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Physical review. B, Condensed matter

Bounds for the ground-state energy in the Emery model determined by cluster calculations

Thomas Wittmann, Joachim Stolze

Published Aug 1, 1993• 7 citations
Publisher

Abstract

We present simple upper and lower bounds for the ground-state energy of the Emery model. These bounds are derived by partitioning the full Hamiltonian into a sum of sub-Hamiltonians that may be treated with the Lanczos algorithm. The energy bounds are used to derive bounds on the magnetization as a function of magnetic field and the hole density as a function of chemical potential. We improve upon the recent variational calculation of Carstensen and Dichtel [Phys. Rev. B 46, 11 040 (1992)], which suggested the existence of phase separation. Our results contain no evidence for phase separation.

DOI
10.1103/physrevb.48.3479

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We present simple upper and lower bounds for the ground-state energy of the Emery model. These bounds are derived by partitioning the full Hamiltonian into a sum of sub-Hamiltonians that may be treated with the Lanczos algorithm. The energy bounds are used to derive bounds on the magnetization as a function of magnetic field and the hole density as a function of chemical potential. We improve upon the recent variational calculation of Carstensen and Dichtel [Phys. Rev. B 46, 11 040 (1992)], which suggested the existence of phase separation. Our results contain no evidence for phase separation.
We show that upper and lower bounds on the ground-state energy of models describing correlated Fermi systems may be combined to produce bounds on the ground-state magnetization and chemical potential. Such bounds are obtainable through standard variational techniques and through recently developed methods involving exact diagonalization of finite-size clusters. For the Hubbard model on the square lattice, we give rigorous bounds for the magnetization at nonzero magnetic field B and for the chemical potential at nonzero hole density 1-n. The quality of these bounds degrades as B\ensuremath{\rightarrow}0 and n\ensuremath{\rightarrow}1, precluding rigorous statements about the stability of the ferromagnetic state or the existence of a Mott-Hubbard gap. Nevertheless, the tendency towards large-U ferromagnetism and localization is evident. We discuss ways of improving these bounds, including the use of kinetic frustration, nonuniform clusters, and averaging over boundary conditions.
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.
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.
We calculate the single-particle excitation spectrum of holes in the Emery model thereby extending and improving previous calculations. The system is considered at half filling (nh{\mathit{n}}_{\mathit{h}}nh​=1, one hole per CuO2{\mathrm{CuO}}_{2}CuO2​ unit) and for hole doping, where the on-site hole-hole repulsions are kept finite. A paramagnetic form of the ground state is used. For the determination of the retarded Green's functions of copper and oxygen holes, the projection technique is applied solving the resulting equations of motions self-consistently. At half filling, the excitation spectrum exhibits a charge-transfer gap bounded by Zhang-Rice singlet states and the upper Hubbard band. Upon hole doping the flat singlet band crosses the Fermi level giving rise to a large Fermi surface at a hole concentration of nh{\mathit{n}}_{\mathit{h}}nh​=1.25. Moreover, spectral weight is shifted from the upper Hubbard band to the states near the Fermi energy. The calculated spectral densities, the singlet dispersion for the doped system, and the transfer of spectral weight are in good quantitative agreement with exact diagonalization results for 2\ifmmode\times\else\texttimes\fi{}2 CuO2{\mathrm{CuO}}_{2}CuO2​ cluster.
The exact phase diagrams of the one-dimensional Hubbard model, both attractive and repulsive, in the presence of an arbitrary magnetic field h for various electron concentrations n and on-site interaction strengths U <0 or U >0 are calculated and investigated. The exact ground-state properties, namely, the ground-state energy, the average spin (magnetization), the concentration of the doubly occupied sites, the kinetic energy, the chemical potential, the spin (magnetic) susceptibility and the charge compressibility, are calculated and examined over a wide range of interaction strengths U for various h and n . It is found that the spin susceptibility at half-filling is non-analytic and changes discontinuously as U →0. The exact theory shows the absence of a charge energy gap in the U ⩽0 region for all n and provides, for the chemical potential, the rigorous upper and lower bounds for half-filled and empty bands respectively. The analytical results derived for the weak-coupling limit and asymptotic expansions for strong coupling are in full agreement with the numerical calculations.