Heavy Fermion Quantum Criticality
2015国际青年物理学家锦标赛(IYPT)赛题翻译
Problems for the 28th IYPT 20151. PackingThe fraction of space occupied by granular particles depends on their shape. Pour non-spherical particles such as rice, matches, or M&M’s candies into a box. How do characteristics like coordination number, orientational order, or the random close packing fraction depend on the relevant parameters?1.堆积(Packing)被颗粒状物体(particles)占据的小部分空间取决于它们的形状。
将例如米、火柴或M&M糖果的非球状物体倾倒进一个盒子里，相关参量如何影响配位数(coordination number)、秩序性排列(orientational order)和随机紧密堆积分数(random close packing fraction)这样的特征？2. Plume of SmokeIf a burning candle is covered by a transparent glass, the flame extinguishes and a steady upward stream of smoke is produced. Investigate the plume of smoke at various magnifications.2.羽状的烟/烟羽(Plume of smoke)如果一支燃烧着的蜡烛被一块透明玻璃板覆盖，火焰会熄灭，并且产生一缕稳定的向上流动的轻烟。
研究在各种放大倍数下的羽状的烟。
Singularity of the density of states in the two-dimensional Hubbard model from finite size
a r X i v :c o n d -m a t /9503139v 1 27 M a r 1995Singularity of the density of states in the two-dimensional Hubbard model from finitesize scaling of Yang-Lee zerosE.Abraham 1,I.M.Barbour 2,P.H.Cullen 1,E.G.Klepfish 3,E.R.Pike 3and Sarben Sarkar 31Department of Physics,Heriot-Watt University,Edinburgh EH144AS,UK 2Department of Physics,University of Glasgow,Glasgow G128QQ,UK 3Department of Physics,King’s College London,London WC2R 2LS,UK(February 6,2008)A finite size scaling is applied to the Yang-Lee zeros of the grand canonical partition function for the 2-D Hubbard model in the complex chemical potential plane.The logarithmic scaling of the imaginary part of the zeros with the system size indicates a singular dependence of the carrier density on the chemical potential.Our analysis points to a second-order phase transition with critical exponent 12±1transition controlled by the chemical potential.As in order-disorder transitions,one would expect a symmetry breaking signalled by an order parameter.In this model,the particle-hole symmetry is broken by introducing an “external field”which causes the particle density to be-come non-zero.Furthermore,the possibility of the free energy having a singularity at some finite value of the chemical potential is not excluded:in fact it can be a transition indicated by a divergence of the correlation length.A singularity of the free energy at finite “exter-nal field”was found in finite-temperature lattice QCD by using theYang-Leeanalysisforthechiral phase tran-sition [14].A possible scenario for such a transition at finite chemical potential,is one in which the particle den-sity consists of two components derived from the regular and singular parts of the free energy.Since we are dealing with a grand canonical ensemble,the particle number can be calculated for a given chem-ical potential as opposed to constraining the chemical potential by a fixed particle number.Hence the chem-ical potential can be thought of as an external field for exploring the behaviour of the free energy.From the mi-croscopic point of view,the critical values of the chemical potential are associated with singularities of the density of states.Transitions related to the singularity of the density of states are known as Lifshitz transitions [15].In metals these transitions only take place at zero tem-perature,while at finite temperatures the singularities are rounded.However,for a small ratio of temperature to the deviation from the critical values of the chemical potential,the singularity can be traced even at finite tem-perature.Lifshitz transitions may result from topological changes of the Fermi surface,and may occur inside the Brillouin zone as well as on its boundaries [16].In the case of strongly correlated electron systems the shape of the Fermi surface is indeed affected,which in turn may lead to an extension of the Lifshitz-type singularities into the finite-temperature regime.In relating the macroscopic quantity of the carrier den-sity to the density of quasiparticle states,we assumed the validity of a single particle excitation picture.Whether strong correlations completely distort this description is beyond the scope of the current study.However,the iden-tification of the criticality using the Yang-Lee analysis,remains valid even if collective excitations prevail.The paper is organised as follows.In Section 2we out-line the essentials of the computational technique used to simulate the grand canonical partition function and present its expansion as a polynomial in the fugacity vari-able.In Section 3we present the Yang-Lee zeros of the partition function calculated on 62–102lattices and high-light their qualitative differences from the 42lattice.In Section 4we analyse the finite size scaling of the Yang-Lee zeros and compare it to the real-space renormaliza-tion group prediction for a second-order phase transition.Finally,in Section 5we present a summary of our resultsand an outlook for future work.II.SIMULATION ALGORITHM AND FUGACITY EXPANSION OF THE GRAND CANONICALPARTITION FUNCTIONThe model we are studying in this work is a two-dimensional single-band Hubbard HamiltonianˆH=−t <i,j>,σc †i,σc j,σ+U i n i +−12 −µi(n i ++n i −)(1)where the i,j denote the nearest neighbour spatial lat-tice sites,σis the spin degree of freedom and n iσis theelectron number operator c †iσc iσ.The constants t and U correspond to the hopping parameter and the on-site Coulomb repulsion respectively.The chemical potential µis introduced such that µ=0corresponds to half-filling,i.e.the actual chemical potential is shifted from µto µ−U412.(5)This transformation enables one to integrate out the fermionic degrees of freedom and the resulting partition function is written as an ensemble average of a product of two determinantsZ ={s i,l =±1}˜z = {s i,l =±1}det(M +)det(M −)(6)such thatM ±=I +P ± =I +n τ l =1B ±l(7)where the matrices B ±l are defined asB ±l =e −(±dtV )e −dtK e dtµ(8)with V ij =δij s i,l and K ij =1if i,j are nearestneigh-boursand Kij=0otherwise.The matrices in (7)and (8)are of size (n x n y )×(n x n y ),corresponding to the spatial size of the lattice.The expectation value of a physical observable at chemical potential µ,<O >µ,is given by<O >µ=O ˜z (µ){s i,l =±1}˜z (µ,{s i,l })(9)where the sum over the configurations of Ising fields isdenoted by an integral.Since ˜z (µ)is not positive definite for Re(µ)=0we weight the ensemble of configurations by the absolute value of ˜z (µ)at some µ=µ0.Thus<O >µ= O ˜z (µ)˜z (µ)|˜z (µ0)|µ0|˜z (µ0)|µ0(10)The partition function Z (µ)is given byZ (µ)∝˜z (µ)N c˜z (µ0)|˜z (µ0)|×e µβ+e −µβ−e µ0β−e −µ0βn (16)When the average sign is near unity,it is safe to as-sume that the lattice configurations reflect accurately thequantum degrees of freedom.Following Blankenbecler et al.[1]the diagonal matrix elements of the equal-time Green’s operator G ±=(I +P ±)−1accurately describe the fermion density on a given configuration.In this regime the adiabatic approximation,which is the basis of the finite-temperature algorithm,is valid.The situa-tion differs strongly when the average sign becomes small.We are in this case sampling positive and negative ˜z (µ0)configurations with almost equal probability since the ac-ceptance criterion depends only on the absolute value of ˜z (µ0).In the simulations of the HSfields the situation is dif-ferent from the case of fermions interacting with dynam-ical bosonfields presented in Ref.[1].The auxilary HS fields do not have a kinetic energy term in the bosonic action which would suppress their rapidfluctuations and hence recover the adiabaticity.From the previous sim-ulations on a42lattice[3]we know that avoiding the sign problem,by updating at half-filling,results in high uncontrolledfluctuations of the expansion coefficients for the statistical weight,thus severely limiting the range of validity of the expansion.It is therefore important to obtain the partition function for the widest range ofµ0 and observe the persistence of the hierarchy of the ex-pansion coefficients of Z.An error analysis is required to establish the Gaussian distribution of the simulated observables.We present in the following section results of the bootstrap analysis[17]performed on our data for several values ofµ0.III.TEMPERATURE AND LATTICE-SIZEDEPENDENCE OF THE YANG-LEE ZEROS The simulations were performed in the intermediate on-site repulsion regime U=4t forβ=5,6,7.5on lat-tices42,62,82and forβ=5,6on a102lattice.The ex-pansion coefficients given by eqn.(14)are obtained with relatively small errors and exhibit clear Gaussian distri-bution over the ensemble.This behaviour was recorded for a wide range ofµ0which makes our simulations reli-able in spite of the sign problem.In Fig.1(a-c)we present typical distributions of thefirst coefficients correspond-ing to n=1−7in eqn.(14)(normalized with respect to the zeroth power coefficient)forβ=5−7.5for differ-entµ0.The coefficients are obtained using the bootstrap method on over10000configurations forβ=5increasing to over30000forβ=7.5.In spite of different values of the average sign in these simulations,the coefficients of the expansion(16)indicate good correspondence between coefficients obtained with different values of the update chemical potentialµ0:the normalized coefficients taken from differentµ0values and equal power of the expansion variable correspond within the statistical error estimated using the bootstrap analysis.(To compare these coeffi-cients we had to shift the expansion by2coshµ0β.)We also performed a bootstrap analysis of the zeros in theµplane which shows clear Gaussian distribution of their real and imaginary parts(see Fig.2).In addition, we observe overlapping results(i.e.same zeros)obtained with different values ofµ0.The distribution of Yang-Lee zeros in the complexµ-plane is presented in Fig.3(a-c)for the zeros nearest to the real axis.We observe a gradual decrease of the imaginary part as the lattice size increases.The quantitative analysis of this behaviour is discussed in the next section.The critical domain can be identified by the behaviour of the density of Yang-Lee zeros’in the positive half-plane of the fugacity.We expect tofind that this density is tem-perature and volume dependent as the system approaches the phase transition.If the temperature is much higher than the critical temperature,the zeros stay far from the positive real axis as it happens in the high-temperature limit of the one-dimensional Ising model(T c=0)in which,forβ=0,the points of singularity of the free energy lie at fugacity value−1.As the temperature de-creases we expect the zeros to migrate to the positive half-plane with their density,in this region,increasing with the system’s volume.Figures4(a-c)show the number N(θ)of zeros in the sector(0,θ)as a function of the angleθ.The zeros shown in thesefigures are those presented in Fig.3(a-c)in the chemical potential plane with other zeros lying further from the positive real half-axis added in.We included only the zeros having absolute value less than one which we are able to do because if y i is a zero in the fugacity plane,so is1/y i.The errors are shown where they were estimated using the bootstrap analysis(see Fig.2).Forβ=5,even for the largest simulated lattice102, all the zeros are in the negative half-plane.We notice a gradual movement of the pattern of the zeros towards the smallerθvalues with an increasing density of the zeros nearθ=πIV.FINITE SIZE SCALING AND THESINGULARITY OF THE DENSITY OF STATESAs a starting point for thefinite size analysis of theYang-Lee singularities we recall the scaling hypothesis forthe partition function singularities in the critical domain[11].Following this hypothesis,for a change of scale ofthe linear dimension LLL→−1),˜µ=(1−µT cδ(23)Following the real-space renormalization group treatmentof Ref.[11]and assuming that the change of scaleλisa continuous parameter,the exponentαθis related tothe critical exponentνof the correlation length asαθ=1ξ(θλ)=ξ(θ)αθwe obtain ξ∼|θ|−1|θ|ναµ)(26)where θλhas been scaled to ±1and ˜µλexpressed in terms of ˜µand θ.Differentiating this equation with respect to ˜µyields:<n >sing =(−θ)ν(d −αµ)∂F sing (X,Y )ν(d −αµ)singinto the ar-gument Y =˜µαµ(28)which defines the critical exponent 1αµin terms of the scaling exponent αµof the Yang-Lee zeros.Fig.5presents the scaling of the imaginary part of the µzeros for different values of the temperature.The linear regression slope of the logarithm of the imaginary part of the zeros plotted against the logarithm of the inverse lin-ear dimension of the simulation volume,increases when the temperature decreases from β=5to β=6.The re-sults of β=7.5correspond to αµ=1.3within the errors of the zeros as the simulation volume increases from 62to 82.As it is seen from Fig.3,we can trace zeros with similar real part (Re (µ1)≈0.7which is also consistentwith the critical value of the chemical potential given in Ref.[22])as the lattice size increases,which allows us to examine only the scaling of the imaginary part.Table 1presents the values of αµand 1αµδ0.5±0.0560.5±0.21.3±0.3∂µ,as a function ofthe chemical potential on an 82lattice.The location of the peaks of the susceptibility,rounded by the finite size effects,is in good agreement with the distribution of the real part of the Yang-Lee zeros in the complex µ-plane (see Fig.3)which is particularly evident in the β=7.5simulations (Fig.4(c)).The contribution of each zero to the susceptibility can be singled out by expressing the free energy as:F =2n x n yi =1(y −y i )(29)where y is the fugacity variable and y i is the correspond-ing zero of the partition function.The dotted lines on these plots correspond to the contribution of the nearby zeros while the full polynomial contribution is given by the solid lines.We see that the developing singularities are indeed governed by the zeros closest to the real axis.The sharpening of the singularity as the temperature de-creases is also in accordance with the dependence of the distribution of the zeros on the temperature.The singularities of the free energy and its derivative with respect to the chemical potential,can be related to the quasiparticle density of states.To do this we assume that single particle excitations accurately represent the spectrum of the system.The relationship between the average particle density and the density of states ρ(ω)is given by<n >=∞dω1dµ=ρsing (µ)∝1δ−1(32)and hence the rate of divergence of the density of states.As in the case of Lifshitz transitions the singularity of the particle number is rounded at finite temperature.However,for sufficiently low temperatures,the singular-ity of the density of states remains manifest in the free energy,the average particle density,and particle suscep-tibility [15].The regular part of the density of states does not contribute to the criticality,so we can concentrate on the singular part only.Consider a behaviour of the typedensity of states diverging as the−1ρsing(ω)∝(ω−µc)1δ.(33)with the valueδfor the particle number governed by thedivergence of the density of states(at low temperatures)in spite of thefinite-temperature rounding of the singu-larity itself.This rounding of the singularity is indeedreflected in the difference between the values ofαµatβ=5andβ=6.V.DISCUSSION AND OUTLOOKWe note that in ourfinite size scaling analysis we donot include logarithmic corrections.In particular,thesecorrections may prove significant when taking into ac-count the fact that we are dealing with a two-dimensionalsystem in which the pattern of the phase transition islikely to be of Kosterlitz-Thouless type[23].The loga-rithmic corrections to the scaling laws have been provenessential in a recent work of Kenna and Irving[24].In-clusion of these corrections would allow us to obtain thecritical exponents with higher accuracy.However,suchanalysis would require simulations on even larger lattices.The linearfits for the logarithmic scaling and the criti-cal exponents obtained,are to be viewed as approximatevalues reflecting the general behaviour of the Yang-Leezeros as the temperature and lattice size are varied.Al-though the bootstrap analysis provided us with accurateestimates of the statistical error on the values of the ex-pansion coefficients and the Yang-Lee zeros,the smallnumber of zeros obtained with sufficient accuracy doesnot allow us to claim higher precision for the critical ex-ponents on the basis of more elaboratefittings of the scal-ing behaviour.Thefinite-size effects may still be signifi-cant,especially as the simulation temperature decreases,thus affecting the scaling of the Yang-Lee zeros with thesystem rger lattice simulations will therefore berequired for an accurate evaluation of the critical expo-nent for the particle density and the density of states.Nevertheless,the onset of a singularity atfinite temper-ature,and its persistence as the lattice size increases,areevident.The estimate of the critical exponent for the diver-gence rate of the density of states of the quasiparticleexcitation spectrum is particularly relevant to the highT c superconductivity scenario based on the van Hove sin-gularities[25],[26],[27].It is emphasized in Ref.[25]thatthe logarithmic singularity of a two-dimensional electrongas can,due to electronic correlations,turn into a power-law divergence resulting in an extended saddle point atthe lattice momenta(π,0)and(0,π).In the case of the14.I.M.Barbour,A.J.Bell and E.G.Klepfish,Nucl.Phys.B389,285(1993).15.I.M.Lifshitz,JETP38,1569(1960).16.A.A.Abrikosov,Fundamentals of the Theory ofMetals North-Holland(1988).17.P.Hall,The Bootstrap and Edgeworth expansion,Springer(1992).18.S.R.White et al.,Phys.Rev.B40,506(1989).19.J.E.Hirsch,Phys.Rev.B28,4059(1983).20.M.Suzuki,Prog.Theor.Phys.56,1454(1976).21.A.Moreo, D.Scalapino and E.Dagotto,Phys.Rev.B43,11442(1991).22.N.Furukawa and M.Imada,J.Phys.Soc.Japan61,3331(1992).23.J.Kosterlitz and D.Thouless,J.Phys.C6,1181(1973);J.Kosterlitz,J.Phys.C7,1046(1974).24.R.Kenna and A.C.Irving,unpublished.25.K.Gofron et al.,Phys.Rev.Lett.73,3302(1994).26.D.M.Newns,P.C.Pattnaik and C.C.Tsuei,Phys.Rev.B43,3075(1991);D.M.Newns et al.,Phys.Rev.Lett.24,1264(1992);D.M.Newns et al.,Phys.Rev.Lett.73,1264(1994).27.E.Dagotto,A.Nazarenko and A.Moreo,Phys.Rev.Lett.74,310(1995).28.A.A.Abrikosov,J.C.Campuzano and K.Gofron,Physica(Amsterdam)214C,73(1993).29.D.S.Dessau et al.,Phys.Rev.Lett.71,2781(1993);D.M.King et al.,Phys.Rev.Lett.73,3298(1994);P.Aebi et al.,Phys.Rev.Lett.72,2757(1994).30.E.Dagotto, A.Nazarenko and M.Boninsegni,Phys.Rev.Lett.73,728(1994).31.N.Bulut,D.J.Scalapino and S.R.White,Phys.Rev.Lett.73,748(1994).32.S.R.White,Phys.Rev.B44,4670(1991);M.Veki´c and S.R.White,Phys.Rev.B47,1160 (1993).33.C.E.Creffield,E.G.Klepfish,E.R.Pike and SarbenSarkar,unpublished.Figure CaptionsFigure1Bootstrap distribution of normalized coefficients for ex-pansion(14)at different update chemical potentialµ0for an82lattice.The corresponding power of expansion is indicated in the topfigure.(a)β=5,(b)β=6,(c)β=7.5.Figure2Bootstrap distributions for the Yang-Lee zeros in the complexµplane closest to the real axis.(a)102lat-tice atβ=5,(b)102lattice atβ=6,(c)82lattice at β=7.5.Figure3Yang-Lee zeros in the complexµplane closest to the real axis.(a)β=5,(b)β=6,(c)β=7.5.The correspond-ing lattice size is shown in the top right-hand corner. Figure4Angular distribution of the Yang-Lee zeros in the com-plex fugacity plane Error bars are drawn where esti-mated.(a)β=5,(b)β=6,(c)β=7.5.Figure5Scaling of the imaginary part ofµ1(Re(µ1)≈=0.7)as a function of lattice size.αm u indicates the thefit of the logarithmic scaling.Figure6Electronic susceptibility as a function of chemical poten-tial for an82lattice.The solid line represents the con-tribution of all the2n x n y zeros and the dotted line the contribution of the six zeros nearest to the real-µaxis.(a)β=5,(b)β=6,(c)β=7.5.。
Quantum entanglement
Quantum entanglementMaciej LewensteinMaciej Lewenstein has obtained his degree in Physics from Warsaw University. From 1980 he worked at the Center for Theoretical Physics of the Polish Academy of Sciences. He received his doctoral degree in 1983 at the University of Essen and habilitation in 1986 in Warsaw. He became a full Professor in Poland in 1993. In 1995 he joined “Service de Photones, Atomes et Molecules” of CEA in Saclay. In 1998 he became a full professor and a head of the quantum optics theory group at the University of Hannover. In 2005 he started a new theory group at the “Insitut de Ciencias Fotoniques” in Barcelona. His research interests include: quantum optics, quantum information and statistical physics.Chiara MacchiavelloChiara Macchiavello finished her degree in Physics in 1991 and her PhD in 1995 at the University of Pavia. She held a post-doctoral for two years at the University of Oxford. Since 1998 she has been an Assistant Professor at the University of Pavia.Her research interests include quantum information processing and quantum optics.Dagmar BrussSince 2003 Dagmar Bruss is a professor at the Institute of Theoretical Physics at the University of Duesseldorf, Germany. Her research interests include the foundations of quantum information theory, classification of entanglement and quantum optical implementations of quantum computation.AbstractEntanglement is a fundamental resource in quantum information theory. It allows performing new kinds of communication, such as quantum teleportation and quantum dense coding. It is an essential ingredient in some quantum cryptographic protocols and in quantum algorithms. We give a brief overview of the concept of entanglement in quantum mechanics, and discuss the major results and open problems related to the recent scientific progress in this field.IntroductionEntanglement is a property of the states of quantum systems that are composed of many parties, nowadays frequently called Alice, Bob, Charles etc. Entanglement expresses particularly strong correlations between these parties, persistent even in the case of large separations among the parties, and going beyond simple intuition.Historically, the concept of entanglement goes back to the famous Einstein-Podolski-Rosen (EPR) “paradox”. Einstein, who discovered relativity theory and the modern meaning of causality, was never really happy with quantum mechanics. In his opinion every reasonable physical theory should exhibit a so called local realism.Suppose that we consider two particles, one of which is sent to Alice and one to Bob, and we perform independent local measurements of “reasonable” physical observables on these particles. Of course, the results might be correlated, because the particles come from the same source. But Einstein wanted really to restrict the correlations for “reasonable” physical observables to the ones that result from statistical distributions of some hidden (i.e. unknown to us and not controlled by us) variables that characterize the source of the particles. Since quantum mechanics did not seem to produce correlations consistent with a local hidden variable (LHV) model, Einstein concluded that quantum mechanics is not a complete theory. Erwin Schrödinger, in answer to Einstein’s doubts, introduced in 1935 the term “Verschränkung” (in English “entanglement”) in order to describe these particularly strong quantum mechanical correlations.Entanglement was since then a subject of intense discussions among experts in the foundations of quantum mechanics and philosophers of science (and not only science). It took, however, nearly 30 years until John Bell was able to set the framework for experimental investigations on the question of local realism. Bell formulated his famous inequalities, which have to be fulfilled in any multiparty system described by a LHV model. Alain Aspect and coworkers in Paris have demonstrated in their seminal experiment in 1981 that quantum mechanical states violate these inequalities. Recent very precise experiments of Anton Zeilinger’s group in Vienna confirmed fully Aspect’s demonstrations. All these experiments indicate the correctness of quantum mechanics, and despite various loopholes, they exclude the possibility of LHV models describing properly the physics of the considered systems.Entanglement has become again the subject of cover pages news in the 90’s, when quantum information was born. It was very quickly realized that entanglement is one of the most important resources for quantum information processing. Entanglement is a necessary ingredient for quantum cryptography, quantum teleportation, quantum densecoding, and if not necessary, then at least a much desired ingredient for quantum computing.At the same time the theory of entanglement is related to some of the open questions of mathematics, or more precisely linear algebra and functional analysis. A solution of the entanglement problem could help to characterize the so called positive linear maps, i.e. linear transformations of positive definite operators (or physically speaking quantum mechanical density matrices, see below) into positive definite operators.Entanglement of pure statesIn quantum mechanics (QM) a state of a quantum system corresponds to a vector |Psi> in some vector space, called Hilbert space. Such states are called pure states. One of the most important properties of QM is that linear superpositions of state-vectors are also legitimate state-vectors. This superposition principle lies at the heart of the matter-wave dualism and of quantum interference phenomena.Entanglement is also a result of superposition, but in the composite space of the involved parties. Let us for the moment focus on two parties, Alice and Bob. It is then easy to define states which are not entangled. Such states are product states of the form |Φ>= |a>|b>, i.e. Alice has at her disposal |a>, while Bob has |b>. Product states obviously carry no correlations between Alice and Bob. Entangled pure states may be now defined as those which are superpositions of at least two product states, such as|Φ> = α1|a1>|b1> + α2|a2>|b2> + etc.but cannot be written as a single product state in any other basis. All entangled pure states contain strong quantum mechanical correlations, and do not admit LHV models.Entanglement of mixed states and the separability problemVerify whether a given state-vector is a product state or not is a relatively easy task. In practice, however, we often either do not have full information about the system, or are not able to prepare a desired state perfectly. In effect in everyday situations we deal practically always with statistical mixtures of pure states. There exists a very convenient way to represent such mixtures as so called density operators, or matrices. A density matrix rho corresponding to a pure state-vector |Φ> is a projector onto this state. More general density matrices can be represented as sums of projectors onto pure state-vectors weighted by the corresponding probabilities.The definition of entangled mixed states for composite systems has been formulated by Reinhard Werner from Braunschweig in 1989. In fact, this definition determines which states are not entangled. Non-entangled states, called separable states, are mixtures of pure product states, i.e. convex sums of projectors onto product vectors:ρ = Σι pi|ai>|bi><ai|<bi|, (*)where 0 ≤ pi ≤ 1 are probabilities, i.e. Σιpi= 1. The physical interpretation of thisdefinition is simple: a separable state can be prepared by Alice and Bob by using local operations and classical communication. Checking whether a given state is separable or not is a notoriously difficult task, since one has to check whether the decomposition (*) exists or not. This difficult problem is known under the name of “separability or entanglement problem”, and has been a subject of intensive studies in the recent years.Simple entanglement criteriaThe difficulty of the separability problem comes from the fact that rho admits in general an infinite number of decompositions into a mixture of some states, and one has to check whether among them there exists at least one of the form (*). One of the most powerful necessary conditions for separability has been found by one of the fathers of quantum information, the late Asher Peres. Peres (Technion, Haifa) observed that since Alice and Bob may prepare separable states using local operations, Alice may safely reverse the time arrow in her system, which will change the state, but will not produce something unphysical. In general, such a partial time reversal is not a physical operation, and can transform a density operator (which is positive definite) into an operator that is no more positive definite. In fact this is what happens with all pure entangled states. Mathematically speaking partial time reversal corresponds to partial transposition of the density matrix (only on Alice's side). We arrive in this way at the Peres criterion: If a stateρis separable then its partial transposition has to be positive definite.This criterion is usually called positive partial transpose condition, or shortly PPT condition. Amazingly, the PPT condition is not only necessary for separability, but it is also a sufficient condition for low dimensional systems such as two qubits (dimension 2x2)and a system composed of one qubit and one qutrit (dimension 2x3). In higher dimensions, starting from 2x4 and 3x3, this is no longer true: there exist entangled states with positive partial transpose, which are called PPT entangled states.There exist several other necessary or sufficient separability criteria which have been established and frequently discussed in recent years. For example, states that are close to the completely chaotic state (whose density operator is equal to the normalized identity) are necessarily separable. There exist also other criteria that employ entropic inequalities, uncertainty relations, or an appropriate reordering of the density matrix (so called realignment criterion) etc. There exists, however, no general simple operational criterion of separability that would work in systems of arbitrary dimension.Entanglement witnessesThe set of all states P is obviously compact and convex. If ρ1 and ρ2are legitimate states,so is their convex mixture. The set of separable states S is also compact and convex (seeFigure 1). From the theory of convex sets and Hahn-Banach theorem we conclude that for any entangled state there exists a hyperplane in the space of operators separating rhofrom S. Such a hyperplane defines uniquely a Hermitian operator W (observable) which has the following properties: The expectation value of W on all separable states, <W> ≥ 0, whereas its expectation value on ρ is negative, i.e. <W>ρ< 0.Figure 1Such an observable is for obvious reasons called entanglement witness, since it “detects” the entanglement of ρ. Every entangled state has its witnesses; the problem obviously is to find appropriate witnesses for a given state. To find out whether a given state is separable one should check whether its expectation value is non-negative for all witnesses. Obviously this is a necessary and sufficient separability criterion, but unfortunately it is not operational, in the sense that there is no simple procedure to test for all witnesses.Nevertheless, witnesses provide a very useful tool to study entanglement, especially if one has some knowledge about the state in question. They provide a sufficient entanglement condition, and may be obviously optimized (see Figure 2) by shifting the hyperplane in a parallel way towards S.Figure 2Bell inequalitiesAfter introducing the concept of separability and entanglement for mixed states, it is legitimate to ask what is the relation of mixed state entanglement and the existence of a LHV model, which requires that the state cannot violate any of the Bell-like inequalities. Let us discuss an example of such inequalities, the so called Clauser-Horne-Shimony- Holt inequality for two qubits. Let us assume that Alice and Bob measure two binary observables each, namely A 1, A 2, and B 1, B 2. The observables are random variables taking the values +1 or − 1, correlated possibly through some dependence on local hidden variables. It is easy to see that in the classical world, if B 1 + B 2 is zero, then B 1 − B 2 is either +2 or −2, and vice versa. Therefore if we define s = A 1(B 1 + B 2 ) + A 2 (B 1 − B 2 ) , we obtain that 2 ≥ s ≥ −2. This inequality holds also after averaging over various realizations. On the other hand, it can be shown that by taking suitable sets of observables for Alice and Bob we can find pure and even mixed quantum states that violate this inequality.Are Bell-like inequalities similar in this respect to witnesses, i.e. for a given entangled state can one always find a Bell-like inequality that “detects” it? The answer to this question is no, and has been already given by R. Werner in 1989. Even for two qubits there exist entangled states that admit an LHV model, i.e. cannot violate any Bell-like inequality.This observation indicates already that there is more structure in the “eggs” of Figure 1 and Figure 2. Separable states are evidently inside the PPT egg, according to the Peres condition. They admit an LHV model, i.e. they are also inside the LHV egg. But what about PPT entangled states? Do they violate some Bell-like inequality? Peres has formulated a conjecture that this not the case, and there is a lot of evidence that this conjecture is correct, although a rigorous proof is still missing.The distillability problem and bound entanglement Above we have classified quantum states according to the property of being either separable or entangled. An alternative classification approach is based on the possibility of distilling the entanglement of a given state. In a distillation protocol the entanglement of a given state is increased by performing local operations and classical communication on a set of identically prepared copies. In this way one obtains fewer, but “more entangled”, copies. This kind of technique was originally proposed in 1996 by Bennett and coworkers in the context of quantum teleportation, in order to achieve faithful transmission of quantum states over noisy channels. It also has applications in quantum cryptography as a method for quantum privacy amplification in entanglement based protocols in the presence of noise, as pointed out by David Deutsch and coworkers from Oxford.The distillability problem poses the question whether a given quantum state can be distilled or not. A separable state can never be distilled because the average entanglement of a set of states cannot be increased by local operations. Furthermore, the positivity of the partial transpose ensures that no distillation is possible. Thus, a given PPT entangled state is not distillable, and is therefore called bound entangled. There mayeven exist undistillable entangled states which do not have the PPT property. However, this conjecture is not proved at the moment.The first example of a PPT entangled state has been found by Pawel Horodecki from Gdansk in 1997. These states are so called edge states, which means that they cannot be written as a mixture of a separable state and a PPT entangled state. Particularly simple families of states have been suggested by Charles Bennett and coworkers at IBM, New York. They have found the so called unextendible product bases (UPB), i.e. sets of orthogonal product state-vectors, with the property that the space orthogonal to this set does not contain any product vector. It turns out that the projector onto this space is a PPT state, which obviously has to be entangled since it does not contain any product vector in its range (note that all state-vectors in the decomposition of a separable state ρinto a mixture of product states belong automatically to the range of ρ).The existence of bound entanglement is a mysterious invention of Nature. It is an interesting question to ask whether bound entanglement is a useful resource to perform quantum information processing tasks. It was shown so far that this is not the case for communication protocols such as quantum teleportation and quantum dense coding (i.e.a protocol that allows to enhance the transmission of classical information, using entanglement). However, surprisingly, it is possible to distill a secret key in quantum cryptography, starting from certain bound entangled states.Entanglement detectionAs discussed above, entanglement is a precious resource in quantum information processing. Typically in a real world experiment noise is always present and it leads to a decrease of entanglement in general. Thus, it is of fundamental interest for experimental applications to be able to test the entanglement properties of the generated states. A traditional method to this aim is represented by the Bell inequalities, a violation of which indicates the presence of entanglement. However, as mentioned above, not every entangled state violates a Bell inequality. So, not all entangled states can be detected by using this method.Another possibility is to perform complete state tomography, which allows determining all the elements of the density matrix. This is a useful method to get a complete knowledge of the density operator of a quantum system, but to detect entanglement it is an expensive process as it requires an unnecessary large number of measurements. If one has certain knowledge about the state the most appropriate technique is the measurement of the witness observable, which can be achieved by few local measurements. A negative expectation value clearly indicates the presence of entanglement.All these methods have been successfully implemented in various experiments. Recently another method for the detection of entanglement was suggested based on the physical approximation of the partial transpose. It remains a challenge to implement this idea in the laboratory because it requires the implementation of non local measurements.Entanglement measuresWhen classifying a quantum state as being entangled, a natural question is to quantify the amount of entanglement it contains. For pure quantum states there exists a well defined entanglement measure, namely the von Neumann entropy of the density operator of a subsystem of the composite state. For mixed states the situation is more complicated. There are several different possibilities to define an entanglement measure. The so called entanglement cost describes the amount of entanglement one needs in order to generate a given state. An alternative measure is the entanglement of formation, which is a more abstract definition. A further possibility to quantify entanglement is given by the minimum distance to separable states. Finally, motivated by physical applications, one can introduce the distillable entanglement which quantifies the extractable amount of entanglement.Unfortunately all of these quantities are very difficult to compute in general. For example, in order to determine the entanglement of formation one has to find the decomposition of the state that leads to the minimum average von Neumann entropy of a subsystem and this is a very challenging task. So far a complete analytical formula for the entanglement of formation only exists for composite systems of two qubits.Entanglement in multipartite systemsSo far, we have restricted ourselves to the case of composite systems with two subsystems, so called bipartite systems. When considering more than two parties, i.e multipartite systems, the situation becomes much more complex. For example, for the most simple tripartite case of three qubits, a pure state can be either completely separable, or biseparable (i.e. one of the three parties is not entangled with the other two), or genuinely entangled among all three parties. The latter class again consists of inequivalent subclasses, the so called GHZ and W states. This concept can be generalized to mixed states. For more than three parties it is easy to imagine that the number of subclasses grows fast.In recent years there has been much progress in the creation of multipartite entangled states in the laboratory. The existence of genuine multipartite entanglement has also been demonstrated experimentally by using the concept of witness operators.Even if the full classification of multipartite entanglement is a formidable task, certain classes of states, the so called graph states, have been completely characterized and shown to be useful both for quantum computational and quantum error correction protocols. Moreover, a deeper understanding of entanglement has proved to be very fruitful in connection with statistical properties of physical systems. All of these problems are discussed in more details in other sections of this publication.References[1] Einstein, P. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935)[2] J.S. Bell, Physics 1, 195 (1964)[3] P. Horodecki, Phys. Lett. A 232, 333 (1997)[4] M. Lewenstein et al., J. Mod. Opt. 47, 2481 (2000)[5] A. Peres, Phys. Rev. Lett. 77, 1413 (1996)[6] E. Schrödinger, Naturwissenschaften 23, 807 (1935)[7] R.F. Werner, Phys. Rev. A 40, 4277 (1989) Contact information of the author of this article Maciej LewensteinInstitut de Ciènces Fotòniques (ICFO)C/Jordi Girona 29, Nexus 2908034 BarcelonaSpainEmail: maciej.lewenstein@icfo.esChiara MacchiavelloIstituto Nazionale di Fisicadella Materia, Unita' di Pavia Dipartimento di Fisica "A. Volta"via Bassi 6I-27100 PaviaItalyEmail: chiara@unipv.itProf. Dr. Dagmar BrussInst. fuer Theoretische Physik IIIHeinrich-Heine-Universitaet Duesseldorf Universitaetsstr. 1, Geb. 25.32D-40225 Duesseldorf,GermanyEmail: bruss@thphy.uni-duesseldorf.de。
量子相变中的量子纠缠度量研究-英文版
RG flow
• c-theorem:
cUV cIR
fixed point
(Zamolodchikov, 1986)
UV
fixed point
IR
• Massive theory (off critical) Block entropy saturation
fixed point
RG flow
UV
Irreversibility of RG trajectories
Loss of entanglement
Open Systems & Quantum Information Milano, 10 Marzo 2006
• Local Entropy: when the subsystem A is a single site.
• The concurrence measures the entanglement between two sites after having traced out the remaining sites. • The transition is signaled by the first derivative of the concurrence, which diverges logarithmically (specific heat).
h 1 lnh 1
S1 has the same singularity as
Local (single site) entropy: 1 ρ1 = I + σ z σ z S1 = Tr ρ1lnρ1 2
σz
Local measures of entanglement based on the 2-site density matrix depend on 2-point functions
超导材料发展历程及现行理论解释与应用.pptx
6
发展历程
• 1911年，荷兰科学家H. K. Ones 利用低温技术研究金属的电阻特性时发现Hg在温度 低至4.2K时电阻降为零。后人把这种状态叫超导态。并把电阻突然降为零的温度 称为临界温度，记为Tc。
• 但由于早期的超导体存在于液氦极低温度条件下，极大地限制了超导材料的应用。 人们一直在探索高温超导体，从1911年到1986年，75年间从水银的4.2K提高到铌 三锗的23．22K，才提高了19K,科学家们用乌龟来形容这个程度。
• 一个比较形象的理解：当一个电子在晶格中运动时，由于异性电荷相吸而导 致局域晶格畸变，当另外一个电子通过时，会感受到第一个电子通过时导致 的晶格畸变的影响，从而在两个电子之间产生间接吸引相互作用，这就是 “库珀对”，其总动量和总自旋为零。所有电子对在运动过程中能够保持 “步调一致”（相位相干，即相位相同），即使受到杂质等散射也将保持总 动量不变，从而在外加电场作用下能够不损失能量而运动——这种现象就是 超导。所以说，超导是微观量子凝聚态的宏观表现。
• （3）临界电流密度Jc：当通过超导体的电流密度超过临界电流密度Jc时， 超导体由超导体恢复为正常状态。临界电流密度Jc与温度、磁场强度有关。
4
实验检验
为了证实（超导体）电阻为零，科学家将一 个铅制圆环，放入温度低于Tc=7.2K的空间， 利用电磁感应使环内激发起感应电流。结果 发现，环内电流能持续下去，从1954年3月16 日始，到1956年9月5日止，在两年半的时间 内的电流一直没有衰减，这说明圆环内的电 能没有损失，当温度升到高于Tc时，圆环由 超导状态变正常态，材料的电阻骤然增大， 感应电流立刻消失，这就是著名的昂尼斯持 久电流实验。
的崔田教授组在“传统
高温超导体”的研究上
基于微波消解-_ICP-OES_法测定植物中5_种重金属元素
宁夏农林科技，Ningxia Journal of Agri.and Fores.Sci.&Tech.2023，64（06）：54-58基金项目：宁夏回族自治区重点研发计划项目（2021BEG03017）、中央引导地方科技发展专项（2021FDG020）、宁夏绿创生态修复创新中心科研项目“贺兰山乡土植物引选及在矿山生态修复中的应用研究”、宁夏回族自治区312人才计划资助。
作者简介：马丽慧（1978—），女，宁夏隆德人，硕士，工程师，主要从事化学分析及质量管理研究。
收稿日期：2022-08-04修回日期：2022-11-17基于微波消解-ICP-OES 法测定植物中5种重金属元素马丽慧1，2，许浩1，2，张蕾蕾1，2，王文帆1，2，吴旭东1，21.宁夏农林科学院林业与草地生态研究所，宁夏银川7500022.宁夏防沙治沙与水土保持重点实验室，宁夏银川750002摘要：土壤重金属是城市环境污染的一个重要方面，部分植物对土壤重金属具有吸收富集功能，是土壤重金属污染修复的有效途径。
采用HNO 3-H 2O 2微波消解法对植物样品进行消解处理，然后采用电感耦合等离子体发射光谱仪（ICP-OES）测定植物样品中的铜、锌、镉、铬、铅5种重金属元素含量。
结果表明，该方法的线性范围为0～1mg/L 时，相关系数均在0.999以上，加标回收率在88.8%～104.0%，可满足植物中重金属元素测定的要求。
对银川市不同绿化树种枝干与叶片中重金属含量检测结果显示，各种金属元素最高值出现在不同植物中，红瑞木和垂柳枝条Cd 含量最高（0.51mg/kg），小檗叶片中Cr 含量最高（40.72mg/kg±6.25mg/kg），白蜡枝条Cu 含量最高（16.26mg/kg±2.52mg/kg），小檗叶片Pb 含量最高（4.49mg/kg±1.01mg/kg），垂柳枝条Zn 含量最高（97.41mg/kg±13.22mg/kg），表明这几种植物对相应的重金属元素具有较高的吸收积累能力。
Two-dimensional Quantum Field Theory, examples and applications
Abstract The main principles of two-dimensional quantum field theories, in particular two-dimensional QCD and gravity are reviewed. We study non-perturbative aspects of these theories which make them particularly valuable for testing ideas of four-dimensional quantum field theory. The dynamics of confinement and theta vacuum are explained by using the non-perturbative methods developed in two dimensions. We describe in detail how the effective action of string theory in non-critical dimensions can be represented by Liouville gravity. By comparing the helicity amplitudes in four-dimensional QCD to those of integrable self-dual Yang-Mills theory, we extract a four dimensional version of two dimensional integrability.
2 48 49 52 54 56
5 Four-dimensional analogies and consequences 6 Conclusions and Final Remarks
The-Universe
Passage 5The Universe's Invisible HandBy Christopher J. ConseliceDark energy (暗能量) does more than hurry along the expansion of the universe. It also has a stranglehold on the shape and spacing of galaxiesWhat took us so long? Only in 1998 did astronomers discover we had been missing nearly three quarters of the contents of the universe, the so-called dark energy--an unknown form of energy that surrounds each of us, tugging at us ever so slightly, holding the fate of the cosmos in its grip, but to which we are almost totally blind. Some researchers, to be sure, had anticipated that such energy existed, but even they will tell you that its detection ranks among the most revolutionary discoveries in 20th-century cosmology. Not only does dark energy appear to make up the bulk of the universe, but its existence, if it stands the test of time, will probably require the development of new theories of physics.Scientists are just starting the long process of figuring out what dark energy is and what its implications are. One realization has already sunk in: although dark energy betrayed its existence through its effect on the universe as a whole, it may also shape the evolution of the universe's inhabitants--stars, galaxies, galaxy clusters. Astronomers may have been staring at its handiwork for decades without realizing it.暗能量不仅仅会加速宇宙膨胀。
遇事不决量子力学 英语
遇事不决量子力学英语Quantum Mechanics in Decision-MakingIn the face of complex and uncertain situations, traditional decision-making approaches often fall short. However, the principles of quantum mechanics, a field of physics that explores the behavior of matter and energy at the subatomic level, can provide valuable insights and a new perspective on problem-solving. By understanding and applying the fundamental concepts of quantum mechanics, individuals and organizations can navigate challenging scenarios with greater clarity and effectiveness.One of the key principles of quantum mechanics is the idea of superposition, which suggests that particles can exist in multiple states simultaneously until they are observed or measured. This concept can be applied to decision-making, where the decision-maker may be faced with multiple possible courses of action, each with its own set of potential outcomes. Rather than prematurely collapsing these possibilities into a single decision, the decision-maker can embrace the superposition and consider the various alternatives in a more open and flexible manner.Another important aspect of quantum mechanics is the principle of uncertainty, which states that the more precisely one property of a particle is measured, the less precisely another property can be known. This principle can be applied to decision-making, where the decision-maker may be faced with incomplete or uncertain information. Instead of trying to eliminate all uncertainty, the decision-maker can acknowledge and work within the constraints of this uncertainty, focusing on making the best possible decision based on the available information.Furthermore, quantum mechanics introduces the concept of entanglement, where two or more particles can become inextricably linked, such that the state of one particle affects the state of the other, even if they are physically separated. This idea can be applied to decision-making in complex systems, where the actions of one individual or organization can have far-reaching and unpredictable consequences for others. By recognizing the interconnectedness of the various elements within a system, decision-makers can better anticipate and navigate the potential ripple effects of their choices.Another key aspect of quantum mechanics that can inform decision-making is the idea of probability. In quantum mechanics, the behavior of particles is described in terms of probability distributions, rather than deterministic outcomes. This probabilistic approach can be applied to decision-making, where the decision-maker canconsider the likelihood of different outcomes and adjust their strategies accordingly.Additionally, quantum mechanics emphasizes the importance of observation and measurement in shaping the behavior of particles. Similarly, in decision-making, the act of observing and gathering information can influence the outcomes of a situation. By being mindful of how their own observations and interventions can impact the decision-making process, decision-makers can strive to maintain a more objective and impartial perspective.Finally, the concept of quantum entanglement can also be applied to the decision-making process itself. Just as particles can become entangled, the various factors and considerations involved in a decision can become deeply interconnected. By recognizing and embracing this entanglement, decision-makers can adopt a more holistic and integrated approach, considering the complex web of relationships and dependencies that shape the outcome.In conclusion, the principles of quantum mechanics offer a unique and compelling framework for navigating complex decision-making scenarios. By embracing the concepts of superposition, uncertainty, entanglement, and probability, individuals and organizations can develop a more nuanced and adaptable approach to problem-solving. By applying these quantum-inspired strategies, decision-makers can navigate the challenges of the modern world with greater clarity, resilience, and effectiveness.。
量子纠缠 双缝干涉 英语 范例
量子纠缠双缝干涉英语范例Engaging with the perplexing world of quantum entanglement and the double-slit interference phenomenon in the realm of English provides a fascinating journey into the depths of physics and language. Let's embark on this exploration, delving into these intricate concepts without the crutchesof conventional transition words.Quantum entanglement, a phenomenon Albert Einstein famously referred to as "spooky action at a distance," challengesour conventional understanding of reality. At its core, it entails the entwining of particles in such a way that the state of one particle instantaneously influences the stateof another, regardless of the distance separating them.This peculiar connection, seemingly defying the constraints of space and time, forms the bedrock of quantum mechanics.Moving onto the enigmatic realm of double-slit interference, we encounter another perplexing aspect of quantum physics. Imagine a scenario where particles, such as photons or electrons, are fired one by one towards a barrier with twonarrow slits. Classical intuition would suggest that each particle would pass through one of the slits and create a pattern on the screen behind the barrier corresponding tothe two slits. However, the reality is far more bewildering.When observed, particles behave as discrete entities, creating a pattern on the screen that aligns with the positions of the slits. However, when left unobserved, they exhibit wave-like behavior, producing an interferencepattern consistent with waves passing through both slits simultaneously. This duality of particle and wave behavior perplexed physicists for decades and remains a cornerstoneof quantum mechanics.Now, let's intertwine these concepts with the intricate fabric of the English language. Just as particles become entangled in the quantum realm, words and phrases entwineto convey meaning and evoke understanding. The delicate dance of syntax and semantics mirrors the interconnectedness observed in quantum systems.Consider the act of communication itself. When wearticulate thoughts and ideas, we send linguistic particles into the ether, where they interact with the minds of others, shaping perceptions and influencing understanding. In this linguistic entanglement, the state of one mind can indeed influence the state of another, echoing the eerie connectivity of entangled particles.Furthermore, language, like quantum particles, exhibits a duality of behavior. It can serve as a discrete tool for conveying specific information, much like particles behaving as individual entities when observed. Yet, it also possesses a wave-like quality, capable of conveying nuanced emotions, cultural nuances, and abstract concepts that transcend mere words on a page.Consider the phrase "I love you." In its discrete form, it conveys a specific sentiment, a declaration of affection towards another individual. However, its wave-like nature allows it to resonate with profound emotional depth, evoking a myriad of feelings and memories unique to each recipient.In a similar vein, the act of reading mirrors the double-slit experiment in its ability to collapse linguistic wave functions into discrete meanings. When we read a text, we observe its words and phrases, collapsing the wave of potential interpretations into a singular understanding based on our individual perceptions and experiences.Yet, just as the act of observation alters the behavior of quantum particles, our interpretation of language is inherently subjective, influenced by our cultural background, personal biases, and cognitive predispositions. Thus, the same text can elicit vastly different interpretations from different readers, much like the varied outcomes observed in the double-slit experiment.In conclusion, the parallels between quantum entanglement, double-slit interference, and the intricacies of the English language highlight the profound interconnectedness of the physical and linguistic worlds. Just as physicists grapple with the mysteries of the quantum realm, linguists navigate the complexities of communication, both realmsoffering endless opportunities for exploration and discovery.。
量子力学英语词汇
量子力学专业英语词汇1、microscopic world 微观世界2、macroscopic world 宏观世界3、quantum theory 量子[理]论4、quantum mechanics 量子力学5、wave mechanics 波动力学6、matrix mechanics 矩阵力学7、Planck constant 普朗克常数8、wave-particle duality 波粒二象性9、state 态10、state function 态函数11、state vector 态矢量12、superposition principle of state 态叠加原理13、orthogonal states 正交态14、antisymmetrical state 正交定理15、stationary state 对称态16、antisymmetrical state 反对称态17、stationary state 定态18、ground state 基态19、excited state 受激态20、binding state 束缚态21、unbound state 非束缚态22、degenerate state 简并态23、degenerate system 简并系24、non-deenerate state 非简并态25、non-degenerate system 非简并系26、de Broglie wave 德布罗意波27、wave function 波函数28、time-dependent wave function 含时波函数29、wave packet 波包30、probability 几率31、probability amplitude 几率幅32、probability density 几率密度33、quantum ensemble 量子系综34、wave equation 波动方程35、Schrodinger equation 薛定谔方程36、Potential well 势阱37、Potential barrien 势垒38、potential barrier penetration 势垒贯穿39、tunnel effect 隧道效应40、linear harmonic oscillator 线性谐振子41、zero proint energy 零点能42、central field 辏力场43、Coulomb field 库仑场44、δ-function δ-函数45、operator 算符46、commuting operators 对易算符47、anticommuting operators 反对易算符48、complex conjugate operator 复共轭算符49、Hermitian conjugate operator 厄米共轭算符50、Hermitian operator 厄米算符51、momentum operator 动量算符52、energy operator 能量算符53、Hamiltonian operator 哈密顿算符54、angular momentum operator 角动量算符55、spin operator 自旋算符56、eigen value 本征值57、secular equation 久期方程58、observable 可观察量59、orthogonality 正交性60、completeness 完全性61、closure property 封闭性62、normalization 归一化63、orthonormalized functions 正交归一化函数64、quantum number 量子数65、principal quantum number 主量子数66、radial quantum number 径向量子数67、angular quantum number 角量子数68、magnetic quantum number 磁量子数69、uncertainty relation 测不准关系70、principle of complementarity 并协原理71、quantum Poisson bracket 量子泊松括号72、representation 表象73、coordinate representation 坐标表象74、momentum representation 动量表象75、energy representation 能量表象76、Schrodinger representation 薛定谔表象77、Heisenberg representation 海森伯表象78、interaction representation 相互作用表象79、occupation number representation 粒子数表象80、Dirac symbol 狄拉克符号81、ket vector 右矢量82、bra vector 左矢量83、basis vector 基矢量84、basis ket 基右矢85、basis bra 基左矢86、orthogonal kets 正交右矢87、orthogonal bras 正交左矢88、symmetrical kets 对称右矢89、antisymmetrical kets 反对称右矢90、Hilbert space 希耳伯空间91、perturbation theory 微扰理论92、stationary perturbation theory 定态微扰论93、time-dependent perturbation theory 含时微扰论94、Wentzel-Kramers-Brillouin method W. K. B.近似法95、elastic scattering 弹性散射96、inelastic scattering 非弹性散射97、scattering cross-section 散射截面98、partial wave method 分波法99、Born approximation 玻恩近似法100、centre-of-mass coordinates 质心坐标系101、laboratory coordinates 实验室坐标系102、transition 跃迁103、dipole transition 偶极子跃迁104、selection rule 选择定则105、spin 自旋106、electron spin 电子自旋107、spin quantum number 自旋量子数108、spin wave function 自旋波函数109、coupling 耦合110、vector-coupling coefficient 矢量耦合系数111、many-particle system 多子体系112、exchange forece 交换力113、exchange energy 交换能114、Heitler-London approximation 海特勒-伦敦近似法115、Hartree-Fock equation 哈特里-福克方程116、self-consistent field 自洽场117、Thomas-Fermi equation 托马斯-费米方程118、second quantization 二次量子化119、identical particles 全同粒子120、Pauli matrices 泡利矩阵121、Pauli equation 泡利方程122、Pauli’s exclusion principle泡利不相容原理123、Relativistic wave equation 相对论性波动方程124、Klein-Gordon equation 克莱因-戈登方程125、Dirac equation 狄拉克方程126、Dirac hole theory 狄拉克空穴理论127、negative energy state 负能态128、negative probability 负几率129、microscopic causality 微观因果性。
Quantum Mechanics
Quantum MechanicsQuantum mechanics is a fundamental theory in physics that describes the behavior of particles at the smallest scales. It has revolutionized our understanding of the universe and has led to numerous technological advancements. However, it is also a source of confusion and controversy, with many peoplefinding its concepts difficult to grasp. In this response, I will explore the various perspectives on quantum mechanics, including its scientific significance, philosophical implications, and practical applications. From a scientific perspective, quantum mechanics has had a profound impact on our understanding of the physical world. It has provided a framework for understanding the behavior of particles at the atomic and subatomic levels, leading to the development of technologies such as lasers, transistors, and MRI machines. Quantum mechanics has also led to the development of new materials and technologies with applications in fields such as computing, communication, and energy production. The theory has been tested and confirmed through countless experiments, and its predictions have been found to be incredibly accurate. However, despite its successes, quantum mechanics is also a source of confusion and debate. Its concepts, such as superposition, entanglement, and wave-particle duality, are counterintuitive and difficult to reconcile with our everyday experiences. This has led to a range of interpretations and philosophical debates about the nature of reality and the role of consciousness in quantum phenomena. Some scientists and philosophers have even suggested that quantum mechanics may require a radical rethinking of our understanding of the universe. From a practical perspective, quantum mechanics has the potential to revolutionize technology in the coming decades. Quantum computing, for example, has the potential to solve complex problems that are currently intractable for classical computers, leading to advances in fields such as cryptography, drug discovery, and materials science. Quantum communication technologies could also enable secure communication over long distances, with applications in fields such as finance, defense, and telecommunications. These practical applications have led to significant investment and research in thefield of quantum technology, with the potential to transform numerous industries in the near future. From a philosophical perspective, quantum mechanics raisesprofound questions about the nature of reality and our place in the universe. The theory's implications for determinism, causality, and the nature of measurement have led to a range of interpretations and debates about the fundamental nature of the physical world. Some interpretations, such as the Copenhagen interpretation and the many-worlds interpretation, have sparked intense philosophical and scientific debates about the nature of reality and the role of observation in quantum phenomena. These debates have led to a rich and diverse field of research at the intersection of physics, philosophy, and metaphysics, with implications for our understanding of the universe and our place within it. In conclusion, quantum mechanics is a complex and multifaceted theory with profound implications for science, philosophy, and technology. Its scientific significance, philosophical implications, and practical applications have led to a range of perspectives and debates about the nature of reality and our place in the universe. While it has revolutionized our understanding of the physical world and has the potential to transform technology in the coming decades, it also raises profound questions about the fundamental nature of reality and our place within it. As we continue to explore and develop our understanding of quantum mechanics, it is likely that these debates and perspectives will continue to evolve, leading to new insights and discoveries about the nature of the universe.。
奥克托今3硝基124三唑5酮共晶炸药晶体缺陷的分子动力学研究
第40卷第1期2019年1月兵工学报ACTA ARMAMENTARIIVol.40No.1Jan.2019奥克托今/3⁃硝基⁃1,2,4⁃三唑⁃5⁃酮共晶炸药晶体缺陷的分子动力学研究杭贵云,余文力,王涛,王金涛,苗爽(火箭军工程大学核工程学院,陕西西安710025) 摘要:为研究晶体缺陷对奥克托今(HMX )/3⁃硝基⁃1,2,4⁃三唑⁃5⁃酮(NTO )共晶炸药稳定性㊁感度㊁爆轰性能与力学性能的影响,分别建立了 完美”型与含有掺杂㊁空位与位错缺陷的共晶炸药模型㊂采用分子动力学方法,预测了 完美”型与缺陷模型的性能,得到了不同模型的结合能㊁引发键键长分布㊁键连双原子作用能㊁内聚能密度㊁爆轰参数与力学性能参数,并进行了比较㊂结果表明:由于晶体缺陷的影响,造成炸药结合能减小,稳定性变差;缺陷晶体的引发键最大键长增大,键连双原子作用能与内聚能密度减小,炸药感度升高,安全性减弱;缺陷晶体的密度㊁爆速与爆压减小,能量密度与威力降低;与 完美”型晶体相比,缺陷晶体的拉伸模量㊁体积模量与剪切模量减小,柯西压力增大,炸药刚性与硬度减弱,柔性与延展性增强㊂ 关键词:共晶炸药;晶体缺陷;稳定性;感度;爆轰性能;力学性能;分子动力学 中图分类号:TQ564.4+2;O641.3文献标志码:A 文章编号:1000⁃1093(2019)01⁃0049⁃09 DOI :10.3969/j.issn.1000⁃1093.2019.01.007 收稿日期:2018⁃05⁃09基金项目:武器装备预先研究项目(403020302)作者简介:杭贵云(1989 ),男,博士研究生㊂E⁃mail:1910319052@ 通信作者:余文力(1967 ),男,教授,博士生导师㊂E⁃mail:wlyu888@Molecular Dynamics Investigation on Crystal Defect ofHMX /NTO Cocrystal ExplosiveHANG Guiyun,YU Wenli,WANG Tao,WANG Jintao,MIAO Shuang(School of Nuclear Engineering,Rocket Force University of Engineering,Xi’an 710025,Shaanxi,China)Abstract :The perfect”and defective cocrystal models with adulteration,vacancy and dislocation are established to investigate the influence of crystal defect on stability,sensitivity,detonation performance and mechanical property of octahydro⁃1,3,5,7⁃tetranitro⁃1,3,5,7⁃tetrazocine (HMX)/3⁃nitro⁃1,2,4⁃triazol⁃5⁃one (NTO)cocrystal explosive.Molecular dynamics method is applied to predict the properties of the proposed crystal models.The binding energy,bond length distribution of trigger bond,interaction energy of trigger bond,cohesive energy density,detonation parameters and mechanical properties were obtained and compared.The results show that the binding energy of explosive is declined and its stability is weakened due to the influence of crystal defect.The maximum trigger bond length of defective crystal model is increased,while the interaction energy of trigger bond and the cohesive energy density are de⁃creased,meaning that the sensitivity of explosive is increased and its safety is worsened.The density,detonation velocity and detonation pressure of defective crystal model are declined,indicating that the ener⁃gy density and power are pared with the perfect”crystal model,the tensile modulus,bulk兵 工 学 报第40卷modulus and shear modulus of defective crystal model are decreasaed,and Cauchy pressure is increased, namely,the rigidity and stiffness are declined and the plastic property and ductility are increased. Keywords:cocrystal explosive;crystal defect;stability;sensitivity;detonation performance;mechani⁃cal property;molecular dynamics0 引言奥克托今(HMX)是一种常见的高能炸药,也是目前综合性能最好的单质炸药之一,因此长期以来备受关注㊂但HMX的机械感度较高,从而导致其发展应用受到限制㊂3⁃硝基⁃1,2,4⁃三唑⁃5⁃酮(NTO)是一种高能钝感炸药,能量接近于黑索今(RDX),感度与1,3,5⁃三氨基⁃2,4,6⁃三硝基苯(TATB)相当㊂近年来,共晶成为一种改善含能材料性能的有效途径,并在含能材料领域得到发展应用[1-4]㊂2013年,Lin等[5]研究了HMX/NTO共晶炸药的性能,结果表明,HMX/NTO共晶炸药的感度低于HMX,安全性较好,且共晶具有较高的能量密度㊂因此,HMX/NTO共晶炸药有望成为一种新型高能炸药㊂HMX通常采用改进型乙酸酐法制备,在产物中会存在杂质成分RDX.在HMX/NTO共晶炸药制备过程中,RDX也会进入到共晶炸药中,从而导致制备的炸药不纯,即存在掺杂缺陷㊂此外,在晶体的生长过程中,由于外界因素的干扰,晶体的生长过程可能会受到影响,从而使晶体中存在缺陷㊂晶体缺陷的存在会影响炸药的性能,如稳定性㊁感度㊁能量特性与力学性能等,从而进一步影响武器弹药的威力与安全性[6-10]㊂因此,研究晶体缺陷对炸药性能的影响,具有一定的军事意义与实际应用价值㊂本文分别建立了 完美”型与含有晶体缺陷(掺杂㊁空位与位错)的HMX/NTO共晶炸药模型㊂在材料计算软件Materials Studio7.0(以下简称MS软件)中,采用分子动力学方法,预测了各种模型的稳定性㊁感度㊁爆轰性能与力学性能并进行了比较㊂研究成果可以为炸药的性能评估提供相关的理论指导㊂1 计算模型与计算方法1.1 单个分子的建立HMX/NTO共晶炸药由HMX与NTO组成,摩尔比为1∶1.在MS软件中,分别建立HMX与NTO 的分子模型,如图1所示㊂图1 HMX与NTO的分子模型Fig.1 Molecule structures of HMX and NTO1.2 HMX/NTO共晶初始模型的建立本文中使用的HMX/NTO共晶炸药初始结构为模拟结构㊂HMX/NTO共晶炸药属于三斜晶系,空间群为Pī,晶格参数a=0.9060nm,b= 0.8190nm,c=1.0270nm,α=81.94°,β=98.42°,γ=82.03°,单个晶胞中包含2个HMX分子与2个NTO分子[5]㊂HMX/NTO共晶炸药的单个晶胞模型如图2(a)所示,而后将单个晶胞模型扩展为36(4×3×3)的超晶胞模型,其中包含72个HMX分子与72个NTO分子,一共144个分子,如图2(b)所示㊂为了便于与含有晶体缺陷的模型进行比较,将 完美”型晶体模型记作模型1.图2 HMX/NTO共晶炸药的单个晶胞与超晶胞模型Fig.2 Single unit cell and supercell of HMX/NTOcocrystal explosive1.3 HMX/NTO共晶缺陷模型的建立用4个RDX分子替换 完美”型晶体中的4个HMX分子(图3(a)中标记为黄色),得到掺杂率为2.78%的缺陷晶体模型,如图3(b)所示,将所得缺陷模型标记为模型2.类似地,分别用6个RDX分子替换初始模型中的6个HMX分子,用8个RDX分子替换8个HMX05 第1期奥克托今/3⁃硝基⁃1,2,4⁃三唑⁃5⁃酮共晶炸药晶体缺陷的分子动力学研究图3 初始晶胞模型与含有掺杂缺陷的晶胞模型Fig.3 Initial crystal model and defective crystal modelwith adulteration分子,得到掺杂率分别为4.17%㊁5.56%的缺陷晶体模型,分别标记为模型3㊁模型4.删除超晶胞模型中的4个HMX 分子(图4(a)中标记为黄色),得到空位率为2.78%的缺陷晶体模型,如图4(b)所示,将缺陷晶体模型标记为模型5.图4 初始晶胞模型与含有空位缺陷的晶胞模型Fig.4 Initial crystal model and defective crystal modelwith vacancy采用同样的方法,分别删除初始模型中的4个NTO 分子㊁2个HMX 与2个NTO 分子㊁8个HMX 分子㊁8个NTO 分子㊁4个HMX 与4个NTO 分子,将所得空位缺陷的模型分别标记为模型6~模型10.将初始模型中最上层的HMX 分子(图5(a)中标记为黄色)向上方移动0.2nm 的距离,得到含位错缺陷的晶体模型,如图5(b)所示,将缺陷晶体模型标记为模型11.类似地,将初始模型中最上层的HMX 分子向下方移动0.2nm 的距离,得到位错缺陷晶体模型,标记为模型12.1.4 计算条件设置分别对HMX /NTO 共晶炸药的 完美”型与含有晶体缺陷的模型进行能量最小化,对其结构进行优化,而后进行分子动力学计算,其中温度设置为295K,压力设置为0.0001GPa,选择恒温恒压(NPT)系综与COMPASS 力场[11-12]㊂之所以选择图5 初始晶胞模型与含有位错缺陷的晶胞模型Fig.5 Initial crystal model and defective crystal modelwith dislocationCOMPASS 力场进行分子动力学模拟,是因为该力场适用于凝聚相和不同类型物质相互作用的研究,能在较大范围内对处于孤立体系和凝聚态体系的多种物质的性能进行准确预测㊂采用Andersen 控温方法[13]与Parrinello 控压方法[14],范德华力(vdW)的计算采用atom⁃based 方法,静电作用的计算采用Ewald 方法,时间步长为1fs,总模拟步数为2×105步,其中前105步用于热力学平衡计算,后105步用于统计分析㊂模拟过程中,每103fs 保存一次轨迹,共得100帧轨迹文件㊂2 结果分析2.1 HMX /NTO 共晶的晶胞参数表1给出了 完美”型的HMX /NTO 共晶模型在295K 下NPT 模拟所得晶胞参数和密度㊂为方便比较,表1中还给出了理论计算结果㊂从表1可以看出,计算得到的HMX /NTO 晶胞参数㊁密度与理论值非常接近,吻合较好,表明COMPASS 力场对HMX /NTO 晶体具有较好的适用性㊂表1 HMX /NTO 晶胞参数与密度的计算值与理论值Tab.1 Calculated and theoretical results of HMX /NTOlattice parameters and density参数理论值[5]计算值a /nm 0.90600.9065b /nm 0.81900.8194c /nm 1.02701.0276α/(°)81.9481.87β/(°)98.4298.48γ/(°)82.0382.10ρ/(g ㊃cm -3)1.9201.91715兵 工 学 报第40卷2.2 平衡判别和平衡结构在提取计算结果时,需要让体系达到平衡状态,而体系平衡时必须同时满足温度平衡与能量平衡㊂通常认为当温度与能量波动范围在5%~10%时,体系已经达到热力学平衡状态㊂以掺杂率为4.17%的缺陷模型(模型3)为例,模拟过程中混合体系的温度T 变化曲线与能量E 随时间t 变化曲线,如图6所示㊂图6 温度与能量的变化曲线Fig.6 Temperature and energy versus time从图6可以看出:在模拟计算初期,体系的温度与能量均有所上升,并且波动幅度较大;随着时间的推移,温度与能量的波动幅度逐渐减小,最终温度波动幅度为±15K 左右,能量波动幅度为±5%左右,偏差相对较小,表明混合体系已达到热力学平衡状态㊂对于其他晶体模型,分子动力学计算时,均以温度与能量变化曲线来判别混合体系是否达到平衡状态㊂2.3 结合能结合能E b 定义为分子之间相互作用力E i 的负值,主要用来预测体系的稳定性㊂结合能越大,表明分子之间的相互作用力越强,体系的稳定性越好㊂对于HMX /NTO 共晶及其缺陷晶体炸药,结合能的计算公式为E b =-E i =-[E tot -(E NTO +E o )],(1)式中:E tot 为混合体系达到热力学平衡状态时对应的总能量;E NTO 为去掉体系中的其他组分后,NTO 分子对应的总能量;E o 为去掉晶体模型中的NTO 分子后,体系中的HMX 与其他组分对应的总能量㊂根据分子动力学计算得到的各组分能量,通过计算得到不同模型的结合能,结果如图7所示㊂图7 不同模型的结合能Fig.7 Binding energies of different crystal models从图7可以看出: 完美”型晶体模型(模型1)对应的结合能最大(362.7kJ /mol),表明HMX 与NTO 分子之间的相互作用力最强,炸药的稳定性最好;缺陷晶体的结合能均有不同程度的减小,其中掺杂缺陷晶体模型(模型2)的结合能最大(351.6kJ /mol),而空位缺陷晶体模型(模型8)的结合能最小(313.9kJ /mol),结合能减小的幅度为3.06%~13.45%.同时,结合能的变化趋势也表明掺杂缺陷对结合能的影响相对较小,而空位缺陷对结合能的影响最大㊂此外,从图7还可以看出,对于同种类型的晶体缺陷,随着缺陷数量的增加,结合能逐渐减小,表明分子之间作用力的强度逐渐减弱,炸药的稳定性逐渐减弱,即晶体缺陷会对炸药的稳定性产生不利影响㊂结合能减小的原因可能是由于缺陷的影响,晶体的结构遭到破坏,分子的排列方式发生了变化,从而使分子之间的作用力减弱㊂2.4 感度感度定义为含能材料在受到外界刺激时发生分解或者爆炸的难易程度,是含能材料安全性的指标,也是含能材料最重要的性能之一㊂目前,国内外通常采用试验测试与理论分析的方法来预测含能材料的感度㊂根据热点理论[15]与引发键思想[16],同时参考以往的研究工作[17-21],本文中选用引发键键长㊁引发键键连双原子作用能与内聚能密度(CED)25 第1期奥克托今/3⁃硝基⁃1,2,4⁃三唑⁃5⁃酮共晶炸药晶体缺陷的分子动力学研究来预测炸药的感度,并评价其安全性㊂2.4.1 引发键键长所谓引发键,是指含能材料中能量最低㊁最容易发生断裂的化学键㊂在外界刺激下,引发键更容易发生断裂破坏,从而使含能材料发生分解或爆炸㊂在HMX /NTO 共晶及其缺陷晶体中,HMX 所占的比重最高,而HMX 的引发键是N NO 2键中的N N 键[16,22-23]㊂因此,选择HMX 分子中的N N 键作为引发键来预测体系的感度㊂以模型5为例,图8给出了体系达到平衡状态时引发键的键长分布情况,其中L 为键长,P 为键长的分布概率㊂在热力学平衡状态下,不同模型中引发键的最可几键长L p ㊁平均键长L a 与最大键长L max 如表2所示㊂图8 引发键的键长分布Fig.8 Bond length distribution of trigger bond表2 不同模型中引发键的键长Tab.2 Trigger bond lengths of different crystal models模型L p /nmL a /nmL max /nm 10.13950.13940.154320.13950.13950.155130.13950.13940.155240.13950.13950.155950.13960.13950.158260.13960.13960.158570.13970.13960.159180.13970.13980.160490.13970.13970.1598100.13960.13970.1594110.13960.13950.1570120.13960.13960.1577 从图8可以看出,在平衡状态下,体系中引发键(N NO 2键)的键长分布呈近似对称的高斯分布㊂从表2可以看出,对于 完美”型与含有晶体缺陷的模型,最可几键长与平均键长近似相等,并且变化范围很小,表明晶体缺陷对最可几键长与平均键长的影响很小,而最大键长变化很明显,且不同模型之间差异较大㊂ 完美”型晶体(模型1)的最大键长最小(0.1543nm),而缺陷晶体的最大键长均大于初始模型对应的键长值,其中掺杂缺陷模型(模型2)的键长最小(0.1551nm ),而空位缺陷模型(模型8)的键长最大(0.1604nm),与 完美”型晶体相比,最大键长的增大幅度为0.52%~3.95%.最大键长增大,表明引发键的键能降低,预示含能材料的感度增大,安全性降低,即晶体缺陷使得炸药的感度升高㊂此外,表2也表明,空位缺陷的晶体模型对应的引发键键长最大,且随着缺陷数量的增加,最大键长逐渐增大,即炸药的感度逐渐升高,安全性逐渐降低㊂2.4.2 键连双原子作用能键连双原子作用能主要用来反映键的强度,键连双原子作用能越大,表明键的强度越大,键断裂时需要的能量越多,含能材料的感度越低,安全性越好㊂键连双原子作用能E N -N 的计算公式为E N -N =(E T -EF )/n ,(2)式中:E T 为体系达到热力学平衡状态时对应的总能量;E F 为固定晶体中HMX 分子中所有的N 原子后体系的总能量;n 为体系中HMX 分子包含的N N 键的数量㊂通过计算,得到不同模型的键连双原子作用能,如图9所示㊂图9 不同模型的键连双原子作用能Fig.9 Interaction energies of trigger bonds of different models从图9可以看出,在不同的晶体模型中,键连双原子作用能的差异较大㊂其中, 完美”型晶体模型(模型1)的键连双原子作用能最大35兵 工 学 报第40卷(160.77kJ/mol),而缺陷晶体的键连双原子作用能均小于 完美”型晶体模型对应的键的能量㊂在缺陷晶体中,掺杂缺陷模型(模型2)的键能最大(156.57kJ/mol),而空位缺陷模型(模型9)的键能最小(135.53kJ/mol),键连双原子作用能减小的幅度为2.61%~15.70%.键连双原子作用能减小,表明键的强度减弱,键断裂时需要的能量减小,即含能材料的感度增大,安全性降低㊂由此可见,晶体缺陷会对炸药的安全性产生不利影响㊂此外,空位缺陷的模型对应的键连双原子作用能最小,键的强度最弱,感度最高,且随着缺陷数量的增加,键的能量逐渐降低,预示空位缺陷对炸药的感度影响更为显著㊂之前的研究也表明,由于空位缺陷的影响,在晶体内部容易形成 热点”,从而使含能材料的感度升高,安全性降低[6,24-25]㊂2.4.3 CED计算CED定义为单位体积内1mol物质由凝聚态变为气态时克服分子间作用力所做的功㊂CED属于非键力,在数值上等于vdW与静电力之和㊂根据计算结果,得到不同模型的CED㊁vdW与静电力,结果如表3所示㊂表3 不同模型的CED㊁vdW与静电力Tab.3 CED,vdW and electrostatic forces ofdifferent models模型CED/(kJ㊃cm-3)vdW/(kJ㊃cm-3)静电力/(kJ㊃cm-3) 10.8830.2520.631 20.8720.2480.624 30.8670.2460.621 40.8530.2380.615 50.8360.2280.608 60.8210.2220.599 70.8250.2240.601 80.8080.2140.594 90.7970.2080.589 100.8130.2170.596 110.8450.2340.611 120.8460.2350.611 从表3可以看出, 完美”型晶体模型(模型1)的能量最高,其中CED㊁vdW与静电力分别为0.883kJ/cm3㊁0.252kJ/cm3㊁0.631kJ/cm3,而缺陷晶体的能量均有不同程度的减小㊂在缺陷晶体中,掺杂缺陷模型(模型2)对应的CED最大(0.872kJ/cm3),而空位缺陷模型(模型9)的CED 最小(0.797kJ/cm3),CED减小的幅度为1.25%~ 9.74%.CED减小,表明炸药由凝聚态变为气态时吸收的能量减小,预示炸药的感度增大,安全性降低㊂此外,表3也进一步表明,空位缺陷晶体模型的CED最低,且随着缺陷数量的增加,CED逐渐减小,表明炸药的感度逐渐增大,安全性呈下降趋势㊂2.5 爆轰性能爆轰性能是含能材料威力与能量密度的直接体现,也是武器弹药毁伤效果的直接反映,通常用爆轰参数进行表征㊂常见的爆轰参数主要有爆速D与爆压p等㊂本文中采用修正氮当量法[26]来计算炸药的爆轰参数并预测其能量密度㊂爆速D与爆压p的计算公式为D=(690+1160ρ)∑N c,p=(1.106ρ∑N)ch2-0.84,∑N c=100M(r d i N d i+∑B K N BK+∑G j N G)j ìîíïïïïïï,(3)式中:∑N c为炸药的修正氮当量;d i为1mol炸药爆炸时生成第i种爆轰产物的摩尔数;N d i为第i种爆轰产物的氮当量系数;B K为炸药分子中第K种化学键出现的次数;N BK为第K种化学键的氮当量系数;G j为炸药分子中第j种基团出现的次数;N G j为第j种基团的氮当量系数㊂根据修正氮当量理论,通过计算得到不同模型的爆轰参数,结果如表4所示,其中炸药密度可以根据分子动力学计算结果从平衡体系中直接提取得到㊂表4 不同模型的爆轰参数Tab.4 Density and detonation parameters of different models 模型ρ/(g㊃cm-3)D/(m㊃s-1)p/GPa11.917897037.6821.897889636.8631.891887436.6141.883884436.2851.832865634.2661.862878535.5971.859876335.3981.802853833.0391.811861333.71101.818861733.81111.881884236.25121.904892437.16 从表4可以看出,对于不同的晶体模型, 完45 第1期奥克托今/3⁃硝基⁃1,2,4⁃三唑⁃5⁃酮共晶炸药晶体缺陷的分子动力学研究美”型晶体模型(模型1)对应的密度㊁爆速与爆压最大,分别为1.917g/cm3㊁8970m/s㊁37.68GPa,表明 完美”型晶体的威力最大,能量密度最高㊂在缺陷晶体中,位错缺陷模型(模型12)的密度㊁爆速与爆压最大,分别为1.904g/cm3㊁8924m/s㊁37.16GPa,而空位缺陷模型(模型8)的密度㊁爆速与爆压最小,分别为1.802g/cm3㊁8538m/s㊁33.03GPa,密度㊁爆速与爆压的减小幅度分别为0.68%~6.00%㊁0.51%~4.82%㊁1.38%~12.34%.密度㊁爆速与爆压减小,表明炸药的威力减小,能量密度降低,因此晶体缺陷会对炸药的能量特性产生不利影响㊂此外,在3种类型的晶体缺陷中,空位缺陷模型对应的密度㊁爆速与爆压最小,表明其能量密度最低,也进一步表明空位缺陷对能量密度的影响更为显著㊂在缺陷晶体中,随着缺陷数量的增加,炸药的密度㊁爆速与爆压逐渐减小,能量密度逐渐降低㊂2.6 力学性能力学性能主要包括拉伸模量E㊁剪切模量G㊁体积模量K㊁泊松比γ与柯西压C12⁃C44,其中E㊁K㊁G 主要用来预测体系的刚性与硬度,其值越大,表明体系的刚性越强,硬度越大㊂柯西压力主要反映体系的延展性,柯西压力为正值,表明体系的延展性较好;柯西压力为负值,表明体系呈脆性㊂力学参数可通过弹性系数矩阵进行描述,表达式[27-28]为σi=C ijεj,i,j=1,2, ,6,(4)式中:σ为应力;ε为应变;C ij为弹性系数,满足C ij= C ji,因此独立的弹性常数只有21个,对于完全的各向同性体,独立的弹性常数只有2个(C11,C22)㊂ 体积模量K与剪切模量G的计算公式为K R=[S11+S22+S33+2(S12+S23+S31)]-1,(5) G R=15[4(S11+S22+S33)-4(S12+S23+S31)+3(S44+S55+S66)]-1,(6)式中:下标R表示Reuss平均;柔量系数矩阵S= [S ij]等于弹性系数矩阵C的逆矩阵,即S=C-1= [C ij]-1.力学参数之间存在如(7)式的关系:E=2G(1+γ)=3K(1-2γ).(7)基于(7)式,可以求得拉伸模量E与泊松比γ的表达式为E=9GK3K+G,(8)γ=3K-2G2(3K+G).(9)通过计算,得到不同模型的力学性能参数,结果如表5与图10所示㊂从表5与图10可以看出: 完美”型模型(模型1)的拉伸模量㊁体积模量与剪切模量的值最大,分别为14.441GPa㊁8.773GPa㊁5.891GPa,而柯西压的值最小(0.198GPa),表明炸药的刚性最强,延展性与塑性较差;对于缺陷晶体,由于炸药的晶体结构发生了变化,E㊁K㊁G减小,而柯西压增大,表明炸药的刚性与硬度减弱,延展性与塑性增强㊂在外界作用下,炸药更容易发生形变㊂在缺陷晶体中,掺杂缺陷模型(模型2)对应的E㊁K㊁G最大,分别为13.853GPa㊁8.518GPa㊁5.634GPa,而空位缺陷模型(模型10)的模量最小,分别为9.325GPa㊁5.687GPa㊁3.801GPa.因此,空位缺陷对炸药力学 表5 不同模型的弹性系数与力学参数Tab.5 Elastic coefficients and mechanical properties of different crystal models模型C11/GPa C22/GPa C33/GPa C44/GPa C55/GPa C66/GPa C12/GPa C13/GPa C23/GPa 116.89816.32314.9197.1387.0215.2327.3367.0256.917 216.73016.41115.1176.8066.8185.4087.2126.9277.004 316.40216.21714.9026.9226.8305.1957.1876.9696.768 415.57415.73114.4676.6166.5235.0027.0986.4266.518 514.73313.97813.1045.6366.0035.1476.1095.3485.515 615.23614.73013.6615.9525.7835.2456.4125.9176.115 715.18515.02314.8306.1305.9365.1136.8186.4386.023 814.51414.30811.7734.7365.7305.1135.5824.7085.136 914.63014.21312.8755.1175.8535.0046.3494.9785.917 1013.92912.62310.3184.4275.1384.0255.1744.5264.835 1114.03813.47111.9724.4185.4344.3705.3334.4195.211 1216.01715.54815.0036.5176.6355.3347.2306.6186.42755兵 工 学 报第40卷续表5 模型C 15/GPa C 25/GPa C 35/GPa C 46/GPa E /GPaγK /GPa G /GPa C 12⁃C 44/GPa 10.0980.112-0.2300.04414.4410.2268.7735.8910.19820.126-0.1060.2180.11513.8530.2298.5185.6340.40630.1210.0340.168-0.21213.4680.2238.1195.5040.2654-0.2070.3150.117-0.18212.6320.2277.7035.1490.4825-0.3020.2690.1500.03611.8600.2277.2324.8340.47360.108-0.2260.3130.12411.9430.2267.2564.8720.46070.1200.158-0.0590.15712.1130.2287.4154.9330.6888-0.2150.2080.0390.16811.0600.2256.7064.5140.8469-0.1180.3100.2280.14911.1480.2306.8714.5331.23210-0.1140.230-0.020-0.2159.3250.2275.6873.8010.747110.177-0.196-0.0450.17610.4430.2286.3934.2520.915120.1110.2350.303-0.09812.8760.2267.8345.2510.713图10 不同模型的力学性能参数Fig.10 Mechanical properties of different crystal models性能的影响更为显著㊂此外,从图10还可以看出,对于同种类型的晶体缺陷(掺杂㊁空位),随着缺陷数量的增加,E ㊁K ㊁G 逐渐减小,而柯西压呈现出增大的变化趋势,预示炸药的刚性减弱,塑性与延展性增强㊂3 结论本文采用分子动力学方法,分别预测了 完美”型与含有晶体缺陷的HMX /NTO 共晶炸药的稳定性㊁感度㊁爆轰性能与力学性能,研究并评估了晶体缺陷对炸药性能的影响情况㊂得出以下结论:1)由于晶体缺陷的影响,结合能减小幅度为3.06%~13.45%,分子之间的作用力减弱,炸药的稳定性降低㊂空位缺陷晶体的结合能最小,稳定性最差,且随着缺陷数量的增加,炸药的稳定性逐渐减弱㊂2)与 完美”型晶体相比,缺陷晶体的引发键键长增大0.52%~3.95%,而键连双原子作用能与CED 分别减小2.61%~15.70%㊁1.25%~9.74%,表明炸药的感度增大,安全性减弱,其中空位缺陷晶体的感度最高㊂随着晶体缺陷数量的增加,炸药的感度逐渐增大,安全性逐渐减弱㊂3)缺陷晶体的密度㊁爆速与爆压减小幅度分别为0.68%~6.00%㊁0.51%~4.82%㊁1.38%~12.34%,表明其威力减小,能量密度降低,其中空位缺陷晶体的能量密度最低,预示空位缺陷对能量密度的影响更为显著㊂随着缺陷数量的增加,炸药的密度与爆轰参数逐渐减小,能量密度逐渐降低㊂4)缺陷晶体的拉伸模量㊁体积模量与剪切模量减小,柯西压增大,表明炸药的刚性减弱,柔性与延展性增强,其中空位缺陷晶体的模量最小,掺杂缺陷晶体的模量最大,即空位缺陷对力学性能的影响更加显著,而掺杂缺陷的影响相对较小㊂参考文献(References )[1] BOLTON O,SIMKE L R,PAGORIA P F,et al.High power ex⁃plosive with good sensitivity:a 2∶1cocrystal of CL⁃20:HMX[J].Crystal Growth &Design,2012,12(9):4311-4314.[2] LIU K,ZHANG G,LUAN J Y,et al.Crystal structure,spectrumcharacter and explosive property of a new cocrystal CL⁃20/DNT [J].Journal of Molecular Structure,2016,1110:91-96.[3] 侯聪花,刘志强,张园萍,等.TATB /HMX 共晶炸药的制备及性能研究[J].火炸药学报,2017,40(4):44-49.HOU C H,LIU Z Q,ZHANG Y P,et al.Study on preparation and properties of TATB /HMX cocrystal explosive [J].ChineseJournal of Explosives &Propellants,2017,40(4):44-49.(in Chinese)[4] GAO H,DU P,KE X,et al.A novel method to prepare nano⁃sized CL⁃20/NQ cocrystal:vacuum freeze drying[J].Propellants,Explosives,Pyrotechnics,2017,42(8):889-895.[5] LIN H,ZHU S G,ZHANG L,et al.Intermolecular interactions,thermodynamic properties,crystal structure,and detonation per⁃65 第1期奥克托今/3⁃硝基⁃1,2,4⁃三唑⁃5⁃酮共晶炸药晶体缺陷的分子动力学研究formance of HMX/NTO cocrystal explosive[J].International Jour⁃nal of Quantum Chemistry,2013,113(10):1591-1599. [6] ZHOU T T,HUANG F L.Effects of defects on thermal decompo⁃sition of HMX via ReaxFF molecular dynamics si⁃mulations[J].The Journal of Physical Chemistry B,2011,115(2):278-287.[7] 陈科全,蓝林钢,路中华,等.含预制缺陷PBX炸药裂纹扩展过程的试验研究[J].兵器装备工程学报,2017,38(1): 134-136.CHEN K Q,LAN L G,LU Z H,et al.Experimental study on the crack growth of polymer bonded explosives with prefabricated de⁃fects[J].Journal of Ordnance Equipment Engineering,2017, 38(1):134-136.(in Chinese)[8] 王洪波,王旗华,卢永刚,等.PBX炸药细观孔洞缺陷对其冲击点火特性的影响[J].火炸药学报,2015,38(5):31-36.WANG H B,WANG Q H,LU Y G,et al.Effect of meso⁃defect holes on the shock⁃to⁃ignition characteristics of PBX explosives [J].Chinese Journal of Explosives&Propellants,2015,38(5): 31-36.(in Chinese)[9] XUE X G,WEN Y S,LONG X P,et al.Influence of dislocationson the shock sensitivity of RDX:molecular dynamics simulations by reactive force field[J].The Journal of Physical Chemistry C, 2015,119(24):13735-13742.[10] 吕鹏博,王伟力,刘晓夏,等.含装药缺陷的侵爆战斗部穿甲过程装药安定性的数值模拟[J].海军航空工程学院学报,2017,32(4):389-394.LYU P B,WANG W L,LIU X X,et al.Numerical simulation ofthe stability of the charge of invasion of explosive warheads con⁃taining defects in armor⁃piercing process[J].Journal of NavalAeronautical and Astronautical University,2017,32(4):389-394.(in Chinese)[11] SUN H,REN P,FRIED J R.The COMPASS force field:para⁃meterization and validation for polyphosphazenes[J].Computa⁃tional and Theoretical Polymer Science,1998,8(1/2):229-246.[12] SUN PASS:an ab initio force⁃field optimized for con⁃densed⁃phase applications⁃overview with details on alkane andbenzene compound[J].The Journal of Physical Chemistry B,1998,102(38):7338-7364.[13] ANDERSEN H C.Molecular dynamics simulations at constantpressure and/or temperature[J].Journal of Chemical Physics,1980,72(4):2384-2393.[14] PARRINELLO M,RAHMAN A.Polymorphic transitions in sin⁃gle crystals:a new molecular dynamics method[J].Journal ofApplied Physics,1981,52(12):7182-7190. [15] BOWDEN F P,YOFFE A D.Initiation and growth of explosionin liquids and solids[M].Cambridge,UK:Cambridge Universi⁃ty Press,1952.[16] KAMLET M J,ADOIPH H G.The relationship of impact sensi⁃tivity with structure of organic high explosives[J].Propellants,Explosives,Pyrotechnics,1979,4(2):30-34. [17] 朱伟,刘冬梅,肖继军,等.多组分高能复合体系的感度判别㊁热膨胀和力学性能的MD研究[J].含能材料,2014,22(5):582-587.ZHU W,LIU D M,XIAO J J,et al.Molecular dynamics studyon sensitivity criterion,thermal expansion and mechanical proper⁃ties of multi⁃component high energy system[J].Chinese Journalof Energetic Materials,2014,22(5):582-587.(in Chinese) [18] 刘强,肖继军,张将,等.CL⁃20/TNT共晶炸药的分子动力学研究[J].高等学校化学学报,2016,37(3):559-566.LIU Q,XIAO J J,ZHANG J,et al.Molecular dynamics simula⁃tion on CL⁃20/TNT cocrystal explosive[J].Chemical Journal ofChinese Universities,2016,37(3):559-566.(in Chinese) [19] SUN T,XIAO J J,LIU Q,et parative study on struc⁃ture,energetic and mechanical properties of aε⁃CL⁃20/HMX co⁃crystal and its composite with molecular dynamics simulation[J].Journal of Materials Chemistry A,2014,2(34):13898-13904.[20] XIAO J J,ZHAO L,ZHU W,et al.Molecular dynamics studyon the relationships of modeling,structural and energy propertieswith sensitivity for RDX⁃based PBXs[J].Science China Chemis⁃try,2012,55(12):2587-2594.[21] ZHU W,WANG X J,XIAO J J,et al.Molecular dynamics si⁃mulations of AP/HMX composite with a modified force field[J].Journal of Hazardous Materials,2009,167(1):810-816.[22] MULLAY J.Relationship between impact sensitivity and molecu⁃lar electronic structure[J].Propellants,Explosives,Pyrotech⁃nics,1987,12(4):121-124.[23] BRILL B T,JAMES J K.Kinetics and mechanisms of thermaldecomposition of nitroaromatic explosives[J].Chemical Re⁃views,1993,93(8):2667-2692.[24] 彭亚晶,蒋艳雪.分子空位缺陷对环三亚甲基三硝胺含能材料几何结构㊁电子结构及振动特性的影响[J].物理学报,2015,64(24):243102⁃1-243102⁃9.PENG Y J,JIANG Y X.Analyses of the influences of molecularvacancy defect on the geometrical structure,electronic structure,and vibration characteristics of hexogeon energetic material[J].Acta Physica Sinica,2015,64(24):243102⁃1-243102⁃9.(inChinese)[25] DUAN X H,LI W P,PEI C H,et al.Molecular dynamics simu⁃lations of void defects in the energetic material HMX[J].Journalof Molecular Modeling,2013,19(9):3893-3899. [26] 国迂贤,张厚生.炸药爆轰性质计算的氮当量公式及修正氮当量公式:炸药爆速的计算[J].爆炸与冲击,1983,3(3):56-66.GUO Y X,ZHANG H S.Nitrogen equivalent(NE)and modi⁃fied nitrogen equivalent(MNE)equations for predicting detona⁃tion parameters of explosives⁃prediction of detonation velocity ofexplosives[J].Explosion and Shock Waves,1983,3(3):56-66.(in Chinese)[27] 吴家龙.弹性力学[M].上海:同济大学出版社,1993.WU J L.Mechanics of elasticity[M].Shanghai:Tongji Univer⁃sity Press,1993.(in Chinese)[28] WEINER J H.Statistical mechanics of elasticity[M].New York,NY,US:John Wiley&Sons,1983.75。
太空中的失重原理作文
太空中的失重原理作文英文回答：The principle of weightlessness in space is a fascinating concept that involves the absence of gravitational forces. In space, there is no gravity pulling objects towards the center of the Earth, which creates a sensation of weightlessness. This phenomenon allows astronauts to float freely and perform tasks that would be impossible on Earth.One way to understand the principle of weightlessnessis by considering the concept of freefall. When an objectis in freefall, it is only subject to the force of gravity. In space, astronauts and objects are constantly falling towards the Earth, but they are also moving forward at a high speed. This combination of falling and forward motion creates a curved trajectory known as an orbit.Imagine throwing a ball horizontally while standing onthe edge of a tall building. As the ball moves forward, it also falls towards the ground due to gravity. If you throw the ball fast enough, it will fall at the same rate as the curvature of the Earth, resulting in a circular path. This is similar to how objects in space orbit around the Earth.Another way to understand weightlessness in space is by considering the concept of microgravity. Although there is no gravity in space, there is still a small amount of gravitational force present. This force is known as microgravity and is about one-millionth of the gravity on Earth. In microgravity, objects and astronauts experience a sensation of weightlessness because the gravitational force is so weak.To illustrate this, imagine being inside an elevator that is accelerating upwards at the same rate as the force of gravity. As you stand inside the elevator, you wouldfeel weightless because the upward acceleration cancels out the downward force of gravity. This is similar to how astronauts in space feel weightless because the force of gravity is effectively canceled out by the constantfreefall motion.中文回答：太空中的失重原理是一个令人着迷的概念，它涉及到没有重力的存在。
三体系统量子纠缠度特性研究
三体系统量子纠缠度特性研究杨富社;王菊霞【摘要】Using the method of quantum calculation, the entanglement degrees of the three-body system, any two-body among the them, and the Tavis-Cummings model system are calculated in detail. The evolution property of three-body entangtement degree is showed via solving the Schrodinger equation of the atom-light field system. It is found that the entanglement degree of the atom-light field system shows the periodic oscillations with the time and the cycle is closely related to the initial state of the system. The entanglement degree in the three-body system has the same cycle as that of the any two-body in the three-body system. The system will evolve to a three-body maximum or partly entangled state from the initial non-entangled state.%应用量子计算方法,对三原子体系中三体问、两两间以及Tavis-Cummings模型系统的纠缠度进行了详细计算.通过求解原子与光场相互作用系统态矢满足的薛定谔方程,得出了该模型三体问纠缠度随时间的演化规律.结果表明:原子光场相互作用系统的纠缠度呈现随时问变化的周期振荡性,周期大小与系统初始状态有关.系统中三体间的纠缠度与三体中两两间纠缠度的振荡周期有关.随着时间的变化,体系由开始的非纠缠状态可演化为三体最大或部分纠缠态.【期刊名称】《陕西师范大学学报（自然科学版）》【年(卷),期】2011(039)001【总页数】5页(P33-37)【关键词】量子光学;纠缠态;三体纠缠;纠缠度【作者】杨富社;王菊霞【作者单位】长安大学,理学院,陕西,西安,710064;渭南师范学院,物理与电子工程系,量子光学与光子学研究所,陕西,渭南,714000【正文语种】中文【中图分类】O413.1自从纠缠态概念提出以来,人们已认识到纠缠态的重要性.为了定量描述纠缠的程度,引进纠缠度的概念,它是指纠缠态携带纠缠量的多少[1],从而可使不同纠缠态之间建立了可比关系.迄今,对纠缠度的具体量化定义形形色色,尚无普遍认可的统一化形式[2-11].1995年,Schlienz和M ahler提出了关于三体纠缠态纠缠量以及三体中两两间的纠缠量的计算公式[2];之后查新未[3]尝试给出相互独立的多体系统对应的量子纯态纠缠度定义;石名俊等人[4]又给出量子纯态纠缠度的几何解释;美国科学家Bennett提出了两种形式化但物理意义十分鲜明的定义[5],即生成纠缠EF(ρAB)与蒸馏纠缠ED(ρAB);波兰科学家 Horodecki和他的儿子提出了定义纠缠度应遵从的原则[6],即应遵从明显假定、基本假设、渐进性假设;Vedral等人给出几何意义非常鲜明的定义:纠缠度就是密度矩阵距离非纠缠集合的最小距离[7].关于量子纠缠态的度量虽然已提出一些相关描述的物理量,如 Von Neumann熵、纠缠相对熵[8]、密度算符之间的距离[9]、Rènyi纠缠度[10]和共生纠缠度(Concurrence)[11]等,但是人们仍未放弃寻找意义鲜明同时表达简单、易于求解的普遍的统一纠缠度定义.最先研究纠缠态判别问题并取得重要进展的是Peres[12],他给出了判别两个子系统的量子态为可分的必要条件;Ho rodecki还发现在束缚纠缠态中存在一种“纠缠激活”的有趣现象[13];Schlienz[14]和 Keller[15]也给出了不同的解释;阮曼奇、徐岗、曾谨言等提出[16]:对于二粒子体系而言,判断一个自旋态是否为纠缠态,最确切的判据是通过计算该自旋的单粒子约化密度的秩来确定.本文利用一种典型的纠缠度计算法,对三原子纠缠态、原子与光场相互作用系统的三体纠缠度进行较详细的计算,并定性分析纠缠度演化特性.1995年,Schlienz和M ahler提出的关于三体纠缠态纠缠量以及三体中两两间的纠缠量的计算公式[2]为可求解其相应状态的三原子间纠缠量以及三原子中任意两两间的纠缠量.在相互作用绘景中,T-C模型[17]中两个全同定态二能级原子与单模光场体系的相互作用哈密顿量为[18]从两种不同初态下的纠缠度的分析可以看出,原子与光场相互作用体系的纠缠量都呈现出明显的随时间变化的周期性振荡,周期的大小与体系的初始状态有关:初始状态不同,随时间变化过程中振荡的周期也不同.三体中两两间的纠缠量也呈现出周期性振荡,振荡周期与三体间纠缠量的振荡周期相等.随着时间的变化,体系由开始三体所处的非纠缠态(即t=0时,E3=0)可演化为三体最大纠缠态(即 E=1)以及部分纠缠态(即0＜E＜1).在计算三原子体系间的纠缠量以及三原子中任意两个原子间纠缠量大小的基础上,对原子与光场三体纠缠量的演化特性进行了分析.结果表明:无论初始原子处于EPR 纠缠态或非纠缠态,体系都将演化为三体最大或部分纠缠态,原子与光场体系的纠缠量呈现出明显的周期性振荡现象,周期的大小与光场及原子构成体系的状态有关,体系初始状态不同,演化过程中的周期大小不同.三体中两两间的纠缠量也呈现出周期性振荡,并且体系的三体纠缠量与三体中两两间的纠缠量的振荡周期相等.这对于多体纠缠程度量化问题起到了基础性指导作用,以便于有效的提高纠缠度,这将广泛应用于量子信息编码、量子信息传送和量子计算等量子信息光学和光场量子统计领域的研究之中.【相关文献】[1]Zhou Zhengwei,Guo Guangcan.Quantum entangled states[J].Physics,2000,29(11):695-699.[2]Schlienz J,Mahler G.Descrip tion of entanglement[J].Physical Review:A,1995,52(6):4396-4404.[3]Cha Xinwei.Entanglement of quantum pure states[J].西安邮电学院学报,2003,8(1):56-58.[4]Shi M ingjun,Du Jiangfeng,Zhu Dongpei.Entanglement of quantum pure states[J].物理学报,2000,49(5):825-829.[5]Bennett C H,Vincenzo D P,Smolin J A,et al.M ixedstate entanglement and quantum erro r correction[J].Physical Review:A,1996,54:3824-3851.[6]Horodecki M,Horodecki P,Horodecki R.Separability of mixed states:necessary and sufficient conditions[J].Physics Letters:A,1996,223(25):1-8.[7]Vedral V,Plenin M B,Rippin M A,et al.Quantifying entanglement[J].Physical Review Letters,1997,78:2275-2279.[8]Zheng Shibiao,Guo Guangcan.Efficient scheme for twoatom entanglement and quantum information p rocessing in cavity QED[J].Physical Review Letters,2000,85:2392-2395.[9]Knöll L,Orlow ski A.Distance between density operato r:App lication to the Jaynes-Commings model[J].Physical Review:A,1995,51:1622-1630.[10]Zheng Shibiao.One-step synthesis of multiatom Greenberger-Horne-Zeilinger states[J].Physical Review Letters,2001,87:2304041-2304044.[11]Christian F.R,Mark Riebe,Hartmut Häffner,et al.Control and Measurement of Three-Qubit Entangled States[J].Science,2004,304(4):1478-1480.[12]Peres A.Separability Criterion for Density Matrices[J].Physical ReviewLetters,1996,77:1413-1415.[13]Ho rodecki P,Ho rodecki M,Ho rodecki R.Bound Entanglement Can BeActivated[J].Physical Review Letters,1999,82:1056-1059.[14]Schlienz J,Mahler G.Descrip tion of entanglement[J].Physical Review:A,1995,52:4396-4404.[15]Keller T E,Rubin M H,Shih Y,et al.Theo ry of the three-photon entangledstate[J].Physical Review:A,1998,57:2076-2079.[16]Ruan Manqi,Xu Gang,Zeng Jinyan.The three kinds of types spin-maximum entangled state in two-particle system[J].中国科学:G辑,2003,33(5):411-415.[17]Peng Jinsheng,Li Gaoxiang.Introduction of Modern Quantum Op tics[M].北京:科学出版社,1996:81.[18]Zuo Zhanchun,Xia Yunjie.The evolution p roperty of three-body entanglement measure in Tavis-Cummings model[J].物理学报,2003,52(11):2687-2693.。
弱hopf量子yang-baxter h-模基本定理(英文)
弱hopf量子yang-baxter h-模基本定理(英文)The Weak Hopf Quantum Yang-Baxter Homological Theory (WHQYBHT) is a set of mathematical principles that provide a theoretical foundation for the study of quantum entanglement and related topics. The theory is based on two main ideas, the weak Hopf algebra and the quantum Yang-Baxter equation.The weak Hopf algebra is a generalization of the notion of Hopf algebras that allow for a non-commutative or non-associative structure of certain objects within a certain algebraic structure. This makes weak Hopf algebras a useful tool in understanding the structure of quantum systems and the various interactions that can occur within them.The quantum Yang-Baxter equation (YBE) is a mathematical statement that relates two groups of quantities, known as‘spin variables’ and ‘couplings’. It provides a means of expressing the physical interactions that occur within a quantum system, and it is an example of a linear operator equation. The YBE is also able to describe the way that different quantum systems interact with each other, and this can be used to explain the behavior of entangled particles.The WHQYBHT combines the weak Hopf algebra and the YBE to provide a theoretical foundation for the study of entanglement and other related topics. It is based on the premise that quantum entanglement is the result of the exchange of certain information between the entangledparticles. This information can be represented by certain operators known as ‘spin variables’, which are used in the YBE equation.The WHQYBHT is used to study the effects of entanglement and its related phenomena, such as teleportation and quantum computing. The theory is also used to analyze the effects of non-commutative quantum systems and to study the propertiesof quantum bits.The WHQYBHT is a powerful and useful tool in many areas of quantum physics. It is an important part of thetheoretical foundations of quantum entanglement and the study of the various phenomena such as teleportation and quantum computing. The theory also provides a means of analyzing the structure of quantum systems and their interactions, and it has the potential to provide significant insights into the behavior of entangled particles.。
具有Dzyaloshinskii-Moriya相互作用的XY模型的量子相干性
具有Dzyaloshinskii-Moriya相互作用的XY模型的量子相干性伊天成;丁悦然;任杰;王艺敏;尤文龙【摘要】研究了具有Dzyaloshinskii-Moriya(DM)相互作用的一维横场XY自旋链的量子相变和量子相干性.采用约旦-维格纳变换严格求解了哈密顿量,并描绘了体系的关联函数和相图,相图包含反铁磁相、顺磁相和螺旋相.利用相对熵和Jensen-Shannon熵讨论了XY模型的量子相干性.研究发现,相对熵与Jensen-Shannon 熵所表现的行为都可以很好地表征该模型的量子相变.非螺旋相中量子相干性不依赖DM相互作用,而在螺旋相DM相互作用对量子相干性有显著影响.此外,指出了在带有DM相互作用的这一类反射对称破缺体系中关联函数计算的常见问题.【期刊名称】《物理学报》【年(卷),期】2018(067)014【总页数】12页(P31-42)【关键词】量子相变;量子相干性;约旦-魏格纳变换【作者】伊天成;丁悦然;任杰;王艺敏;尤文龙【作者单位】苏州大学物理与光电·能源学院, 苏州 215006;苏州大学物理与光电·能源学院, 苏州 215006;常熟理工学院物理系, 常熟 215500;陆军工程大学通信工程学院, 南京 210007;苏州大学物理与光电·能源学院, 苏州 215006;苏州大学, 江苏省薄膜材料重点实验室, 苏州 215006【正文语种】中文1 引言量子相干和量子纠缠是量子力学最基本的两个特性,其根源在于量子态的叠加原理,即单个量子态可由多个不同态以不同的方式相干叠加得到.量子相干和量子纠缠被认为是量子物理和量子信息的核心,是实现各种量子任务（如量子远程通信和量子计算等）的量子资源,是量子信息科学的优势所在.这两种现象尽管概念完全不同,却有着紧密的联系.相干性和纠缠可以通过运算等效[1].目前,作为一种对量子叠加性的量化,量子相干性已经成为处理量子信息的一种重要的手段[2].研究表明,量子纠缠可以被认为是相干性的一种体现,反之则不成立.例如,两量子比特的直乘态没有纠缠,只有相干. 物理学家对量子纠缠的研究已有几十年,成果斐然[3].量子纠缠在量子科技中发挥着不可或缺的作用,比如可以实现密钥共享[4]、量子密码术[5]和量子克隆[6]等.与量子纠缠相比,人们对量子相干领域的研究还较少.而借鉴量子纠缠领域的研究成果可以大大加深对量子相干性的研究.近年来,人们应用量子信息的概念,如量子纠缠[7−11]、量子失谐[12,13]、保真率[14,15]来描述量子相变,拓展了对量子临界行为的认识.人们发现量子纠缠在非临界区满足面积定理,而在临界点表现出对数发散行为[16].为此,利用量子相干性来研究强关联体系的量子相变,有助于理解凝聚态系统中更广泛而深刻的行为.同时,对量子相干性的深入研究不仅对量子信息的发展有着重大的意义,也可以促进对量子力学基本问题的理解.本文研究具有Dzyaloshinskii-Moriya(DM)相互作用的自旋1/2的XY模型的量子相干性.XY模型是低维量子系统的一个重要模型,它在1961年就被Lieb等[17]严格求解.XY模型具有非常广阔的变化范围,它包含了Ising模型、各向同性XX模型以及各向异性XY模型,具有非常丰富的物理内涵.许多复杂模型都可以借助一系列的方法映射到此模型上来解决问题,它等价于二聚化的XX模型[18].不仅如此,众多材料也能实现有效的XY模型.例如,Cs2CoCl4是沿着b轴的自旋1/2的准一维XY自旋链[19];将钴原子链蒸发在Cu2N/Cu(100)衬底上可以形成有效的XY模型[20].近年来,人们对DM相互作用具有浓厚的兴趣,对于具有DM相互作用的反铁磁性的材料进行了广泛的研究.DM相互作用是存在于磁性材料中的一种非对称的交换作用,源于自旋轨道耦合[21,22].在一级近似下,DM相互作用可以写成Dij·(σi× σj),这里Dij称为DM矢量.DM 相互作用可以使一些材料表现出新的性质,比如引起亚铁磁绝缘体Cu2OSeO3的螺旋自旋基态和铁电性质的变化[23−25],诱发多铁材料BiFeO3的摆线状磁结构[26].DM相互作用的存在使得电子自旋共振[27]和电场调制[28]等技术可以应用在磁性化合物上.这些神奇的现象激发了理论研究者的广泛兴趣[29−36].本文的结构如下:第2部分介绍具有DM相互作用的一维XY模型,计算该模型的能谱和基态性质,并给出XY模型在不同情况下的相图;第3部分讨论XY模型的关联函数;第4部分利用相对熵和Jensen-Shannon熵讨论XY模型的量子相干性;第5部分给出最终的结论.2 模型和相图在此引入XY模型的哈密顿量:其中γ表示描述各向异性程度的参数,h表示磁场的大小,D表示沿着z轴的DM相互作用强度,表示位于格点i上自旋的泡利矩阵的α分量,N为总格点数.本文选择周期边界条件,即σN+1=σ1.当γ=0时,模型(1)为各向同性的XX模型;当γ=1时,模型(1)简化成Ising模型.利用约旦-维格纳变换做进一步处理:这里需要借助产生算符σ+和湮灭算符σ−,其定义为σ±≡(σx±iσy)/2.由此,上述哈密顿量(1)式可以被写成关于无自旋费米子的产生算符和湮灭算符的二次型:由于哈密顿量(1)式具有平移不变性,可以通过傅里叶变换将方程(5)转换到动量表象.最后,利用波戈留波夫变换再进行对角化处理,这样可以得到基态能量的精确解.最终哈密顿量可以变换为对角化的形式:其中图1显示了h=0.5,γ=1时D取不同值时的激发谱ϵk,十分清楚地反映了能谱随DM相互作用变化的趋势.当D=0时,能带存在能隙∆≡ minkϵk.这时基态|ψ0中所有动量k上的电子占据数为0,即bk|ψ0=0.随着D的增加,能隙∆变得越来越小.当D=0.5时,ϵk在某一个非公度的动量kc等于0,能隙关闭.当D>0.5时,部分能谱变成负数,能隙消失,费米面上有两个交点kL和kR.对于一般情况,不难从(7)式得到图1 h=0.5,γ=1时的激发谱Fig.1.Excitation spectra for h=0.5,γ=1.换句话说,对于kL6 k 6 kR,激发谱ϵk是负的,基态中所有负能态被占据,即[33].众所周知,当DM相互作用不存在的时候,h=1对应反铁磁相-顺磁相的Ising相变,此时能隙在动量kc=0上闭合.γ=0且h 6 1则是一条各向异性相变线.以上相变都属于二级量子相变.根据能隙关系,在图2中给出了γ=1,γ=0.5,γ=0时对应的相图.从图2中可以看出,在一般情况下XY模型存在三个相.区域I和区域II是有能隙相,能隙在h=1关闭.区域III为无能隙相,其与区域I和区域II的相变线所对应的解析式分别为γ=2D和不失一般性,下面主要关注γ=0和γ=1的情况,因为后者代表06 γ 6 1的一般情况.图2 不同γ所对应的相图(a)γ=1;(b)γ=0.5;(c)γ=0;区域I,II,III分别代表反铁磁相、顺磁相和螺旋相Fig.2.Phase diagram with respect to γ:(a) γ =1;(b) γ =0.5;(c)γ =0.Region I,II and III correspond to antiferromagnetic phase,paramagneticphase and chiral phase,respectively.3 关联函数为了表征不同相,选取两格点之间的关联函数作为序参量来描述系统基态的性质,定义关联函数≡其中α,β=x,y,z.由于该系统具有平移不变性,因此关联函数的大小仅与i,j两格点之间的相对距离有关,而与两个格点所在的具体位置无关,所以可以将简记为其中r=i−j.对于一般的展开形式可以表示为法夫式(Pfaffian)[37,38].换句话说,可以写成2n×2n(n≡|j−i|)维反对称矩阵的行列式,更进一步的细节性的讨论见附录A. 首先考虑各个分量的最近邻关联.不失一般性,在图3中展示了γ=1的两条参数路径:D=0和D=1.在图3(a)中,可以看出x分量的最近邻关联随着磁场强度h的增加从−1渐渐趋近于0,而y和z分量的最近邻关联随h的增加从零开始呈增长之势,这表明区域I为反铁磁相.三种最近邻关联在量子相变点处都发生了突然的变化.如图3(a)插图所示,的一阶导数在h=1有一个明显的尖峰,意味着此处发生了二级量子相变.经过相变点后,和都逐渐趋近于0,而则渐渐地趋近于1,说明区域II为顺磁相.如图3(b)所示,由于DM相互作用的出现,临界磁场hc的位置发生了明显的变化,从D=0时的hc=1移动到D=1时的hc=2.类似地,的一阶导数在hc=2也呈现了一个尖峰(见图3(b)插图),这一结果与图2中的相变线一致.值得注意的是,当h<hc,三种最近邻关联的绝对值都比较小,且均未起到主导作用，即不存在某个方向的关联远大于另外两个方向.从图4中可以看出,和在区域III皆不为0,=且绝对值随着h的增大而减小,这表明区域III是DM相互作用引起的螺旋相.在非螺旋相中,和都为0.因此,和可以被认为是发掘螺旋相很好的序参量,螺旋相可以根据图4体现出来.除了最近邻的两点关联,接下来考虑长程的两点关联.Barouch和McCoy[38]研究了横场XY模型的磁化强度和关联函数.Its等[39]考虑横场XX模型的非零温关联.研究表明关联函数的渐进行为(r→∞)可以写成图3 最近邻关联函数随着h的变化(a)γ=1,D=0;(b)γ=1,D=1;插图为的一阶导数Fig.3.The nearest correlation functionwith respect to hfor:(a)γ=1,D=0;(b)γ=1,D=1.Insets show the first-derivative of图4 (a)γ=1时随着h和D变化的情况;(b)γ=1时随着h和D变化的情况Fig.4.(a)a s a function of h and D for γ =1;(b)as a function of h and D for γ =1.图5 关联函数的绝对值随着距离r的变化(a)D=0,γ=0;(b)D=0,γ=1;(c)D=1,γ=0;(d)D=1,γ=1Fig.5.The absolute value of the correlation functionwith respect tor:(a)D=0,γ=0;(b)D=0,γ=1;(c)D=1,γ=0;(d)D=1,γ=1.其中A是一个常数,为x方向上的磁化强度,ξ为关联长度.当|h|>1时,体系处于顺磁态,x方向上的磁化强度消失,即=0,此时limr→∞∼ (−1)rr−2exp(−r/ξ). 当|h|6 1时,limr→∞=(−1)r2[γ2(1 − h2)/(1+γ). 这意味着当γ≠0时,体系存在长程序.当γ=0时,长程序不存在.从图5(a)和图5(b)可以看出,由于反铁磁相有x方向的长程序,随着r的变化基本保持不变.在顺磁相中随着r的增加迅速趋向于0.在Ising临界点上,即h=1时,∼r−1/4.在各向异性相变线γ=0上且h=0时,∼r−1/2[40].这与图5(a)和图5(b)给出的趋势相符合.图5(c)和图5(d)显示了D=1的相关情况,选取了处于螺旋相的两点h=0,h=1所对应的两组参数.从图5(c)和图5(d)中可以看出,DM相互作用使得关联函数随着距离r的增加呈现振荡性地下降,γ=0时(图5(c))的关联函数在随r变化时振荡比γ=1时(图5(d))厉害.图5(c)和图5(d)中关联函数随r变化的曲线振荡包络面的上边界近似为∼ r−1/2,下边界近似为∼r−3/2,所以螺旋相中体系也不存在长程序.4 量子相干性在凝聚态理论中,量子相变需要构建合适的序参量来描述,如上文提及的自旋关联函数,但是往往序参量是事先不知道的.量子相变发生在零温时体系的量子关联随着参数的调节发生了突然的变化,对系统的量子关联进行量化和表征已经成为研究热点之一.随着量子信息的发展,一些量子信息的概念被移植到凝聚态领域,如量子纠缠、量子失谐、保真率也被用来描述量子相变[41−43].自然而然,量子相干性是一种典型的量子关联的刻度.两个量子比特乘积态的表象下,选取如下为基:其中代表自旋泡利算符σz向上(下)的本征态.为此,两个量子比特的密度矩阵可以表示为:当a=1,2,3时分别对应泡利矩阵中的代表一个单位矩阵.最终可以把一个两体的密度矩阵化为如下的形式:其中值得一提的是,很多先前的研究[44,45]在考虑(13)式和(14)式时默认了 = 0和=0,这与图4的结论是不符合的.这里要强调=0和=0只在纯实数哈密顿量的条件下才成立,对于哈密顿量中含有DM相互作用或者XZY-YZX类三体相互作用是不成立的[31,46,47],甚至一些文献[48]忽略了此时计算要利用原始公式(A10)(见附录A),而不是(A13)式.相对熵是Baumgratz等[49]依据资源理论引入的相干性的量化指标,其计算公式为其中S(ρ)= −Tr(ρlog2ρ).S(ρ)代表密度矩阵ρ的冯·诺依曼熵,S(ρdiag)则代表密度矩阵清除非对角项之后得到的新密度矩阵的冯·诺依曼熵.研究表明,XY模型的量子相变临界点可以用量子相干率(coherence susceptibility)的奇异性来描述[50],其定义为相对熵的一阶导数图6显示了在不同γ的情况下不同近邻程度的相对熵CRE以及相干率χRE随着磁场h变化的情况.当γ=0的时候,不同距离的两量子比特的相对熵具有相似的变化趋势(见图6(a)).随着h的增加,它们都呈现出减小的趋势.距离r越大,相对熵越小.不同的是,最近邻的相对熵衰减相对比较明显,次近邻以及更远近邻的相对熵起初保持一定的鲁棒性.当h靠近1的时候,所有距离的两量子比特相对熵都急剧衰减为0.在顺磁相中相对熵一直保持为0.图6(c)中,当γ=1时两量子比特的相对熵随着h的增长而减小,由于长程序的存在,不同距离r差别不大.同样在h=1处CRE有一个急剧的下降.与γ=0的情况不同,进入顺磁相后χRE仍然存在,且绝对值随着h的增加而减小.从图6(b)和图6(d)中可以看出,在h=1处,量子相干率χRE出现了一个奇点,表明在此处体系的相干性有一个剧烈的变化,即发生了量子相变.图6 D=0时相对熵CRE和量子相干率χRE随h的变化(a)γ=0时的CRE;(b)γ=0时的χRE;(c)γ=1时的CRE;(d)γ=1时的χREFig.6.The relative entropy CREand the quantum coherence susceptibility χREfor D=0:(a)CREfor γ =0;(b) χREfor γ =0;(c)CREfor γ =1;(d) χREfor γ =1.图7 D=0时Jensen-Shannon熵CJS和量子相干率χJS随h的变化(a)γ=0时的CJS;(b)γ=0时的χJS;(c)γ=1时的CJS;(d)γ=1时的χJSFig.7.Jensen-Shannon entropy CJSand the quantum coherence susceptibility χJSfor D=0:(a)CJSfor γ =0;(b) χJSfor γ =0;(c)CJSfor γ =1;(d) χJSfor γ =1.图8 γ=0时不同D的情况下相对熵CRE和量子相干率χRE随着h的变化 (a)最近邻情况下的CRE;(b)最近邻情况下的χRE;(c)D=1时不同近邻情况下的CRE;(d)D=1时不同近邻情况下的χREFig.8.The relative entropy CREand the quantum coherence susceptibility χREwith respect to D for γ =0:(a)CREfor the nearest-neighbor qubits;(b)χREfor the neares t-neighborqubits;(c)CREfor Different distances r between two qubits withD=1;(d)χREfor Different distances r between two qubits with D=1.除利用相对熵以外,还可利用Jensen-Shannon熵讨论XY模型的量子相干性[51].Jensen-Shannon熵的定义与相对熵类似:同样,其相干率亦定义为图7显示了在不同γ的情况下Jensen-Shannon熵CJS以及所对应的相干率χJS 随着磁场h变化的情况.Jensen-Shannon熵所表现的行为与相对熵定性上十分类似,都可以体现出在h=1处,系统发生了量子相变,所以接下来主要通过相对熵来研究量子相干性.图8展示了γ=0时两量子比特的相对熵在不同DM相互作用下的情况,同时给出了各自对应的量子相干率的变化规律.结果表明有限的D只能改变临界磁场hc的大小,而不改变临界行为.图8(a)中最近邻两量子比特的相对熵CRE随着h的增加而单调减少.从图8(c)可以看出D=1时,不同近邻程度的两量子比特的相对熵CRE有着十分类似的变化趋势,都是先随着磁场的增加缓慢下降,然后在相变点的位置急速下降.在远离相变点的位置,最近邻情况下的相对熵随磁场的变化相对比较明显,而次近邻和次次近邻则相对比较缓慢.当磁场h驱使系统进入顺磁相,CRE都消失为0.图9展示了γ=1时与图8相对应的情况,同样可以看出,两量子比特的相对熵CRE随着h的增加而减少.令人惊讶的是,当D=0时,CRE与h的依赖关系和D=0.4时重合.确切地说,非螺旋相中CRE不依赖D.而在螺旋相中,D对CRE有显著影响,见图8(a).图9 γ=1时不同D的情况下相对熵CRE和量子相干率χRE随着h的变化 (a)最近邻情况下的CRE;(b)最近邻情况下的χRE;(c)D=1时不同近邻情况下的CRE;(d)D=1时不同近邻情况下的χREFig.9.The relative entropy CREand the quantum coherence susceptibility χREwith respect to D for γ =1:(a)CREfor the nearest-neighbor qubits;(b)χREfor the nearest-neighborqubits;(c)CREfor Different distances r between two qubits withD=1;(d)χREfor Different distances r between two qubits with D=1.5 结论本文研究了含有DM相互作用的一维XY模型.严格求解出了此模型的基态和关联函数,给出了它的相图,发现关联函数和量子相干性可以很好地标志出量子相变临界点的位置.量子相干性不仅仅存在于最近邻格点之间,也存在距离更远的两格点之间.发现DM相互作用会移动量子相变临界点的位置,甚至诱发无能隙螺旋相.指出了一些计算细节,这些细节在很多文献里没有得到适当的处理,本文对此做了进一步的补充和诠释.附录A 自旋关联函数这里引入两个新的算符,它们满足如下代数关系:于是,泡利矩阵可以改写成:首先考虑x分量的自旋关联函数:类似地,y分量和z分量的关联可以表示为：此外,也可以计算交叉关联项：根据维克定理,可以对多算符连乘的平均值进行分解,变成二次形算符平均值的连乘,不难发现,对于展开形式可以表示为法夫式(Pfaffian)[38],即=pf(),其中α,β=x,y.换句话说,其可以写成2n×2n(n≡|j−i|)维反对称矩阵的行列式,即而对于z分量,可以发现,对于任意k,当其电子能谱ϵk总是正的时候,准粒子bk表象下基态没有元激发,即=0.此时, 换句话说,不同格点i和j上的同种算符期待值为零,即于是,关联函数可以简化成n×n维Toeplitz矩阵的行列式表示:参考文献【相关文献】[1]Alexander S,Uttam S,Himadri S D,Manabendra N B,Gerardo A 2015 Phys.Rev.Lett.115 020403[2]Alexander S,Gerardo A,Martin B P 2017 Rev.Mod.Phys.89 041003[3]Amico L,Fazio R,Osterloh A,Vedral V 2008 Rev.Mod.Phys.80 517[4]Shan C J,Man Z X,Xia Y J,Liu T K 2007 rm.5 335[5]Ekert A K 1991 Phys.Rev.Lett.67 661[6]Wooters W K,Zurek W H 1982 Nature 299 802[7]Osterloh A,Amico L,Falci G,Fazio R 2002 Nature 416 608[8]Osborne T J,Nielsen M A 2002 Phys.Rev.A 66 032110[9]Gu S J,Lin H Q,Li Y Q 2003 Phys.Rev.A 68 042330[10]Vidal G,Latorre G I,Rico E,Kitaev A 2003 Phys.Rev.Lett.90 227902[11]Vidal J,Palacios G,Mosseri R 2004 Phys.Rev.A 69 022107[12]Ollivier H,Zurek W H 2001 Phys.Rev.Lett.88 017901[13]Modi K,Brodutch A,Cable H,Paterek T,Vedral V 2012 Rev.Mod.Phys.84 1655[14]You W L,Li Y W,Gu S J 2007 Phys.Rev.E 76 022101[15]Gu S J,Int J 2010 Mod.Phys.B 24 4371[16]Eisert J,Cramer M,Plenio M B 2010 Rev.Mod.Phys.82 277[17]Lieb E,Schultz T,Mattis D 1961 Ann.Phys.16 407[18]Lorenzo C V,Marco R 2010 Phys.Rev.A 81 060101[19]Kenzelmann M,Coldea R,Tennant D A,Visser D,Hofmann M,Smeibidl P,Tylczynski Z 2002 Phys.Rev.B 65 144432[20]Toskovic R,van-den Berg R,Spinelli A,Eliens I S,vanden Toorn B,Bryant B,Caux J S,OtteA F 2016 Nat.Phys.12 656[21]Dzyaloshinskii I 1958 J.Phys.Chem.Solids 4 241[22]Moriya T 1960 Phys.Rev.Lett.4 288[23]Seki S,Yu X Z,Ishiwata S,Tokura Y 2012 Science 336 198[24]Adams T,Chacon A,Wagner M,Bauer A,Brandl G,Pedersen B,Berger H,LemmensP,Pfleiderer C 2012 Phys.Rev.Lett.108 237204[25]Yang J H,Li Z L,Lu X Z,Whangbo M H,Wei S H,Gong X G,Xiang H J 2012Phys.Rev.Lett.109 107203[26]Matsuda M,Fishman R S,Hong T,Lee C H,Ushiyama T,Yanagisawa Y,Tomioka Y,Ito T 2012 Phys.Rev.Lett.109 067205[27]Povarov K Y,Smirnov A I,Starykh O A,Petrov S V,Shapiro A Y 2011 Phys.Rev.Lett.107 037204[28]Zhang X F,Liu T Y,Flatté M E,Tang H X 2014 Ph ys.Rev.Lett.113 037202[29]You W L,Dong Y L 2010 Eur.Phys.J.D 57 439[30]You W L,Dong Y L 2011 Phys.Rev.B 84 174426[31]You W L,Liu G H,Horsch P,Oleś A M 2014 Phys.Rev.B 90 094413[32]Shan C J,Cheng W W,Liu T K,Huang Y X,Li H 2008 Acta Phys.Sin.57 2687(in Chinese)[单传家,程维文,刘堂昆,黄燕霞,李宏2008物理学报57 2687][33]Zhong M,Xu H,Liu X X,Tong P Q 2013 Chin.Phys.B 22 090313[34]Song J L,Zhong M,Tong P Q 2017 Acta Phys.Sin.66 180302(in Chinese)[宋加丽,钟鸣,童培庆2017物理学报66 180302][35]Brockmann M,Klumper A,Ohanyan V 2013 Phys.Rev.B 87 054407[36]Derzhko O,Verkholyak T,Krokhmalskii T,Büttner H 2006 Phys.Rev.B 73 214407[37]Barouch E,McCoy B M 1970 Phys.Rev.A 2 1075[38]Barouch E,McCoy B M 1971 Phys.Rev.A 3 786[39]Its A R,Izergin A G,Korepin V E,Slavnov N A 1993 Phys.Rev.Lett.70 1704[40]Bunder J E,McKenzie R H 1999 Phys.Rev.B 60 344[41]Vedral V 2002 Rev.Mod.Phys.74 197[42]Horodecki R,Horodecki P,Horodecki M,Horodecki K 2009 Rev.Mod.Phys.81 865[43]Modi K,Brodutch A,Cable H,Paterek T,Vedral V 2012 Rev.Mod.Phys.84 1655[44]Liu B Q,Shao B,Li J G,Zou J,Wu L A 2011 Phys.Rev.A 83 052112[45]Radhakrishnan C,Ermakov I,Byrnes T 2017 Phys.Rev.A 96 012341[46]You W L,Qiu Y C,Oleś A M 2016 Phys.Rev.B 93 214417[47]You W L,Zhang C J,Ni W,Gong M,Oleś A M 2017 Phys.Rev.B 95 224404[48]Lei S,Tong P 2015 Physica B 463 1[49]Baumgratz T,Cramer M,Plenio M B 2014 Phys.Rev.Lett.113 140401[50]Chen J,Cui J,Zhang Y,Fan H 2016 Phys.Rev.A 94 022112[51]Lamberti P W,Majtey A P,Borras A,Casas M,Plastino A 2008 Phys.Rev.A 77 052311。
2019年夸克星-暗星黑洞可能是宇宙暗物质的主要形态by张海鹏.doc
暗星（“黑洞”）-夸克星可能是暗物质的主要形态——兼及宇宙学对粒子物理的启示*张海鹏河北涿鹿县医院应用物理学研究所，河北张家口桑干河 075600[摘要] 通过分析大量文献，基于万有引力定律、量子力学和相对论力学理论，根据银心“黑洞”等观测结果，提出暗星（“黑洞”）-夸克星可能是宇宙中数倍于可见物质的暗物质的主要形态；基于琼斯假说的宇宙早期神秘的“消电离作用”作用及暗物质晕假说存在的问题，提出±e/3的夸克通过长距电磁作用与短程强相互作用构成的原始夸克星是宇宙早期的暗物质星体，其碰撞是产生恒星等可见物质之源，漩涡星系碰撞并合为一个椭圆星系可佐证之；已在γ暴941017的余晖中观测到的0.1TeV超高能γ光子，因其波长与0.1飞米级的夸克之径相符，亦佐证夸克星的暗物质星体假说。
上述观点与宇宙微波背景、可见物质的元素丰度均无矛盾，但有待宇宙观测（宇宙极早期声子的探测以及2.7K是否存在于所假设的宇宙诞生之际原始夸克星形成可能释放的更高频率的辐射背景）证实或证伪。
基于普朗克能标和上述宇宙学探讨的启示，提出±e/3的夸克可能是不能再分的基本粒子。
[关键词]：暗物质；黑洞，暗星；夸克星；γ射线暴；普朗克能标1. 关于暗物质与暗星之关联的可能性1.1 关于“黑洞”尽管“暗物质”问题可能最早源于1970年代后期宇宙微波背景的新发现，但，事实上，很早即已发现的宇宙膨胀现象衬托出的恒星等星体如何产生(物质如何在“种子”引力的局部缩聚基础上成星)的问题，本是宇宙学的一个基本问题；而当1998年发现膨胀的加速性之后，这个问题变得则更加尖锐。
按照2003年前后的资料，“暗能量”约占宇宙全部质-能约73%，“暗物质”约占23%，可见的重子等物质仅占4%[1]，即“暗物质”乃宇宙物质（不包括特纳假设的“暗能量”）的主体。
在众多的暗物质模型中，冷暗粒子（弱相互作用中的中间玻色子之类）比较突出，而加速器产生的弱力玻色子类均系寿命极短的粒子（而弱力玻色子尚不具备强相互作用的条件）；而分散的暗物质（“暗物质晕”）构成宇宙物质的主体，不仅尚未见较有说服力的理论依据或较有力的观测提示，而且似乎与星体“种子”不无矛盾:如其位于星际而非星核，不一定利于物质聚集成星；而观测方面，经对2009年钱德拉塞卡、牛顿两个牌号的X线太空望远镜的观测结果分析，已找到的在约4亿光年外的一大型星系处原“丢失”（注：此前应属所谓暗物质）的温热星系际（而不是星系中及恒星内）物质——属于重子物质[2]，亦或为重要提示。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
a r X i v :0807.1325v 2 [c o n d -m a t .s t r -e l ] 13 J u l 2008Heavy Fermion Quantum CriticalityZaira Nazario †and David I.Santiago ⋆†Max Planck Institute for the Physics of Complex systems,N¨o thnitzer Strasze 38,01187Dresden,Germany⋆Instituut-Lorentz for Theortical Physics,Universiteit Leiden,P.O.Box 9506,NL-2300RA Leiden,The Netherlands(Dated:July 13,2008)During the last few years,investigations of Rare-Earth materials have made clear that not only the heavy fermion phase in these systems provides interesting physics,but the quantum criticality where such a phase dies exhibits novel phase transition physics not fully understood.Moreover,attempts to study the critical point numerically face the infamous fermion sign problem,which limits their accuracy.Effective action techniques and Callan-Symanzik equations have been very popular in high energy physics,where they enjoy a good record of success.Yet,they have been little exploited for fermionic systems in condensed matter physics.In this work,we apply the RG effective action and Callan-Symanzik techiques to the heavy fermion problem.We write for the first time the effective action describing the low energy physics of the system.The f -fermions are replaced by a dynamical scalar field whose nonzero expected value corresponds to the heavy fermion phase.This removes the fermion sign problem,making the effective action amenable to numerical studies as the effective theory is bosonic.Renormalization group studies of the effective action can be performed to extract approximations to nonperturbative effects at the transition.By performing one-loop renormalizations,resummed via Callan-Symanzik methods,we describe the heavy fermion criticality and predict the heavy fermion critical dynamical susceptibility and critical specific heat.The specific heat coefficient exponent we obtain (0.39)is in excellent agreement with the experimental result at low temperatures (0.4).PACS numbers:71.27.+a,73.43.Nq,71.10.HfFor a couple of decades,heavy fermion materials have attracted the focus of a large part of the experimental and theoretical condensed matter community[1,2].There are many reasons for such a spotlight on these materi-als.They exhibit exotic superconductivity,interesting magnetism,but most importantly heavy quasiparticles with an enlarged Fermi surface.This heavy quasipar-ticle phase perishes into a quantum critical point with interesting,puzzling and not yet understood nature[3,4].It is quite striking to see the fermionic quasiparticle with masses from tenths to about thousandths of an el-ementary electron mass.This has been understood as arising from Kondo-like physics of the almost localized f −electrons when they hybridize with the lower atomic angular momentum bands of the material[2,5].This hy-bridization gives rise to an enlarged Fermi surface as the f −electrons now contribute to the Fermi volume,and to the large quasiparticle mass and large specific heat coeffi-cient,through enhanced collective Kondo-like low energy scattering.Of course,the f −electrons have a strong ten-dency to localize due to their large U which fights the hybridization V .Some of these materials can be tuned (by apply-ing pressure,etc.)so that a critical value (U/V )c is reached such that,for values larger than the critical value,the f −electrons localize and there is no heavy fermion phase.Instead,there is a small Fermi surface metal that usually exhibits magnetic order mediated via RKKY interactions[6].For subcritical values of U/V ,the system is in the heavy fermion phase with large Fermi surface and no magnetism.At the critical value,a continous quantum phase tran-sition occurs as corroborated via scaling experiments.This transition is not understood.The lack of under-standing is a barrier to the full characterization of the physical properties,phase diagram and experimental re-sponse features of these materials.In this work,we turn our attention to understanding this heavy fermion quan-tum criticality[3,7].We start from the partition function for f −electron hybridizing with metallic c −electrons,i.e.the periodic Anderson ModelZ =D c †D c D f †D f e −SS =Mdω(2π)3−ω f †M (ω, k )f M (ω, k )+c †M (ω, k )c M (ω, k ) +(ǫ k−µ)+ Mdω(2π)3E 0−UMdω(2π)3f †M (ω, k )c M (ω, k )+h.c.+U(2π)3d 3kd 3 k 2d 3 k 3idτ n 2i −n i .We can decouple this interaction term by using the Hubbard-Stratonovichidentity e−R dτP i U [ϕ2i +2iϕi n i ]which in Euclidean time and momentum spacereads D ϕe−R dτd 3 k [ϕk ϕ− k +2in k ϕ−k ].Using n k =d 3k(2π)3f †kM ∂f † kM fkM +U∂τc kM +(ǫ k −µ)(2π)3VMdτd 3kd 3 q 2πd 3 k 2πd 3 q+Λ−2g 4dω2dω3(2π)6˜ϕ0(ω, k )˜ϕ0(ω2, k 2)˜ϕ0(ω3, k 3)˜ϕ0(−ω−ω2−ω3,− k − k 2− k 3)where˜ϕ0(ω, k )=πD 3ϕ0(ω, k )g 2=D 24NU Λ (|E 0|+U )g 3=2Λ2(E 0−U )5;g 4=πD 32πd 3 k 2πd 3 q (2π)2d 3 k 2d 3 k 38π4(1−µ/Λ)1−(ω−k r )2π(1−µ/Λ)(2π)4(1−µ/Λ)1g 2−1−4g 22Λ+(ω−k r )2g 2Λ(g 2−1)2lng 2+14into −µ−1/2g R 3Z ϕZ 2c =−Λ−1/2(g 3+Γc ),whereΓc =g 23(g 42−1)−2πg 2(g 22−1)2arctan124π4Λ2(1−µ/Λ)+......g 24g 2+12+164π4(1−µ/Λ),Z k =g 23µ1/2≃g 3(2π)3g3µ2≃g 4(2π)4Λ2−g 43m2(2π)4Λ2(1−µ/Λ).To this order of approximationβ3=µ∂g R 32g 3−g 238π−1g c 3=4π 1+∂µµ=Λ=2g 4−6g 248π4+3g 23g 44+(2π)433g c 4336+(2π)4∂µµ=Λ=g 232π21+π2+∂µ+γϕωγϕ/2∼11+2π2]/(4π2)F (ω, k =0)∼i ϕ(ω, k =0) =iG (1)(ω)χ(ω)=lim ǫ→0Im [F (ω+iǫ)]∼1Tγϕ/2(6)We obtain a specific heat coefficient exponent γϕ/2=0.39.This is in excellent agreement withthe exponent 0.4found for YbRh 2Si 2at low temperatures[3].Via renormalization group studies and effective ac-tion techniques common to field theories of particle physics,we have obtained the effective field theory for heavy fermion quantum criticality.This marks impor-tant progress as the effective field theory is bosonic,viti-ating the fermion sign problem and thus being amenable to numerical studies and high order ǫexpansion studies.The critical field theory can be studied using the renor-malization group.We did so via one-loop renormalization studies,improved by means of Callan-Symanzik resuma-tions to access some of the nonpertrubative effects.We thus make predictions for the exponents that character-ize the critical divergence of the specific heat coefficient and the critical charge susceptibility.Our specific heat coefficient exponent of 0.39is in excellent agreement with the 0.4found in experiments at low temperatures[3].[1]Stewart,G.R.Heavy-fermion systems.Rev.Mod.Phys.56,755-787(1984).[2]P.Coleman,Phys.Rev.B 29,3035(1984);N.Read,J.Phys.C:Solid State Phys 18,2651(1985).[3]P.Gegenwart,Q.Si and F.Steglich,Nature Physics 4,186(2008).[4]H.V.L¨o hneysen,T.Pietrus,G.Portisch,H.G.Schlager,A.Schr¨o der,M.Sieck,and T.Trappmann,Phys.Rev.Lett.72,3262(1994);A.Schr¨o der,G.Aeppli,E.Bucher,5R.Ramazashvili and P.Coleman,Phys.Rev.Lett.80, 5623(1998); A.Schr¨o der,G.Aeppli,R.Coldea,M.Adams,O.Stockert,H.V.L¨o hneysen,E.Bucher,R.Ramazashvili and P.Coleman,Nature407,351(2000);P.Gegenwart,J.Custers,C.Geibel,K.Neumaier,T.Tayama,K.Tenya,O.Trovarelli,and F.Steglich,Phys.Rev.Lett.89,056402(2002);J.Custers,P.Gegenwart,H.Wilhelm,K.Neumaier,Y.Tokiwa,O.Trovarelli,C.Geibel,F.Steglich,C.P´e pin and P.Coleman,Nature 424,524(2003).[5]N.Nagaosa,Quantum Field Theory in Strongly Cor-related Electronic Systems,Chapter4,Springer-Verlag, Berlin-Heidelberg,Germany(1999).[6]C.Kittel,Solid State Phys.22,1(1968);Solid StatePhys.23,184(1969);A.J.Heeger,Solid State Phys.23,248(1969).[7]P.Coleman et.al.,J.Cond.Matt.13,R723(2001).[8]K.Wilson and J.Kogut,Phys.Rep.C12,75(1974).[9]D.J.Gross and F.Wilczek,Phys.Rev.Lett.30,1343(1973);H.D.Politzer,Phys.Rev.Lett.30,1346(1973);D.J.Gross and F.Wilczek,Phys.Rev.D8,3633(1973).[10]D.J.Gross and A.Neveu,Phys.Rev.D10,3235(1974).[11]J.Zinn-Justin,Quantum Field Theory and Critical Phe-nomena,Fourth Edition,Oxford Univ.Press,Oxford, UK(2002).[12]S.Weinberg,Phys.Rev.D8,3497(1973).[13]C.G.Callan,Phys.Rev.D2,1541(1970);K.Symanzik,Comm.Math.Phys.18,227(1970).。