• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    A Renormalized-Hamiltonian-Flow Approach to Eigenenergies and Eigenfunctions?

    2019-07-25 02:01:58WenGeWang王文閣
    Communications in Theoretical Physics 2019年7期
    關(guān)鍵詞:文閣

    Wen-Ge Wang (王文閣)

    Department of Modern Physics,University of Science and Technology of China,Hefei 230026,China

    Abstract We introduce a decimation scheme of constructing renormalized Hamiltonian flows,which is useful in the study of properties of energy eigenfunctions,such as localization,as well as in approximate calculation of eigenenergies.The method is based on a generalized Brillouin-Wigner perturbation theory.Each flow is specific for a given energy and,at each step of the flow,a finite subspace of the Hilbert space is decimated in order to obtain a renormalized Hamiltonian for the next step.Eigenenergies of the original Hamiltonian appear as unstable fixed points of renormalized flows.Numerical illustration of the method is given in the Wigner-band random-matrix model.

    Key words: generalized Brillouin-Wigner perturbation theory,Hamiltonian flow,eigenfunction structure,eigenvalue

    1 Introduction

    Properties of energy eigenvalues and eigenfunctions are of central importance in a variety of fields,from nuclei physics,atomic physics,to condensed matter physics,and so on.[1?12]In particular,they are of relevance to thermalization,[13?21]a topic which has attracted renewed interest in recent years.An important method of studying these properties is the renormalization group method.Various versions of this method have been developed.For example,in calculating energy eigenfunctions in the low energy region,wide use has been made of Wilson’s numerical renormalization group[22?23]and of the density-matrix renormalization group method.[24?25]

    Localization of wavefunctions is one of the most important phenomena discovered in the field of condensed matter physics[26?30]and in the field of quantum chaos.[31?32]A real-space renormalization-group method and its modified versions[33?38]have been found quite successful in the study of localization properties in one-dimensional systems with (effectively) finite range of coupling; while in the case of two and more than two-dimensional systems,they have been found successful only in some special cases such as the Fibonacci quasi-lattices (see,e.g.,Refs.[39–40]).Moreover,recently,the phenomenon of many-body location has attracted wide attention (see,e.g.,Refs.[41–44]).

    Different schemes of constructing renormalized Hamiltonian flows are usually suitable for different types of problems.No scheme has been found universally useful.Hence,it is always of interest to find new schemes of constructing renormalized Hamiltonian flows.

    In this paper we introduce a new method of constructing renormalized Hamiltonian flow,based on a generalized Brillouin-Wigner perturbation theory (GBWPT).[45?49]The GBWPT shows that an arbitrary eigenfunction of a Hamiltonian can be divided into two parts,a perturbative part and a non-perturbative part,with the perturbative part expanded in a convergent perturbation expansion in terms of the non-perturbative part.Making use of this result of the GBWPT,we show that a subspace of the Hilbert space,which is associated with a perturbative part of the eigenfunction,can be decimated.This decimation scheme produces a renormalized Hamiltonian and,following this procedure,a renormalized Hamiltonian flow can be constructed.

    We show that,for a renormalized Hamiltonian flow constructed by the method mentioned above,eigenenergies of the original Hamiltonian appear as(unstable)fixed points of a property of the flow.Furthermore,those eigenfunctions of the renormalized Hamiltonians in the flow,which share the same eigenenergy,have related components.These two properties of the renormalized Hamiltonian flow may be made use of in approximate calculation of eigenenergies and in the study of properties of the eigenfunctions of the original Hamiltonian,e.g.,their localization properties.These predictions are checked numerically in the Wigner-band random-matrix model.

    2 General Theory

    2.1 Generalized Brillouin-Wigner Perturbation Theory

    In this section,we discuss the basic contents of GBWPT.It is a direct generalization of the ordinary Brillouin-Wigner perturbation theory,which can be found in textbooks,e.g.,in Ref.[50].Consider a perturbed HamiltonianH=H0+V,whereH0is an unperturbed Hamiltonian andVis a generic perturbation.In the normalized eigenbasis ofH0,denoted by

    For an energy eigenstate|α〉,let us divide the setinto two subsets,denoted bySαand,respectively.This gives two projection operatorsPandQ,

    Here we useto indicate basis statesinSαandforCorrespondingly,the stateis divided into two parts,

    Multiplying both sides of Eq.(4) byQand noticing thatQH0=H0Q,one has

    where

    Substituting Eq.(5) into Eq.(3) and doing iteration,one finds thatcan be expanded in a convergent perturbation expansion,

    when the following condition is satisfied

    When the setSαincludes only one basis vector,the expansion in Eq.(7) gives the ordinary Brillouin-Wigner perturbation expansion.Since the exact eigenenergyEαappears in the expansion,the expansion can not be immediately employed in numerical computation.However,noticing thathas one component only in the basisand that the componentsCαkshould satisfy certain normalization condition,this problem can be overcome.For example,taking the normalization condition=1 for normalizedand multiplying Eq.(4) from left byone can write the exact energy asEα=E0i+Then,one can writeEαandin the form of two related iterative expansions.[50]

    In the case thatSαincludes more than one vectorsEq.(7) gives a generalization of the (ordinary) Brillouin-Wigner perturbation theory(GBWPT).In this case,since there are at least two componentsCαiin,merely making use of the normalization condition,one can not writeEαandin two related iterative expansions.Therefore,in the GBWPT,Eαandcan not be calculated in a way similar to that discussed above in the ordinary Brillouin-Winger perturbation theory.

    Several applications of the GBWPT have been found.The condition (8) determines the separation ofinto two parts,and.In systems with band structure of the Hamiltonian,usuallycorresponds to the main body of,whilecorresponds to the tail part ofwith small components.[45,48]It has been shown that the expansion in Eq.(7) is useful in deriving analytical expressions for the decaying behavior of the tails of[45,48]This separation ofhas also been found useful in approximate calculation of eigenstates in certain energy region.[47]Further numerical investigation reveals that this separation of energy eigenstates is useful in the study of phenomenon like dynamical localization[46,48]and in the study of the distribution of components of wave functions in quantum chaotic systems.[49]

    In this paper,we discuss a new application of the GBWPT,namely,a general scheme of constructing renormalized Hamiltonian flow.Before doing this,it is useful to give further discussion for the condition of separating an energy eigenstate into the two parts discussed above.A sufficient (unnecessary) condition for Eq.(8) to hold is

    In order to understand better the condition (9),we insert the expression of the projection operatorQgiven in Eq.(2) into Eq.(6) and get

    It is seen that only basis statesgive contribution to the denominator ofTα.Therefore,as long as the setis chosen such that allE0jofare far enough fromEα,Eq.(9) and hence Eq.(7) hold.This gives a convenient way of doing the separation of

    An advantage of using Eq.(9)is that one does not need to know the exact statein advance.Equation (9) is also useful when we treat a HamiltonianHwith a degenerate spectrum.As well known,degenerate spectrum ofHmay bring problem to the ordinary perturbation theory.However,in the GBWPT,Eq.(7)can still hold whenHhas a degenerate spectrum.In fact,since Eq.(9) does not contain any eigenstate,for eigenstates with the same eigenenergyEα,this equation gives the same separation of the basis states,i.e.,the setSα.For such a separation,Eq.(7) holds for all the eigenstates with the eigenenergyEα.In this case,Sαincludes more than one basis vectors.Different eigenstateswith the same eigenvalueEαhave different componentsCαiin,hence,have differentdetermined by Eq.(7).

    For the above reasons,in what follows,we use Eq.(9)to determine the separation ofinto the two partsand

    2.2 Renormalized Hamiltonian

    A renormalized Hamiltonian can be constructed for an eigenstateofH,by decimation of the statesinFor this purpose,making use of Eq.(7),we writeas

    where

    replacingCαjby the right hand side of Eq.(11),one has

    where

    This suggests that a renormalized Hamiltoniancan be introduced,

    which is an operator in the subspace spanned by states∈Sα.The most important relation betweenHandis that the stateis an eigenstate ofwith the eigenenergyEα,as shown in Eq.(14).Note that the elementsare functions ofEα.

    WhenHhas a degenerate spectrum,as discussed in the previous section,degenerate eigenstates with the same eigenenergyEαshare the same separationSα,hence,they have the same quantitiesAα(j →i′).As a result,degenerate eigenstatesare eigenstates of the same renormalized HamiltonianTherefore,the above scheme also works in the case of degenerate spectrum.

    The structure of non-zero off-diagonal elements ofHin the basisis usually different from that ofin.Indeed,Eqs.(12)and(15)show thatis typically nonzero when eitherHii′≠0 or there is a path of coupling fromtothrough statesin the set.Therefore,the number of basis stateswhich are coupled tobyis equal to or larger than that byH.

    We remark that the condition (8),which guarantees the expansion in Eq.(7),can not completely fix the setSα.Hence,one usually has much free space in choosingSαin constructing a renormalized Hamiltonian.

    2.3 Renormalized Hamiltonian Flow

    Repeating the procedure discussed in the previous section,withplaying the role ofH,one can obtain a new renormalized Hamiltonian from ?H.Following this,a renormalized Hamiltonian flow can be constructed,which is specific for the eigenstatewith eigenenergyEαof the original HamiltonianH.However,this method of constructing Hamiltonian flow has a drawback,namely,andEαare usually unknown.(The purpose of constructing a renormalized Hamiltonian flow is usually just to study properties ofandEα.) To avoid this drawback,in what follows we propose a more general method of constructing renormalized Hamiltonian flow,which is not specific for any eigensolution ofH.

    Let us denote byH(0)the original HamiltonianH,byEα(0)andits eigenenergies and eigenstates,respectively.For a set of basis states in the Hilbert space ofH(0),denoted by{|k(0)〉},H(0)is divided into two parts as in Eq.(1),The set of basis states is also divided into two partsS(0)and,with∈S(0)and∈; correspondingly,two projection operatorsP(0)andQ(0)can be introduced in the same way as in Eq.(1).The components ofare denoted by

    In considering the condition for a division of{|k(0)〉},let us write Eq.(9) in the following form,

    where

    HereEis a parameter with energy dimension,which is used in the construction of the renormalized Hamiltonian flow.Note that Eq.(17) gives Eq.(9) forE=Eα.

    Then,we can decimate the basis states inand,similar toin Eq.(16),introduce the first renormalized Hamiltonianin the flow,

    where

    Here

    ForE=Eα(0),similar to Eq.(14),we have

    hence,Eα(0)is an eigenenergy ofH(1)EwithE=Eα(0).IfEis not equal to any ofEα(0),it is usually not an eigenenergy of.Note thatis an operator in the Hilbert space spanned by(0).

    In the above procedure,with the superscript (0) replaced by Eq.(1),the second renormalized Hamiltonianin the flow can be constructed for the same parameterE.Then,with the superscript(1)replaced by Eq.(2),and so on,a renormalized Hamiltonian flowcan be constructed,withn=1,2,...

    IfE=Eα(0)for a Hamiltonian flow thus obtained,an equation similar to Eq.(22)holds with 0 replaced byn?1 and 1 byn.This implies the following important relation betweenandH(0),that is,an eigenstateofhas the following relation to|α(0)〉,

    whereis the same basis state asbut in the original labelling.This equation shows that some information in properties ofmay be obtained from properties of the corresponding eigenstateofIn the general case withEnot necessarily equal to any ofEα(0),let us denote byE(n)the closest eigenenergy oftoE.(Forn=0,takeH(0)).With increasingn,E(n)form a sequence with the flow,(E(0),E(1),E(2),...).IfE=Eα(0),Eq.(23) shows thatE(n)=Eα(0)for all values ofn; on the other hand,ifE≠Eα(0),E(n)are usually not equal toEα(0).Hence,Eα(0)are fixed points of the sequenceE(n),under the choice ofE=Eα(0).One may also consider the sequence of the deviation|Eα(n)?E|,for which zero is the fixed point corresponding to the choiceE=Eα(0).

    2.4 An Efficient Method of Constructing Renormalized Hamiltonian Flow

    The condition(17)with 0 replaced bynmust be satisfied,in order to constructfromHE(n)by decimating basis statesin.For a given choice of,it is usually not easy to prove whether the condition is satisfied or not.In fact,for an arbitrarily chosen setand an arbitrary value ofE,the condition is usually not satisfied.Therefore,it would be useful,if a general method can be found for decimation of an arbitrarily chosen set.In what follows,we introduce such a method.For brevity,in the following part of this section,we omit the superscript“(n)”,i.e.,all quantities should have the superscript“(n)”,except for the parameterE.

    The technique is to first carry out a rotation in the subspace spanned by states∈,such thatHis diagonalized in the subspace.We assume that the number of states inis not large and it is not difficult to diagonalize numerically the sub-matrix of the HamiltonianHin this subspace.Let us denote bythe obtained eigenstates of the sub-matrix ofHin the subspace and byEjathe corresponding eigenenergies.

    Now take the set ofas a new subset.Correspondingly,the HamiltonianHis divided into two parts,H0andV,in the same way as discussed in previous sections.In particular,by definition,is an eigenstate of,

    Then,making use of the expression ofQin Eq.(2),we can writeTEin Eq.(18) as (with the superscript (0) replaced by (n) and then omitted)

    whenEis not equal to any ofEja.Equation (25) implies that (TE)2=0,since there is no coupling among,namely,=0.As a result,Eq.(17) holds with 0 replaced byn.When it happens thatEis equal to one ofEja,one may change a little the two original subsetsSofandofby exchanging a few states in them; this may change the values ofEjaand makeE≠Eja.

    Finally,by the method discussed in the previous section,the set of(equivalently,that of) can be decimated and a renormalized Hamiltonian can be obtained.In particular,AE(ja→i) has a quite simple expression,

    since (TE)2=0 for the choice of the set of.It is not difficult to see that the above schemes can work for a degenerate spectrum,as well.

    3 Some Applications

    In this section,we show that the method presented in this paper supplies a useful approach to properties of energy eigenvalues and eigenfunctions.

    3.1 Eigenenergies as Unstable Fixed Points

    As discussed in Subsec.2.3,the eigenenergiesEα(0)of the original HamiltonianH(0)are fixed points of the sequenceE(n),whereE(n)is the eigenenergy ofwhich is the closest toE.As a result of this property,the difference|E ?E(n)|as a function ofE(withnfixed)has local minima at the positionsE=Eα(0).Hence,the eigenenergiesEα(0)can be calculated by finding out the local minima.In fact,numerical evaluation of eigenenergies of large-scale Hamiltonian matrices is a very important topic in many fields in physics.Various methods have been developed in dealing with this problem (see,e.g.,Refs.[47,51–57]).The renormalization group method discussed above supplies an alternative approach to this important problem.

    To test the above predictions,we consider a banded random matrix model.Banded random matrix models have applications in several fields and are still under investigation (see,e.g.,Refs.[58–62]).Here we consider the so-called Wigner Band Random Matrix (WBRM) model,which was first introduced by Wigner more than 50 years ago for the description of complex quantum systems as nuclei.[63]It is still of interest (see,e.g.,Refs.[46,48,64–70]),since it is believed to provide an adequate description also for some other complex systems,e.g.,the Ce atom[71]and as well as dynamical conservative systems possessing chaotic classical limits.

    We consider the following form of the Hamiltonian matrix in the WBRM model,

    whereE0k=k(k=1,...,N),off-diagonal matrix elementsvkk′=vk′kare random numbers with Gaussian distribution forand are zero otherwise,andλis a running parameter for adjusting the perturbation strength.Herebis the band width of the Hamiltonian matrix andNis its dimension.

    The theory discussed above predicts that the pointsE=Eα(0)are fixed points for the propertyE(n)of the renormalized Hamiltonian flow.To check this numerically,we consider original HamiltoniansH(0)as given in Eq.(27),whose dimensions are not very large such that they can be diagonalized directly by using ordinary diagonalization methods.For eachH(0)thus obtained,we diagonalize it to obtain its eigenenergiesEα(0).Then,we takeE=Eα(0)and construct a (finite) renormalized Hamiltonian flowby making use of the method discussed in Subsec.2.4,with a number of arbitrarily chosen basis statesk(n)decimated at each step.Numerically,all the renormalized Hamiltonianshave been found sharing the same eigenenergyEα(0)and having related eigenfunctions,as predicted in Eq.(23).

    There are two types of fixed points: stable and unstable.We perform further numerical investigation to see whether the fixed pointsEα(0)are stable or unstable.For this,we take a value ofE,which deviates a little from an exact eigenvalueEα(0),say byδE=|E ?Eα(0)|.Variation of|E(n)?E|withncan show whether the fixed pointE=Eα(0)is stable or unstable.Our numerical simulations show that they are unstable.An example is given in Fig.1,which shows that the value of|E(n)?E|increases withn,indicating thatEα(0)is an unstable fixed point.In our numerical computation for this figure,at each step of the renormalization flow,we decimated 30 basis stateswith successive labellingk(n)and with the firstk(n)chosen arbitrarily.

    Now we study variation of|E(n)?E|as a function ofE,withnfixed.The theory predicts that this quantity has local minima of zero at the values ofE=Eα(0).Our numerical simulations indeed reveal this phenomenon.As shown in Fig.2,the positions of the local minima with the value of zero indeed correspond to positions of the exact eigenenergiesEα(0),which are indicated by the vertical dotted lines.This shows that the eigenenergies of the original Hamiltonian can be evaluated by numerical calculation of the local minima of|E(n)?E|.

    Fig.1 Variation of |E(n) ?E| with n,for the param-eters N=1000,b=100,and λ=10,where E(n) is the eigenenergy of HE(n )which is the closest to E.The value of E has a little deviation from an arbitrarily chosen exact eigenenergy Eα(0) of the original Hamiltonian H(0).For the solid curve,δE=|E ?Eα(0)|=0.01.At each step of the flow,an arbitrarily chosen set of 30 basis states with successive labelling are decimated.The value of |E(n)?E| increases with n,implying that Eα(0) is an unstable fixed point.The circles represent |E(n)?E|/10 for δE=0.001.The agreement of the solid curve and the circles show that for these small values of δE,|E(n)?E|is in the linear region of δE.

    Fig.2 Variation of |E(n) ?E| (circles connected by dashed lines) with E for n=5,the parameters N=300,b=100,λ=10,and E=247 + 0.06m with m=1,2,...,100.At each step of the renormalized Hamiltonian flow,30 basis states are decimated.Within the energy region shown in this figure,the original Hamiltonian has three eigenenergies with positions indicated by the three vertical dotted lines.Approximate values of the eigenenergies can be get from extrapolation of the circles close to the local minima of |E(n)?E|.

    3.2 Localization of Eigenfunctions

    Based on Eq.(23),the theory here can also be used in the study of properties of energy eigenfunctions ofH(0),namely,the components ofin.For this,one should first know the eigenenergyEα(0),which may be obtained by the method discussed in the previous section or by some other method.Next,one can useE=Eα(0)to construct a finite renormalized Hamiltonian flow,untilwhose dimension is small enough for direct numerical diagonalization.Then,one can perform direct numerical diagonalization for this Hamiltonian and findwhich give the corresponding components ofinby the relation (23).In this way,some information about the wavefunctione.g.,its localization properties,may be obtained.In fact,if data for the construction ofofm=1,...,nhave been stored,it is even possible to obtain all the components

    We also employ the WBRM model discussed in the previous section to check the applicability of the method discussed above.Consider,e.g.,the parametersN=100,b=4,andλ=10.Hamiltonians with these parameters have localized eigenfunctions,e.g.,the one shown in Fig.3 by the solid curve.To check the validity of Eq.(23),we first diagonalizeH(0)directly and obtain its eigenenergiesEα(0)numerically.Then,we construct a(finite)renormalized Hamiltonian flowwithE=Eα(0),by making use of the method discussed in Subsec.2.4 with 10 basis states decimated at each step.Our numerical results indeed confirm the prediction of Eq.(23).An example is given in Fig.3 forn=5,which shows that the values ofagree well with the corresponding ones ofeven whenis as small as e?20.

    Fig.3 Values of the componentsfor n=0 and 5 in a renormalized Hamiltonian flow of .The original Hamiltonian is a realization of the Hamiltonian matrix in the WBRM model with parameters N=100,b=4,and λ=10.In the construction of the renormalized Hamiltonians,E=Eα(0) and 10 basis states are decimated at each step of the flow. and are eigenstates of H(0) and,respectively,with the same eigenenergy Eα(0).The two eigenfunctions agree well,as predicted in Eq.(23).

    3.3 A Discussion of Computation Time

    In this section,we give a brief discussion for the dependence of the computation time required by the method here on the dimensionNof the original Hamiltonian.This is to be compared with the corresponding dependence in ordinary direct diagonalization methods,in which the computation time usually scales asN3.

    When using the method here to calculate eigenenergies,as discussed in Subsec.3.1,one first needs to choose the energy region of interest and divide the region into consecutive segments,say,to (Ns?1) segments.Then,one can take theNsends of the segments as the parameterEand construct renormalized Hamiltonian flows.Suppose at each step totallymbasis states are decimated,withm ?N.This requires diagonalization of anm×mmatrix,which takes a time scaling asm3.After decimation of thembasis states,one obtains a new renormalized Hamiltonian and needs to calculate its new elements.(Some elements of the renormalized Hamiltonian may remain unchanged in the decimation process.) If there areM1new elements to be calculated and the time of calculating each new element scales asM2,then,calculation of the new elements needs a time scaling asM1M2.The values ofM1andM2depend on the structure of the original Hamiltonian.For example,for a 1-dimensional chain with nearest-neighbor coupling,it is possible for bothM1andM2to be quite small;on the other hand,for a full original Hamiltonian,(N2?m2) matrix elements are changed in the first step of the flow.

    Suppose one performsnsteps of the renormalization procedure and at last obtains a final renormalized Hamiltonian of dimension(N?nm).Diagonalization of the final Hamiltonian needs a time scaling as (N ?nm)3.Summarizing the above results,the total computation time scales asZ=Nsn(m3+M1M2)+Ns(N ?nm)3,where for simplicity in discussion,we assume thatM1M2can be taken as a constant.

    The method here is useful when a narrow energy region is of interest,because in this caseNsis not large.Usually,one may choose the value ofnsuch thatnmis close toN.This givesZ~NNs(m2+M1M2/m).Comparing it withN3for direct diagonalization method,we see that the method here is more efficient ifNs(m2+M1M2/m)?N2.In fact,the method here has another advantage,that is,it needs a relatively small memory for diagonalization.Specifically,it needs to diagonalize matrices with dimensionsmand (N ?nm),respectively,which can be small even for largeN.In contrast,a direct diagonalization method usually requires a memory scaling asN2,which is much larger thanm2and (N ?nm)2.

    4 Conclusions and Discussions

    In summary,based on the GBWPT,we propose a general method of constructing renormalized Hamiltonian flow with the energyEof interest as a parameter.Eigenenergies of the original Hamiltonian appear as (unstable) fixed points of some property of the renormalized Hamiltonian flow.WhenEis chosen as an eigenenergy of the original Hamiltonian,all the renormalized Hamiltonians in the same flow share the same eigenenergy asE,with the corresponding eigenfunctions possessing related components.we introduce a useful technique,by which an arbitrary set of basis states in the Hilbert space can be decimated in the construction of a renormalized Hamiltonian.We also discuss potential applications of the method in numerical evaluation of eigenenergies as well as in the study of localization of eigenfunctions,and illustrate them numerically in the WBRM model.In particular,by considering the scaling behavior of computation time,we find some situations in which the method here may be more efficient than the ordinary numerical diagonalization methods.

    As is known,localization in the WBRM model can be related to localization in another band-random-matrix model,by making use of a renormalization technique based on the GBWPT.[48]The method discussed in this paper can be used to improve the method in Ref.[48],specifically,by partial diagonalization of the Hamiltonian in the subspace spanned by states in,without rotation in the subspace spanned by states inSα.

    Finally,we give some remarks on the relation of the method discussed in this paper to some other methods of constructing renormalized Hamiltonians.The realspace renormalization-group method used in Refs.[33-34]for the one-dimensional tight-binding model with nearestneighbor-hopping,is in fact a special case of the method here,with the setincluding only one basis stateat each step of decimation.Its modified versions for 1D or quasi-1D systems,e.g.,those in Refs.[36–38],have some technical difference from the method here.A merit of the theory here is that it supplies a general approach to the construction of renormalized Hamiltonian flow,not restricted to some special types of models.

    猜你喜歡
    文閣
    難忘雷鋒的關(guān)愛(ài)
    婦女(2023年2期)2023-03-27 10:41:57
    男旦“頭牌”胡文閣的雙面人生
    梅葆玖送衣
    做人與處世(2022年3期)2022-05-26 00:18:36
    梅葆玖送衣
    窩棚、紙信、500元錢(qián):京漂導(dǎo)演有顆天真的心
    Similar Early Growth of Out-of-time-ordered Correlators in Quantum Chaotic and Integrable Ising Chains?
    Supercontinuum generation of highly nonlinear fibers pumped by 1.57-μm laser soliton?
    高文閣:堅(jiān)韌不拔揮灑筆墨苦研多年運(yùn)筆勁健
    僑園(2019年4期)2019-04-28 23:50:54
    胡文閣:男旦的憂傷
    北廣人物(2018年40期)2018-11-14 09:00:02
    胡文閣男旦的憂傷
    考比视频在线观看| 亚洲av.av天堂| 亚洲内射少妇av| 亚洲 欧美一区二区三区| 热99国产精品久久久久久7| 啦啦啦啦在线视频资源| 国产精品麻豆人妻色哟哟久久| 在线精品无人区一区二区三| 亚洲av国产av综合av卡| 有码 亚洲区| 亚洲国产欧美网| 国产亚洲一区二区精品| 午夜影院在线不卡| 久久国产精品大桥未久av| 亚洲精品,欧美精品| 各种免费的搞黄视频| 日本av手机在线免费观看| 曰老女人黄片| 国产成人精品一,二区| 在线天堂中文资源库| 久久免费观看电影| 免费久久久久久久精品成人欧美视频| 99久国产av精品国产电影| 中文字幕制服av| 秋霞伦理黄片| 欧美精品一区二区大全| 中文字幕最新亚洲高清| 我要看黄色一级片免费的| 999精品在线视频| 一区二区日韩欧美中文字幕| 国产成人精品久久二区二区91 | 99热网站在线观看| 毛片一级片免费看久久久久| 成人二区视频| 国产毛片在线视频| 电影成人av| 午夜日本视频在线| 最新的欧美精品一区二区| 成人手机av| 欧美bdsm另类| 69精品国产乱码久久久| 亚洲av成人精品一二三区| av片东京热男人的天堂| 国产精品人妻久久久影院| 亚洲精品国产一区二区精华液| 夫妻午夜视频| 久久青草综合色| 9191精品国产免费久久| 国产精品99久久99久久久不卡 | 日本免费在线观看一区| 女人精品久久久久毛片| 亚洲综合精品二区| 一个人免费看片子| 女人久久www免费人成看片| 只有这里有精品99| 成人18禁高潮啪啪吃奶动态图| 国产片特级美女逼逼视频| 高清在线视频一区二区三区| 色网站视频免费| 国产欧美日韩综合在线一区二区| 色视频在线一区二区三区| 一边摸一边做爽爽视频免费| 免费在线观看完整版高清| 母亲3免费完整高清在线观看 | 成年av动漫网址| 亚洲久久久国产精品| 麻豆av在线久日| 国产精品免费大片| 在现免费观看毛片| 精品亚洲成国产av| 久久人妻熟女aⅴ| 国产一区二区在线观看av| 看免费av毛片| 久久久久久人人人人人| 五月开心婷婷网| 成人黄色视频免费在线看| 国产一区二区 视频在线| 国产精品免费视频内射| 亚洲人成网站在线观看播放| 国产黄色免费在线视频| 成人影院久久| 免费看av在线观看网站| 伦精品一区二区三区| 91午夜精品亚洲一区二区三区| 自拍欧美九色日韩亚洲蝌蚪91| 美国免费a级毛片| 亚洲av在线观看美女高潮| 国产探花极品一区二区| 成人黄色视频免费在线看| 人人澡人人妻人| 国产乱来视频区| 国产午夜精品一二区理论片| 免费高清在线观看视频在线观看| 午夜免费鲁丝| 午夜精品国产一区二区电影| www.av在线官网国产| 免费人妻精品一区二区三区视频| 黑人欧美特级aaaaaa片| a级毛片黄视频| 国产成人精品一,二区| 日本-黄色视频高清免费观看| av片东京热男人的天堂| 久久午夜福利片| 伊人久久大香线蕉亚洲五| 欧美黄色片欧美黄色片| 欧美日韩亚洲国产一区二区在线观看 | 欧美成人午夜免费资源| 午夜福利在线免费观看网站| 国产成人91sexporn| 黄色毛片三级朝国网站| 亚洲美女搞黄在线观看| 国产av码专区亚洲av| 日日撸夜夜添| 亚洲国产精品成人久久小说| 久久久欧美国产精品| a级毛片在线看网站| 国产精品免费大片| 热re99久久精品国产66热6| 一边摸一边做爽爽视频免费| 一本一本久久a久久精品综合妖精 国产伦在线观看视频一区 | 日日摸夜夜添夜夜爱| 国产精品久久久久久久久免| 国产免费现黄频在线看| 久久这里只有精品19| 美女福利国产在线| 国产精品麻豆人妻色哟哟久久| 亚洲第一区二区三区不卡| 色吧在线观看| 国产片特级美女逼逼视频| 久久精品熟女亚洲av麻豆精品| 永久免费av网站大全| 国产激情久久老熟女| 国产老妇伦熟女老妇高清| 国产免费又黄又爽又色| 亚洲av国产av综合av卡| 亚洲精品av麻豆狂野| 在线亚洲精品国产二区图片欧美| 国产av码专区亚洲av| 18禁裸乳无遮挡动漫免费视频| 超碰成人久久| kizo精华| 亚洲精品久久久久久婷婷小说| 国产高清国产精品国产三级| 韩国高清视频一区二区三区| 久久久久久久久久久久大奶| 免费av中文字幕在线| 久久精品夜色国产| 十八禁网站网址无遮挡| 色婷婷av一区二区三区视频| 国产片特级美女逼逼视频| 久久狼人影院| 日韩av在线免费看完整版不卡| av一本久久久久| 精品国产超薄肉色丝袜足j| 伦精品一区二区三区| 午夜日本视频在线| 如日韩欧美国产精品一区二区三区| 国产一级毛片在线| 青春草国产在线视频| 亚洲精品一区蜜桃| 亚洲第一青青草原| 成年动漫av网址| 久久久精品免费免费高清| 亚洲成色77777| 亚洲精品视频女| 亚洲av.av天堂| 国产视频首页在线观看| 嫩草影院入口| 日韩 亚洲 欧美在线| 啦啦啦啦在线视频资源| 成年动漫av网址| 精品国产乱码久久久久久男人| 亚洲一区二区三区欧美精品| 日韩中字成人| 亚洲内射少妇av| 日日爽夜夜爽网站| 国产 精品1| 国产深夜福利视频在线观看| 国产淫语在线视频| 日韩熟女老妇一区二区性免费视频| 成人毛片60女人毛片免费| 久久精品久久久久久噜噜老黄| av天堂久久9| 国产精品嫩草影院av在线观看| 狠狠精品人妻久久久久久综合| 日本欧美视频一区| 免费大片黄手机在线观看| 国产精品 国内视频| 极品人妻少妇av视频| 久久青草综合色| 久久精品夜色国产| 美女午夜性视频免费| 国产男女超爽视频在线观看| 如日韩欧美国产精品一区二区三区| 伦理电影免费视频| 精品少妇内射三级| 国产 精品1| 午夜久久久在线观看| 男女高潮啪啪啪动态图| 日日摸夜夜添夜夜爱| av女优亚洲男人天堂| 伊人久久国产一区二区| 国产成人精品一,二区| av有码第一页| 熟女电影av网| 日韩av在线免费看完整版不卡| 亚洲国产欧美在线一区| 91在线精品国自产拍蜜月| 高清欧美精品videossex| 亚洲美女搞黄在线观看| 99国产精品免费福利视频| 伊人亚洲综合成人网| 亚洲伊人久久精品综合| 午夜日韩欧美国产| 免费看av在线观看网站| 亚洲精品久久成人aⅴ小说| 国产av精品麻豆| 中文字幕人妻熟女乱码| 国产精品女同一区二区软件| 青春草国产在线视频| 国产又爽黄色视频| 国产有黄有色有爽视频| 精品国产乱码久久久久久小说| 亚洲精品国产一区二区精华液| 丝袜美足系列| 下体分泌物呈黄色| 午夜影院在线不卡| 天天影视国产精品| 90打野战视频偷拍视频| 观看av在线不卡| 中国国产av一级| 一本大道久久a久久精品| 久久久久久久大尺度免费视频| 91精品三级在线观看| 久久久亚洲精品成人影院| av又黄又爽大尺度在线免费看| 99久久综合免费| 天天躁夜夜躁狠狠躁躁| 伦精品一区二区三区| 国产精品国产三级专区第一集| 男女边吃奶边做爰视频| 超碰成人久久| 在线观看一区二区三区激情| 夫妻性生交免费视频一级片| 99香蕉大伊视频| 黑人欧美特级aaaaaa片| 一边摸一边做爽爽视频免费| av视频免费观看在线观看| 久久久久久久久久久久大奶| 国产成人91sexporn| 午夜福利视频精品| 黄色 视频免费看| 深夜精品福利| 日产精品乱码卡一卡2卡三| 国产激情久久老熟女| 五月天丁香电影| 在线观看美女被高潮喷水网站| 两性夫妻黄色片| av又黄又爽大尺度在线免费看| 欧美少妇被猛烈插入视频| 在线观看美女被高潮喷水网站| 欧美国产精品va在线观看不卡| 999久久久国产精品视频| 精品国产一区二区三区久久久樱花| 日韩一本色道免费dvd| 国产亚洲av片在线观看秒播厂| 久久毛片免费看一区二区三区| 你懂的网址亚洲精品在线观看| 久久久久精品久久久久真实原创| 久久久久久久精品精品| 午夜精品国产一区二区电影| 女人久久www免费人成看片| 久久精品久久久久久噜噜老黄| 久久热在线av| 国产 精品1| 国产一区有黄有色的免费视频| 欧美日韩一区二区视频在线观看视频在线| 亚洲国产av新网站| av卡一久久| 女人高潮潮喷娇喘18禁视频| 成人亚洲欧美一区二区av| av免费在线看不卡| 午夜91福利影院| 伊人久久国产一区二区| 国产av精品麻豆| 各种免费的搞黄视频| 春色校园在线视频观看| 久久久a久久爽久久v久久| 免费女性裸体啪啪无遮挡网站| 久久久国产一区二区| 国产一级毛片在线| 这个男人来自地球电影免费观看 | a 毛片基地| 九草在线视频观看| 国产高清国产精品国产三级| 999久久久国产精品视频| 久久久久久久久久久免费av| 亚洲av成人精品一二三区| 男女高潮啪啪啪动态图| 最近2019中文字幕mv第一页| 国产成人精品一,二区| 亚洲av欧美aⅴ国产| 大片电影免费在线观看免费| 最近2019中文字幕mv第一页| 午夜福利乱码中文字幕| 电影成人av| 国产亚洲最大av| 一二三四在线观看免费中文在| 80岁老熟妇乱子伦牲交| 超色免费av| 久久ye,这里只有精品| 青春草国产在线视频| 深夜精品福利| 免费观看a级毛片全部| 我的亚洲天堂| 中文字幕亚洲精品专区| 老司机影院毛片| 桃花免费在线播放| 成年人午夜在线观看视频| h视频一区二区三区| 亚洲一区二区三区欧美精品| 两个人看的免费小视频| 久热久热在线精品观看| 捣出白浆h1v1| 亚洲国产av新网站| av不卡在线播放| 日韩欧美一区视频在线观看| 亚洲av欧美aⅴ国产| 久久久久人妻精品一区果冻| 亚洲精品在线美女| 国产精品二区激情视频| 日本色播在线视频| 国产成人精品婷婷| 日本av手机在线免费观看| 纯流量卡能插随身wifi吗| 夫妻性生交免费视频一级片| 十分钟在线观看高清视频www| 不卡av一区二区三区| 亚洲三级黄色毛片| 成人毛片a级毛片在线播放| av网站在线播放免费| 下体分泌物呈黄色| 亚洲欧美成人精品一区二区| 国产一区二区三区av在线| 久久韩国三级中文字幕| 99久久人妻综合| 在线天堂最新版资源| 亚洲国产av影院在线观看| av不卡在线播放| 另类亚洲欧美激情| 日本色播在线视频| 又黄又粗又硬又大视频| 精品一区二区三卡| 免费高清在线观看视频在线观看| 久久青草综合色| 新久久久久国产一级毛片| 久久久久久久久久久久大奶| 9色porny在线观看| 香蕉精品网在线| 久久精品人人爽人人爽视色| 中国三级夫妇交换| 日韩 亚洲 欧美在线| 91精品三级在线观看| 亚洲伊人久久精品综合| 亚洲国产精品成人久久小说| 91精品三级在线观看| 最黄视频免费看| a级片在线免费高清观看视频| 国产成人精品无人区| 在线观看三级黄色| 我要看黄色一级片免费的| 亚洲少妇的诱惑av| 成年美女黄网站色视频大全免费| 女人精品久久久久毛片| 久久 成人 亚洲| 国产一区亚洲一区在线观看| 国语对白做爰xxxⅹ性视频网站| 亚洲欧美色中文字幕在线| 亚洲精品视频女| 国产av国产精品国产| 丁香六月天网| 日韩av不卡免费在线播放| 观看美女的网站| 久久久a久久爽久久v久久| 免费高清在线观看视频在线观看| 啦啦啦视频在线资源免费观看| 丁香六月天网| freevideosex欧美| 国产成人免费无遮挡视频| 亚洲三级黄色毛片| 久久精品亚洲av国产电影网| 色播在线永久视频| 亚洲国产看品久久| 两性夫妻黄色片| av电影中文网址| 黄片小视频在线播放| 视频区图区小说| www日本在线高清视频| 高清视频免费观看一区二区| 亚洲少妇的诱惑av| 伊人亚洲综合成人网| 亚洲情色 制服丝袜| 女人高潮潮喷娇喘18禁视频| 18在线观看网站| 久久97久久精品| 日本欧美视频一区| 看非洲黑人一级黄片| 久久国内精品自在自线图片| 国产毛片在线视频| 菩萨蛮人人尽说江南好唐韦庄| 久久97久久精品| 久久婷婷青草| 菩萨蛮人人尽说江南好唐韦庄| 久久精品夜色国产| 黄色配什么色好看| 欧美日韩亚洲国产一区二区在线观看 | 日韩成人av中文字幕在线观看| 国产日韩欧美亚洲二区| av.在线天堂| 两个人看的免费小视频| 精品亚洲成国产av| 久久精品亚洲av国产电影网| 精品久久久精品久久久| 亚洲国产毛片av蜜桃av| 91aial.com中文字幕在线观看| 成人漫画全彩无遮挡| av在线老鸭窝| 国产精品成人在线| 欧美人与性动交α欧美软件| 宅男免费午夜| 一级毛片 在线播放| 在线免费观看不下载黄p国产| 久久女婷五月综合色啪小说| 亚洲欧洲精品一区二区精品久久久 | 免费久久久久久久精品成人欧美视频| 女人被躁到高潮嗷嗷叫费观| 亚洲av欧美aⅴ国产| 91精品国产国语对白视频| 80岁老熟妇乱子伦牲交| 美女国产视频在线观看| 下体分泌物呈黄色| 亚洲经典国产精华液单| 欧美少妇被猛烈插入视频| 热re99久久精品国产66热6| 亚洲av日韩在线播放| 久久久久国产一级毛片高清牌| www日本在线高清视频| 欧美激情极品国产一区二区三区| 18禁国产床啪视频网站| 菩萨蛮人人尽说江南好唐韦庄| 国产乱来视频区| 日本-黄色视频高清免费观看| 91在线精品国自产拍蜜月| 青春草视频在线免费观看| 26uuu在线亚洲综合色| 欧美日韩精品网址| 精品国产一区二区三区久久久樱花| 久久久久久久久免费视频了| 国产成人精品一,二区| 黄片无遮挡物在线观看| av不卡在线播放| 国产探花极品一区二区| 国产男女内射视频| 亚洲伊人色综图| 一区二区三区精品91| 成年人午夜在线观看视频| 免费看不卡的av| 日韩精品免费视频一区二区三区| 巨乳人妻的诱惑在线观看| 日本欧美视频一区| 丝袜喷水一区| 日韩人妻精品一区2区三区| 午夜日韩欧美国产| 日韩免费高清中文字幕av| 亚洲伊人色综图| 男人操女人黄网站| 国产黄色视频一区二区在线观看| 久久女婷五月综合色啪小说| 美女xxoo啪啪120秒动态图| 欧美精品高潮呻吟av久久| 日韩伦理黄色片| 九色亚洲精品在线播放| 日本免费在线观看一区| 多毛熟女@视频| 国产欧美日韩一区二区三区在线| 国产黄色免费在线视频| 亚洲国产最新在线播放| 熟女av电影| 日韩av免费高清视频| 精品亚洲乱码少妇综合久久| av免费观看日本| 97精品久久久久久久久久精品| av.在线天堂| 午夜日韩欧美国产| 欧美老熟妇乱子伦牲交| 亚洲精品国产一区二区精华液| av在线app专区| 哪个播放器可以免费观看大片| 精品少妇一区二区三区视频日本电影 | 亚洲精品国产一区二区精华液| 999久久久国产精品视频| 欧美精品一区二区免费开放| 亚洲欧美中文字幕日韩二区| 精品国产乱码久久久久久男人| 亚洲,一卡二卡三卡| 1024香蕉在线观看| 亚洲精品国产一区二区精华液| 国产乱人偷精品视频| 欧美日韩一级在线毛片| 麻豆乱淫一区二区| 老鸭窝网址在线观看| 2021少妇久久久久久久久久久| 亚洲第一青青草原| 欧美精品国产亚洲| 在线观看人妻少妇| 亚洲伊人久久精品综合| 纯流量卡能插随身wifi吗| 亚洲人成电影观看| 国产精品久久久久久精品电影小说| 香蕉丝袜av| 午夜福利视频精品| 捣出白浆h1v1| 人人澡人人妻人| 蜜桃在线观看..| 欧美另类一区| 亚洲欧美色中文字幕在线| 国产成人91sexporn| 国产爽快片一区二区三区| 亚洲色图 男人天堂 中文字幕| 99热全是精品| 亚洲欧美成人综合另类久久久| 韩国高清视频一区二区三区| 亚洲人成电影观看| 秋霞在线观看毛片| 麻豆av在线久日| 国语对白做爰xxxⅹ性视频网站| 久久精品国产亚洲av涩爱| 日本猛色少妇xxxxx猛交久久| 一级爰片在线观看| 99久久中文字幕三级久久日本| 香蕉精品网在线| 婷婷色麻豆天堂久久| 免费在线观看视频国产中文字幕亚洲 | 考比视频在线观看| 亚洲精品国产一区二区精华液| 日本欧美视频一区| 成年女人毛片免费观看观看9 | 久热这里只有精品99| 亚洲一级一片aⅴ在线观看| 亚洲美女搞黄在线观看| 免费高清在线观看视频在线观看| 午夜91福利影院| 黄色一级大片看看| 欧美日韩视频高清一区二区三区二| 欧美激情 高清一区二区三区| 欧美另类一区| 天堂俺去俺来也www色官网| 超碰成人久久| 欧美+日韩+精品| 日韩av免费高清视频| 亚洲人成网站在线观看播放| 成人影院久久| 亚洲精品久久成人aⅴ小说| 国产精品亚洲av一区麻豆 | 黑人欧美特级aaaaaa片| 秋霞伦理黄片| 日韩不卡一区二区三区视频在线| 国产欧美日韩一区二区三区在线| 69精品国产乱码久久久| 久久久久久久大尺度免费视频| 最新的欧美精品一区二区| 女人被躁到高潮嗷嗷叫费观| 免费播放大片免费观看视频在线观看| 纯流量卡能插随身wifi吗| 在线亚洲精品国产二区图片欧美| 交换朋友夫妻互换小说| 亚洲四区av| 黄频高清免费视频| 性色avwww在线观看| 久久女婷五月综合色啪小说| av国产精品久久久久影院| 不卡视频在线观看欧美| 男女高潮啪啪啪动态图| 久久久久久久亚洲中文字幕| 精品午夜福利在线看| 日韩电影二区| 日本猛色少妇xxxxx猛交久久| 午夜免费观看性视频| 激情五月婷婷亚洲| 国产午夜精品一二区理论片| 男女高潮啪啪啪动态图| a级片在线免费高清观看视频| 国产综合精华液| 亚洲精品美女久久av网站| 中文字幕av电影在线播放| 国产成人精品福利久久| 国精品久久久久久国模美| 制服丝袜香蕉在线| 丝袜在线中文字幕| 日韩中文字幕视频在线看片| 精品国产一区二区三区四区第35| 在线观看美女被高潮喷水网站| 黄频高清免费视频| 少妇人妻精品综合一区二区| 欧美bdsm另类| 午夜免费观看性视频| 日本爱情动作片www.在线观看| 成人亚洲欧美一区二区av| 欧美成人午夜精品| 日日撸夜夜添| 国产亚洲av片在线观看秒播厂| 爱豆传媒免费全集在线观看| 美女午夜性视频免费| 国产精品人妻久久久影院| 国产又爽黄色视频|