Chiral Symmetry Breaking in 2+1 dimensions due to Sphalerons

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The Subleading Isgur-Wise Form Factor $chi_3(vcdot v')$ to Order $alpha_s$ in QCD Sum Rules

a rXiv:h ep-ph/9212266v116Dec1992SLAC–PUB–6017WIS–92/99/Dec–PH December 1992T/E The Subleading Isgur-Wise Form Factor χ3(v ·v ′)to Order αs in QCD Sum Rules Matthias Neubert Stanford Linear Accelerator Center Stanford University,Stanford,California 94309Zoltan Ligeti and Yosef Nir Weizmann Institute of Science Physics Department,Rehovot 76100,Israel We calculate the contributions arising at order αs in the QCD sum rule for the spin-symmetry violating universal function χ3(v ·v ′),which appears at order 1/m Q in the heavy quark expansion of meson form factors.In particular,we derive the two-loop perturbative contribution to the sum rule.Over the kinematic range accessible in B →D (∗)ℓνdecays,we ﬁnd that χ3(v ·v ′)does not exceed the level of ∼1%,indicating that power corrections induced by the chromo-magnetic operator in the heavy quark expansion are small.(submitted to Physical Review D)I.INTRODUCTIONIn the heavy quark eﬀective theory(HQET),the hadronic matrix elements describing the semileptonic decays M(v)→M′(v′)ℓν,where M and M′are pseudoscalar or vector mesons containing a heavy quark,can be systematically expanded in inverse powers of the heavy quark masses[1–5].The coeﬃcients in this expansion are m Q-independent,universal functions of the kinematic variable y=v·v′.These so-called Isgur-Wise form factors characterize the properties of the cloud of light quarks and gluons surrounding the heavy quarks,which act as static color sources.At leading order,a single functionξ(y)suﬃces to parameterize all matrix elements[6].This is expressed in the compact trace formula[5,7] M′(v′)|J(0)|M(v) =−ξ(y)tr{(2)m M P+ −γ5;pseudoscalar meson/ǫ;vector mesonis a spin wave function that describes correctly the transformation properties(under boosts and heavy quark spin rotations)of the meson states in the eﬀective theory.P+=1g s2m Q O mag,O mag=M′(v′)ΓP+iσαβM(v) .(4)The mass parameter¯Λsets the canonical scale for power corrections in HQET.In the m Q→∞limit,it measures theﬁnite mass diﬀerence between a heavy meson and the heavy quark that it contains[11].By factoring out this parameter,χαβ(v,v′)becomes dimensionless.The most general decomposition of this form factor involves two real,scalar functionsχ2(y)andχ3(y)deﬁned by[10]χαβ(v,v′)=(v′αγβ−v′βγα)χ2(y)−2iσαβχ3(y).(5)Irrespective of the structure of the current J ,the form factor χ3(y )appears always in the following combination with ξ(y ):ξ(y )+2Z ¯Λ d M m Q ′ χ3(y ),(6)where d P =3for a pseudoscalar and d V =−1for a vector meson.It thus eﬀectively renormalizes the leading Isgur-Wise function,preserving its normalization at y =1since χ3(1)=0according to Luke’s theorem [10].Eq.(6)shows that knowledge of χ3(y )is needed if one wants to relate processes which are connected by the spin symmetry,such as B →D ℓνand B →D ∗ℓν.Being hadronic form factors,the universal functions in HQET can only be investigated using nonperturbative methods.QCD sum rules have become very popular for this purpose.They have been reformulated in the context of the eﬀective theory and have been applied to the study of meson decay constants and the Isgur-Wise functions both in leading and next-to-leading order in the 1/m Q expansion [12–21].In particular,it has been shown that very simple predictions for the spin-symmetry violating form factors are obtained when terms of order αs are neglected,namely [17]χ2(y )=0,χ3(y )∝ ¯q g s σαβG αβq [1−ξ(y )].(7)In this approach χ3(y )is proportional to the mixed quark-gluon condensate,and it was estimated that χ3(y )∼1%for large recoil (y ∼1.5).In a recent work we have reﬁned the prediction for χ2(y )by including contributions of order αs in the sum rule analysis [20].We found that these are as important as the contribution of the mixed condensate in (7).It is,therefore,worthwhile to include such eﬀects also in the analysis of χ3(y ).This is the purpose of this article.II.DERIV ATION OF THE SUM RULEThe QCD sum rule analysis of the functions χ2(y )and χ3(y )is very similar.We shall,therefore,only brieﬂy sketch the general procedure and refer for details to Refs.[17,20].Our starting point is the correlatord x d x ′d ze i (k ′·x ′−k ·x ) 0|T[¯q ΓM ′P ′+ΓP +iσαβP +ΓM+Ξ3(ω,ω′,y )tr 2σαβ2(1+/v ′),and we omit the velocity labels in h and h ′for simplicity.The heavy-light currents interpolate pseudoscalar or vector mesons,depending on the choice ΓM =−γ5or ΓM =γµ−v µ,respectively.The external momenta k and k ′in (8)are the “residual”oﬀ-shell momenta of the heavy quarks.Due to the phase redeﬁnition of the eﬀective heavy quark ﬁelds in HQET,they are related to the total momenta P and P ′by k =P −m Q v and k ′=P ′−m Q ′v ′[3].The coeﬃcient functions Ξi are analytic in ω=2v ·k and ω′=2v ′·k ′,with discontinuities for positive values of these variables.They can be saturated by intermediate states which couple to the heavy-light currents.In particular,there is a double-pole contribution from the ground-state mesons M and M ′.To leading order in the 1/m Q expansion the pole position is at ω=ω′=2¯Λ.In the case of Ξ2,the residue of the pole is proportional to the universal function χ2(y ).For Ξ3the situation is more complicated,however,since insertions of the chromo-magnetic operator not only renormalize the leading Isgur-Wise function,but also the coupling of the heavy mesons to the interpolating heavy-light currents (i.e.,the meson decay constants)and the physical meson masses,which deﬁne the position of the pole.1The correct expression for the pole contribution to Ξ3is [17]Ξpole 3(ω,ω′,y )=F 2(ω−2¯Λ+iǫ) .(9)Here F is the analog of the meson decay constant in the eﬀective theory (F ∼f M√m QδΛ2+... , 0|j (0)|M (v ) =iF2G 2tr 2σαβΓP +σαβM (v ) ,where the ellipses represent spin-symmetry conserving or higher order power corrections,and j =¯q Γh (v ).In terms of the vector–pseudoscalar mass splitting,the parameter δΛ2isgiven by m 2V −m 2P =−8¯ΛδΛ2.For not too small,negative values of ωand ω′,the coeﬃcient function Ξ3can be approx-imated as a perturbative series in αs ,supplemented by the leading power corrections in 1/ωand 1/ω′,which are proportional to vacuum expectation values of local quark-gluon opera-tors,the so-called condensates [22].This is how nonperturbative corrections are incorporated in this approach.The idea of QCD sum rules is to match this theoretical representation of Ξ3to the phenomenological pole contribution given in (9).To this end,one ﬁrst writes the theoretical expression in terms of a double dispersion integral,Ξth 3(ω,ω′,y )= d νd ν′ρth 3(ν,ν′,y )1Thereare no such additional terms for Ξ2because of the peculiar trace structure associated with this coeﬃcient function.possible subtraction terms.Because of theﬂavor symmetry it is natural to set the Borel parameters associated withωandω′equal:τ=τ′=2T.One then introduces new variables ω±=12T ξ(y) F2e−2¯Λ/T=ω0dω+e−ω+/T ρth3(ω+,y)≡K(T,ω0,y).(12)The eﬀective spectral density ρth3arises after integration of the double spectral density over ω−.Note that for each contribution to it the dependence onω+is known on dimensionalgrounds.It thus suﬃces to calculate directly the Borel transform of the individual con-tributions toΞth3,corresponding to the limitω0→∞in(12).Theω0-dependence can be recovered at the end of the calculation.When terms of orderαs are neglected,contributions to the sum rule forΞ3can only be proportional to condensates involving the gluonﬁeld,since there is no way to contract the gluon contained in O mag.The leading power correction of this type is represented by the diagram shown in Fig.1(d).It is proportional to the mixed quark-gluon condensate and,as shown in Ref.[17],leads to(7).Here we are interested in the additional contributions arising at orderαs.They are shown in Fig.1(a)-(c).Besides a two-loop perturbative contribution, one encounters further nonperturbative corrections proportional to the quark and the gluon condensate.Let usﬁrst present the result for the nonperturbative power corrections.WeﬁndK cond(T,ω0,y)=αs ¯q q TT + αs GG y+1− ¯q g sσαβGαβq√y2−1),δn(x)=1(4π)D×1dλλ1−D∞λd u1∞1/λd u2(u1u2−1)D/2−2where C F=(N2c−1)/2N c,and D is the dimension of space-time.For D=4,the integrand diverges asλ→0.To regulate the integral,we assume D<2and use a triple integration by parts inλto obtain an expression which can be analytically continued to the vicinity of D=4.Next we set D=4+2ǫ,expand inǫ,write the result as an integral overω+,and introduce back the continuum threshold.This givesK pert(T,ω0,y)=−αsy+1 2ω0dω+ω3+e−ω+/T(16)× 12−23∂µ+3αs9π¯Λ,(17)which shows that divergences arise at orderαs.At this order,the renormalization of the sum rule is thus accomplished by a renormalization of the“bare”parameter G2in(12).In the9π¯Λ 1µ2 +O(g3s).(18)Hence a counterterm proportional to¯Λξ(y)has to be added to the bracket on the left-hand side of the sum rule(12).To evaluate its eﬀect on the right-hand side,we note that in D dimensions[17]¯Λξ(y)F2e−2¯Λ/T=3y+1 2ω0dω+ω3+e−ω+/T(19)× 1+ǫ γE−ln4π+2lnω+−ln y+12T ξ(y) F2e−2¯Λ/T=αsy+1 2ω0dω+ω3+e−ω+/T 2lnµ6+ y r(y)−1+ln y+1According to Luke’stheorem,theuniversalfunction χ3(y )vanishes at zero recoil [10].Evaluating (20)for y =1,we thus obtain a sum rule for G 2(µ)and δΛ2.It reads G 2(µ)−¯ΛδΛ224π3ω00d ω+ω3+e −ω+/T ln µ12 +K cond (T,ω0,1),(21)where we have used that r (1)=1.Precisely this sum rule has been derived previously,starting from a two-current correlator,in Ref.[16].This provides a nontrivial check of our ing the fact that ξ(y )=[2/(y +1)]2+O (g s )according to (19),we ﬁnd that the µ-dependent terms cancel out when we eliminate G 2(µ)and δΛ2from the sum rule for χ3(y ).Before we present our ﬁnal result,there is one more eﬀect which has to be taken into account,namely a spin-symmetry violating correction to the continuum threshold ω0.Since the chromo-magnetic interaction changes the masses of the ground-state mesons [cf.(10)],it also changes the masses of higher resonance states.Expanding the physical threshold asωphys =ω0 1+d M8π3 22 δ3 ω032π2ω30e −ω0/T 26π2−r (y )−ξ(y ) δ0 ω096π 248T 1−ξ(y ).It explicitly exhibits the fact that χ3(1)=0.III.NUMERICAL ANALYSISLet us now turn to the evaluation of the sum rule (23).For the QCD parameters we take the standard values¯q q =−(0.23GeV)3,αs GG =0.04GeV4,¯q g sσαβGαβq =m20 ¯q q ,m20=0.8GeV2.(24) Furthermore,we useδω2=−0.1GeV from above,andαs/π=0.1corresponding to the scale µ=2¯Λ≃1GeV,which is appropriate for evaluating radiative corrections in the eﬀective theory[15].The sensitivity of our results to changes in these parameters will be discussed below.The dependence of the left-hand side of(23)on¯Λand F can be eliminated by using a QCD sum rule for these parameters,too.It reads[16]¯ΛF2e−2¯Λ/T=9T4T − ¯q g sσαβGαβq4π2 2T − ¯q q +(2y+1)4T2.(26) Combining(23),(25)and(26),we obtainχ3(y)as a function ofω0and T.These parameters can be determined from the analysis of a QCD sum rule for the correlator of two heavy-light currents in the eﬀective theory[16,18].Oneﬁnds good stability forω0=2.0±0.3GeV,and the consistency of the theoretical calculation requires that the Borel parameter be in the range0.6<T<1.0GeV.It supports the self-consistency of the approach that,as shown in Fig.2,weﬁnd stability of the sum rule(23)in the same region of parameter space.Note that it is in fact theδω2-term that stabilizes the sum rule.Without it there were no plateau.Over the kinematic range accessible in semileptonic B→D(∗)ℓνdecays,we show in Fig.3(a)the range of predictions forχ3(y)obtained for1.7<ω0<2.3GeV and0.7<T< 1.2GeV.From this we estimate a relative uncertainty of∼±25%,which is mainly due to the uncertainty in the continuum threshold.It is apparent that the form factor is small,not exceeding the level of1%.2Finally,we show in Fig.3(b)the contributions of the individual terms in the sum rule (23).Due to the large negative contribution proportional to the quark condensate,the terms of orderαs,which we have calculated in this paper,cancel each other to a large extent.As a consequence,ourﬁnal result forχ3(y)is not very diﬀerent from that obtained neglecting these terms[17].This is,however,an accident.For instance,the order-αs corrections would enhance the sum rule prediction by a factor of two if the ¯q q -term had the opposite sign. From thisﬁgure one can also deduce how changes in the values of the vacuum condensates would aﬀect the numerical results.As long as one stays within the standard limits,the sensitivity to such changes is in fact rather small.For instance,working with the larger value ¯q q =−(0.26GeV)3,or varying m20between0.6and1.0GeV2,changesχ3(y)by no more than±0.15%.In conclusion,we have presented the complete order-αs QCD sum rule analysis of the subleading Isgur-Wise functionχ3(y),including in particular the two-loop perturbative con-tribution.Weﬁnd that over the kinematic region accessible in semileptonic B decays this form factor is small,typically of the order of1%.When combined with our previous analysis [20],which predicted similarly small values for the universal functionχ2(y),these results strongly indicate that power corrections in the heavy quark expansion which are induced by the chromo-magnetic interaction between the gluonﬁeld and the heavy quark spin are small.ACKNOWLEDGMENTSIt is a pleasure to thank Michael Peskin for helpful discussions.M.N.gratefully acknowl-edgesﬁnancial support from the BASF Aktiengesellschaft and from the German National Scholarship Foundation.Y.N.is an incumbent of the Ruth E.Recu Career Development chair,and is supported in part by the Israel Commission for Basic Research and by the Minerva Foundation.This work was also supported by the Department of Energy,contract DE-AC03-76SF00515.REFERENCES[1]E.Eichten and B.Hill,Phys.Lett.B234,511(1990);243,427(1990).[2]B.Grinstein,Nucl.Phys.B339,253(1990).[3]H.Georgi,Phys.Lett.B240,447(1990).[4]T.Mannel,W.Roberts and Z.Ryzak,Nucl.Phys.B368,204(1992).[5]A.F.Falk,H.Georgi,B.Grinstein,and M.B.Wise,Nucl.Phys.B343,1(1990).[6]N.Isgur and M.B.Wise,Phys.Lett.B232,113(1989);237,527(1990).[7]J.D.Bjorken,Proceedings of the18th SLAC Summer Institute on Particle Physics,pp.167,Stanford,California,July1990,edited by J.F.Hawthorne(SLAC,Stanford,1991).[8]M.B.Voloshin and M.A.Shifman,Yad.Fiz.45,463(1987)[Sov.J.Nucl.Phys.45,292(1987)];47,801(1988)[47,511(1988)].[9]A.F.Falk,B.Grinstein,and M.E.Luke,Nucl.Phys.B357,185(1991).[10]M.E.Luke,Phys.Lett.B252,447(1990).[11]A.F.Falk,M.Neubert,and M.E.Luke,SLAC preprint SLAC–PUB–5771(1992),toappear in Nucl.Phys.B.[12]M.Neubert,V.Rieckert,B.Stech,and Q.P.Xu,in Heavy Flavours,edited by A.J.Buras and M.Lindner,Advanced Series on Directions in High Energy Physics(World Scientiﬁc,Singapore,1992).[13]A.V.Radyushkin,Phys.Lett.B271,218(1991).[14]D.J.Broadhurst and A.G.Grozin,Phys.Lett.B274,421(1992).[15]M.Neubert,Phys.Rev.D45,2451(1992).[16]M.Neubert,Phys.Rev.D46,1076(1992).[17]M.Neubert,Phys.Rev.D46,3914(1992).[18]E.Bagan,P.Ball,V.M.Braun,and H.G.Dosch,Phys.Lett.B278,457(1992);E.Bagan,P.Ball,and P.Gosdzinsky,Heidelberg preprint HD–THEP–92–40(1992).[19]B.Blok and M.Shifman,Santa Barbara preprint NSF–ITP–92–100(1992).[20]M.Neubert,Z.Ligeti,and Y.Nir,SLAC preprint SLAC–PUB–5915(1992).[21]M.Neubert,SLAC preprint SLAC–PUB–5992(1992).[22]M.A.Shifman,A.I.Vainshtein,and V.I.Zakharov,Nucl.Phys.B147,385(1979);B147,448(1979).FIGURESFIG.1.Diagrams contributing to the sum rule for the universal form factorχ3(v·v′):two-loop perturbative contribution(a),and nonperturbative contributions proportional to the quark con-densate(b),the gluon condensate(c),and the mixed condensate(d).Heavy quark propagators are drawn as double lines.The square represents the chromo-magnetic operator.FIG.2.Analysis of the stability region for the sum rule(23):The form factorχ3(y)is shown for y=1.5as a function of the Borel parameter.From top to bottom,the solid curves refer toω0=1.7,2.0,and2.3GeV.The dashes lines are obtained by neglecting the contribution proportional toδω2.FIG.3.(a)Prediction for the form factorχ3(v·v′)in the stability region1.7<ω0<2.3 GeV and0.7<T<1.2GeV.(b)Individual contributions toχ3(v·v′)for T=0.8GeV and ω0=2.0GeV:total(solid),mixed condensate(dashed-dotted),gluon condensate(wide dots), quark condensate(dashes).The perturbative contribution and theδω2-term are indistinguishable in thisﬁgure and are both represented by the narrow dots.11。

SrVO3 Hubbard band 和氧空位

PHYSICAL REVIEW B94,241110(R)(2016)Hubbard band versus oxygen vacancy states in the correlated electron metal SrVO3S.Backes,1T.C.R¨o del,2,3F.Fortuna,2E.Frantzeskakis,2P.Le F`e vre,3F.Bertran,3M.Kobayashi,4R.Yukawa,4 T.Mitsuhashi,4M.Kitamura,4K.Horiba,4H.Kumigashira,4R.Saint-Martin,5A.Fouchet,6B.Berini,6Y.Dumont,6A.J.Kim,1F.Lechermann,7,8H.O.Jeschke,1M.J.Rozenberg,9R.Valent´ı,1,*and A.F.Santander-Syro2,†1Institut f¨u r Theoretische Physik,Goethe-Universit¨a t Frankfurt,Max-von-Laue-Strasse1,60438Frankfurt am Main,Germany 2CSNSM,Univ.Paris-Sud,CNRS/IN2P3,Universit´e Paris-Saclay,91405Orsay Cedex,France3Synchrotron SOLEIL,L’Orme des Merisiers,Saint-Aubin-BP48,91192Gif-sur-Yvette,France 4Photon Factory,Institute of Materials Structure Science,High Energy Accelerator Research Organization(KEK),1-1Oho,Tsukuba305-0801,Japan5SP2M-ICMMO-UMR-CNRS8182Universit´e Paris-Sud,Universit´e Paris-Saclay,91405Orsay Cedex,France 6GEMaC,Universit´e de Versailles St.Quentin en Y.-CNRS,Universit´e Paris-Saclay,Versailles,France7Institut f¨u r Theoretische Physik,Universit¨a t Hamburg,Jungiusstrasse9,20355Hamburg,Germany8Institut f¨u r Keramische Hochleistungswerkstoffe,TU Hamburg-Harburg,D-21073Hamburg,Germany 9Laboratoire de Physique des Solides,CNRS,Univ.Paris-Sud,Universit´e Paris-Saclay,91405Orsay Cedex,France(Received22February2016;published19December2016)We study the effect of oxygen vacancies on the electronic structure of the model strongly correlatedmetal SrVO3.By means of angle-resolved photoemission spectroscopy(ARPES)synchrotron experiments,we investigate the systematic effect of the UV dose on the measured spectra.We observe the onset of a spuriousdose-dependent prominent peak at an energy range where the lower Hubbard band has been previously reportedin this compound,raising questions on its previous interpretation.By a careful analysis of the dose-dependenteffects we succeed in disentangling the contributions coming from the oxygen vacancy states and from the lowerHubbard band.We obtain the ARPES spectrum in the limit of a negligible concentration of vacancies,wherea clear signal of a lower Hubbard band remains.We support our study by means of state-of-the-art ab initiocalculations that include correlation effects and the presence of oxygen vacancies.Our results underscore therelevance of potential spurious states affecting ARPES experiments in correlated metals,which are associatedwith the ubiquitous oxygen vacancies as extensively reported in the context of a two-dimensional electron gas atthe surface of insulating d0transition metal oxides.DOI:10.1103/PhysRevB.94.241110Introduction.A major challenge of modern physics is to understand the fascinating phenomena in strongly correlated transition metal oxides(TMOs),which emerge in the neighbor-hood of the Mott insulator state.Some preeminent examples that have gathered interest for almost30years are high temperature superconductivity,colossal magnetoresistance, heavy fermion physics,and,of course,the Mott metal-insulator transition itself[1].Signiﬁcant theoretical progress was made with the introduction of dynamical meanﬁeld theory (DMFT)and its combination with ab initio density functional methods[local density approximation(LDA)+DMFT],which allows treatment of the interactions promoting itinerancy and localization of electrons on equal footing[2–4].Among the most emblematic achievements of DMFT is the prediction of a Hubbard satellite,which separates from the conduction band of a metal.This satellite results from the partial localization of conduction electrons due to their mutual Coulomb repulsion. Early DMFT studies also showed that it is the precursor of the localized electronic states of a Mott insulator[5].Since then,these predictions promoted a large number of studies using photoemission spectroscopy,which is a technique to directly probe the presence of Hubbard bands.In this context, the TMO system SrVO3has emerged as the drosophila model system to test the predictions of strongly correlated electron *valenti@itp.uni-frankfurt.de†andres.santander@csnsm.in2p3.fr theories.In fact,SrVO3is arguably the simplest correlated metal.It is a simple cubic perovskite,with nominally one electron per V site,which occupies a threefold degenerate t2g conduction band.While the presence of a satellite in the photoemission spectra of Ni metal was already well known,in the context of correlated TMOs,the Hubbard band was originally reported in a systematic investigation of Ca1−x Sr x VO3[6],which was followed by many subsequent studies,including angle-resolved photoemission spectroscopy (ARPES)[7–9]and comparisons with theoretical predictions (see,for instance,Refs.[10–20],among others).One of the most salient features in SrVO3is the observation of a broad peak at an energy of about−1.5eV in angle integrated photoemission spectra(upper black curve in Fig.1), which is interpreted as a Hubbard satellite linked to the V t2g electrons.This feature is also seen in a large range of 3d1materials[21,22].The ratio of spectral strength between the quasiparticle(QP)state and the incoherent satellite in SrVO3is an important indicator of the magnitude of electron correlations[1,2].However,photoemission experiments using different photon energies or light brilliance have reported very dissimilar values for such a ratio[11],making the quantitative benchmarking of realistic ab initio theories for correlated electron systems difﬁcult[6,11,18,23,24].Moreover,as shown in Fig.1,a broad peak at about the same energy is also observed in several d0TMO cubic perovskites,such as SrTiO3, KTaO3,or anatase TiO2.Nevertheless,in all these cases the feature has been clearly linked to the presence of oxygenS.BACKES et al.PHYSICAL REVIEW B94,241110(R)(2016)FIG.1.Integrated UV photoemission spectra for various per-ovskite oxides,showing a quasiparticle peak at E F and an in-gap state at energies between1and1.5eV.For SrVO3(upper black curve),a correlated-electron metal,the QP peak corresponds to the bulk conduction band,and as will be shown further,the in-gap state is a superposition of the lower Hubbard band and localized electronic states associated with oxygen vacancies.For the other d0 oxides,such as KTaO3(blue curve),anatase TiO2(green curve), or SrTiO3(red curve),the QP peak and in-gap state correspond respectively to a conﬁned quasi-2D electron gas at the sample surface and to localized states,all formed by oxygen vacancies.The crystal orientation(normal to the samples’surface)is indicated in all cases. defects[25–32].Interestingly,recent ab initio calculations show that the spectral weight at−1.3eV in SrTiO3most likely is not of Ti t2g orbital character,but should be understood as an in-gap defect state with Ti e g character[33–36].Thus,we are confronted with the fact that at about1.5eV below the Fermi level(E F),weﬁnd the lower Hubbard bands of d1systems as well as the in-gap states of oxygen-deﬁcient d0systems.In view of these observations one may unavoidably wonder(and worry),despite the great success of DMFT methods,whether the putative Hubbard satellite of SrVO3might also originate from oxygen vacancy states.Moreover,one should also worry about the possibility of these extrinsic states affecting the features of the conduction band dispersion.In the present Rapid Communication we resolve these issues in a thorough manner.We present a systematic photoemission study of SrVO3,to demonstrate dramatic consequences in the spectra due to the creation of oxygen ing ARPES,we directly show that the UV or x-rays used for measurements can produce a large enhancement, of almost an order of magnitude,of the peak at−1.5eV, similar to the effect observed in d0oxide insulators[25–28,37].Despite these signiﬁcant effects on the energy states around the Mott-Hubbard band,we are able to determine the bulk SrVO3photoemission spectrum in the limit of a negligible concentration of vacancies,where a clear signal of the dispersive correlated Hubbard band remains.We support the interpretation of the experimental data by means of state-of-the-art LDA+DMFT calculations on SrVO3with oxygen vacancies.Consistent with our experimental data,the calculations show that oxygen vacancies produce states(of e g symmetry)at energies near the Hubbard satellite.While our study provides deﬁnite evidence of a correlated Hubbard band in SrVO3as predicted by DMFT,it also underlines the signiﬁcant effects due to oxygen vacancies,which may also affect photoemission data in other TMOs.Methods.The bulk-like relaxed,crystalline(001)oriented SrVO3thinﬁlms were grown by pulsed laser deposition (PLD)either at the GEMaC laboratory,then measured at the CASSIOPEE beamline of Synchrotron SOLEIL,or in a PLD chamber directly connected to the ARPES setup at beamline2A of KEK-Photon Factory(KEK-PF)[9,38,39]. To clean the surfaces in UHV prior to ARPES experiments at SOLEIL,the SrVO3thinﬁlms were annealed at550◦C for t=5–20min at pressures lower than2×10−8Torr.At KEK-PF,the PLD growth was performed under a pressure below10−7Torr,to obtain UHV-clean surfaces,using a Sr2V2O7target,which has excess oxygen with respect to SrVO3,thus minimizing the formation of vacancies during the growth.In all cases,the surface quality was conﬁrmed right before ARPES measurements by low-energy electron diffraction(LEED).The thinﬁlms measured at KEK-PF showed a c(4×4)surface reconstruction,which does not affect the analysis and conclusions of this work.For the ARPES measurements we used linearly polarized photons in the energy range30–110eV and hemispherical electron analyzers with vertical slits at SOLEIL and horizontal slits at KEK-PF.The angular and energy resolutions were0.25◦and 15meV.The mean diameter of the incident photon beam was smaller than100μm.The UV light brilliance,measured using calibrated photodiodes,was≈5×109photons s−1μm−2at SOLEIL,and about100times smaller at KEK-PF.The samples were cooled down to T=20K before measuring.Unless speciﬁed otherwise,all data were taken at that temperature. The results were reproduced on more thanﬁve samples.All through this Rapid Communication,directions and planes are deﬁned in the cubic unit cell of SrVO3.We denote [hkl]the crystallographic directions in real space, hkl the corresponding directions in reciprocal space,and(hkl)the planes orthogonal to those directions.The indices h,k,and l of hkl correspond to the reciprocal lattice vectors of the cubic unit cell of SrVO3.The Supplemental Material[40]presents further details about the sample growth and measurements.Experimental results.Figure2(a)shows the integrated photoemission spectra of SrVO3as a function of the UV dose, measured at CASSIOPEE SOLEIL under the same conditions of light brilliance of any standard ARPES experiment at a third-generation synchrotron.The measurements were done by continuously irradiating the sample with hν=33eV photons while recording the spectra as a function of irradiation time, with an accumulation time of about2min per spectrum.The blue and black curves show spectra for the lowest and highest measured doses,obtained respectively after∼2min and∼2h of irradiation.These data clearly demonstrate that the very UV or x-rays used for photoemission experiments can produce radical changes in the measured spectra of SrVO3.Note in fact that a similar effect has been observed for VO2[41].In particular,from Fig.2(a)we observe that the amplitude ofHUBBARD BAND VERSUS OXYGEN V ACANCY STATES IN THE...PHYSICAL REVIEW B94,241110(R)(2016)FIG.2.(a)Photoemission spectra of SrVO3as a function of UV dose,measured at Synchrotron SOLEIL(hν=33eV).The energy distribution curves(EDCs)were extracted from raw ARPES data around the 002point integrated along the k= 010 direction.(b)Corresponding momentum distribution curves(MDCs)integrated over50meV below E F.Peaks in the MDCs indicate the Fermi momenta.(c),(d)Same as(a),(b)for SrTiO3(hν=47eV).Theﬁlling of a2DEG upon UV irradiation is evidenced by the formation of QP peaks in the EDCs and MDCs at E F[inset of(c)and(d),respectively].All data were taken at20K.the in-gap state at−1.5eV,and,more signiﬁcantly,the ratio of in-gap to quasiparticle(QP)amplitudes,strongly increase with increasing UV dose,going from about1:3in a pristine sample to more than2:1in a heavily irradiated sample. Importantly,note that the QP peak position remains basically dose independent,implying that the carriers created by the UV or x irradiation do not signiﬁcantly dope the conduction band,and form dominantly localized states.This is conﬁrmed in Fig.2(b),which shows that the Fermi momenta of the QP band,given by the peak positions in the momentum distribution curves(MDCs)at E F,are also dose independent. Additional data presented in the Supplemental Material further demonstrate that our measurements yield the expected3D bulk Fermi surface of SrVO3.Thus,the observed increase in intensity of the in-gap state upon UV irradiation cannot be ascribed to a change inﬁlling of the conduction band,which could have affected the electron correlations.Instead,this unambiguously shows the light-assisted formation of localized defect states at essentially the same energy as that of the expected intrinsic lower Hubbard band—which should then resemble the in-gap peak observed at the lowest UV doses.In fact,as mentioned previously,it is well established that strong doses of UV or x-rays create a large concentration of oxygen vacancies in several d0perovskites[25–32,42]. As illustrated in Figs.2(c)and2(d)for the case of SrTiO3, the progressive doping of the surface region with oxygen vacancies,due to synchrotron UV irradiation,has two effects: the formation of a very intense in-gap state at about−1.3eV, and,in contrast to SrVO3,the simultaneous creation of a sharp QP peak at E F corresponding to a conﬁned quasi-2D electron gas(2DEG)at the samples’surface.The effective mass of such2DEG,precisely determined by ARPES,matches the mass expected from density functional theory calcula-tions[25,26,43,44].Thus,as in SrVO3,the increase in intensityof the in-gap state observed in SrTiO3upon UV or x irradiationcannot be due to an onset or increase of electron correlations,and should be ascribed to an extrinsic effect.We therefore conclude that,in SrVO3,exposure to syn-chrotron UV or x-rays creates oxygen vacancies,which are inturn responsible for the extrinsic increase in intensity of thein-gap state evidenced by our measurements.This effect canseriously obscure the determination of the spectral function ofthis model system,thus hampering the advancement of validtheories for correlated electron systems.Theﬁndings described above imply that the correct ex-perimental determination of the vacancy-free photoemissionspectrum of SrVO3should(i)use samples that from thebeginning have the lowest possible concentration of oxygenvacancies,and(ii)use doses of UV or x-ray light low enoughto avoid light-induced changes in the measured spectra.Tothis end,we measured SrVO3thinﬁlms grown directly insitu at beamline2A of KEK-PF.As mentioned before,the growth protocol of such thinﬁlms minimizes the formationof vacancies,while the UV light brilliance at KEK-PF is ∼100times smaller than the one in Figs.2(a)and2(b)from measurements at SOLEIL.We checked(see the SupplementalMaterial)that under these conditions the spectra did notchange with time,even after several hours of measurements.The resulting energy-momentum ARPES map,and its secondderivative,are presented in Figs.3(a)and3(b).One clearlyobserves the dispersing QP band along with an also dispersivein-gap state of weaker intensity,corresponding to the intrinsiclower Hubbard band,as reported in previous works[7].Theintrinsic spectral function of SrVO3will then be given by such aphotoemission spectrum,which approaches the vacancy-freelimit,modulo dipole-transition matrix elements,inherent toS.BACKES et al.PHYSICAL REVIEW B94,241110(R)(2016)FIG.3.(a)Energy-momentum ARPES intensity map measured at KEK-PF with a low UV dose on a SrVO3sample prepared in situ,using a well-established protocol to minimize the formation of oxygen vacancies(see the main text and Supplemental Material).Note that due to the choice of light polarization,the heavy bands along(100)are not observed and only the contribution of the light d xy band is detected.The data were acquired at hν=88eV around ¯103.(b)Second derivative(negative values)of the map in(a).The use of second derivatives allows a better visualization of the dispersion of both the quasiparticle and Mott-Hubbard bands on the same color plot.The dispersionless feature at E F is a spurious effect of such a second derivative on the Fermi-Dirac cutoff.(c),(d)Same as(a),(b)after a strong UV irradiation dose,measured at SOLEIL(hν=33eV),typical of modern third-generation synchrotrons.The measurements were done at hν=33eV close to 002.All data were taken at20K.Note that at constant photon energy,ARPES maps out the electronic structure at a spherical surface of three-dimensional (3D)k space,which can be locally approximated to a plane for all our measurements(details in the Supplemental Material).The different choice of photon energies and k-space positions for measurements at KEK-PF[(a)and(b)]and SOLEIL[(c)and(d)]was dictated by the different geometrical conﬁgurations and constraints of the beamlines in both synchrotrons.the photoemission process,which can still modulate the intensity of the QP peak relative to the Hubbard peak.A calculation of such matrix elements requires a full one-step calculation of the photoemission process,which is beyond the scope of this work.By contrast,Figs.3(c)and3(d) show the momentum-resolved electronic structure of a sample, measured at SOLEIL,that was intensively irradiated.There, the peak at−1.5eV becomes broader,more intense,and nondispersive—all characteristic signatures of a high random concentration of oxygen vacancies.Numerical calculations.To rationalize from a microscopic point of view the inﬂuence of oxygen vacancies on the electronic structure of SrVO3,we performed charge self-consistent LDA+DMFT calculations for bulk SrVO3and var-ious relaxed oxygen-deﬁcient SrVO3supercells.The latter are computationally demanding calculations.We shall focus here on the case of a2×2×3supercell with two oxygen vacancies located at opposite apical sites of one vanadium atom,as shown in the inset of Fig.4(b).We use such a vacancy arrangement as it is the prototypical one for d0compounds[43].For our LDA+DMFT calculations we chose values of U= 2.5eV and J=0.6eV for vanadium and included the effects of bandwidth renormalization due to dynamically screened Coulomb interactions by following the prescription suggested in Ref.[45](the LDA+DMFT unrenormalized data are shown in the Supplemental Material).In Figs.4(a)and4(c)we show, respectively,the results of the k-integrated and k-resolved spectral functions for bulk SrVO3without oxygen vacancies. Weﬁnd the expected features of a t2g quasiparticle peak at the Fermi level and a lower Hubbard band at negative energies of the same t2g nature,in agreement with the photoemission spectra in Fig.2(a)and Figs.3(a)and3(b).The light band at E F along k 100 [Fig.4(c)]consists of two degenerate bands of d xy and d xz characters,while the heavy band along the same direction has d yz character.While comparing with the measured k-resolved spectral function[Figs.3(a)and3(b)],HUBBARD BAND VERSUS OXYGEN V ACANCY STATES IN THE...PHYSICAL REVIEW B94,241110(R)(2016)FIG.4.LDA+DMFT results for SrVO3including bandwidth renormalization effects[45].(a)k-integrated spectral function for bulk SrVO3.The V t2g orbitals show a quasiparticle peak at E F and a lower Hubbard band at−1.6eV.(b)Spectral function for the2×2×3supercell of SrVO3with two oxygen vacancies.An additional nondispersive V e g vacancy state originating from the V atom neighboring the oxygen vacancies leads to a sharp peak below the Fermi level at∼−1.0eV.The V t2g orbitals show a quasiparticle peak at E F and a lower Hubbard band at−1.8eV.(c)and(d)show the corresponding spectral functions(multiplied by a Fermi-Dirac function at20K)along the X- -X path.one should bear in mind that along -X(or -Y)the heavy d yz(or d xz)bands are silenced by dipole-transition selection rules in the experiment[25].Inclusion of bandwidth renormalization[45]renders the lower Hubbard band at an energy(−1.6eV)in reasonable agreement with experiment (−1.5eV).We adopted typical values for U and J from the literature.We did not attempt to further optimize the values to get a better quantitative agreement with the experimental data, for two reasons:First is the heavy numerical cost,and second, as we show next in the calculations with oxygen vacancies, the adopted values facilitate the distinct visualization of the contributions from the Hubbard and localized states to the incoherent peak at∼−1.5eV.The removal of oxygen atoms in the system leads to the donation of two electrons per oxygen to its surrounding. Already at the level of density functional theory(DFT)in the local density approximation(LDA)(see the Supplemental Material),weﬁnd that most of the charge coming from the additional electrons is transferred to the3d z2orbitals of the neighboring V atom,developing into a sharp peak of e g symmetry located around−1.0eV,i.e.,at an energy close to the position of the experimentally observed oxygen vacancy states.In analogy to the experimental average over many lattice sites,note that averaging among various supercells with differentoxygen vacancy locations and concentrations(which is beyondthe scope of the present work)would result in a wider in-gape g band,as demonstrated for the case of SrTiO3(see Fig.3ofRef.[34])and for some cases in SrVO3(see the SupplementalMaterial,Fig.S7).By including electronic correlations within(bandwidth renormalized)LDA+DMFT we then see that allthe experimental observations qualitatively emerge.In fact,the conducting t2g orbitals develop a lower Hubbard bandpeaked at energies about−1.8eV[Figs.4(b)and4(d)],similarto the bulk case without oxygen vacancies.Most notably,this lower Hubbard satellite does not increase in amplitudewith the introduction of vacancies,but rather broadens.Inaddition,the oxygen vacancy defect states situated at about −1eV remain qualitatively unchanged by the correlation effects,but experience a broadening with respect to the pureLDA case.This is in agreement with the photoemission data,evidencing that the increase in intensity of the in-gap statein the oxygen-deﬁcient SrVO3is not to be attributed to anincrease in population of the lower Hubbard satellite,butinstead to the manifestation of vacancy states of e g character.Conclusions.In summary,we performed a detailed study of the effects of oxygen vacancies in the spectroscopy of the archetypal strongly correlated electron system SrVO3.We found that oxygen vacancy states,which are created by UV or x-ray irradiation,occur at energies close to the Hubbard satellite.This dramatically affects the measured line shape of the Mott-Hubbard band and the ratio of intensities between the quasiparticle and the Mott-Hubbard peaks.By means of a systematic study under a controlled irradiation dose, using samples directly grown in situ,we were able to obtain the photoemission spectrum of the bulk SrVO3system in the limit of a negligible concentration of oxygen vacancies. Our experimental interpretation is supported by LDA+DMFT calculations,which provided further insight into the likely nature of the oxygen vacancy states.Acknowledgments.We thank Silke Biermann,Ralph Claessen,Marc Gabay,and Michael Sing for discussions. This work was supported by public grants from the French National Research Agency(ANR),project LACUNES No. ANR-13-BS04-0006-01,and the“Laboratoire d’Excellence Physique Atomes Lumi`e re Mati`e re”(LabEx PALM project ELECTROX)overseen by the ANR as part of the“Investisse-ments d’Avenir”program(reference:ANR-10-LABX-0039). 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axions

AXIONSGEORG RAFFELTMax-Planck-Institut für Physik(Werner-Heisenberg-Institut),Föhringer Ring6,80805München,Germany(e-mail:*****************.de)(Received7August2001;accepted29August2001)Abstract.Axions are one of the few particle-physics candidates for dark matter which are well motivated independently of their possible cosmological role.A brief review is given of the theoreticalmotivation for axions,their possible role in cosmology,the existing astrophysical limits,and thestatus of experimental searches.1.IntroductionDespite its uncanny success,the particle-physics standard model has many looseends,among them the CP problem of quantum chromodynamics(QCD).The non-trivialﬁeld structure of the QCD ground state(‘ -vacuum’)and a phase of thequark mass matrix each induce a non-perturbative CP-violating term in the QCDLagrangian which is proportional to the coefﬁcient = QCD+arg det M quark, where could lie anywhere between0and2π.The experimental upper limit to aputative neutron electric dipole moment,a CP-violating quantity,informs us that 10−9,a severeﬁne-tuning problem given that is a sum of two unrelated terms which would be expected to be of order unity each.One particularly elegant solution was proposed by Peccei and Quinn,where theparameter is re-interpreted as a dynamical variable, →a(x)/f a,where a(x)isthe axionﬁeld and f a an energy scale called the Peccei-Quinn scale or axion decayconstant(Peccei and Quinn,1977a,b;Weinberg,1977;Wilczek,1977).The previ-ous CP-violating term automatically includes a potential for the axionﬁeld whichdrives it to its CP-conserving minimum(dynamical symmetry restoration).Whilethis may sound complicated,Sikivie(1996)has constructed a beautiful mechanicalanalogy which nicely explains the main features of axion physics.While axions would be very weakly interacting,they are still a QCD phenom-enon.They share their quantum numbers with neutral pions;all generic axionproperties are roughly determined by those ofπ0,scaled with fπ/f a where fπ= 93MeV is the pion decay constant.For example,the axion mass is roughly given by m a f a=mπfπ,and the coupling to photons or nucleons is roughly suppressed by fπ/f a relative to the pion couplings.Axions have not been found during the quarter century since they wereﬁrstproposed,but the interest in this hypothesis is well alive because other proposed Space Science Reviews100:153–158,2002.©2002Kluwer Academic Publishers.Printed in the Netherlands.154G.RAFFELTsolutions of the strong CP problem are not clearly superior,and mainly because axions are one of the few well-motivated particle candidates for the cold dark matter which apparently dominates the dynamics of the universe.The current status of axions physics and astrophysics was reviewed at a recent conference(Sikivie,1999).Particle-physics aspects,the status of astrophysical limits,and that of current search experiments are summarized in three separate mini-reviews in the Review of Particle Physics(Groom et al.,2000).Chapters on axions are also found in some textbooks(Kolb and Turner,1990;Raffelt,1996).For theoretical reviews see Kim(1987)and Cheng(1988),for a review of experimental searches see Rosenberg and van Bibber(2000).2.Stellar-Evolution LimitsThe main argument which proves that the Peccei-Quinn scale f a must be very large, corresponding to a very small axion mass m a,is related to stellar evolution.Axions would be produced by various processes in the hot and dense interior of stars and would thus carry away energy directly,much in analogy to the standard thermal neutrino losses.The strength of the axion interaction with photons,electrons,and nucleons can be constrained from the requirement that stellar-evolution time scales are not modiﬁed beyond observational limits(Raffelt,1996).For example,the helium-burning lifetime of horizontal-branch stars inferred from number counts in globular clusters reveals that the Primakoff processγ+Ze→Ze+a must not be too efﬁcient in these stars,leading to a limit of m a 0.4eV(Figure1).Very restrictive limits arise from the observed neutrino signal of the supernova (SN)1987A.After collapse,the SN core is so hot and dense that neutrinos are trapped and escape only by diffusion so that it takes several seconds to cool a roughly solar-mass object the size of a few ten kilometers.The emission of axions would remove energy from the deep inner core which should show up in late-time neutrinos.Therefore,the observed duration of the SN1987A neutrino signal provides the most restrictive limits on the axion-nucleon coupling(Figure1).In the early papers on this topic,the difﬁculty of calculating the axion emission from a dense and hot nuclear medium had been underestimated;the most recent discussions attempt an inclusion of dense-medium effects(Janka et al.,1996).If axions are too‘strongly’interacting,they are trapped in a SN core,inval-idating the energy-loss argument and implying a mass above which axions are not excluded by the SN1987A signal(Turner,1988;Burrows et al.,1990).They would still carry away some of the energy and would cause excess counts in the water Cherenkov detectors which registered the neutrinos,allowing one to exclude another interval of axion masses(Engel et al.,1990).Probably there is a small crack of allowed axion masses between these two SN1987A arguments(Figure1), sometimes called the‘hadronic axion window’.Therefore,inﬁne-tuned axion models where the tree-level coupling to photons nearly vanishes,eV-mass axionsAXIONS155Figure1.Astrophysical and cosmological exclusion regions(hatched)for the axion mass m a or the Peccei–Quinn scale f a.An‘open end’of an exclusion bar means that it represents a rough estimate. The globular cluster limit depends on the axion-photon coupling;it was assumed that E/N=83as in GUT models or the DFSZ model.The SN1987A limits depend on the axion-nucleon couplings; the shown case corresponds to the KSVZ model and approximately to the DFSZ model.The dot-ted‘inclusion regions’indicate where axions could plausibly be the cosmic dark matter.Most of the allowed range in the inﬂation scenario requiresﬁne-tuned initial conditions.Also shown is the projected sensitivity range of the search experiments for galactic dark-matter axions.may be allowed and could thus play the role of a cosmological hot dark matter component(Moroi and Murayama,1998).The axion coupling to electrons can be constrained from the properties of glob-ular-cluster stars and the white-dwarf luminosity function.However,the tree-level existence of such a coupling is not generic,and the resulting limits on m a and f a do not extend the range covered by the previous arguments.3.CosmologyIn the early universe,axions come into thermal equilibrium only if f a 108GeV, a region excluded by the stellar-evolution limits.For f a 108GeV cosmic axions are produced nonthermally.If inﬂation occurred after the Peccei-Quinn symmetry breaking or if T reheat<f a,the‘misalignment mechanism’(Preskill et al.,1983; Abbott and Sikivie,1983;Dine and Fischler,1983;Turner,1986)leads to a contri-bution to the cosmic critical density of a h2≈1.9×3±1(1µeV/m a)1.175 2i F( i)156G.RAFFELTwhere h is the Hubble constant in units of100km s−1Mpc−1.The stated range re-ﬂects recognized uncertainties of the cosmic conditions at the QCD phase transition and of the temperature-dependent axion mass.The function F( )with F(0)=1 and F(π)=∞accounts for anharmonic corrections to the axion potential.Be-cause the initial misalignment angle i can be very small or very close toπ,there is no real prediction for the mass of dark-matter axions even though one wouldexpect 2i F( i)∼1to avoidﬁne-tuning the initial conditions.A possibleﬁne-tuning of i is limited by inﬂation-induced quantumﬂuctu-ations which in turn lead to temperatureﬂuctuations of the cosmic microwave background(Lyth,1990;Turner and Wilczek,1991;Linde,1991).In a broad class of inﬂationary models one thusﬁnds an upper limit to m a where axions could be the dark matter.According to the most recent discussion(Shellard and Battye,1998) it is about10−3eV(Figure1).If inﬂation did not occur at all or if it occurred before the Peccei-Quinn symme-try breaking with T reheat>f a,cosmic axion strings form by the Kibble mechanism (Davis,1986).Their motion is damped primarily by axion emission rather than gravitational waves.After axions acquire a mass at the QCD phase transition they quickly become nonrelativistic and thus form a cold dark matter component.The axion density is similar to that from the misalignment mechanism,but in detail the calculations are difﬁcult and somewhat controversial between one group of authors(Davis,1986;Davis and Shellard,1989;Battye and Shellard,1994a,b)and another(Harari and Sikivie,1987;Hagmann and Sikivie,1991;Hagmann et al., 2001).Taking into account the uncertainty in various cosmological parameters one arrives at a plausible range for dark-matter axions as indicated in Figure1.4.Experimental SearchIf axions are indeed the dark matter of our galaxy one can search for them by the ‘haloscope’method(Sikivie,1983).The generic two-photon vertex which axions posess in analogy to neutral pions allows for the Primakoff conversion a↔γin the presence of external electromagneticﬁelds.Therefore,the galactic axions should excite a microwave resonator which is placed in a strong magneticﬁeld, i.e.,one expects a narrow line above the thermal noise of the cavity.While this line would not be difﬁcult to identify once it has been found,searching for it requires to step a tunable cavity through many resonance intervals in order to cover a given m a range.In the late1980s,this method was pioneered in two pilot experiments (Wuensch et al.,1989;Hagmann et al.,1990).At the present time two full-scale ‘second generation’axion haloscopes are in operation,one in Livermore,Califor-nia(Hagmann et al.,1998,2000)and one in Kyoto,Japan(Ogawa et al.,1996; Yamamoto et al.,2001),the latter one using a beam of Rydberg atoms as a low-noise microwave detector.The projected sensitivity shown in Figure1covers the lower end of the plausible mass range for dark-matter axions.If axions are indeedAXIONS157 the galactic dark matter,these experiments for theﬁrst time are in a position to actually detect them.Axions or axion-like particles are currently also searched by the‘helioscope’method(Sikivie,1983;van Bibber et al.,1989).Axions would be produced in the Sun by the Primakoff effect,and could be back-converted into X-rays in a long dipole magnet oriented toward the Sun.A dedicated experiment of this sort in Tokyo has recently reported new limits(Inoue et al.,2000)while a much larger ef-fort using a decommissioned LHC test magnet,the CAST experiment,is currently under construction at CERN(Zioutas et al.,1999).It should be noted,however, that these searches are unrelated to axion dark matter,i.e.,if axions were to show up at CAST they almost certainly could not provide the cosmic dark matter.The evidence for the reality of dark matter has mounted for several decades,and most recently culminated with the determination of the cosmological parameters by cosmic-microwave precision experiments and other arguments.On the other hand, the physical nature of dark matter remains as mysterious as it was two decades ago. Therefore,the direct search experiments for particle candidates such as axions are among the most important efforts in the area of experimental cosmology.AcknowledgementsThis research was supported,in part,by the Deutsche Forschungsgemeinschaft under grant No.SFB-375and by the ESF network Neutrino Astrophysics.ReferencesAbbott,L.and Sikivie,P.:1983,‘A Cosmological Bound on the Invisible Axion’.Phys.Lett.B120, 133–136.Battye,R.A.and Shellard,E.P.S.:1994a,‘Global String Radiation’.Nucl.Phys.B423,260–304. Battye,R.A.and Shellard,E.P.S.:1994b,‘Axion String Constraints’.Phys.Rev.Lett.73,2954–2957;(E)ibid.76,2203–2204(1996).Burrows,A.,Ressel,T.and Turner,M.:1990,‘Axions and SN1987A:Axion trapping’.Phys.Rev.D42,3297–3309.Cheng,H.-Y.:1988,‘The Strong CP Problem Revisited’.Phys.Rept.158,1–89.Davis,R.L.:1986,‘Cosmic Axions from Cosmic Strings’.Phys.Lett.B180,225–230.Davis,R.L.and Shellard,E.P.S.:1989,‘Do Axions Need Inﬂation?’.Nucl.Phys.B324,167–186. Dine,M.and Fischler,W.:1983,‘The Not So Harmless Axion’.Phys.Lett.B120,137–141. Engel,J.,Seckel,D.and Hayes,A.C.:1990,‘Emission and Detectability of Hadronic Axions from SN1987A’.Phys.Rev.Lett.65,960–963.Groom,D.E.et al.:2000,‘The Review of Particle Physics’.Eur.Phys.J.C15,1–878.See also /Hagmann,C.and Sikivie,P.:1991,‘Computer Simulation of the Motion and Decay of Global Strings’.Nucl.Phys.B363,247–280.Hagmann,C.,Chang,S.and Sikivie,P.:2001,‘Axion Radiation from Strings’.Phys.Rev.D63, 125018(12pp).158G.RAFFELTHagmann,C.et al.:1990,‘Results from a Search for Cosmic Axions’.Phys.Rev.D42,1297–1300. Hagmann,C.et al.:1998,‘Results from a High-Sensitivity Search for Cosmic Axions’.Phys.Rev.Lett.80,2043–2046.Hagmann,C.et al.:2000,‘Cryogenic Cavity Detector for a Large-Scale Cold Dark-Matter Axion Search’.Nucl.Instrum.Meth.A444,569–583.Harari,D.and Sikivie,P.:1987,‘On the Evolution of Global Strings in the Early Universe’.Phys.Lett.B195,361–365.Inoue,Y.et al.:2000,‘Recent Results from the Tokyo Axion Helioscope Experiment’.astro-ph/0012338.Janka,H.-T.,Keil,W.,Raffelt,G.and Seckel,D.:1996,‘Nucleon Spin Fluctuations and the Supernova Emission of Neutrinos and Axions’.Phys.Rev.Lett.76,2621–2624.Kim,J.E.:1987.‘Light Pseudoscalars,Particle Physics and Cosmology’.Phys.Rept.150,1–177. Kolb,E.W.and Turner,M.S.:1990,‘The Early Universe’.Addison-Wesley,Reading,Mass. Linde,A.:1991,‘Axions in Inﬂationary Cosmology’.Phys.Lett.B259,38–47.Lyth,D.H.:1990,‘A Limit on the Inﬂationary Energy Density from Axion Isocurvature Fluctua-tions’.Phys.Lett.B236,408–410.Moroi,M.and Murayama,H.:1998,‘Axionic Hot Dark Matter in the Hadronic Axion Window’.Phys.Lett.B440,69–76.Ogawa,I.,Matsuki,S.and Yamamoto,K.:1996,‘Interactions of Cosmic Axions with Rydberg Atoms in Resonant Cavities Via the Primakoff Process’.Phys.Rev.D53,R1740–R1744. Peccei,R.D.and Quinn,H.R.:1977a,‘CP Conservation in the Presence of Instantons’.Phys.Rev.Lett.38,1440–1443.Peccei,R.D.and Quinn,H.R.:1977b,‘Constraints Imposed by CP Conservation in the Presence of Instantons’.Phys.Rev.D16,1791–1797.Preskill,J.,Wise,M.and Wilczek,F.:1983,‘Cosmology of the Invisible Axion’.Phys.Lett.B120, 127–132.Raffelt,G.G.:1996,‘Stars as Laboratories for Fundamental Physics’.University of Chicago Press. Rosenberg,L.and van Bibber,K.:2000,‘Searches for Invisible Axions’.Phys.Rept.325,1–39. Sikivie,P.:1983,‘Experimental Tests of the“Invisible”Axion’.Phys.Rev.Lett.51,1415–1417;(E) ibid.52,695(1984).Sikivie,P.:1996,‘The Pool Table Analogy to Axion Physics’.Physics Today49,22–27. Sikivie,P.(ed.):1999,‘Proceedings Axion Workshop’.Nucl.Phys.B(Proc.Suppl.)72,1–238. Shellard,E.P.S.and Battye,R.A.:1998,‘Cosmic Axions’.astro-ph/9802216.Turner,M.S.:1986,‘Cosmic and Local Mass Density of“Invisible”Axions’.Phys.Rev.D33,889–896.Turner,M.S.:1988,‘Axions from SN1987A’.Phys.Rev.Lett.60,1797–1800.Turner,M.S.and Wilczek,F.:1991,‘Inﬂationary Axion Cosmology’.Phys.Rev.Lett.66,5–8.van Bibber,K.et al.:1989,‘Design for a Practical Laboratory Detector for Solar Axions’.Phys.Rev.D39,2089–2099.Weinberg,S.:1978,‘A New Light Boson?’.Phys.Rev.Lett.40,223–226.Wilczek,F.:1978,‘Problem of Strong P and T Invariance in the Presence of Instantons’.Phys.Rev.Lett.40,279–282.Wuensch,W.U.et al.:1989,‘Results of a Laboratory Search for Cosmic Axions and Other Weakly Coupled Light Particles’.Phys.Rev.D40,3153–3167.Yamamoto,K.et al.:2001,‘The Rydberg Atom Cavity Axion Search’.hep-ph/0101200. Zioutas,K.et al.:1999,‘A Decommissioned LHC Model Magnet as an Axion Telescope’.Nucl.Instrum.Meth.A425,482–489.See also http://axnd02.cern.ch/CAST/。

核子中奇异夸克分布不对称性与轻味夸克碎裂效应

29 10 2005 10HIGH ENERGY PHYSICS AND NUCLEAR PHYSICSVol.29,No.10Oct.,2005*( 100871)– , . , .D ..1, –[1—3]. ,. ,, –(DIS) (global analysis)[4,5], ,(intrinsic sea theory) ,µ CCFR NuTeV,[6,7]. ,. ,[4,8—12]NuTeVWeinberg [13,14],.. ,Fν2 F¯ν2,:Fν2−F¯ν2=2x[s(x)−¯s(x)]. ,,. c.CCFR NuTeV µ [6,15,16].νµs→µ−c νµd→µ−c,Cabibbo ,c ;, ¯c.CCFR NuTeV µµ+(µ−) c(¯c) ,c→H(c¯q)→µ+X. µ,µ ,. ,CCFR νµ(¯νµ) ,c(¯c) µ+(µ−) ¯B c(¯B¯c):¯Bc−¯B¯c0.1147∼0−20%[6]., ,µ . ,CCFR NuTeV µ,.( c ¯c10 965dξd y =G2s2|V cd|2].(1)s=2MEν ,r2≡(1+Q2/M2W)2.ξ . , c ,ξ Bjorken:ξ≈x(1+m2c /Q2). (1)f c≡1−m2c/2MEνξ c, [18]., ¯cd2σ¯νµN→µ+¯c Xπr2f c•ξ[¯s(ξ)|V cs|2+¯d(ξ)+¯u(ξ)dξd y−d2σ¯νµN→µ+¯c Xπr2f c•ξ (s(ξ)−¯s(ξ))|V cs|2+d v(ξ)+u v(ξ)2ξ[d v(ξ)+u v(ξ)] ,|V cs|2≃0.95 |V cd|2≃0.05[19] .12S−|V cs|2+Q V|V cd|2,(4)S−≡ ξ[s(ξ)−¯s(ξ)]dξ,Q V≡ ξ[d v(ξ)+u v(ξ)]dξ., NuTeV.[9—12],NuTeV ., (4) ,c ¯c P SA( 1).1 NuTeV c ¯c P SADing-Ma[9]Q2030%—80%0.007—0.01812%—26% Alwall-Ingelman[10]20GeV230%0.00915% Ding-Xu-Ma[11]Q2060%—100%0.014—0.02221%—29% Wakamatsu[12]16GeV270%—110%0.022—0.03530%—40%966 (HEP&NP) 29 2S+|V cs|2+(Q V+2Q S)|V cd|2.(5)S+≡ ξ[s(ξ)+¯s(ξ)]dξ,Q S≡ ξ[¯u(ξ)+¯d(ξ)]dξ.CTEQ5 Q2=16GeV2 S+,Q V,Q S, |V cs|2=0.95,|V cd|2=0.05,1 2S−/Q V 0.007(0.022), R 20%(25%). ,c ¯c, c¯c.3 µ, , c,( µ) ( ) . cH+d3σνµN→µ−H+Xdξd y D H+q(z),(6)D H+q(z) q H+ ,z H+ q . H+ c D+(c¯d) D0(c¯u) ,H− D−(¯c d) ¯D0(¯c u).c H+ H+ . , Lund , q¯qexp(−bm2q)[20], s¯s λ∼0.3[21,22], c¯c 10−5., . µ [17]. . , e+e− . , , c ¯c , D , c , , , c(¯c) , µ . , c ¯c D(c¯q) ¯D(¯c q). , , , :u→cu),d→D−(dξd y d z=G2s2|V cd|2]+δ dσνN→µ−µ+Xdξd y d z LQF=G2s2|V ud|2(1−y)2,(8)D q(z)≡D Dq(z)+D D∗q(z), D Dq(z)≡D¯D0u(z)=D D0¯u(z)=D D−d(z)=D D+¯d(z),D D∗q(z)≡D¯D∗0u(z)=D D∗0¯u(z)=D D∗−d(z)=D D∗+¯d(z). , D q(z) q , . (8) ,¯BD(∗)+=1dξd y d z=G2s2|V cd|2]+δ dσ¯νN→µ+µ−Xdξd y d z LQF=G2s2•|V ud|2(1−y)2.(10)10 967(σνN→µ−µ+X−σ¯νN→µ+µ−X)total≈−1Q V|V cd|2+2S−|V cs|2•D q¯BD(∗)+¯f c ¯Bc.D q¯BD(∗)+d x d y d z=G2s2|V ud|2D q(z)B¯D0,(12),B¯D0 ¯D0 µ− , ¯D∗0¯D0 , B¯D0 .,µ+µ+d3σ¯νN→µ+µ+Xπr2x¯u(x)+¯d(x)σµ−µ+≈Q ud|V ud|2¯fc¯Bc,(14)Q ud≡1¯fc¯Bc,D qσµ−µ+.(15)CDHSW[26] ( )µ µ σµ−µ−/σµ−µ+(σµ+µ+/σµ+µ−). 2E vis 100—200GeV ,3 .2 , , σµ−µ−/σµ−µ+σµ−µ−/σµ− ,, σµ−µ−/σµ−µ+, ,.2CDHSW 100<E vis<200GeV µ [26]pµ>6GeV(3.5±1.6)%(1.6±0.74)×10−4(4.5±2.0)%(2.2±1.0)×10−4 pµ>9GeV(2.9±1.2)%(1.05±0.43)×10−4(4.4±1.8)%(1.7±0.7)×10−4 pµ>15GeV(2.3±1.0)%(0.52±0.22)×10−4(4.1±2.3)%(0.8±0.45)×10−4968 (HEP&NP) 29dξd y d z −d3σ¯νµN→µ+H−Xπr2f cξ[(s(ξ)−¯s(ξ))|V cs|2+ d v(ξ)+u v(ξ)πr2xd v(x)+u v(x)πr2xd v(x)+u v(x)10 969(References)1Brodsky S J,MA B-Q.Phys.Lett.,1996,B381:3172Signal A I,Thomas A W.Phys.Lett.,1987,B191:2053Burkardt M,Warr B J.Phys.Rev.,1992,D45:9584Olness F et al.hep-ph/03123235Barone V et al.Eur.Phys.J.,2000,C12:2436Bazarko A O et al(CCFR Collaboration).Z.Phys.,1995, C65:1897Mason D(NuTeV Collaboration).hep-ex/04050378Kretzer S et al.Phys.Rev.Lett.,2004,93:0418029DING Y,MA B-Q.Phys.Lett.,2004,B590:216;DING Yong,L¨U Zhun,MA Bo-Qiang.HEP&NP,2004,28(9): 947(in Chinese)( , , . ,2004,28(9):947) 10Alwall J,Ingelman G.Phys.Rev.,2004,D70:111505.11DING Y,XU R-G,MA B-Q.Phys.Lett.,2005,B607:101 12Wakamatsu M.hep-ph/041120313Zeller G P et al.Phys.Rev.Lett.,2002,88:09180214Zeller G P et al.Phys.Rev.,2002,D65:11110315Rabinowitz S A et al.Phys.Rev.Lett.,1993,70:13416Goncharov M et al.Phys.Rev.,2001,D64:11200617Godbole R M,Roy D P.Z.Phys.,1984,C22:39;Z.Phys., 1989,C42:21918Astier P et al(NOMAD Collaboration).Phys.Lett.,2000, B486:3519Eidelman S et al(Particle Data Group).Phys.Lett.,2004, B592:120Andersson B et al.Nucl.Phys.,1981,B178:24221Laﬀerty G D.Phys.Lett.,1995,B353:54122Abe K et al(SLD Collaboration).Phys.Rev.Lett.,1997, 78:334123Smith J,Valenzuela G.Phys.Rev.,1983,D28:107124Aitala E M et al(Fermilab E791Collaboration).Phys.Lett.,1996,B371:15725Dias de Deus J,Dur˜a es F.Eur.Phys.J.,2000,C13:647 26Burkhardt H et al.Z.Phys.,1986,C31:3927Sandler P H et al.Z.Phys.,1993,C57:1,and References Therein.28Jonker M et al.Phys.Lett.,1981,B107:24129de Lellis G et al.Phys.Rep.,2004,399:22730Kayis-Topaksu A et al(CHORUS Collaboration).Phys.Lett.,2002,B549:4831¨Oneng¨u t G et al(CHORUS Collaboration).Phys.Lett., 2004,B604:145Nucleon Strange Asymmetry and the Light QuarkFragmentation Eﬀect*GAO Pu-Ze MA Bo-Qiang(School of Physics,Peking University,Beijing100871,China)Abstract Nucleon strange asymmetry is an important non-perturbative eﬀect in the study of nucleon structure,but it has not been checked by experiments yet.For eﬀectively measuring the nucleon strange asymmetry,we investigate the light quark fragmentation eﬀect that may aﬀect the measurement of the strange asymmetry.We suggest an inclusive measurement of charged and neutral charmed hadrons by using an emulsion target in the neutrino and antineutrino in-duced charged current deep inelastic scattering,in which the strange asymmetry eﬀect and the light quark fragmentation eﬀect can be separated.Key words strange asymmetry,light quark fragmentation,charged current deep inelastic scattering。

固体物理英语

固体物理英语固体物理基本词汇（汉英对照）一画一维晶格 One-dimensional crystal lattice一维单原子链 One-dimensional monatomic chain 一维双原子链 One-dimensional diatomic chain 一维复式格子One-dimensional compound lattice 二画二维晶格 Two-dimensional crystal lattice二度轴 Twofold axis二度对称轴 Twofold axis of symmetry几何结构因子 Geometrical structure factor三画三斜晶系 Triclinic system三方晶系 Trigonal system三斜晶系 Triclinic system刃位错 Edge dislocation小角晶界 Low angle grain boundary马德隆常数 Madelung constant四画元素晶体 Element crystal元素的电负性 Electronegativities of elements元素的电离能 Ionization energies of the elements 元素的结合能 Cohesive energies of the elements 六方密堆积 Hexagonal close-packed六方晶系 Hexagonal system反演 Inversion分子晶体 Molecular Crystal切变模量 Shear module双原子链 Diatomic linear chain介电常数 Dielectric constant化学势 Chemical potential内能 Internal energy分布函数 Distribution function夫伦克耳缺陷 Frenkel defect比热 Specific heat中子散射 Neutron scattering五画布喇菲格子 Bravais lattice布洛赫函数 Bloch function布洛赫定理 Bloch theorem布拉格反射 Bragg reflection布里渊区 Brillouin zone布里渊区边界 Brillouin zone boundary 布里渊散射 Brillouin scattering正格子 Direct lattice正交晶系 Orthorhombic crystal system正则振动 Normal vibration正则坐标 Normal coordinates立方晶系 Cubic crystal system立方密堆积 Cubic close-packed四方晶系 Tetragonal crystal system对称操作 Symmetry operation对称群 Symmetric group正交化平面波 Orthogonalized plane wave电子-晶格相互作用 Electron-lattice interaction 电子热容量 Electronic heat capacity电阻率 Electrical resistivity电导率 Conductivity电子亲合势 Electron affinity电子气的动能 Kinetic energy of electron gas 电子气的压力 Pressure of electron gas电子分布函数 Electron distribution function 电负性 Electronegativity电磁声子 Electromagnetic phonon功函数 Work function长程力 Long-range force立方晶系 Cubic system平面波方法 plane wave method平移对称性 Translation symmetry平移对称操作 Translation symmetry operator 平移不变性 Translation invariance石墨结构 Graphite structure闪锌矿结构 Blende structure六画负电性 Electronegativity共价结合 Covalent binding共价键 Covalent bond共价晶体 Covalent crystals共价键的饱和 Saturation of covalent bonds 光学模 Optical modes光学支 Optical branch光散射 Light scattering红外吸收 Infrared absorption压缩系数 Compressibility扩散系数 Diffusion coefficient扩散的激活能 Activation energy of diffusion 共价晶体 Covalent Crystal价带 Valence band导带 Conduction band自扩散 Self-diffusion有效质量 Effective mass有效电荷 Effective charges弛豫时间 Relaxation time弛豫时间近似 Relaxation-time approximation扩展能区图式 Extended zone scheme自由电子模型 Free electron model自由能 Free energy杂化轨道 Hybrid orbit七画纯金属 Ideal metal体心立方 Body-centered cubic体心四方布喇菲格子 Body-centered tetragonal Bravais lattices 卤化碱晶体 Alkali-halide crystal劳厄衍射 Laue diffraction间隙原子 Interstitial atom间隙式扩散 Interstitial diffusion肖特基缺陷 Schottky defect位错 Dislocation滑移 Slip晶界 Grain boundaries伯格斯矢量 Burgers vector杜隆-珀替定律 Dulong-Petit’s law粉末衍射 Powder diffraction里查孙-杜师曼方程 Richardson-Dushman equation 克利斯托夫方程 Christofell equation克利斯托夫模量Christofell module位移极化 Displacement polarization声子 Phonon声学支 Acoustic branch应力 Stress 应变 Strain切应力 Shear stress切应变 Shear strain八画周期性重复单元 Periodic repeated unit底心正交格子 Base-centered orthorhombic lattice 底心单斜格工 Base-centered monoclinic lattices 单斜晶系 Monoclinic crystal system金刚石结构 Diamond structure金属的结合能 Cohesive energy of metals金属晶体 Metallic Crystal转动轴 Rotation axes转动-反演轴 Rotation-inversion axes转动晶体法 Rotating crystal method空间群 Space group空位 Vacancy范德瓦耳斯相互作用 Van der Waals interaction 金属性结合 Metallic binding单斜晶系 Monoclinic system单电子近似 Single-erection approximation极化声子 Polarization phonon拉曼散射 Raman scattering态密度 Density of states铁电软模 Ferroelectrics soft mode空穴 Hole万尼尔函数 Wannier function平移矢量 Translation vector非谐效应 Anharmonic effect周期性边界条件 Periodic boundary condition九画玻尔兹曼方程 Boltzman equation点群 Point groups迪. 哈斯－范. 阿耳芬效应 De Hass－Van Alphen effect胡克定律Hooke’s law氢键 Hydrogen bond亲合势 Affinity重迭排斥能 Overlap repulsive energy结合能 Cohesive energy玻恩-卡门边界条件 Born-Karman boundary condition费密-狄喇克分布函数 Fermi-Dirac distribution function费密电子气的简并性 Degeneracy of free electron Fermi gas 费密 Fermi费密能 Fermi energy费密能级 Fermi level费密球 Fermi sphere费密面 Fermi surface费密温度 Fermi temperature费密速度 Fermi velocity费密半径 Fermi radius恢复力常数 Constant of restorable force绝热近似 Adiabatic approximation十画原胞 Primitive cell原胞基矢 Primitive vectors倒格子 Reciprocal lattice倒格子原胞 Primitive cell of the reciprocal lattice 倒格子空间 Reciprocal space倒格点 Reciprocal lattice point倒格子基矢Primitive translation vectors of the reciprocal lattice倒格矢 Reciprocal lattice vector倒逆散射 Umklapp scattering粉末法 Powder method原子散射因子 Atomic scattering factor配位数 Coordination number原子和离子半径 Atomic and ionic radii原子轨道线性组合 Linear combination of atomic orbits离子晶体的结合能 Cohesive energy of inert crystals离解能 Dissociation energy离子键 Ionic bond离子晶体 Ionic Crystal离子性导电 Ionic conduction洛伦兹比 Lorenz ratio魏德曼－佛兰兹比 Weidemann－Franz ratio 缺陷的迁移 Migration of defects缺陷的浓度 Concentrations of lattice defects 爱因斯坦 Einstein爱因斯坦频率 Einstein frequency爱因斯坦温度 Einstein temperature格波 Lattice wave格林爱森常数 Gruneisen constant索末菲理论 Sommerfeld theory热电子发射 Thermionic emission热容量 Heat capacity热导率 Thermal conductivity热膨胀 Thermal expansion能带 Energy band能隙 Energy gap能带的简约能区图式 Reduced zone scheme of energy band 能带的周期能区图式 Repeated zone scheme of energy band 能带的扩展能区图式 Extended zone scheme of energy band 配分函数 Partition function准粒子 Quasi- particle准动量 Quasi- momentum准自由电子近似 Nearly free electron approximation十一画第一布里渊区 First Brillouin zone密堆积 Close-packing密勒指数 Miller indices接触电势差 Contact potential difference基元 Basis基矢 Basis vector弹性形变 Elastic deformation排斥能Repulsive energy弹性波 Elastic wave弹性应变张量 Elastic strain tensor弹性劲度常数 Elastic stiffness constant弹性顺度常数 Elastic compliance constant 弹性模量 Elastic module弹性动力学方程 Elastic-dynamics equation 弹性散射 Elastic scattering十二画等能面 Constant energy surface晶体 Crystal晶体结构 Crystal structure晶体缺陷 Crystal defect晶体衍射 Crystal diffraction晶列 Crystal array晶面 Crystal plane晶面指数 Crystal plane indices晶带 Crystal band晶向 direction晶格 lattice晶格常数 Lattice constant晶格周期势 Lattice-periodic potential 晶格周期性 Lattice-periodicity晶胞 Cell, Unit cell晶面间距 Interplanar spacing晶系 Crystal system晶体 Crystal晶体点群 Crystallographic point groups晶格振动 Latticevibration晶格散射 Lattice scattering散射 Scattering等能面 surface of constant energy十三画隋性气体晶体的结合能 Cohesive energy of inert gas crystals 滑移 Slip滑移面 Slip plane简单立方晶格 Simple cubic lattice简单晶格 Simple lattice简单单斜格子 Simple monoclinic lattice简单四方格子 Simple tetragonal lattice简单正交格子 Simple orthorhombic lattice简谐近似 Harmonic approximation简正坐标 Normal coordinates简正振动 Normal vibration简正模 Normal modes简约波矢 Reduced wave vector简约布里渊区 Reduced Brillouin zone禁带 Forbidden band紧束缚方法 Tight－binding method零点振动能 Zero-point vibration energy 雷纳德－琼斯势 Lenard－Jones potential 满带 Filled band十四画磁致电阻 Magnetoresistance模式密度 Density of modes漂移速度 Drift velocity漂移迁移率 Drift mobility十五至十七画德拜 Debye德拜近似 Debye approximation德拜截止频率 Debye cut-off frequency 德拜温度 Debye temperature霍耳效应 Hall effect螺位错 Screw dislocation赝势 Pseudopotential。

Two-Dimensional Gas of Massless Dirac Fermions in Graphene

Two-Dimensional Gas of Massless Dirac Fermions in Graphene K.S. Novoselov1, A.K. Geim1, S.V. Morozov2, D. Jiang1, M.I. Katsnelson3, I.V. Grigorieva1, S.V. Dubonos2, A.A. Firsov21Manchester Centre for Mesoscience and Nanotechnology, University of Manchester, Manchester, M13 9PL, UK2Institute for Microelectronics Technology, 142432, Chernogolovka, Russia3Institute for Molecules and Materials, Radboud University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, the NetherlandsElectronic properties of materials are commonly described by quasiparticles that behave as nonrelativistic electrons with a finite mass and obey the Schrödinger equation. Here we report a condensed matter system where electron transport is essentially governed by the Dirac equation and charge carriers mimic relativistic particles with zero mass and an effective “speed of light” c∗ ≈106m/s. Our studies of graphene – a single atomic layer of carbon – have revealed a variety of unusual phenomena characteristic of two-dimensional (2D) Dirac fermions. In particular, we have observed that a) the integer quantum Hall effect in graphene is anomalous in that it occurs at halfinteger filling factors; b) graphene’s conductivity never falls below a minimum value corresponding to the conductance quantum e2/h, even when carrier concentrations tend to zero; c) the cyclotron mass mc of massless carriers with energy E in graphene is described by equation E =mcc∗2; and d) Shubnikov-de Haas oscillations in graphene exhibit a phase shift of π due to Berry’s phase.Graphene is a monolayer of carbon atoms packed into a dense honeycomb crystal structure that can be viewed as either an individual atomic plane extracted from graphite or unrolled single-wall carbon nanotubes or as a giant flat fullerene molecule. This material was not studied experimentally before and, until recently [1,2], presumed not to exist. To obtain graphene samples, we used the original procedures described in [1], which involve micromechanical cleavage of graphite followed by identification and selection of monolayers using a combination of optical, scanning-electron and atomic-force microscopies. The selected graphene films were further processed into multi-terminal devices such as the one shown in Fig. 1, following standard microfabrication procedures [2]. Despite being only one atom thick and unprotected from the environment, our graphene devices remain stable under ambient conditions and exhibit high mobility of charge carriers. Below we focus on the physics of “ideal” (single-layer) graphene which has a different electronic structure and exhibits properties qualitatively different from those characteristic of either ultra-thin graphite films (which are semimetals and whose material properties were studied recently [2-5]) or even of our other devices consisting of just two layers of graphene (see further). Figure 1 shows the electric field effect [2-4] in graphene. Its conductivity σ increases linearly with increasing gate voltage Vg for both polarities and the Hall effect changes its sign at Vg ≈0. This behaviour shows that substantial concentrations of electrons (holes) are induced by positive (negative) gate voltages. Away from the transition region Vg ≈0, Hall coefficient RH = 1/ne varies as 1/Vg where n is the concentration of electrons or holes and e the electron charge. The linear dependence 1/RH ∝Vg yields n =α·Vg with α ≈7.3·1010cm-2/V, in agreement with the theoretical estimate n/Vg ≈7.2·1010cm-2/V for the surface charge density induced by the field effect (see Fig. 1’s caption). The agreement indicates that all the induced carriers are mobile and there are no trapped charges in graphene. From the linear dependence σ(Vg) we found carrier mobilities µ =σ/ne, whichreached up to 5,000 cm2/Vs for both electrons and holes, were independent of temperature T between 10 and 100K and probably still limited by defects in parent graphite. To characterise graphene further, we studied Shubnikov-de Haas oscillations (SdHO). Figure 2 shows examples of these oscillations for different magnetic fields B, gate voltages and temperatures. Unlike ultra-thin graphite [2], graphene exhibits only one set of SdHO for both electrons and holes. By using standard fan diagrams [2,3], we have determined the fundamental SdHO frequency BF for various Vg. The resulting dependence of BF as a function of n is plotted in Fig. 3a. Both carriers exhibit the same linear dependence BF = β·n with β ≈1.04·10-15 T·m2 (±2%). Theoretically, for any 2D system β is defined only by its degeneracy f so that BF =φ0n/f, where φ0 =4.14·10-15 T·m2 is the flux quantum. Comparison with the experiment yields f =4, in agreement with the double-spin and double-valley degeneracy expected for graphene [6,7] (cf. caption of Fig. 2). Note however an anomalous feature of SdHO in graphene, which is their phase. In contrast to conventional metals, graphene’s longitudinal resistance ρxx(B) exhibits maxima rather than minima at integer values of the Landau filling factor ν (Fig. 2a). Fig. 3b emphasizes this fact by comparing the phase of SdHO in graphene with that in a thin graphite film [2]. The origin of the “odd” phase is explained below. Another unusual feature of 2D transport in graphene clearly reveals itself in the T-dependence of SdHO (Fig. 2b). Indeed, with increasing T the oscillations at high Vg (high n) decay more rapidly. One can see that the last oscillation (Vg ≈100V) becomes practically invisible already at 80K whereas the first one (Vg <10V) clearly survives at 140K and, in fact, remains notable even at room temperature. To quantify this behaviour we measured the T-dependence of SdHO’s amplitude at various gate voltages and magnetic fields. The results could be fitted accurately (Fig. 3c) by the standard expression T/sinh(2π2kBTmc/heB), which yielded mc varying between ≈ 0.02 and 0.07m0 (m0 is the free electron mass). Changes in mc are well described by a square-root dependence mc ∝n1/2 (Fig. 3d). To explain the observed behaviour of mc, we refer to the semiclassical expressions BF = (h/2πe)S(E) and mc =(h2/2π)∂S(E)/∂E where S(E) =πk2 is the area in k-space of the orbits at the Fermi energy E(k) [8]. Combining these expressions with the experimentally-found dependences mc ∝n1/2 and BF =(h/4e)n it is straightforward to show that S must be proportional to E2 which yields E ∝k. Hence, the data in Fig. 3 unambiguously prove the linear dispersion E =hkc∗ for both electrons and holes with a common origin at E =0 [6,7]. Furthermore, the above equations also imply mc =E/c∗2 =(h2n/4πc∗2)1/2 and the best fit to our data yields c∗ ≈1⋅106 m/s, in agreement with band structure calculations [6,7]. The employed semiclassical model is fully justified by a recent theory for graphene [9], which shows that SdHO’s amplitude can indeed be described by the above expression T/sinh(2π2kBTmc/heB) with mc =E/c∗2. Note that, even though the linear spectrum of fermions in graphene (Fig. 3e) implies zero rest mass, their cyclotron mass is not zero. The unusual response of massless fermions to magnetic field is highlighted further by their behaviour in the high-field limit where SdHO evolve into the quantum Hall effect (QHE). Figure 4 shows Hall conductivity σxy of graphene plotted as a function of electron and hole concentrations in a constant field B. Pronounced QHE plateaux are clearly seen but, surprisingly, they do not occur in the expected sequence σxy =(4e2/h)N where N is integer. On the contrary, the plateaux correspond to half-integer ν so that the first plateau occurs at 2e2/h and the sequence is (4e2/h)(N + ½). Note that the transition from the lowest hole (ν =–½) to lowest electron (ν =+½) Landau level (LL) in graphene requires the same number of carriers (∆n =4B/φ0 ≈1.2·1012cm-2) as the transition between other nearest levels (cf. distances between minima in ρxx). This results in a ladder of equidistant steps in σxy which are not interrupted when passing through zero. To emphasize this highly unusual behaviour, Fig. 4 also shows σxy for a graphite film consisting of only two graphene layers where the sequence of plateaux returns to normal and the first plateau is at 4e2/h, as in the conventional QHE. We attribute this qualitative transition between graphene and its two-layer counterpart to the fact that fermions in the latter exhibit a finite mass near n ≈0 (as found experimentally; to be published elsewhere) and can no longer be described as massless Dirac particles. 2The half-integer QHE in graphene has recently been suggested by two theory groups [10,11], stimulated by our work on thin graphite films [2] but unaware of the present experiment. The effect is single-particle and intimately related to subtle properties of massless Dirac fermions, in particular, to the existence of both electron- and hole-like Landau states at exactly zero energy [912]. The latter can be viewed as a direct consequence of the Atiyah-Singer index theorem that plays an important role in quantum field theory and the theory of superstrings [13,14]. For the case of 2D massless Dirac fermions, the theorem guarantees the existence of Landau states at E=0 by relating the difference in the number of such states with opposite chiralities to the total flux through the system (note that magnetic field can also be inhomogeneous). To explain the half-integer QHE qualitatively, we invoke the formal expression [9-12] for the energy of massless relativistic fermions in quantized fields, EN =[2ehc∗2B(N +½ ±½)]1/2. In QED, sign ± describes two spins whereas in the case of graphene it refers to “pseudospins”. The latter have nothing to do with the real spin but are “built in” the Dirac-like spectrum of graphene, and their origin can be traced to the presence of two carbon sublattices. The above formula shows that the lowest LL (N =0) appears at E =0 (in agreement with the index theorem) and accommodates fermions with only one (minus) projection of the pseudospin. All other levels N ≥1 are occupied by fermions with both (±) pseudospins. This implies that for N =0 the degeneracy is half of that for any other N. Alternatively, one can say that all LL have the same “compound” degeneracy but zeroenergy LL is shared equally by electrons and holes. As a result the first Hall plateau occurs at half the normal filling and, oddly, both ν = –½ and +½ correspond to the same LL (N =0). All other levels have normal degeneracy 4B/φ0 and, therefore, remain shifted by the same ½ from the standard sequence. This explains the QHE at ν =N + ½ and, at the same time, the “odd” phase of SdHO (minima in ρxx correspond to plateaux in ρxy and, hence, occur at half-integer ν; see Figs. 2&3), in agreement with theory [9-12]. Note however that from another perspective the phase shift can be viewed as the direct manifestation of Berry’s phase acquired by Dirac fermions moving in magnetic field [15,16]. Finally, we return to zero-field behaviour and discuss another feature related to graphene’s relativistic-like spectrum. The spectrum implies vanishing concentrations of both carriers near the Dirac point E =0 (Fig. 3e), which suggests that low-T resistivity of the zero-gap semiconductor should diverge at Vg ≈0. However, neither of our devices showed such behaviour. On the contrary, in the transition region between holes and electrons graphene’s conductivity never falls below a well-defined value, practically independent of T between 4 and 100K. Fig. 1c plots values of the maximum resistivity ρmax(B =0) found in 15 different devices, which within an experimental error of ≈15% all exhibit ρmax ≈6.5kΩ, independent of their mobility that varies by a factor of 10. Given the quadruple degeneracy f, it is obvious to associate ρmax with h/fe2 =6.45kΩ where h/e2 is the resistance quantum. We emphasize that it is the resistivity (or conductivity) rather than resistance (or conductance), which is quantized in graphene (i.e., resistance R measured experimentally was not quantized but scaled in the usual manner as R =ρL/w with changing length L and width w of our devices). Thus, the effect is completely different from the conductance quantization observed previously in quantum transport experiments. However surprising, the minimum conductivity is an intrinsic property of electronic systems described by the Dirac equation [17-20]. It is due to the fact that, in the presence of disorder, localization effects in such systems are strongly suppressed and emerge only at exponentially large length scales. Assuming the absence of localization, the observed minimum conductivity can be explained qualitatively by invoking Mott’s argument [21] that mean-free-path l of charge carriers in a metal can never be shorter that their wavelength λF. Then, σ =neµ can be re-written as σ = (e2/h)kFl and, hence, σ cannot be smaller than ≈e2/h per each type of carriers. This argument is known to have failed for 2D systems with a parabolic spectrum where disorder leads to localization and eventually to insulating behaviour [17,18]. For the case of 2D Dirac fermions, no localization is expected [17-20] and, accordingly, Mott’s argument can be used. Although there is a broad theoretical consensus [18-23,10,11] that a 2D gas of Dirac fermions should exhibit a minimum 3conductivity of about e2/h, this quantization was not expected to be accurate and most theories suggest a value of ≈e2/πh, in disagreement with the experiment. In conclusion, graphene exhibits electronic properties distinctive for a 2D gas of particles described by the Dirac rather than Schrödinger equation. This 2D system is not only interesting in itself but also allows one to access – in a condensed matter experiment – the subtle and rich physics of quantum electrodynamics [24-27] and provides a bench-top setting for studies of phenomena relevant to cosmology and astrophysics [27,28].1. Novoselov, K.S. et al. PNAS 102, 10451 (2005). 2. Novoselov, K.S. et al. Science 306, 666 (2004); cond-mat/0505319. 3. Zhang, Y., Small, J.P., Amori, M.E.S. & Kim, P. Phys. Rev. Lett. 94, 176803 (2005). 4. Berger, C. et al. J. Phys. Chem. B, 108, 19912 (2004). 5. Bunch, J.S., Yaish, Y., Brink, M., Bolotin, K. & McEuen, P.L. Nanoletters 5, 287 (2005). 6. Dresselhaus, M.S. & Dresselhaus, G. Adv. Phys. 51, 1 (2002). 7. Brandt, N.B., Chudinov, S.M. & Ponomarev, Y.G. Semimetals 1: Graphite and Its Compounds (North-Holland, Amsterdam, 1988). 8. Vonsovsky, S.V. and Katsnelson, M.I. Quantum Solid State Physics (Springer, New York, 1989). 9. Gusynin, V.P. & Sharapov, S.G. Phys. Rev. B 71, 125124 (2005). 10. Gusynin, V.P. & Sharapov, S.G. cond-mat/0506575. 11. Peres, N.M.R., Guinea, F. & Castro Neto, A.H. cond-mat/0506709. 12. Zheng, Y. & Ando, T. Phys. Rev. B 65, 245420 (2002). 13. Kaku, M. Introduction to Superstrings (Springer, New York, 1988). 14. Nakahara, M. Geometry, Topology and Physics (IOP Publishing, Bristol, 1990). 15. Mikitik, G. P. & Sharlai, Yu.V. Phys. Rev. Lett. 82, 2147 (1999). 16. Luk’yanchuk, I.A. & Kopelevich, Y. Phys. Rev. Lett. 93, 166402 (2004). 17. Abrahams, E., Anderson, P.W., Licciardello, D.C. & Ramakrishnan, T.V. Phys. Rev. Lett. 42, 673 (1979). 18. Fradkin, E. Phys. Rev. B 33, 3263 (1986). 19. Lee, P.A. Phys. Rev. Lett. 71, 1887 (1993). 20. Ziegler, K. Phys. Rev. Lett. 80, 3113 (1998). 21. Mott, N.F. & Davis, E.A. Electron Processes in Non-Crystalline Materials (Clarendon Press, Oxford, 1979). 22. Morita, Y. & Hatsugai, Y. Phys. Rev. Lett. 79, 3728 (1997). 23. Nersesyan, A.A., Tsvelik, A.M. & Wenger, F. Phys. Rev. Lett. 72, 2628 (1997). 24. Rose, M.E. Relativistic Electron Theory (John Wiley, New York, 1961). 25. Berestetskii, V.B., Lifshitz, E.M. & Pitaevskii, L.P. Relativistic Quantum Theory (Pergamon Press, Oxford, 1971). 26. Lai, D. Rev. Mod. Phys. 73, 629 (2001). 27. Fradkin, E. Field Theories of Condensed Matter Systems (Westview Press, Oxford, 1997). 28. Volovik, G.E. The Universe in a Helium Droplet (Clarendon Press, Oxford, 2003).Acknowledgements This research was supported by the EPSRC (UK). We are most grateful to L. Glazman, V. Falko, S. Sharapov and A. Castro Netto for helpful discussions. K.S.N. was supported by Leverhulme Trust. S.V.M., S.V.D. and A.A.F. acknowledge support from the Russian Academy of Science and INTAS.43µ (m2/Vs)0.8c4P0.4 22 σ (1/kΩ)10K0 0 1/RH(T/kΩ) 1 2ρmax (h/4e2)1-5010 Vg (V) 50 -10ab 0 -100-500 Vg (V)50100Figure 1. Electric field effect in graphene. a, Scanning electron microscope image of one of our experimental devices (width of the central wire is 0.2µm). False colours are chosen to match real colours as seen in an optical microscope for larger areas of the same materials. Changes in graphene’s conductivity σ (main panel) and Hall coefficient RH (b) as a function of gate voltage Vg. σ and RH were measured in magnetic fields B =0 and 2T, respectively. The induced carrier concentrations n are described by [2] n/Vg =ε0ε/te where ε0 and ε are permittivities of free space and SiO2, respectively, and t ≈300 nm is the thickness of SiO2 on top of the Si wafer used as a substrate. RH = 1/ne is inverted to emphasize the linear dependence n ∝Vg. 1/RH diverges at small n because the Hall effect changes its sign around Vg =0 indicating a transition between electrons and holes. Note that the transition region (RH ≈ 0) was often shifted from zero Vg due to chemical doping [2] but annealing of our devices in vacuum normally allowed us to eliminate the shift. The extrapolation of the linear slopes σ(Vg) for electrons and holes results in their intersection at a value of σ indistinguishable from zero. c, Maximum values of resistivity ρ =1/σ (circles) exhibited by devices with different mobilites µ (left y-axis). The histogram (orange background) shows the number P of devices exhibiting ρmax within 10% intervals around the average value of ≈h/4e2. Several of the devices shown were made from 2 or 3 layers of graphene indicating that the quantized minimum conductivity is a robust effect and does not require “ideal” graphene.ρxx (kΩ)0.60 aVg = -60V4B (T)810K12∆σxx (1/kΩ)0.4 1ν=4 140K 80K B =12T0 b 0 25 50 Vg (V) 7520K100Figure 2. Quantum oscillations in graphene. SdHO at constant gate voltage Vg as a function of magnetic field B (a) and at constant B as a function of Vg (b). Because µ does not change much with Vg, the constant-B measurements (at a constant ωcτ =µB) were found more informative. Panel b illustrates that SdHO in graphene are more sensitive to T at high carrier concentrations. The ∆σxx-curves were obtained by subtracting a smooth (nearly linear) increase in σ with increasing Vg and are shifted for clarity. SdHO periodicity ∆Vg in a constant B is determined by the density of states at each Landau level (α∆Vg = fB/φ0) which for the observed periodicity of ≈15.8V at B =12T yields a quadruple degeneracy. Arrows in a indicate integer ν (e.g., ν =4 corresponds to 10.9T) as found from SdHO frequency BF ≈43.5T. Note the absence of any significant contribution of universal conductance fluctuations (see also Fig. 1) and weak localization magnetoresistance, which are normally intrinsic for 2D materials with so high resistivity.75 BF (T) 500.2 0.11/B (1/T)b5 10 N 1/2025 a 0 0.061dmc /m00.04∆0.02 0c0 0 T (K) 150n =0e-6-3036Figure 3. Dirac fermions of graphene. a, Dependence of BF on carrier concentration n (positive n correspond to electrons; negative to holes). b, Examples of fan diagrams used in our analysis [2] to find BF. N is the number associated with different minima of oscillations. Lower and upper curves are for graphene (sample of Fig. 2a) and a 5-nm-thick film of graphite with a similar value of BF, respectively. Note that the curves extrapolate to different origins; namely, to N = ½ and 0. In graphene, curves for all n extrapolate to N = ½ (cf. [2]). This indicates a phase shift of π with respect to the conventional Landau quantization in metals. The shift is due to Berry’s phase [9,15]. c, Examples of the behaviour of SdHO amplitude ∆ (symbols) as a function of T for mc ≈0.069 and 0.023m0; solid curves are best fits. d, Cyclotron mass mc of electrons and holes as a function of their concentration. Symbols are experimental data, solid curves the best fit to theory. e, Electronic spectrum of graphene, as inferred experimentally and in agreement with theory. This is the spectrum of a zero-gap 2D semiconductor that describes massless Dirac fermions with c∗ 300 times less than the speed of light.n (1012 cm-2)σxy (4e2/h)4 3 2 -2 1 -1 -2 -3 2 44Kn7/ 5/ 3/ 1/2 2 2 210 ρxx (kΩ)-4σxy (4e2/h)0-1/2 -3/2 -5/2514T0-7/2 -4 -2 0 2 4 n (1012 cm-2)Figure 4. Quantum Hall effect for massless Dirac fermions. Hall conductivity σxy and longitudinal resistivity ρxx of graphene as a function of their concentration at B =14T. σxy =(4e2/h)ν is calculated from the measured dependences of ρxy(Vg) and ρxx(Vg) as σxy = ρxy/(ρxy + ρxx)2. The behaviour of 1/ρxy is similar but exhibits a discontinuity at Vg ≈0, which is avoided by plotting σxy. Inset: σxy in “two-layer graphene” where the quantization sequence is normal and occurs at integer ν. The latter shows that the half-integer QHE is exclusive to “ideal” graphene.。

向涛-General Introduction

– quasiparticles, phonon, spin waves, plasmons, excitons, spinons and holons, …
• Elementary interactions:
– el-el, el-ph, el-mag, …
• Elementary structures and phases:
• Development of theoretical methods
– Analytical: path integral, Green’s function, mean-field theory … – Numerical: density functional, Monte Carlo, DMRG, DMFT …
2. Band Theory
– 1900: 金属电导和热传导的经典自由电子理论(Drude) – 1924: 基于Fermi统计的自由电子理论(Pauli 和 Sommerfield) – 1907: 铁磁性相变的分子场理论(Weiss) – 金属导电的能带理论(Bloch)
Standard Picture of Metel and Insulator
Fermi Statistics：Foundation of Chemistry
Bosons have no Chemical Bonds
Fermions can form Chemical Bonds
“With a heavy heart, I have been converted to the idea that Fermi-Dirac, not Einstein-Bose is the correct statistics” (for electrons) Pauli, letter to Schrodinger, Dec 1926

仿射对称空间SU(1，2)

第 63 卷第 1 期2024 年 1 月Vol.63 No.1Jan.2024中山大学学报（自然科学版）（中英文）ACTA SCIENTIARUM NATURALIUM UNIVERSITATIS SUNYATSENI仿射对称空间SU(1，2)/SO(1，2)上的Plancherel定理*金四海，范兴亚新疆大学数学与系统科学学院，新疆乌鲁木齐 830017摘要：研究了Hilbert空间L2(Z，μ)上酉表示的不可约分解，其中Z=SU(1，2)/SO(1，2)是Hermitian型仿射对称空间，μ是群SU(1，2)作用在Z上不变的Haar测度. 利用SO(1，2)不变的分布函数，具体的构造了缠结算子，进而得到了L2(Z，μ)上的离散序列表示. 在此基础上，结合离散序列表示的正交补部分，证明了L2(Z，μ)上的Plancherel公式.关键词：仿射对称空间；离散序列表示；Plancherel定理中图分类号：O152.5 文献标志码：A 文章编号：2097 - 0137（2024）01 - 0173 - 08 Plancherel theorem for the affine symmetric space SU(1，2)/SO(1，2)JIN Sihai， FAN XingyaCollege of Mathematics and System Sciences，Xinjiang University，Urumqi 830017， ChinaAbstract：The irreducible decomposition of unitary representations is investigated on the Hilbert spaceL2(Z，μ)，where Z=SU(1，2)/SO(1，2)，and μdenotes an SU(1，2)-invariant Haar measure on Z.By using the SO(1，2)-invariant distribution functions， the intertwining operators is constructed concrete‐ly， and then the discrete series representations on L2(Z，μ) are obtained. On this basis， combined with the orthogonal complement parts of the discrete series representations，the Plancherel formula onL2(Z，μ) is proved.Key words：affine symmetric space； discrete series； Plancherel theorem李群表示理论是近现代分析学的主要研究领域之一，隶属于抽象调和分析范畴，是Fourier级数和Fourier变换理论的深远推广，此方面的研究成果极大地推动了数学物理等相关学科的发展. 悉知，酉表示中不可约表示的分解是李群表示理论研究的主要问题之一，目的是给出抽象的Plancherel公式，进而构造缠结算子来解决相关实际问题. 在此分解中，离散谱的分解是一种极端的情况，此情况可以利用分析的工具来处理. 基于此想法，本文在Hermitian型仿射对称空间上考虑其离散谱的具体构造. 从几何结构方面来讲，Hermitian型仿射对称空间是伪黎曼对称空间，此空间隶属于半单对称空间，据此赋予了Hermite结构的齐性空间. 20世纪80年代初，Flensted-Jensen（1980）系统地研究了半单对称空间上的离散序列表示，并得到离散序列表示存在的必要条件. Flensted-Jensen的这一工作引起了许多表示论专家的关注，究其原因是此理论极大地继承并发展了Harish-Chandra的经典离散序列表示（Harish-Chandra，1965）.近年来，Hermitian型仿射对称空间上的离散序列表示受到了国内外广大学者的关注. 该领域主要关心的是Gelfand-Gindikin问题（Gelfand et al.，1977）. 最近，Delorme et al.（2021）研究了扭曲离散序列表示，此DOI：10.13471/ki.acta.snus.2022A091*收稿日期：2022 − 10 − 19 录用日期：2023 − 02 − 20 网络首发日期：2023 − 11 − 15基金项目：国家自然科学基金（12161083）；国家自然科学基金天元数学访问学者项目（12126360）；新疆维吾尔自治区自然科学基金（2020D01C048，2021D01C071）作者简介：金四海（1997年生），男；研究方向：李群表示论；E-mail：****************通信作者：范兴亚（1986年生），男；研究方向：李群表示论；E-mail：*****************.cn第 63 卷中山大学学报（自然科学版）（中英文）离散序列表示旨在给出Gelfand-Gindikin 问题的回答，主要的方法是通过Bersentan 态射来构造Plancherel 公式. 此方法简化了离散序列表示与主序列表示的复杂计算，主要判别工具是限制球根系是否为零向量空间，进而给出抽象的Plancherel 公式. 从理论方面来讲，此方法值得推广（韩威等，2023）. 但是在Hermite 情形下，Delorme et al.（2021）的方法不能具体的构造扭曲离散序列表示，究其原因是限制球根系对应的极大交换子空间是一个零向量空间. 至于0-扭曲离散序列表示，可以通过'Olafsson et al.（1988）的缠结算子来构造此序列.根据Flensted-Jensen （1980）提出的必要条件，容易判定Hermitian 对称空间SU ()1，2/SO ()1，2上存在离散序列表示. 但没有给出具体的形式，因此有必要去研究它的具体构造. 基于'Olafsson et al.（1988）的思想，本文具体地构造缠结算子，进而展开研究SU ()1，2/SO ()1，2上的离散序列表示，并在限制球根系上，找到了判别离散序列表示存在的一个新的等价条件. 在Plancherel 公式中Plancherel 测度的计算是一个核心问题. 在Duistermaat et al.（1979）的基础上，本文精确地计算了此具体情形下的Plancherel 测度，并得到了对应的Plancherel 公式.1 符号准备及主要引理设SU ()1，2是SL ()3，C 的子群，此群保不定内积[]z ，z =||z 12-||z 22-||z 32，其中z =(z 1，z 2，z 3)∈C 3.群SU ()1，2对应的李代数为su ()1，2. 定义su ()1，2上的Cartan 对合为θ(X )=-X *，其中X *表示X 的共轭转置. 在此对合意义下，su ()1，2有Cartan 分解k⊕p ，其中k ={}X ∈su ()1，2 | θ()X =X ，p ={}X ∈su ()1，2 | θ()X =-X . 此外，在su ()1，2上引进另外一种对合τ(X )=X ，其中X 为X 的共轭，则有相应的τ分解h⊕q ，其中h ={}X ∈su ()1，2 | τ()X =X ，q ={}X ∈su ()1，2 | τ()X =-X .易证θτ=τθ，结合此对合交换关系，得到su ()1，2关于θτ的正交分解su(1，2)= (k ∩h)⊕(p ∩q)g +⊕(h ∩p)⊕(k ∩q)g -=g +⊕g -.易证g +是su ()1，2的子空间，且对su ()1，2上的李括积运算封闭，则g +是su ()1，2的李子代数. 设a p ∩q 是p ∩q 的极大交换子空间，记作a ≔a p ∩q =ìíîïïï|||||||i ()0t 0-t000 t ∈R üýþïïïï.注意，此处a 的取法还有另外一种形式ìíîïïï|||||||i ()00t000-t t ∈R üýþïïïï.此取法在下文计算根系时，所得根系与上面对应的结果一致，故此全文仅考虑第一种情形.利用a 在su ()1，2上的伴随作用，可得根系±2α与±α所对应的特征向量分别如下()i 101-i 0000， ()i -10-1-i 0000， ()00±i 00-1∓i10，()00±i 001∓i -10， ()0100i 1i0， ()00100-i 1-i 0.令a *是a 的对偶空间，定义SU ()1，2在a 下的根系为Σ(su(1，2)，a)={±2α，±α}，其中α∈a *且α(X )=t ，X ∈a . Σ(su(1，2)，a)的正根系为Σ+(su(1，2)，a)={}2α，α. 由根系的反射可得Weyl 群，此群同构于对称群S 2.定义1 设R 是由{}2α，α生成的半群，且R ⊂a */{0}. 设S ={α}⊂R ，定义压缩锥为a -≔{X ∈a |∀γ∈S ，γ(X )≤0}.注 1 设a E ={X ∈a |∀γ∈R ，γ(X )=0}，利用定义1，可知a E =a -∩(-a -)≡{0}. 值得一提的是，174第 1 期金四海，等：仿射对称空间SU(1，2)/SO(1，2)上的Plancherel 定理a E ≡{0}为仿射对称空间SU ()1，2/SO ()1，2上存在离散序列表示的必要条件，这是Harish-Chandra 定理（离散序列表示存在性定理）的推广. 事实上，设a=a +a E =a. 利用条件a E ={0}，可知商空间a /a 没有非平凡对偶，从而SU ()1，2/SO ()1，2中存在离散序列表示. 这种离散序列表示称为0-扭曲离散序列表示（Krötz et al.，2020），此0-扭曲离散序列表示恰好对应的是SU ()1，2/SO ()1，2中的离散序列表示，具体见文献（Krötz et al.，2020）. 结合Flensted-Jensen （1980）的定理1.1，只需验证条件：rank(SU(1，2)/SO(1，2))=rank ()SO(3)/()SO(3)∩SO(1，2).此条件可根据SU ()1，2的极大环面的维数证实，从而SU ()1，2/SO ()1，2满足Flensted-Jensen 条件.引理1 g +的根系为Σ()g +，a ={±α}，其对应的Weyl 群同构于对称群S 2.证明由k ，h ，p 和q 的定义，可知k ∩h =ìíîïïï|||||||()0000k 230-k 230 k 23∈R üýþïïïï， p ∩q =ìíîïïïi |||||||()0x 1x 2-x 100-x 2x 1，x 2∈R üýþïïïï.于是g +=(h ∩k)⊕(p ∩q)≔ìíîïïï|||||||()0i x 1i x 2-i x 10a 23-i x 2-a 23x 1，x 2，a 23∈R üýþïïïï.令h =i ()10-1000，则方程[]X ，h =±X 对应的特征向量为()00±i 00-1∓i10，()00±i001∓i -10的常数倍. 利用上式，结合g +的定义，可知Σ()g +，a ={±α}. g +的Weyl 群由根系Σ()g +，a ={±α}的反射组成，此群同构于对称群S 2，引理证毕.注2 利用引理1，得到Σ()g +，a 的素根系为I ={α}. 定义a I ≔{X ∈a | α(X )=0}，易证a I ={0}，此条件暗示了在限制根系意义下，SU ()1，2/SO ()1，2上不存在非零扭曲离散序列表示.2 主要结果设n =n ()Σ+()su ()1，2，a ，定义P =LN 为SU ()1，2中τθ不变的极小抛物子群，其中N =exp (n )，L 是a 在SU ()1，2上的中心化子，即L =ìíîïïï|||||||()l 1l 20-l 2l 1000-1l 1，l 2∈C ，||l 12-||l 22=1üýþïïïï.设L =MA ，其中M 是τ稳定子群，A =exp (a)=ìíîïïïï|||||()cosh (t )i sinh (t )0-i sinh (t )cosh (t )0001 t ∈R üýþïïïï.定义2 子群P 诱导SU ()1，2的主序列表示定义为：H σ，λ=Ind SU(1，2)P(σ⊗λ)={f | f (g ，m ，a ，n )=σ(m )-1(a )-λ+ρf (g )，g ∈G ，m ⋅a ⋅n ∈P }，其中ρ为正根和的一半，σ是M 的离散序列表示，λ∈ia *.现具体构造M 的离散序列表示，其记号为()σ，V σ.175第 63 卷中山大学学报（自然科学版）（中英文）引理2 M /(M ∩H )≃SU ()1,1/SO ()1,1，其中H =SO ()1，2 .证明由于M 是L 的τ稳定子群，则M ≃SU ()1，1. 设H =ìíîüýþ|||()db t cA ∈SL (3，R ) b ，c ∈R 2，d ∈R ，A t A -c t c =1，A t b =dc ，b 2-d 2=-1，从而M ∩H =ìíîïïï|||||||()d b 0-bd 000-1b ，d ∈R ，d 2-b 2=1üýþïïïï≃SO(1，1).引理证毕.利用Harish-Chandra 嵌入，可知SU ()1，1/SO(2)≃D ={}z ∈C | 1-||z 2>0 . 假定ω∈R ，考虑加权平方可积空间L 2，ω(D )：L 2，ω(D )=ìíîïïF (z ) |||| F 2L 2，ω(D )=c ω∫D ||F (z )2[1-||z 2]ω-2d z <+∞üýþïï，其中c ω是一个亚纯系数，且满足 1L 2，ω(D )=1，d z 是C 上的Lebesgue 测度. 定义L 2，ω(D )在群SU ()1，1作用下的酉表示为[πω(g )f ](z )=()b z +a -ω-2f (g -1⋅z )，其中g -1=()ab b a ∈SU ()1，1，且g -1⋅z =(az +b )()b z +a -1.当n =ω≥2且n ∈Z +时，在L 2，ω(D )上定义一个加权Bergman 空间H n (D )≔ìíîïïf ∈O (D )|||| f 2L 2，ω(D )≔n -1π∫D ||f (z )2()1-||z 2n -2d z <+∞üýþïï，其中O (D )是D 上全纯函数构成的集合. 易证πn 是SU ()1，1在H n (D )上的全纯离散序列表示. 对于任意的g ∈SU ()1，1，设H ∞n (D )是πn (g )作用在H n (D )上关于g 的无穷可微的函数构成的空间. 定义H -∞n (D )是H ∞n (D )的前对偶空间，根据空间的嵌入关系可知H ∞n (D )⊂H n (D )⊂H -∞n (D ).引理3（'Olafsson et al.，1988）设n ≥2且n ∈Z +，定义π∨n 为πn 的逆步表示. 设x 0=eH ，g =k θa t x 0，其中k θ∈K ，a t ∈A . 对于任意的h (z )∈H ∞n (D )，则存在一个SO ()1，1不变的分布Φn ∈H -∞n (D )使得缠结算子I n h (z )(g )=h ，π∨n (g )Φn 可将H n (D )等距地延拓到V n ，其中V n 是L2()SU ()1, 1/SO ()1, 1的闭子空间.当n ≥2且n ∈Z +时，则π-n 为SU ()1，1的共轭全纯离散系列表示，其表示空间为闭子空间V -n . 结合引理2和引理3，可知M 的离散序列表示有同构关系σ≃π±n ，记作σ±n .为了得到本文的主要定理，首先需要计算Plancherel 测度. 由于{}α，2α为a *中的一组基，于是对于∀β∈a *，β=a 1(α)+a 2(2α)，其中a 1，a 2∈R . 令a 1+2a 2=λ1，将a *中的元素记作λ=(λ1，0).命题 1 ia *上的Plancherel 测度为d λ=||||||||||27Γ(λ1)Γ()12+λ12Γ()52Γ()748Γ(1+λ1)Γ()1+λ12Γ()32Γ()54||||||||||-2d λ1.证明结合文献（Duistermaat et al.，1979），可知ia *上的Plancherel 测度为d μ(λ)=||c (λ)-2d λ.（1）设ρ为Σ()g ，a 的半根系，其中c (λ)=I ()λI ()ρ， I (λ)=I (α；λ).176第 1 期金四海，等：仿射对称空间SU(1，2)/SO(1，2)上的Plancherel 定理利用正特征向量的重数，得到n (α)=dim (g α)=2， n (2α)=dim (g 2α)=1， d (α)=n (α)+n (2α)=3，从而I (α；λ)=α，α-12d (α)Γ()λ，αα，αΓ()14n (α)+12λ，αα，αΓ()12n (α)+λ，αα，αΓ()12n (2α)+14n (α)+12λ，αα，α，其中Γ(⋅)定义为Gamma 函数，⋅，⋅为a *中的Killing 形式. 代入λ和ρ得I (α；λ)=Γ(λ1)Γ()12+λ12Γ(1+λ1)Γ()1+λ12， I (ρ；λ)=8Γ()32Γ()5427Γ()52Γ()74.于是c (λ)=27Γ(λ1)Γ()12+λ12Γ()52Γ()748Γ(1+λ1)Γ()1+λ12Γ()32Γ()54.现将c (λ)代入式（1），此命题得证.定理1（Plancherel 公式）仿射对称空间Z =SU ()1, 2/SO ()1, 2上的Plancherel 公式为L 2(Z )=∑n ≥2∫ia *⊕H σ±n， λd λ⊕∑||ν-22<r ≤||ν-2F 2r +4-||ν，其中σ±n 和d λ如上所述，F 2r +4-||ν见下文.3 Plancherel 公式的证明首先构造定理1中的离散序列表示部分. 本部分的构造需要一些准备和技巧. 提醒读者的是，如下所定义的符号和第2节定义的单位圆盘不同，因此空间的定义域和指标也不尽相同.定义B ≔{}z =(z 1，z 2)∈C 2 |||z 12+||z 22<1. 悉知，B 的自同构群为SU ()1, 2，且SU ()1, 2通过分式线性变换可迁地作用在B 上. 定义Hilbert 空间L 2(B ，d μν(z ))=ìíîïïf |||| ∫B ||f (z )2()1-||z 2||ν-3d z <+∞üýþïï，其中d z 是Lebesgue 测度，ν∈R . 如果|ν|>2，||ν∈Z +，定义Hilbert 空间H ν(B )=ìíîïïf 是全纯函数 |||| f 2=∫B ||f (z )2()1-||z 2||ν-3d z <+∞üýþïï.特别地，如果||ν=3，那么H 3(B )是经典的Bergman 空间. 定义群SU ()1, 2在H ν(B )上的酉表示为[πν(g )f ](z )=[det (cz +d )]-||νf (g -1⋅z )，（2）其中g -1=()abcd∈SU ()1, 2，g -1⋅z =(az +b )(cz +d )-1. 如果||ν>2且||ν∈Z +，那么πν是关于SU ()1, 2的离散序列，如果||ν>2，||ν∉Z +，则πν是万有覆盖群~SU ()1, 2的射影表示. 具体例子见文献（'Olafsson etal.，1988）.引理4（Zhang ，1992）设L 2(B ，d μν(z ))d 是L 2(B ，d μν(z ))的闭子空间，且为离散序列表示，则L 2(B ，d μν(z ))d 有以下不可约分解L 2(B ，d μν(z ))d ≃∑||ν-22<r ≤||ν-2H 2r +4-||ν .177第 63 卷中山大学学报（自然科学版）（中英文）注3 当||ν≤2时，不存在离散序列表示；当||ν-2是正的偶整数时，有||ν-22-1个离散序列表示；当||ν-2是正的奇整数时，有||ν-12个离散序列表示.引理5（钟家庆，1989）设l ≥0，a l =Γ(ρ+l )Γ(ρ)Γ(l +1)，且Re ρ>0，则(1-w )-ρ=∑l ≥0a lw l(w ∈C ， ||w <1).设()λ1，λ2为2l 的拆分，即(2l ，0)，(2l -1，1)，⋯，(l ，l )共有l +1种取法. 定义两个集合Λ+和Λ-满足ìíîλi ∈Λ+⇔λi -i 为奇数；λi ∈Λ-⇔λi -i 为偶数.为了行文方便，分别记Λ+以及Λ-中的元素为λi 1，λi 2，其中i 1，i 2∈{1，2}且i 1≠i 2.引理6（钟家庆，1989）设l >0，z 1，z 2∈C ，则(z21+z 22)l=l ！∑λ1≥λ2≥0λ1+λ2=2lb ()λ1，λ2S ()λ1，λ2(z 1，z 2)，其中S (λ1，λ2)(z 1，z 2)=z λ1+11z λ22-z λ12z λ2+11z λ11z λ22-z λ12z λ21， b (λ1，λ2)=(-1)i 1-1()λi 1-i 1+12！()λi 2-i 2+22！，λi 1∈Λ+，λi 2∈Λ-.在引理6中，函数S (λ1，λ2)(z 1，z 2)称为Schur 函数，具体实例见文献（钟家庆，1989）. 定义Laplace-Beltrami 算子为L =∑j =12(z j∂∂z j )2+2∑i ≠j ()z 2i z i -z j∂∂z j .类似MacDonald （2015）的证明，对于2l =∑i =12λi ，有L S (λ1，λ2)(z 1，z 2)=C (λ1，λ2)S (λ1，λ2)(z 1，z 2)， C (λ1，λ2)=∑j =12λj (λj -2j )+3l ，且L 2S (λ1，λ2)(z 1，z 2)=C 2(λ1，λ2)S (λ1，λ2)(z 1，z 2).对于任意的z ∈B ，通过Laplace 方程的正则化理论，我们可定义H m (B )上的光滑闭子空间H ∞m (B )≔ìíîïïï∑l ≥0 ∑λ1≥λ2≥0λ1+λ2=la (λ1，λ2)S (λ1，λ2)(z 1，z 2) |||| ∑λ1≥λ2≥0||a(λ1，λ2)2C2(λ1，λ2)C (λ1，λ2)<+∞üýþïïïï，这里C (λ1，λ2)=∏j =12∏k =1λj(m -(j -1)+k -1).令m >2且m ∈2Z +，定义在B 上的分布向量为v 0(z )≔[]det (I 2-z tz )-m2.易证v 0(z )=(1-zzt)-m2=(1-z 21-z22)-m 2.命题2 设z ∈B ，m =2ρ. 记H -∞m (B )为H ∞m (B )的前对偶空间. 如果Re ρ>0，那么v 0(z )∈H -∞m (B ).证明对于z ∈B ，利用引理2，引理3以及S (λ1，λ2)(z 1，z 2)的缓增系数，易得v 0(z )可以展开为收敛的幂级数形式. 从而在H ∞m (B )上赋予一些缓增系数得到其对应的连续线性泛函，有v 0(z )∈H -∞m (B )，命题得证.对于任意的g ∈SU ()1, 2和f ∈H -∞m (B )，定义逆步表示178第 1 期金四海，等：仿射对称空间SU(1，2)/SO(1，2)上的Plancherel 定理π∨m (g )f (⋅)=------------πm(g -1)f (⋅).（3）结合表示πν（见式（2）），可知v 0(z )是一个SO ()1, 2-不变的分布向量.命题3 设A p =exp ()a p ，对于任意的a t ∈A p ，有--------------------π∨m (a t )v 0(0)=1，π∨m (a t )v 0= cosh -m2(2t ).证明设t ∈R ，且a t =()cosh ti sinh t 0-i sinh tcosh t 0001∈a p .利用式（2）~（3），可知--------------------π∨m (a t )v 0(0)=1，π∨m (a t )v 0=(cosh 2t +sinh 2t )-m 2=cosh -m2(2t ).引理7 对于任意g ∈SU ()1, 2，有Cartan-Berger 分解g =kah ，其中k ∈K ，a ∈A p ，h ∈SO ()1, 2.设x 0=e SO ()1, 2. 根据引理6、文献（'Olafsson et al.，1988），以及SU ()1，2/SO ()1，2=K exp (a +p )⋅x 0，有∫SU(1，2)/SO(1，2)f (x )d μ(x )=∫K ∫a+pf (kat⋅x 0)d μ0(t )d k ，其中d k 是K 上的规范Haar 测度，且d μ0(t )=4cosh (t )||sinh (t )cosh (2t )d t .（4）定理2 设m >2，则∫a +pcosh -m ()2t d μ0(t )=12Γ(1)Γ()m -22Γ()m 2.证明利用式（4），有∫a +pcosh -m()2t d μ0(t )=∫∞2cosh (t )sinh (t )cosh1-m(2t )d (2t )=∫∞sinh ()2t cos h 1-m ()2t d ()2t .此外，对于Re(α)>-1且Re(β-α)>0，定义Beta 函数∫0∞sinh α(x )cosh -β(x )d x =12Γ()α+12Γ()β-α2Γ()β+12.取α=1，β=m -1，有∫a +pcosh -m ()2t d μ0(t )=12Γ(1)Γ()m -22Γ()m 2.利用Gamma 函数的性质，可知m >2，定理证毕.由引理3，定理2，可得以下结果：定理3 对于任意的h (z )∈H ∞2r +4-||ν(B )，则存在一个SO ()1, 2-不变的分布函数v 0∈H -∞2r +4-||ν(B )使得缠结算子I 2r +4-||ν(h )(g )=h ，π∨2r +4-||ν(g )v 0将H 2r +4-||ν(B )等距的延拓为F 2r +4-||ν，其中2r +4-||ν>2.证明设m =2r +4-||ν. 对于ka t ∈KA p ，仅需找到Flensted-Jensen 函数1，π∨m ()ka t v 0，将其缠结化算子定义为I m 1(x )≔1，π∨m ()x v 0，179第 63 卷中山大学学报（自然科学版）（中英文）其中x ∈SU ()1，2/SO ()1，2. 根据文献（'Olafsson et al.，1988），容易得到空间F m ≔span{}|1，π∨m(ka t )v 0 ka t ∈KA p .此空间关于SU ()1，2的表示是不可约的. 于是，对于k ∈K ，a t ∈A p ，存在一个非零复数c 使得I m 1(ka t ⋅x 0)=cχ(k )cosh (t )-m 2，其中χ为K 的不可约特征标.现只需证明I m 1∈L 2()SU ()1，2/SO ()1，2. 为了证明这一事实，下文需要用到Bergmann 空间的一些结构，即循环向量的技术来表明此事实. 悉知，常值函数1在表示πm 意义下是H m (B )的循环向量，即对于任意的g ∈SU ()1，2，πm (g )1线性张成的空间在H m (B )中稠密. 对于1∈H m (B )，利用式（3）、引理4和定理3，可知m >2且m ∈2Z +，即F m 的指标与H m (B )的指标一致，可知F m ≃H m (B ). 因为特征标函数χ(k )是一个类函数，从而I m 1∈L 2()SU ()1，2/SO ()1，2，定理证毕.归结起来，由引理6以及定理3，定理1得证.参考文献：韩威，范兴亚， 2023. 关于仿射对称空间SU （2，2）/SL （2，C ）+R 非紧分歧离散谱的一点注记［J ］. 新疆大学学报（自然科学版）（中英文）， 40（1）： 17-22+29.钟家庆， 1989. Schur 函数的一个展式及其在计数几何中的应用［J ］. 中国科学（A 辑），（10）： 1018-1029.DELORME P ， KNOP F ， KRÖTZ B ， et al ， 2021. Plancherel theory for real spherical spaces ： Construction of the Bernstein mor ‐phisms ［J ］. J Amer Math Soc ， 34（3）： 815-908.DUISTERMAAT J J ， KOLK J A C ， VARADARAJAN V S ， 1979. Spectra of compact locally symmetric manifolds of negative cur ‐vature ［J ］. Invent Math ， 52（1）： 27-93.FLENSTED-JENSEN M ， 1980. Discrete series for semisimple symmetric spaces ［J ］. Ann Math ， 111（2）： 253-311.GEL'FAND I M ， GINDIKIN S G ， 1977. Complex manifolds whose skeletons are semisimple real Lie groups ， and analytic discreteseries of representations ［J ］. Funct Anal Its Appl ， 11（4）： 258-265.HARISH-CHANDRA ， 1965. Discrete series for semisimple Lie groups I ： Construction of invariant eigendistributions ［J ］. ActaMath ， 113： 241-318.KRÖTZ B ， KUIT J J ， OPDAM E M ， et al ， 2020. The infinitesimal characters of discrete series for real spherical spaces ［J ］. GeomFunct Anal ， 30（3）： 804-857.MACDONALD I G ， 2015. Symmetric functions and Hall polynomials ［M ］. 2th ed. New York ： Oxford University Press.'OLAFSSON G ， ØRSTED B ， 1988. The holomorphic discrete series for affine symmetric spaces ， I ［J ］. J Funct Anal ， 81（1）：126-159.ZHANG G ， 1992. A weighted Plancherel formula II ： The case of the ball ［J ］. Studia Math ， 102（2）： 103-120.（责任编辑冯兆永）180。

一类动力学方程及流体力学方程解的Gevrey类正则性

Boltzmann 方程 . . . . . . . . . . . . . . . . . . . . . . . . 碰撞算子 Q(f, f ) 的基本性质 . . . . . . . . . . . . . . . . . Fokker-Planck 方程、Landau 方程以及 Boltzmann 方程线性化模型 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Navier-Stokes 方程 . . . . . . . . . . . . . . . . . . . . . . . Gevrey 函数空间 . . . . . . . . . . . . . . . . . . . . . . . .
研究现状及本文主要结果 . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 1.2.2 1.2.3 1.2.4 存在性及唯一性 . . . . . . . . . . . . . . . . . . . . . . . . . 动力学方程的正则性理论: 空间齐次情形 . . . . . . . . . . . 动力学方程的正则性理论: 空间非齐次情形 . . . . . . . . . . Navier-Stokes 方程的正则性理论 . . . . . . . . . . . . . . .
第二章预备知识 2.1 2.2 2.3 基本记号
Fourier 变换 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 基本函数空间及常用不等式 . . . . . . . . . . . . . . . . . . . . . . 2.3.1 2.3.2 Lp 空间及其性质 . . . . . . . . . . . . . . . . . . . . . . . . Sobolev 空间及其性质 . . . . . . . . . . . . . . . . . . . . .

结晶学

[SM(〗地球科学大辞典结晶学结晶学【结晶学】crystallography又称晶体学。

研究晶体的外部形貌、化学组成、内部结构、物理性质、生成和变化，以及它们相互间关系的一门科学。

它诞生于17世纪下半叶，但早期只是作为矿物学的一个分支而存在，其研究对象亦局限于天然的矿物晶体。

直到19世纪，随着其研究范围逐步扩大到矿物以外的各种晶体，结晶学才逐渐脱离矿物学而成为一门独立的学科。

近代结晶学主要包括晶体发生学、几何结晶学、晶体结构学、晶体化学及晶体物理学等分支。

它们阐明晶体各个方面的性质和规律，并可用以指导对晶体的利用和人工培养。

【晶体发生学】crystallogeny又称晶体生成学。

结晶学的一个分支。

研究晶体的发生、成长、变化等方面的现象、机理和规律。

它对指导人工制备晶体以及解释晶体的某些现象、特性和成岩、成矿作用的一系列问题等方面均具有重要意义。

【几何结晶学】geometrical crystallography结晶学的一个分支。

是早期结晶学的主要内容，也是矿物学的基本内容之一。

研究具有天然规则多面体外形晶体的几何形貌、几何要素(晶面、晶棱等)以及其间的对称性和各种几何关系。

它对晶体的描述、分类和矿物的鉴定均具有重要意义。

【晶体结构学】crystallology又称结构晶体学。

结晶学的一个分支。

研究晶体内部结构中质点排布的各种规律和晶体结构的具体测定，以及实际晶体结构的不完善性。

它对从根本上阐明晶体的一系列现象和性质起着重要的作用。

【晶体化学】crystal chemistry又称结晶化学。

结晶学的一个分支。

是结晶学与化学之间的边缘科学。

主要研究晶体的化学组成与晶体结构之间的关系和规律。

对于阐明晶体的一系列现象和性质及它们相互的内部联系等方面有着重要的意义。

【晶体物理学】crystallophysics结晶学的一个分支。

是结晶学与固体物理学之间的边缘科学。

主要研究晶体的各项物理性质及其形成机理和规律。

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Chiral Symmetry Breaking in 2+1 dimensions due to Sphalerons

The Subleading Isgur-Wise Form Factor $chi_3(vcdot v')$ to Order $alpha_s$ in QCD Sum Rules

SrVO3 Hubbard band 和 氧空位

axions

核子中奇异夸克分布不对称性与轻味夸克碎裂效应

固体物理英语

Two-Dimensional Gas of Massless Dirac Fermions in Graphene

向涛-General Introduction

仿射对称空间SU(1，2)

一类动力学方程及流体力学方程解的Gevrey类正则性

结晶学

SrVO3 Hubbard band 和氧空位