Matrix Theory in Curved Space

We study curved-space versions of matrix stringtheory taking as a definition of the theory a gaugedmatrix sigma model. By computing the contribution to theone-loop divergent terms in the effective action coming from the diagonal matrix elements weshow that these versions of matrix theory in curvedspace reproduce the string equations of motion and theR⁴ correction to the Hilbert-Einstein action.It is then demonstrated that the divergences due tothe nondiagonal elements induce terms in the effectiveaction that cannot be removed by appropriatecounterterms. This implies that the model can only beconsistent for Ricci flat manifolds with vanishingsix-dimensional Euler density.     

Scattering Matrix in Quantum Field Theory

We show that a relativistically covariant, unitary, and causative scattering matrix, which is finite in each order of perturbation theory, can be constructed using fundamental postulates of quantum field theory and give a physical interpretation of the results obtained.     

The Orbit Matrix and The Orbit Diagram in Glauber Multiple Scattering Theory

In the present article the Glauber multiple scattering theory (GMST) is briefly reviewed, and a practical method of erecting the whole set of orbit diagram/matrix including the trick of compiling the computer program is proposed. It is a rather difficult problem which has never been solved before in the application of GMST. Taken as an example, the α-α collision process is calculated, and the result well fits thetexisting experimental values.     

A Random Matrix Model From Two Dimensional Yang-Mills Theory

We use fat graphs to give a unified treatment of various asymptotic freeness results. In particular, a random matrix model from two dimensional Yang-Mills Theory on the plane is presented. Received: 10 June 1996 / Accepted: 6 April 1997     

A Theory of Electronic Energy Transfer in Complex Molecular Systems

The Förster theory for energy transfer is critically reviewed in the context of the present-day theory of nonradiative transitions, and the fundamental tenets of the Förster theory are shown to be erroneous. A new theory of electronic energy transfer is constructed taking into account the electronic transition theory. The intermolecular interaction between donor and acceptor molecules is assumed to perturb electron states of isolated molecules before the donor-molecule excitation. A distinguishing characteristic of the intermolecular interaction is spatial delocalization of the wave functions of electron states of the interacting molecules. It is this fact that has made it possible to realize ordinary photophysical processes between electron states of various molecules in a bimolecular system. In the experiments under study, the result of the intermolecular nonradiative photoprocess is given as evidence of electronic energy transfer from the donor molecule to the acceptor molecule.     

量子不可积性和随机矩阵理论

本文概述了量子不可积性的拓扑观.按拓扑观点探讨了量子体系完全可积的条件,通向不可积的机制,以及在量子不可积条件下的能谱性质,从动力学角度阐明了随机矩阵理论的基础,还简要讨论了这些理论在核结构问题中的应用. Quantum nonintegrability is studied in a topological view.The condition for complete integrability of quantum systems and the mechanism leading to global nonintegrability are investigated.The basis of random matrix theory for describing energy spectra of nonintegrable quantum systems is clarified.Applications to nuclear structure are briefly discussed.     

Density Matrix Theory for the BDS-Hamiltonian

In the present paper, we have extended the usual uncertainty relations, as well as the entropic uncertainty relations to the mixed states, particularly to the thermal states, using the density matrix formalism. As quantum model we have choosen the quantum mechanical ideal gas with the harmonic oscillator-like Hamiltonian H, introduced by Beckers, Debergh and Szafraniec, generically named BDS-Hamiltonian.     

Random Matrix Theory and Entanglement in Quantum Spin Chains

We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians — those that are related to quadratic forms of Fermi operators — between the first N spins and the rest of the system in the limit of infinite total chain length. We show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given Hamiltonian and those characterizing the Haar measure of the associated group. These averages are either Toeplitz determinants or determinants of combinations of Toeplitz and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture are used to compute the leading order asymptotics of the entropy as N. This is shown to grow logarithmically with N. The constant of proportionality is determined explicitly, as is the next (constant) term in the asymptotic expansion. The logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal-field-theoretic calculations. In these calculations the constant of proportionality was determined in terms of the central charge of the Virasoro algebra. Our results therefore lead to an explicit formula for this charge. We also show that the entropy is related to solutions of ordinary differential equations of Painlevé type. In some cases these solutions can be evaluated to all orders using recurrence relations.Acknowledgement We gratefully acknowledge stimulating discussions with Estelle Basor, Peter Forrester and Noah Linden. We are also grateful for the kind hospitality of the Isaac Newton Institute for the Mathematical Sciences, Cambridge, while this research was completed. Francesco Mezzadri was supported by a Royal Society Dorothy Hodgkin Research Fellowship.     

Random Matrix Theory and the Anderson Model

This paper is devoted to a discussion of possible strategies to prove rigorously the existence of a metal-insulator Anderson transition for the Anderson model in dimension d≥3. The possible criterions used to define such a transition are presented. It is argued that at low disorder the lowest order in perturbation theory is described by a random matrix model. Various simplified versions for which rigorous results have been obtained in the past are discussed. It includes a free probability approach, the Wegner n-orbital model and a class of models proposed by Disertori, Pinson, and Spencer, Comm. Math. Phys. 232:83–124 (2002). At last a recent work by Magnen, Rivasseau, and the author, Markov Process and Related Fields 9:261–278 (2003) is summarized: it gives a toy modeldescribing the lowest order approximation of Anderson model and it is proved that, for d=2, its density of states is given by the semicircle distribution. A short discussion of its extension to d≥3 follows.     

Matrix Product States, Random Matrix Theory and the Principle of Maximum Entropy

Using random matrix techniques and the theory of Matrix Product States we show that reduced density matrices of quantum spin chains have generically maximum entropy.     

关于分子磁性理论的研究

已建立的一种研究过渡族元素化合物分子磁性的DSF理论方法 ,在一定范围内解决了分子磁性理论研究中如何将物理模型付诸理论计算的问题 ,由此研究了一些弱共价的同核分子体系 ,首次揭示了这些分子的铁磁性是起源于分子内相邻过渡金属离子轨道间的交叉相互作用 ,反铁磁性则源于平行相互作用 .通过发展GJK模型 ,又建立了能描述异核分子和共价分子的磁性理论 ,并由此分析和解释了多种无机、有机化合物和生物蛋白分子中的磁性起源 ;特别是对核糖核酸还原酶中的强共价体系的强反铁磁现象给出了合理的理论解释. Recently, we have established a DSF theoretical method suitable for researching molecular magnetism of the compounds consisting of transition group elements. By this method, we have revealed that the ferromagnetism of molecules is due to the cross-interaction between d orbitals of adjacent transition-metal ions, and that the antiferromagnetism is due to the parallel interactions. Further more, we have also established a magnetism theory for heterodinuclear...     

Random Matrix Theory and the Boson Peak in Two-Dimensional Systems

Physics of the Solid State - The random matrix theory is used to describe the vibrational properties of two-dimensional disordered systems with a large number of degrees of freedom. It is shown...     

Combinatorics of Dispersionless Integrable Systems and Universality in Random Matrix Theory

It is well-known that the partition function of the unitary ensembles of random matrices is given by a τ-function of the Toda lattice hierarchy and those of the orthogonal and symplectic ensembles are τ-functions of the Pfaff lattice hierarchy. In these cases the asymptotic expansions of the free energies given by the logarithm of the partition functions lead to the dispersionless (i.e. continuous) limits for the Toda and Pfaff lattice hierarchies. There is a universality between all three ensembles of random matrices, one consequence of which is that the leading orders of the free energy for large matrices agree. In this paper, this universality, in the case of Gaussian ensembles, is explicitly demonstrated by computing the leading orders of the free energies in the expansions. We also show that the free energy as the solution of the dispersionless Toda lattice hierarchy gives a solution of the dispersionless Pfaff lattice hierarchy, which implies that this universality holds in general for the leading orders of the unitary, orthogonal, and symplectic ensembles. We also find an explicit formula for the two point function F _nm which represents the number of connected ribbon graphs with two vertices of degrees n and m on a sphere. The derivation is based on the Faber polynomials defined on the spectral curve of the dispersionless Toda lattice hierarchy, and \frac1nmF_nm{\frac{1}{nm}F_{nm}} are the Grunsky coefficients of the Faber polynomials.     

Matrix Isolation Laser Raman Spectroscopy: The Raman Spectrum and Molecular Structure of Xenon Dichloride

The technique of matrix isolation infrared spectroscopy has proven on numerous occasions to be an extremely powerful tool for determining vibrational frequencies and molecular structures of highly reactive species. The need for reliable Raman data on comparable systems is obvious.     

Random Tensor Theory: Extending Random Matrix Theory to Mixtures of Random Product States

We consider a problem in random matrix theory that is inspired by quantum information theory: determining the largest eigenvalue of a sum of p random product states in \({(\mathbb {C}^d)^{\otimes k}}\), where k and p/d ^k are fixed while d → ∞. When k = 1, the Mar?enko-Pastur law determines (up to small corrections) not only the largest eigenvalue (\({(1+\sqrt{p/d^k})^2}\)) but the smallest eigenvalue \({(\min(0,1-\sqrt{p/d^k})^2)}\) and the spectral density in between. We use the method of moments to show that for k > 1 the largest eigenvalue is still approximately \({(1+\sqrt{p/d^k})^2}\) and the spectral density approaches that of the Mar?enko-Pastur law, generalizing the random matrix theory result to the random tensor case. Our bound on the largest eigenvalue has implications both for sampling from a particular heavy-tailed distribution and for a recently proposed quantum data-hiding and correlation-locking scheme due to Leung and Winter.Since the matrices we consider have neither independent entries nor unitary invariance, we need to develop new techniques for their analysis. The main contribution of this paper is to give three different methods for analyzing mixtures of random product states: a diagrammatic approach based on Gaussian integrals, a combinatorial method that looks at the cycle decompositions of permutations and a recursive method that uses a variant of the Schwinger-Dyson equations.     

有机场致发光中能带模型与分子理论的讨论

在有机场致发光中,能带模型及分子理论从20世纪就存在尖锐的矛盾。在分层优化方案中,经SiO₂加速后的电子能量可以到达10eV,这足以激发发光材料发光,将分层优化方案应用到有机场致发光材料中。发现了固态阴极射线发光(SSCL),经过对它的交叉证明、普适性的验证,肯定了固态阴极射线的发光确实是在发光二极管,无机及有机场致发光之外的一种完全新型的电场诱导的发光。SSCL的特征是在它的光谱中出现短波发光峰,实验证明长波发光峰的减弱是由于电场离化效应。研究了这种效应出现的电压阈值并和SSCL的短波峰出现的电压相比,发现短波峰的出现是在激子的电场离化之后,从而找出了电子处于局域态与扩展态的分水岭,解释了在有机场致发光中能带模型和分子理论并不矛盾,只是适用的条件不同。激子的离化是随电场强度而渐进的变化,因此会有一个两种过程并存的范围。