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Physical review. B, Condensed matter

Exact numerical study of the ground-state magnetic properties of clusters

Florentino Lopéz‐Urías, G. M. Pastor

Published Feb 15, 1999• 30 citations
Publisher

Abstract

The ground states of bcc, fcc, hcp, and icosahedral clusters having $N=12--14$ atoms are calculated exactly in the framework of the Hubbard Hamiltonian by using Lanczos's numerical diagonalization method. The total spin S shows a diversity of remarkable, sometimes surprising behaviors as a function of Coulomb interaction strength $U/t$ and band filling $\ensuremath{\nu}/N.$ The magnetic properties are analyzed, in particular, by relating them to the corresponding single-particle spectra. Changes of symmetry close to real-space shell closings at $N=13$ are discussed. The relative stability of the different cluster geometries is determined and the corresponding structural and magnetic diagrams are derived. Electron correlation effects close to half-band filling result in changes of structure that are often accompanied by important changes in the magnetic behavior. Ferromagnetism is found to be particularly stable for $\ensuremath{\nu}/N\ensuremath{\simeq}1.2--1.4.$ The main trends as a function of $N, \ensuremath{\nu}$ and $U/t$ are compared with previous exact diagonalization studies on smaller clusters.

DOI
10.1103/physrevb.59.5223

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The ground states of bcc, fcc, hcp, and icosahedral clusters having N=12−−14N=12--14N=12−−14 atoms are calculated exactly in the framework of the Hubbard Hamiltonian by using Lanczos's numerical diagonalization method. The total spin S shows a diversity of remarkable, sometimes surprising behaviors as a function of Coulomb interaction strength U/tU/tU/t and band filling \ensuremathν/N.\ensuremath{\nu}/N.\ensuremathν/N. The magnetic properties are analyzed, in particular, by relating them to the corresponding single-particle spectra. Changes of symmetry close to real-space shell closings at N=13N=13N=13 are discussed. The relative stability of the different cluster geometries is determined and the corresponding structural and magnetic diagrams are derived. Electron correlation effects close to half-band filling result in changes of structure that are often accompanied by important changes in the magnetic behavior. Ferromagnetism is found to be particularly stable for \ensuremathν/N\ensuremath≃1.2−−1.4.\ensuremath{\nu}/N\ensuremath{\simeq}1.2--1.4.\ensuremathν/N\ensuremath≃1.2−−1.4. The main trends as a function of N,\ensuremathνN, \ensuremath{\nu}N,\ensuremathν and U/tU/tU/t are compared with previous exact diagonalization studies on smaller clusters.
The electronic, magnetic, and structural properties of small clusters are studied in the framework of the single-band Hubbard Hamiltonian. Results for various ground-state and excited-state many-body properties are presented, which were calculated exactly by means of Lanczos's numerical diagonalization method. A full geometry optimization is performed for N\ensuremath{\le}8 atoms by considering all possible nonequivalent cluster structures with fixed nearest-neighbor bond lengths. The most stable structure and the corresponding total spin S are obtained rigorously as a function of the Coulomb interaction strength U/t and number of electrons \ensuremath{\nu}. The resulting interplay between electron correlations, magnetism, and cluster structure is analyzed and the main trends as a function of N, U/t, and \ensuremath{\nu} are derived. The stability of cluster ferromagnetism is studied from two complementary points of view. First, for N\ensuremath{\le}8 and \ensuremath{\nu}=N+1, we determine exactly the stability of the ferromagnetic ground state with respect to electronic excitations and structural changes. It is shown that in small clusters the structural changes can be as important to the temperature dependence of the magnetization as the purely electronic excitations. Second, we determine the stability of the saturated ferromagnetic state with respect to single spin flips as a function of the band filling \ensuremath{\nu}/N. In this case a few selected larger clusters (7\ensuremath{\le}N\ensuremath{\le}43) in the strongly correlated limit (U/t\ensuremath{\rightarrow}+\ensuremath{\infty}) are considered. It is shown that the \ensuremath{\nu}/N dependence of the spin-flip energy \ensuremath{\Delta}${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{sf}}$ shows interesting electronic-shell-like oscillations, which reflect the characteristics of the single-particle energy-level structure and its dependence on the symmetry and size of the cluster. Finally, we conclude by discussing some of the limitations of the model together with relevant extensions. \textcopyright{} 1996 The American Physical Society.
Many-body properties of N-atom clusters having N\ensuremath{\le}8 are calculated exactly in the framework of the Hubbard model for all possible nonequivalent cluster structures. The most stable structure and the corresponding total spin S are obtained rigorously as a function of the Coulomb-interaction strength U/t and the number of electrons. Results for the stability of cluster ferromagnetism against electronic excitations and structural changes are obtained for the first time. The resulting interplay between electron correlations, magnetism, and cluster structure is determined.
We analyze the ground-state properties of the U=\ensuremath{\infty} Hubbard model on a 4\ifmmode\times\else\texttimes\fi{}4 cluster using a Lanczos technique. The quantum numbers of the ground-state, its momentum distribution, the magnetic structure factor, some 〈nˆk{\mathrm{\^{}}}_{\mathbf{k}}ˆk​n${\mathrm{\^{}}}_{\mathbf{k}\ensuremath{'}}$〉 dynamic correlation functions, and the overlap of the ground state with the noninteracting one (both before and after a Gutzwiller projection) are given for all fillings. As the number of electrons approaches half-filling the system undergoes a transition from a conventional ground state that overlaps appreciably with the noninteracting one to a new ground state that shows local ferromagnetic correlations and negligible overlap with the corresponding U=0 ground state. A detailed analysis of the wave function corresponding to the ground state of 14 electrons (two holes) proves the existence of a finite shift of the Fermi surface. Our results support the existence of a stable magnetic phase at small but finite hole doping. This new magnetic phase is locally ferromagnetic although the total spin is not maximum.
The ground-state magnetic properties of clusters with N = 3 atoms are studied by sohing lhe single band Hubbard Hamiltonian in the unrestricted Hartree-Fock approximation. Three-dimensional arrangements oi spins are considcrcd in the solution. Results for the total energy and magnetic arder, as a function of the Coulomb interaction strength Ujt and num- ber of eleetrons v, are prcsented. For half-band filling an antiferromagnetic nOll-collinear spin arrangement is the most stable for the trianglc structurc. For v = 2 ami Uji ~ 16.2, v = 4 and U/t ~ 2.3 the linear chain is the most stable stcucture. In particular, it is found that the inclusion oi non-collinear spins improves in the Hartrce-Fock ground-state cncrgy as compared with collinear spins. Our rcsults are compared and discusscd with exact calculations. RESU1.IEN. Las propiedades magneticas del estado base de cumulos con N = 3 atomos son estu- diadas resolviendo el hamiltoniano de Hubbard para una banda en la aproximacion no restringida de Hartree-Fock. Se consideran soluciones magneticas con arreglos de espines no colineales. Se presentan los resultados de la energia total y el orden magnetico en funcion de la intensidad de la interaccion de Coulomb y el numero de electrones. Para la banda semillena la solucion mas estable para la estructura triangular, corresponde a un arreglo antiferromagnetico no colineal de espines. Para v = 2 Y U/t 2: lG.2, v = 4 Y U/t ~ 2.3 la cadena lineal es la estructura mas estable. En particular 1 se encuentra que considerar espines no colineales mejora la energia del estado base de Hartree-Fock comparada con espines colineales. Se discuten y comparan nuestros resultados con soluciones exactas.
The single-band Hubbard model (both with and without first-neighbor interactions) is studied by exact diagonalization on several one- and two-dimensional clusters, ranging from four to nine sites, including rings and arrangements of squares. Occupancies include the half-filled band (one electron per site) and configurations with one and two holes in the half-filled band. Ground-state properties are determined for a wide range of parameters. We discuss the phase diagram in the (U,V) plane and consider spin, charge, and pairing correlations. The binding of a pair of holes is investigated. Circumstances in which binding and pairing are enhanced by first-neighbor interactions are described.
The authors have analysed finite clusters with different degrees of frustration on the basis of the conventional Hubbard model. In particular they have calculated the ground-state energy and certain correlation functions as a function of the interaction energy U. As U varies, a sharp transition is observed between two different structures for half-filled bands. The authors have studied these clusters not only for the case of a half-filled band but also for different numbers of holes. They found that the structure of the ground state depends critically on the total spin and the number of holes.