Exactly solvable two-dimensional quantum spin models

A method is proposed for constructing an exact ground-state wave function of a two-dimensional model with spin 1/2. The basis of the method is to represent the wave function by a product of fourth-rank spinors associated with the nodes of a lattice and the metric spinors corresponding to bonds between nearest neighbor nodes. The function so constructed is an exact wave function of a 14-parameter model. The special case of this model depending on one parameter is analyzed in detail. The ground state is always a nondegenerate singlet, and the spin correlation functions decay exponentially with distance. The method can be generalized for models with spin 1/2 to other types of lattices. Zh. éksp. Teor. Fiz. 115, 249–267 (January 1999)     

Variational approach to spin fluctuations in the Hubbard model

We study a one parameter variational wave function to improve the spin density wave ground state of the Hubbard model by inclusion of quantum spin fluctuations. Using a perturbative approach and novel lattice summation techniques we present analytical as well as numerical results for the correlation energies and the staggered magnetizations in one and two dimensions. We find ground state energies which are satisfyingly close to known exact results and are significantly lower than those of existing Gutzwiller and numerical treatments.     

Cluster-variational calculations for extended nuclear systems

The energy as a function of density is calculated for neutron matter and for symmetrical nuclear matter, based on Jastrow trial wave functions. The energy expectation value is truncated in low cluster order. A detailed analysis of the two- and three-body cluster contributions and a special portion of the four-body contribution is given. Variation of a parameterized two-body correlation function is subjected to constraints designed to confine the trial wave function to the domain corresponding to rapid cluster convergence. Results are presented for a variety of model central potentials containing hard cores, for different sets of constraints, and for two- and three-parameter correlation functions. Calculations constrained by the “average Pauli condition” are found to yield results very close to those constrained by the “normalization” or “unitarity” condition, and the two-parameter correlation function appears to be quite adequate. The convergence of the cluster expansion, as reflected in the low orders, is good except at the highest densities considered. The three-body cluster contribution displays, in all cases, a remarkable internal cancellation between its “two-correlation-line” addends.     

Electrons in a strong magnetic field on a disk

The problem of interacting electrons moving under the influence of a strong magnetic field in two dimensions on a finite disk is reconsidered. First, the results of exact diagonalizations for up to N = 9 electrons for Coulomb as well as for a short-range interaction are used in the search for a peculiar ground state corresponding to filling factor 1/3. Not for the Coulomb, but only for the short-range interaction, can the 1/3-state be safely identified amongst the spectra of various filling factors close to 1/3. Second, the propositions of the concept of quasiparticles, as used in the hierarchical theory, are examined in view of the exact results for the disk geometry. Whereas the theory for the quasiholes is in complete accordance with the spectra, for the quasielectrons, finite size corrections make an analysis difficult. For the quasielectron energy, an extrapolation to N→ ∞ is given and compared with the corresponding extrapolations of three different proposals for trial wave functions. While the limiting value for the best trial wave function is very close to the limit of the exact results, the behavior of the finite size corrections of the exact energies and of the trial wave functions, respectively, is qualitatively rather different.     

Annihilation of positronium atoms in an oscillatory field

Annihilation parameters (the angular correlation curve and positronium lifetime) are calculated for annihilation of positronium atoms in a three-dimensional oscillatory well. This well can serve as a model of a positron trap in rigid bodies. The model lends itself to exact mathematical analysis, and it admits separation of variables of the center-of-mass motion and the relative motion of the particle. A calculation is given for the wave functions of a positronium, which oscillatory field. The wave function of the relative motion of the particles at small distances is similar to the wave function of free positronium, which gives us a basis to speak of the annihilation of an electron-positron pair as the annihilation of a positronium atom in an oscillatory field. With a decrease in the size of the trap, broadening of the correlation curve occurs, which has Gaussian form for the annihilation of positronium from the minimum condition for center-of-mass motion. The bound state of the electron and positron exists even in traps of the size of the Bohr radius. The model permits evaluation of the size of traps over an experimentally chosen narrow component of the correlation curve.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 82–87, March, 1989.     

Can the eigenstates of a many-body Hamiltonian be represented exactly using a general two-body cluster expansion?

It is proved that a recent conjecture that the exact ground-state wave function of an arbitrary many-fermion system with one- and two-body interactions may be represented by an exponential cluster expansion employing finite two-body operators, starting from any reference function sufficiently close to the exact eigenfunction, is not valid. We show that the space of initial reference functions which lead to the exact ground state is of dimension equal to the number of two-body operators. If the dimension of the multiparticle space is greater than the number of two-body operators, then the space of good reference functions is of measure zero in it.     

Lagrange method and boundary conditions for the intensity correlation functions in turbulent media

An exact functional integral representation for the two-point intensity correlation function was previously obtained by the author for a collimated beam wave by solving the moment equation. The variable functions of integration involved therein can be effectively limited to a set of functions determined so that the entire phase term of the integrand becomes stationary against arbitrary variation of the variable functions, exactly according to the Lagrange variational principle in dynamics. The result is free from any expansion and is presented with a set of unperturbed equations of closed form. When making a formal expansion, it leads to the zeroth- and first-order expressions similar to those obtained by an improved two-scale method. With exactly the same procedure, the three-point intensity correlation and the two-frequency intensity correlation were also obtained.The Lagrange method leads to the 'equation of motion' subjected to boundary conditions to continue the phase term from the incident beam wave. The boundary conditions were previously found based on a physical reasoning, while the same conditions are found here purely based an the variational principle. A focused beam wave is assumed for the incident wave, including both spherical and plane waves as special cases.     

浅海波导中目标散射的边界元方法

提出了一种计算三维散射体在声速剖面随深度变化、距离无关浅海波导中散射声场的数值方法波导边界元方法。当散射体不十分靠近波导界面因而边界多次散射可以忽略时,在边界元计算中可以用自由场格林函数近似波导格林函数。应用镜像法和球波函数加法定理推导了理想波导中球体散射声场的解析解,用来验证波导边界元方法的计算精度,证明该数值方法是准确的。对浅海波导中水下潜器散射声场数值模拟的结果表明,浅海波导海面、海底界面反射、声速剖面等对目标散射声场的幅值和方向性都有很大的影响。      

On the calculation of densities of light nuclei

For certain classes of wave functions one can calculate the exact expressions of the charge distribution and similar densities. One has to make use of the asymptotic behaviour of these wave functions (whose radial parts have to be of the form \(r^{2k} e^{ - \beta r^2 } \) , in order to obtain a closed form forp(r); the problem is then reduced to the computation of moments, which can be evaluated by a generating function technique. No complications due to a spurious center-of-mass motion do appear. The method is most appropriate for the calculation of densities of refined cluster model functions or translationally invariant shell model functions (both for a nucleon or an alpha-particle shell model) including short-range correlation factors that take care of a soft core in nuclear potentials.     

Hole states in helium-like ions

An analysis is made of the poles of the single-particle Green's function of helium-like ions in which electron correlation is taken into account on the basis of the model of singly filled orbitals. The energy of Hartree-Fock pole calculated through the use of the hydrogen-like wave functions for the orbitals turns out to be approximately equal to the exact value. The second pole, which corresponds to a different hole state, has a residue which vanishes as z increases and must be considered fortuitous.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 90–93, April, 1971.     

Some results in the theory of interacting electron systems

The expressions for the longitudinal dielectric function, spin and orbital susceptibilities in the static, long wavelength limit are evaluated by solving the corresponding linearized vertex functions exactly in this limit. The plasma dispersion relation to leading order in the long wave limit is similarly obtained. These are compared with the corresponding results obtained previoulsy by us by a variational solution to the same vertex equations. It is established that the variational method gives the exact results in the static, zero wave vector limit, involving the proper renormalizations. The plasma dispersion relation is found to be the same as in the exact calculation whereas the coefficient of q² in the static density correlation function has an important additional contribution to the variational result. Applications of these results are briefly discussed.     

Lagrange method and boundary conditions for the intensity correlation functions in turbulent media*

Abstract

An exact functional integral representation for the two-point intensity correlation function was previously obtained by the author for a collimated beam wave by solving the moment equation. The variable functions of integration involved therein can be effectively limited to a set of functions determined so that the entire phase term of the integrand becomes stationary against arbitrary variation of the variable functions, exactly according to the Lagrange variational principle in dynamics. The result is free from any expansion and is presented with a set of unperturbed equations of closed form. When making a formal expansion, it leads to the zeroth- and first-order expressions similar to those obtained by an improved two-scale method. With exactly the same procedure, the three-point intensity correlation and the two-frequency intensity correlation were also obtained.The Lagrange method leads to the ‘equation of motion’ subjected to boundary conditions to continue the phase term from the incident beam wave. The boundary conditions were previously found based on a physical reasoning, while the same conditions are found here purely based an the variational principle. A focused beam wave is assumed for the incident wave, including both spherical and plane waves as special cases.     

Tunneling and energy splitting in an asymmetric double-well potential

An asymmetric double-well potential is considered, assuming that the minima of the wells are quadratic with a frequency ω and the difference of the minima is close to a multiple of ?ω. A WKB wave function is constructed on both sides of the local maximum between the wells, by matching the WKB function to the exact wave functions near the classical turning points. The continuities of the wave function and its first derivative at the local maximum then give the energy-level splitting formula, which not only reproduces the instanton result for a symmetric potential, but also elucidates the appearance of resonances of tunneling in the asymmetric potential.     

Comparison of variational approaches for the exactly solvable 1/r-Hubbard chain

We study Hartree-Fock, Gutzwiller, Baeriswyl, and combined Gutzwiller-Baeriswyl wave functions for the exactly solvable one-dimensional 1/r-Hubbard model. We find that none of these variational wave functions is able to correctly reproduce the physics of the metal-to-insulator transition which occurs in the model for halffilled bands when the interaction strength equals the bandwidth. The many-particle problem to calculate the variational ground state energy for the Baeriswyl and combined Gutzwiller-Baeriswyl wave function is exactly solved for the 1/r-Hubbard model. The latter wave function becomes exact both for small and large interaction strength, but it incorrectly predicts the metal-to-insulator transition to happen at infinitely strong interactions. It is thus seen that neither Hartree-Fock nor an energetically excellent Jastrow-type wave function yield a reliable prediction on the zero temperature phase transition in the one-dimensional 1/r-Hubbard chain.     

The extended Hubbard model in the ionic limit

In this paper, we study the Hubbard model with intersite Coulomb interaction in the ionic limit (i.e. no kinetic energy). It is shown that this model is isomorphic to the spin-1 Ising model in presence of a crystal field and an external magnetic field. We show that for such models it is possible to find, for any dimension, a finite complete set of eigenoperators and eigenvalues of the Hamiltonian. Then, the hierarchy of the equations of motion closes and analytical expressions for the relevant Green's functions and correlation functions can be obtained. These expressions are formal because these functions depend on a finite set of unknown parameters, and only a set of exact relations among the correlation functions can be derived. In the one-dimensional case we show that by means of algebraic constraints it is possible to obtain extra equations which close the set and allow us to obtain a complete exact solution of the model. The behavior of the relevant physical properties for the 1D system is reported.     

Maximum entropy approach to the coupled quadrupole system

The spectral functions of the spin pair correlation functions in a coupled quadrupole system are evaluated using the maximum entropy formalism starting from the exact derivation of the frequency moments to fourth order at the high temperature limit. The numerical computations are carried out for different wave vectors and ratios of the anisotropic field to the quadrupolar interaction. The effects of these parameters on the spectral function are discussed.     

Resonating valence bond wave functions for strongly frustrated spin systems

The resonating-valence-bond (RVB) theory for two-dimensional quantum antiferromagnets is shown to be the correct paradigm for large enough "quantum frustration." This scenario, proposed a long time ago but never confirmed by microscopic calculations, is strongly supported by a new type of variational wave function, which is extremely close to the exact ground state of the J(1)-J(2) Heisenberg model for 0.4 less than approximately J(2)/J(1) less than approximately 0.5. This wave function is proposed to represent the generic spin-half RVB ground state in spin liquids.     

Comparison of several equivalent local potentials for microscopic nonlocal Nα potentials

The phase shifts obtained in the Perey-Saxon, Horiuchi and Peierls-Vinh Mau approximations for the microscopic nonlocal Nα interaction of Lassaut and Vinh Mau have been compared at 10 and 20 MeV to the exact results, without the customary approximation of Perey and Buck, which neglects part of the angular dependence of the nonlocality and is inadequate. We have also compared the equivalent local potentials with the corresponding local wave functions and damping factors to their exact wronskian counterparts and obtained the corresponding nonlocal wave functions as a product of the approximate local wave function and the damping factor in all cases. The results show that of all the approximations the Peierls-Vinh Mau approximation is the most accurate one for s-waves and that the Perey-Saxon approximation is inadequate for Nα scattering. The accuracy of all the approximations is dependent on the degree of repulsiveness of the effective nuclear force. For the exact wronskian equivalent local potential and damping factor however, we reproduce the exact nonlocal wave function to high accuracy in all cases, confirming the accuracy of our numerical methods. The implications for nuclear reactions where the off-shell behaviour of the Nα interaction is important are discussed.     

Probability of boundary conditions in quantum cosmology

One of the main interest in quantum cosmology is to determine boundary conditions for the wave function of the universe which can predict observational data of our universe. For this purpose, we solve the Wheeler–DeWitt equation for a closed universe with a scalar field numerically and evaluate probabilities for boundary conditions of the wave function of the universe. To impose boundary conditions of the wave function, we use exact solutions of the Wheeler–DeWitt equation with a constant scalar field potential. These exact solutions include wave functions with well known boundary condition proposals, the no-boundary proposal and the tunneling proposal. We specify the exact solutions by introducing two real parameters to discriminate boundary conditions, and obtain the probability for these parameters under the requirement of sufficient e-foldings of the inflation. The probability distribution of boundary conditions prefers the tunneling boundary condition to the no-boundary boundary condition. Furthermore, for large values of a model parameter related to the inflaton mass and the cosmological constant, the probability of boundary conditions selects an unique boundary condition different from the tunneling type.