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AIP conference proceedings

Exact diagonalization study of quantum spin clusters

Sanjubala Sahoo, Alfred Hucht, Shreekantha Sil, P. Entel

Published Jan 1, 2012• 0 citations
Publisher

Abstract

The magnetic and thermodynamic properties of 13-atom clusters with decahedral and layered geometries have been studied with spin-1/2 using the quantum Heisenberg model. The numerical exact diagonalization technique is applied to solve the quantumspin Hamiltonian for both ferro (FM) and anti-ferromagnetic (AFM) interactions of the clusters. The ground state energies for the FM case are found to be all degenerate while for the AFM case, they are non-degenerate. In the presence of external magnetic field, we found discontinuities in the field-dependent magnetization accompanied by plateaus for the AFM interaction. The thermodynamic entities such as entropy and specific heat as a function of external magnetic field are smeared out with increasing temperature. They also show a strong dependence on the cluster geometry.

DOI
10.1063/1.4736886

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Physical review. B, Condensed matter•Apr 15, 1996•55 citations
The magnetic and thermodynamic properties of 13-atom clusters with decahedral and layered geometries have been studied with spin-1/2 using the quantum Heisenberg model. The numerical exact diagonalization technique is applied to solve the quantumspin Hamiltonian for both ferro (FM) and anti-ferromagnetic (AFM) interactions of the clusters. The ground state energies for the FM case are found to be all degenerate while for the AFM case, they are non-degenerate. In the presence of external magnetic field, we found discontinuities in the field-dependent magnetization accompanied by plateaus for the AFM interaction. The thermodynamic entities such as entropy and specific heat as a function of external magnetic field are smeared out with increasing temperature. They also show a strong dependence on the cluster geometry.
The spin -½ Heisenberg antiferromagnet Hamiltonian with exchange integral and applied external magnetic field is studied with the help of direct exact diagonalization technique. Two site cluster lattices with open boundary conditions are considered and the criteria for the occurrence of a transition from antiferromagnetic (AFM) phase to a ferromagnetic (FM) phase are discussed.However, the results obtained shows that the temperature dependence of the uniform magnetization depends on the strength of the applied magnetic field h. If the applied magnetic field is smaller than the critical field strength, the zero temperature magnetization vanishes, otherwise it is activated.
The thermodynamics of low-dimensional spin-12\frac{1}{2}21​ Heisenberg ferromagnets (HFMs) in an external magnetic field is investigated within a second-order two-time Green function formalism in the wide temperature and field ranges. A crucial point of the proposed scheme is a proper account of the analytical properties for the approximate transverse commutator Green function obtained as a result of the decoupling procedure. A good quantitative description of the correlation functions, magnetization, susceptibility, and heat capacity of the HFMs on a chain and square and triangular lattices is found for both infinite- and finite-sized systems. The dependences of the thermodynamic functions of two-dimensional HFMs on the cluster size are studied. The obtained results agree well with the corresponding data found by Bethe ansatz, exact diagonalization, high temperature series expansions, and quantum Monte Carlo simulations.
The nearest-neighbor antiferromagnetic Heisenberg model is considered for spins si=12{s}_{i}=\frac{1}{2}si​=21​ and 1 located on the vertices of the dodecahedron and the icosahedron, which belong to the point symmetry group Ih{I}_{h}Ih​. Taking into account the permutational and spin-inversion symmetries of the Hamiltonian results in a drastic reduction of the dimensionality of the problem, leading to full diagonalization for both clusters. There is a strong signature of the frustration present in the systems in the low-energy spectrum for si=12{s}_{i}=\frac{1}{2}si​=21​, where the first excited states are singlets. Frustration also results in a doubly peaked specific heat as a function of temperature for the dodecahedron. Furthermore, there is a discontinuity in the magnetization as a function of magnetic field for the dodecahedron, where a specific total spin sector never becomes the ground state in a field. This discontinuity is accompanied by a magnetization plateau. For si=1{s}_{i}=1si​=1 both systems again have singlet excitations. The magnetization of the dodecahedron has now two discontinuities in an external field and also magnetization plateaux, and the specific heat of the icosahedron, a two-peak structure as a function of temperature. The similarities between the two systems suggest that the antiferromagnetic Heisenberg model on a larger cluster with the same symmetry, the 60-site cluster, will have similar properties.
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 $\ensuremath{\nu}/N.$ 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=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 $\ensuremath{\nu}/N\ensuremath{\simeq}1.2--1.4.$ The main trends as a function of $N, \ensuremath{\nu}$ and U/tU/tU/t are compared with previous exact diagonalization studies on smaller clusters.
Cluster molecular magnets prove their potential for applications in quantum technologies, encouraging studies of quantum entanglement in spin systems. In the paper we discuss quantum entanglement properties of pentamer cluster composed of spins S=1/2 forming a tetrahedron with additional spin in its center, with geometry reproducing the smallest nonplanar graph. We model the system with isotropic Heisenberg Hamiltonian including external magnetic field and use exact diagonalization approach to explore the ground-state phase diagram and thermodynamic properties within canonical ensemble formalism. We focus the interest on two-spin entanglement quantified by Wootters concurrence. For ground state, we find two states with total cluster spin equal to 3/2 exhibiting entanglement, occurring preferably for antiferromagnetic interactions. For finite temperatures, we predict the presence of magnetic-field-induced entanglement as well as temperature-induced entanglement.
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.