The irreducible (physically relevant) representations selected in the previous section have a further important property, calledCluster property cluster property.
This article is the first of a trilogy that addresses various aspects of the perturbative response of general quantum systems, with possibly nontrivial ground state geometry, beyond linear order. Here, we use group theoretical considerations to investigate the structure of retarded correlators, and demonstrate how they decompose according to irreducible representations of a `time-reversal group' and relevant permutation groups, with the former probing dissipative and time-reversal properties, and the latter discerning configurational properties -- longitudinal, transverse and their generalizations. We establish second order fluctuation-dissipation and fluctuation-reaction theorems, and connect them to well-known second order transport effects such as the shift and injection currents. Exploiting the Schur-Weyl duality between irreducible representations of general linear groups and irreducible representations of permutation groups, we show how to decide which terms in the decomposition based on the latter can be supported by different crystals described via the 32 point groups, and perform the full point group classification for rank 3 and 4 polar and axial tensors. Our results provide a formal basis for the extraction of uniquely differing physical effects from retarded correlators. Applications to second order charge current responses in selected cases are given in part III.
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.
In this paper we describe and apply integral quantization, a procedure based on operator-valued measures and the resolution of the identity.We explore its covariance properties in the context of group representation theory.Three applications based on group representations are carried out.The first one concerns the covariant integral quantization based on the spin one-half irreducible representation of SU(2).In this case we show that the quantization of both the quadratic Hopf variables and the Euler angles reproduces the SU(2) Lie algebra.The second example revisits integral quantization based on the Weyl-Heisenberg group.By completing previous results, we show the universality of the canonical commutation relation in such quantizations.In the last example we revisit and enrich the integral quantization based on the affine group of the real line and we give a short account of a relevant application.