory. A construction of such a duality was given by relating the Gromov-Witten theory of Calabi-Yau orbifolds to FJRW theory an analogous Landau-Ginzburg "FJRW" theory in The Witten Equation, Mirror Symmetry and Quantum Singularity Theory. Witten's sigma models were later used to describe the low energy dynamics of 4-dimensional gauge theories with monopoles as well as brane constructions. Gaiotto, Gukov & Seiberg (2013)See also[edit]•Domain wall (magnetism)•Flux pinning•Gross–Pitaevskii equation•Husimi Q representation•Landau theory•Magnetic domain•Magnetic flux quantum•Reaction–diffusion systems•Quantum vortex•Topological defectReferences[edit]1.^ Jump up to:a b Ginzburg VL (July 2004). "On superconductivity and superfluidity (what I have and havenot managed to do), as well as on the 'physical minimum' at the beginning of the 21 st century". Chemphyschem.5 (7): 930–945. doi:10.1002/cphc.200400182. PMID15298379.2.Jump up^ Lev D. Landau; Evgeny M. Lifschitz (1984). Electrodynamics of Continuous Media. Course ofTheoretical Physics8. Oxford: Butterworth-Heinemann. ISBN0-7506-2634-8.3.Jump up^ David J. E. Callaway (1990). "On the remarkable structure of the superconductingintermediate state". Nuclear Physics B344 (3): 627–645. Bibcode:1990NuPhB.344..627C.doi:10.1016/0550-3213(90)90672-Z.Papers[edit]•V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz.20, 1064 (1950). English translation in: L. D. Landau, Collected papers (Oxford: Pergamon Press, 1965) p. 546• A.A. Abrikosov, Zh. Eksp. Teor. Fiz.32, 1442 (1957) (English translation: Sov. Phys. JETP5 1174 (1957)].) Abrikosov's original paper on vortex structure of Type-II superconductors derived as a solution of G–L equations for κ > 1/√2•L.P. Gor'kov, Sov. Phys. JETP36, 1364 (1959)• A.A. Abrikosov's 2003 Nobel lecture: pdf file or video•V.L. Ginzburg's 2003 Nobel Lecture: pdf file or video•Gaiotto, David; Gukov, Sergei; Seiberg, Nathan (2013), "Surface Defects and Resolvents" (PDF), Journal of High Energy Physics。

旅行者-凝聚态(30215148)LDA+DMFT应用于强关联体系报告(2012-12-02)

菜刀___宇宙(1725100166) 20:29:19欢迎“旅行者-凝聚态”作报告！虚云居士-光学(350887807) 20:29:52旅行者-凝聚态(30215148) 20:30:02谢谢。

物理界的又一个创新者(1501865379) 20:30:23夜鸮-物理(1900543431) 20:30:46旅行者-凝聚态(30215148) 20:30:55首先很高兴能有机会能跟各位交流。

旅行者-凝聚态(30215148) 20:32:08首先我介绍一下LDA+DMFT的理论背景旅行者-凝聚态(30215148) 20:32:49凝聚态中处理强关联的体系是一个极为复杂的问题，可是说是整个凝聚态最难处理的问题。

旅行者-凝聚态(30215148) 20:33:17我们知道固体的哈密顿量在做了绝热近似之后为旅行者-凝聚态(30215148) 20:33:22旅行者-凝聚态(30215148) 20:33:39这个式子包含三部分旅行者-凝聚态(30215148) 20:33:47第一项Te电子动能，第二项Ven电子在核库仑场中的势能，以及第三项Vee电子之间相互作用的库仑。

旅行者-凝聚态(30215148) 20:34:14其中第一项和第二项应该是容易处理的，第三项最难处理旅行者-凝聚态(30215148) 20:34:57电子之间的库仑相互作用项，在进一步做了Hatree-Fock自洽场近似之后可以将最后一项写为旅行者-凝聚态(30215148) 20:35:12公式中NZ为电子数目旅行者-凝聚态(30215148) 20:35:34可以想想直接求和存在库仑发散的困难。

旅行者-凝聚态(30215148) 20:36:04这个式子就是说把其中一个电子i所感受到的其他电子对它的作用用一个自洽的平均势场ve表示。

旅行者-凝聚态(30215148) 20:36:32所谓的自洽就是指由于平均势场对i电子的作用会改变i电子的运动状态，而改变了运动状态的i电子显然会反过来影响平均势场，从而进一步改变i电子的状态。

蛋白质HP模型的改进和PERM算法求解

１引言生物信息科学在上世纪末到目前取得重大突破和进展，
完成了水稻基因组和人类基因组测序的伟大工程。随着研究的深入，目前的研究重点从对基因的研究逐步转移到对生物蛋白质结构和功能研究上来。这就是后基因组学。
生物蛋白质是生物体内的基本活性功能单元，是生物机能的最基本的载体。对于蛋白质的生物活性功能，目前一个广泛接受的基本假设是１９６１年Ａｎｆｉｎｓｅｎ提出的蛋白质折叠过程的热力学假设：蛋白质的自然结构形式完全包含在相应分子的氨基酸链的序列信息中，并且自然界存在的蛋白质形态是最稳定的形态（自由能最低）。蛋白质链的空间折叠结构决定了蛋白质分子的功能。就是常说的“序列决定结构，
图１ＨＰ正方格点模型缺陷“ｐａｒｉｔｙｃｏｎｓｔｒａｉｎｔ”：蛋白质链“ＰＨＰＨＰＨＰＨＰ”在方形格点一定中没有“ＨＨ”相切的势能。在右图的三角格点中。同一个链的“ＨＨ”能量是５
图１中说明的缺陷，文献［３］文称之为“ｐａｒｉｔｙｃｏｎ· ｓｔｒａｉｎｔ”，即一条仅由Ｎ对“ＰＨ”串构成的长蛋白质链，在正方形格点模型中的最低自由能永远为０，没有“ＨＨ”相切点。这与该链在实际空问中的能量不符合。由于ｐａｒｉｔｙｃｏｎｓｔｒａｉｎｔ的存在，ＨＰ格点模型存在缺陷。这是由于单位球紧密排列在单位正方形格点空间中，本身具有不紧密性。
第２４卷第５期文章编号：１００６—９３４８（２００７）０５—０２８７—０４
计算机仿真
２００７年５月
蛋白质ＨＰ模型的改进和ＰＥＲＭ算法求解
陈凤飞，黄文奇
（华中科技大学计算机科学与技术学院，武汉４３００７４）
摘要：ＨＰ模型是一种被广泛研究的简化的蛋白质折叠模型。在蛋白质螺旋结构的预测上有很高的可信度。但是ＨＰ的正
当囊‰。＝０时，表示此ｉ长度的构型不能进行第ｉ＋１步生长，该链的生长终止。

具有不同自旋的电子在对称双δ势垒中的量子隧穿

摘要：Ｓｏｃｗｋ的自由电子理论模型下，虑对称双势垒中具有不同自旋的电子问相互作用，用量在ｌｎｚｓｉ考利子隧穿原理计算了铁磁／缘层／导体／缘层／磁（Ｍ／／Ｍ／／Ｍ）隧道结中电子的反射系数与透射绝半绝铁ＦＩＳＩＦ双
Ｋｅｏｄ：ｓｍｍｅｒｃｌｏｂｅｐｔｎｉｌａｒｅ；ｓｉｙｗｒｓｙｔｉａｕｌｏｅｔｒｉｒｐｎ；ｒｆｅｔｏｏｆｃｅｔｒｎｍｉｓｎｃｅｆｉｎｄａｂｅｌｃｉｎｃｅｆｉｎ；ｔａｓｓｉｏｆｉｅｔｉｏｃ
Ｓｐ。０１ｅ２０
文章编号：１０－３３２１）３Ｏ３一３０４４５（Ｏ０Ｏ一２２Ｏ
具有不同自旋的电子在对称双．势垒中的量子隧穿６ｉ
卢祁，郭振平
（延边大学理学院物理系，吉林延吉１３０３０２）
（Ｖ≈ Ｏ，）ＴＭＲ与２铁磁电极中的电子在费米面上的自旋极化率的乘积成正比，个因而２个铁磁电极的自旋极化率越大，ＭＲ也越大；ｌｎｚｗｋ提出的自由电子模型是指，ＴＳｏｃｅｓｉ在铁磁金属中，自旋向上和
Ａｓｒｃ：ＴｈｖｓｉａｉｎｒａｅｎｔｅＳｏｅｗｓｉｄｌｃｎｉｅｉｇｉｔｒｃｉｎｂｔｒｎｅｅｔｏｓｗｉｂｔａｔｅｉｅｔｇｔｓａｅｂｓｄｏｈｌｎｚｋｎｏｍｏｅ，ｏｓｄｒｅａｔｅｘｅｌｃｒｎ隧穿概率与自旋方向和波矢都有关，而且势垒高度对ＴＭＲ也有重

Manuscript_H-VII-4

PAPER REFERENCE: H-VII.4 a-Si:H BASED TWO-DIMENSIONAL PHOTONIC CRYSTALSE.Bennici*1, S. Ferrero1,F.Giorgis1, C.F.Pirri1, R.Rizzoli2, P.Schina3, L.Businaro4, E.Di Fabrizio41 INFM - Dipartimento di Fisica del Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129Torino, Italy2 CNR - IMM Sez. Bologna, Via Gobetti 101, I-40129 Bologna, Italy3 Olivetti i-jet S.p.a., Loc. Les Vieux – Arnad (Aosta), Italy4 INFM - TASC at Elettra Synchrotron Light Source - Lilit Beam-line, S.S.14 Km 163.5, AreaScience Park, 34012 Basovizza - Trieste (Italy)AbstractWe describe the fabrication processes of silicon based two-dimensional photonic crystals (2D-PCs) with a photonic band gap in the near-IR range. The procedures involve electron beam lithography followed by an anisotropic etching step of hydrogenated amorphous silicon thin films deposited by Plasma Enhanced Chemical Vapor Deposition. Micrometric and submicrometric arrays of cylindrical holes are transferred using a poly-methylmethacrylate resist layer as a mask. A careful comparison between standard parallel plate Reactive Ion Eching and Inductively Coupled Plasma Etching techniques is performed, aimed at obtaining periodic structures with high aspect ratio and good profile sharpness.Keywords: Photonic crystals; amorphous Silicon; Anisotropic etching*CorrespondingAuthor:Tel.+390115647381:Fax.+390115647399:E-mail:*****************1. IntroductionPhotonic crystals (PCs, also known as photonic band gap materials) are optical materials with periodic changes in the dielectric constant at a periodicity on a wavelength scale. As demonstrated several years ago, this influences the propagation of light in a way which is analogous to the effect that crystalline potentials have on electrons [1]. PCs were investigated in the range of microwave frequencies in the past, but they are currently being pursued to obtain a range of forbidden frequencies (a photonic band gap, PBG) in the near infrared-visible region of the electromagnetic spectrum.Recently, we are developing fabrication processes for the realization of two-dimensional photonic crystals starting from silicon based layers. The analysed structures consist of squared and triangular sub-micrometric sized lattices of holes drilled on a homogeneous matrix with a high refractive index. In our case the high refractive index layer (~3.5 in the near infrared) is realized growing a hydrogenated amorphous silicon (a-Si:H) thin film on a crystalline silicon substrate by Plasma Enhanced Chemical Vapor Deposition (PECVD). The lattices of holes create an in-plane periodic variation in the refractive index, yielding forbidden energy gaps in the photon dispersion relations [2].Actually, 2D photonic crystals patterned on homogeneous layers do not have a full three-dimensional photonic band gap, so that the propagation of an electromagnetic wave can be inhibited only in the plane of the slab. On the other hand the optical modes below the light line suffer a total internal reflection at the air/slab interface and cannot phase match to the radiation modes [3]. Taking into account these features, several applications can be addressed for a-Si:H photonic crystal slabs.A first application can be a cap layer for the enhancement of the vertical extraction of the radiation pattern yielded by an underlying light source. Actually, a 2D photonic crystal, folds the guided modes at the two-dimensional Brillouin zone center, allowing phase matching to the radiation modes that lie above the air light line. Thus, the optical modes that phase match to radiation modes and satisfy the same symmetry of the electromagnetic field, become leaky resonances of the photonic crystal leading to high extraction efficiencies at particular wavelengths [4].A second application of a-Si:H based 2D-PCs slabs can be a planar waveguide. For this aim the a-Si:H films can be deposited on a pregrown SiO2 layer. A linear defect, constituted by the absence of holes, is introduced in the lattice of holes patterned on the a-Si:H slab. In such a mode, the planar photon confinement is assured by the 2D-PBG, while the vertical one is induced by internalreflection at the a-Si:H/SiO2 and a-Si:H/air interfaces. Thus, an efficient guided mode can be created through the linear defect.Finally, a possible structure towards a three-dimensional photonic crystal can be a planar Fabry-Perot (F-P) microcavity patterned with a lattice of holes running along all the layer of the F-P stack. In detail, F-P microcavities were recently grown by PECVD by alternating quarterwave a-SiN multilayers with different N content, with a central ‘defect’ constituted by a λ-thick luminescent N-rich film [5]. This structures yield a resonance inside of the 1D-PBG. By etching the multilayer structure with a lattice of holes having a ‘puntual defect’ (represented by the absence of a hole, or with a hole with different radius), a resonance within the 2D-PBG can be created. If the resonance of the etched cavity overlaps with that of the 2D structure, a sort of three-dimensional cavity would be fabricated.In this work, the results dealing with the fabrication of a-Si:H based 2D-PCs operating in the near-IR range, are presented and discussed. In particular we focus on the experimental procedures which will be devoted to the fabrication of the above mentioned prototypes.2. Experimentala-Si:H layers were grown by 13.56 MHz PECVD on c-Si (100) substrates using SiH4 as a gas source with a flow of 20 sccm and a substrate temperature of 220 °C; the pressure in the deposition chamber and the r.f. power were fixed at 0.6 mbar and 4 W, respectively. The resulting deposition rate was 2 Å/s.Since PBGs appear around a wavelength of ~2a (where a is the period of the lattice), the period must be reduced to few hundreds nanometers in order to obtain a photonic band gap in the near-IR-visible range. The pattern of cylindrical holes was defined in poly-methylmethacrylate (PMMA) by direct-write electron-beam lithography. The electron sensitive resist, with molecular weight of 950 K, was spinned on the sample and then baked by a hot plate at 180°C for 5 minutes. The spinning speed was calibrated on 4000 rpm in order to obtain a thickness of 170 nm. The exposures were performed by scanning electron microscopes (Jeol 6400 and Cambridge S90) equipped with electron beam pattern generator systems (Raith Elphy Quantum and Elphy Plus) using a beam energy of 25 kV. The PMMA development was performed in a 1:3 mixture of Methyl-isobutyl ketone (MIBK) and isopropyl alcohol (IPA) for 30s, finally rinsed in IPA [6].The designed patterns were transferred to the layer beneath the PMMA mask by dry etching techniques. With the aim to achieve a good control in the etching rate of a-Si:H and in the selectivity with respect to the PMMA, we compared two different etching systems and some different reactive gas mixtures such as CF4, CF4 + O2 and SF6. The etching procedures were performed both in a standard parallel plate Reactive Ion Etching system capacitively coupled to a r.f. power supply (RIE, Tegal 901 etching system), and in an inductively coupled plasma system (ICP, STS Multiplex system). In the ICP system two independent r.f. power supplies are respectively devoted to the substrate bias and to the plasma generation (trough a linked coil). Such a control can yield low ion energies and thus low ion bombardment of the substrate in a high-density plasma. Consequently, an ICP system can provide high chemical etch selectivities with high etching rates. In the used system, interfaced by a standard personal computer, the process pressure was controlled by a butterfly valve that is operable in fixed and automatic mode. The wafers was automatically loaded into the process chamber from a load-lock module.3. Results and DiscussionFig.1 show a SEM micrograph of one of the 2D-structures (array of submicrometric holes with triangular symmetry) realized by direct-write electron beam lithography on a PMMA coating spinned onto an a-Si:H film. It is worth to underline that important parameters for the optical properties of a 2D-photonic crystal are the refractive index of the dielectric slab, the lattice constant a (inter-hole spacing) and the radius r of the drilled holes. The structure here shown consists of holes with a radius, r = 0.41a, where a=350 nm realized on a matrix of a-Si:H (with a refractive index of 3.5 in the low absorption region). Such a structure gives a band gap from 800 nm up to 1.4 μm for TE modes (electric field in the plane of the slab) and two narrow band gaps, from 900nm up to 950nm and from 1.32 µm to 1.52 µm, for TM modes (electric field perpendicular to the slab), as verified by a freeware software aimed at computing the dispersion relations of periodic dielectric structures [7].A prerequisite for dimensional control in etching small features is the maintenance of a good anisotropy and the maximization of etch selectivity and uniformity. The experimental data concerning with the etching of a-Si:H films performed by the RIE system, have been compared with that obtained by the ICP. Etch rate and selectivity have been examined as a function of type of gas, flow rate, pressure and r.f. power.Focusing on the standard reactive ion etching system (RIE), the process optimisations led to the results summarized in Table 1. Under the same process conditions, the etch rate of a SF6 plasma ishigher than that obtainable with CF4 and CF4 + O2 gas mixtures; moreover, also the selectivity with respect to the PMMA mask results to be much higher. On the other side, the best results come from the etching performed by ICP, where it is possible to achieve higher etching rates and selectivity values enhanced by more than one order of magnitude, with a good control in the profile sharpness. Etching parameters dealing with ICP system are summarized in Table 2. It is evident that an independent control of the substrate bias voltage with respect to the r.f. power coupled to the plasma gives the possibility to obtain high aspect ratio structures since the low ion energy prevents substrate damage and mask erosion. SEM micrographs with the comparison between etching profiles achieved with RIE and ICP systems are shown in Fig. 2.At present, we are also experimenting a new recipe to improve the sharpness of the hole profiles, using a binary gas mixture in the ICP system: SF6 as etching gas, and C4F8 as passivation gas. The addition of C4F8 is aimed at realizing vertical sidewalls. In fact, in a plasma regime C4F8 creates a polymer on the etched surface, thus preventing vertical walls from undesired undercut.4. Conclusions and future prospectivesIn summary we developed a fabrication processes for the realization of amorphous silicon based two-dimensional photonic crystals. Such procedures involve electron beam lithography and RIE or ICP dry etching. With ICP technique it is possible to achieve high aspect ratio and high selectivity values of a-Si:H with respect to PMMA.The same studies are going to be performed on a-SiN:H with the aim to realize PC cap layers on luminescent silicon nitride thin films and arrays of cylindrical holes patterned on a-SiN:H based Distributed Bragg Reflectors and Fabry-Perot microcavities. Planar PCs-based waveguides will be also developed starting from SOI structures (a-Si:H on SiO2 thin films).AcknowledgmentsThis work has been supported by MURST through the PRIN 2002-2003 project “Plasmachemical technologies in the deposition of Silicon-based nanostructured films for photonic and photovoltaic applications“ and by INFM through the project PHOBOS “Photonic Band gap Materials for Si-based Optoelectronic Structures”References[1] J. D. Joannopoulos, R. D. Meade, J. N. Winn, "Photonic Crystals", Ed. Princeton University Press 1995.[2] M. Loncar, T. Doll, J. Vuckovic, A. Sherrer, J. Lighwave Techn., Vol. 18, No 10, October 2000.[3] A. A. Erchak, Daniel J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, L. A. Kolodziejski, Appl. Phys. Lett. 78 (2001) 563.[4] S. Fan, P. R. Villeneuve, J. D. Joannopoulos, Phys. Rev. Lett. 78 (1997) 3294.[5] V. Ballarini, G. Barucca, E. Bennici, C.F. Pirri, C. Ricciardi, E. Tresso, F. Giorgis, published in the same volume.[6] J.R. Sheats, B.W. Smith, “Microlithography – Science and Technology”, Ed. Marcel Dekker.[7] MIT Photonic-Bands (MPB) /mpb/Figure CaptionsFig.1SEM micrograph of a 2D-structure fabricated by electron beam lithography on a PMMA coating spinned onto an a-Si:H film. The structure consists of cylindrical holes with a radius, r=0.41a, where a=350nm is the inter-hole spacing.Fig.2Cross-sectional SEM viewgraphs of a-Si:H etched structures: comparison between etching profiles obtained by RIE (a) and ICP (b) systems.Table 1Etch rate and selectivity data – etching processes performed on a-Si:H films by a standard parallelplate reactive ion etching system (RIE):Reactive gas Pressure r.f. Power Etch rate Selectivity a-Si/PMMASF6 [30sccm] 300mTorr 70W 840nm/min 9:1CF4 [40sccm] 300mTorr 100W 80nm/min 1:1CF4 [40sccm]O2 [5sccm]300mTorr 100W 540nm/min 2:1Table 2Etch rate and selectivity data – etching processes performed on a-Si:H films by an inductivelycoupled plasma etching system (ICP):Reactive gas Pressure Power Etch rate Selectivity a-Si/PMMASF6 [70sccm] 25mTorr 100W coil10W platen1400nm/min 94:1PAPER REFERENCE: H-VII.4 Fig. 19PAPER REFERENCE: H-VII.4 Fig. 2a Fig. 2b0.5µm10。

声学双曲构型超材料的负折射特性研究

第53卷第2期2024年2月人㊀工㊀晶㊀体㊀学㊀报JOURNAL OF SYNTHETIC CRYSTALS Vol.53㊀No.2February,2024声学双曲构型超材料的负折射特性研究刘㊀松1,赵仁洁1,杜一帆1,吴㊀芳2,宋和滨3,高㊀鹏3(1.大连理工大学工业装备结构分析优化与CAE 软件全国重点实验室,大连㊀116024;2.大连船舶重工集团有限公司,大连㊀116011;3.中国船级社(CCS)大连分社,大连㊀116013)摘要:声学双曲超材料是具有双曲色散特性的人工材料,具有极强的各向异性,其负折射特性是研究实现高分辨率聚焦型超透镜的理论依据㊂针对远场噪声源识别受制于0.5倍波长声波瑞利衍射识别分辨率问题,结合声学超材料对声波的优异调控效果,引进可以实现亚波长超分辨率成像的双曲超材料,利用其负折射特性设计了一种用于工作频率为2271.5Hz 的声学双曲结构㊂分析了该构型的双曲结构色散特性及负折射特性,结果表明声波在该双曲超材料中传播的群速度方向垂直于波矢,并沿着色散曲线的法线方向㊂本文的研究为实现对声波和弹性波的任意调控,以及噪声源的聚焦定位㊁识别放大等提供了一定的设计参考㊂关键词:声学超材料;声学透镜;弹性波带隙特性;负折射;双曲色散;声聚焦中图分类号:O735;TL375.2㊀㊀文献标志码:A ㊀㊀文章编号:1000-985X (2024)02-0246-06Negative Refraction Characteristics of Acoustic Hyperbolic Configuration MetamaterialsLIU Song 1,ZHAO Renjie 1,DU Yifan 1,WU Fang 2,SONG Hebin 3,GAO Peng 3(1.State Key Laboratory of Structural Analysis,Optimization and CAE Software for Industrial Equipment,Dalian University of Technology,Dalian 116024,China;2.Dalian Shipbuilding Industry Company,Dalian 116011,China;3.China Classification Society (CCS)Dalian Branch,Dalian 116013,China)Abstract :Acoustic hyperbolic metamaterials are artificial materials with hyperbolic dispersion characteristics and strong anisotropy.Their negative refractive properties are the theoretical basis for studying the implementation of high-resolution focused superlenses.In response to the problem that the recognition of far-field noise sources is limited by the resolution of 0.5times wavelength acoustic Rayleigh diffraction recognition,combined with the excellent control effect of acoustic metamaterials on sound waves,a hyperbolic metamaterial that can achieve sub wavelength super-resolution imaging is introduced,and its negative refractive characteristics are used to design an acoustic hyperbolic structure for working at a frequency of 2271.5Hz.The dispersion and negative refraction characteristics of the hyperbolic structure of this configuration were analyzed,and the results show that,the group velocity direction of sound waves propagating in this hyperbolic metamaterial is perpendicular to the wave vector and follows the normal direction of the dispersion curve.The research in this paper provides some design references for realizing arbitrary regulation of sound wave and elastic wave,as well as focusing,locating,identifying and amplifying noise sources.Key words :acoustic metamaterial;acoustic lens;elastic wave bandgap characteristic;negative refraction;hyperbolic dispersion;acoustic focusing㊀㊀收稿日期:2023-08-21㊀㊀基金项目:国家自然科学基金(51609037)㊀㊀作者简介:刘㊀松(1982 ),男,吉林省人,博士,高级工程师㊂E-mail:liusong@0㊀引㊀㊀言负折射率材料是某一特定频段下折射率为负数的新型超材料,当入射波与折射波位于法线的同侧时被称作负折射㊂正常聚焦透镜只聚焦传输波的能量,但是负折射材料可以在聚焦传输波的基础上继续聚焦倏㊀第2期刘㊀松等:声学双曲构型超材料的负折射特性研究247㊀逝波的能量,可以突破衍射极限,形成完美透镜㊂该类材料最早在电磁波领域被提出,前苏联物理学家Veselago[1]通过大量的理论推导设想了一种介电常数和磁导率均为负数的材料,具有负的折射率,当电磁波通过具有该特性的材料后出现负折射效应和声聚焦特性㊂Pendry[2]根据负折射的理论制备出具有等效负介电常数的周期性特性的超材料㊂Smith等[3]与Shelby等[4]设计了一种棱镜,首次从实验角度证实了负折射现象的真实存在,并由此实验证明当光线入射到负折射率介质表面时,折射光线与入射光线分布在分界面一侧㊂声学超材料具有与电磁材料通过周期性结构来调控电磁波传播的相似性[5]㊂声学超材料的特殊属性,尤其是负折射特性研究可为声场聚焦和声源定位提供支撑[6]㊂声学双曲超材料是具有双曲色散特性的人工材料,具有极强的各向异性,通过改变双曲材料结构尺寸㊁分布规律能够完成对声波强度和传播方向的控制[7]㊂对于声场聚焦问题,需要根据声源特性或设定的带宽对超材料进行详细的拓扑优化设计㊂王涵[8]从声学超材料的波衰减特性和双负特性这两个重要性质入手,提出三种新型蜂窝声学超材料均具有双负特性,但并未证实其负折射线现象㊂宋刚永[9]设计了基于变换声学理论的浸没式声学放大透镜,通过实验验证该透镜可在5650~6350Hz实现声场聚焦㊂杨帅等[10]在空气中将工字钢排列为正方形实现了负折射率,但只有在特定的频率范围,如5000Hz左右,声波在Z型线性波导中才能够较好地传播㊂整体来讲,目前双曲超材料的带隙频段较高,随着声源特性频率的降低,需设计可用于中低频段的超材料㊂本文基于拓扑优化方法,设计了一种可用于中低频段声场调控的双曲构型,从平面波入射三角棱镜声场分布研究入手,通过数值模拟方法分析了该构型的双曲结构色散特性及负折射特性㊂1㊀声学双曲构型设计在能够保持晶格对称性的前提下,构成晶体的最小的周期性结构单元称为晶体的单胞㊂本文设计双曲构型单胞示意图如图1所示,晶格常数a=26mm,交错分布的结构臂长d为22.5mm,壁厚t为1mm,尺寸构型由两个对称分形组成,两部分间隔c为2mm㊂图1㊀双曲构型单胞示意图Fig.1㊀Schematic diagram of the acoustic hyperbolic configuration metamaterial 本文设计亮点在于声子晶体内部有交错分布的结构臂,当声波通过此结构后能够延长声波的传递路径,进而延长了声波总的传播时间,最终实现对声波的相位调控㊂2㊀能带图分析结合双曲构型单胞特点,采用正方形晶格计算其能带特性,正方形晶格不可约布里渊区示意如图2所示㊂数值模拟双曲构型能带结构时,边界条件选为Floquet周期,根据Bloch定理可知,将波矢k沿着倒格矢空间内不可约布里渊区边界进行扫掠,即可得到能带结构,同时得到各能带对应的结构振动模态㊂扫掠方向为M-Γ-X-M㊂材料参数为:泊松比σ=0.41㊁弹性模量E=2450MPa㊁波速c1=716m/s,密度ρ1=1300kg/m3㊂空气参数为:密度ρ2=1.21kg/m3,速度c2=343m/s㊂计算得到的能带结构如图3所示㊂图中横坐标为波矢k的扫掠方向,即形成一个完整的扫掠回路;纵坐标为扫掠所对应的频率,频率范围为0~6000Hz㊂通过对所设计构型的能带结构模拟研究,从能带结构图中可以发现,第二能带的带顶较平,并且关于Γ248㊀研究论文人工晶体学报㊀㊀㊀㊀㊀㊀第53卷点附近对称性较好,共振频率为2271.5Hz㊂这表明声学超材料拥有更多的分支㊁更长的传播路径,且在此频率附近将出现负折射现象㊂图2㊀正方形晶格不可约布里渊区示意图[11]Fig.2㊀Schematic diagram of irreducible Brillouin region of a square lattice[11]图3㊀设计的双曲超材料的能带结构图Fig.3㊀Energy band structure diagram of designed hyperbolic metamaterials 3㊀色散特性在二维空间内,声学双曲超材料的等频线分布情况可以用声波色散方程描述:k 2x ρx +k 2y ρy =ω2B (1)式中:k x 为x 方向的波矢分量,k y 为y 方向的波矢分量,ρx 为x 方向的等效密度分量,ρy 为y 方向的等效密度分量,ω为声波波数,B 为等效模量㊂本文设计的双曲构型在2271.5Hz 时的等频线为双曲分布,双曲色散曲线如图4(a)所示,入射波由自由空间入射到双曲媒质,其入射波的传播方向为k i ㊁折射方向为k r 及双曲媒质中的群速度方向为v g ,其中群速度与频率关系式由v g =Δk ω可知其群速度的方向垂直于波矢,即沿着色散曲线的法线方向㊂尽管折射波的相速度为正,但入射波与折射波的能流都在法线同侧,因而此时出现负折射㊂图4(b)为通过COMSOL Multiphysics 软件提取的双曲构型的二维等频色散分布图㊂图4㊀双曲构型色散曲线示意图Fig.4㊀Diagram of hyperbolic configuration dispersion curve 4㊀折射率计算双曲构型超材料折射率计算示意图如图5所示,计算区域由五大部分组成:完美匹配层(perfectly matched layer,PML)-背景压力场-双曲构型-周期性边界-完美匹配层㊂背景压力场区域提供幅值为1Pa 的㊀第2期刘㊀松等:声学双曲构型超材料的负折射特性研究249㊀平面波,模拟声波入射环境;图中四个红点表示提取双曲构型前后声压值的位置点,在声波入射方向布置两个传声器1和2,分别在距离单胞左边界30和0mm 处,在构型右侧同样布置2个传声器3和4,提取四个位置点的声压值便于后续计算㊂在COMSOL Multiphysics 压力声学频域中进行计算㊂进行网格划分时,完美匹配层划分5层网格,其余部分按照四边形网格划分,单元尺寸选为1mm㊂在折射率图中横坐标为扫频频率,范围为2000~3000Hz,纵坐标为各个频率计算得到对应的折射率㊂计算的双曲构型单胞尺寸选为26mm ˑ26mm,背景压力场尺寸选为26mm ˑ150mm,周期性边界尺寸选为26mm ˑ150mm,完美匹配层尺寸选为26mm ˑ130mm,图5中可看到计算域的划分以及各区域的名称㊂为保证超材料出现中低频带隙特性,将构型在2000~3000Hz 的频率范围内在空气场中作扫频分析,计算声场的尺寸为510mm ˑ26mm,随后提取构型左边界的入射声压及左边界的出射声压随之得到该双曲材料的透射系数与反射系数,得到透射系数和反射系数后可以得到折射率的表达式n =-i lg x +2πm ka (2)式中:k =ωC 0为声波波数,ω为圆频率,C 0为声波声速,ω=2πf ,f 为频率;m 为反余弦函数分支,仅能取整数,由于实际声传播方向不存在周期性结构,故m =0;i 为虚数㊂x =1-R 2P +T 2P +r 2T P (3)r =ʃ(R 2P -T 2P -1)-4T 2P(4)式中:T P 为透射系数,R P 为反射系数,n 为折射率,a 为晶格常数,取值为26mm㊂计算得本双曲构型折射率如图6所示,可以清晰看到本构型在2271.5Hz 时的折射率为负数㊂图5㊀COMSOL 计算折射率示意图Fig.5㊀COMSOL calculation of refractive index diagram 图6㊀双曲构型折射率计算示意图Fig.6㊀Schematic diagram for calculating refractive index of hyperbolic configuration 5㊀负折射特性数值仿真验证通过有限元分析方法对双曲构型的负折射特性进行仿真验证㊂将双曲构型排列为边长尺寸为461mm 的三角棱镜(见图7),置于自由场中,四周采用完美匹配层营造良好的吸声效果,避免回波干扰;在棱镜左侧设置介质为空气的背景压力场平面波幅值为1Pa,三角棱镜右侧部分设为空气域,如图8所示㊂为对比双曲构型负折射特性仿真的效果,在保证计算域条件相同的情况下,分别探讨了有无三角棱镜的声场传播特性㊂首先给出平面波在空气域中的传播特性,可以看到平面波在声场中均匀传播,如图9所示㊂然后在声场中添加三角棱镜,在棱镜左侧施加平面波完成激励,平面波的入射方向沿棱镜左侧向右(如图250㊀研究论文人工晶体学报㊀㊀㊀㊀㊀㊀第53卷10)㊂观察平面波穿过声学双曲介质排布而成三角棱镜后的声场分布可以清晰看出,当声波在经过声学双曲超材料三角棱镜的声波调控后,入射波与经过声学双曲超材料调控后的折射波位于法线同侧,即出现了负折射现象㊂为了使负折射效果更加明显,该部分放大了显示倍数,且在完美匹配层的声学边界中对声压进行计算,证明了声压呈现衰减状态,声波传播无反射㊂图7㊀双曲构型棱镜示意图Fig.7㊀Schematic diagram of a hyperbolic configurationprism图8㊀COMSOL负折射验证示意图Fig.8㊀Schematic diagram of negative refractionverification图9㊀平面波无棱镜声场分布示意图Fig.9㊀Schematic diagram of plane wave sound field withoutprism图10㊀平面波入射三角棱镜声场分布示意图Fig.10㊀Schematic diagram of plane wave incidentsound field of triangular prism 从平面波有无三角棱镜耦合的声场分布图对比来看,负折射率的双曲介质对平面波的传播起到了调控作用,当声波穿过透镜后声波的传播方向得到了改变,入射声波经声学双曲超材料调控后的折射波与入射波位于法线的同侧㊂由斯涅耳定律可知,棱镜的折射率为负数,该材料存在负折射现象,经过有限元仿真验证了所设计声学双曲超材料的负折射属性㊂这说明本文所设计的声学双曲超材料单元可以对相应声波进行调控,该双曲构型后续可以用于声学透镜的聚焦㊂6㊀结㊀㊀论本文提出一种声学双曲超材料构型,分析了该构型的负折射特性,基于有限元分析软件对该构型的折射率进行了计算,折射率为负数,该构型满足负折射的条件㊂从能带结构图中可以发现第二能带的带顶较平,并且关于Γ点附近对称性较好,共振频率为2271.5Hz㊂这表明本文设计的声学双曲超材料拥有较多分支,对声波的传播路径可以有效延长,且在此频率附近出现了负折射现象,可用于声波调控及声场聚焦㊂本文的研究为实现对声波和弹性波的任意调控,以及远场噪声源的聚焦定位㊁识别放大等方面提供了一定的设计参考㊂参考文献[1]㊀TANASHYAN M M,LAGODA O V,VESELAGO O V,et al.A pathogeneteic approach to the treatment of vestibular disorders in angioneurology[J].Zhurnal Nevrologii i Psikhiatrii Im S S Korsakova,2019,119(5):32.[2]㊀PENDRY J B.Transfer matrices and conductivity in two-and three-dimensional systems.I.Formalism[J].Journal of Physics:CondensedMatter,1990,2(14):3273-3286.㊀第2期刘㊀松等:声学双曲构型超材料的负折射特性研究251㊀[3]㊀SMITH D R,PADILLA W J,VIER D C,et posite medium with simultaneously negative permeability and permittivity[J].PhysicalReview Letters,2000,84(18):4184-4187.[4]㊀SHELBY R A,SMITH D R,SCHULTZ S.Experimental verification of a negative index of refraction[J].Science,2001,292(5514):77-79.[5]㊀李丽萍.分形声学超材料声学特性研究[D].长沙:湖南大学,2018.LI L P.Study on acoustic characteristics of fractal acoustic metamaterials[D].Changsha:Hunan University,2018(in Chinese).[6]㊀LI J,FOK L,YIN X B,et al.Experimental demonstration of an acoustic magnifying hyperlens[J].Nature Materials,2009,8(12):931-934.[7]㊀SHEN C,XIE Y B,SUI N,et al.Broadband acoustic hyperbolic metamaterial[J].Physical Review Letters,2015,115(25):254301.[8]㊀王㊀涵.蜂窝型声学超材料带隙特性与双负特性的数值模拟研究[D].秦皇岛:燕山大学,2022.WANG H.Numerical simulation study on bandgap and double negative characteristics of honeycomb acoustic metamaterials[D].Qinhuangdao: Yanshan University,2022(in Chinese).[9]㊀宋刚永.声学超材料对声波的调控理论与实验研究[D].南京:东南大学,2019.SONG G Y.Theoretical and experimental study on the regulation of acoustic metamaterials on sound waves[D].Nanjing:Southeast University, 2019(in Chinese).[10]㊀杨㊀帅,李昌清,赖虹君,等.流固混合声子晶体中负折射与导波特性研究[J].哈尔滨工程大学学报,2022,43(9):1370-1375.YANG S,LI C Q,LAI H J,et al.Study on negative refraction and guided wave characteristics in liquid-solid mixed phononic crystals[J].Journal of Harbin Engineering University,2022,43(9):1370-1375(in Chinese).[11]㊀LIU J,LI L P,XIA B Z,et al.Fractal labyrinthine acoustic metamaterial in planar lattices[J].International Journal of Solids and Structures,2018,132/133:20-30.。

String Theory in Two Dimensions

These notes are an expanded version of the lectures I gave at the 1991 ICTP Spring School on String Theory and Quantum Gravity. Here I have attempted to review, from my own personal viewpoint, some of the exciting developments in twodimensional string theory that have taken place over the last year and a half. Because of the multitude of new results, and since the ﬁeld is still developing rapidly, a comprehensive review must await a later date. These notes are mainly devoted to the matrix model approach1 that has truly revolutionized the two-dimensional Euclidean quantum gravity. Recently this approach has led to the exact solution of quantum gravity coupled to conformal matter systems with c ≤ 1.2−6 These lectures are about the c = 1 model5,6 that is both the richest physically and among the most easily soluble. This model is deﬁned by the sum over geometries

原子核的三轴超形变

原子核的三轴超形变
邢正;陈星蕖
【期刊名称】《原子核物理评论》
【年(卷),期】2001(18)4
【摘要】利用粒子 -转子模型研究三轴超形变核态 ,讨论了区别轴对称超形变和三轴超形变可能的实验信息 .为了直接从实验上识别三轴超形变带 ,必须同时测量能谱和电磁跃迁几率 .
【总页数】3页(P282-284)
【关键词】三轴超形变;原子核结构;粒子-转子模型;三轴超形变带;能谱;电磁跃迁几率
【作者】邢正;陈星蕖
【作者单位】兰州大学现代物理系;中国科学院上海原子核研究所
【正文语种】中文
【中图分类】O571.21
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2.157Tm原子核三轴超形变的研究 [J], 图雅;拉布敦;于少英
3.三轴超形变原子核的预言 [J], 沈彩万;陈永寿
4.175Hf原子核的三轴超形变的探寻 [J], 李晓伟;于少英;沈彩万;陈永寿
5.轴对称超形变和三轴超形变核态性质的研究 [J], 邢正;徐进章;陈星蕖;王子兴
因版权原因，仅展示原文概要，查看原文内容请购买。

一份关于二维电子气(2DEG)的讲义

Ψ(r, z) = χ(z) ψ(r)
where r is the vector in plane of 2DEG. Throughout our considerations we will assume that all the distances are much larger than interatomic distance and thus we will use the eﬀective
Density of States
The density of states g( ) is deﬁned as number of states per the energy interval , + d . It is clear that
g( ) = δ( − α)
α
where α is the set of quantum numbers characterizing the states. In the present case it includes the subband quantum number n, spin quantum number σ, valley quantum number
v (for n-type materials), and in-plane quasimomentum k. If the spectrum is degenerate
with respect to spin and valleys one can deﬁne the spin degeneracy νs and valley degeneracy
n-AlGaAs i-GaAs
EC
EF EC
EF
EV
EV
EF

固体物理 8_reciprocal-lattice-notes

j j bj :
the reciprocal lattice vectors.
b 1 · (b 2 × b 3 ) = (2π )3 /a1 · (a2 × a3 ): The volume of a primitive cell of the reciprocal lattice is (2π )3 divided by the volume of a primitive cell of the direct lattice.
Chapter 8
Reciprocal Lattice
1
Reciprocal Lattice
Derivation of the Reciprocal Lattice The condition for nonvanishing X-ray scattering q · R n = 2mπ, m = 0, ±1, ±2 · · · . q = k − k : The wave vector transfer. Using R n = q·
Fuxiang Han Chapter 8 Reciprocal Lattice 6
Reciprocal Lattice
Derivation of the Reciprocal Lattice, cont’d Rederive the reciprocal lattice through Fourier transforming the Bravais lattice in real space (the direct lattice) into k -space (the reciprocal space). (q ) = V (2π )3 N V = (2π )3 N V = (2π )3 N d r ( r ) e − i q ·r d r δ (r − R n )e−i q ·r

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arXiv:cond-mat/0703391v2 [cond-mat.stat-mech] 19 Jul 2007Typesetwithjpsj2.clsLetterEﬀectiveFieldTheoryofTriangular-LatticeThree-SpinInteractionModelHiromiOtsuka∗DepartmentofPhysics,TokyoMetropolitanUniversity,Tokyo192-0397,Japan(ReceivedFebruary6,2008)Wediscussaneﬀectiveﬁeldtheoryofatriangular-latticethree-spininteractionmodeldeﬁnedbytheZpvariables.Basedonthesymmetrypropertiesandtheideal-stategraphconcept,weshowthatthevectordualsine-Gordonmodeldescribesthelong-distancepropertiesforp≥5;wethencompareitspredictionswiththepreviousargument.Toprovidetheevidences,wenumericallyanalyzetheeigenvaluestructureofthetransfermatrixforp=6,andwecheckthecriticalitywiththecentralchargec=2oftheintermediatephaseandthequantizationconditionofthevectorcharges.

KEYWORDS:triangular-latticethree-spininteractionmodel,eﬀectiveﬁeldtheory,ideal-stategraph,vectordualsine-Gordonmodel,two-dimensionalmelting

TheXYmodelconsistingoftheinnerproductsoftwoneighboringplanarspinswiththeZpsymmetry-breakingﬁeldhasbeenextensivelyresearched;itprovidestheba-sicunderstandingofmorecomplicatedmodelsandalsoofrealmaterials.Especially,forthetwo-dimensional(2D)case,theeﬀectivetheoryforthelong-distancebehav-iorswasestablishedbasedonthe2DCoulombgas(CG)pictureandtherenormalization-group(RG)concepts,therebyenablingitscompleteunderstanding.1Inthispa-per,weinvestigateitsextension,i.e.,thethree-spinin-teractionmodel(TSIM)thatwasintroducedalongtimeagobyAlcarazetal.2,3Supposethat󰀈k,l,m󰀉denotesthreesitesatthecornersofeachelementaryplaquetteofthetriangularlatticeΛ(consistingofinterpenetratingsublatticesΛa,Λb,andΛc),thenthefollowingreducedHamiltonianexpressesaclassofTSIM:

H=−βJ󰀁󰀃k,l,m󰀄cos(ϕk+ϕl+ϕm).(1)

Angleϕk=2πnk/p,nk∈[0,p−1]deﬁnestheZpvari-able.Forp=2,eq.(1)istheexactlysolvedBaxter-Wumodelwiththree-spin-productinteractions,4butforlargerp,itbecomespuzzlinginitsexpressionintermsofthespinvariables.However,forp≥5onwhichwewillconcentrate,itseﬀectivetheorypossessesaremarkablysimplestructureandcanprovideauniﬁedunderstandinginawideareaofresearches,includingthoseonthe2Dmeltingphenomena.Therefore,ouraimistoformulateitbasedonarecentapproachandtoconﬁrmitspredictionsquantitativelyvianumericalcalculations.Weshallbeginwiththesymmetryproperties.Addingtothetranslationsandspaceinversions,themodelisinvariantundertheglobalspinrotations:ϕk→ϕk+󰀃ρ=a,b,c󰀃

l∈Λρ

(2πnρ/p)δk,l,ifthesublatticedependent

numberssatisfyaconditionna+nb+nc=0(modp).2,3Thissymmetryoperation—denotedas(na,nb,nc)—canbegeneratedfromtwoofthethreefundamentalopera-tionswiththefollowingminimalspinrotations:

ˆRa:(1,p−1,0),ˆRb:(0,1,p−1),ˆRc:(p−1,0,1).(2)2J.Phys.Soc.Jpn.LetterAuthorName000150240330420510000105255345435525015105204354444534024114204303453543033123213303402552042132222312402501051141231321411501000150240330420510000

02π+0.8−0.2−1.2−2.2

RbRc

Fig.1.Theideal-stategraphIforp=6.Tripletsofnumbersex-hibit36orderedstates.Thecontourplotofthelockingpotentialandthetransformationeq.(3)(inset)arealsogiven.

e1M

6e1: Repeat Lattice R: Reciprocal Lattice R*

Fig.2.TherepeatlatticeRanditsreciprocallatticeR∗.Thefundermentallatticevectorsarealsogivenbyarrows.

betweenthediscretesymmetryoperationsforspinsandtheﬁeldtransformations,Φ→Φ+δΦ;especially,thecorrespondenceoftheminimalspinrotationseq.(2)tothefollowingδΦ(seetheinsetofFig.1):

ˆRa:2πpe1+2πpe2,(3)

whichretainrelations,e.g.,ˆRcˆRbˆRa=ˆ1.OurnexttaskistoderivetheLagrangiandensitybywhichthelow-energyandlong-distancebehaviorscanbecaptured.Forthispurpose,weshallﬁrstfocusonthelow-temperatureregion,atwhichtherelevantpotentialmayperturbthekineticenergypartrepresentingthespatialﬂuctuationofΦ(x).Asweshallsee,thelatterisalsoresponsibleforthedescriptionoftheintermediatecriticalphase.2,3DuetotheperiodicityofΦ,thevectorchargeMisquantizedtotakethevaluesonthereciprocallatticeofR,R∗(Fig.2).Then,thelocaldensitiesrelatingtothespindegreesoffreedommaybeexpandedtotheFourierseriesbyusingthevertexoperatorsasQ(Φ)=󰀃

M∈R∗hMeiM·Φ.Whiletheinner-productformensures

acoordinateinvarianceinR2,welookintoitsexpressionbyusingtheirelementstogetsomefamiliarity.Writ-

ingthereciprocallatticevectorsas{eα}(α=1,2),theinnerproductbetweenM=mαeαandΦ=√

2mαφα,wherethecovariantandcontravari-antelementssatisfymα∈Zand√2φα+2π,respectively(thesummationconventionisusedandthefactor√

mαmα,wherethemetrictensorisdeﬁnedbygαβ≡eα·eβandthecontravariantelementsbymα≡gαβmβ,asusual.7Toobtainanexplicitformofthelockingpotential,thefollowingthreeissuesneedtobeaddressed:5,6(i)SincethepotentialisapartoftheLagrangiandensity,itshouldbeinvariantundereq.(3).(ii)ItissuﬃcienttokeepthemostrelevanttermsintheRGsense.Sincethedimensionofavertexoperatorisproportionaltothesquarednormofitsvectorcharge(seebelow),itissuf-ﬁcienttokeepthosewiththeshortestones.(iii)ThepointgroupsymmetryofthetriangularlatticeforI,whichstemsfromthesublatticeandthespin-rotationalsymmetries,shouldbetakenintoaccount.Consequently,theserequirementscanberealizedastheexpressionV(Φ)=hp󰀃󰀂M󰀂=pa∗eiM·Φ,wherea∗≡󰀂