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arXiv:cond-mat/9511113 · Physical review. B, Condensed matter

Control of the finite-size corrections in exact diagonalization studies

Claudius Gros

Published Mar 15, 1996• 47 citations
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Abstract

The finite-size corrections in exact diagonalization studies of the one- and two-dimensional Hubbard model can be reduced systematically by a grand-canonical integration over boundary conditions. We present results for ground-state properties of the two-dimensional (2D) Hubbard model and an evaluation of the specific heat for the 1D and 2D Hubbard model. We find the reduction of the finite-size corrections to be substantial, especially in two dimensions. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.6865

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The finite-size corrections in exact diagonalization studies of the one- and two-dimensional Hubbard model can be reduced systematically by a grand-canonical integration over boundary conditions. We present results for ground-state properties of the two-dimensional (2D) Hubbard model and an evaluation of the specific heat for the 1D and 2D Hubbard model. We find the reduction of the finite-size corrections to be substantial, especially in two dimensions. \textcopyright{} 1996 The American Physical Society.
We investigate, by numerically calculating the charge stiffness, the effects of random diagonal disorder and electron-electron interaction on the nature of the ground state in the 2D Hubbard model through the finite-size exact diagonalization technique. By comparing with the corresponding 1D Hubbard model results and by using heuristic arguments we conclude that it is unlikely that there is a 2D metal-insulator quantum phase transition, although the effect of interaction in some range of parameters is to substantially enhance the noninteracting charge stiffness.
We use the exact-diagonalization results for a one-dimensional finite-size boson Hubbard-like Hamiltonians to initialize the renormalization-group equations in the vicinity of the superfluid-insulator transition and demonstrate that this provides a rather accurate method of extrapolation of the finite-size results to larger system sizes, and in particular, pinpointing the critical parameters of the Hamiltonians. With our approach we reproduce with the accuracy \ensuremath{\sim}2.5% the known analytical result for the transition point in the half-filled system of hard-core bosons, obtain with a controllable accuracy the critical ratio (t/U)c{)}_{\mathit{c}})c​=0.304\ifmmode\pm\else\textpm\fi{}0.002 for the boson Hubbard model, and find out that, in contrast to the general belief, the reduced Hamiltonian (with the constraint that the site occupation numbers be less than 3) is far from being a good approximation to the full model. \textcopyright{} 1996 The American Physical Society.
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The ground state of the two-dimensional (2D) t-J model is studied by exact diagonalization for seven different unit cells with N=10, 16, 18, 20, 26, 32, and 36 sites on a square lattice. In the limit of low electron densities (up to six electrons) we calculate the lowest eigenvalues and classify the ground state according to wave vector K, total spin S, and point-group symmetry both in the limit J=0 (U=\ensuremath{\infty} Hubbard model) and for finite J>0. The transition to a completely phase-separated state for J>JPS{\mathit{J}}_{\mathrm{PS}}JPS​(N) is examined by calculating the free energy as a function of the electron concentration. We find that JPS{\mathit{J}}_{\mathrm{PS}}JPS​(N) increases with the lattice size N. An extrapolation of our finite-size results to the thermodynamic limit gives JPS{\mathit{J}}_{\mathrm{PS}}JPS​(\ensuremath{\infty})/t=4.1 (\ifmmode\pm\else\textpm\fi{}0.1).
We present simple lower bounds on the ground-state energy of the two-dimensional (2D) Hubbard and t-J models for arbitrary values of band filling and coupling constant. For the Hubbard model we derive two types of bounds, both based on decomposing the model Hamiltonian into a sum of sub-Hamiltonians. For a decomposition into local cluster sub-Hamiltonians, we perform a generalized Legendre transform on previously derived bounds for the grand-canonical potential. For a decomposition into spin-up and spin-down parts, previous results on the spinless Falicov-Kimball model may be used to obtain bounds for the Hubbard model, generalizing a result of Langer and Mattis to arbitrary filling. For the 2D t-J model we have only considered the decomposition into clusters. The 1D Hubbard model is used as a test case. The bounds may be improved by diagonalizing the Hamiltonian for larger clusters.