Optimal truncation procedures in renormalization-group calculations

Renormalization or rescaling transformations generally produce more complicated interactions than are present in the initial Hamiltonian. After each rescaling it is necessary to truncate the Hamiltonian to make the next rescaling mathematically tractable. One is faced with the problem of choosing the coupling constants of the truncated Hamiltonian to obtain the best approximation. Following ideas of McMillan, we consider truncation procedures which give lower and upper bounds to the free energy. Conditions for optimal lower- and upper-bound truncations are derived. These optimal truncations are seen to yield exact results for the free energy in both the high- and low-temperature limits. Some of the problems inherent in all renormalization transformations that incorporate an optimal lower- or upper-bound truncation are discussed. Calculations for the twodimensional Ising model based on renormalization transformations which combine decimation and an optimal truncation are described. Even in the simplest approximation in which only nearest-neighbor interactions are retained the free energy is obtained to an accuracy of better than 1% for all temperatures if an optimal truncation rather than an ordinary truncation with no readjustment of the coupling constants is made. However, the simplest calculations involving optimal truncations are less successful in predicting derivatives of the free energy and critical exponents than the free energy itself.     

Perturbative order analysis of the similarity transformed Hamiltonian in Fock-space coupled cluster theory: difference energy and electric response properties

A perturbative analysis of the ground-state similarity transformed Hamiltonian and its effect on the various Fock-space coupled cluster (FSCC) sectors is presented through calculation of ionisation potential, electron affinity, excitation energies and response properties. Various truncation schemes of the effective Hamiltonian are presented with explicit form of the defining equations. Based on such a truncation, the approximate methods are labelled as FSCC(n), where n represents the correlation energy of the ionised, electron attached or excited states corrected at least up to nth order within coupled cluster singles and doubles scheme (CCSD). A lower scaling CC2 type of approach (abbreviated as FS-CC2) is compared against the group of FSCC(n) methods for energies. Electric response properties have been compared and contrasted for the two lower scaling methods: FSCC(2) and FS-CC2. The various truncated methods are tested for a number of small molecules. The results obtained from a range of truncated methods are compared against full FSCCSD calculations.     

Anti-symmetry consideration on the preservation of entanglement of spin system

In this work we offer an approach to protect the entanglement based on the anti-symmetric property of the Hamiltonian. Our main objective is to protect the entanglement of a given initial three-qubit state which is governed by Hamiltonian of a three-spin Ising chain in site-dependent transverse fields. We show that according to anti-symmetric property of the Hamiltonian with respect to some operators mimicking the time reversal operator, the dynamics of the system can be effectively reversed. It equips us to control the dynamics of the system. The control procedure is implemented as a sequence of cyclic evolution; accordingly the entanglement of the system is protected for any given initial state with any desired accuracy and long-time. Using this approach we could control not only the multiparty entanglement but also the pairwise entanglement. It is also notable that in this paper although we restrict ourselves mostly within a three-spin Ising chain in site-dependent transverse fields, our approach could be applicable to any n$n$-qubit spin system models.     

Approach to Equilibrium for the Stochastic NLS

We study the approach to equilibrium, described by a Gibbs measure, for a system on a d-dimensional torus evolving according to a stochastic nonlinear Schrödinger equation (SNLS) with a high frequency truncation. We prove exponential approach to the truncated Gibbs measure both for the focusing and defocusing cases when the dynamics is constrained via suitable boundary conditions to regions of the Fourier space where the Hamiltonian is convex. Our method is based on establishing a spectral gap for the non self-adjoint Fokker-Planck operator governing the time evolution of the measure, which is uniform in the frequency truncation N. The limit N →∞ is discussed.     

Multiple-quantum dynamics of one-dimensional nuclear spin systems in solids

Multiple-quantum spin dynamics is studied using analytic and numerical methods for one-dimensional finite linear chains and rings of nuclear spins 1/2 coupled by dipole-dipole interactions. An approximation of dipole-dipole interaction between nearest neighbors having the same constants is used to obtain exact expressions for the intensities of the multiple-quantum coherences in the spin systems studied, which are initially in thermal equilibrium and whose evolution is described by a two-spin two-quantum Hamiltonian. An approximation of nearest neighbors with arbitrary dipole-dipole interaction constants is used to establish a simple relationship between the multiple-quantum dynamics and the dynamics of spin systems with an XY Hamiltonian. Numerical methods are developed to calculate the intensities of multiple-quantum coherences in multiple-quantum NMR spectroscopy. The integral of motion is obtained to expand the matrix of the two-spin two-quantum Hamiltonian into two independent blocks. Using the nearest-neighbor approximation the Hamiltonian is factorized according to different values of the projection operator of the total spin momentum on the direction of the external magnetic field. Results of calculations of the multiple-quantum dynamics in linear chains of seven and eight nuclear spins and a six-spin ring are presented. It is shown that the evolution of the intensities of the lowest-order multiple-quantum coherences in linear chains is accurately described allowing for dipole-dipole interaction of nearest and next-nearest neighbors only. Numerical calculations are used to compare the contributions of nearest and remote spins to the intensities of the multiple-quantum coherences.     

Many-body spin interactions and the ground state of a dense Rydberg lattice gas

We study a one-dimensional atomic lattice gas in which Rydberg atoms are excited by a laser and whose external dynamics is frozen. We identify a parameter regime in which the Hamiltonian is well approximated by a spin Hamiltonian with quasilocal many-body interactions which possesses an exact analytic ground state solution. This state is a superposition of all states of the system that are compatible with an interaction induced constraint weighted by a fugacity. We perform a detailed analysis of this state which exhibits a crossover between a paramagnetic phase with short-ranged correlations and a crystal. This study also leads us to a class of spin models with many-body interactions that permit an analytic ground state solution.     

Variations on the Kepler problem

The elliptical orbits resulting from Newtonian gravitation are generated with a multifaceted symmetry, mainly resulting from their conservation of both angular momentum and a vector fixing their orientation in space—the Laplace or Runge-Lenz vector. From the ancient formalisms of celestial mechanics, I show a rather counterintuitive behavior of the classical hydrogen atom, whose orbits respond in a direction perpendicular to a weak externally-applied electric field. I then show how the same results can be obtained more easily and directly from the intrinsic symmetry of the Kepler problem. If the atom is subjected to an oscillating electric field, it enjoys symmetry in the time domain as well, which is manifest by quasi-energy states defined only modulo ħω. Using the Runge-Lenz vector in place of the radius vector leads to an exactly-solvable model Hamiltonian for an atom in an oscillating electric field—embodying one of the few meaningful exact solutions in quantum mechanics, and a member of an even more exclusive set of exact solutions having a time-dependent Hamiltonian. I further show that, as long as the atom suffers no change in principal quantum number, incident radiation will produce harmonic radiation with polarization perpendicular to the incident radiation. This unusual polarization results from the perpendicular response of the wavefunction, and is distinguished from most usual harmonic radiation resulting from a scalar nonlinear susceptibility. Finally, I speculate on how this radiation might be observed.     

From coherently excited highly correlated states to incoherent relaxation processes in semiconductors

Recent theories of highly excited semiconductors are based on two formalisms, referring to complementary experimental conditions, the real-time nonequilibrium Green's function techniques and the coherently controlled truncation of the many-particle problem. We present a novel many-particle theory containing both of these methods as limiting cases. As a first example of its application, we investigate four-particle correlations in a strong magnetic field including dephasing resulting from the growth of incoherent one-particle distribution functions. Our results are the first rigorous solution concerning formation and decay of four-particle correlations in semiconductors. They are in excellent agreement with experimental data.     

Observer theories based on stueckelberg quations of motion

Stueckelberg dynamics is regarded as providing a basis for the construction of observer centered theories of particle motions. The approach involves the use of a generalized Jacobi principle to replace the four-dimensional dynamical theory of Stueckelberg by a four-dimensional geometrical theory, and then a three-dimensional dynamics is constructed from this. The causal difficulties encountered by Stueckelberg for curves which reverse direction in time appear to be absent in the present scheme.Our purpose has been to make more concrete, in a simple context, some of the ideas involved in the (conventional) causal framework recently constructed by us to deal with causal difficulties associated with hyperlight phenomena. Some insight is gained into the possible roles to be played by tachyons in a particle theory and interesting results are found involving classical Lagrangian and canonical formalisms for lightlike particles.     

Effects of long-range interactions on the dynamics of ions in aqueous solution

Molecular dynamics simulations have been carried out for a Br⁻ ion in aqueous solution in order to establish the effect of truncation of long-range interactions on the dynamical properties of the ion. Simulations using smooth truncation of the potential at different cutoff radii were carried out and compared to results using the Ewald summation method. It is shown that when small cutoffs are applied (i.e. R_c = 8Å), the calculations yield low ionic diffusion coefficients relative to experiment, as well as short-time dynamical behavior which is inconsistent with the Ewald calculations. As the cutoff is increased, the results approach both the Ewald and the experimental results. In contrast to the results with the truncated potential, the short-time dynamical behavior of the hydrated bromide ion obtained with the Ewald method can be described by simple Langevin dynamics.     

Irreversible quantum mechanics in the neutralK-system

The neutral kaon system is used to test the quantum theory of resonance scattering and decay phenomena. The two dimensional Lee-Oehme-Yang theory with complex Hamiltonian is obtained by truncating the complex basis vector expansion of the exact theory in rigged Hilbert space. This can be done forK ₁ andK ₂ as well as forK _S andK _L , depending upon whether one chooses the (self-adjoint, semibounded) Hamiltonian as commuting or noncommuting withCP. As an unexpected curiosity, one can show that the exact theory (without truncation) predicts long-time 2π decays of the neutral kaon system even if the Hamiltonian conservesCP.     

New Accurate Fit of an Extended Set of Saturation Data for the nu(3) Band of SF(6): Comparison of Hamiltonians in the Spherical and Cubic Tensor Formalisms

An extended set of 321 frequencies of vibration-rotation lines of the nu(3) band of SF(6) has been measured by saturation spectroscopy using various isotopic species of CO(2). A least-squares fit of these data has been performed using an effective Hamiltonian written either with a spherical tensor or with a cubic tensor formalism. We have derived correspondence formulas between the parameters in the two approaches and checked that both formalisms give the same results up to the seventh order. Corrected parameters are given for the fit with a fifth-order Hamiltonian. An accurate representation of the band is obtained at the tenth order (standard deviation approximately 12 kHz) with a remarkable predictive power (better than 40 kHz) for J values 相似文献   

Variational approach to configuration interaction

A variational method is developed to determine configuration interaction wave functions. The method is straightforward and is applied to a pairing Hamiltonian with constant matrix elements, for which exact eigenvalues are available. Comparisons are made with the exact results. Calculations can be carried out to any desired degree of accuracy. The method is also applied to a Hamiltonian that has neutron-neutron, proton-proton, and neutron-proton pairing. No difficulties are found in extending the method to this Hamiltonian that has many collective modes. In practice, the method scales linearly with , where is the number of variational basis states.     

耗散子运动方程组的可控截断

为耗散子运动方程组提出精度可控的预截断方案来模拟和研究非微扰的量子耗散动力学．发展了具有类似Caldeira-Leggett主方程形式、在终止阶层采取马尔科夫和高温近似方法的截断方案,根据马尔科夫性质的分析给出确定终止层数的精度判据,并通过电荷转移体系的动力学模拟进行了检验．     

Diagonalization of the Rabi Hamiltonian by means of a unitary transformation

A unitary transformation is derived which diagonalizes the Rabi Hamiltonian. While the solution of this diagonalization problem by standard methods has long been known, it is found that the unitary operator is of a form which has previously not been used in the connexion with spin-boson problems. The transformed Hamiltonian as well as some other characteristic transformed operators are calculated. A comparison is made between the exact unitary operator and a weak coupling transformation which is equivalent to second order perturbation theory.     

电子储存环中插入元件的非线性束流动力学的Lie代数方法

介绍了用能保证哈密顿系统“辛”性质的Lie映射方法来研究电子储存环中的插入元件对束流动力学的影响.并以此方法对合肥光源中将要安装的波荡器UD?–?1对储存环的影响作了初步的研究     

Cosmological solutions of time varying speed of light theories

We consider scalar–tensor theory for describing varying speed of light in a spatially flat FRW space–time. We find some exact solutions in the metric and Palatini formalisms. Also we examine the dynamics of this theory by dynamical system method assuming a ΛCDM background and we find some exact solutions by considering the character of critical points of the theory in both formalisms. We show that for any attractor the form of non-minimal coupling coefficient is quadratic in terms of the scalar field Ψ. Also we show that only attractors of the de Sitter era satisfy the horizon criteria.     

Convergence properties of the cluster expansion for equal- time Green functions in scalar theories

We investigate the convergence properties of the cluster expansion of equal-time Green functions in scalar theories with quartic self-coupling in (0 + 1), (1 + 1), and (2 + 1) space-time dimensions. The computations are carried out within the equal-time correlation dynamics approach, which consists in a closed set of coupled equations of motion for connected Green functions as obtained by a truncation of the BBGKY hierarchy. We find that the cluster expansion shows good convergence as long as the system is in a localized state (single phase configuration) and that it breaks down in a non-localized state (two phase configuration), as one would naively expect. Furthermore, in the case of dynamical calculations with a time dependent Hamiltonian for the evaluation of the effective potential we find two timescales determining the adiabaticity of the propagation; these are the time required for adiabaticity in the single phase region and the time required for tunneling into the non-localized lowest energy state in the two phase region. Our calculations show a good convergence for the effective potentials in (1 + 1) and (2+1) space-time dimensions since tunneling is suppressed in higher space-time dimensions.