Stochastic diagonalization

Hans A. De Raedt, Kristel F. L. Michielsen

Dean Lee

Werner Fettes, I. Morgenstern, Thomas Husslein

We present exact and stochastic diagonalization results for a BCS-reduced Hubbard model. The kinetic Hamiltonian is the same as in the single band Hubbard model with additional next nearest neighbor hopping. The interaction of this model is designed to inhibit superconductivity in the d x2-y2 channel. The ground state of this model is studied by exact and stochastic diagonalization technique. We present a review of the technical details of the application of the stochastic diagonalization algorithm on this problem. To verify our results obtained with the stochastic diagonalization, they are compared with the exact diagonalization results. In order to show the convergence of the stochastic diagonalization we give a detailed analysis of the behavior of physical properties with increasing number of states. Finally we study superconductivity in this BCS-reduced Hubbard model. As an indicator of superconductivity we use the occurrence of Off Diagonal Long Range Order. We study the scaling behavior of this model for various attractive interactions and in addition the dependence of the superconducting correlation functions from the filling of the system.

Daniel F. Walls, Gerard J Milburn

Dean Lee

We briefly review the diagonalization of quantum Hamiltonians using the quasisparse eigenvector (QSE) method. We also introduce the technique of stochastic error correction, which systematically removes the truncation error of the QSE result by stochastically sampling the contribution of the remaining basis states. 1 Quasi-sparse eigenvector method Quasi-sparse eigenvector (QSE) diagonalization is a new computational method which finds approximate low-lying eigenvalues and eigenvectors for a general quantum field Hamiltonian H [1]. It handles the exponential increase in the dimension of Fock space by exploiting the sparsity of the Hamiltonian. The method is most effective when the splitting between low-lying eigenvalues is not too small compared to the size of the Hamiltonian off-diagonal entries. In such cases the low-lying eigenvectors are quasi-sparse, meaning that the vector is dominated by a small fraction of its largest components. The QSE algorithm is applied using the following steps: (1) Select a subset of orthonormal basis vectors {ei1, · · · , ein} and call the

Daniel F. Walls, Gerard J Milburn