Projected Wave Functions Research Articles

Gutzwiller–RVB theory of high-temperature superconductivity: Results from renormalized mean-field theory and variational Monte Carlo calculations

We review the resonating valence bond (RVB) theory of high-temperature superconductivity using Gutzwiller projected wave functions that incorporate strong correlations. After a general overview of the phenomenon of high-temperature superconductivity, we discuss Anderson's RVB picture and its implementation by renormalized mean-field theory (RMFT) and variational Monte Carlo (VMC) techniques. We review RMFT and VMC results with an emphasis on recent developments in extending VMC and RMFT techniques to excited states. We compare results obtained from these methods with angle-resolved photoemission spectroscopy (ARPES) and scanning tunnelling microscopy (STM). We conclude by summarizing recent successes of this approach and discuss open problems that need to be solved for a consistent and complete description of high-temperature superconductivity using Gutzwiller projected wave functions.

Quantum algebra ofNsuperspace

We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincar\'e algebra in the $N>1$ and $D=4$ superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for $D=4$, $N=1$ reported recently. For the case $N=2$ with central charges, we present the equivalent results when the central charge and the mass are different. For the $\ensuremath{\kappa}$-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra.

Beyond the relativistic mean-field approximation: Configuration mixing of angular-momentum-projected wave functions

We report the first study of restoration of rotational symmetry and fluctuations of the quadrupole deformation in the framework of relativistic mean-field models. A model is developed which uses the generator coordinate method to perform configuration mixing calculations of angular momentum projected wave functions, calculated in a relativistic point-coupling model. The geometry is restricted to axially symmetric shapes, and the intrinsic wave functions are generated from the solutions of the constrained relativistic mean-field + BCS equations in an axially deformed oscillator basis. A number of illustrative calculations are performed for the nuclei 194Hg and 32Mg, in comparison with results obtained in non-relativistic models based on Skyrme and Gogny effective interactions.

Evaluation of matrix elements in partially projected wave functions

We generalize the Gutzwiller approximation scheme to the calculation of nontrivial matrix elements between the ground state and excited states. In our scheme, the normalization of the Gutzwiller wave function relative to a partially projected wave function with a single non projected site (the reservoir site) plays a key role. For the Gutzwiller projected Fermi sea, we evaluate the relative normalization both analytically and by variational Monte-Carlo (VMC). We also report VMC results for projected superconducting states that show novel oscillations in the hole density near the reservoir site.

Projected wave functions for fractionalized phases of quantum spin systems

The Gutzwiller projection allows the construction of an assortment of variational wave functions for strongly correlated systems. For quantum spin $S=1/2$ models, Gutzwiller-projected wave functions have resonating-valence-bond structure and may represent states with fractional quantum numbers for the excitations. Using insights obtained from field-theoretical descriptions of fractionalization in two dimensions, we construct candidate wave functions of fractionalized states by projecting specific superconducting states. We explicitly demonstrate the presence of topological order in these states.

Projected wave functions and high temperature superconductivity.

We study the Hubbard model with parameters relevant to cuprates using variational Monte Carlo for projected d-wave states. For doping 0<x approximately less than or equal to 0.35 we obtain a superconductor whose order parameter Phi(x) tracks the observed nonmonotonic T(c)(x). The variational parameter Delta(var)(x) scales with the (pi,0) "hump" and T* seen in photoemission. Projection leads to incoherence in the spectral function and from the singular behavior of its moments we obtain the nodal quasiparticle weight Z approximately x though the Fermi velocity remains finite as x-->0. The Drude weight D(low) and superfluid density are consistent with experiment and D(low) approximately Z.

Configuration mixing of mean-field wave functions projected on angular momentum and particle number: Application to [formula omitted]Mg

We present in this paper the general framework of a method which permits to restore the rotational and particle number symmetries of wave functions obtained in Skyrme HF + BCS calculations. This restoration is nothing but a projection of mean-field intrinsic wave functions onto good particle number and good angular momentum. The method allows us also to mix projected wave functions. Such a configuration mixing is discussed for sets of HF + BCS intrinsic states generated in constrained calculations with suitable collective variables. This procedure gives collective states which are eigenstates of the particle number and the angular momentum operators and between which transition probabilities are calculated. An application to 24 Mg is presented, with mean-field wave functions generated by axial quadrupole constraints. Theoretical spectra and transition probabilities are compared to the experiment.

Spin-charge separation at small length scales in the two-dimensional t-J model.

We consider projected wave functions for the two-dimensional t-J model. For various wave functions, including correlated Fermi-liquid and Luttinger-type wave functions, we present the static charge-charge and spin-spin structure factors. Comparisons with recent results from a high-temperature expansion by Putikka et al. indicates spin-charge separation at small length scales.

Criterion for a good variational wave function.

The variance of the Hamiltonian in a given variational wave function measures how good an eigenstate this wave function is. In some instances, as for the two-dimensional antiferromagnetic Heisenberg Hamiltonian (2D AFH), the energy expectation value is not enough to distinguish between different trial Ansa$iuml---tze. Here we propose the variance as a simple criterion, which allows for further differentiation between degenerate trial wave functions. We show that this criterion establishes the projected wave functions as candidates for the ground state of the 2D AFH. A strong interference effect is discovered in computer experiment.

Variational evaluation of correlation functions for lattice electrons in high dimensions

We present a general formalism for the diagrammatic calculation of correlation functions for Hubbard-type models in terms of projected wave functions. It is shown that in the limit of high spatial dimensionsd only diagrams with bubble-structure remain. This causes correlation functions to have an overall RPA-type form ind→∞. Exact evaluations are performed for the Gutzwiller wave function. Nearest neighbor correlations are shown to be proportional to their value in the non-interacting case, i.e. are renormalized. However, their absolute value is only of order 1/d. Hence this wave function does not describe spin correlations adequately in high dimensions. The asymptotic behavior of the spin-correlation function is extracted and is found to have a scaling form similar tod=1. Assuming this form to hold in all dimensions we show that the Brinkman-Rice transition only occurs ind=∞. Finite orders of perturbation theory in 1/d around this singular point are not sufficient to remove the transition.

Projected evolution superoperators and the density operator: theory and applications to inelastic scattering

The projection operator method of Zwanzig and Feshbach is used to construct the time dependent density operator associated with a binary scattering event. The formula developed to describe this time dependence involves time-ordered cosine and sine projected evolution (memory) superoperators. Both Sehrödinger and interaction picture results are presented. The former is used to demonstrate the equivalence of the time dependent solution of the von Neumann equation and the more familiar, frequency dependent Laplaee transform solution. For two particular classes of projection superoperators projected density operators arc shown to be equivalent to projected wave functions. Except for these two special eases, no projected wave function analogs of projected density operators exist. Along with the decoupled-motions approximation, projected interaction picture density operators arc applied to inelastic scattering events. Simple illustrations arc provided of how this formalism is related to previously established results for two-state processes, namely, the theory of resonant transfer events, the first order Magnus approximation, and the Landau–Zener theory.

Algebraic approach to the projected deformed oscillator model

Abstract A new method of calculation in terms of the projected deformed oscillator model is proposed. The method involves expansion of its wave functions in terms of the wave functions of an isotropic oscillator potential. Only overlap integrals between projected wave functions and reduced probabilities B (E2) of E2 transitions are examined. B (E2) values are expressed as a series containing the corresponding values of the Elliott SU(3) scheme. The convergence of these expansions is shown to be fairly good. The expectation values of operators ( QQ ) and ( QQQ ), which characterize the effective internal non-sphericity and non-axiality of the nucleus, are also calculated and discussed.

On the interaction between the ground and S bands in the CHFB model

Abstract The interaction between the ground configuration and the first excited 2 qp (s) configuration in the i sol13 2 CHFB model is eliminated in order to investigate how the interaction is built up by the different terms in the hamiltonian. The changes of sign of the interaction can be understood from the particle number projected wave functions. Oscillations are still present after projection.

Microscopic calculations of high-spin rotational states

The high-spin rotational states of /sup 162/Er, /sup 168/Yb, /sup 174/Hf, and /sup 238/U are calculated microscopically by diagonalizing the cranking Hamiltonian H-..omega..J/sub x/ using both BCS and fully particle-number projected wave functions. The computational effort in the latter case is greatly reduced due to a newly derived compact formula for the residuum integral. The results show that pairing collapse does not occur in all four nuclei up to spin 20. The moderate increase of the moment of inertia at low spin is due to both higher-order cranking and Coriolis-antipairing effects. The crossing of the decoupled two-quasiparticle band with the ground band is responsible for the rapid increase of the moment of inertia at high spin. The present calculations are able to produce the rotational energies fairly well in general, but the Nilsson single-particle levels have to be adjusted in order to reproduce the backbending behavior in /sup 162/Er.

High energy elastic and inelastic p- 12C scattering and nuclear deformation

Starting from a microscopic deformed picture of the 12C nucleus, and using angular momentum projected wave functions, the elastic and inelastic 1 GeV p-scattering differential cross sections have been analysed within the framework of the Glauber theory. From a comparison of the results obtained in a full Glauber calculation and in the optical limit, it has been shown that the elastic scattering and, to a lesser extent, the transition to the 2 +(4.44 MeV) level are only weakly affected by the long-range correlations. In contrast the scattering to the 4 +(14.08 MeV) state includes crucial multi-step contributions which affect the differential cross section both in shape and magnitude. Similarly the corrections to the DWIA are essential for the transition to the 3 −(9.64 MeV) state and a satisfactory explanation of the qualitative differences observed between the 2 + and 3 − inelastic cross sections in 12C is given. The great sensitivity of the cross sections to the nuclear deformation is shown and a generally good agreement with experiment for both the electron form factors and 1 GeV p cross sections has been obtained for the ground-state intraband transitions using a single oblate intrinsic state. The importance of a correct treatment of the rotational motion through the angular momentum projection is underlined and the use of the adiabatic approxmation is critically examined.

Electron scattering and α-cluster description of 12C and 20Ne

The elastic and inelastic electron scattering form factors in the ground-state band of 12C and 20Ne have been calculated using the Brink α-cluster model projected wave functions. From a systematic comparison with experiment it is concluded that usual central forces like the Brink-Boeker B1 force give an unrealistically large clustering while that obtained using density dependent interactions like the Skyrme S1 force is too low. For intermediate values of the clustering it is possible in 12C to reproduce simultaneously the elastic as well as the inelastic form factors to the 2 1 +(4.44 MeV), 3 1 −(9.64 MeV) and 4 1 +(14.08 MeV) states for not too large transfers. Noticeable discrepancies between calculated and experimental 2 1 + and 3 1 − inelastic form factors at transfers higher than 1.5 fm −1 reveal inherent weaknesses of the α-cluster wave functions which are not corrected after inclusion of the θ-vibrations.

Influence of the particle-number fluctuation on theE1ΔK=±1transitions in deformed odd-mass nuclei

Dipole electric transitions with $\ensuremath{\Delta}K=\ifmmode\pm\else\textpm\fi{}1$ are studied theoretically for odd-mass nuclei in the rare-earth region (i) in the frame work of an elaborated Nilsson model with rotation-particle coupling and with isospin-dependent parameters, (ii) with the usual BCS wave function, and (iii) with sharp particle-number-projected BCS wave functions. These projected wave functions are the limit of very speedily converging sequences of functions. General formulas for the pairing reduction factors with strict nucleon number conservation are established. The unphysical effects due to particle fluctuation in the usual BCS wave functions are found to be very important in the theoretical BCS evaluation of reduction factors and half-lives. In most of the analyzed $E1$ transitions, the hindrance factors are improved by the elimination of the unphysical effects due to particle fluctuation. The blocking effect is systematically taken into account. The importance and validity of the physical and mathematical approximations used is discussed.RADIOACTIVITY $^{153,155}\mathrm{Eu}$; $^{159,161}\mathrm{Tb}$; $^{175,177}\mathrm{Lu}$; $^{179,181}\mathrm{Ta}$; $^{155,157}\mathrm{Gd}$; $^{161}\mathrm{Dy}$; $^{169,171,173,177}\mathrm{Yb}$; $^{175,177,179}\mathrm{Hf}$; $E1$ $\ensuremath{\Delta}K=\ifmmode\pm\else\textpm\fi{}1$ particle-number-projected pairing reduction factor and ${T}_{\frac{1}{2}}$.

On the generalized Brillouin theorem for spin projected wave functions

The generalized Brillouin theorem in its most general form is shown to be fully equivalent to the variation principle for spin projected Slater determinant wave functions.

An analysis of α-particle inelastic scattering using Hartree-Fock densities

Differential cross sections for the elastic and inelastic scattering of α-particles from 20Ne and 24Mg were calculated, using HF wave functions that had previously been used for (p, p') calculations. We do not use projected wave functions, but we do require consistency between the elastic and inelastic form factors. The result is that although the HF solutions are unable to fit the data, the discrepancy is significantly less than that found in (p, p') calculations.

Multipole deformations, static quadrupole moments and electromagnetic transitions in light nuclei

Hartree-Fock calculations have been performed in light 4 n nuclei from 8Be to 28Si using central interactions. The rms radii, static quadrupole moments and electromagnetic transition rates have been deduced exactly from the angular momentum projected wave functions obtained in a projection before variation (PBV) procedure. The quadrupole and hexadecapole deformations of the intrinsic states are compared with the experimental results deduced recently from the analysis of electron, proton and α-particle inelastic scattering data. The whole system of A nucleons is treated and consequently we do not introduce any additional effective charge. A detailed comparison with experiment is made and good agreement is obtained, with some exceptions in 12C and 28Si. Our projected HF description is compared with other microscopic investigations of light nuclei and with the phenomenological rotational-model predictions. The importance of the PBV procedure is displayed and remarks are also made concerning the usual approximations used to deduce the nuclear properties from the HF wave functions.