The four-spin operators are suspected to have a significant amplitude and an impact on the collective properties in spin ladders and magnetic two-dimensional lattices. They originate from cyclic circulation of electrons (ring currents). Starting from a simple Hubbard Hamiltonian one may establish their form from a fourth order expansion of quasidegenerate perturbation theory. This form slightly deviates from the generally assumed biquadratic expression. From a quantitative point of view, their amplitude can be assessed from accurate ab initio explicitly correlated wave functions and energies on embedded clusters. The present work shows that symmetry-broken density functional theory calculations may also provide estimates of the four-body operator amplitudes, but reliable results require the use of a large Fock component in the exchange functional.
The four-spin operators are suspected to have a significant amplitude and an impact on the collective properties in spin ladders and magnetic two-dimensional lattices. They originate from cyclic circulation of electrons (ring currents). Starting from a simple Hubbard Hamiltonian one may establish their form from a fourth order expansion of quasidegenerate perturbation theory. This form slightly deviates from the generally assumed biquadratic expression. From a quantitative point of view, their amplitude can be assessed from accurate ab initio explicitly correlated wave functions and energies on embedded clusters. The present work shows that symmetry-broken density functional theory calculations may also provide estimates of the four-body operator amplitudes, but reliable results require the use of a large Fock component in the exchange functional.
In this thesis we discuss in detail the density-matrix renormalization group (DMRG) for simulating low-energy properties of quantum spin models. We implement an original DMRG routine on the S=1/2 antiferromagnetic Heisenberg chain and benchmark its efficiency against exact results (energies, correlation functions, etc.) as well as conformal field-theoretical calculations due to finite-size scaling (ground-state energy and spin gap logarithmic corrections). Moreover, we apply the DMRG to a two-leg square ladder system, where in addition to bilinear exchange terms, we also consider an additional cyclic four-spin ring-exchange. The transposition of four spins gives rise to biquadratic exchange terms which are non-trivial to implement in the DMRG. Intermediate results of the ring-exchange are presented along with the difficulties presently encountered.
Since the seminal works in the '50s, it has been known that a bath of noninteracting electrons mediates an interaction between local moments (such as nuclear spins or magnetic impurities) coupled to distant sites of the lattice. Recent efforts in simultaneous control of multiqubit arrays rely on defining quantum dots in environments which likewise contain electrons capable of mediating effective interactions. It is therefore interesting to analyze systems with more than two impurities in higher orders of perturbation theory. Here we derive and discuss the significance of the effective spin interactions between four impurities in fourth order of the coupling between the impurities and the substrate, mediated by conduction electrons of a two-dimensional square lattice. This effective interaction resembles the ring exchange---a coupling of four spins appearing in perturbative treatment of the Hubbard model. Despite it being a higher-order effect, it can dramatically influence the magnetic ordering. We show that as an effective interaction between impurity spins, the ring exchange can also compete with the more familiar two-spin interactions.
The subject of the present work are dimerized spin 1/2 systems in one, two and three dimensions.The models are investigated perturbatively by means of cluster expansion.This method makes it possible to study systems even in two and three dimension up to high orders.Starting with a generic HAMILTON-Operator which includes both bilinear and biquadratic interaction terms, the groundstate energies and the low lying excitations of the alternating spin chain und the spin ladder are calculated.Well-known results from BETHE ansatz and bosonization are used to verify the series.Based on this model, two speci£c one-dimensional problems are investigated in more detail: the £rst one is the incommensurate regime of the generalized spin ladder, which was previously analyzed by other approximation techniques.This regime is characterized by a minimum of the dispersion between 0 and π.The second problem deals with a quantum phase transition in the spin ladder with additional cyclic four-spin interaction (ring exchange).The excitation gap which is always present in the regular spin ladder can be reduced by taking the ring exchange into account.Within the region which is accessible by perturbation theory a transition line is presented on which the gap vanishes.For further characterization of the phase transition critical exponents are calculated which give a connection between exchange strength and gap size.Within the range of the higher-dimensional spin dimer systems the compounds KCuCl 3 , TlCuCl 3 (3d) and (C 4 H 12 N 2 )Cu 2 Cl 6 (2d) are studied.To clarify the microscopic interactions, dispersion relations which take all reasonable (i.e.evident from the chemical structure) interactions into account are computed.In the case of KCuCl 3 and TlCuCl 3 there are nine and in the case of (C 4 H 12 N 2 )Cu 2 Cl 6 there are ten spin spin interactions considered.The strength of the exchange parameters is determined by comparison with inelastic neutron scattering data.For the above-mentioned substances the interaction constants are evaluated to match the dispersion relation with the experimental data to a high accuracy.Finally, the dynamical structure factor which is limited to the singlet-triplet transition is calculated perturbatively for the compounds KCuCl 3 and TlCuCl 3 .Since it is proportional to the differential cross section, one is able to establish a relationship to inelastic neutron scattering experiments.
Various methods and approximation schemes are used to study many-electron interacting systems. Two important many-particle models, the Anderson model and the Hubbard model, and their electromagnetic properties have been investigated in many parameter regimes, and applied to physical systems. An Anderson single-impurity model Hamiltonian based calculation of the magnetic susceptibility is performed for YbN in the presence of crystal fields using an alteration of the Non-Crossing Approximation proposed by Zwicknagl et.al., incorporating parameters obtained from ab initio band structure calculations. It yields good agreement with experimental data. For the Anderson lattice model, a variational scheme which uses specific many-electron wavefunctions as basis is applied to both one- and two-dimensional systems represented by symmetric Anderson lattice Hamiltonians. Without much computational effort, the ground state energy is well approximated, especially in strong-coupling limit. Some electronic properties are examined using the variational ground state wavefunction. The one-dimensional Hubbard model has been solved exactly for small-size clusters by diagonalizing the Hamiltonian in the basis of many-electron Bloch states. The results for the energy spectrum and eigenfunctions of the ground state and low-lying excited states are presented. Also, mean field calculations of the two-dimensional single-band Hubbard model and Cu-O lattice model (three-band Hubbard model) are carried out for various physical quantities including the energy, occupation probability, staggered magnetization, momentum distribution Fermi surface and density of states, by using a projection operator formalism. To develop a systematic approach to solving many-electron problems, the many-particle partition function for the free electron gas system is explored using a cumulant expansion scheme. Starting from the ground state, the partition function can be approximated to any order in terms of excitation energy. Its application to interacting systems such as the Anderson model and the Hubbard model is briefly discussed.
For the CuO2 square lattice, as a parent system of high-Tc superconductors, a numerical study has suggested that the effective-spin model contains a large cyclic four-spin exchange interaction Jc. This paper investigates the effects of Jc upon magnetic Raman scattering and upon properties of the ground state by an exact numerical method mainly for a 16-site cluster. It is found that the main Raman peak with the B1g symmetry is shifted by 20% to the lower-energy side with a realistic magnitude for Jc, which is about one fourth the size of the nearest-neighbor exchange J. Accordingly a value of J extracted from analyses of experimental data becomes larger by 10% than that estimated with the use of the Heisenberg model. Furthermore the four-spin exchange enhances a shoulder due to multimagnon states at an energy of about 4J, and this result compares favorably with the experimental line shape. Detailed discussion is given on the effect of the four-spin exchange upon the ground-state properties such as the staggered magnetization, the ground-state energy, weight of various spin configurations, and the spin-spin correlation function.
Spin-only descriptions of the half-filled one-band Hubbard model are relevant for a wide range of Mott insulators. In addition to the usual Heisenberg exchange, many other types of interactions, including ring exchange, appear in the effective Hamiltonian in the intermediate coupling regime. In order to improve on the quantitative description of magnetic excitations in the insulating antiferromagnetic phase of copper-oxide (cuprate) materials, and to be consistent with band-structure calculations and photoemission experiments on these systems, we include second- and third-neighbor hopping parameters, ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$, into the Hubbard Hamiltonian. A unitary transformation method is used to find systematically the effective Hamiltonian and any operator in the spin-only representation. The results include all closed four-hop electronic pathways in the canonical transformation. The method generates many ring exchange terms that play an important role in the comparison with experiments on La2CuO4. Performing a spin-wave analysis, we calculate the magnon dispersion as a function of U, t, ${t}^{\ensuremath{'}}$, and ${t}^{\ensuremath{''}}$. The four parameters are estimated by fitting the magnon dispersion to the experimental results of Coldea et al. [Phys. Rev. Lett. 86, 5377 (2001)] for La2CuO4. The ring exchange terms are found essential, in particular to determine the relative sign of ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$, with the values found in good agreement with independent theoretical and experimental estimates for other members of the cuprate family. The zero-temperature sublattice magnetization is calculated using these parameters and also found to be in good agreement with the experimental value estimated by Lee et al. [Phys. Rev. B 60, 3643 (1999)]. We find a value of the interaction strength $U\ensuremath{\simeq}8t$ consistent with Mott insulating behavior.