Quantum Probabilistic Subroutines and Problems in Number Theory

We present a quantum version of the classical probabilistic algorithms $\grave{a}$ la Rabin. The quantum algorithm is based on the essential use of Grover's operator for the quantum search of a database and of Shor's Fourier transform for extracting the periodicity of a function, and their combined use in the counting algorithm originally introduced by Brassard et al. One of the main features of our quantum probabilistic algorithm is its full unitarity and reversibility, which would make its use possible as part of larger and more complicated networks in quantum computers. As an example of this we describe polynomial time algorithms for studying some important problems in number theory, such as the test of the primality of an integer, the so called 'prime number theorem' and Hardy and Littlewood's conjecture about the asymptotic number of representations of an even integer as a sum of two primes.Comment: 9 pages, RevTex, revised version, accepted for publication on PRA: improvement in use of memory space for quantum primality test algorithm further clarified and typos in the notation correcte

Dispersion of a single hole in the t-J model

The dispersion of a single hole in the t-J model obtained by the exact result of 32 sites and the results obtained by self-consistent Born approximation and the Green function Monte Carlo method can be simply derived by a mean-field theory with d-RVB and antiferromagnetic order parameters. In addition, it offers a simple explanation for the difference observed between those results. The presence of the extended van Hove region at (pi,0) is a consequence of the d-RVB pairing independenct of the antiferromagnetic order. Results including t' and t" are also presented and explained consistently in a similar way.Comment: LaTex file, 5 pages with 5 embedded eps figure

Doping dependence of the superconducting gap in Bi2Sr2CaCu2O{8 + delta}

Bi2Sr2CaCu2O{8 + \delta} crystals with varying hole concentrations (0.12 < p < 0.23) were studied to investigate the effects of doping on the symmetry and magnitude of the superconducting gap. Electronic Raman scattering experiments that sample regions of the Fermi surface near the diagonal (B_{2g}) and principal axes (B_{1g}) of the Brillouin Zone have been utilized. The frequency dependence of the Raman response function at low energies is found to be linear for B_{2g} and cubic for B_{1g} (T< T_c). The latter observations have led us to conclude that the doping dependence of the superconducting gap is consistent with d_{x^2-y^2} symmetry, for slightly underdoped and overdoped crystals. Studies of the pair-breaking peak found in the B_{1g} spectra demonstrate that the magnitude of the maximum gap decreases monotonically with increasing hole doping, for p > 0.12. Based on the magnitude of the B_{1g} renormalization, it is found that the number of quasiparticles participating in pairing increases monotonically with increased doping. On the other hand, the B_{2g} spectra show a weak "pair-breaking peak" that follows a parabolic-like dependence on hole concentration, for 0.12 < p < 0.23.Comment: 9 pages REvTex document including 8 eps figures; new table II; changes to Fig. 5 and tex

The Electron Spectral Function in Two-Dimensional Fractionalized Phases

We study the electron spectral function of various zero-temperature spin-charge separated phases in two dimensions. In these phases, the electron is not a fundamental excitation of the system, but rather ``decays'' into a spin-1/2 chargeless fermion (the spinon) and a spinless charge e boson (the chargon). Using low-energy effective theories for the spinons (d-wave pairing plus possible N\'{e}el order), and the chargons (condensed or quantum disordered bosons), we explore three phases of possible relevance to the cuprate superconductors: 1) AF*, a fractionalized antiferromagnet where the spinons are paired into a state with long-ranged N\'{e}el order and the chargons are 1/2-filled and (Mott) insulating, 2) the nodal liquid, a fractionalized insulator where the spinons are d-wave paired and the chargons are uncondensed, and 3) the d-wave superconductor, where the chargons are condensed and the spinons retain a d-wave gap. Working within the $Z_2$ gauge theory of such fractionalized phases, our results should be valid at scales below the vison gap. However, on a phenomenological level, our results should apply to any spin-charge separated system where the excitations have these low-energy effective forms. Comparison with ARPES data in the undoped, pseudogapped, and superconducting regions is made.Comment: 10 page

Higgs algebraic symmetry of screened system in a spherical geometry

The orbits and the dynamical symmetries for the screened Coulomb potentials and isotropic harmonic oscillators have been studied by Wu and Zeng [Z. B. Wu and J. Y. Zeng, Phys. Rev. A 62,032509 (2000)]. We find the similar properties in the responding systems in a spherical space, whose dynamical symmetries are described by Higgs Algebra. There exists a conserved aphelion and perihelion vector, which, together with angular momentum, constitute the generators of the geometrical symmetry group at the aphelia and perihelia points $(\dot{r}=0)$.Comment: 8 pages, 1 fi

Anisotropic two-dimensional Heisenberg model by Schwinger-boson Gutzwiller projected method

Two-dimensional Heisenberg model with anisotropic couplings in the $x$ and $y$ directions ($J_x \neq J_y$) is considered. The model is first solved in the Schwinger-boson mean-field approximation. Then the solution is Gutzwiller projected to satisfy the local constraint that there is only one boson at each site. The energy and spin-spin correlation of the obtained wavefunction are calculated for systems with up to $20 \times 20$ sites by means of the variational Monte Carlo simulation. It is shown that the antiferromagnetic long-range order remains down to the one-dimensional limit.Comment: 15 pages RevTex3.0, 4 figures, available upon request, GWRVB8-9

Structure and properties of a novel fulleride Sm6C60

A novel fulleride Sm6C60 has been synthesized using high temperature solid state reaction. The Rietveld refinement on high resolution synchrotron X-ray powder diffraction data shows that Sm6C60 is isostructural with body-centered cubic A6C60 (A=K, Ba). Raman spectrum of Sm6C60 is similar to that of Ba6C60, and the frequencies of two Ag modes in Sm6C60 are nearly the same as that of Ba6C60, suggesting that Sm is divalent and hybridization between C60 molecules and the Sm atom could exist in Sm6C60. Resistivity measurement shows a weak T-linear behavior above 180 K, the transport at low temperature is mainly dominated by granular-metal theory.Comment: 9 pages, 3 figures, submitted to Phys. Rev. B (March 12, 1999

Hubbard model versus t-J model: The one-particle spectrum

The origin of the apparent discrepancies between the one-particle spectra of the Hubbard and t-J models is revealed: Wavefunction corrections, in addition to the three-site terms, should supplement the bare t-J. In this way a quantitative agreement between the two models is obtained, even for the intermediate-$U$ values appropriate for the high-Tc cuprate superconductors. Numerical results for clusters of up to 20 sites are presented. The momentum dependence of the observed intensities in the photoemission spectra of Sr2CuO2Cl2 are well described by this complete strong-coupling approach.Comment: 4 two-column RevTeX pages, including 4 Postscript figures. Uses epsf. Accepted for publication in Physical Review B, Rapid Communicatio

Numerical renormalization group study of the 1D t-J model

The one-dimensional (1D) $t-J$ model is investigated using the density matrix renormalization group (DMRG) method. We report for the first time a generalization of the DMRG method to the case of arbitrary band filling and prove a theorem with respect to the reduced density matrix that accelerates the numerical computation. Lastly, using the extended DMRG method, we present the ground state electron momentum distribution, spin and charge correlation functions. The $3k_F$ anomaly of the momentum distribution function first discussed by Ogata and Shiba is shown to disappear as $J$ increases. We also argue that there exists a density-independent $J_c$ beyond which the system becomes an electron solid.Comment: Wrong set of figures were put in the orginal submissio

The Numerical Renormalization Group Method for correlated electrons

The Numerical Renormalization Group method (NRG) has been developed by Wilson in the 1970's to investigate the Kondo problem. The NRG allows the non-perturbative calculation of static and dynamic properties for a variety of impurity models. In addition, this method has been recently generalized to lattice models within the Dynamical Mean Field Theory. This paper gives a brief historical overview of the development of the NRG and discusses its application to the Hubbard model; in particular the results for the Mott metal-insulator transition at low temperatures.Comment: 14 pages, 7 eps-figures include