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In strongly correlated systems, some of the low-energy excitations are enhanced. In geometrically frustrated systems, due to the absence of long-range order, there remains (pseudo-) degeneracy associated with the low-energy excitations at low temperatures. Lifting of degeneracy in both Mott insulating systems and itinerant electron systems are discussed by considering the pyrochlore lattice as an example.

Yb4As3 is a unique 4f compound having Yb3+ chains below the charge-ordering temperature T c =290 K . The Yb3+ chains are believed to be responsible for the most part of anomalous properties this material shows at low temperatures. First a microscopic Hamiltonian is derived for the Yb3+ chains from its structure and the crystal-field wave functions to show that a staggered g tensor and a staggered Dzyaloshinsky–Moriya (DM) term are present. This Hamiltonian naturally explains the energy gap formed for the magnetic field applied perpendicular to the chain. There are some experimental features, which cannot be explained by the single-chain Hamiltonian and may be due to interchain couplings. It is pointed out that the interchain DM interaction should be nonzero and may lead to an upturn of susceptibility observed at low temperatures for the magnetic field parallel to the chain.

In the conference various problems in strongly correlated electron systems were discussed both theoretically and experimentally. Significant progress has been achieved since the last SCES conference in Paris and a few problems have been solved. However, many important issues are left to be investigated or require further studies. From my personal point of view, some important theoretical questions for future studies are summarized.

The existence of the first order phase transition in the ferromagnetic phase is important for the occurrence of the superconductivity in UGe 2 . In order to clarify the mechanism of the first order phase transition, we have constructed a microscopic Hamiltonian of UGe 2 taking account of the orbital degeneracy of the 5-f electrons and investigated the ground state phase diagram of the Hamiltonian by the Hartree–Fock approximation. A rich ground state phase diagram which includes an SDW, a CDW, and a ferromagnetic phase modulated by an SDW component is obtained because of the orbital degeneracy. We have found two kinds of uniform ferromagnetic phases, which are separated by a first order phase transition, by adjusting the number of f electrons per uranium atom to be about 2.60. We discuss differences of the physical properties between the two ferromagnetism by comparing the Fermi surfaces and other properties.

The paramagnetic metallic phase of the one-dimensional Kondo lattice model is studied by the density matrix renormalization group method. We calculate spin excitation gap and Friedel oscillations in finite systems. The finite size corrections of the spin gap and the long-range Friedel oscillations are consistent with a Tomonaga-Luttinger liquid. The Friedel oscillations reflect the charge-charge and spin-spin correlation functions. From the period of the Friedel oscillations, it is concluded that the Fermi surface is large including both the conduction electrons and the localized spins, kF = π(1 + nc)/2 where nc is the density of conduction electrons.

We study a theoretical model for CaV3O7: the 14-depleted square spin-12 Heisenberg model which includes both the nearest-neighbor coupling (J) and the next-nearest-neighbor coupling (J′), where J and J′ are antiferromagnetic. Recent experiments of the neutron diffraction by Harashima et al. report a magnetic collinear ordering at low temperatures, which may be called as a stripe phase. Most remarkable feature we find in CaV3O7 is that the stripe phase, which is not stable in the classical theory, is stabilized by the quantum fluctuations. The modified spin wave theory predicts that the stripe phase is realized for J′/J > 0.69. In CaV3O7, the coupling constants are estimated as J ∼ J′ by comparing the theoretical and experimental results.

Recently, there has been great progress in the theoretical understanding of the Kondo lattice model which is a basic model for the heavy electron phenomena. The ground-state phase diagram of the Kondo lattice model has been completed in one dimension, where there are three different phases: the ferromagnetic metallic phase, the paramagnetic metallic one and the insulating spin liquid phase. The last one is a theoretical prototype for the Kondo insulators. The ferromagnetic metallic phase is an example of magnetic phases with reduced moment. The paramagnetic metallic phase is the region where we expect the typical heavy electron phenomena. In one dimension it belongs to the universality class of the Tomonaga-Luttinger liquids. From the period of the Friedel oscillations it is concluded that kF is determined by the density of conduction electrons and the f-spins.

In bulk, an unconventional superconductor with two components of the order parameter can have the anisotropic internal vortex structure, since it is possible for the two components to have their own cores at different places. In a thin film there is a critical thickness dc for such vortices where the first critical magnetic field Hc1 has an anomaly. The vortex below the critical thickness is fractionally quantized where the core of one component of the order parameter is pushed out from the surface. The zero supercurrent at one point on the surface is a characteristic of the fractionally quantized vortex.

The one-dimensional Kondo-lattice model with frustrating next-nearest-neighbor hopping has a paramagnetic ground state in the strong coupling limit. This new fixed point of the Kondo-lattice model belongs to the universality class of Luttinger liquids. The Luttinger liquid phase has a large Fermi surface whose area is determined by the total number of conduction electrons and localized spins.

We study the ground-state properties of the spin-1 pyrochlore antiferromagnet by applying the method of the valence-bond-solid-type spin break-up to the pyrochlore lattice. The tetrahedral spin-singlet states with E representations create the many-fold degeneracy for low lying energy states, which are lifted by some couplings. We find that the magneto-elastic couplings lead to the spontaneous lattice distortion and the inter-tetrahedral couplings stabilize the parity-broken ground state.

We show that the structural phase transition which occurs in PrFe 4 P 12 and PrRu 4 P 12 coincides with anti-quadrupolar ordering of the Pr 4f 2 electrons. Coupling between phonon modes of the displacements of the Fe/Ru ions and the Pr 4f 2 ground state doublet leads to a symmetry lowering (1,1,−2) type distortion of the Fe/Ru positions and a simultaneous splitting of the 4f 2 doublet into eigenstates with quadrupole moments ±8. The sign of both the distortion and the quadrupole moment alternate from site to site, resulting in anti-quadrupolar ordering.

The Kondo lattice model can be characterized by two parameters: one is the exchange coupling constant J t and the other is the density of conduction electrons ρ= N L . In the low-density limit the Kondo lattice model has a ferromagnetic ground state for any J. In one dimension one can show further that the ground state is ferromagnetic also in the strong coupling limit at any ρ, while it has a spin liquid ground state at the half-filling. The phase diagram of the Kondo lattice model in one dimension is discussed in the entire space of the parameters based on exact diagonalization.

We measured the magnetic susceptibility and the specific heat of a single-domain sample of Yb4As3 under the magnetic field along the trigonal axis. The magnetic susceptibility in the charge ordered phase is very anisotropic and the upturn at low temperatures becomes smaller than that of the multi-domain sample. The suppression in the specific heat by applied fields parallel to the trigonal axis becomes weak compared with that of the multi-domain sample. These results indicate that this anisotropy is explained by the alternating Dzyaloshinsky–Moriya interaction and the effect of the alternating off-diagonal g-tensors expected from the symmetry of Yb3+ sites on the one-dimensional chains.

The ground-state phase diagram of the one-dimensional Kondo lattice model with an anisotropic exchange is investigated. Four different phases are identified, two types of Neel phases, the Haldane phase, and the Kondo spin liquid phase. The boundaries between these phases are determined by using mapping to effective models in the strong coupling region combined with numerical methods of the exact diagonalization and the density matrix renormalization group.

Complex magnetic phase diagrams of the trivalent cerium monopnictides are widely known as an example of the devil's staircase. We present an effective Hamiltonian for CeX which explains their complex magnetic phase diagrams in a unified way. The effective Hamiltonian consists of semimetallic conduction bands and localized f spins. A new feature is the inter-band transitions with spin exchange which frustrates with the usual intra-band Kondo couplings.

Dynamical correlation functions of the one-dimensional Kondo lattice model (1D KLM) are calculated as a reference for dynamic properties of Kondo insulators. In the strong coupling limit of the 1D KLM, the antiferromagnetic dynamic spin susceptibility consists of a peak at the lowest spin excitation energy Δ s and multiple peaks in the higher energy region. The optical conductivity has a dominant peak just above the charge excitation energy Δ c . The ratio Δ c /Δ s is larger than 3/2, and is a useful quantity to measure the electron correlation in a Kondo insulator.

It is shown by exact diagonalization that the one-dimensional Kondo lattice has a spin liquid ground state at half filling for all nonzero Kondo couplings. The antiferromagnetic and ferromagnetic coupling regions are characterized by the universality classes of the coherent Kondo singlet and the Haldane gap state respectively.

We discuss structures of the differential conductance in the limit of the large bias voltage and magnetic field by employing the self-consistent perturbation theory up to the fourth order in the Coulomb interaction. It is found that the dips for the strong correlation regime and the peaks for the weak correlation regime, respectively, appear in the differential conductance when the Zeeman energy corresponds to the bias voltage.

All possible couplings between a half-integral localised angular momentum in a cubic environment and lattice distortions are examined. When the cubic symmetry is broken the result can only be quadrupolar ordering. Application to NdFe4P12, SmRu4P12 and GdRu4P12 is discussed.

SrCu2(BO3)2 is a unique two-dimensional spin gap system. Furthermore, two magnetization plateaus corresponding to 1 4 and 1 8 of the full Cu moment are observed. We present a theoretical model of these plateaus. In the Heisenberg model for this material triplet excitations are almost localized, which makes them easy to crystallize, leading to the magnetization plateaus. For small magnetization we can map this Heisenberg model to an Ising model and the magnetization curve is calculated. We find 1 2 -, 1 3 -, 1 4 - and 1 8 -plateaus.

In high- T c cuprates, the Hall coefficient ( R H ) and the resistivity ( ρ ) are known to show interesting anomalous temperature dependences: R H ∝ T −1 and ρ ∝ T in a wide temperature range. Here, we give the theoretical explanation of these long-standing problems from the standpoint of the nearly antiferromagnetic Fermi liquid. We study intensively the vertex corrections for the current based on the conserving approximation, and find out that they give the anomalous behaviors both for R H and ρ in the under-doped region. The obtained numerical results for R H and ρ by using the fluctuation-exchange (FLEX) approximation are highly consistent with experiments.

We study the one–dimensional SU(4) spin–orbital model by means of the abelian bosonization as well as the Lanczos and the density matrix renormalization group methods. Starting from the SU(4) gapless model, we introduce the SU(2) Heisenberg-type perturbations for the spin and the orbital degrees of freedom. We investigate how the orbital gap and/or the spin gap are generated by these perturbations. We find the novel phases which have gapless spin (or orbital) excitations in addition to the known phases with both excitations being gapful or gapless.

We numerically study several lower symmetric spin–orbit coupled models in one dimension as the generalization of the mostly simplified SU(4) model. First, the SU(4) model under uni-axial magnetic fields is investigated to observe the crossover phenomena between the SU(4) and SU(2) symmetric limits, which correspond to zero and fully polarized magnetizations, respectively. As another example, we study the SU(2)⊗SU(2) symmetric model, deviating from the SU(4) symmetric condition. We show the existence of the new critical phases around the SU(4) symmetric point, spreading both sides of the previously known gapless line.

The groundstate phase diagram of the one dimensional Kondo lattice model is obtained by combining exact results at various limiting cases and numerial exact diagonalization study. In particular it is shown that the groundstate at half filling is a spin liquid state for any exchange coupling. Crossover behavior in the Kondo spin liquid state among its limiting cases (the hybridization gap state, the local singlet and the collective singlet) is discussed in connection with the Kondo insulators.

The structure of vortices of an unconventional superconductor is studied in a thin film using a Ginzburg-Landau theory for a two-component order parameter. It is found that vortices can decay into quasi-independent fractional vortices.

The low-concentration limit of the Kondo lattice model is analyzed. The ground state for one conduction electron is ferromagnetic in any dimension (rigorous result). A spin wave analysis in 3D shows that the ferromagnetic state is suppressed by spin fluctuations above a critical band filling n c ≈0.6%.

The one-dimensional Kondo lattice is diagonalized numerically. It is shown that the ground state is ferromagnetic at low densities of conduction electrons. On the other hand at half-filling the ground state is a singlet with strong antiferromagnetic correlations which are the main mechanism to compensate the localized spins.