In irreversible aggregation processes without a gelation transition the cluster size distribution approaches a scaling form, c"(t) -s 2@(k/s). Usking Smoluchowski's coagulation equation we determine the exponents in the mean cluster size s(t) -t' (t~) and in the small-and large-x behavior of the scaling function @(x). Depending on certain characteristics of the coagulation coefficients, @(x) -x ' (x 0) or $(x) -exp( -x") (x 0) with p, some negative constant. In aggregation processes with gelation a similar scaling form is obtained as t approaches the gel point.PACS numbers: 64.60. -i, 05.50. +q, 64.75. +g, 82.35.+t To study the kinetics of irreversible aggregation and clustering phenomena, in particular the time evolution of the cluster size distribution ck(t), Smoluchowski s coagulation equation is one of the few available, and also one of the most widely used, theoretical tools in many fields of physics, astronomy, polymer physics, colloid chemistry, atmospheric physics, biology, and technology. ' 5 It reads Ck= 2 g K(l,J)ctcJ Ck XK(k,J) Jc, i+ j=k j=l where the coagulation kernel K (i j ) represents the rate coefficient for a specific clustering mechanism between clusters of sizes i and j. We distinguish gelling and nongelling mechanisms.In the former the mean cluster size s(t) diverges as t approaches the gel point t, ; in the latter s (t) keeps increasing with time.It is known from exact solutions, ' coagulation experiments, and computer simulations that the size distribution approaches a scaling form, c"(t) -s '@(k/s), as soon as s(t) becomes large compared to the characteristic size at the initial time. The important point is that the k and t dependence of ck (t) is given through a universal function of a single variable, k/s(t), that does not depend on the initial distribution. For a limited number of coagulation mechanisms, all belonging to class III (see below), Friedlander's theory of selfpreserving spectra (SPS theory) gave a satisfactory explanation of the experimental observations on Brownian coagulation in the hydrodynamic and molecular regime, although the experimental data at large k and t are rather poor.By generalizing the SPS theory we can give a unifying description of the scaling behavior occurring in gelling and nongelling systems, described by Smoluchowski's equation.Our generalization covers all coagulation kernels K (i j) that are homogeneous functions of i and j, and includes large classes of models, for which the original SPS theory is not valid, e.g. , K(ij) =i +j.Since Smoluchowski's equation with a homogeneous kernel is invariant under a group of similarity transformations, it admits exact similarity or scaling solutions, 2 3 that can be solved from a nonlinear integral equation and whose properties are analyzed in this Letter. The basic assumption of our method is that the solutions for general initial distributions indeed approach the special similarity solution. With the help of the integral equation we can determine scaling functions and related exponents, analytically or nu...

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The anisotropic degenerate two-orbital Hubbard model is studied within dynamical mean-field theory at low temperatures. High-precision calculations on the basis of a refined quantum Monte Carlo (QMC) method reveal that two distinct orbital-selective Mott transitions occur for a bandwidth ratio of 2 even in the absence of spin-flip contributions to the Hund exchange. The second transition -not seen in earlier studies using QMC, iterative perturbation theory, and exact diagonalization -is clearly exposed in a low-frequency analysis of the self-energy and in local spectra. PACS numbers: 71.30.+h, 71.10.Fd, 71.27.+a The Mott-Hubbard metal-insulator transition -a nonperturbative correlation phenomenon -has been a subject of fundamental interest in solid state theory for decades. 1 Recently, this field became even more exciting by the discovery 2,3 of a two-step metal-insulator transition in the effective 3-band system Ca 2−x Sr x RuO 4 , for which the name orbital-selective Mott-transition (OSMT) was coined. 4 The Ca 2−x Sr x RuO 4 system was investigated theoretically in detail by Anisimov et al. 4 within the local density approximation (LDA and LDA+U) and within dynamical mean-field theory 5 (DMFT) solved using the non-crossing approximation (NCA). The underlying assumption of a correlation (rather than lattice-distortion) induced OSMT found support in further band structure calculations 6,7 and strong-coupling expansions for the localized electrons in the orbital-selective Mott phase. 8 Microscopic studies of the OSMT usually consider the 2-band Hubbard model H = H 1 + H 2 , wherehopping between nearest-neighbor sites i, j with amplitude t m for orbital m ∈ {1, 2}, intra-and interorbital Coulomb repulsion parametrized by U and U ′ , respectively, and Ising-type Hund's exchange coupling; n imσ = c † imσ c imσ for spin σ ∈ {↑, ↓}. In addition,contains spin-flip and pair-hopping terms (with1 ≡ 2, ↑ ≡↓ etc.). In cubic lattices, the Hamiltonian is invariant under spin rotation, J z = J ⊥ ≡ J; furthermore U ′ = U − 2J. In the following, we refer to H 1 + H 2 in this spin-isotropic case as the J-model and to the simplified Hamiltonian H 1 as the J z -model. Liebsch 9,10,11 questioned the OSMT scenario for Ca 2−x Sr x RuO 4 on the basis of finite-temperature quantum Monte Carlo (QMC) calculations (within DMFT) for the J z -model using J z = U/4, U ′ = U/2, and semielliptic "Bethe" densities of states with a bandwidth ratio W 2 /W 1 = 2. Additional studies using iterative perturbation theory (IPT) 11 seemed 12 to confirm his conclusion of a single Mott transition of both bands at the same critical U -value. Meanwhile, Koga et al. found an OSMT using exact diagonalization (ED), applied to the full J-model, 13 but not for the J z -model. 14 Consequently, the OSMT scenario was attributed to spin-flip and pairhopping processes.Very recently, four preprints appeared, 15,16,17,18 in which the OSMT was investigated in detail within the DMFT framework. Ref. 15 applied the Gutzwiller variational approach and ED to the J-model at...

The extended Hubbard model with on-site interaction U and nearest-neighbor interaction V/d is studied at half filling in high dimensions (^^1). At small U and K the critical temperature and the order parameter are calculated including the \/d corrections. The results diifer from those obtained in the Hartree approximation by a factor of order unity. At K/t/ = l/2 a transition is found from an antiferromagnetic to a charge-density-wave phase. A similar transition is found at large U and K, suggesting that the transition is present for all U and K satisfying K/t/==l/2.