Elementary Excitations of One-Dimensional t-J Model with Inverse-Square Exchange

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一维无限深势阱英语

A one-dimensional infinite-depth potential well is a mathematical model used to describe the motion of particles in a region of space where the potential energy is constant and uniform. It is a popular model used in many physical systems, such as quantum mechanics, solid-state physics, and particle physics.The one-dimensional infinite-depth potential well is represented by a function that has a constant value of zero at all points. This means that, although the particles can move freely in the region, they cannot escape since the potential energy remains constant. This means that any particle in the region will be in a state of equilibrium.In quantum mechanics, the one-dimensional infinite-depth potential well is used to describe the behavior of particles in the so-called “quantum confinement”. This means that the particles are confined to a finite region of space, and the total energy of the system is constant. The energy levels of particles in the potential well are quantized, meaning they can only take on certain discrete values.In solid-state physics, the one-dimensional infinite-depth potential well is used to describe the behavior of electrons in a material. The potential well helps to explain the behavior of electrons in solids, and how they interact with each other. For example, the potential well is used to explain why certain materials are insulators and others are conductors.In particle physics, the one-dimensional infinite-depth potential well is used to describe the behavior of subatomic particles. It is used to describe the behavior of quarks, for example, and how they interact with each other. The potential well can also be used to explain the behavior of particles in the so-called “Standard Model” of particle physics.In conclusion, the one-dimensional infinite-depth potential well is a powerful mathematical model used to describe the behavior of particles in many physical systems. It is used to explain the behavior of particles in quantum mechanics, solid-state physics, and particle physics. The potential well is also used to explain the behavior of particles in the Standard Model of particle physics.。

Linear and Nonlinear Waves

Chapter 2 Linear and Nonlinear Waves
To begin exploring the vast mathematical continent that is partial diﬀerential equations, our ﬁrst task is to understand simple ﬁrst order equations. In applications, ﬁrst order partial diﬀerential equations are most commonly used to describe dynamical processes, and so time, t, is one of the independent variables. Most of our discussion will focus on dynamical models in a single space dimension, bearing in mind that most of the methods can be readily extended to higher dimensional situations. First order partial diﬀerential equations and systems model a wide variety of wave phenomena, including transport of solvents in ﬂuids, ﬂood waves, acoustics, gas dynamics, glacier motion, chromatography, traﬃc ﬂow, and also a variety of biological and ecological systems. As always in mathematical analysis, one must be able to handle relatively tame linear examples before venturing into the nonlinear wilderness. A basic solution technique relies on an inspired change of variables, rewriting the equation in a moving coordinate frame. In this manner, we are naturally led to the fundamental concept of a characteristic curve. Their physical relevance comes from the fact that signals and disturbances propagate along the characteristic curves in space-time. Indeed, their multi-dimensional counterparts are the light cones of special relativity, [ 84, 88 ]. The characteristic method solves a ﬁrst order linear partial diﬀerential equation by reducing it to a ﬁrst order nonlinear ordinary diﬀerential equation! In the nonlinear regime, the most important new phenomenon is the possible breakdown of solutions in ﬁnite time, resulting in the formation of discontinuous shock waves. A familiar example is the supersonic boom produced by an airplane that breaks the sound barrier. Signals continue propagate along the characteristics, but in the nonlinear case, characteristic curves may cross each other, precipitating the onset of a shock discontinuity. The characterization of the ensuing shock dynamics is not speciﬁed by the partial diﬀerential equation alone, but relies on additional physical information, in the form of a conservation law and entropy condition. Having attained a basic understanding of ﬁrst order wave dynamics, we then focus our attention on the second order wave equation in a single space dimension, used to model waves and vibrations in a violin string, a column of air in a clarinet, or an elastic bar. The wave equation is one of the fundamental partial diﬀerential equations that must be mastered in an introductory course. Its multi-dimensional counterparts serve to model vibrations of membranes, solid bodies, water waves, electromagnetic waves, including light, radio and micro-waves, acoustic waves, and many other physical systems. The one-dimensional wave equation is one of a small handful of physically relevant partial differential equations to be favored with an explicit solution formula, originally discovered by 1/13/08 12

Coulomb gap in a model with finite charge transfer energy

a r X i v :c o n d -m a t /0003311v 1 17 M a r 2000Coulomb gap in a model with ﬁnite charge transfer energy.S.A.Basylko 1,P.J.Kundrotas 2,3,V.A.Onischouk 1,2,E.E.Tornau 2,4and A.Rosengren 21Joint Institute of Chemical Physics of Russian Academy of Sciences,117977Kosygin Str.4,Moscow,Russia 2Department of Physics/Theoretical physics,Royal Institute of Technology,SE–10044Stockholm,Sweden3Faculty of Physics,Vilnius University,Sauletekio al.9,LT–2040,Vilnius,Lithuania4Semiconductor Physics Institute,Goˇs tauto 11,LT–2600Vilnius,Lithuania(Received )The Coulomb gap in a donor-acceptor model with ﬁnite charge transfer energy ∆describing theelectronic system on the dielectric side of the metal-insulator transition is investigated by means of computer simulations on two-and three-dimensional ﬁnite samples with a random distribution of equal amounts of donor and acceptor sites.Rigorous relations reﬂecting the symmetry of the model presented with respect to the exchange of donors and acceptors are derived.In the immediate neighborhood of the Fermi energy µthe the density of one-electron excitations g (ε)is determined solely by ﬁnite size eﬀects and g (ε)further away from µis described by an asymmetric power law with a non-universal exponent,depending on the parameter ∆.PACS numbers:71.23.-k,71.30.+h,71.45.GmI.INTRODUCTIONDoping of solids might lead to drastic qualitative changes in their properties.The metal-insulator tran-sition (MIT)is a spectacular manifestation of this.The understanding of the driving forces of the MIT is a long-standing problem.In the early seventies,the prediction 1was made that on the dielectric side of the MIT the long-range Coulomb interactions deplete the density of one-electron excitations (DOE)g (ε)near the Fermi energy µ.Further,analytical calculations of g (ε)with Coulomb correlation taken into consideration have been performed on the metallic side of the MIT.Altshuler and Aronov 2showed that for the metallic case g (ε)in three dimensions has a cusp-like dependence g (ε)∼|ε−µ|1/2near µ.This was later conﬁrmed in electron tunneling experiments for amorphous alloys 3and granular metals 4.On the insulating side of the MIT charge transport occurs via inelastic electron tunneling hopping between states localized on the impurity sites with one-electron energies close to µ.Mott 5demonstrated that at low temperatures electrons seek accessible energy states by hopping distances beyond the localization length,lead-ing to a hopping conductivity σ(T )∼exp(−T 0/T )νwith T 0being a characteristic temperature depending on lo-calization length and with the hopping exponent ν=1/4for the non-interacting case in three dimensions.Efros and Shklovskii 6(ES)argued that the ground state of a system with long-range Coulomb interactions is stable with respect to one-particle excitations only if g (ε)in the vicinity of µhas the symmetric shapeg (ε)∼|ε−µ|D −1(1)with the universal exponent D −1depending only on the dimensionality D of the system.In particular,ES pre-dicted that in D =3g (ε)=3e 2 3(ε−µ)2,where χis the dielectric constant and e is the electron charge.Because g (ε)vanishes only at ε=µ,this is called a “soft”Coulomb correlation gap with a width ∆ε∼e 3(N 0/χ3)1/2,where N 0is the DOE far away from µ.The power law (1)gives 7a hopping exponent ν=D/(D +3)at low temperatures,so for three-dimensional system with long-range Coulomb interactions ν=1/2.The intriguing hypothesis about universality of (1)has stimulated further theoretical research,both analytical 8and numerical 9–13.To establish the hypothesis (1)Efros 14used the ground-state stability conditions for lo-calized electrons (LES)with respect to charge transferεj −εi −e 2localized on all the sites of a D-dimensional lattice andthe negative charge from k×N acceptors is uniformlysmeared over the lattice sites so that each site i has acharge e(n i−k),where n i=1if a donor on the site i isionized and n i=0if a donor is neutral.Disorder in thismodel is ensured by introducing randomly distributedone-site potentials.Monte Carlo simulations12on verylarge specimens of the lattice d-a model,however,havegiven rise to doubts about the universality of the g(ε)behavior.Another hint about possible non-universal behavior ofg(ε)has come from the intriguing and still not com-pletely unfolded problem whether the so called spin-glassphase does exist in the classical d-a model(see,e.g.Ref.18–20).Grannan and Yu18studied the classical three-dimensional d-a model with k=0.5but with the totalacceptor charge uniformly distributed over donor sitesas in the lattice d-a model.In this case,the classical d-amodel is equivalent to a model of Ising spins,localized onrandomly distributed sites,with pairwise Coulomb inter-actions,a model in which a transition into the spin-glassstate was found18to occur at non-zero temperature.Itwas then concluded that such a transition should existin all d-a models(with and without smearing of negativecharge,deﬁned on a lattice or on a continuous sample)aswell because of the Efros universality hypothesis.Voitaand Schreber20,however,have shown that the spin glasstransition does not exist in the lattice d−a model14.Besides,in recent work by one of us19it was unequivo-cally demonstrated that the ground state of the classicald-a model and that of the model studied in Ref.18arequalitatively diﬀerent.An analysis of histograms H[Qαβ]of the so called overlaps Qαβ=12 i=j n a(i)n a(j)2 k=l(1−n d(k))(1−n d(l))− i,k(1−n d(k))n a(i)r a−d ik +1r a−aij++1r d−dkl,(4)whereεa(i)is the one-electron excitation(OEE)energy for the acceptorsεa(i)≡δE(n a,n d)r a−aij− k1−n d(k)r a−aij≥0,(8)whereε1(0)a(i)denotesεa(i)if n(i)=1(0).The stabilityconditions with respect to the other three manners of thecharge transfer are obtainable in the similar manner.The relation(8)implies thatεa’s for the neutral ac-ceptors are,in general,larger thanεa’s for the chargedacceptors.Furthermore,the pair of neutral and chargedacceptors might be located on any distance and thereforein the thermodynamic limit the chemical potential for theacceptors(i.e.an energy level which separates the ener-gies of the neutral and charged acceptors)is determinedasµa=min{ε0a(i)}=max{ε1a(i)}.(9)Alike,there exist the chemical potentialµd for the donorsas well.Moreover,the stability relations with respect tothe ionization and recombination lead toµa=µd=µ.(10)Despite theﬁnite size of samples we investigated,the re-lation(10)with the chemical potentials calculated from(9)is valid within the limits of accuracy of our calcula-tions(see Sect.III).A macroscopic state of the sample R is characterizedby degree of acceptor ionizationC a(R)=1N iδ(ε−εa(i))(12)and by the corresponding DOE g d(εd,R)for the donors.Note,that for theﬁnite samples(especially for the rel-ative small systems we were able to investigate)C a(R),g a(εa,R)and g d(εd,R)depend essentially on the partic-ular implementation R of the spatial distributions of thedonor and acceptor sites(if a sample would be big enoughall quantities would be self-averaging).Therefore,inorder to obtain reliable results,one has to work withthe quantities C a≡ C a(R) ,g a(ε)≡ g a(εa,R) andg d(ε)≡ g d(εd,R) ,where ... denotes the average overa number of R’s.Note,that the values g a(d)(εa(d),R)dεobtained for independent R’s are scattered according tothe Gaussian distribution with the mean g a(d)(ε)dεandthe standard deviationB.Acceptor-donor symmetryLet us rewrite the energy(3)in terms of the OEE energies(5)E(n a,n d)=12 kεd(k)(1−n d(k))−∆2.(20) Thus,the Fermi energy of our model system in the ther-modynamic limit is a fundamental quantity depending only on the energy of charge transfer from an acceptor to a donor.III.METHODA.Algorithm of energy minimizationWe start from a random allocation of N donor andN acceptor sites in the continuous D-dimensional sys-tem(generate a sample R)with the density n=1,sothat the system has a linear size L=N1/D and then charge randomly chosen C a×N both donors and accep-tors(usually we take C a=0.7),i.e.generate an initial microscopic state(IMS)(n a,n d)of the sample R.Fur-ther,we search for such microscopic state(n0a,n0d)which obeys the stability conditions(8)with respect to the four mechanisms of the charge transfer allowed in our model. We used an algorithm which is an extension of the al-gorithm proposed in Ref.9to the case∆=∞.The algorithm consists of the three main steps.In order to save computer time,ﬁrst,we look for pairs a0−a−(d0−d+)for which the“crude”stability relation ∆ε≡ε0a(d)−ε1a(d)>0is violated.Then,the energy of the system is decreased by transferring an electron be-tween such pair of sites for which∆εhas its minimal non-positive value.This process is repeated until a state is reached,in which∆ε>0for all possible a0−a−and d0−d+pairs(step I).In the similar manner,we further minimize the energy of the system with respect to the “true”stability relations(8)for the charge transfer be-tween the a0−a−and d0−d+pairs(step II).And,ﬁnally, in the step III we diminish the energy of the system with respect to the stability relations for ionization and recom-bination processes.Since ionization and recombination processes change the degree C a of the system ionization, each time after one of these processes takes place dur-ing calculations,we go back to the step II.Repeating the steps II and III,weﬁnally arrive at a microscopic state(n0a,n0d)for which all four stability conditions are fulﬁlled.We name the procedure(n a,n d)→(n0a,n0d)via above steps I,II and III as“a single descent”.It should be noted,however,that the state(n0a,n0d)isnot necessarily the ground state of the sample R since for the ground state,in general,not only the simplest re-lations(8)with only pairs of sites included,but the more complicated relations involving quadruplets,sextets,etc. of sites have to be fulﬁlled.Therefore,the state(n0a,n0d) (after Ref.9)hereafter will be referred to as the pseudo-ground state(PGS)of the sample R.Then,two questions naturally arise:How close the PGS and the ground state of the given sample are and how this may inﬂuence the output of our calculations?In order to answer theﬁrst question,we calculate and analyze the histograms H for the so-called overlapsQαβ=1with diﬀerent IMS(n a,n d).If two PGS’s are identical then Qαβ=1.We calculated for the D=2systemwith N=500at∆=0the mean Q(R)= Qαβ αβfor the sequence of100PGS’s generated by single descentsfrom the diﬀerent IMS of the same sample R.We further acquire Q(R)for100diﬀerent samples and obtain that the mean¯Q≡ Q(R) R=0.96.It means that in PGS generated by the single descent only20acceptors out of 500are,in average,in the“wrong”states compared to those in the true ground state of the sample.In order to evaluate how the“erroneousness”of PGS inﬂuences the outcome of our calculations we perform an analysis of ground states obtained by means of the so called multirank descents.Descent of rank m comprises of a consequence of the single descents on the same sam-ple with diﬀerent IMS when calculations are stopped af-ter the lowest observed PGS energy repeats m times.We calculate¯Q(all other parameters were the same as de-scribed in the previous paragraph,where actually the case m=0was explored)for descents with diﬀerent ranks m=5,10,15and found that,for instance,for m=15(which implies drastic increase in the compu-tation time)¯Q=0.990.g a(ε)and g d(ε)obtained from the PGS’s generated by means of the single descents and by means of descents with m=10,say,do not diﬀer within the limits of statistical errors.So,we conclude, that reliable results can be obtained by means of single descents already,thereby saving a lot of computer time and resources.B.Finite-size eﬀectsDue to constraints in computer resources,the largest samples,we were able to deal with,comprise up to N= 2000donor and N=2000acceptor sites(L∼45for D=2and L∼12for D=3).Such relative small sizes of the samples investigated might inﬂuence the outcome of calculations.Detailed analysis ofﬁnite size eﬀects on the results obtained will be presented in Section IV and here we want to make two remarks about inherentﬁnite size eﬀects in the model system considered.First,as follows from(8),the energiesε0a for the neutral acceptors andε1a for the charged ones inﬁnite samples at T=0cannot be further away than(L×√D)−1(22) Of course,the same holds for donors as well.The rela-tion(22)gives the estimation how close toµdata on the energy spectrum are,in principle,obtainable from the calculations onﬁnite samples.Secondly,as follows from(5)the energiesεa andεd for theﬁnite samples are sensitive to the location of the donor and acceptor sites.Therefore,the Fermi energyµforﬁnite samples does diﬀer,in general,from sample to0.00020.0004ε−µg(ε−µ)FIG.1.Density of one electron excitations g a(ε−µ) in the vicinity of the Fermi energyµobtained for the two-dimensional model(3)with N=1500at∆=0(cir-cles),2(squares),4(diamonds)and10(triangles).Data points presented in theﬁgure are calculated as the average over10.000(∆=0),5.100(∆=2),3.700(∆=4)and2.200 (∆=10)diﬀerent samples.Insert shows double logarithmic plot of g a(ε−µ)forε>µin the regionε−µ 0.05.sample.A straightforward averaging of g(ε)over diﬀer-ent samples might thus lead to a distortion of the g(ε) shape especially in the region where the Coulomb gap is observed.In order to avoid this undesired eﬀect,we used a trickﬁrst proposed in Ref.9.During accumulation of the results for g(ε)we added together g(ε)for the same values ofε−µ(R)rather than for the same values ofε. Hereµ(R)denotes the Fermi energy for aﬁnite sample R calculated asµ(R)=10.010.030.10.310−710−610−510−410−710−610−510−40.010.030.10.3(a)(c)(b)(d)∆ = 0∆ = 0∆ = 4∆ = 4123123123123|ε − µ|g(ε−µ)FIG.2.Density of one electron excitations g a (ε−µ)for ε>µ(a,b)and ε<µ(c,d)obtained for the two-dimensional model (3)at ∆=0(a,c)and 4(b,d),with N =500(curves numbered 1),1000(2)and 1500(3).The dashed lines are least-squares power-law ﬁts g a (ε−µ)∼|ε−µ|γwith γ=0.9(a),0.55(b),0.98(c)and 0.78.Data presented in the ﬁgure are calculated as the average over 10.000diﬀerent samples (except the case N =1500and ∆=4with the average over 3700diﬀerent samples).width of the Coulomb gap ∆εand the energy scale in our model E 0=e 2n 1/D /χare of the same order of mag-nitude.Fig.1shows g a (ε−µ)in the vicinity of the Fermi energy µobtained for the two-dimensional samples with N =1000and various values of ∆.As it is seen,g a (ε−µ)depends considerably on ∆except for a narrow window |ε−µ| 0.05,where all data merge into some “uni-versal”curve symmetric with respect to µ,the curve which can be anticipated to obey the Efros universal-ity hypothesis (1).However,a double-logarithmic plot of the “universal”g a (ε−µ)(insert in the Fig.1),reveals that the behavior of g a (ε−µ)in the “universality”region is not even a power law.The width of this “universality”region is comparable to the width of the region where g a (ε−µ)=0due to the ﬁnite size eﬀects (for the data presented in Fig.1relation (22)gives |ε−µ|<0.011),so it is plausible to suggest that the “universal”behavior of g a (ε−µ)is governed by the ﬁnite-size eﬀects.This is clearly demonstrated in Fig.2where g a (ε−µ)are shown for several sizes of the samples investigated.The εwindow where ﬁnite size eﬀects are severe,shrinks considerably with increasing N for all values of ∆we investigated.For instance,g a (ε−µ)for N =500and N =1000at ∆=0(see Fig.2a,c)merge when |ε−µ| 0.2while corresponding curves for N =1000and N =1500are indistinguishable already at |ε−µ| 0.1.The statistical noise observed for the curves in Fig.2is quite small even close to µand hence,the inﬂuence of insuﬃcient large statistics on the results obtained is ex-0246810∆0.20.40.60.81γFIG.3.The exponent γof the power law g a (ε−µ)∼|ε−µ|γas a function of the charge-transfer energy ∆.The data are obtained from least-squares ﬁts of g a (ε−µ)for the two-dimensional model (3)with N =1500within the region 0.2 |ε−µ| 0.7.Circles represent the positive values of ε−µwhile diamonds stand for the negative values of ε−µ.Lines are guides to the eye.cluded.Note,that the “universal”behavior of g (ε)in the vicinity of µobtained for the classical d −a model (see Fig.3in Ref.11)is most likely due to the ﬁnite size eﬀects as well.In the region |ε−µ| 0.2,where the curves for all N collapse into a single curve (and where we believe the thermodynamic limit is reached),the behavior of g a (ε−µ)is described by a power law g a (ε−µ)∼|ε−µ|γ.The deviation from the power-law observed far away from µ(|ε−µ| 0.7)is due to the boundaries of the Coulomb gap which,as was mentioned above,are ∼1in units of E 0.One can see from a comparison of the data shown in Fig.2for diﬀerent ∆,that the exponent γdepends considerably on ∆.Furthermore,values of γin the region ε−µ>0and those in the region ε−µ<0diﬀer as well with this diﬀerence increasing with increasing ∆.The data for γobtained for the two-dimensional MCDAM are summarized in Fig.3where a signiﬁcant deviation of γfrom the value D −1predicted by the hypothesis (1)is observed at all values of ∆investigated except for the case ∆=0when γ≈1within the limits of statistical accuracy.Note,that the deviation of γfrom its predicted value grows monotonically with increasing ∆.At ∆=10where the features of the MCDAM are expected to be nearly the same as those of the classical d −a model with all the acceptors being ionized (indeed,the degree of the acceptor ionization C a ∼0.9for the two-dimensional MCDAM at ∆=10,see Fig.6below)the deviation from the Efros exponent is very large.The main results for g a (ε−µ)obtained for the three-ε−µ00.00040.0008g(ε−µ)FIG.4.Density of one electron excitations g a (ε−µ)in the vicinity of the Fermi energy µobtained for the three-dimensional model (3)with N =1000at ∆=0(cir-cles),2(squares),4(diamonds)and 10(triangles).Data points presented in the ﬁgure are calculated as the av-erage over 10.000diﬀerent samples.Inserts show dou-ble-logarithmic plots of g a (ε−µ)at ∆=2,for N =500(curves numbered 1),1000(2)and 2000(3),in the regions ε>µ(a)and for ε<µ(b).The dashed lines in the in-serts are least-squares power-law ﬁts g a (ε−µ)∼|ε−µ|γwith γ=1.16(a),1.29(b),0246810∆0.511.5γFIG. 5.The exponent γof the power lawg a (ε−µ)∼|ε−µ|γas a function of the charge-transfer energy ∆.The data are obtained from least-squares ﬁts of g a (ε−µ)for the three-dimensional model (3)with N =1000within the region 0.4 |ε−µ| 0.8.Circles represent the positive values of ε−µwhile diamonds stand for the negative values of ε−µ.Lines are guides to the eye.0246810∆0.70.80.91C aFIG.6.The degree of acceptor ionization C a as a function of the charge-transfer energy ∆.The data are obtained for the model (3)in two (circles)and three (diamonds)dimensions with N =500as an average over 1000diﬀerent samples.The solid lines are third-degree polynomial ﬁts.dimensional MCDAM are summarized in Figs.4and 5.It is seen,that the behavior of g a (ε−µ)in three di-mensions does not diﬀer qualitatively from the behavior of g a (ε−µ)in two dimensions.Some quantitative dif-ferences observed arise from the fact that at given N (the parameter which determines the amount of com-puter memory needed for the calculations)the linear size of a two-dimensional sample with a given density of sites is larger than that of a three-dimensional sample with the same density of sites and thereby,the ﬁnite size eﬀects for three-dimensional samples with given N are more pro-nounced compared to those for the two-dimensional sam-ples with the same N .For example,the lower boundary of the region where g a (ε−µ)can be described by the power law |ε−µ|γshifts towards larger |ε−µ| 0.4values (see inserts in Fig.4).Remarkably,the exponent γdoes not reach the value D −1predicted by the uni-versality hypothesis (1)even at ∆=0(Fig.5).Unlike g a (ε−µ)in the vicinity of the Coulomb gap,the density of ionized acceptors C a (11)describes the state of the entire sample and therefore reaches the thermody-namic limit much faster than g a (ε−µ).This allows us to obtain quite accurate results for C a from data on a relatively small amount of samples with N =500only.Fig.6shows the variations of C a with ∆both for two and three dimensions.In three dimensions almost all accep-tors become ionized (C a ∼1)rather soon while for two dimensions even for the largest ∆investigated around 10%of the acceptors remain neutral.So,one can say,that the three-dimensional MCDAM at ∆ 7reduces already to the classical d −a model.It is known that the classicalTABLE I.The means ¯µand standard deviations ∆µoftheFermienergy calculatedforthethree-dimensionalmodel(3)withN =1000and various ∆.∆¯µ∆µ2ε.I.e.,sites with energies ε1i ∈[−ε,0]cannot be inside a D-dimensional sphere of radius R sp =12εDεg (ε′)dε′(24)where S (D )is the volume of a D-dimensional sphere with the radius equal to unity.Since V sp cannot exceed the total volume V of a sample (V =N at n =1)we arrive at the inequalityεg (ε′)dε′≤(2ε)DS (D )|ε|D −1(26)The universality hypothesis (1)then is a limit case of (26).The density of sites with energies ε1i ∈[−ε,0]indeed decreases when ε→0,so the assumption (24)for the spheres with ﬁnite radii seems to be plausible.However,simultaneously R sp →∞and consequently the plausibility of the assumption (24)and thereby of the hypothesis (1)becomes questionable.And ﬁnally,the universality hypothesis (1)can be also obtained as the asymptotic behavior of a non-linear in-tegral equation for g (ε)as ε→0,the equation which,in turn,is heuristically obtained from the stability condition (2).The derivation of this integral equation (given,for example,in Ref.17)is based on the implicit assump-tion that the sites with charged donors are randomly distributed in space according to the Poisson statistics.However,it was unequivocally demonstrated in computer studies of the Coulomb gap 11that charged donor sites with energies close to µtend to form clusters (Ref.11,Fig.6).We conclude that g a (ε−µ)in the region of the Coulomb gap in model (3)has a power law behavior for all energies down to µand that the universality hypothesis of Efros (1)is questionable.Note,that our results are in contra-diction not only to the universality hypothesis (1),but to the inequality (26)as well.Up to now,all exponents found are in good agreement with this inequality.E.g.in Ref.12specimens of 40000and 125000sites for two-and three-dimensional samples were investigated in the Efros’lattice model 14and the power law g a (ε−µ)∼|ε−µ|γwas found with γ=1.2±0.1and γ=2.6±0.2for two and three dimensions,respectively.The main conclusionTABLE II.Some donor–acceptor pairs for which the diﬀerence between the donor and acceptor energy levels does not exceed 10meV.E g,E v and E c are,respectively,the energy gap,the top of the valence band and the bottom of the conductivity band. If the solubilities of both donor and acceptor are known,the parameter E0is calculated using the data for the less soluble of the pair.Donor Acceptor Solubility,cm−3E0,meV E j,meV∆,meVmin max min maxGe(E g=740meV,χ=15.9)S no reliable data E c−2964 Ni4.8×10158×1015 1.5 1.8E c−300ACKNOWLEDGEMENTSThis research was supported by The Swedish Natural Science Council and by The Swedish Royal Academy of Sciences.。

One-Particle Excitation of the Two-Dimensional Hubbard Model

KEYWORDS: Hubbard model, two dimensions, t-matrix approximation, forward scattering
The nature of the low-energy excitation of the two-dimensional system is of great interest recently. The Fermi-liquid picture was considered to be valid from diagramatic studies,1) while it was suggested by Anderson that the anomalous behavior of the forward scattering phase shift leads to the renormalization factor Z = 0, i.e., the breakdown of the Fermi-liquid.2, 3) This remarkable suggestion has attracted much interest,4, 5) and several calculations of the self-energy of the twodimensional Hubbard model have been carried out based on the t-matrix approximation,6, 7, 8, 9) in which the self-energy is approximated by the summation of ladder diagrams of the particleparticle process. In these calculations, however, only the imaginary part of the self-energy has been considered, and the real part has not been studied in detail. In this paper, we will calculate explicitly the real part of the self-energy of the two-dimensional Hubbard model by the t-matrix approximation, and show that the singularity of the t-matrix in the forward scattering region gives rise to an anomalous term to the real part of the self-energy, which leads to the renormalization factor Z = 0. This result is in accordance with the claim by Anderson.

Fermi liquids and non--Fermi liquids

3 Transport properties and the metal–insulator transition . . . . . . . . 4 Spin–charge separation . . . . . . . . . . . . . . . . . . . . . . . . . . VII Conclusion and outlook
I. INTRODUCTION
51 53 54
Much of the current understanding of solid state physics is based on a picture of non– interacting electrons. This is clearly true at the elementary level, but in fact extends to many areas of current research, examples being the physics of disordered systems or mesoscopic physics. The most outstanding examples where the non–interacting electron picture fails are provided by electronic phase transitions like superconductivity or magnetism. However, more generally one clearly has to understand why the non–interacting approximation is successful, for example in understanding the physics of metals where one has a rather dense gas (or liquid) of electrons which certainly interact via their mutual Coulomb repulsion. A ﬁrst answer is provided by Landau’s theory of Fermi liquids [1–3] which, starting from the (reasonable but theoretically unproven) hypothesis of the existence of quasiparticles shows that the properties of an interacting system of fermions are qualitatively similar to that of a non–interacting system. A brief outline of Landau’s theory in its most elementary aspects will be given in the following section, and a re–interpretation as a ﬁxed point of a renormalization group will be discussed in sec.III. A natural question to ask is whether Fermi liquid like behavior is universal in many– electron systems. The by far best studied example showing that this is not the case is the one–dimensional interacting electron gas. Starting with the early work of Mattis and Lieb [4], of Bychkov et al. [5] and of others is has become quite that in one dimension Landau type quasiparticles do not exist. The unusual one–dimensional behavior has now received the name “Luttinger liquid”. This is still a very active area of research, and the rest of these notes is devoted to the discussion of various aspects of the physics of one–dimensional interacting fermions. The initial plan for these lectures was considerably wider in scope. It was in particular considered to include a discussion of the Kondo eﬀect and its non–Fermi–liquid derivatives, as well as possibly current theories of strongly correlated fermions in dimension larger than one. This plan however turned out to be overly ambitious and it was decided to limit the scope to the current subjects, allowing for suﬃciently detailed lectures. Beyond this limitation on the scope of the lectures, length restrictions on the lecture notes imposed further cuts. In view of the fact that there is a considerable and easily accessible literature on Fermi liquid theory, it seemed best to remain at a rather elementary level at this point and to retain suﬃcient space for the discussion of the more unusual one–dimensional case. It is hoped that the references, especially in the next and the last section will allow the interested reader to ﬁnd sources for further study.

Quantum Deformations of the One-Dimensional Hubbard Model

♠
Abstract The centrally extended superalgebra psu(2|2) R3 was shown to play an important role for the integrable structures of the one-dimensional Hubbard model and of the planar AdS/CFT correspondence. Here we consider its quantum deformation Uq (psu(2|2) R3 ) and derive the fundamental R-matrix. From the latter we deduce an integrable spin chain Hamiltonian with three independent parameters and the corresponding Bethe equations to describe the spectrum on periodic chains. We relate our Hamiltonian to a two-parametric Hamiltonian proposed by Alcaraz and Bariev which can be considered a quantum deformation of the one-dimensional Hubbard model.
† ∼ c† 1 c2 |◦ .
(1.1)
The middle two states are considered fermionic while the outer two states are overall bosonic. This model is exciting because it shows some characteristics of superconductivity, and therefore it is very desirable to understand its foundations well. Integrability was established by Lieb and Wu who also solved the spectrum by means of the Bethe ansatz [3]. An R-matrix which encodes the integrable structure was later found by Shastry [4]. On the one hand, the R-matrix is the foundation for much of the integrable machinery, such as the algebraic Bethe ansatz [5, 6]. On the other hand, this particular R-matrix is rather exceptional because unlike most other known R-matrices it cannot be written as a function of the diﬀerence of two spectral parameters. Altogether, the 1

erlangen纲领

A COMPARATIVE REVIEW OF RECENT RESEARCHES INGEOMETRY.1(PROGRAMME ON ENTERING THE PHILOSOPHICAL FACULTY AND THE SENATE OFTHE UNIVERSITY OF ERLANGEN IN1872.)BY PROF.FELIX KLEIN.Prefatory Note by the Author.-My1872Programme,appearing as a separate publication (Erlangen,A.Deichert),had but a limited circulation atﬁrst.With this I could be satisﬁed more easily,as the views developed in the Programme could not be expected atﬁrst to receive much attention.But now that the general development of mathematics has taken,in the meanwhile, the direction corresponding precisely to these views,and particularly since Lie has begun the publication in extended form of his Theorie der Transformationsgruppen(Liepzig,Teubner,vol.I. 1888,vol.II.1890),it seems proper to give a wider circulation to the expositions in my Programme. An Italian translation by M.Gina Fano was recently published in the Annali di Matematica,ser.2, vol.17.A kind reception for the English translation,for which I am much indebted to Mr.Haskell, is likewise desired.The translation is an absolutely literal one;in the two or three places where a few words are changed,the new phrases are enclosed in square brackets[].In the same way are indicated a number of additional footnotes which it seemed desirable to append,most of them having already appeared in the Italian translation.-F.KLEIN.1.Translated by Dr.M.W.HASKELL,Assistant Professor of Mathematics in the University of California.Publi-shed in Bull.New York Math.Soc.2,(1892-1893),215-249.Among the advances of the lastﬁfty years in theﬁeld of geometry,the development of projective geometry2occupies theﬁrst place.Although it seemed atﬁrst as if the so-called metrical relations were not accessible to this treatment,as they do not remain unchanged by projection,we have nevertheless learned recently to regard them also from the projective point of view,so that the projective method now embraces the whole of geometry.But metrical properties are then to be regarded no longer as characteristics of the geometricalﬁgures per se,but as their relations to a fundamental conﬁguration,the imaginary circle at inﬁnity common to all spheres.When we compare the conception of geometricalﬁgures gradually obtained in this way with the notions of ordinary(elementary)geometry,we are led to look for a general principle in accor-dance with which the development of both methods has been possible.This question seems the more important as,beside the elementary and the projective geometry,are arrayed a series of other methods,which albeit they are less developed,must be allowed the same right to an individual exis-tence.Such are the geometry of reciprocal radii vectores,the geometry of rational transformations, etc.,which will be mentioned and described further on.In undertaking in the following pages to establish such a principle,we shall hardly develop an essentially new idea,but rather formulate clearly what has already been more or less deﬁnitely conceived by many others.But it has seemed the more justiﬁable to publish connective observations of this kind,because geometry,which is after all one in substance,has been only too much broken up in the course of its recent rapid development into a series of almost distinct theories3,which are advancing in comparative independence of each other.At the same time I was inﬂuenced especially by a wish to present certain methods and views that have been developed in recent investigation by Lie and myself.Our respective investigations,diﬀerent as has been the nature of the subjects treated,have led to the same generalized conception here presented;so that it has become a sort of necessity to thoroughly discuss this view and on this basis to characterize the contents and general scope of those investigations.Though we have spoken so far only of geometrical investigations,we will include investigations on manifoldnesses of any number of dimensions4,which have been developed from geometry by making abstraction from the geometric image,which is not essential for purely mathematical in-vestigations5.In the investigation of manifoldnesses the same diﬀerent types occur as in geometry; and,as in geometry,the problem is to bring out what is common and what is distinctive in inves-tigations undertaken independently of each other.Abstractly speaking,it would in what follows be suﬃcient to speak throughout of manifoldnesses of n dimensions simply;but it will render the exposition simpler and more intelligible to make use of the more familiar space-perceptions.In proceeding from the consideration of geometric objects and developing the general ideas by using these as an example,we follow the path which our science has taken in its development and which it is generally best to pursue in its presentation.A preliminary exposition of the contents of the following pages is here scarcely possible,as it can hardly be presented in a more concise form6;the headings of the sections will indicate the general course of thought.2.See Note I of the appendix.3.See Note II.4.See Note IV.5.See Note III.6.This very conciseness is a defect in the following presentation which I fear will render the understanding of it essentially more diﬃcult.But the diﬃculty could hardly be removed except by a very much fuller exposition,in which the separate theories,here only touched upon,would have been developed at length.At the end I have added a series of notes,in which I have either developed further single points, wherever the general exposition of the text would seem to demand it,or have tried to deﬁne with reference to related points of view the abstract mathematical one predominant in the observationsof the text.1GROUPS OF SPACE-TRANSFORMATIONS.PRINCIPAL GROUP.FORMULATION OF A GENERAL PROBLEM.The most essential idea required in the following discussion is that of a group of space-transformations.The combination of any number of tranformations of space7is always equivalent to a single transformation.If now a given system of transformations has the property that any transformation obtained by combining any tranformations of the system belongs to that system,it shall be calleda group of transformations8.An example of a group of transformations is aﬀorded by the totality of motions,every motion being regarded as an operation performed on the whole of space.A group contained in this groupis formed,say,by the rotations about one point9.On the other hand,a group containing the group of motions is presented by the totality of the collineations.But the totality of the dualistic transformations does not form a group;for the combination of two dualistic transformations is equivalent to a collineation.A group is,however,formed by adding the totality of the dualistic tothat of the collinear transformations10.Now there are space-transformations by which the geometric properties of conﬁgurations in space remain entirely unchanged.For geometric properties are,from their very idea,independentof the position occupied in space by the conﬁguration in question,of its absolute magnitude,andﬁnally of the sense11in which its parts are arranged.The properties of a conﬁguration remain therefore unchanged by any motions of space,by transformation into similar conﬁgurations,by transformation into symmetrical conﬁgurations with regard to a plane(reﬂection),as well as byany combination of these transformations.The totality of all these transformations we designateas the principal group12of space-transformations;geometric properties are not changed by the transformations of the principal group.And,conversely,geometric properties are characterized by7.We always regard the totality of conﬁgurations in space as simultaneously aﬀected by the transformations,and speak therefore of transformations of space.The transformations may introduce other elements in place of points,like dualistic transformations,for instance;there is no distinction in the text in this regard.8.[This deﬁnition is not quite complete,for it has been tacitly assumed that the groups mentioned always includethe inverse of every operation they contain;but,when the number of operations is inﬁnite,this is by no means a necessary consequence of the group idea,and this assumption of ours should therefore be explicitly added to thedeﬁnition of this idea given in the text.]The ideas,as well as the notation,are taken from the theory of substitutions,with the diﬀerence merely that there instead of the transformations of a continuous region the permutations of aﬁnite number of discrete quantities are considered.9.Camille Jordan has formed all the groups contained in the general group of motions:Sur les groupes de mouve-ments,Annali di Matematica,vol.2.10.It is not at all necessary for the transformations of a group to form a continuous succession,although the groupsto be mentioned in the text will indeed always have that property.For example,a group is formed by theﬁnite series of motions which superpose a regular body upon itself,or by the inﬁnite but discrete series which superpose asine-curve upon itself.11.By“sense”is to be understood that peculiarity of the arrangement of the parts of aﬁgure which distinguishesit from the symmetricalﬁgure(the reﬂected image).Thus,for example,a right-handed and a left-handed helix areof opposite“sense”.12.The fact that these tranformations form a group results from their very idea.their remaining invariant under the transformations of the principal group.For,if we regard space for the moment as immovable,etc.,as a rigid manifoldness,then everyﬁgure has an individual character;of all the properties possessed by it as an individual,only the properly geometric ones are preserved in the transformations of the principal group.The idea,here formulated somewhat indeﬁnitely,will be brought out more clearly in the course of the exposition.Let us now dispose with the concrete conception of space,which for the mathematician is not essential,and regard it only as a manifoldness of n dimensions,that is to say,of three dimensions, if we hold to the usual idea of the point as space element.By analogy with the transformations of space we speak of transformations of the manifoldness;they also form groups.But there is no longer,as there is in space,one group distinguished above the rest by its signiﬁcation;each group is of equal importance with every other.As a generalization of geometry arises then the following comprehensive problem:Given a manifoldness and a group of transformations of the same;to investigate the conﬁ-gurations belonging to the manifoldness with regard to such properties as are not altered by the transformations of the group.To make use of a modern form of expression,which to be sure is ordinarily used only with reference to a particular group,the group of all the linear transformation,the problem might be stated as follows:Given a manifoldness and a group of transformations of the same;to develop the theory of invariants relating to that group.This is the general problem,and it comprehends not alone ordinary geometry,but also and in particular the more recent geometrical theories which we propose to discuss,and the diﬀerent methods of treating manifoldnesses of n dimensions.Particular stress is to laid upon the fact that the choice of the group of transformations to be adjoined is quite arbitrary,and that consequently all the methods of treatment satisfying our general condition are in this sense of equal value.2GROUPS OF TRANSFORMATIONS,ONE OF WHICH INCLUDES THE OTHER, ARE SUCCESIVELY ADJOINED.THE DIFFERENT TYPES OF GEOMETRICAL INVESTIGATION AND THEIR RELATION TO EACH OTHER.As the geometrical properties of conﬁgurations in space remain unaltered under all the trans-formations of the principal group,it is by the nature of the question absurd to inquire for such properties as would remain unaltered under only a part of those transformations.This inquiry be-comes justiﬁed,however,as soon as we investigate the conﬁgurations of space in their relation to elements regarded asﬁxed.Let us,for instance,consider the conﬁgurations of space with reference to one particular point,as in spherical trigonometry.The problem then is to develop the properties remaining invariant under the transformations of the principal group,not for the conﬁgurations taken independently,but for the system consisting of these conﬁgurations together with the given point.But we can state this problem in this other form:to examine conﬁgurations in space with regard to such properties as remain unchanged by those transformations of the principal group which can still take place when the point is keptﬁxed.In other words,it is exactly the same thing whether we investigate the conﬁgurations of space taken in connection with the given point from the point of view of the principal group or whether,without any such connection,we replace the principal group by that partial group whose transformations leave the point in question unchanged.This is a principle which we shall frequently apply;we will therefore at once formulate it generally,as follows:Given a manifoldness and a group of transformations applying to it.Let it be proposed to examine the conﬁgurations contained in the manifoldness with reference to a given conﬁguration. We may,then,either add the given conﬁguration to the system,and then we have to investigate the properties of the extended system from the point of view of the given group,or we may leave the system unextended,limiting the transformations to be employed to such transformations of the given group as leave the given conﬁguration unchanged.(These transformations necessarily form a group by themselves.)Let us now consider the converse of the problem proposed at the beginning of this section.This is intelligible from the outset.We inquire what properties of the conﬁgurations of space remain unaltered by a group of transformations which contains the principal group as a part of itself. Every property found by an investigation of this kind is a geometric property of the conﬁguration itself;but the converse is not true.In the converse problem we must apply the principle just enunciated,the principal group being now the smaller.We have then:If the principal group be replaced by a more comprehensive group,a part only of the geometric properties remain unchanged.The remainder no longer appear as properties of the conﬁgurations of space by themselves,but as properties of the system formed by adding to them some particular conﬁguration.This latter is deﬁned,in so far as it is a deﬁnite13conﬁguration at all,by the following condition:The assumption that it isﬁxed must restrict us to those transformations of the given group which belong to the principal group.In this theorem is to be found the peculiarity of the recent geometrical methods to be discussed here,and their relation to the elementary method.What characterizes them is just this,that they base their investigations upon an extended group of space-transformations instead of upon the principal group.Their relation to each other is deﬁned,when one of the groups includes the other, by a corresponding theorem.The same is true of the various methods of treating manifoldnesses of n dimensions which we shall take up.We shall now consider the separate methods from this point of view,and this will aﬀord an opportunity to explain on concrete examples the theorems enunciated in a general form in this and the preceding sections.3PROJECTIVE GEOMETRY.Every space-transformation not belonging to the principal group can be used to transfer the properties of known conﬁgurations to new ones.Thus we apply the results of plane geometry to the geometry of surfaces that can be represented(abgebildet)upon a plane;in this way long before the origin of a true projective geometry the properties ofﬁgures derived by projection from a given ﬁgure were inferred from those of the givenﬁgure.But projective geometry only arose as it became customary to regard the originalﬁgure as essentially identical with all those deducible from it by projection,and to enunciate the properties transferred in the process of projection in such a way as to put in evidence their independence of the change due to the projection.By this process the group of all the projective transformations was made the basis of the theory in the sense of§1,and that is just what created the antithesis between projective and ordinary geometry.A course of development similar to the one here described can be regarded as possible in the case of every kind of space-transformation;we shall often refer to it again.It has gone on still further in two directions within the domain of projective geometry itself.On the one hand,the concep-13.Such a conﬁguration can be generated,for instance,by applying the transformations of the principal group to any arbitrary element which cannot be converted into itself by any transformation of the given group.tion was broadened by admitting the dualistic transformations into the group of the fundamental transformation.From the modern point of view two reciprocalﬁgures are not to be regarded as two distinctﬁgures,but as essentially one and the same.A further advance consisted in extending the fundamental group of collinear and dualistic transformations by the admission in each case of the imaginary transformations.This step requires that theﬁeld of true space-elements has previously been extended so as to include imaginary elements,-just exactly as the admission of dualistic transformations into the fundamental group requires the simultaneous introduction of point and line as space-elements.This is not the place to point out the utility of introducing imaginary ele-ments,by means of which alone we can attain an exact correspondence of the theory of space with the established system of algebraic operations.But,on the other hand,it must be remembered that the reason for introducing the imaginary elements is to be found in the consideration of al-gebraic operations and not in the group of projective and dualistic transformations.For,just as we can in the latter case limit ourselves to real transformations,since the real collineations and dualistic transformations form a group by themselves,so we can equally well introduce imaginary space-elements even when we are not employing the projective point of view,and indeed must do so in strictly algebraic investigations.How metric properties are to be regarded from the projective point of view is determined by the general theorem of the preceding section.Metrical properties are to be considered as projective relations to a fundamental conﬁguration,the circle at inﬁnity14,a conﬁguration having the property that it is transformed into itself only by those transformations of the projective group which belong at the same time to the principal group.The proposition thus broadly stated needs a material modiﬁcation owing to the limitation of the ordinary view taken of geometry as treating only of real space-elements(and allowing only real transformations).In order to conform to this point of view,it is necessary expressly to adjoin to the circle at inﬁnity the system of real space-elements (points);properties in the sense of elementary geometry are projectively either properties of the conﬁgurations by themselves,or relations to this system of the real elements,or to the circle at inﬁnity,orﬁnally to both.We might here make mention further of the way in which von Staudt in his“Geometrie der Lage”(N¨u rnberg,1847)develops projective geometry,-i.e.,that projective geometry which is based on the group containing all the real projective and dualistic transformations15.We know how,in his system,he selects from the ordinary matter of geometry only such features as are preserved in projective transformations.Should we desire to proceed to the consideration of metrical properties also,what we should have to do would be precisely to introduce these latter as relations to the circle at inﬁnity.The course of thought thus brought to completion is in so far of great importance for the present considerations,as a corresponding development of geometry is possible for every one of the methods we shall take up.4TRANSFER OF PROPERTIES BY REPRESENTATIOINS(ABBILDUNG).Before going further in the discussion of the geometrical methods which present themselves beside the elementary and the projective geometry,let us develop in a general form certain conside-14.This view is to be regarded as one of the most brilliant achievements of[the French school];for it is precisely what provides a sound foundation for that distinction between properties of position and metrical properties,which furnishes a most desirable starting-point for projective geometry.15.The extended horizon,which includes imaginary transformations,wasﬁrst used by von Staudt as the basis of his investigation in his later work,“Beitr¨a ge zur Geometrie der Lage”(N¨u rnberg,1856-60).rations which will continually recur in the course of the work,and for which a suﬃcient number of examples are already furnished by the subjects touched upon up to this point.The present section and the following one will be devoted to these discussions.Suppose a manifoldness A has been investigated with reference to a group B.If,by any trans-formation whatever,A be then converted into a second manifoldness A ,the group B of trans-formations,which transformed A into itself,will become a group B ,whose transformations are performed upon A .It is then a self-evident principle that the method of treating A with reference to B at once furnishes the method of treating A with reference to B ,i.e.,every property of a conﬁgu-ration contained in A obtained by means of the group B furnishes a property of the corresponding conﬁguration in A to be obtained by the group B .For example,let A be a straight line and B the∞3linear transformations which transform A into itself.The method of treating A is then just what modern algebra designates as the theory of binary forms.Now,we can establish a correspondence between the straight line and a conic section A in the same plane by projection from a point of the latter.The linear transformations B of the straight line into itself will then become,as can easily be shown,linear transformations B of the conic into itself,i.e.,the changes of the conic resulting from those linear transformations of the plane which transform the conic into itself.Now,by the principle stated in§216,the study of the geometry of the conic section is the same, whether the conic be regarded asﬁxed and only those linear transformations of the plane which transform the conic into itself be taken into account,or whether all the linear transformations of the plane be considered and the conic be allowed to vary too.The properties which we recognized in systems of points on the conic are accordingly projective properties in the ordinary bining this consideration with the result just deduced,we have,then:The theory of binary forms and the projective geometry of systems of points on a conic are one and the same,i.e.,to every proposition concerning binary forms corresponds a proposition concerning such systems of points,and vice versa.17Another suitable example to illustrate these considerations is the following.If a quadric surface be brought into correspondence with a plane by stereographic projection,the surface will have one fundamental point,-the centre of projection.In the plane there are two,-the projections of the generators passing through the centre of projection.It then follows directly:the linear transformations of the plane which leave the two fundamental points unaltered are converted by the representation(Abbildung)into linear transformations of the quadric itself,but only into those which leave the centre of projection unaltered.By linear transformations of the surface into itself are here meant the changes undergone by the surface when linear space-transformations are performed which transform the surface into itself.According to this,the projective investigation of a plane with reference to two of its points is identical with the projective investigation of a quadric surface with reference to one of its points.Now,if imaginary elements are also taken into account,the former is nothing else but the investigation of the plane from the point of view of elementary geometry. For the principal group of plane transformations comprises precisely those linear transformations which leave two points(the circular points at inﬁnity)unchanged.We obtain thenﬁnally: Elementary plane geometry and the projective investigation of a quadric surface with reference to one of its points are one and the same.16.The principle might be said to be applied here in a somewhat extended form.17.Instead of the plane conic we may equally well introduce a twisted cubic,or indeed a corresponding conﬁguration in an n-dimensional manifoldness.These examples may be multiplied at pleasure18;the two here developed were chosen because we shall have occasion to refer to them again.5ON THE ARBITRARINESS IN THE CHOICE OF THE SPACE-ELEMENT.HESSE’S PRINCIPLE OF TRANSFERENCE.LINE GEOMETRY.As element of the straight line,of the plane,of space,or of any manifoldness to be investigated, we may use instead of the point any conﬁguration contained in the manifoldness,-a group of points, a curve or surface19,etc.As there is nothing at all determined at the outset about the number of arbitrary parameters upon which these conﬁgurations shall depend,the number of dimensioins of our line,plane,space,etc.,may be anything we like,according to our choice of the element.But as long as we base our geometrical investigation upon the same group of transformations,the substance of the geometry remains unchanged.That is to say,every proposition resulting from one choice of the space-element will be a true proposition under any other assumption;but the arrangement and correlation of the propositions will be changed.The essential thing is,then,the group of transformations;the number of dimensions to be assigned to a manifoldness appears of secondary importance.The combination of this remark with the principle of the last section furnishes many interesting applications,some of which we will now develop,as these examples seem betterﬁtted to explain the meaning of the general theory than any lengthy exposition.Projective geometry on the straight line(the theory of binary forms)is,by the last section, equivalent to projective geometry on the conic.Let us now regard as element on the conic the point-pair instead of the point.Now,the totality of the point-pairs of the conic may be brought into correspondence with the totality of the straight lines in the plane,by letting every line correspond to that point-pair in which it intersects the conic.By this representation(Abbildung)the linear transformations of the conic into itself are converted into those linear transformations of the plane (regarded as made up of straight lines)which leave the conic unaltered.But whether we consider the group of the latter,or whether we base our investigation on the totality of the linear transformations of the plane,always adjoining the conic to the plane conﬁgurations under investigation,is by§2 one and the same thing.Uniting all these considerations,we have:The theory of binary forms and projective geometry of the plane with reference to a conic are identical.Finally,as projective geometry of the plane with reference to a conic,by reason of the equality of its group,coincides with that projective metrical geometry which in the plane can be based upon a conic20,we can also say:The theory of binary forms and general projective metrical geometry in the plane are one and the same.In the preceding consideration the conic in the plane might be replaced by the twisted cubic, etc.,but we will not carry this out further.The correlation here explained between the geometry of the plane,of space,or of a manifoldness of any number of dimensions is essentially identical with the principle of transference proposed by Hesse(Borchardt’s Journal,vol.66).18.For other examples,and particularly for the extension to higher dimensions of which those here presented are capable,let me refer to an article of mine:Ueber Liniengeometrie und metrische Geometrie(Mathematische Annalen, vol.5),and further to Lie’s investigations cited later.19.See Note III.20.See Note V.。

大连理工大学物理学院研究生培养方案

物理与光电工程学院研究培养方案大连理工大学2012年6月目录物理学一级学科培养方案 (3)博士研究生 (3)学术型硕士研究生 (13)光学工程一级学培养方案 (31)博士研究生 (31)学术型硕士研究生 (15)全日制专业学位硕士 (37)电子科学与技术一级学科培养方案 (44)博士研究生 (49)学术型硕士研究生 (49)大连理工大学博士研究生培养方案物理学一级学科（一级学科（专业）代码：0702授予理学博士学位）一、培养目标本学科专业培养能够从事理论物理学、等离子体物理学、凝聚态物理学、光学，原子分子物理，生物物理学、神经信息学方面的教学、科研及管理工作的高层次人才。

学位获得者应具备坚实的物理理论基础和系统深入的专门知识；能熟练地运用一门外语进行专业学术交流；具备严谨的治学态度和刻苦钻研精神；具备独立从事科学研究工作的能力，并能够在科学或专门技术上取得创造性的成果。

二、学科群、专业及研究方向简介本学科点包括国家重点学科：等离子体物理；辽宁省重点学科：理论物理,等离子体物理。

2003年获得物理学一级学科博士学位授予权，设有物理学一级学科博士后流动站；•等离子体物理1998年获得博士授予权, 2001年被评为“国家重点学科”,2007年再次被评为国家重点学科；理论物理1986年获得硕士授予权，2000年获得博士授予权，2008年获批辽宁省重点建设学科。

本学科群依托教育部三束材料改性重点实验室和辽宁省先进光电技术重点实验室，设有理论物理研究所等十余个研究机构，拥有先进大型研究设备近百台（套），在“十一五”期间，共承担了各类课题100余项，包括国家自然科学基金重点项目，面上项目，青年基金项目，国际重大合作项目；“973”重点基础研究计划（课题）、国家重大研究计划ITER专项（课题），02专项（课题），“863”科技攻关项目；国防项目及预研项目；教育部骨干教师和博士点基金项目；辽宁省自然科学基金项目以及企业委托项目。

λ ring

Chapter1λ-RingsThe deﬁnition of aλ-ring involves certain polynomials that arise in the theory of symmetric functions.We will,therefore,begin by discussing symmetric functions in§1.1.This material is fairly standard and can be found in most books on abstract algebra,e.g.,Chapter2in[Jacobson(1985)].The deﬁnition of aλ-ring and some of its basic properties are given in§1.2.In §1.3we discuss the freeλ-ring on one generator.This is used in§1.4to prove the Veriﬁcation Principle,which states that a natural operation onλ-rings is uniquely determined by its actions on sums of one-dimensional elements.The Splitting Prin-ciple is discussed in§1.5.This result allows one to write an n-dimensional element in aλ-ring as a sum of n one-dimensional elements.The materials on the freeλ-ring, the Veriﬁcation Principle,and the Splitting Principle originated in the paper[Atiyah and Tall(1969)]by Atiyah and Tall.Another reference is Knutson’s book[Knut-son(1973)].The original sources forλ-rings are[Berthelot(1971);Grothendieck (1971)].Throughout this book,we use the following standard notations:•Z,the integers,•Z(p),the p-local integers for a prime p,•Q,the rationals,•C,the complex numbers.Unless otherwise speciﬁed,by a ring we always mean an associative and commu-tative ring with a multiplicative identity1=0.Ring homomorphisms are required to preserve the multiplicative identity.An object R that satisﬁes all the ring ax-ioms except for the existence of a multiplicative identity is called a non-unital ring. Non-unital rings will be used in§3.3.1.1Symmetric FunctionsFix a ring R.Let R[x1,...,x n]be the polynomial ring over R in n independent variables x1,...,x n.If f∈R[x1,...,x n]andπis a permutation on{1,...,n},then12Lambda-Ringsπf is deﬁned as the polynomialπf=f(xπ(1),...,xπ(n)).A polynomial f∈R[x1,...,x n]is called a symmetric function ifπf=f for every permutationπon{1,...,n}.Example1.1.For1≤k≤n,let s k∈R[x1,...,x n]be the polynomials k= 1≤i1<···<i k≤n x i1x i2···x i k.For example,we haves1=x1+···+x n,s2=x1x2+x1x3+···+x n−1x n,s n=x1x2···x n.Each s k is a symmetric function,since s k is the coeﬃcient of t n−k innk=1(t−x k)=(t−x1)(t−x2)···(t−x n)=t n−s1t n−1+s2t n−2−···+(−1)n s n.(1.1)The polynomial s k is called the k th elementary symmetric function on x1,...,x n. We will sometimes write s k(x1,...,x n)for s k to avoid any ambiguity.Another way to deﬁne the elementary symmetric functions is by looking at their generating function:nk=1(1+x i t)=1+s1t+s2t2+···+s n t n.(1.2) Note thats k(x1,...,x n,0)=s k(x1,...,x n).Example1.2.For any positive integer r,the r th power sumx r1+···+x r nis a symmetric function.Example1.3.For any positive integers m and n,consider the polynomialg(t)=1≤i1<···<i m≤mn (1+x i1···x imt)Then the coeﬃcient of each t j in g is a symmetric function in the variables x1,...,x mn.A fundamental property of symmetric functions is that they are all generated by the elementary symmetric functions.λ-Rings3 Theorem1.4(The Fundamental Theorem of Symmetric Functions). The elementary symmetric functions s1,...,s n in R[x1,...,x n]are algebraically independent over R.Moreover,every symmetric function f in R[x1,...,x n]can be written uniquely as a polynomial with coeﬃcients in R in the elementary symmetric functions.Before giving the proof,we need the following terminology.The degree of a monomialx k11···x k n n(1.3)is k1+···+k n.A polynomial f∈R[x1,...,x n]is said to be homogeneous of degree m if every non-zero monomial in f has degree m.We order monomials lexicographically.In other words,we say thatx k11···x k n n is higher than x l11···x l n n(1.4)if there exists s∈{1,...,n}such that k i=l i for i<s and k s>l s.Example1.5.The elementary symmetric function s k=s k(x1,...,x n)is homo-geneous of degree k.The highest monomial in s k is x1x2···x k.In general,given non-negative integers l1,...,l n,the polynomial s l11···s l n n is homogeneous of degreel1+2l2+···+nl n=n k=1kl k.Its highest monomial isx l1+···+l n 1x l2+···+l n2···x l n n.Moreover,if s j11···s j n n has the same highest monomial as that of s l11···s l n n,then j n=l n from the exponents of x n.Therefore,j n−1=l n−1from the exponents of x n−1,and so forth.So j i=l i for i=1,...,n.Proof.[Proof of Theorem1.4]First we prove that every symmetric function f in R[x1,...,x n]can be written as a polynomial with coeﬃcients in R in the ele-mentary symmetric functions.It suﬃces to prove this when f is homogenous,say, of degree m.Suppose that x k11···x k n n is the highest monomial in f with non-zerocoeﬃcient.Since f is symmetric,it has a monomial of the form x k1π(1)···x k nπ(n)forany permutationπ.Since x k11···x k n n is the highest monomial in f,it follows that k i≥k i+1for i=1,...,n−1.From Example1.5,the polynomial s k1−k21s k2−k32···s k n n is homogeneous of degreek1+···+k n=m and has highest monomial x k11···x k n n.If the coeﬃcient of x k11···x k n n in f is r,then the polynomialf1=f−rs k1−k21s k2−k32···s k n nis homogeneous of degree m,whose highest monomial is less than x k11···x k n n.Now we apply the same procedure to f1,and so forth.The procedure has to stop after a4Lambda-Ringsﬁnite number of steps,since there are only ﬁnitely many monomials in R [x 1,...,x n ]of degree m .This shows that every symmetric function f ∈R [x 1,...,x n ]can be written as a polynomial in the elementary symmetric functions.Next we show that the elementary symmetric functions are algebraically inde-pendent over R .Indeed,if they are not algebraically independent,then there is a non-trivial relation r (d 1,...,d n )s d 11···s d n n =0in which every coeﬃcient r (d 1,...,d n )=0in R .Among the multi-indexes (d 1,...,d n ),choose the one for which s d 11···s d n n has the highest degree and the highest monomial,say,x k 11···x k n n .From Example 1.5again,this monomial only appears once.Thisis a contradiction to the algebraic independence of the variables x 1,...,x n .This proves that the elementary symmetric functions are algebraically independent.The uniqueness assertion in the Theorem now follows from the algebraic inde-pendence of the elementary symmetric functions. Example 1.6.The square sum can be expressed as x 21+···+x 2n =(x 1+···+x n )2−2i<jx i x j =s 21−2s 2.In general,there is a polynomial Q r with integer coeﬃcients such thatx r 1+···+x r n =Q r (s 1,...,s n ).These polynomials Q r will be used in Theorem 3.9to express Adams operations in terms of λ-operations.Example 1.7.In the polynomial 1≤i<j ≤4(1+x i x j t ),the coeﬃcient of t is s 2and the coeﬃcient of t 6is s 34.In general,there is a universal polynomial P n,m in nm variables with integer coeﬃcients such that the coeﬃcient of t n in g (t )= 1≤i 1<···<i m ≤nm(1+x i 1···x i m t )is P n,m (s 1,...,s nm ).These polynomials P n,m will appear in the deﬁnition of a λ-ring (Deﬁnition 1.10).For example,P n,1(s 1,...,s n )is the coeﬃcient of t n in 1≤i ≤n(1+x i t )=(1+x 1t )···(1+x n t ),which is x 1···x n =s n .In other words,we haveP n,1(s 1,...,s n )=s n .(1.5)Similarly,P 1,m (s 1,...,s m )is the coeﬃcient of t in 1≤i 1<···<i m ≤m (1+x i 1···x i m t )=1+(x 1···x m )t.So we haveP 1,m (s 1,...,s m )=s m .(1.6)λ-Rings51.1.1Symmetric functions in two sets of variablesWe will need a slight generalization of Theorem1.4.Let y1,...,y n be another set of variables,and letσ1,...,σn be their elementary symmetric functions.Let f be an element in the polynomial ring R[x1,...,x n;y1,...,y n].We say that f is a symmetric function iff(x1,...,x n;y1,...,y n)=f(xπ(1),...,xπ(n);yτ(1),...,yτ(n))for every pair of permutationsπandτon{1,...,n}.Theorem1.8.Every symmetric function f in R[x1,...,x n;y1,...,y n]can be writ-ten uniquely as a polynomial with coeﬃcients in R in the elementary symmetric functions s1,...,s n andσ1,...,σn.e Theorem1.4twice.The details are left to the reader as an easy exercise. Example1.9.The coeﬃcient of each t k inh(t)=ni,j=1(1+x i y j t)is a symmetric function on the x i’s and the y j’s.Therefore,it must be a polynomial in the elementary symmetric functions s1,...,s n andσ1,...,σn.For example,a little bit of computation shows that2i,j=1(1+x i y j t)=1+(s1σ1)t+ s21σ2+s2σ21−2s2σ2 t2+(s1s2σ1σ2)t3+ s22σ22 t4. The reader should check this.In general,the coeﬃcient of t n in h(t)= n i,j=1(1+x i y j t)is of the formP n(s1,...,s n;σ1,...,σn)for some universal polynomial P n in2n variables with integer coeﬃcients.For example,we haveP1(x;y)=xy,P2(x1,x2;y1,y2)=x21y2+x2y21−2x2y2.These polynomials P n will appear in the deﬁnition of aλ-ring(Deﬁnition1.10). 1.2λ-RingsThe deﬁnition of aλ-ring involves certain axioms about the operationsλ0,λ1,λn(x+y),λn(xy),andλm(λn(x)).Before giving the exact deﬁnition,let us provide some motivation for the axioms.6Lambda-Rings1.2.1Theλ-ring ZThe simplestλ-ring is the ring of integers Z,in whichλi(n)= n i =n!λ-Rings71.2.2Basic properties ofλ-ringsDeﬁnition1.10.Aλ-ring is a ring R together with functionsλn:R→R(n≥0),calledλ-operations,such that for all x,y∈R,the following axioms are satisﬁed:(1)λ0(x)=1,(2)λ1(x)=x,(3)λn(1)=0for n≥2,(4)λn(x+y)= i+j=nλi(x)λj(y),(5)λn(xy)=P n(λ1(x),...,λn(x);λ1(y),...,λn(y)),(6)λn(λm(x))=P n,m(λ1(x),...,λnm(x)).Here P n and P n,m are the universal polynomials with integer coeﬃcients from Exam-ple1.9and Example1.7,respectively.If only axioms(1),(2),and(4)are satisﬁed, then we call R a pre-λ-ring.In a(pre-)λ-ring R,we writeλt(x)=∞n=0λn(x)t n,(1.7)considered as a formal power series in t with coeﬃcients in ing this notation, we can rewrite axiom(4)above asλt(x+y)=λt(x)λt(y).(1.8)Deﬁnition1.11.Ifλt(x)is a polynomial of degree n,then we say that x has dimension n and write dim(x)=n.If every element in R is the diﬀerence ofﬁnite dimensional elements,then we say that R isﬁnite dimensional.Remark1.12.What we call aλ-ring(pre-λ-ring)here used to be called a special λ-ring(λ-ring)in[Atiyah and Tall(1969);Berthelot et al.(1971);Fulton and Lang (1985);Grothendieck(1971)].Our terminology follows that in[Knutson(1973)]. Also,the dimension of an element in aλ-ring was called the degree of that element in[Knutson(1973)].In Proposition2.8and Corollary2.10below,we will provide some criteria that make a pre-λ-ring into aλ-ring.Here are some of the most basic properties of aλ-ring.Proposition1.13.Let R be aλ-ring,and let x and y be elements in R.Then the following statements hold.(1)λt(1)=1+t.(2)λt(0)=1.(3)λt(x+y)=λt(x)λt(y).8Lambda-Rings(4)λt(−x)=λt(x)−1.(5)dim(x+y)≤dim(x)+dim(y).(6)If x and y are both1-dimensional,then so is xy.So aﬁnite product of1-dimensional elements is again1-dimensional.Proof.All the statements are immediate from theλ-ring axioms.For example, assuming(2)and(3)we have1=λt(0)=λt(x−x)=λt(x)λt(−x),which proves(4).For statement(6),suppose that n≥2.In the producth(t)=ni,j=1(1+x i y j t),set x2,...,x n,y2,...,y n to0.Then direct inspection shows that the coeﬃcient of t n is0.This implies thatP n(s1,0,0,...;σ1,0,0,...)=0,which in turn implies thatλn(xy)=P n(λ1(x),0,0,...;λ1(y),0,0,...)=0.Therefore,the product xy is1-dimensional.The proofs of the other statements are left as exercises. Proposition1.14.The following statements hold in aλ-ring R for k≥1and a,a1,...,a n∈R:λk(a1+···+a n)=ni=1λk(a i)+(polynomial inλj(a l),j<k,1≤l≤n),λk(−a)=−λk(a)+(polynomial inλj(a),j<k).Here by“polynomial”we mean a polynomial with integer coeﬃcients.Proof.Theﬁrst assertion follows from the equalityλk(a1+···+a n)= k1+···+k n=kλk1(a1)···λk n(a n)and the axiomλ0(x)=1.For the second assertion,we have0=λk(a+(−a))=λk(a)+λk(−a)+(polynomial inλi(a),λi(−a),i<k).Thus,we haveλk(−a)=−λk(a)+(polynomial inλi(a),λi(−a),i<k).Now apply the same procedure to the termsλi(−a)(i<k),and so forth.The second assertion of the Lemma follows by aﬁnite downward induction.λ-Rings9 1.2.3Examples ofλ-ringsExample1.15.The ring Z of integers is aﬁnite-dimensionalλ-ring withλt(n)=(1+t)nfor n∈Z,i.e.,λi(n)=coeﬃcient of t i in(1+t)n.Indeed,we have already seen that axioms(1)to(4)are satisﬁed.Axioms(5)and (6)are certain universal identities about the binomial coeﬃcients.It is possible to prove them directly.However,it is easier to verify them using the universalλ-ring (Corollary2.11).Notice that an integer n≥0has dimension n,while negative integers do not haveﬁnite dimensions.Example1.16.The topological K-theory K(X)of any nice-enough space X(e.g., para-compact Hausdorﬀspace)is aλ-ring[Atiyah(1989);Atiyah and Tall(1969)],in whichλi is induced by the i th exterior power on vector bundles over X.Moreover, the K-theory of a point(or any contractible space)is K(pt.)∼=Z as aλ-ring. Moreover,if X is aﬁnite CW complex,then K(X)is aﬁnite-dimensionalλ-ring.Example1.17.For a group G,the complex representation ring R(G)is aλ-ring [Atiyah and Tall(1969)],in whichλi is induced by the i th exterior power on rep-resentations of G.Theλ-ring R(G)isﬁnite-dimensional if G is aﬁnite group. Moreover,if G={e}is the trivial group,then R({e})∼=Z as aλ-ring.As we will see in§5.5,the trivial group is the onlyﬁnite group whose representation ring is isomorphic to Z.This is a consequence of Knutson’s Theorem5.36.A lot more examples ofλ-rings will be given in the following chapters when we have the universalλ-ring and Adams operations at our disposal.Here is a preview of some of them.Example1.18.Let R be a ring.Then its universalλ-ringΛ(R)is always aλ-ring (Theorem2.6).Example1.19.Let R be a ring.Then the ring W(R)of big Witt vectors on R is aλ-ring,which is isomorphic toΛ(R)(Theorem4.16)via the Artin-Hasse Exponential isomorphism.Example1.20.The ringˆZ p of p-adic integers is aλ-ring(Corollary3.63),as is the ring Z(p)of p-local integers(Corollary3.59).Example1.21.Let G be a group,and let k be aﬁeld of characteristic0.Let C G be the set of conjugacy classes of G,and let k(C G)be the ring of central functions on G with values in k.Then k(C G)is aλ-ring(Chapter3Exercise(7d)).10Lambda-RingsExample1.22.Every binomial ring is aλ-ring(Theorem5.3).In particular,every ring Int(Z X)of integer-valued polynomials is aλ-ring(Theorem5.21).Example1.23.For a binomial ring R,its necklace ring Nr(R)is aλ-ring(Corol-lary5.45).Example1.24.Many quotients of the power series ring Z[[x1,...,x n]]areλ-rings (Theorem6.13).1.2.4Homomorphisms and idealsDeﬁnition1.25.Let R and S be(pre-)λ-rings.(1)A(pre-)λ-homomorphism f:R→S is a ring homomorphism such that fλi=λi f for i≥0.(2)A(pre-)λ-ideal in R is an ideal I in R such thatλi(x)∈I for i≥1and x∈I.(3)A(pre-)λ-subring of R is a subring R of R such thatλi(x)∈R for all i≥0and x∈R .Example1.26.(1)If f:X→Y is a map of topological spaces,then the induced mapK(f):K(Y)→K(X)on K-theory is aλ-homomorphism.(2)Ifϕ:G→H is a group homomorphism,then the induced map R(ϕ):R(H)→R(G)on the representation rings is aλ-homomorphism.The usual properties and constructions of rings extend in an obvious way to λ-rings.Proposition 1.27.Let R and S beλ-rings,and let f:R→S be aλ-homomorphism.(1)The kernel of f is aλ-ideal in R.(2)The image of f is aλ-subring in S.(3)The quotient R/I of R by aλ-ideal I is naturally aλ-ring,and the projectionmap R→R/I is aλ-homomorphism.(4)The direct product R×S is aλ-ring,in whichλt(r,0)=(1,1)+ n≥1(λn(r),0)t n,λt(0,s)=(1,1)+ n≥1(0,λn(s))t n,λt(r,s)=λt(r,0)λt(0,s).(5)The tensor product R⊗S is aλ-ring,in whichλn(r⊗1)=λn(r)⊗1,λn(1⊗s)=1⊗λn(s),λn(r⊗s)=λn((r⊗1)(1⊗s)).(6)If{Rα}is an inverse system ofλ-rings,then the inverse limit ring lim←−αRαis naturally aλ-ring.Proof.Straightforward exercise.An ideal in a ring is often described by a set of generators.To check that an ideal I in aλ-ring R is aλ-ideal,one would have to check thatλn(x)∈I for n≥1and all x∈I.The following result reduces this process to just checking the generators. Proposition1.28.Let R be aλ-ring,and let I be an ideal in R generated by {z j}j∈J.Then I is aλ-ideal if and only ifλn(z j)∈I for n≥1and j∈J. Proof.The“only if”part is obvious.For the other direction,ﬁrst consider an element in I of the form rz for some r∈R and z∈{z j}j∈J.In the polynomialh(t)=ni,j=1(1+x i y j t),if one sets all the y j to0,then the coeﬃcient of t n is0.This implies thatP n(s1,...,s n;0,0,...,0)=0,which in turn implies that every non-zero term in P n(s1,...,s n;σ1,...,σn)contains a factor ofσk for some k.It follows that the elementλn(rz)=P n(λ1(r),...,λn(r);λ1(z),...,λn(z))is aﬁnite sum of terms,each one containing a factor ofλk(z)∈I for some k.So λn(rz)lies in I as well.A general element in I is a linear combination w= m k=1r k z j k with coeﬃcients in R.To show thatλn(w)∈I for n≥1,note that by Proposition1.13(3)one hasλt(w)= m k=1λt(r k z j k).Therefore,eachλn(w)is aﬁnite sum of products of λp(r k z jk)∈I for various p and k.This shows thatλn(w)lies in I.It was mentioned earlier that Z is the simplestλ-ring.The following result is the reason for that assertion.Proposition1.29.The following statements hold.(1)The ring Z has a uniqueλ-ring structure withλt(m)=(1+t)m for m∈Z.(2)Everyλ-ring has characteristic0.(3)Everyλ-ring contains aλ-subring that is isomorphic to Z.Proof.In anyλ-ring,one hasλt(1)=1+t.Therefore,for a positive integer m one has thatλt(m)=λt(1+···+1m times)=λt(1)m=(1+t)m.Moreover,by Proposition1.13(4)one hasλt(−m)=λt(m)−1=(1+t)−m,which shows that the statedλ-ring structure on Z is the only one.In aλ-ring R,for a positive integer m,a similar calculation as above shows thatλt(m·1)=λt(1+···+1m times)=(1+t)m=1+···+t m.This is a polynomial of degree m over any ring,which shows that m=0in R.Therefore,R has characteristic0and has aλ-subring isomorphic to Z.1.2.5Augmentedλ-ringsManyλ-rings in practice come with aλ-homomorphism to Z.We give theseλ-rings a special name.Deﬁnition1.30.An augmentedλ-ring is aλ-ring R that comes equipped with a λ-homomorphismε:R→Z,called the augmentation.Example1.31.If x0is a base point of a topological space X,then the K-theory K(X)of X is an augmentedλ-ring.The augmentationε:K(X)→Z is induced by the map that sends an actual vector bundle over X to the dimension of theﬁber over x0.Example1.32.The representation ring R(G)of a group G is an augmentedλ-ring.The augmentationε:R(G)→Z is induced by the map that sends an actual representation to its dimension.Augmentedλ-rings admit a natural decomposition with Z as a factor. Proposition1.33.The following statements hold.(1)If R is an augmentedλ-ring with augmentationε,then0≤ε(x)≤dim(x)for anyﬁnite dimensional element x∈R.(2)Aλ-ring R is augmented if and only if there exists aλ-ideal I such that R=Z⊕I as an abelian group.(3)If R and S are augmentedλ-rings,then R⊗S is also an augmentedλ-ring.Proof.If x∈R is ofﬁnite dimension d,then for any integer n>d one has thatλn(ε(x))=ε(λn(x))=ε(0)=0.This shows thatε(x)≤d and is non-negative.For the second statement,suppose that R is augmented with augmentationε. Let I denote the kernel ofε.Given any element x∈R,one can always write it uniquely asx=ε(x)+(x−ε(x)),in whichε(x)∈Z and(x−ε(x))∈I.This shows that R decomposes as Z⊕I. Conversely,if there exists aλ-ideal I such that R=Z⊕I as an abelian group,then R is augmented by the projection map onto theﬁrst factor.The tensor product R⊗S is aλ-ring by Proposition1.27.If the augmentations of R and S areεR andεS,respectively,then R⊗S has augmentationεR⊗εS.Given aλ-ring R,it is convenient to know how to put aλ-ring structure on the polynomial ring R[x]or the power series ring R[[x]].This is illustrated by the following result.Proposition1.34.Let R be aλ-ring.Then there exists a uniqueλ-ring structure on the polynomial ring R[x]such thatλt(x)=1+xt,i.e.,dim(x)=1.If R is augmented,then so is R[x]withε(x)=0or1.The same holds for the power series ring R[[x]].Proof.We takeλt(x)=1+xt as a deﬁnition and extend it to all of R[x]or R[[x]] as follows.Theλ-operations are extended to the powers x i using theλ-ring axiom forλn(xy)repeatedly.For an element r∈R,we use this axiom again to extend theλ-operations to monomials of the form rx i.For a polynomial(or formal power series)f= i r i x i with coeﬃcients in R,we deﬁneλt(f)= iλt(r i x i)using the axiom forλn(x+y).The same reasoning shows that theλ-ring structure is uniquely determined by the conditionλt(x)=1+xt.The assertion about the augmentation is clear.We will use this result below in the construction of the freeλ-ring on one gen-erator.Applying Proposition1.34repeatedly,we obtain the following result. Corollary1.35.Let R be aλ-ring.Then there exists a uniqueλ-ring structure on the polynomial ring R[x1,...,x n]on n variables such thatλt(x i)=1+x i t for1≤i≤n,i.e.,dim(x i)=1for each i.If R is augmented,then so is R[x1,...,x n]with ε(x i)=0or1for each i.The same holds for the power series ring R[[x1,...,x n]] on n variables.Proof.Apply Proposition1.34n times.1.3Freeλ-RingsIn this section we construct the freeλ-ring U on one generator.This ring U also occurs in(at least)two other places.First,U is also the ring of operations on λ-rings,which we will discuss in§1.4.Second,in representation theory,the ring U is often called the ring of symmetric functions.In fact,U is isomorphic to the graded ringR(S)=∞ n=0R(S n),in which R(S n)is the representation ring of the symmetric group S n on n letters [Knutson(1973)].Before we construct U,let us provide some motivation for its construction.Since U is to be the freeλ-ring on one generator,say,x,it should have the following universal property.Given anyλ-ring R and an element r∈R,there exists a unique λ-homomorphism u r:U→R such that u r(x)=r.Because U is aλ-ring itself,it should also containλn(x)for n≥1.Therefore,it contains all the polynomials with integer coeﬃcients in x,λ2(x),λ3(x),etc.,and hence also the ringU =Z[x,λ2(x),λ3(x),...].This latter ring U is already aλ-ring because of theλ-ring axiom forλm(λn(x)), which expresses it as a polynomial with integer coeﬃcients inλi(x)(1≤i≤mn). So U is aλ-subring of U containing x,which implies that U should be equal to U .It turns out that the elementsλi(x)(i≥1)in U must also be algebraically independent.In other words,U is a polynomial ring over Z in countably inﬁnitely many variables y1,y2,...such thatλn(y1)=y n.We now give the exact construction of U.1.3.1The freeλ-ring on one generatorLet x1,x2,...be an inﬁnite sequence of algebraically independent variables.Con-sider the polynomial ringΩr=Z[x1,...,x r]in r≥0variables.Since Z is aλ-ring(Proposition1.29),by Corollary1.35there exists a uniqueλ-ring structure onΩr in whichλt(x i)=1+x i t,i.e.,each x i has dimension1.Theseλ-ringsﬁt into an inverse systemZ=Ω0p1←−Ω1p2←−Ω2p3←−Ω3p4←−···,(1.9) in which theλ-homomorphismp r+1:Ωr+1→Ωr(1.10)is deﬁned byp r+1(x i)= x i if1≤i≤r,0if i=r+1.DeﬁneΩ=lim←−Ωr,(1.11) the inverse limit of the inverse system(1.9).Letφr:Ω→Ωr be the structure map that comes with the inverse limit.By Proposition1.27,Ωis aλ-ring,andφr is a λ-homomorphism.From the deﬁnition of inverse limit,an element inΩis a power series f with integer coeﬃcients in the variables x i(i≥1)such that for each r≥1,φr(f)=f(x1,...,x r,0,0,...)is a polynomial in x1,...,x r.Let s n(x1,...,x r)be the n th elementary symmetric function on x1,...,x r(Ex-ample1.1).Sinces n(x1,...,x r,0)=p r+1(s n(x1,...,x r+1))=s n(x1,...,x r),the sequence{s n(x1,...,x r)}r≥0determines an elements n=lim←−rs n(x1,...,x r)inΩ.Thinking of s n as a power series,we haves n= i1<···<i n x i1···x i n,φr(s n)=s n(x1,...,x r)= 1≤i1<···<i n≤r x i1···x i n.(1.12)For example,s1is the inﬁnite sum s1=x1+x2+x3+···,andφr(s1)=x1+···+x r for each r≥1.In view of the above discussion,we think of s n as the n th elementary symmetric function on a countable set of variables x i(i≥1).Therefore,we can write down the generating function for these elementary symmetric functions formally as1+∞n=1s n t n=∞ i=1(1+x i t).(1.13)This is the analog of(1.2)involving countably inﬁnitely many variables.There is an obvious analog of The Fundamental Theorem of Symmetric Functions(Theorem 1.4)that involves an inﬁnite sequence of variables x i(i≥1),with s n(1.12)playing the role of the n th elementary symmetric function.Proposition1.36.In theλ-ringΩ,the elements s n(n≥1)are algebraically independent.Moreover,one hasλn(s1)=s n for n≥1.Proof.Suppose that f(s1,...,s n)=0is a polynomial relation involving s1,...,s n.Then0=φn f(s1,...,s n)=f(φn(s1),...,φn(s n))=f(s1(x1,...,x n),...,s n(x1,...,x n)).The algebraic independence of s i(x1,...,x n)(1≤i≤n)(Theorem1.4)then shows that f is the0polynomial.This proves that the elements s n(n≥1)are algebraically independent.For the second assertion,note that for r≥n,one hasφrλn(s1)=λn(φr(s1))=λn(x1+···+x r)= 1≤i1<···<i n≤r x i1···x i n=φr(s n).The third equality is a consequence of theλ-ring axiom forλn(x+y)and the fact that dim(x i)=1for each i.Since this holds for all r≥n,it follows that λn(s1)=s n,as desired. Deﬁnition1.37.Let U be the smallestλ-subring ofΩthat contains s1.Theλ-ring U is called the freeλ-ring on one generator.Proposition1.38.The freeλ-ring on one generator is the polynomial ringU=Z[s1,s2,s3,...]on the algebraically independent elements s i(i≥1).Proof.Since s1∈U,so isλn(s1)=s n(Proposition1.36).It follows that U contains the stated polynomial ring.The latter is closed under theλ-operations by theλ-ring axiom onλm(λn(x)),so it is aλ-ring.Therefore,it must be equal to the ring U.We now show that U has the right universal property.Theorem1.39.Let R be aλ-ring,and let r be an element in R.Then there exists a uniqueλ-homomorphism u r:U→R such thatλn(r)=u r(s n)for n≥1. Proof.Deﬁne a ring homomorphism u r:U→R byﬁrst setting u r(s1)=r.This extends uniquely to aλ-homomorphism on U,since,to be such,u r must satisfyλn(r)=λn(u r(s1))=u r(λn(s1))=u r(s n).So we must choose u r(s n)to beλn(r).One can construct the freeλ-ring on any set of generators following the procedure above.We leave it to the reader to work out the details in the exercises.。

物理学外文翻译

Effect of Quantum Confinement on Electrons and Phonons in Semiconductors We have studied the Gunn effect as an example of negative differential resistance(NDR).This effect is observed in semiconductors,such as GaAs,whose conduction band structure satisfies a special condition,namely,the existence of higher conduction minima separated from the band edge by about 0.2-0.4eV..As a way of achieving this condition in any semiconductor,Esaki and Tsu proposed in 1970 [9.1]the fabrication of an artificial periodic structure consisting of alternate layers of two dissimilar semiconductors with layer superlattice.They suggested that the artificial periodicity would fold the Brillouin zone into smaller Brillouin zones or “mini-zones”and therefore create higher conduction band minima with the requisite energies for Gunn oscillations.iWith the development of sophisticated growth techniques such as molecular beam epitaxy（MBE）and metal-organic chemical vapor deposition(MOCVD)discussed in Sect.1.2,it is now possible to fabricate the superlattices(to be abbreviated as SLs)envisioned by Esaki and Tsu[9.1].In fact,many other kinds of nanometer scale semiconductor structures(often abbreviated as nanostructures)have since been grown besides the SLs.A SL is only one example of a planar or two-dimensional nanostructure .Another example is the quantum well (often shortened to QW).These terms were introduced ｉｎＳｅｃｔｓ．１．２ａｎｄ７．１５ｂｕｔｈａｖｅｎｏｔｙｅｔｂｅｅｎｄｉｓｃｕｓｓｅｄｉｎｄｅｔｉａｌ．Ｔｈｅｐｒｏｐｏｓｅof this chapter is to study the electronic and vibrational properties of these two-dimensional nanostructures.Structures with even lower dimension than two have also been fabricated successfully and studied. For example,one-dimensional nanostructures are referred to as quantum wires.In the same spirit,nanometer-size crystallites are known as quantum dots.There are so many different kinds of nanostructures and ways to fabricate them that it is impossible to review them all in this introductory book. In some nanostructures strain may be introduced as a result of lattice mismatch between a substrate and its overlayer,giving rise to a so-called strained-layer superlattice.In this chapter we shall consider only the best-study nanostructures.Our purpose is to introduce readers to this fast growing field.One reason why nanostructures are of great interest is that their electronic and vibrational properties are modified as a result of their lower dimensions and symmetries.Thus nanostructures provide an excellent opportunity for applying the knowledge gained in the previous chapters to understand these new developments in the field of semiconductors physics.Due to limitations of space we shall consider in this chapter only the effects of spatial confinement on the electronic and vibrational properties of nanostructures and some related changers in their optical and transport properties.Our main emphasis will be on QWs,since at present they can be fabricated with much higher degrees of precision and perfection than all other structures.We shall start by defining the concept of quantum confinement and discuss its effect on the electrons and phonons in a crystal.This will be followed by a discussion of the interaction between confined electrons and phonons.Finally we shall conclude with a study of a device(known as a resonant tunneling device)based on confined electrons and the quantum Hall effect(QHE)in a two-dimensional electron gas.The latter phenomenon was discoveredby Klaus von Klitzing and coworkers in 1980 and its significance marked by the award of the 1985 Nobel Prize in physics to ｖｏｎ Klitzing for this discovery.Together with the fractional quantum Hall effect it is probably the most important development in semiconductor physics within the last two decades.Quantum Confinement and Density of StatesIn this book we have so far studied the properties of electrons ,phonons and excitons in either an infinite crystal or one with a periodic boundary condition(the cases of surface and interface states )In the absence of defects, these particles or excitations are described in terms of Bloch waves,which can propagate freely throughout the crystal.Suppose the crystal is finite and there are now two infinite barriers,separated by a distance L,which can reflect the Bloch waves along the z direction.These waves are then said to be spatially confined.A classical example of waves confined in one dimension by two impenetrable barriers is a vibrating string held fixed at two ends.It is well-known that the normal vibration modes of this string are standing waves whose wavelength λ takes on the discrete values given by Another classical example is a Fabry-Perot interferometer (which has been mentioned already in Set.7.2.6 in connection with Brillouin scattering).As a result of multiple reflections at the two end mirrors forming the cavity ，electromagnetic waves show maxima and minima in transmission through the interferometer at discrete wavelengths.If the space inside the cavity is filled with air,the condition for constructive interference is given by (9.1).At a transmission minimum the wave can be considered as “confined ”inside the interferometer.n λ=2L/n, n=1,2,3… .(9.1)For a free particle with effective mass *m confined in a crystal by impenetrablebarriers(i.e.,infinite potential energy)in the z direction,the allowed wavevectors z k of the Bloch waves are given byzn κ=2∏/n λ=n ∏/L, n=1,2,3… (9.2)And its ground state energy is increased by the amount E relative to the unconfined case:))(2(2222212Lm m k E z ∏==∆** (9.3)This increase in energy is referred to as the confinement energy of the particle.It is a consequence of the uncertainty principle in quantum mechanics. When the particle is confined within a distance L in space(along the z direction in this case)the uncertainty in the z component of its momentum increases by an amount of the order of /L.The corresponding increase in the particle ’s kinetic energy is then givenby(9.3).Hence this effect is known also as quantum confinement.In addition to increasing the minimum energy of the particle,confinement also causes its excited state energies to become quantized.We shall show later that for an infinite one-dimensional”square well”potential the excited state energies are given by n E∆2,where n=1,2,3…as in (9.2).It is important to make a distinction between confinement by barriers and localization via scattering with imperfections。

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a r X i v :c o n d -m a t /9406059v 1 14 J u n 1994IASSNS-HEP-94/42Elementary Excitations of One-Dimensional t -J Modelwith Inverse-Square ExchangeZ.N.C.Ha (1)and F.D.M.Haldane (2)(1)Schoolof Natural Sciences,Institute for Advanced Study,Princeton,New Jersey 08540.(2)Department of Physics,Princeton University,Princeton,New Jersey 08544.(June 14,1994)Abstract We identify exact excitation content of the intermediate states for the one-particle Green’s functions,spin-spin and (charge)density-density correlation functions of the periodic one-dimensional t -J model with inverse square ex-change.The excitations consist of neutral S =1/2spinons and spinless (charge −e )holons with semionic fractional statistics,and bosonic (charge +2e )“anti-holons”which are excitations of the holon condensate.Due to the supersymmetric Yangian quantum symmetry of this model,only the excitedstates with ﬁnite number of elementary excitations contribute to the spectralfunctions.We ﬁnd a set of selection rules,and this allows us to map out theregions of non-vanishing spectral weight in the energy-momentum space forthe various correlation functions.PACS 05.30-d,71.10.+x,71.27.+aTypeset using REVT E XRecently,there have been many developments in understanding the family of Calogero-Sutherland models(CSM)which are identiﬁed with their peculiar inverse-square exchange (ISE)[1–3,6,8–10,17].An important feature of these models is that they belong to the same low-energy universality class as the family of Bethe-ansatz solvable models and may provide a new fully soluble paradigm next to the non-interacting models[2].The one-dimensional supersymmetric ISE t-J model[3]represents aﬁxed point model where the elementary excitations form an ideal gas obeying fractional statistics.In contrast to this model,the NNE t-J model[4,5],which has essentially the same low energy spec-tra spanned by the same elementary excitations,obscures the simple low energy structure intrinsic to this class of models.We rediscover the spinons,the holons and the antiholons—the elementary excitations of the NNE t-J model[5]—in the context of the supersymmetric Yangian of the ISE model.Furthermore,weﬁnd that only the states withﬁnite number of these elementary excitations contribute to the spectral functions of the one-particle Green’s functions(G(1)),the charge density-density(C(c))and the spin-spin correlation functions (C(s)).First,we examine the symmetry in the ISE supersymmetric t-J model.The model with periodic boundary conditions possesses,in addition to the global SU(m|n)supersymmetry, a hidden dynamical“quantum group”symmetry algebra called the supersymmetric Yangian [2,6,7].This symmetry is responsible for the“supermultiplets”in the energy spectrum and the ideal gas-like features of the elementary excitations and,furthermore,provides us with a simple numerical way to identify the exact content of the elementary excitations relevant for the various correlation functions.The supersymmetric generalization of the SU(n)Haldane-Shastry model Hamiltonian [8–10]is given byH=t i<j P ijαa†iαa iα=1,the exchange operator can be written as P ij= αβa†iαa†jβa iβa jα.If m of the species labeled byαare bosons,and n are fermions,the model(1)has a global SU(m|n) supersymmetry with generators given by the traceless part of Jαβ0= i a†iαa iβ.The Yangian symmetry generator of the periodic ISE model isJαβ1= i>j,γw ij a†iαa†jγa iγa jβ,(2) where w ij=cot(π(i−j)/N a).The higher order generators of the Yangian are obtained recursively from various commutators involving only J0and J1[6,7].If we specialize to SU(1|2)supersymmetry,withα∈{0,↑,↓},we can rewrite the Hamil-tonian in terms of the SU(2)fermionic operators c†iσ=a†iσa i0as P H0P,where H0(up to a shift in total energy and in chemical potential)is− i=j,σt ij c†iσc jσ+ i<j(J ij S i·S j+V ij n i n j),(3) where t ij=J ij/2=−2V ij=t/d2(i−j)and n i=n i↑+n i↓;P is the projection operator that projects out all states with doubly-occupied sites.The ground state|Ψ0 of this model is known[3,9]to be{x,σ}i>j(z i−z j)δσi,σj(i)sgn(σi−σj) k z J0k j c†x jσi|0 ,(4)where z j=exp(i2πx j/N a),J0=−(N/2−1)/2,N is the total number of particles,and|0 the electron vacuum(empty state).In order to have a non-degenerate ground state,we take N/2to be odd.Note that this wave function is just the full Gutzwiller projection of a free electron state[11].A remarkable feature of this model is that the eigenstates of(1)form degenerate“su-permultiplets”[8]with multiplicities much higher than those expected from the global su-persymmetry.All supermultiplets on the SU(m|n)model with m,n>0are present(with the same energy and momentum,but multiplicity reduced to2)in the spinless free fermion SU(1|1)model[2].This means that they can be represented by a binary sequence of N a−1 ones and zeroes,representing(in the spinless fermion model)the occupations of Bloch stateswith non-zero momentum(the zero-momentum orbital has zero energy,which is the super-symmetry,and its occupation is notﬁxed).There are thus2N a−1distinct supermultiplets.In the SU(1|2)case,the“occupation number”sequence describes a supermultiplet span-ning a large range of possible fermion charges N.The state of minimum charge in the supermultiplet is given by the number of zeroes in the sequence;the maximum charge is N a minus the number of times two consecutive ones occur.The ground state of the model with t>0has a sequence111...111,so its minimum charge is N=0and its maximum charge is N a−(N a−2)=2.The multiplet represented by the alternating sequence10101 (10101)has a maximum charge state N=N a,which is the spin-singlet ground state of the antifer-romagnetic S=1/2Haldane-Shastry chain,and a minimum charge(N a−2)/2.We study the model(1)with t>0and a chemical potential that maximizes N,so the ground-state has0<N<N a.Then,only intermediate states with the maximum value of charge in their supermultiplet contribute to the thermodynamic limit of the ground-state correlation functions.To determine the excitation content of these maximal charge states, it is convenient to add a zero to each end of the binary sequence,expanding its length to N a+1.The ground state sequence is then of the form0101010...1111111...0101010,with a central section of consecutive ones,with equal-length wings of the alternating sequence.In the limit N=N a,the excitations of the S=1/2antiferromagnet are neutral spin-1/2spinons[12–14]represented by consecutive zeroes(e.g....010********...)and spinless charge−e holons by consecutive ones(e.g....010*********...).At intermediate densities, the central region...1111111...may be considered as a holon condensate or“pseudo-Fermi-sea”.However,the holons and spinons are not fermions,but semions,or particles with“half-fractional”statistics,resulting from the spin-charge separation of a hole,which is a spin-1/2,charge−e fermion.A conﬁguration...11111110111111...has a“hole in the holon condensate”which we will call an“antiholon”;because of the semionic statistics of the holons,we identify it as a charge+2e,spinless boson.Using concepts from Chern-Simons theory,as applied to the fractional quantum Hall eﬀect[15],if condensed particles have charge q and statisticsΘ=πλ,vortices or holes inthe condensate have charge−q/λand statisticsΘ′=π/λ.Here holons have charge−e andΘ=π/2(a semion),so the vortex or hole in the holon condensate(antiholon)then has charge2e andΘ=2π(a boson).The applicability of such“2D”concepts to1D ISE-type models has recently been demonstrated:the holon(antiholon)corresponds to particle (hole)excitations of theλ=1/2Calogero-Sutherland model where the particle excitations are semions and the holesλ=2bosons[2,16].The main results of this paper can be summarized in Table I,which lists all the pos-sible elementary excitations for the corresponding local perturbations of the ground state. The quantum symmetry prevents the injected electron or hole from breaking up into more than a very simple set of elementary excitations consisting of the left(right)spinons(s L(R)), holons(h L(R)),and antiholons(¯h).As a result,the spectral functions of the various dynam-ical correlation functions vanish except in certain regions of the energy-momentum plane (i.e.,has“compact support”).Figs.1-3show the regions of compact support formed by theﬁnite number of elementary excitations contributing to the intermediate states for G(1),C(c),and C(s),respectively.If the correlation functions are given by the following integral,C(x,t)= (Q,E)∈σdQ dE S(Q,E)e i(Qx−Et),(5) theﬁgures show the regionσwhere the spectral function S(Q,E)is non-zero;this is de-termined by combining the energies and(Bloch)momenta of theﬁnite set of elementary excitations contributing to S(Q,E).The dispersion relations for the spinon(E s),holon(E h)and antiholon(E¯h)in the ther-modynamic limit are given byE sR(L)/t=−q(q∓v0s),0≤|q|≤π¯n2,(6b)E¯h/t=(v0c)2−qwhere v0s=π(spin-wave velocity),v0c=π(1−¯n)(sound velocity)and¯n the density of electrons.The right(left)movers take the upper(lower)signs and are allowed only in q≥0 (q≤0)relative to the Q=0ground state.The curvature of the antiholon dispersion is half that of holon,indicating that¯h is made by destroying two holons.It is then natural to assign charge C=+2e and S=0to the antiholon while C=0and S=1were allowed then states of the type(s L,h L)+2¯h+2(s R,h R)would contribute.Our numerical study indicates that states with two antiholons do not contribute.In fact,the observed states listed in Table I are the simplest possible states satisfying the charge(spin)conservation with at most one antiholon.In Fig.2,only holon-antiholon branches are present at4k F(2π−4k F)while the spinon-holon branches show up at2k F(2π−2k F).At¯n=0.1,the spin-charge separation is hardly visible.In Fig.3weﬁnd that the pure spinon excitations are possible only if they both belong to the same sector,otherwise they are accompanied by two holons and an antiholon. The excitation content weﬁnd for S z i(=(n i↑−n i↓)/2)should be same for S±i since the ground state is a spin singlet.As¯n→0we recover the two spinon spectrum for the S=1/2 spin chain.Finally,we have examined how the ISE results for the charge of the elementary excitations change if we interpolate between the ISE and NNE t−J models,which are respectively theγ=0andγ=∞limits of the integrable family of hyperbolic models with exchange ∝1/sinh2γ(i−j)[17].Away from the ISE limit,the charge carried by the holon and antiholon excitations vary with their velocity,and become equal in magnitude(and opposite in sign)as the velocities approach the sound velocity v0c.In the ISE limit,however the holon charge(|v|>v0c)is always−e,and the antiholon charge(|v|<v0c)is always+2e.The“dressed charge”carried by the excitations can be calculated using the asymptotic Bethe Ansatz[18].The charge enhancement of the test holon is measured by the diﬀerence in the phase shifts of the holon condensate at the pseudo-Fermi points and will in general depend on where the holon is with respect to the condensate.The total charge C(the bare plus the enhanced)is plotted in Fig.4as a function of the momentum of the test holon at a ﬁxed density of electrons(¯n=0.5)for various values ofγ.The pseudo-Fermi points of the condensate for eachγare labeled by“x”.The ISE limit is given by the solid line.The curve with the smallest charge enhancement in the condensate corresponds to the NNE model.In the ISE limit,there is a clear jump in the holon charge from−e to−2e at the pseudo-Fermi pointπ(1−¯n).Therefore,if a holon is taken out the condensate,the hole excitation carriescharge+2e independent of where it is in the condensate.We call this hole an antiholon. For all the other values ofγ,there is a considerable charge enhancement of the test holon in the condensate,and asγ→0the charge approaches−2e.In conclusion,we have devised simple rules for constructing the motifs for the excited states of the1D ISE t-J model and identiﬁed the exact excitation content of the interme-diate states for the one-particle Green’s function,the charge density-density and spin-spin correlation functions.We believe that this model is in the same universality class as the NNE model,and that the most relevant states for the ground state correlation functions of the NNE model are also given by Table I.Finally,the presence of spinons,holons,and antiholons in two-dimensional models and their role in the high T c superconductivity is an amusing possibility.ZNCH is supported by DOE grant#DE-FG02-90ER40542and FDMH by NSF-DMR-922407.REFERENCES[1]F.Calogero,J.Math.Phys.10,2191(1962);10,2197(1969);B.Sutherland,J.Math.Phys.12,246(1971);12251(1971);Phys.Rev.A4,2019(1971);5,1372(1972). [2]F.D.M.Haldane,in the Proceeding of the16th Taniguchi Symposium,Kashikojima,Japan,October26-29,1993.[3]Y.Kuramoto and H.Yokoyama,Phys.Rev.Lett.67,1338(1991).[4]B.Sutherland,Phys.Rev.B12,3795(1975);P.Schlottmann,Phys.Rev.B36,5177(1987);S.Sarkar,J.Phys.A:Math.Gen.23,L409(1990).[5]P.A.Bares and G.Blatter,Phys.Rev.Lett.64,2567(1990).[6]F.D.M.Haldane,Z.N.C.Ha,J.C.Talstra,D.Bernard,and V.Pasquier,Phys.Rev.Lett.69,2021(1992).[7]V.G.Drinfel’d,Sov.Math.Dokl.32,254(1985).[8]F.D.M.Haldane,Phys.Rev.Lett.60,635(1988);B.S.Shastry,Phys.Rev.Lett.60,639(1988).[9]Z.N.C.Ha and F.D.M.Haldane,Phys.Rev.B46,9359(1992);47,12459(1993);Z.N.C.Ha,PhD Thesis(Princeton Univ.),1992.[10]N.Kawakami,Phys.Rev.B46,3191(1992).[11]W.Metzner and D.Vollhardt,Phys.Rev.Lett.59,121(1987);Phys.Rev.B37,7382(1988).[12]H.Bethe,Z.Phys.71,205(1931).[13]L.D.Faddeev and L.A.Takhtajan,Phys.Lett.85A,375(1981).[14]D.A.Tennant,T.G.Perring,R.A.Cowley,and S.E.Nagler,Phys.Rev.Lett.70,4003(1993).[15]S.C.Zhang,Int.J.Mod.Phys.B6,25(1992);and references therein.[16]Z.N.C.Ha,(to be published).[17]V.I.Inozemtsev,J.Stat.Phys.59,1143(1990).[18]H.Frahm and V.E.Korepin,Phys.Rev.B42,10553(1990);and references therein.pact support for the one-particle Green’s function.The momentum is in units of πand the excitation energy E inπ2/t.The contributing elementary excitations to this region are(h L,s L)+¯h+2(h R,s R)for the positive energy part(i.e.c iσ|Ψ0 )and(s L,h L)+¯h for the negative part(i.e.c†iσ|Ψ0 ).Their mirror states(i.e.L and R exchanged)also contribute.The four momenta at which E=0is allowed are k F,2π−3k F,3k F,and2π−k F where k F=π¯n/2.pact support for the density-density correlation function.(s L,h L)+¯h+(s R,h R),¯h+2hR and their mirror states contribute.E=0is allowed at0(2π),2k F,2π−4k F,4k F,2π−2k F. Only holon-antiholon branches are present at4k F(2π−4k F)indicating that4k F is the holon Fermi point.pact support for the spin-spin correlation function.(s L,h L)+¯h+(s R,h R),2s L and their mirror states contribute.E=0allowed at0(2π),2k F,2π−2k F.This indicates that 2k F is the spinon Fermi point.FIG.4.Charge of a test holon versus its momentum in the vicinity of the holon condensate for γ=0,0.2,0.3,0.5,1.0,2.0,∞.The charge is in units of−e where e is the electron charge and the momentum of the test holon in units ofπ.The pseudo-Fermi points are labeled by“x”for each γ.The step function corresponds to the ISE model(γ=0).The NNE model has the smallest but still considerable charge enhancement in the condensate(γ=∞).TABLE I.List of all the possible excitations from the ground state perturbed by the local operators c iσ(c†iσ)(G(1)),n i↑+n i↓(C(c)),and n i↑−n i↓(C(s)).The mirror states(L↔R)not listed are also allowed.The spinon(v s),holon(v h),antiholon(v¯h),spin-wave(v0s)and sound(v0c) velocities always satisfy:(i)v0c<v0s,(ii)v0c≤|v h|(|v s|)≤v0s,(iii)|v¯h|≤v0c,and(iv)for a givenspinon-holon pair(s R,h R),|v sR |≥|v hR|.Local OperatorˆO i Excitation contents ofˆO i|Ψ0 c†iσ(s L,h L)+¯hn i↑−n i↓(s L,h L)+¯h+(s R,h R)2s L00.20.40.60.81Momentum (in units of Pi)11.21.41.61.82C x x xx x xx。