Collective density wave excitations in two-leg Sr_{14-x}Ca_xCu_{24}O_{41} ladders

电荷密度波（CDW）介绍电荷密度波（Charge Density Wave，CDW）是一种凝聚态物理现象，指的是电子在晶体中形成的周期性密度调制。

CDW广泛存在于各种材料中，包括低维材料、高温超导体和有机分子晶体等。

研究CDW可以揭示物质的电子结构、输运性质和相互作用效应，对于理解材料的性质和开发新型功能材料具有重要意义。

形成机制CDW的形成机制涉及电子-电子相互作用、电子-晶格相互作用和电子自旋相互作用等多种因素。

一般来说，CDW的形成可以分为两个步骤：零温下的费米面形变和有限温度下的相变。

在费米面形变阶段，电子-电子相互作用导致费米面的形状发生变化，出现密度波的倾向。

而在相变阶段，费米面的形变进一步发展，形成周期性的电荷密度波。

实验观测实验上，CDW可以通过多种技术手段进行观测和研究。

其中，X射线衍射是最常用的方法之一。

通过测量晶体在CDW相和非CDW相的X射线衍射图样的差异，可以确定CDW的存在和性质。

此外，还可以利用电子能谱、电导率和磁化率等物理性质的测量，来研究CDW的形成和演化过程。

CDW的性质和应用CDW对材料的性质和行为有着重要影响。

首先，CDW可以改变材料的电导率，使其呈现出特殊的电子输运性质。

其次，CDW还可以影响材料的磁性和超导性等性质。

因此，研究CDW不仅可以深化对材料的理解，还可以为新型电子器件和能源材料的设计与开发提供指导。

CDW在低维材料中的研究低维材料是研究CDW的重要平台之一。

由于低维材料的特殊结构和性质，CDW在其中的形成和演化过程可能呈现出与三维材料不同的行为。

近年来，石墨烯、二维过渡金属二卤化物和有机分子单层等低维材料中的CDW现象引起了广泛关注。

这些研究不仅有助于理解低维材料的基本物理机制，还为低维电子器件的开发提供了新的思路和方法。

CDW的理论模型和计算方法理论模型和计算方法在研究CDW中起着重要作用。

常用的理论模型包括强耦合模型、弱耦合模型和自旋密度波模型等。

abandon?抛弃abaxial?离轴的ablation?烧蚀abnormal liquid?异常液体abrasion?磨耗abrasion hardness?耐磨硬度abrasion test?磨损试验abrasion tester?磨损试验机abrasives?研磨材料abrupt perturbation?突然扰动abscissa?横坐标absence of collision?无碰撞absence of gravity?失重absence of stress?无应力absence of vortices?无旋涡absolute?绝对的absolute acceleration?绝对加速度absolute angular momentum?绝对角动量absolute ceiling?绝对升限absolute coordinates?绝对坐标absolute deviation?绝对偏差absolute elastic body?完全弹性体absolute equilibrium?绝对平衡absolute extremum?绝对极值absolute flying height?绝对飞行高度absolute frequency?绝对频率absolute geopotential?绝对位势absolute humidity?绝对湿度absolute instability?绝对不稳定性absolute motion?绝对运动absolute path?绝对路径absolute perturbation?绝对扰动absolute pressure?绝对压力absolute rest?绝对静止absolute rotation?绝对旋转absolute similarity?完全相似性absolute stability margin?绝对稳定性限度absolute temperature?绝对温度absolute thermometer?绝对温度表absolute velocity?绝对速度absolute viscosity?绝对粘度absolute vorticity?绝对涡度absolute weight?绝对重量absolute zero?绝对零度absolute zero point?绝对零度absorbability?吸收性absorbing agent?吸收剂absorbing medium?吸收媒质absorbing resistance?吸收阻抗absorptiometer?吸收率计absorption?吸收;缓冲absorption curve?吸收曲线absorption dynamometer?吸收功率计absorption heat?吸收热absorption loss?吸收损失absorption model?吸收模型absorption potential?吸收势absorption tube?吸收管absorption wave meter?吸收式波长计absorptive?吸收的absorptive power?吸收本领absorptivity?吸收率accelerated creep?加速蠕变accelerated motion?加速运动accelerating ability?加速能力accelerating field?加速场accelerating gradient?加速梯度accelerating impact?加速撞击accelerating period?加速周期accelerating tube?加速管accelerating unit?加速装置acceleration?加速度acceleration diagram?加速度图acceleration energy?加速度能量acceleration field?加速度场acceleration meter?加速度计acceleration nozzle?加速喷嘴acceleration of creep?蠕变加速度acceleration of gravity?重力加速度acceleration of moving space?牵连加速度acceleration potential?加速势acceleration pressure?加速压acceleration time graph?加速度时间图acceleration vector?加速度矢量acceleration wave?加速波acceleration work?加速功accelerationless?无加速的accelerograph?加速度记录仪accelerometer?加速度计accessibility?可达性accessible point?可达点accidental?偶然的accidental coincidence?偶然符合accidental error?随机误差accidental resonance?偶然共振accommodation?适应accommodation coefficient?适应系数accumulated error?累积误差accumulator?蓄压器储压器accuracy?准确度accuracy of measurement?测量准确度accuracy of readings?读数准确度accurate adjustment?精确蝶accurate measurement?精确测量accurate model?准确模型accurate similarity?准确相似accurate simulation?准确模拟acicular structure?针状组织acoustic?声的acoustic absorption?声吸收acoustic absorptionsound absorption?声吸收acoustic absorptivity?声吸收率acoustic admittance?声导纳acoustic analysis?声学分析acoustic baffle?隔音板acoustic conductivity?声导率acoustic dispersion?声弥散acoustic disturbance?声音干扰acoustic efficiency?声效率acoustic energy?声能acoustic field?声场acoustic filter?滤声器acoustic frequency?音频acoustic impedance?声阻抗acoustic load?声负载acoustic material?隔声材料acoustic mechanical efficiency?声机械效率acoustic power?声功率acoustic pressure?声压acoustic radiation pressure?声辐射压acoustic reactance?声抗acoustic resistance?声阻acoustic resonance?共鸣acoustic velocity?声速acoustical?声的acoustical wave?声波acting force?织力action at distance?超距酌action center?活动中心action integral?酌积分action of deflagration?爆燃酌action of force?力的酌action principle?酌原理action sphere?酌区action turbine?冲唤涡轮action variable?酌变量action wave?酌波activation?活化active earth pressure?织土压active flight?织飞行active force?织力active gas?活性气体active load reaction?有功负载反酌active pressure?织压力activity of force?力功率actual load?实际载荷actual stress?实际应力actual value?实际值actuator?执行器acyclic?非循环的adaptability?适应性adapter?转接器adaptometer?适应计addition?附加additional drag?附加阻力additional load?附加载荷additional mass?附加质量additional pressure?附加压力adhere?粘着adherence?附着adhesion?附着adhesion coefficient?粘着系数adhesive force?粘附力adhesive tension?粘附张力adiabat?绝热线adiabatic?绝热的adiabatic approximation?绝热近似adiabatic atmosphere?绝热大气adiabatic calorimeter?绝热式量热器adiabatic change?绝热变化adiabatic change of state?态的绝热变化adiabatic compression?绝热压缩adiabatic cooling?绝热冷却adiabatic curve?绝热线adiabatic efficiency?绝热效率adiabatic ellipse?绝热椭圆adiabatic equilibrium?绝热平衡adiabatic expansion?绝热膨胀adiabatic factor?绝热因子adiabatic heating?绝热加热adiabatic hypothesis?绝热假设adiabatic invariant?绝热不变量adiabatic invariant of eddy?涡旋的绝热不变量adiabatic invariant of vortex?涡旋的绝热不变量adiabatic perturbation?绝热扰动adiabatic principle?绝热原理adiabatic process?绝热过程adiabatic pulsations?绝热脉动adjoint operator?伴算子adjustable guide vane?可导叶adjusting device?蝶装置adjusting spring?蝶弹簧adjusting valve?蝶阀adjusting wedge?蝶楔块adjustment?蝶adjustment mark?照准标志admissible concentration?容许浓度admissible crror?容许误差admissible load?容许荷载admissible pressure?容许压力admissible stress?容许应力admissible value?容许值admission?进气admission pipe?进气管admission pressure?进气压力admixture?混合物adsorb?吸附adsorbent?吸附剂adsorption?吸附adsorption force?吸附力adsorption layer?吸附层advancing rate?超前速率advancing wave?前进波advection?平流advection fog?平另advective?平聊aeration?充气aerator?通风机aerial?空气的aerial ropeway?架空死aeroballistics?航空弹道学aerodonetics?滑翔学aerodromics?滑翔力学aerodromometry?空气速度测量法aerodynamic action?空气动力酌aerodynamic analogy?空气动力比拟aerodynamic angle of attack?气动迎角aerodynamic angle of incidence?气动迎角aerodynamic balance?风洞天平aerodynamic center?气动中心aerodynamic coefficient?气动力系数aerodynamic derivative?空气动力导数aerodynamic drag?气动阻力aerodynamic force?气动力aerodynamic heating?气动力加热aerodynamic lift?气动升力aerodynamic resistance?气动阻力aerodynamic stability?空气动力稳定性aerodynamical?空气动力学的aerodynamical moment?气动力矩aerodynamics?空气动力学aerodyne?重航空器aeroelasticity?气动弹性aeroelastics?气动弹性力学aeroengine?飞机发动机aerohydrodynamics?空气铃动力学aerology?高空气象学aeromechanics?航空力学aerometer?气体比重计aeromotor?飞机发动机aeronautical?航空的aeronautics?航空学aeroplane?飞机aeropulse engine?脉动式喷气发动机aerosol?气溶胶aerospace engineering?航空航天技术aerostat?轻航空器aerostatics?气体静力学aerothermodynamics?气动热力学aerothermoelasticity?空气热弹性学;气动热弹性aerotonometry?气体张力测量法affine transformation?仿射变换affinity?亲合力after effect?后效after effect function?后效函数afterbody?后部afterburner?加力燃烧室aftershock wave?余震波age hardening?时效硬化ageing?应变时效ageing test?老化试验agglomeration?凝聚aileron?副翼air amount?空气量air brake?空气制动器air bubble?气泡air buffer?空气缓冲器air burst?空中爆炸air cap?空气冠air circulation?空气环流air column?空气柱air compressor?空气压缩机air condenser?空气电容器air conveying?气龄送air coolant?空气冷却剂air cooling?空气冷却air coordinates?空间坐标air current?气流air cushion?气垫air cylinder?压气缸air damper?空气阻尼器air damping?空气阻尼air density?空气密度air drag?空气阻力air duct?风道air eddy?空气涡旋air flow?气流空气量air flow meter?气疗air friction?空气摩擦air gap?空隙air gas?风煤气air hammer?空气锤air hardening?空气淬火air hoist?风动起重滑车air humidity?空气湿度air inflating?充气air injection diesel engine?空气喷射柴油发动机air inrush?空气侵入air intake valve?进气阀air jet?空气喷射air pipe?气管air plasma?空气等离子体air pressure gage?空气压力计air pressure head?空气压头air pressure reducer?空气减压器air proof?气密的air pump?空气泵air register?空气活门air relief cock?放气塞air reservoir?储气筒air resistance?空气阻力air shock wave?空气激波air speed?空气速率air speed indicator?空气速率指示器air streamline?空气吝air tank?空气箱air temperature?气温air tight?气密的air to air missile?空对空导弹air trajectory?空气轨迹air wave?空气波air wedge?空气楔aircraft dynamics?飞机动力学airfoil?机翼airfoil profile?翼型airfoil section?翼剖面airfoil theory?机翼理论airless injection?无气喷射airline?飞行航线airlock?气锁airmechanics?空气力学airplane?飞机airscrew?空气螺旋桨airship?飞艇airway?航路airy stress function?爱里应力函数algebraic stress model?代数应力模型aligned composite material?定向复合材料aligned structure?定向结构allocation?分配allotropic change?同素异形变化allotropic transformation?同素异形变化allowable clearance?容许间隙allowable deflection?容许挠度allowable deviation?容许偏差allowable error?容许误差allowable load?容许荷载allowable pressure?容许压力allowable stress?容许应力allowable temperature?容许温度allowed cross section?容许截面allowed energy band?容许能带allowedness?容许度alternate immersion test?反复浸没试验alternating bending?交变弯曲alternating bending stress?交变弯曲应力alternating bending test?交变弯曲试验alternating direction?交替方向alternating hysteresis?交变滞后alternating impact bending test?交变冲讳曲试验alternating impact test?交变冲辉验alternating load?交替载荷alternating load deformation?交变负载变形alternating pressure?交变压力alternating stress?交替应力alternation?交变alternation of load?负载交变altimeter?高度表altitude?高度altitude circle?等高圈altitude difference?高度差altitude front?高空锋面ambient air?周围空气ambient medium?周围介质ambient pressure?周围压力ambient temperature?周围温度ambipolar diffusion?两极性扩散amendment?修正amoeboid movement?阿米巴式运动amorphous state?无定形状态amount of energy?能量amount of evaporation?蒸发量amplification?放大amplification limit frequency?放大极限频率amplifier?放大器amplitude?振幅amplitude clipping?脉冲幅度限幅amplitude cosine?幅角余弦amplitude discriminator?振幅甄别器amplitude distortion?波幅畸变amplitude frequency diagram?幅频图amplitude function?振幅函数amplitude locus?幅值轨迹amplitude modulation?爹amplitude resonance?振幅共振amplitude response?振幅响应amplitude selector?振幅选择器amplitude spectrum?振幅谱amplitude swing?幅度变动anabatic front?上升锋面anabatic wind?上坡风anafront?上滑锋anallobar?气压上升区analog?模拟analog digital computer?模拟数字计算机analog method?模拟法analogue?模拟analogue computer?模拟计算机analogue method?相似法analogy?模拟analogy of magnetic field?磁场相似analysis?分析学analysis of the oscillation?振荡分析analytic?解析的analytic transformation?解析变换analytical mechanics?分析力学anchoring?系泊anelastic material?滞弹性材料anemogram?风力自记曲线anemograph?风速计anemometer?风速表anemoscope?风向仪angle modulation?相角灯angle of advance?超前角angle of arrival?到达角angle of attack?迎角angle of bank?倾斜角angle of capillarity?毛细角angle of contact?接触角angle of contingence?切线角angle of crossing?交叉角angle of declination?偏斜角angle of departure?起飞角angle of distortion?畸变角angle of downwash?下洗角angle of elevation?仰角angle of emergence?出角angle of flap deflection?襟翼偏转角angle of flow?燎angle of friction?摩擦角angle of impact?碰撞角angle of inclination?倾角angle of internal friction?内摩擦角angle of repose?静止角angle of rotation?转动角angle of sideslip?侧滑角angle of slope?倾斜角angle of twist?扭转角angle of yaw?偏航角angle pipe?弯管angle preserving mapping?保角映射angle straggling?角离散angle variable?角变量angular acceleration?角加速度angular coefficient?角系数angular coordinates?角坐标angular correlation?角相关angular derivation loss?角偏向损失angular derivative?角微商angular dispersion?角分散angular displacement?角位移angular distance?角距angular distribution?角分布angular division?角分度angular frequency?角频率angular impulse?角冲量angular momentum?角动量angular momentum conservation law?角动量守恒定律angular momentum operator?角动量算符angular momentum tensor?角动量张量angular momentum theorem?角动量定理angular motion?角运动angular separation?方向夹角angular speed?角速度angular unit?角单位angular variable?角变量angular velocity?角速度anharmonic?非低的anharmonic oscillation?非谐振动anharmonic oscillator?非谐振子anharmonic ratio?非低比anisentropic flow?非等熵怜anisothermal porous flow?非等温渗透流anisotropic?蛤异性的anisotropic body?蛤异性体anisotropic liquid?蛤异性液体anisotropic material?蛤异性材料anisotropic rock?蛤异性岩石anisotropic turbulence?蛤异性湍流anisotropy?蛤异性anisotropy constant?蛤异性常数anisotropy ratio?蛤异性比annular flow?环流annular focus?环形焦点annular mist flow?环形雾状流anomalous absorption?异常吸收anomalous diffusion?反常扩散anomalous scattering?反常散射anti diffusion?反扩散anticlockwise rotation?反时针方向旋转anticoincidence?反重合anticoincidence method?反符合法anticyclolysis?反气旋消散anticyclonic vorticity?反气旋涡度antinode?波腹antipodal space?对映空间antiresonance?反共振antiresonance frequency?反共振频率antisymmetric tensor?反对称张量antisymmetrical state?反对称态antisymmetry?反对称antitriptic wind?减速风aperiodic?非周期的aperiodic damping?非周期衰减aperiodic motion?非周期运动aperiodicity?非周期性aperture?孔径aperture ratio?孔径比aphelion?远日点apocenter?远心点apogee?远地点apolar?无极的apolune?远月点apparent absorption coefficient?表观吸收系数apparent cohesion?表观粘力apparent diffusivity?表观扩散率apparent elastic limit?表观弹性极限apparent energy?表观能量apparent equilibrium?表观平衡apparent expansion?表观膨胀apparent force?表观力apparent modulus of elasticity?表观弹性模量apparent phase angle?表观相角apparent velocity?表观速度apparent viscosity?表观粘度apparent work?表观功appearance of fatigue?疲劳现象appell equation?阿佩尔方程application point?酌点applied elasticity theory?应用弹性学applied mechanics?应用力学approach velocity?驶近速度approximate analysis?近似解析approximate model?近似模型approximate similarity?近似相似approximate simulation?近似模拟approximate solution?近似解approximation?近似approximation calculus?近似计算approximation of lagrange function?拉格朗日函数近似approximation theory?近似理论approximative value?近似值arbitrary unit?任意单位arch?拱arch bridge?拱桥arch dam?拱坝arch gravity dam?拱形重力坝arch structure?拱形结构arch truss?拱形桁架arched beam?拱形梁arched girder?拱形梁archimedes principle?阿基米德原理area curve?面积曲线area load?表面负载area moment?面积矩area of contact?接触面积area ratio?面积比areal acceleration?面加速度areal coordinates?面积坐标areal velocity?面积速度areometer?比重计arm?边arm of couple?力偶臂arrest?制动器arrhenius equation?阿雷尼厄斯方程arrhenius law?阿雷尼厄斯定律artesian head?自廉头artesian pressure?自廉压artesian water?自廉artesian well?自廉artificial compression?人工压缩artificial earth's satellite?人造地球卫星artificial earthquake?人工地震artificial gravity?人造重力artificial perturbation?人工扰动artificial satellite?人造卫星artificial ventilation?人工通风ascending air current?向上气流ascending motion?上升运动ascending stroke?上行冲程ascension?升起aseismic?耐震的ash ejector?吹灰器排灰器aspect ratio?展弦比assembly?装配associate cumulation?结合累积associated mass?附加质量associated wave?缔合波astromechanics?天体力学astronavigation?天文导航astronomical azimuth?天文方位角astronomical unit?天文单位asymmetric?非对称的asymmetrical distribution?非对称性分布asymmetrical load?非对称负载asymmetry?非对称性asymptotic model?渐近模型athodyd?冲压喷气发动机atmosphere?大气atmospheric boundary layer?大气边界层atmospheric flight?大气飞行atmospheric humidity?大气湿度atmospheric layer?大气层atmospheric moisture?大气湿度atmospheric perturbation?大气扰动atmospheric pressure?大气压atmospheric pressure chart?气压图atmospheric subsidence?大气下沉atmospheric temperature?大气温度atmospheric turbulence?大气湍流atmospheric wave?大气波atom collision factor?原子碰撞因子atomic blast?原子爆炸atomic crack?原子龟裂atomic distance?原子间距atomic energy?原子能atomic energy level?原子能级atomic hardening?原子硬化atomization?喷雾化atomization pressure?雾化压力attachment?附着attenuation?衰减attenuation vibration?衰减振动attenuator?衰减器attitude?姿态attraction?吸引attraction potential?引力势attractive center?引力中心attractive force?引力audibility?可听性augmentation?增加augmenter?加速室auto spectrum?自乘谱autocontrol?自动控制autocorrelation?自相关autoexcitation?自激autogeneous ignition?自点火autogiro?自动陀螺仪automatic control system?自动控制系统automatic control theory?自动控制理论automatic design?计算机辅助设计automatic frequency control?自动频率控制automatic programming?自动编程序automatic regulator?自动第器autonomous system?自治系统autonomous vibration?自激振荡autopilot?自动导航装置autorotation?自转available energy?有效能available head?有效水头available heat?有效热average?平均;平均的average acceleration?平均加速度average curvature?平均曲率average deviation?平均加速度average diameter?平均直径average error?平均误差average gradient?平均坡度average life?平均寿命average load?平均负载average potential?平均势average power?平均功率average pressure?平均压力average quantity?平均量average speed?平均速度average temperature?平均温度average value?平均值average velocity?平均速度averaged orbital element?平均轨迹元averaging?取平均值averaging method?平均方法avertence?偏斜axes of coordinates?座标轴axial application of force?轴向施加力axial compression?轴向压缩axial elongation?轴向伸长axial flow?轴流轴向气流axial flow pump?轴撩axial force?轴向力axial impact?轴向碰撞axial load?轴向荷载axial magnification?轴向放大率axial mode?轴向模axial moment?轴向力矩axial moment of inertia?轴惯性矩axial pitch?轴向节距axial pressure?轴向压力axial pressure load?轴向压力负载axial stress?轴向应力axial symmetry?轴对称axial tension?轴向拉力axial thrust?轴向推力axial turbine?轴两涡轮axial vector?轴矢量axially symmetric flow?轴对称流轴对称怜axiom of constraint?约束公理axiomatics?公理系统axipetal?向心的axis of abscissas?横坐标轴axis of center of bending?弯曲中心轴axis of curvature?曲率轴axis of ordinates?纵坐标轴axis of revolution?回转轴axis of rotation?转动轴axis of symmetry?对称轴axis of traction?牵引轴axisymmetric?轴对称的axisymmetric element?轴对称单元axisymmetric flow?轴对称流axisymmetric problem?轴对称问题axle load?轴负载azimuth angle?方位角back flow?回流逆流back mixing?反混back pressure?背压力back reaction?逆反应back substitution?回代backpressure?反向压力;反压力backscattering technique?逆散射法backstreaming?逆流backward difference?后向差分backward motion?逆向运动backward precession?逆旋进backward thrust?逆推力backward wave?逆向行波backwash?后涡流backwater?回水backwater curve?回水曲线backwater distance?回水距离backwater function?回水函数backwater surface?回水面backwater surge?回水浪baer law?巴尔定律baeyer strain theory?贝耶尔应变理论balance?平衡balance condition?平衡条件balance equation?平衡方程balance force?平衡力balance weight?平衡锤balanced load?平衡负载balanced system of force?平衡力系balancing?平衡balancing force?补偿力balancing in situ?本机平衡balancing machine?平衡试验机balancing method?平衡法balancing speed?平衡速率ball support?球面支座ballast?压载物ballistic?弹道的ballistic coefficient?弹道系数ballistic constant?弹道常数ballistic curve?弹道曲线ballistic deflection?弹道偏差ballistic error?冲惑差ballistic pendulum?冲悔ballistic rocket?弹道火箭ballistic trajectory?弹道轨迹ballistic wave?弹道波ballistics?弹道学balloon?气球balloon borne rocket?气球发射火箭banded structure?带状结构bandwidth?带宽bank?侧向倾斜bar?巴bar construction?棒构造baric topography?气压形势baric wave?气压波barocline?斜压baroclinic fluid?斜压铃baroclinity?斜压性barodynamics?重结构力学barograph?气压计barometer?气压计barometric altitude?气压高度barometric column?气压柱barometric gradient?气压梯度barometric height?气压高度barosphere?气压层barotropic equation?正压方程barotropic equilibrium?正压平衡barotropic flow?正压流barotropic fluid?正压铃barotropic instability?正压不稳定barotropy?正压性barrel vault?圆柱壳barrier?障碍barycenter?重心barycentric?重心的barycentric coordinates?重心坐标barycentric system?重心系barycentric velocity?重心速度base flow?底流base pressure?底面压力basic equation?基本方程basic load?基本载荷basin?硫basset force?巴塞特力bathyal region?半深海区bauschinger effect?包辛格效应beach?海岸beam?梁beam axle?梁式轴beam balance?杠杆秤beam column?梁杆beam structure?梁结构beam supported of both ends?简支梁beam truss?桁架梁beam with varying section?变截面梁bearing?方位;轴承bearing capacity?承载力bearing friction?支承摩擦bearing friction loss?轴承摩擦损失bearing line?方位线bearing load?轴承荷载bearing power?承重能力bearing pressure?支承压力bearing strain?承载应变bearing strength?承载强度bearing stress?承载应力bearing surface?支承面beat?差拍beat frequency?拍频beat period?拍频周期bed load?推移质beginning of curve?曲线起点behavior?行为bell pressure gage?钟型压力计bell type manometer?钟型压力计bellows?波纹管belt tension?皮带张力bend test?弯曲试验bending?弯曲bending center?弯曲中心bending fatigue limit?弯曲疲劳极限bending line?弯曲线bending load?弯曲载荷bending moment?弯矩bending moment density?弯矩密度bending moment diagram?弯矩图bending radius?弯曲半径bending rigidity?抗弯刚度bending strain?弯曲应变bending strength?抗弯强度bending stress?弯曲应力bending test?弯曲试验bending vibration?弯曲振动bending wave?弯曲波beneding stress?挠应力bernoulli constant?伯努利常数bernoulli equation?伯努利方程bessel function?贝塞耳函数beta ratio?比压biaxial stress?双轴向应力bifurcated shock?分岔激波bifurcation?分支bifurcation point?分支点biharmonic function?双谐函数bilateral?双向的bilateral constraint?双边约束billow?大浪bimoment?双力矩bimotor?双发动机飞机binary collision?双碰撞binary diffusion coefficient?二元扩散系数binder?结合剂binding agent?结合剂binding energy?结合能binding force?结合力bingham body?宾汉物体bingham flow?宾汉怜bingham fluid?宾汉铃bingham model?宾汉模型binodal seiche?双节点驻波振荡biological similarity?生物学相似性biomechanics?生物力学biomechanics of bone?骨力学biomechanics of sports?运动生物力学biorheology?生物龄学biot savart law?毕奥萨伐尔定律biplane?双翼飞机bipolar coordinates?双极坐标birefringence?双折射bivector?双矢black body?绝对黑体;黑体black body radiation?黑体辐射blade?叶片blade efficiency?叶片效率blade exit angle?叶片出口角blade grid?叶栅blade inlet angle?叶片入口角blade loss?叶片损失blade outlet angle?叶片出口角blade pitch?叶片距blading?叶片装置blast?送风blast nozzle?喷气嘴blast of wind?阵风blast pressure?风压blast tuyere?风口blast volume?风量blasting?爆破blasting chamber?起爆室blasting efficiency?爆破效率blimp?软式飞艇blocking phenomenon?堵塞现象blood flow?血流blood visco elasticity?血液粘弹性blood viscosity?血液粘度blower?吹风器blown flap?吹气襟翼bluff body?钝体blunt body?钝头体body centroide?本体极迹body fixed system?物体固定坐标系body force?体力body of revolution?旋转体boger fluid?保格铃boiling?沸腾boiling point?沸点boiling temperature?沸点温度boltzmann constant?玻耳兹曼常数boltzmann distribution?玻耳兹曼分布boltzmann factor?玻耳兹曼因子boltzmann h theorem?玻耳兹曼h 定理boltzmann law?玻耳兹曼定律boltzmann transport equation?玻耳兹曼输运方程bolus flow?团流bond energy?结合能bond strength?附着强度bonding force?耦合力boost?升压booster?加速器bootstrap dynamics?靴袢动力学border?边缘borderline?边线bore?内径borehole?钻孔born green equation?玻陡窳址匠眺born von kormon boundary condition?玻斗肟呓缣跫bottom chord?底弦bottom current?底流bottom standing wave?底层驻波bottom velocity?底层速度bouguer wave number?布格波数bounce?回跳bound energy?束缚能bound state?束缚态bound vector?束缚矢量bound vortex?约束涡boundary?边界boundary collocation?边界配置boundary condition?边界条件boundary effect?边界效应boundary element method?边界元法boundary friction?边界摩擦boundary layer?边界层boundary layer control?边界层控制boundary layer equation?边界层方程boundary layer flow?边界层怜boundary layer method?边界层法boundary layer region?边界层区域boundary layer separation?边界层分离boundary layer thickness?边界层厚度boundary layer transition?边界层转捩boundary method?边界解法boundary of the air mass?气团的边界boundary part?边界部分boundary point?边界点boundary surface?边界面boundary value?边界值boundary value problem?边界值问题boundary wave?界面波bourdon gage?布尔登压力计boussinesq approximation?布辛涅斯克近似boussinesq equation of motion?布辛涅斯克运动方程bow?船首bow shock wave?头波bow wave?头波brachistochrone?最速降线bracket?悬臂梁brake?制动器brake horsepower?制动马力brake parachute?制动伞brake power?制动功率brake pressure?闸压力brake test?闸试验braking?制动braking distance?制动距离braking force?制动功率braking moment?制动力矩braking rocket?制动火箭break through?穿透breakaway?分离breakdown test?断裂试验breaker?破浪breaking angle?断裂角breaking elongation?断裂伸长breaking load?断裂负截breaking of wave?海浪断裂breaking point?断点breaking strain?断裂应变breaking strength?裂断强度breaking stress?破坏应力breaking test?致断试验breaking waves?碎波breaking weight?断裂负截bridge truss?桥桁架brinell hardness?布氏硬度brinell hardness test?布里涅耳硬度试验brittle?脆的brittle behavior?脆性行为brittle coating?脆性涂层brittle creep?脆性蠕变brittle fracture?脆裂brittle material?脆性材料brittle strength?抗脆裂强度brittleness?脆性broken line?折线brownian motion?布朗运动brunt vaisala frequency?布伦特韦伊塞拉频率bubble?气泡bubble center?气泡中心bubble cloud?气泡云bubble density?气泡密度bubble domain?泡畴bubble flow?泡状流bubble formation?气泡形成bubble of turbulence?湍联bubble pressure?泡压bubbling?气泡形成bubbling fluidized bed?鼓泡怜床bucket?叶片buckingham potential?伯金汉姆势buckling?屈曲buckling behavior?翘曲行为buckling coefficient?屈曲系数buckling load?屈曲负载buckling of plate?板的翘曲buckling of shell?壳的皱损buckling strength?抗屈曲强度buckling stress?屈曲应力buckling test?纵弯试验buffer?缓冲器减震器buffer action?缓冲酌buffer beam?缓冲梁buffer solution?缓冲溶液buffer spring?缓冲弹簧buffeting?扰炼震built in arch?固定拱bulge?隆起bulk acceleration?牵连加速度bulk forces?体积力bulk modulus?体积弹性横量bulk modulus of elasticity?体积弹性横量bulk motion?牵连运动bulk potential?体积势bulk resonance?体积共振bulk scattering?体积散射bulk strain?体积应变bulk temperature?总体温度bulk velocity?牵连速度bulk viscosity?体积粘度buoyancy?浮力buoyant frequency?浮力频率burgers material?伯格斯材料burning?燃烧burning load?燃烧负荷burning point?燃点burning temperature?燃点burning velocity?燃烧速度bursting?爆裂bursting pressure?爆炸压力bursting strength?破裂强度bursting stress?破裂应力bursting test?爆破试验busemann relation?布泽曼关系式cable stayed bridge?斜拉桥cad?计算机辅助设计calculator?计算机calculus of approximation?近似计算calculus of finite differences?差分演算calibration?校准calibrator?校准器calorific capacity?热容量calorimetric measurement?量热测量camber changing flap?改变机翼弯度的襟翼canal?管道canal vortex?沟渠涡旋canal wave?沟渠波canard?鸭翼canonical coordinate?正则坐标canonical distribution?正则分布canonical equation of motion?正则运动方程canonical equations?正则方程canonical form?正则形式canonical momentum?正则动量canonical transformation?正则变换canonical variable?正则变量cantilever?悬臂梁caoutchouc elasticity?橡胶弹性capacitive transducer?电容传感器capacity?功率capacity measure?容积量度capacity strain gage?电容应变计capillarity?毛细现象capillary absorption?毛细管吸收capillary action?毛细酌capillary attraction?毛细引力capillary condensation?毛细凝缩capillary constant?毛细常数capillary energy?表面张力能capillary fissure?毛细裂纹capillary flow?毛管流capillary force?毛细力capillary gravity wave?毛细重力波capillary level oscillation?毛细面振动capillary phenomenon?毛细现象capillary pressure?毛细压力capillary rise?毛细升高capillary tension?毛细张力capillary tube?毛细管capillary viscosimeter?毛细管粘度计capillary waves?毛细波capture?俘获carbon fiber?碳纤维cardan angle?卡登角cardan rings?卡登环cardiac dynamics?心脏动力学cardiac work?心脏的工作cargo?货物carrier gas?气体载体carrier inertial force?牵连惯性力carrier liquid?载体液体carrier oscillation?载波振荡carrier rocket?运载火箭carrier velocity?牵连速度carrying capacity?负荷量cartesian coordinates?笛卡儿坐标cartesian vector?笛卡儿矢量cascade excitation?级联激发cascade flow?翼栅怜cascade of aerofoil?翼型叶栅cascade tunnel?叶栅风洞case depth?渗碳层深度case hardening?表面硬化casing?外壳castigliano theorem?卡斯蒂利亚诺定理casting stress?铸造应力catapult?弹射器catenary?悬链线catenoid?悬链曲面cauchy deformation tensor?柯挝变张量cauchy equation of motion?柯嗡动方程cauchy integral theorem?柯锡分定理cauchy law of similarity?柯梧似性定律cauchy residue theorem?柯涡数定理cauchy riemann equations?柯卫杪匠眺cauchy stress tensor?柯桅力张量caudad acceleration?尾向加速度causality?因果律caving?空泡形成cavitating flow?气穴流涡空流cavitation?气蚀现象cavitation bubble?空泡cavitation damage?空化损坏cavitation effect?空化效应cavitation erosion?空蚀cavitation nucleus?空化核cavitation number?空化数cavitation parameter?空化参数cavitation phenomenon?气蚀现象cavitation shock?气蚀冲击cavitation tunnel?空泡试验筒cavity?空腔cavity collapse?空泡破裂cavity drag?空泡阻力cavity flow?气穴流涡空流cavity flow theory?空泡另论cavity formation?空泡形成cavity pressure?空腔压力cavity resonator?空腔共振器cavity vibration?共振腔振动ceiling?上升限度celestial mechanics?天体力学cell model?笼子模型cell reynolds number?网格雷诺数cellular grid?网状栅格cellular structure?栅格结构center?中心center line?中心线center line average height?中线平均高度center line of the bar?杆件轴线center of area?面积中心center of buoyancy?浮心center of curvature?曲率中心center of gravity?重心center of inertia?惯性中心center of inertia system?惯性中心坐标系。

ON ELECTROMAGNETIC WAVE INTERACTION WITH DENSERESONANT ATOM MEDIUMV . DanilovORNL, Oak Ridge, TN 37831, U.S.A.AbstractThe excitation of atomic levels due to interaction with electromagnetic waves became of interest in accelerator physics in relation to high efficiency charge exchange injection into rings for high beam power applications.Usually, the beam density is so small that its influence on the wave is completely neglected. Here we consider the case of dense beams - the beam dimensions are large as compared to light reflection length. This paper shows that the waves can be trapped in the medium under these conditions. Moreover, the atoms with induced dipolemoments start to interact strongly with each other, leading to possibility to create some atomic patterns when the medium is relatively cold. INTRODUCTIONIt was shown in [1], that dense atom medium can trap theelectromagnetic waves in resonance with atomic transitions and act like superconducting loss-free cavity for the waves. If the electromagnetic wave intensity becomes large, it applies a substantial pressure on the medium. This paper shows that one of the forces to counteract the field pressure can be the surface tensionforce that appears due to induced dipole moments of the resonant atoms. We present brief estimations for the energy of dipole interaction as a function of the wavefield, and show that for the small temperature (1000K or less) the atoms may form the string-like or other structures. Therefore, the trapped field transforms themedium into liquid and the liquid form may contain the field inside due to induced surface tension.BASIC PHYSICSImagine now two atoms close to each other in the field of electromagnetic wave. For simplicity, we deal with hydrogen atoms, but it is generally true for all atoms ormolecules having zero dipole moment in their groundstate. The Schrödinger equation for two electrons reads:ψψψ),()(221212r r U mt i +Δ+Δ−=∂∂h h , (1) with the potential function equal to221232121221021)))((3()cos()(),(r e r e rn r n r r r e t z z eE r r U −−−++=r r r r r r ω,where indexes 1, 2 related to the first and second electron,respectively, r ris the vector from the first to the secondnuclei, r r n r r = , and 1r r and 2r r are vectors connectingnuclei with corresponding electrons. We assume the wave functions of the atoms don’t overlap and we seek the solution in the following form:hhh /)(2112422113/222122/2211112121))()()()(()()()()(t E E i t iE t iE e r r C r r C e r r C e r r C +−−−+++=ψψψψψψψψψ, where 21,ψψ are the eigenfunctions and E 1,E 2 are theenergy levels of unperturbed ground and upper states, respectively. After substituting to (1) one yields:,222,222,)(2,)(23212320121021432124201210213262430212161430121r i C e C E i e C E i C r i C e C E i e C E i C C ria e C C E i C C r ia e C C E i C t i t i t i t i ti t i h h h &h h h &h h &h h &μμμμμμμμ−+=−+=−+=−+=ΔΔ−ΔΔ−Δ−Δ(2) where Δ=ω-ω0, the electric field has the form E =E 0cos ωt ,)()(2*13*2112r u ez r u r d r r ∫−==μμ(assuming the light is polarized in the direction z , parallel with theatomic plane), and u 1 and u 2 are the normalized wave functions of the lower and the upper excited states,respectively. In the case of hydrogen, the lower level has primary quantum number n=1. The upper level primaryquantum number is determined by the resonant condition, its angular momentum l=1, and the projection of angular momentum on the z axis m=0. For this case μ=μ1n is realand we omit its subscripts for simplicity. In addition toenergy levels, we added second term of atom-atominteractions; the constants a 1 and a 2 can be found in e.g. [2]. Now the idea is to find stationary solutions, calculate the energy of dipole interaction and find its minimum -we suppose the atoms with low temperatures settle aroundthis minimum. First thing to notice is that there are four eigenmodes for this system of equations, and if t i e C C δω∝43, , then t i e C )(1Δ+∝δω andt i e C )(2Δ−∝δω. We are interested in only symmetric eigenmodes 43C C =. Thus we eliminate one (e.g., last)of the equations. The three remaining eigenmodes have two energy levels close to transition energy and one separated from them by energy of dipole interactions. Assuming 321212,rE h hμμδω<<, we exclude one moreeigenmodes with frequency far from the transition one.One gets )(4213043C C r E C C +==μ. Finally, the equation for two remaining eigenfrequencies reads:),(4)(),(4)(2120362221203611C C E r r a C C C E r ra C +=+Δ−+=+Δ+hh δωδω (5)and 226261261216)2(2h E r r a a r a a +−−Δ±−+Δ≈δω. To calculate the dipole force bond, we have to integrate dipole potential U over 6D two-electron coordinate space with wave functions of the system:321*223132rz z e x d x d U ψψ∫−=. Under our assumptions 2,14,3C C <<, therefore the final expressionfor bond energy is 1*232Re 2C C rU μ−≈. Finally, using(5) and 12121≈+C C one gets2206261220216)2(2hh mEr r a a E U +−−Δ≈μ. (3)For one atom, surrounded by two atom in a chain, we have to multiply (3) by 2, but also, due to time averaging, this factor disappear – time averaging leads tomultiplication by factor 0.5. Besides this, we neglect the transition frequency shift Δ, coming from mostly theDoppler Effect, and we neglect a 2 for estimations. Finally, the bond energy of interaction per atom in a string of atoms is:220626120216)2(2hh E r r a E U +−≈μ. (4)Figure 1 shows how this potential depends on thedistance r between the atoms (the field is taken to be 600 MV/m, a 1=6.5 in atomic units (see [2]) and57322=μin atomic units that corresponds to hydrogen1→2 transition.Figure 1 Potential well.Figure 2 shows how minimum of the potential (in units ofKelvin, 1K ≈11000 eV) depends on the electric field amplitude (in units of 60 MV/m) for hydrogen. The hydrogen has to be cold enough to form chains of the atoms. But if one takes metals or semiconductors with much larger dipole transitions, the strings would form even for hot vapors, creating self sustained formations.Figure 2 Temp versuss electric field.In [1], some solutions for trapped fields were found for 1D and 2D cylindrical symmetries. In the simplestvariant, 2D cylindrically trapped fields can form a torroid: the strings of metal (or other high μ atoms) can hold the field inside (the torroid ball lightning were reported to be seen also). The cross section of possible torroid formation is shown in Figure 3Figure 3 Cross section of possible torroid realization ofball lightning.The final field-atom distribution doesn’t explain, of course, the process of creation of such a formation. We think the resonant field appears in the medium in courseof electric discharge and generated in a process, similar to that of the EM field generation in gas lasers.CONCLUSIONSThe paper describes some possible explanations of self sustained formations, similar or equivalent to the ball lightning.ACKNOWLEDGEMENTResearch sponsored by UT-Batelle, LLC, under contract #DE-AC05-00OR22725 for U.S. Department of Energy.REFERENCES1. V. Danilov, “Resonant Atom Traps for Electromagnetic Waves”, arXiv: 0708.4055 (2007)2. Landau and Lifshitz, “Quantum Mechanics: Nonrelativistic Theory”, in Russian, Nauka, Moscow (1989) p. 403。

Chapter 6 Magnetism of MatterThe history of magnetism dates back to earlier than 600 B.C., but it is only in the twentieth century that scientists have begun to understand it, and develop technologies based on this understanding. Magnetism was most probably first observed in a form of the mineral magnetite called lodestone, which consists of iron oxide-a chemical compound of iron and oxygen. The ancient Greeks were the first known to have used this mineral, which they called a magnet because of its ability to attract other pieces of the same material and iron.The Englishman William Gilbert(1540-1603) was the first to investigate the phenomenon of magnetism systematically using scientific methods. He also discovered that Earth is itself a weak magnet. Early theoretical investigations into the nature of Earth's magnetism were carried out by the German Carl Friedrich Gauss(1777-1855). Quantitative studies of magnetic phenomena initiated in the eighteenth century by Frenchman Charles Coulomb(1736-1806), who established the inverse square law of force, which states that the attractive force between two magnetized objects is directly proportional to the product of their individual fields and inversely proportional to the square of the distance between them.Danish physicist Hans Christian Oersted(1777-1851) first suggested a link between electricity and magnetism. Experiments involving the effects of magnetic and electric fields on one another were then conducted by Frenchman Andre Marie Ampere(1775-1836) and Englishman Michael Faraday(1791-1869), but it was the Scotsman, James Clerk Maxwell(1831-1879), who provided the theoretical foundation to the physics of electromagnetism in the nineteenth century by showing that electricity and magnetism represent different aspects of the same fundamental force field. Then, in the late 1960s American Steven Weinberg(1933-) and Pakistani Abdus Salam(1926-96), performed yet another act of theoretical synthesis of the fundamental forces by showing that electromagnetism is one part of the electroweak force. The modern understanding of magnetic phenomena in condensed matter originates from the work of two Frenchmen: Pierre Curie(1859-1906), the husband and scientific collaborator of Madame Marie Curie(1867-1934), and Pierre Weiss(1865-1940). Curie examined the effect of temperature on magnetic materials and observed that magnetism disappeared suddenly above a certain critical temperature in materials like iron. Weiss proposed a theory of magnetism based on an internal molecular field proportional to the average magnetization that spontaneously align the electronic micromagnets in magnetic matter. The present day understanding of magnetism based on the theory of the motion and interactions of electrons in atoms (called quantum electrodynamics) stems from the work and theoretical models of two Germans, Ernest Ising and Werner Heisenberg (1901-1976). Werner Heisenberg was also one of the founding fathers of modern quantum mechanics.Magnetic CompassThe magnetic compass is an old Chinese invention, probably first made in China during the Qin dynasty (221-206 B.C.). Chinese fortune tellers used lodestonesto construct their fortune telling boards.Magnetized NeedlesMagnetized needles used as direction pointers instead of the spoon-shaped lodestones appeared in the 8th century AD, again in China, and between 850 and 1050 they seemto have become common as navigational devices on ships. Compass as a Navigational AidThe first person recorded to have used the compass as a navigational aid was Zheng He (1371-1435), from the Yunnan province in China, who made seven ocean voyages between 1405 and 1433.有关固体磁性的基本概念和规律在上个世纪电磁学的发展史中就开始建立了。

电荷密度波cdw电荷密度波（Charge Density Wave，简称CDW）是一种固体中的电子密度分布的有序周期性变化现象。

在某些晶体中，由于电子间的相互作用，电子密度呈现出长程有序排列的现象，形成了CDW。

CDW最早在20世纪50年代被发现，并且在近年来引起了广泛的研究兴趣。

CDW的形成与晶格畸变有着密切的关系。

晶格畸变是指晶体中原子周期性排列的畸变现象，而CDW则是电子密度的周期性畸变。

晶体中的CDW通常出现在费米面附近，即电子密度较高的区域。

CDW的形成与电子间的相互作用密切相关。

在晶体中，电子间存在库仑相互作用力和弹性相互作用力。

当这两种相互作用力达到平衡时，CDW就会形成。

库仑相互作用力使得电子倾向于在周期性的电荷密度波中排列，而弹性相互作用力则使得这种波动在晶格中传播。

CDW的形成对晶体的电子输运性质产生了显著影响。

一方面，CDW的形成导致电子密度的周期性变化，使得晶体的导电性降低。

另一方面，CDW的形成还会导致晶格的畸变，进一步影响电子的运动。

因此，CDW的研究对于理解和调控材料的电子输运性质具有重要意义。

除了对于材料性质的影响，CDW还与其他一些物理现象密切相关。

例如，在一些材料中，CDW与超导现象同时存在。

超导是指材料在低温下电阻消失的现象，而CDW的形成往往会抑制超导的出现。

另外，CDW还与磁性和光学性质等方面的变化有关。

近年来，随着实验技术的不断进步，人们对CDW的研究取得了许多重要的进展。

通过使用高分辨率的电子显微镜和散射技术，科学家们能够直接观察到CDW的形成和演化过程。

此外，通过调控材料的结构和成分，人们也能够有效地控制CDW的出现和消失。

CDW作为一种固体中的电子密度分布的有序周期性变化现象，对于材料的性质具有重要影响。

研究CDW不仅有助于理解材料的电子输运性质，还为开发新型材料和器件提供了重要线索。

随着对CDW的深入研究，相信未来将会有更多有趣的发现和应用。

a r X i v :c o n d -m a t /9503139v 1 27 M a r 1995Singularity of the density of states in the two-dimensional Hubbard model from finitesize scaling of Yang-Lee zerosE.Abraham 1,I.M.Barbour 2,P.H.Cullen 1,E.G.Klepfish 3,E.R.Pike 3and Sarben Sarkar 31Department of Physics,Heriot-Watt University,Edinburgh EH144AS,UK 2Department of Physics,University of Glasgow,Glasgow G128QQ,UK 3Department of Physics,King’s College London,London WC2R 2LS,UK(February 6,2008)A finite size scaling is applied to the Yang-Lee zeros of the grand canonical partition function for the 2-D Hubbard model in the complex chemical potential plane.The logarithmic scaling of the imaginary part of the zeros with the system size indicates a singular dependence of the carrier density on the chemical potential.Our analysis points to a second-order phase transition with critical exponent 12±1transition controlled by the chemical potential.As in order-disorder transitions,one would expect a symmetry breaking signalled by an order parameter.In this model,the particle-hole symmetry is broken by introducing an “external field”which causes the particle density to be-come non-zero.Furthermore,the possibility of the free energy having a singularity at some finite value of the chemical potential is not excluded:in fact it can be a transition indicated by a divergence of the correlation length.A singularity of the free energy at finite “exter-nal field”was found in finite-temperature lattice QCD by using theYang-Leeanalysisforthechiral phase tran-sition [14].A possible scenario for such a transition at finite chemical potential,is one in which the particle den-sity consists of two components derived from the regular and singular parts of the free energy.Since we are dealing with a grand canonical ensemble,the particle number can be calculated for a given chem-ical potential as opposed to constraining the chemical potential by a fixed particle number.Hence the chem-ical potential can be thought of as an external field for exploring the behaviour of the free energy.From the mi-croscopic point of view,the critical values of the chemical potential are associated with singularities of the density of states.Transitions related to the singularity of the density of states are known as Lifshitz transitions [15].In metals these transitions only take place at zero tem-perature,while at finite temperatures the singularities are rounded.However,for a small ratio of temperature to the deviation from the critical values of the chemical potential,the singularity can be traced even at finite tem-perature.Lifshitz transitions may result from topological changes of the Fermi surface,and may occur inside the Brillouin zone as well as on its boundaries [16].In the case of strongly correlated electron systems the shape of the Fermi surface is indeed affected,which in turn may lead to an extension of the Lifshitz-type singularities into the finite-temperature regime.In relating the macroscopic quantity of the carrier den-sity to the density of quasiparticle states,we assumed the validity of a single particle excitation picture.Whether strong correlations completely distort this description is beyond the scope of the current study.However,the iden-tification of the criticality using the Yang-Lee analysis,remains valid even if collective excitations prevail.The paper is organised as follows.In Section 2we out-line the essentials of the computational technique used to simulate the grand canonical partition function and present its expansion as a polynomial in the fugacity vari-able.In Section 3we present the Yang-Lee zeros of the partition function calculated on 62–102lattices and high-light their qualitative differences from the 42lattice.In Section 4we analyse the finite size scaling of the Yang-Lee zeros and compare it to the real-space renormaliza-tion group prediction for a second-order phase transition.Finally,in Section 5we present a summary of our resultsand an outlook for future work.II.SIMULATION ALGORITHM AND FUGACITY EXPANSION OF THE GRAND CANONICALPARTITION FUNCTIONThe model we are studying in this work is a two-dimensional single-band Hubbard HamiltonianˆH=−t <i,j>,σc †i,σc j,σ+U i n i +−12 −µi(n i ++n i −)(1)where the i,j denote the nearest neighbour spatial lat-tice sites,σis the spin degree of freedom and n iσis theelectron number operator c †iσc iσ.The constants t and U correspond to the hopping parameter and the on-site Coulomb repulsion respectively.The chemical potential µis introduced such that µ=0corresponds to half-filling,i.e.the actual chemical potential is shifted from µto µ−U412.(5)This transformation enables one to integrate out the fermionic degrees of freedom and the resulting partition function is written as an ensemble average of a product of two determinantsZ ={s i,l =±1}˜z = {s i,l =±1}det(M +)det(M −)(6)such thatM ±=I +P ± =I +n τ l =1B ±l(7)where the matrices B ±l are defined asB ±l =e −(±dtV )e −dtK e dtµ(8)with V ij =δij s i,l and K ij =1if i,j are nearestneigh-boursand Kij=0otherwise.The matrices in (7)and (8)are of size (n x n y )×(n x n y ),corresponding to the spatial size of the lattice.The expectation value of a physical observable at chemical potential µ,<O >µ,is given by<O >µ=O ˜z (µ){s i,l =±1}˜z (µ,{s i,l })(9)where the sum over the configurations of Ising fields isdenoted by an integral.Since ˜z (µ)is not positive definite for Re(µ)=0we weight the ensemble of configurations by the absolute value of ˜z (µ)at some µ=µ0.Thus<O >µ= O ˜z (µ)˜z (µ)|˜z (µ0)|µ0|˜z (µ0)|µ0(10)The partition function Z (µ)is given byZ (µ)∝˜z (µ)N c˜z (µ0)|˜z (µ0)|×e µβ+e −µβ−e µ0β−e −µ0βn (16)When the average sign is near unity,it is safe to as-sume that the lattice configurations reflect accurately thequantum degrees of freedom.Following Blankenbecler et al.[1]the diagonal matrix elements of the equal-time Green’s operator G ±=(I +P ±)−1accurately describe the fermion density on a given configuration.In this regime the adiabatic approximation,which is the basis of the finite-temperature algorithm,is valid.The situa-tion differs strongly when the average sign becomes small.We are in this case sampling positive and negative ˜z (µ0)configurations with almost equal probability since the ac-ceptance criterion depends only on the absolute value of ˜z (µ0).In the simulations of the HSfields the situation is dif-ferent from the case of fermions interacting with dynam-ical bosonfields presented in Ref.[1].The auxilary HS fields do not have a kinetic energy term in the bosonic action which would suppress their rapidfluctuations and hence recover the adiabaticity.From the previous sim-ulations on a42lattice[3]we know that avoiding the sign problem,by updating at half-filling,results in high uncontrolledfluctuations of the expansion coefficients for the statistical weight,thus severely limiting the range of validity of the expansion.It is therefore important to obtain the partition function for the widest range ofµ0 and observe the persistence of the hierarchy of the ex-pansion coefficients of Z.An error analysis is required to establish the Gaussian distribution of the simulated observables.We present in the following section results of the bootstrap analysis[17]performed on our data for several values ofµ0.III.TEMPERATURE AND LATTICE-SIZEDEPENDENCE OF THE YANG-LEE ZEROS The simulations were performed in the intermediate on-site repulsion regime U=4t forβ=5,6,7.5on lat-tices42,62,82and forβ=5,6on a102lattice.The ex-pansion coefficients given by eqn.(14)are obtained with relatively small errors and exhibit clear Gaussian distri-bution over the ensemble.This behaviour was recorded for a wide range ofµ0which makes our simulations reli-able in spite of the sign problem.In Fig.1(a-c)we present typical distributions of thefirst coefficients correspond-ing to n=1−7in eqn.(14)(normalized with respect to the zeroth power coefficient)forβ=5−7.5for differ-entµ0.The coefficients are obtained using the bootstrap method on over10000configurations forβ=5increasing to over30000forβ=7.5.In spite of different values of the average sign in these simulations,the coefficients of the expansion(16)indicate good correspondence between coefficients obtained with different values of the update chemical potentialµ0:the normalized coefficients taken from differentµ0values and equal power of the expansion variable correspond within the statistical error estimated using the bootstrap analysis.(To compare these coeffi-cients we had to shift the expansion by2coshµ0β.)We also performed a bootstrap analysis of the zeros in theµplane which shows clear Gaussian distribution of their real and imaginary parts(see Fig.2).In addition, we observe overlapping results(i.e.same zeros)obtained with different values ofµ0.The distribution of Yang-Lee zeros in the complexµ-plane is presented in Fig.3(a-c)for the zeros nearest to the real axis.We observe a gradual decrease of the imaginary part as the lattice size increases.The quantitative analysis of this behaviour is discussed in the next section.The critical domain can be identified by the behaviour of the density of Yang-Lee zeros’in the positive half-plane of the fugacity.We expect tofind that this density is tem-perature and volume dependent as the system approaches the phase transition.If the temperature is much higher than the critical temperature,the zeros stay far from the positive real axis as it happens in the high-temperature limit of the one-dimensional Ising model(T c=0)in which,forβ=0,the points of singularity of the free energy lie at fugacity value−1.As the temperature de-creases we expect the zeros to migrate to the positive half-plane with their density,in this region,increasing with the system’s volume.Figures4(a-c)show the number N(θ)of zeros in the sector(0,θ)as a function of the angleθ.The zeros shown in thesefigures are those presented in Fig.3(a-c)in the chemical potential plane with other zeros lying further from the positive real half-axis added in.We included only the zeros having absolute value less than one which we are able to do because if y i is a zero in the fugacity plane,so is1/y i.The errors are shown where they were estimated using the bootstrap analysis(see Fig.2).Forβ=5,even for the largest simulated lattice102, all the zeros are in the negative half-plane.We notice a gradual movement of the pattern of the zeros towards the smallerθvalues with an increasing density of the zeros nearθ=πIV.FINITE SIZE SCALING AND THESINGULARITY OF THE DENSITY OF STATESAs a starting point for thefinite size analysis of theYang-Lee singularities we recall the scaling hypothesis forthe partition function singularities in the critical domain[11].Following this hypothesis,for a change of scale ofthe linear dimension LLL→−1),˜µ=(1−µT cδ(23)Following the real-space renormalization group treatmentof Ref.[11]and assuming that the change of scaleλisa continuous parameter,the exponentαθis related tothe critical exponentνof the correlation length asαθ=1ξ(θλ)=ξ(θ)αθwe obtain ξ∼|θ|−1|θ|ναµ)(26)where θλhas been scaled to ±1and ˜µλexpressed in terms of ˜µand θ.Differentiating this equation with respect to ˜µyields:<n >sing =(−θ)ν(d −αµ)∂F sing (X,Y )ν(d −αµ)singinto the ar-gument Y =˜µαµ(28)which defines the critical exponent 1αµin terms of the scaling exponent αµof the Yang-Lee zeros.Fig.5presents the scaling of the imaginary part of the µzeros for different values of the temperature.The linear regression slope of the logarithm of the imaginary part of the zeros plotted against the logarithm of the inverse lin-ear dimension of the simulation volume,increases when the temperature decreases from β=5to β=6.The re-sults of β=7.5correspond to αµ=1.3within the errors of the zeros as the simulation volume increases from 62to 82.As it is seen from Fig.3,we can trace zeros with similar real part (Re (µ1)≈0.7which is also consistentwith the critical value of the chemical potential given in Ref.[22])as the lattice size increases,which allows us to examine only the scaling of the imaginary part.Table 1presents the values of αµand 1αµδ0.5±0.0560.5±0.21.3±0.3∂µ,as a function ofthe chemical potential on an 82lattice.The location of the peaks of the susceptibility,rounded by the finite size effects,is in good agreement with the distribution of the real part of the Yang-Lee zeros in the complex µ-plane (see Fig.3)which is particularly evident in the β=7.5simulations (Fig.4(c)).The contribution of each zero to the susceptibility can be singled out by expressing the free energy as:F =2n x n yi =1(y −y i )(29)where y is the fugacity variable and y i is the correspond-ing zero of the partition function.The dotted lines on these plots correspond to the contribution of the nearby zeros while the full polynomial contribution is given by the solid lines.We see that the developing singularities are indeed governed by the zeros closest to the real axis.The sharpening of the singularity as the temperature de-creases is also in accordance with the dependence of the distribution of the zeros on the temperature.The singularities of the free energy and its derivative with respect to the chemical potential,can be related to the quasiparticle density of states.To do this we assume that single particle excitations accurately represent the spectrum of the system.The relationship between the average particle density and the density of states ρ(ω)is given by<n >=∞dω1dµ=ρsing (µ)∝1δ−1(32)and hence the rate of divergence of the density of states.As in the case of Lifshitz transitions the singularity of the particle number is rounded at finite temperature.However,for sufficiently low temperatures,the singular-ity of the density of states remains manifest in the free energy,the average particle density,and particle suscep-tibility [15].The regular part of the density of states does not contribute to the criticality,so we can concentrate on the singular part only.Consider a behaviour of the typedensity of states diverging as the−1ρsing(ω)∝(ω−µc)1δ.(33)with the valueδfor the particle number governed by thedivergence of the density of states(at low temperatures)in spite of thefinite-temperature rounding of the singu-larity itself.This rounding of the singularity is indeedreflected in the difference between the values ofαµatβ=5andβ=6.V.DISCUSSION AND OUTLOOKWe note that in ourfinite size scaling analysis we donot include logarithmic corrections.In particular,thesecorrections may prove significant when taking into ac-count the fact that we are dealing with a two-dimensionalsystem in which the pattern of the phase transition islikely to be of Kosterlitz-Thouless type[23].The loga-rithmic corrections to the scaling laws have been provenessential in a recent work of Kenna and Irving[24].In-clusion of these corrections would allow us to obtain thecritical exponents with higher accuracy.However,suchanalysis would require simulations on even larger lattices.The linearfits for the logarithmic scaling and the criti-cal exponents obtained,are to be viewed as approximatevalues reflecting the general behaviour of the Yang-Leezeros as the temperature and lattice size are varied.Al-though the bootstrap analysis provided us with accurateestimates of the statistical error on the values of the ex-pansion coefficients and the Yang-Lee zeros,the smallnumber of zeros obtained with sufficient accuracy doesnot allow us to claim higher precision for the critical ex-ponents on the basis of more elaboratefittings of the scal-ing behaviour.Thefinite-size effects may still be signifi-cant,especially as the simulation temperature decreases,thus affecting the scaling of the Yang-Lee zeros with thesystem rger lattice simulations will therefore berequired for an accurate evaluation of the critical expo-nent for the particle density and the density of states.Nevertheless,the onset of a singularity atfinite temper-ature,and its persistence as the lattice size increases,areevident.The estimate of the critical exponent for the diver-gence rate of the density of states of the quasiparticleexcitation spectrum is particularly relevant to the highT c superconductivity scenario based on the van Hove sin-gularities[25],[26],[27].It is emphasized in Ref.[25]thatthe logarithmic singularity of a two-dimensional electrongas can,due to electronic correlations,turn into a power-law divergence resulting in an extended saddle point atthe lattice momenta(π,0)and(0,π).In the case of the14.I.M.Barbour,A.J.Bell and E.G.Klepfish,Nucl.Phys.B389,285(1993).15.I.M.Lifshitz,JETP38,1569(1960).16.A.A.Abrikosov,Fundamentals of the Theory ofMetals North-Holland(1988).17.P.Hall,The Bootstrap and Edgeworth expansion,Springer(1992).18.S.R.White et al.,Phys.Rev.B40,506(1989).19.J.E.Hirsch,Phys.Rev.B28,4059(1983).20.M.Suzuki,Prog.Theor.Phys.56,1454(1976).21.A.Moreo, D.Scalapino and E.Dagotto,Phys.Rev.B43,11442(1991).22.N.Furukawa and M.Imada,J.Phys.Soc.Japan61,3331(1992).23.J.Kosterlitz and D.Thouless,J.Phys.C6,1181(1973);J.Kosterlitz,J.Phys.C7,1046(1974).24.R.Kenna and A.C.Irving,unpublished.25.K.Gofron et al.,Phys.Rev.Lett.73,3302(1994).26.D.M.Newns,P.C.Pattnaik and C.C.Tsuei,Phys.Rev.B43,3075(1991);D.M.Newns et al.,Phys.Rev.Lett.24,1264(1992);D.M.Newns et al.,Phys.Rev.Lett.73,1264(1994).27.E.Dagotto,A.Nazarenko and A.Moreo,Phys.Rev.Lett.74,310(1995).28.A.A.Abrikosov,J.C.Campuzano and K.Gofron,Physica(Amsterdam)214C,73(1993).29.D.S.Dessau et al.,Phys.Rev.Lett.71,2781(1993);D.M.King et al.,Phys.Rev.Lett.73,3298(1994);P.Aebi et al.,Phys.Rev.Lett.72,2757(1994).30.E.Dagotto, A.Nazarenko and M.Boninsegni,Phys.Rev.Lett.73,728(1994).31.N.Bulut,D.J.Scalapino and S.R.White,Phys.Rev.Lett.73,748(1994).32.S.R.White,Phys.Rev.B44,4670(1991);M.Veki´c and S.R.White,Phys.Rev.B47,1160 (1993).33.C.E.Creffield,E.G.Klepfish,E.R.Pike and SarbenSarkar,unpublished.Figure CaptionsFigure1Bootstrap distribution of normalized coefficients for ex-pansion(14)at different update chemical potentialµ0for an82lattice.The corresponding power of expansion is indicated in the topfigure.(a)β=5,(b)β=6,(c)β=7.5.Figure2Bootstrap distributions for the Yang-Lee zeros in the complexµplane closest to the real axis.(a)102lat-tice atβ=5,(b)102lattice atβ=6,(c)82lattice at β=7.5.Figure3Yang-Lee zeros in the complexµplane closest to the real axis.(a)β=5,(b)β=6,(c)β=7.5.The correspond-ing lattice size is shown in the top right-hand corner. Figure4Angular distribution of the Yang-Lee zeros in the com-plex fugacity plane Error bars are drawn where esti-mated.(a)β=5,(b)β=6,(c)β=7.5.Figure5Scaling of the imaginary part ofµ1(Re(µ1)≈=0.7)as a function of lattice size.αm u indicates the thefit of the logarithmic scaling.Figure6Electronic susceptibility as a function of chemical poten-tial for an82lattice.The solid line represents the con-tribution of all the2n x n y zeros and the dotted line the contribution of the six zeros nearest to the real-µaxis.(a)β=5,(b)β=6,(c)β=7.5.。

a r X i v :c o n d -m a t /0312451v 1 [c o n d -m a t .s t r -e l ] 18 D e c 2003Spin Density Wave and D-Wave Superconducting Order Parameter “Coexistence”Zaira Nazario †and David I.Santiago †,⋆†Department of Physics,Stanford University,Stanford,California 94305⋆Gravity Probe B Relativity Mission,Stanford,California 94305We study the properties of a spin-density-wave antiferromagnetic mean-field ground state with d-wave superconducting (DSC)correlations.This ground state always gains energy by Cooper pairing.It would fail to superconduct at half-filling due to the antiferromagnetic gap although its particle-like excitations would be Bogolyubov-BCS quasiparticles consisting of coherent mixtures of electrons and holes.More interesting and relevant to the superconducting cuprates is the case when antiferromagnetic order is turned on weakly on top of the superconductivity.This would correspond to the onset of antiferromagnetism at a critical doping.In such a case a small gap proportional to the weak antiferromagnetic gap opens up for nodal quasiparticles,and the quasiparticle peak would be discernible.We evaluate numerically the absorption by nodal quasiparticles and the local density of states for several ground states with antiferromagnetic and d-wave superconducting correlations.PACS numbers:74.20.-z,74.20.Mn,74.72.-h,71.10.Fd,71.10.PmI.INTRODUCTIONEver since the discovery of high temperature superconductivity 1it was proposed that the supercon-ducting correlations might already exists in the antifer-romagnetic Mott insulator 2.The origin of the supercon-ducting correlations was ascribed to the large Coulombic interactions in the undoped materials.The only other large energy scale in the materials is phononic 3.While the microscopic origin of superconductivity re-mains a matter of debate 4,5,6,7,there is growing experi-mental evidence that the quasiparticles are Bogolyubov-BCS quasiparticles.Bending back of photoemis-sion bands 8fits quantitatively the BCS-Bogolyubov model 9,10.Scanning tunneling microscopy finds coher-ent quasiparticles that disperse as a coherent mixture of particles and holes 11,12.The particle and hole am-plitudes in these experiments and in inverse photoemis-sion experiments 12,13fit accurately to the theoretical Bogolyubov-BCS values calculated from the dispersion and gap measured in the normal and superconducting materials respectively.Regardless of whether the origin of superconducting correlations is exotic Coulombic physics or some more conventional mechanism,it is clear that the cuprates are BCS paired superconductors.This does not mean that the Coulomb interactions do not matter.Rather,the in-teresting and contradictory physics for underdoped ma-terials is the result of Coulomb degradation of the super-fluid density 2,4,5,6and order parameter competition be-tween superconductivity and correlated electron ground states 14,15.The degradation of the superfluid density leads to suppressed T c due to a phase instability of the superconducting order parameter 2,16,17,18,19.There are several Coulomb stabilized competing ground states such as orbital antiferromagnetism 15,stripe or charge density wave ground states 20and perhaps electronic liquid crys-tal phases 21.Regardless of which of these competing ground states are realized,there is strong experimental evidence for incommensurate electronic ordering 22,either0.5 1 1.5 2 2.5-8-6-4-2 0 2 4 6 8D e n s i t y o f S t a t e s [A r b i t r a r y U n i t s ]Energy/Hopping0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-8-6-4-2 0 2 4 6 8D e n s i t y o f S t a t e s [A r b i t r a r y U n i t s ]Energy/HoppingFIG.1:Spectral Function for the SDW ground state without (1a)and with (1b)superconducting correlations built in.static or incipient.The evidence seems more consistent with charge-density-wave or stripe order.In the present work we will study the physics of an antiferromagnet with a strong d-wave Cooper pairing in-teraction.We do not speculate as to the origin of this superconducting interaction except to point out that inFIG.2:In the upper left side we draw two bands separatedby a gap,with the lower band partiallyfilled appropriate toa metal or superconductor.In the upper right side we drawthe situation encountered for the electronfluid under the ac-tion of an electricfield.The lower sketch illustrates the sit-uation appropriate to an insulator,where the lower band iscompletelyfilled,making conduction impossible regardless ofCooper correlations.such a model it competes with the Coulombic antiferro-magnetic physics.Both the superconductor and the anti-ferromagnet are studied in the meanfield approximation.While one can doubt the validity of such an approxima-tion at a phase transition point,it will be qualitativelycorrect within the ordered phases.Cooper pairing leading to a BCS ground state is aninstability of a Fermi liquid ground state.In this studywe apply the BCS approximation to a spin density wave(SDW)insulating ground state as it exists in the cupratesat halffilling.The resulting ground state has Cooperpairing yet it fails to superconduct due to the SDW insu-lating gap.Next we will review some well known facts inorder to understand how a state with Cooper pairs doesnot superconduct.Before doing so we emphasize thatthis only happens as a consequence of having a completelyfilled insulating band.When an electricfield is applied to a metal,it con-ducts dissipatively.The way this happens is that thecenter of mass of the Fermi sea gets displaced upwardin the unfilled metallic band23.Ohmic dissipation oc-curs because newlyfilled electronic states at the top ofthe Fermi sea get scattered into newly empty electronicstates at the bottom of the Fermi sea due to the lack ofrigidity of the Fermi liquid ground state(seefigure2).When there are Cooper correlations,the electron liquidgets displaced upward in the band too,but as long asthe displacement in energy within the band is less thanthe superconducting gap,Cooper pair correlations makethe electron liquid rigid,thus preventing scattering anddissipation.For the case of an SDW ground state at halffilling with Cooper pairing correlations there is no super-conductivity as the electronfluid cannot move upward inthe band for the band is full and there are no electronicstates to befilled unless one excites across the insulatinggap and into the conduction band(seefigure2).That the SDW insulating ground state with d-wavepairing interactions has Cooper pairing in the groundstate can be seen fromfigure1.Infigure1a we plotthe spectral function for the SDW ground state with nosuperconducting correlations.Infigure1b we plot thespectral function for the SDW ground state with super-conducting correlations.In the ground state with bothsuperconductivity and antiferromagnetism,the separa-tion between the coherence peaks is bigger as it gets con-tributions from both the SDW and superconducting gap.A prediction of this model is that the quasiparticles willbe coherent with an electron and a hole component inagreement with the BCS-Bogolyubov model.The SDW ground state with d-wave Cooper pairing(SDW-DSC)will become superconducting when doped.At the mean-field level,without worrying about self con-sistency,the chemical potential will jump to the ap-propriate band and there will be a low superfluid den-sity superconductor.Whether this physics is correctfor the cuprates is controversial.There is experimen-tal evidence for the chemical potential staying pinned atmidgap due to spectral redistribution of states towardmidgap states24.There is also experimental evidence forchemical potential shifts in the cuprates,in the sameway as in regular semiconductor materials25.Indepen-dently of whether the SDW-DSC ground state has chem-ical potential shifts or not,the physics of an insulatorwith Cooper pairing correlations is interesting.For ourstudy we have the cuprates in mind.For these materials,some phenomenology of this form seems to apply2,butit would be interesting if this physics were to be realizedin nature irrespective of the cuprate problem.In the present work we willflip the problem around.We will start with a d-wave superconductor(DSC)andbegin turning on SDW antiferromagnetic order on top ofthe superconductivity.In this limit,the complicationsmentioned in the previous paragraph are nonexistent.Aslow turning on of SDW order on top of the supercon-ductivity will show up as a shift of the antinodal gap anda gapping of the nodal quasiparticles.The latter shouldbe a signal much easier to pick out than the gap shift.The gapping of the nodal quasiparticles is not a uniqueprediction of antiferromagnetic ordering on top of the su-perconductivity,as such a gapping can be produced bydisorder.On the other hand,the coherence of the gapped“nodal”quasiparticles would be nonexistent for a disor-dered gap and is thus a unique signature of antiferromag-netic ordering developing on top of the superconductivity.Therefore,if a quasiparticle peak is discernible,and thebroadening is less than the disorder-induced broadening(>∼ 2/2m∆x2≃350meV for∆x∼1nm,appropriateto the cuprates),then the gap is a long range orderedgap and not a disordered gap.Another unique signatureof an SDW gap is that the gap will open exactly at the10 20 30 40 50 60-6-4-2 0 2 4 6[A r b i t r a r y U n i t s ]5 10 15 20 25 30-6-4-2 0 2 4 65 10 15 20 25 30 35-6-4-2 0 2 4 6S t r e n g t h0.2 SDW gap5 10 15 20 25 30 35-6-4-2 0 2 4 60.3 SDW gap0 5 10 15 20 25 30 35-6-4-20 24 6A b s o r p t i o nEnergy/Hopping0.6 SDW gap1020 30 40 50 60 70-6-4-20 24 6Energy/Hopping0.0 SDW gap 0.1 SDW gap 0.2 SDW gap 0.3 SDW gap 0.6 SDW gapFIG.3:Gapping of the nodal quasiparticles pole as the SDW order develops.doping where the antiferromagnetism starts.There are experimental suggestions of antiferromag-netism competing with superconductivity in the deep un-derdoped regime in the cuprates.For example,measure-ments show the nodal quasiparticle peaks surviving right up to the doping where antiferromagnetism starts.The spectral weight of such peaks diminishes with decreas-ing doping,consistent with spectral weight being robbed from the superconducting long range order by a compet-ing long range order such as antiferromagnetism 26.If one looks in the antiferromagnetically ordered dopings,there are experimental suggestions of a competing order pa-rameter that conducts efficiently.Most strikingly,there5 10 15 20 25 30-6-4-2 0 2 4 6[A r b i t r a r y U n i t s ]5 10 15 20 25 30-6-4-2 0 2 4 60 5 10 15 20 25 30 35-6-4-20 24 6A b s o r p t i o n S t r e n g t hEnergy/Hopping0.6 SDW gap510 15 20 25 30 35-6-4-20 24 6Energy/Hopping0.0 SDW gap 0.3 SDW gap 0.6 SDW gapFIG.4:Shift of the antinodal gap as the SDW order develops.are measurements of metallic conduction even below the Neel ordering temperature 27.The gapping of the nodal quasiparticles pole as the SDW order develops on top of the superconductivity is shown in figure 3for different values of the SDW gap.The reason we only have a quasiparticle sharp pole is that we have not modeled the realistic electronic self energies relevant to the cuprates as they are irrelevant to the pointof principle we are making.Their only effect will be to broaden the quasiparticle peaks and add an incoherent background with the phenomenological features.In fig-ure 4we plot the shift of the antinodal gap as the SDW gap turns on.In figure 5we plot the spectral density of states in a d-wave superconductor as the SDW gap is turned on.The superconductor with no SDW gap does not have a true gap because of its d-wave symmetry.This is seen in the familiar V-shaped collapse at zero energy.As the SDW gap is turned on,we see the V-shape flatten and expand as a signature of the opening of the antifer-romagnetic gap.II.HUBBARD MODEL WITH D-W A VE ATTRACTIVE INTERACTIONSFor the cuprate problem,the two large effects are the antiferromagnetic,or Coulombic,physics and the strong superconductivity.Hence we will start from a phe-nomenological Hamiltonian which is a Hubbard model with a d-wave electronic interaction.This interaction will give rise to d-wave superconductivity when we make the mean-field BCS approximation.The Hamiltonian isH =k,σ(ǫ k −µ)c †k,σck,σ+U0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-8-6-4-22 4 6 8[A r b i t r a r y U n i t s ]0.0 SDW gap0.20.4 0.6 0.8 1 1.2 1.4 1.6-8-6-4-2 0 2 4 6 80.1 SDW gap0 0.2 0.4 0.6 0.8 11.2 1.4 1.6-8-6-4-22 4 6 8S t a t e s0.2 SDW gap0.20.4 0.6 0.8 1 1.2 1.4 1.6 1.8-8-6-4-2 0 2 4 6 80.3 SDW gap0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-8-6-4-2 0 2 4 6 8D e n s i t y o fEnergy/Hopping0.6 SDW gap0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-8-6-4-2 0 2 4 6 8Energy/Hopping0.0 SDW gap 0.1 SDW gap 0.2 SDW gap 0.3 SDW gap 0.6 SDW gapFIG.5:Spectral density of states for a d-wave superconductor as the SDW gap increases.where c †k,σ,ck,σare the electronic creation and destruc-tion operators with momentum k and spin σ,ǫ k is the kinetic energy,µthe chemical potential,and U is the Hubbard repulsion.We are working in a spatial lattice with N sites.The last term is an electronic interaction chosen in the reduced BCS form 9,which will be used to stabilize superconductivity.In order to have d-wavesuperconductivity we choose V ( k 1, k 2)=V 0(cosk 1x −cosk 1y )(cosk 2x −cosk 2y ).This phenomenological Hamil-tonian can have a mean-field SDW ground state and a mean-field DSC ground state.It can be used to study the turning on of DSC correlations on top of an SDW ground state,or the turning on of SDW order on top of the superconductivity.We will analyze this Hamiltonian by imposing an SDW meanfield condition,which is stabilized by the Hubbard term.This will be followed by a DSC mean-field con-dition,which is stabilized by the reduced BCS d-wave interaction.While the use of two mean-field conditions is not common,it has important precedents.It was used by P.W.Anderson28in his study of the role of plasmons in restoring gauge invariance to the BCS ground state.In this work he invented the Anderson-Higgs mechanism29. He solved for the properties of the electron system im-posing a mean-field condition on the electron density,as in the study of electron correlations by Sawada,et al30 and a BCS electron pairing mean-field condition9.The Hubbard interaction stabilizes the mean-field or-derσSN≡ k c† k+ Q,σc k,σ (2)where Q=(π,π)is the commensurate ordering wave vector and S is the average magnetic moment per site. Other ordering wave vectors are possible for spin and/or charge,i.e.stripe,order parameters but we do not con-sider them in our study.When we impose this conditionon the Hamiltonian and neglectfluctuation terms,the Hamiltonian becomesH= k,σ(ǫ k−µ)c† k,σc k,σ+UNS2−US k,σσc† k+ Q,σc k,σ+ k1, k2V( k1, k2)c† k1,↑c†− k1,↓c− k2,↓c k2,↑(3)We see that by ordering antiferromagnetically we gain variational energy−UNS2if self-consistency can be achieved.We next impose the mean-field d-wave Cooper pairing∆k2≡(cosk2x−cosk2y)V0 k1(cosk1x−cosk1y) c† k1,↑c†− k1,↓ ≡∆0(cosk2x−cosk2y)(4) Then the Hamiltonian becomesH= k,σ(ǫ k−µ)c† k,σc k,σ+UNS2−US k,σσc† k+ Q,σc k,σ−∆20V0+ k′∆ k c† k,↑c†− k,↓+c− k,↓c k,↑−c† k+ Q,↑c†− k− Q,↓−c− k− Q,↓c k+ Q,↑ (6)where the prime on the summation sign means that the sum is restricted to the wave vectors in the magneticzone.ǫ+k≡(ǫ k+ǫ k+ Q)/2andǫ− k≡(ǫ k−ǫ k+ Q)/2.The last term in the superconducting interaction is negative because∆k+ Q=−∆k.In order to diagonalize the mag-netic part we define the Bogolyubov operatorsbk,σ=αkck,σ−σβ k c k+ Q,σ(7)bk+ Q,σ=αkck+ Q,σ+σβkck,σ(8) If we chooseα2k=1Ek β2 k=1Ek (9)E2k=(ǫ−k)2+U2S2(10) the Hamiltonian becomesH= k,σ′ (ǫ+ k−µ)(b† k,σb k,σ+b† k+ Q,σb k+ Q,σ)+Ek(b†k,σbk,σ−b† k+ Q,σb k+ Q,σ) +UNS2−∆207Bk+ Q,σ=u−kbk+ Q,σ−σv− k b† k+ Q,¯σ(13)If we choose(u±k )2=1E±k (14)(v±k )2=1E±k (15)(E±k )2=(ǫ+k−µ±E k)2+∆2 k(16)the Hamiltonian then becomesH= k,σ′ E+ k B† k,σB k,σ+E− k B† k+ Q,σB k+ Q,σ+UNS2−∆20U= k ǫ+ k−µ+E k E− k E k (18)from the antiferromagnetic self-consistency condition(2) and−2E+k +12πi ∞−∞dω√πN k(u+ k)2E−E−k+iη−(v+k)2E+E−k−iη(23)The local spectral density function follows from31A( x,E)=−igap to be0.3.The antiferromagnetic gap is chosen any-where between0and0.6,usually with jumps of0.1.We have nearest neighbor hopping only.These values need not be realistic;they are just chosen to illustrate the ef-fect.Similarly,if we Fourier transform the Green’s function (20)in both time and space,we obtain the retarded prop-agator in the wavevector energy representation.G( k,E)=1E−E+k+iη+(u−k)2E+E+k−iη−(v−k)2πIm G( k,E)(26)vs.energy.Energy units,values and uncertainties are chosen as described for the local density of states.V.CONCLUSIONSWe studied a meanfield Hamiltonian with two mean field order parameters.The Hamiltonian contains a spin-density-wave antiferromagnetic meanfield stabilized by a Hubbard interaction and a d-wave Cooper pairing mean field stabilized by a phenomenological d-wave interac-tion.The two order parameters can coexist and the SDW ground state always gains energy by Cooper pairing when the d-wave interaction is attractive and nonzero.The SDW ground state with Cooper pairing fails to super-conduct at half-filling due to the antiferromagnetic gap. Its particle-like excitations are Bogolyubov-BCS quasi-particles consisting of coherent mixtures of electrons and holes.Of greater interest and relevance to the superconduct-ing cuprates is the case when antiferromagnetic order is turned on weakly on top of the superconductivity.This would correspond to the onset of antiferromagnetism at a critical doping.In such a case a small gap proportional to the weak antiferromagnetic gap opens up for nodal quasiparticles,and the quasiparticle peak would be dis-cernible.While the gapping of the nodal quasiparticle could be caused by a large enough disorder,such a dis-order would broaden the quasiparticle peak so much as to make it invisible.A unique signature of antiferromag-netic gapping of the nodal quasiparticles is that it will turn on always at the doping when antiferromagnetism starts while disorder gapping will turn on at different sample dependent dopings.We wrote down the exact expressions for the Green’s function for the system with coexisting SDW and DSC order parameters.These are evaluated numerically in a 1000×1000momentum lattice with.01energy resolu-tion in units of the lattice hopping.From the imaginary parts of the Green’s functions we obtained the absorption by nodal quasiparticles and the local density of states. In our work we did not worry about having self-consistency.This neglect does not affect our results when the two order parameters are nonzero,but it will affect whether the order parameters are nonzero or not,and what the gap values are.Self-consistency will be im-portant in studying how the two order parameters com-pete and if and how they steal spectral weight from each other.Self-consistency might also affect how the SDW-DSC ground state behaves when doped from half-filling. Intuitively one expects chemical potential shifts,but it is not certain that this would be the case.All these is-sues should be studied carefully and we postpone them for future work.Acknowledgments Zaira Nazario is a Ford Founda-tion predoctoral fellow.She was supported by the Ford Foundation and by the School of Humanities and Science at Stanford University.David I.Santiago was supported by NASA Grant NAS8-39225to Gravity Probe B.1G.J.Bednorz and K.A.Muller,Z.Phys.B64,189(1986); M.K.Wu et.al.,Phys.Rev.Lett.,58,908(1987);H. 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a r X i v :c o n d -m a t /0408382v 1 [c o n d -m a t .s t r -e l ] 17 A u g 2004Collective SpinDensity-Wave Response Perpendicular to the Chains of theQuasi One-Dimensional Conductor (TMTSF)2PF 6K.Petukhov and M.Dressel1.Physikalisches Institut,Universit¨a t Stuttgart,Pfaffenwaldring 57,D-70550Stuttgart,Germany(Dated:February 2,2008)Microwave experiments along all three directions of the spin-density-wave model compound (TM-TSF)2PF 6reveal that the pinned mode resonance is present along the a and b ′axes.The collective transport is considered to be the fingerprint of the condensate.In contrast to common quasi one-dimensional models,the density wave also slides in the perpendicular b ′direction.The collective response is absent along the least conducting c ∗direction.PACS numbers:72.15.Nj,75.30.Fv,74.70.KnThe electrodynamic response of quasi one-dimensional materials with a density-wave ground state has been thoroughly explored during past decades.At low tem-peratures the optical conductivity develops an absorption edge in the infrared spectral range due to the opening of the single-particle gap at the Fermi energy.A so-called pinned mode resonance is usually found in the GHz range of frequency;it can be attributed to the collective re-sponse of the condensate pinned to lattice imperfections.At even lower frequencies (in the range of MHz,kHz and even below,depending on temperature)internal deforma-tions and screening by the conduction electrons lead to a broad relaxational behavior.Numerous experimental and theoretical studies performed on model compounds for the formation of charge density waves (CDW),like K 0.3MoO 3,TaS 3,NbSe 3,or (TaSe 4)2I,and the forma-tion of spin-density waves (SDW),like (TMTSF)2PF 6,have been summarized in a number of reviews and mono-graphs [1,2,3,4].The generic conductivity spectrum is plotted in Fig.1for the example of (TMTSF)2PF 6.Frequency f (Hz)10−310−210−11001011021031041010101010−8−6−4−2021010C o n d u c t i v i t y σ (Ω−1 c m −1)Frequency ν (cm −1)FIG.1:Sketch of the frequency dependent SDW conductivity composed using data along the chain axis of (TMTSF)2PF 6[5].The solid arrows indicate the position of single particle gap and the pinned mode resonance in the microwave fre-quency range.The dashed arrow depicts the frequency range of the investigations presented in this work.To our knowledge,all the experiments and models fo-cus on the transport along the highly conducting chains.However,real materials are three dimensional:no mat-ter how anisotropic they are,the interaction between the chains cannot be neglected.In fact it is know that the three-dimensional coupling between the density wave formed on the chains is essential to develop the ordered ground state at finite temperatures [6,7].Some of the most studied density-wave systems,K 0.3MoO 3and (TMTSF)2PF 6,in fact have a tendency toward two-dimensionality;for our example of (TMTSF)2PF 6the transfer integrals (4t a :4t b :4t c )=(1.5:0.1:0.003)eV have been determined from band structure calculations [8].In certain cases the dc transport was measured for the perpendicular directions.Similar to the resistivity along the highly-conducting axis,the density-wave tran-sition can in general also be observed by a sharp in-creases of the resistivity perpendicular to the chains (cf.Fig.2).The explanation is the opening of the single-particle gap over the entire Fermi surface.Nothing,however,is known about the collective response which (besides sophisticated methods like narrow-band noise)can best be observed by a threshold field in the non-linear conductivity or by the pinned-mode resonance.It has been argued,that the density wave is a strictly one-dimensional phenomenon which develops only along the chains.The aim of this study is the search for indica-tions of the collective electrodynamic response of a SDW in the perpendicular directions.Single crystals of the Bechgaard salt tetramethyltetra-selenafulvalene)-hexaflourophosphate,denoted as (TM-TSF)2PF 6,were grown by electrochemical methods as described in [9].The dc resistivity ρ(T )of (TMTSF)2-PF 6along the a -axis was measured on needle-shaped samples with a typical dimension of (2×0.5×0.1)mm 3along the a ,b ′,and c ∗axes,respectively.Due the tri-clinic symmetry (a =7.297˚A ,b =7.711˚A ,c =13.522˚A ,α=83.39◦,β=86.27◦,γ=71.01◦),b ′is perpendicu-lar to a ,and c ∗is normal to the ab ′plane.The b ′-axis conductivity was obtained on a narrow slice cut from a thick crystal perpendicular to the needle axis;the typi-cal dimensions of so-made samples were a ×b ′×c ∗=2101010T(K)ρa (Ω c m )10101010ρb ' (Ω c m )10101010 1/T (K -1)ρc * (Ω c m )FIG.2:Arrhenius plot of the temperature dependent dc re-sistivity of (TMTSF)2PF 6along the a ,b ′and c ∗directions.(0.2×1.3×0.3)mm 3.Due to the large sample geome-try,the b ′-axis resistivity was measured for the first time with basically no influence of the a and c ∗contributions and using standard four-probe technique to eliminate the contact resistances.Also for the c ∗-axis transport,four contacts were applied,two on each side of the crystal.In addition,the microwave conductivity at 24and 33.5GHz was measured in all three directions.The crystals were placed onto a quartz substrate and positioned in the max-imum of the electric field of a cylindrical copper cavity.Along the a -direction the naturally grown needles were used,because this geometry is best for precise microwave measurements.As described above,a slice was cut from a thick single crystal to measure in b ′direction.In order to perform microwave experiments along the c ∗axis,a crystal was chopped into several pieces (approximately cubes of 0.2mm corner size)and arranged up to four as a mosaic in such a way that a needle-shaped sample of about (0.2×0.2×0.8)mm 3was obtained.By record-ing the center frequency and the halfwidth of the reso-nance curve as a function of temperature and comparing them to the corresponding parameters of an empty cav-ity,the complex electrodynamic properties of the sample,like the conductivity and the dielectric constant,can be determined via cavity perturbation theory;further de-tails on microwavemeasurements and the data analysis are summarized in [9,10].10101010T(K)ρa (Ω c m )ρb ' (Ω c m )1/T (K -1)ρc * (Ω c m )FIG.3:Temperature dependent microwave resistivity of (TMTSF)2PF 6along the a ,b ′and c ∗directions measured at 33.5GHz.In Fig.2the temperature dependence of the dc resis-tivity is plotted.When the SDW ground state develops at T SDW =12K a sharp increase of ρ(T )is observed along the a ,b ′and c ∗directions.For T <T SDW an activated behavior ρ(T )∝exp {∆/T }can be identified,with a single-particle energy gap of 27.1K,27.4K,and 20.5K along a ,b ′and c ∗directions,respectively.On cooling down further (somewhat below 6K)the activa-tion energy is slightly reduced giving values of 20.8K,21.3K,and 18.4K in the three orientations.At very low temperatures heating cannot be excluded,leading to a saturation of ρ(T ).While for the a and b ′axes the activation energy is identical within the error bars,a somewhat lower value is observed for the least conduct-ing direction.These data are in good agreement with earlier findings [11,12,13]and estimations by mean field theory:∆(T =0)=3.53T SDW /2≈21K.Also in the microwave data,the SDW transition at 12K is present in all three directions.The tempera-ture dependent resistivity measured at 33.5GHz is plot-ted in Fig.3in the Arrhenius representation.Up to six samples of different batches have been studied for each orientation;the sample-to-sample spread is within the uncertainty to determine the slope.Similar results are obtained at 24GHz,but both frequencies are too close to allow for any conclusions on the frequency dependence.3FIG.4:Schematic Fermi surface nesting of a quasi one-dimensional system with interchain coupling in b -direction.Most surprisingly,the activation energy along the a and b ′axes is much smaller compared to the dc behavior,while for the c ∗orientation the results at microwave fre-quencies perfectly agrees with the dc profile.Right below T SDW the activation energies obtained for the three di-rections are (5.9±0.4)K,(6.0±0.3)K,and (20.7±0.4)K.The significantly reduced values of the activation energy for the a and b ′directions compared to dc data infer a strong frequency dependent response which is associated with the collective mode contribution to the electrical transport.Based on an extensive microwave study along the chain direction,it was proposed [5]that due to impurity pin-ning the collective SDW response in (TMTSF)2PF 6is located around 5GHz.The conductivity below the en-ergy gap decreases exponentially with decreasing temper-ature,except in the range of the pinned mode.As can be seen from Fig.1,the present microwave experiments are performed in the range where the collective mode is still very pronounced,i.e.on the shoulder of the pinned mode resonance.Hence the temperature dependent mi-crowave conductivity is caused by two opposing effects:(i)the exponential freeze-out of the background conduc-tivity caused by the uncondensed conduction electrons,and (ii)the build-up of the collective contribution.It was suggested that this mode does not gain much spec-tral weight as the temperature decreases,but the width and center frequency changes slightly [5].The most sur-prising discovery of our investigation is the presence of the enhanced microwave conductivity no only along the chains,but also perpendicular to them.This implies that the pinned mode resonance is present in the b ′direction in a very similar manner compared to the a axis.Cur-rent models assume that the density wave can slide only along the highly conducting direction;our findings,how-ever,give clear evidence for a collective contribution to the conductivity in the perpendicular direction.No in-dications of a collective response is observed along the c ∗-direction.Hence the sliding density wave has to be considered a two-dimensional phenomenon with severe implications on the theoretical description.These results can be explained by looking at the actual Fermi surface of (TMTSF)2PF 6which is not strictly one-dimensional but shows a warping in the direction of k b (and much less in k c ),as depicted in Fig.4.From NMR experiments [14]it is know that the SDW corresponds to a wavevector Q =(0.5a ∗,0.24±0.03b ∗,−0.06±0.20c ∗);which is incommensurate with the underlying lattice.Most important in this context,there is an appreciable component of Q in the b axis.The tilt of the nesting vector is responsible for the similar collective SDW re-sponse found in the microwave experiments along the a and b ′directions.The density wave does not slide along the highly conducting axis but in direction of the nesting vector.Similar investigations (including studies of the I -V characteristic)on the quasi one-dimensional CDW model compound K 0.3MoO 3are in progress.In conclusion,the enhanced conductivity 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