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Low Temperature Physics

Bound electron pairs in strongly correlated models of high-temperature superconductivity

Hüseyin Boyacı, I. O. Kulik

Published Apr 1, 1998• 2 citations
Publisher

Abstract

The ground-state energy of two electrons on a ring is calculated for the one-dimensional Hubbard model with positive and negative on-site interaction and for the contraction model with additive and multiplicative interaction terms. The hc/2e periodicity of the ground-state energy with respect to a flux Φ threading the loop is derived. The periodicity may serve as an indication of superconductivity. The results are shown to be consistent with the Lieb–Wu solution for Φ=0 limit. In addition, the new states that were missing in the Lieb–Wu solution are derived.

DOI
10.1063/1.593577

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•Jan 1, 2008•0 citations
The ground-state energy of two electrons on a ring is calculated for the one-dimensional Hubbard model with positive and negative on-site interaction and for the contraction model with additive and multiplicative interaction terms. The hc/2e periodicity of the ground-state energy with respect to a flux Φ threading the loop is derived. The periodicity may serve as an indication of superconductivity. The results are shown to be consistent with the Lieb–Wu solution for Φ=0 limit. In addition, the new states that were missing in the Lieb–Wu solution are derived.
By means of the Bethe-ansatz method, the one-dimensional Hubbard model with repulsive on-site coupling in an external magnetic field is solved exactly. It is found that the energy of the ground state depends on the magnetic flux \ensuremath{\varphi} through the ring formed by a periodic lattice chain, and the lowest-energy state corresponds to \ensuremath{\varphi}=n(hc/2e), with n an integer and -e the electron charge. The calculations illustrate that the strongly correlated Hubbard ring with periodic boundary conditions in its ground state is a superconducting ring. It can also be shown that the system possesses an internal SU(2)\ifmmode\times\else\texttimes\fi{}SU(2) symmetry.
Within the basic assumption that oxygen orbitals may play a fundamental role in the superconductive properties of Cu-O high-Tc{\mathit{T}}_{\mathit{c}}Tc​ materials, exact calculations of the ground-state energy and thermodynamic properties of strongly interacting fermions in small oxygen rings have been performed. Persistent currents (mesoscopic effect) and supercurrents have been computed for several microscopic models including repulsive and attractive Hubbard and the occupation-dependent hopping Hamiltonians. It is shown that for a given range of parameters the latter model exhibits features reminiscent of superconductivity: a maximum in the temperature dependence of the specific heat, whose position correlates with the temperature at which the superelectron concentration vanishes and exact hc/2e-periodic dependence of the free energy as a function of magnetic flux.
We present recent theoretical results on superconductivity in correlated-electron systems, especially in the two-dimensional Hubbard model and the three-band d-p model. The mechanism of superconductivity in high-temperature superconductors has been extensively studied on the basis of various electronic models and also electron-phonon models. In this study, we investigate the properties of superconductivity in correlated-electron systems by using numerical methods such as the variational Monte Carlo method and the quantum Monte Carlomethod. The Hubbard model is one of basic models for strongly correlated electron systems, and is regarded as the model of cuprate high temperature superconductors. The d-p model is more realistic model for cuprates. The superconducting condensation energy obtained by adopting the Gutzwiller ansatz is in reasonable agreement with the condensation energy estimated for YBa2Cu3O7. We show the phase diagram of the ground state using this method. We have further investigated the stability of striped and checkerboard states in the under-doped region. Holes doped in a half-filled square lattice lead to an incommensurate spin and charge density wave. The relationship of the hole density x and incommensurability δ, δ~x, is satisfied in the lower doping region, as indicated by the variationalMonte Carlocalculations for the two-dimensional Hubbard model. A checkerboard-like charge-density modulation with a roughly period has also been observed by scanning tunneling microscopy experiments in Bi2212 and Na-CCOC compounds. We have performed a variational Monte Carlo simulation on a two-dimensional t-t′-t″- U Hubbard model with a Bi-2212 type band structure and found that the period checkerboard spin modulation, that is characterized by multi Q vectors, is indeed stabilized. We have further performed an investigation by using a quantumMonte Carlomethod, which is a numerical method that can be used to simulate the behavior of correlated electron systems. We present a new algorithm of the quantum Monte Carlo diagonalization that is a method for the evaluation of expectation value without the negative sign problem. We compute pair correlation functions and show that pair correlation is indeed enhanced with hole doping.
We study the one-band Hubbard model in a three dimensional simple cubic lattice, with periodic boundary conditions, by means of a variational analytic approach. Ground state energies and pairing correlations depend implicitly on the interaction strength (U/41). It is shown that for two electrons, the interaction is always repulsive in the ground state for any positive value of the on-site Coulomb interaction U. Implications of this result for superconductivity are also discussed. Nigerian Journal of Physics Vol. 16(2) 2004: 167-171
We obtain the ground-state energy E0{\mathit{E}}_{0}E0​(\ensuremath{\Phi}) exactly as a function of flux \ensuremath{\Phi} in a one-dimensional ``copper-oxide model'' on rings of finite circumference L, including on-site interactions and a nearest-neighbor interaction V. For V of the order of the charge-transfer gap or larger, the model extrapolated to large L exhibits flux quantization with charge 2e, and a slow algebraic decay of the singlet superconducting correlation function on oxygen sites. The extrapolated superfluid stiffness appears, however, finite only for not too large V. These results suggest a superconductive state at V of order the charge-transfer gap of the model, but a paired and phase-separated state at larger V.
We study the one-dimensional Hubbard model for interacting electrons in a crystal lattice, using two methods: solution of the Lieb-Wu equations in the atomic limit of no intersite hopping, and variational techniques to obtain a lower bound for the ground state energy. From the Lieb-Wu equations we derive an exact expression for the ground state energy in the atomic limit. We obtain our result by exploiting the spin symmetry of the Hubbard Hamiltonian. Our energy is lower than that obtained by previous published results, and we are able to explain why our results differ. Our method also gives information about the magnetic ordering of the ground state as a function of the number of electrons. Our results are valid for arbitrary electron density values. The variational techniques we use are applied to the two-particle reduced-density matrix (2-RDM), because this quantity suffices to obtain expectation values of all one- and two-particle operators, and because it is less computationally intensive than obtaining the complete N -particle wave function. Because the 2-RDM must be positive semidefinite, we use an efficient programming algorithm known as the Semidefinite Programming Algorithm. A naive search through the space of two-particle matrices encounters a difficult problem, known as the " N -representability problem", which is to assure that the trial density matrices actually come from a wave function that has the required exchange symmetry. This problem is ameliorated by imposing constraints on the variational search that eliminate trial matrices with the incorrect exchange symmetry. These constraints express known physical properties of the ground state (for example its total spin and its translational symmetry) and the so-called D, Q , and G two-positivity conditions that express the required exchange symmetry. To calculate matrix elements, we use the two-particle eigenstates of the reduced Hubbard Hamiltonian; this choice of basis significantly decreases the number of matrix elements needed for energy calculations. After an optimal two-particle density matrix has been obtained, we calculate additional properties of the ground state, including spin correlation functions, chemical potentials, and the momentum distribution. We compare all of these properties to values obtained by exact numerical solution of the Schrödinger equation.