Udagawa Masafumi

We study the interplay of topological bottlenecks and energetic barriers to<br /> equilibration in a Coulomb spin liquid where a short-range energetic coupling<br /> between defects charged under an emergent gauge field supplements their<br /> entropic long-range Coulomb interaction. This work is motivated by the<br /> prevalence of memory effects observed across a wide range of geometrically<br /> frustrated magnetic materials, possibly including the spontaneous Hall effect<br /> observed in Pr2Ir2O7. Our model is canonical spin-ice model on the pyrochlore<br /> lattice, where farther-neighbour spin couplings give rise to a nearest-neighbor<br /> interaction between topological defects which can easily be chosen to be<br /> unnatural or not, i.e. attractive or repulsive between defects of equal gauge<br /> charge. Among the novel features of this model are the following. After<br /> applying a field quench, a rich dynamical approach to equilibrium emerges,<br /> dominated by multi-scale energy barriers responsible for long-lived<br /> magnetization plateaux. These even allow for the metastability of a<br /> &quot;fragmented&quot; spin liquid, an elusive regime where partial order co-exists with<br /> a spin liquid. Perhaps most strikingly, the attraction produces clusters of<br /> defects whose stability is due to a combination of energetic barriers for their<br /> break-up and proximity of opposite charges along with an entropic barrier<br /> generated by the topological requirement of annihilating a defect only together<br /> with an oppositely charged counterpart. These clusters may take the form of a<br /> &quot;jellyfish&quot; spin texture, comprising an arrangement of same-sign gauge-charges,<br /> centered on a hexagonal ring with branches of arbitrary length. The ring<br /> carries a clockwise or counterclockwise circular flow of magnetisation. This<br /> emergent toroidal degrees of freedom provides a possibility for time reversal<br /> symmetry breaking with possible relevance to the spontaneous Hall effect<br /> observed in Pr2Ir2O7.

Finite-temperature (T) properties of a Kitaev model defined on a honeycomb lattice are investigated by a quantum Monte Carlo simulation, from the viewpoint of fractionalization of quantum S = 1/2 spins into two types of Majorana fermions, itinerant and localized. In this system, the entropy is released successively at two well-separated T scales, as a clear indication of the thermal fractionalization. We show that the high-T crossover, which is driven by itinerant Majorana fermions, is closely related with the development of nearest-neighbor spin correlations. On the other hand, the low-T crossover originates in thermal fluctuations of fluxes composed of localized Majorana fermions, by which the spectrum of itinerant Majorana fermions is significantly disturbed. As a consequence, in the intermediate-T range between the two crossovers, the system exhibits T-linear behavior in the specific heat and coherent transport of Majorana fermions, which are unexpected for the Dirac semimetallic spectrum in the low-T limit. We also show that the flux fluctuations tend to open an energy gap in the Majorana spectrum near the gapless-gapped phase boundary. Our results indicate that the fractionalization is experimentally observable in the specific heat, spin correlations, and transport properties.

We study the entanglement spectrum of the Hubbard model at half filling on a kagome lattice. The entanglement spectrum is defined by the set of eigenvalues of a reduced thermal density matrix, which is naturally obtained in the framework of the dynamical mean-field theory. Adopting the cluster dynamical mean-field theory combined with continuous-time auxiliary-field Monte Carlo method, we calculate the entanglement spectrum for a three-site triangular cluster in the kagome Hubbard model. We find that the results at the three-particle sector well capture the qualitative nature of the system. In particular, the eigenvalue of the reduced density matrix, corresponding to the chiral degrees of freedom, exhibits a characteristic temperature scale T-chiral, below which a metallic state with large quasiparticle mass is stabilized. The entanglement spectra at different particle number sectors also exhibit characteristic changes around T-chiral, implying the development of inter-triangular ferromagnetic correlations in the correlated metallic regime.

A three-dimensional Kitaev model on a hyperhoneycomb lattice is investigated numerically at finite temperature. The Kitaev model is one of the solvable quantum spin models, where the ground state is given by gapped and gapless spin liquids, depending on the anisotropy of the interactions. This model can be rewritten as a free Majorana fermion system coupled with Z2 variables. The density of states of Majorana fermions shows an excitation gap in the gapped region, while it is semimetallic in the gapless region reflecting the Dirac node. Performing the Monte Carlo simulation, we calculate the temperature dependence of the Majorana spectra. We find that the semimetallic dip is filled as temperature increases in the gapless region, but surprisingly, the spectrum develops an excitation gap in the region near the gapless-gapped boundary. Such changes of the low-energy spectrum appear sharply at the transition temperature from the spin liquid to the paramagnetic state. The results indicate that thermal fluctuations of the Z(2) fields significantly influence the low-energy state of Majorana fermions, especially in the spin liquid formation.

Effects of an open surface on a magnetic Chern insulator are investigated in comparison with those of an interface to a capping magnetic layer. In magnets, an open surface often perturbs the magnetic order by a reconstruction of the magnetic moment directions near the surface. On the other hand, in topological insulators, it leads to the formation of topologically protected surface states. These two contrasting effects may coexist in magnetic Chern insulators, which give rise to nontrivial surface reconstruction. For instance, the chiral edge current is largely enhanced by the edge reconstruction in a two-dimensional magnetic Chern insulator realized in a quarter-filled Kondo lattice model on a triangular lattice. We here show that the edge reconstruction can be described semiquantitatively by a simple junction model between the bulk topological magnetic state and a ferromagnetic capping layer. We further clarify how the chiral edge current is affected by the magnetic structure in the capping layer. Our results indicate that the topological edge state can be controlled magnetically through the junctions.

The quantum spin liquid is an exotic quantum state of matter in magnets. This state is a spin analog of liquid helium that does not solidify down to the lowest temperature due to strong quantum fluctuations. In conventional fluids, the liquid and gas possess the same symmetry and adiabatically connect to each other by bypassing the critical end point. We find that the situation is qualitatively different in quantum spin liquids realized in a three-dimensional Kitaev model; both gapless and gapped quantum spin liquid phases at low temperatures are always distinguished from the high-temperature paramagnet (spin gas) by a phase transition. The results challenge the common belief that the absence of thermodynamic singularity down to the lowest temperature is a symptom of a quantum spin liquid.

We discuss a number of illuminating results for tight-binding models supporting a band with variable Chern numbers and illustrate them explicitly for a simple class of two-banded models. First, for models with a fixed number of bands, we show that the minimal hopping range needed to achieve a given Chern number C increases with C and that the band flattening requires an exponential tail of long-range processes. We further verify that the entanglement spectrum corresponding to a real space partitioning contains C chiral modes and thereby complies with the archetypal correspondence between the bulk entanglement and the edge energetics. Finally, we address the issue of interactions and study the problem of two interacting particles projected to the flattened band as a function of the Chern number. Our results provide valuable insights for the full interacting problem of a partially filled Chern band at variable filling fractions and Chern numbers.

We study the spin-orbit coupling in spin-triplet Cooper pairs and clarify<br /> multiple superconducting (SC) phases in Sr$_2$RuO$_4$. Based on the analysis of<br /> the three-orbital Hubbard model with atomic LS coupling, we show some selection<br /> rules of the spin-orbit coupling in Cooper pairs. The spin-orbit coupling is<br /> small when the two-dimensional $\gamma$-band is the main cause of the<br /> superconductivity, although the LS coupling is much larger than the SC gap.<br /> Considering this case, we investigate multiple SC transitions in the magnetic<br /> fields for both H // [001] and H // [100] using the Ginzburg-Landau theory and<br /> the quasi-classical theory. Rich phase diagrams are obtained because the spin<br /> degree of freedom in Cooper pairs is not quenched by the spin-orbit coupling.<br /> Experimental indications for the multiple phases in Sr$_2$RuO$_4$ are<br /> discussed.

We study the reconstruction of chiral edge states in a magnetic Chern insulator in the Kondo lattice model on a triangular lattice at 1/4 filling. In this state, the spin scalar chirality associated with a four-sublattice noncoplanar magnetic order ensures the topological nature characterized by a nonzero Chern number. Performing a Langevin-based numerical simulation for finite-size clusters with an open boundary condition in one direction, we clarify how the magnetic and electronic properties are modulated near the edges of the system. As a result, we find that the magnetic state near the edges is reconstructed to develop ferromagnetic spin correlations. At the same time, the electronic state is also modified and the chiral edge current is enhanced. We discuss this enhancement from the viewpoint of the electronic band structure for gapless edge states.

We present the benchmark of the polynomial expansion Monte Carlo method to a Kondo lattice model with classical localized spins on a geometrically frustrated lattice. The method enables us to reduce the calculation amount by using the Chebyshev polynomial expansion of the density of states compared to a conventional Monte Carlo technique based on the exact diagonalization of the fermion Hamiltonian matrix. Further reduction is brought about by a real-space truncation of the vector matrix operations. We apply the method to the model with spin-ice type Ising spins on a three-dimensional pyrochlore lattice and carefully examine the convergence in terms of the order of polynomials and the truncation distance. We find that, in a wide range of electron density at a relatively weak Kondo coupling compared to the noninteracting bandwidth, the results by the polynomial expansion method show good convergence to those by the conventional method within reasonable numbers of polynomials. This enables us to study the systems up to 4 x 8(3) = 2048 sites, while the previous study by the conventional method was limited to 4 x 4(3) = 256 sites. On the other hand, the real-space truncation is not helpful in reducing the calculation amount for the system sizes that we reached, as the sufficient convergence is obtained when most of the sites are involved within the truncation distance. The necessary truncation distance, however, appears not to show significant system size dependence, suggesting that the truncation method becomes efficient for larger system sizes. (C) 2013 Elsevier B.V. All rights reserved.

Spin scalar chiral ordering gives rise to nontrivial topological characters and peculiar transport properties. Here, we examine how quantum spin fluctuations affect the spin scalar chiral ordering in itinerant electron systems. We consider the Kondo lattice model on a triangular lattice and perform the linear spin wave analysis in Chern insulator phases with spin scalar chiral ordering obtained in the case that the localized spins are classical. We find that, although quantum fluctuation destabilizes the spin scalar chiral phase at 3/4 filling that originates from the perfect nesting of the Fermi surface, it retains the phase at 1/4 filling that is induced by an effective positive biquadratic interaction. The reduction in the ordered magnetic moment by zero-point quantum fluctuation is considerably smaller than those in spin-only systems. The results suggest that the Chern insulator at 1/4 filling remains robust under quantum fluctuations. © 2013 The Physical Society of Japan.

We show the existence of invariant energy levels in a Kondo lattice model on<br /> an isolated complete graph, such as a triangle and a tetrahedron. These energy<br /> levels always have fixed eigenenergies $t \pm J/2$, irrespective of the<br /> configuration of localized moments ($t$ is the transfer integral of conduction<br /> electrons and $J$ is the spin-charge coupling constant). We also extend the<br /> analysis to geometrically frustrated lattices by using the complete graphs as<br /> basic building blocks. We show that the construction rule for the invariant<br /> energy levels leads to the existence condition of localized states, if the<br /> model is defined on the triangle-based line graphs, such as a kagome lattice.<br /> We further propose a procedure of engineering isolated flat bands with broken<br /> time-reversal symmetry, which are separated from other dispersive bands with<br /> finite energy gaps.

An effective Ising model for the spin-ice type Kondo lattice model is<br /> investigated by the classical Monte Carlo simulation. We clarify the magnetic<br /> phase diagram with four phases: ice-ferro, ice-(0,0,2\pi), 32-sublattice, and<br /> all-in/all-out ordered states. The result well reproduces the phase diagram of<br /> the Kondo lattice model studied previously [H. Ishizuka et al.: J. Phys. Soc.<br /> Jpn. 81 (2012) 113706], which suggests that the RKKY interactions up to third<br /> neighbors are sufficient to describe the magnetic properties of the itinerant<br /> electron model. We discuss the peculiar nature of phase transitions: the<br /> suppression of the critical temperatures down to zero temperature between two<br /> ice phases and the presence of the tricritical points on the phase boundary of<br /> the 32-sublattice ordered state.

We study the anomalous Hall effect due to noncoplanar magnetism on a pyrochlore structure. We focus on the frustration-induced spatial inhomogeneity of different magnetic low-temperature regimes, between which one can efficiently tune using an external magnetic field. We incorporate nonmagnetic scattering on a phenomenological level so that we can distinguish between the effects of short-range correlations and short-range coherence. We obtain a Hall conductivity (σH) as a function of field strength and direction which compares well to the experimental data of Pr2Ir 2O7. In particular, we show that the observed peak in σH for Hâ̂\[111] signals the crossover from zero-field spin ice to kagome ice. © 2013 American Physical Society.

A phase diagram of a spin-ice type Kondo lattice model, potentially relevant to metallic pyrochlore oxides, is obtained by the Monte Carlo simulation implementing the polynomial expansion technique up to a system size with 2048 sites. We identified a new 32-sublattice magnetic phase with concomitant charge disproportionation, along with other phases such as two-in two-out and all-in/all-out orders. The spin and charge pattern can be switched by an external magnetic field to a different one accompanied by the formation of a half-magnetization plateau.

The ground state of the periodic Anderson model on a triangular lattice is systematically investigated by mean-field approximation. We found that the model exhibits two different types of partially disordered states: one is at half filling and the other is at other commensurate fillings. In the latter case, the kinetic energy is lowered by forming an extensive network involving both magnetic and nonmagnetic sites, in sharp contrast to the former case in which nonmagnetic sites are rather isolated. This spatially extended nature of nonmagnetic sites yields a metallic partially disordered state by hole doping. We discuss the mechanism of the metal-insulator transition by the change in electronic structure.

We present a mechanism of resistivity minimum in conduction electron systems coupled with localized moments, which is distinguished from the Kondo effect. Instead of the spin-flip process in the Kondo effect, electrons are elastically scattered by local spin correlations which evolve in a particular way under geometrical frustration as decreasing temperature. This is demonstrated by the cellular dynamical mean-field theory for a spin-ice-type Kondo lattice model on a pyrochlore lattice. Peculiar temperature dependences of the resistivity, specific heat, and magnetic susceptibility in the non-Kondo mechanism are compared with the experimental data in metallic Ir pyrochlore oxides.

We reveal the significance of kinetic-driven multiple-spin interactions hidden in geometrically frustrated Kondo lattice models. Carefully examining the perturbation in terms of the spin-charge coupling up to the fourth order, we find that a positive biquadratic interaction is critically enhanced and plays a crucial role on stabilizing a spin scalar chiral order near 1/4 filling in a triangular lattice case. This is a generalized Kohn anomaly, appearing only when the second-order perturbation is inefficient because of the degeneracy under frustration. The mechanism is potentially common to frustrated spin-charge coupled systems, leading to emergence of unusual magnetic orders.

We present the results of Monte Carlo simulation for a Kondo lattice model in which itinerant electrons interact with Ising spins with spin-ice type easy-axis anisotropy on a pyrochlore lattice. We demonstrate the efficiency of the truncated polynomial expansion algorithm, which enables a large scale simulation, in comparison with a conventional algorithm using the exact diagonalization. Computing the sublattice magnetization, we show the convergence of the data with increasing the number of polynomials and truncation distance.

We investigate the effect of hole and electron doping to half-filling in the periodic Anderson model on a triangular lattice by the Hartree-Fock approximation at zero temperature. At half-filling, the system exhibits a partially disordered insulating state, in which a collinear antiferromagnetic order on an unfrustrated honeycomb subnetwork coexists with nonmagnetic state at the remaining sites. We find that the carrier doping destabilizes the partially disordered state, resulting in a phase separation to a doped metallic state with different magnetic order. The partially disordered state is restricted to the half-filled insulating case, while its metallic counterpart is obtained as a metastable state in a narrow electron doped region.

We investigate the ground state of the Kondo necklace model on<br /> geometrically-frustrated lattices by the variational Monte Carlo simulation. To<br /> explore the possibility of a partially-ordered phase, we employ an extension of<br /> the Yosida-type wave function as a variational state, which can describe a<br /> coexistence of spin-singlet formation due to the Kondo coupling and magnetic<br /> ordering by the Ruderman-Kittel-Kasuya-Yosida interaction. We show the<br /> benchmark of the numerical simulation to demonstrate the high precision brought<br /> by the optimization of a large number of variational parameters. We discuss the<br /> ground-state phase diagram for the model on the kagome lattice in comparison<br /> with that for the triangular-lattice case.

We report our theoretical results on the emergence of a partially disordered state at zero temperature and its detailed nature in the periodic Anderson model on a triangular lattice at half-filling. The partially disordered state is characterized by the coexistence of a collinear antiferromagnetic order on an unfrustrated honeycomb subnetwork and nonmagnetic state at the remaining sites. This state appears with the opening of a charge gap between a noncollinear antiferromagnetic metal and Kondo insulator while the hybridization and Coulomb repulsion are changed. We also find a characteristic crossover in the low-energy excitation spectrum as a result of the coexistence of magnetic order and nonmagnetic sites. The result demonstrates that the partially disordered state is observed distinctly even in the absence of spin anisotropy, in marked contrast to the partial Kondo screening state found in the previous study for the Kondo lattice model.

We present our theoretical results on the ground states in the layered triangular-lattice compounds ANiO(2) (A = Na, Li, Ag). To describe the interplay between charge, spin, orbital, and lattice degrees of freedom in these materials, we study a doubly degenerate Hubbard model with electron-phonon couplings by the Hartree-Fock approximation combined with the adiabatic approximation. In a weakly correlated region, we find a metallic state accompanied by root 3 x root 3 charge ordering. On the other hand, we obtain an insulating phase with spin-ferro and orbital-ferro ordering in a wide range from intermediate to strong correlation. These phases share many characteristics with the low-temperature states of AgNiO2 and NaNiO2, respectively. The charge-ordered metallic phase is stabilized by a compromise between Coulomb repulsions and effective attractive interactions originating from the breathing-type electron-phonon coupling as well as the Hund's-rule coupling. The spin-orbital-ordered insulating phase is stabilized by the cooperative effect of electron correlations and the Jahn-Teller coupling, while the Hund'-rule coupling also plays a role in the competition with other orbital-ordered phases. The results suggest a unified way of understanding a variety of low-temperature phases in ANiO(2). We also discuss a keen competition among different spin-orbital-ordered phases in relation to the puzzling behavior observed in LiNiO2.

We report the results of our theoretical and numerical study on electronic and transport properties of fermion systems with charge frustration. We consider an extended Falicov-Kimball model in which itinerant spinless fermions interact repulsively by U with localized particles whose distribution satisfies a local constraint under geometrical frustration, the so-called ice rule. Electronic states of the itinerant fermions are studied by approximating the statistical average by the arithmetic mean over different configurations of localized particles under the constraint. We numerically calculate the density of states, optical conductivity, and inverse participation ratio for models on the pyrochlore, checkerboard, and kagome lattices, and discuss the nature of metal-insulator transitions at commensurate fillings. The results are compared with exact solutions for models on Husimi cacti as well as with numerical results for completely random distributions of localized particles. As a result, we show that the ice-rule local constraint leads to several universal features in the electronic structure common to different lattice structures; a charge gap opens at a considerably small U compared to the bandwidth, and the energy spectrum approaches a characteristic form in the large-U limit, that is, the noninteracting tight-binding form in one dimension or a d-functional peak. In the large-U region, the itinerant fermions are confined in the macroscopically degenerate ice-rule configurations, which consist of a bunch of one-dimensional loops: We call this insulating state the charge ice. On the other hand, transport properties are much affected by the geometry and dimensionality of the lattices; e.g., the pyrochlore lattice model exhibits a transition from a metallic to the charge-ice insulating state with increasing U, while the checkerboard lattice model appears to show Anderson localization before opening a gap. Meanwhile, in the kagome lattice case, we do not obtain clear evidence of Anderson localization. Our results elucidate the universality and diversity of phase transitions to the charge-ice insulator in fully frustrated lattices.

We have studied the effects of electron correlation on Van Vleck susceptibility (chi vv) in transition metal compounds. A typical crossover behavior is found for the correlation effect on chi vv as sweeping spin orbit interaction, lambda. For a small lambda, orbital fluctuation plays a dominant role in the correlation enhancement of chi vv; however, the enhancement rate is rather small. In contrast, for an intermediate lambda, chi vv shows a substantial increase, accompanied by the development of spin fluctuation. We will discuss the behavior of chi vv in association with the results of Knight-shift experiments on Sr2RuO4 and an anomalously large magnetic susceptibility observed for 5d Ir compounds.

We investigate the effect of geometrical frustration on the competition between the Kondo coupling and the Ruderman-Kittel-Kasuya-Yosida interaction in Kondo lattice systems. By variational Monte Carlo simulations, we reveal an emergent quantum phase with partial ordering in which the frustration is relieved by forming a magnetic order on a sublattice and leaving the rest in the Kondo screening with spin-singlet formation. The role of quantum fluctuations and spin-charge interplay is elucidated.

An exact solution is obtained for a model of itinerant electrons coupled to ice-rule variables on the tetrahedron Husimi cactus, an analogue of the Bethe lattice of corner-sharing tetrahedra. It reveals a quantum critical point with the emergence of non-Fermi-liquid behavior in melting of the "charge ice" insulator. The electronic structure is compared with the numerical results for the pyrochlore-lattice model to elucidate the physics of electron systems interacting with the tetrahedron ice rule.

We investigate the quasiparticle-mass enhancement in the Hubbard model on the frustrated kagome lattice by using a cluster extension of the dynamical mean-field theory. By analyzing the cluster density matrix, we find a hierarchy of energy scale among charge, spin, and chirality degrees of freedom. A large amount of entropy associated with the chirality is released at a much lower temperature than other energy scales for spin and charge fluctuations, leading to a sharp peak in the specific heat and the single-particle spectrum. The results manifest a generic mechanism of mass enhancement driven by an emergent composite degree of freedom under geometrical frustration.

Effect of geometrical frustration in strongly-correlated metallic region is studied for the Hubbard model on the kagome lattice at half filling by a cluster extension of the dynamical mean-field theory combined with a continuous-time auxiliary-field quantum Monte Carlo method. We find that the electron correlation enhances the spin chirality in both vector and scalar channels. The chirality grows as decreasing temperature and exhibits a peak at a low temperature, indicating a new energy scale under strong correlation. The peak temperature is considerably lower than that for the local spin moment, namely, the characteristic temperatures for the chirality and the local moment are well separated. This is a signature of separation between spin and chiral degrees of freedom in the correlated metallic regime under geometrical frustration

We study current-induced dynamics of a magnetic domain wall by solving a time-dependent Schrodinger equation combined with Landau-Lifshitz-Gilbert equation in a one-dimensional electron system coupled to localized spins. Two types of domain-wall motions are observed depending on the hard-axis anisotropy, K-perpendicular to, of the localized spin system. For small values of K-perpendicular to, the magnetic domain wall shows a streaming motion driven by spin transfer. In contrast, for large values of K-perpendicular to, a stick-slip motion driven by momentum transfer is obtained. We clarify the origin of these characters of domain-wall motions in terms of the dynamics of one-particle energy levels and distribution functions.

We study the Hubbard model on a geometrically-frustrated hyperkagome lattice by a cluster extension of the dynamical mean field theory. We calculate the temperature (T) dependences of the specific heat (C) and the spin-lattice relaxation time (T-1) in correlated metallic region. C/T shows a peak at T = T-p1 and rapidly decreases as T -&gt; 0. On the other hand, 1/T1T has a peak at a higher temperature T-p2 than T-p1, and largely decreases below T-p2, followed by the Korringa law 1/T-1 proportional to T as T -&gt; 0. Both peak temperatures are suppressed and the peaks become sharper as electron correlation is increased. These behaviors originate from strong renormalization of the energy scales in the peculiar electronic structure in this frustrated system; a pseudo-gap like feature, the van-Hove singularity, and the flat band. The results are discussed in comparison with the experimental data in the hyperkagome material, Na4Ir3O8.

We present a scenario for the peculiar coexistence of charge fluctuations observed in quasi-2D 1/4-filled organic conductors theta-(BEDT-TTF)(2)X in the quantum critical regime where the charge ordering is suppressed down to zero temperature. The scenario is explored in the extended Hubbard model including electron-phonon couplings on an anisotropic triangular lattice. We find that the coexisting fluctuations emerge from two different instabilities, the "Wigner crystallization on a lattice" driven by the off-site Coulomb repulsion and the charge-density-wave formation due to the nesting of the Fermi surface, not from phase competition or real-space inhomogeneity. This mechanism explains the contrastive temperature dependence of two fluctuations in experiments.

We have studied the superconducting state of Sr2RuO4 under a magnetic field parallel to the superconducting plane. We show that due to a weak spin-orbit coupling, a nonunitary k(y)(z - i alpha y) state is formed right at H-c2, which then changes to a unitary k(y)z state as the magnetic field is lowered. On the basis of this crossover, we address the origin of the observed double peaks of specific heat and the disappearance of the double peaks at low fields.

We have studied the field-orientational dependence of zero-energy density of<br /> states (FODOS) for a series of systems with different Fermi surface and<br /> superconducting gap structures. Instead of phenomenological Doppler-shift<br /> method, we use an approximate analytical solution of Eilenberger equation<br /> together with self-consistent determination of order parameter and a<br /> variational treatment of vortex lattice. First, we compare zero-energy density<br /> of states (ZEDOS) when a magnetic field is applied in the nodal direction<br /> ($\nu_{node}(0)$) and in the antinodal direction ($\nu_{anti}(0)$), by taking<br /> account of the field-angle dependence of order parameter. As a result, we found<br /> that there exists a crossover magnetic field $H^*$ so that $\nu_{anti}(0) &gt;<br /> \nu_{node}(0)$ for $H &lt; H^*$, while $\nu_{node}(0) &gt; \nu_{anti}(0)$ for $H &gt;<br /> H^*$, consistent with our previous analyses. Next, we showed that $H^*$ and the<br /> shape of FODOS are determined by contribution from the small part of Fermi<br /> surface where Fermi velocity is parallel to field-rotational plane. In<br /> particular, we found that $H^*$ is lowered and FODOS has broader minima, when a<br /> superconducting gap has point nodes, in contrast to the result of the<br /> Doppler-shift method. We also studied the effects of in-plane anisotropy of<br /> Fermi surface. We found that in-plane anisotropy of quasi-two dimensional Fermi<br /> surface sometimes becomes larger than the effects of Doppler-shift and can<br /> destroy the Doppler-shift predominant region. In particular, this tendency is<br /> strong in a multi-band system where superconducting coherence lengths are<br /> isotropic. Finally, we addressed the problem of cusp-like singularity in<br /> YNi$_2$B$_2$C and present a possible explanation of this phenomenon.

We study the vortex state of a layered superconductor with vertical line<br /> nodes on its Fermi surface when a magnetic field is applied in the ab-plane<br /> direction. We rotate the magnetic field within the plane, and analyze the<br /> change of low-energy excitation spectrum. Our analysis is based on the<br /> microscopic<br /> Bogoliubov-de Gennes equation and a convenient approximate analytical method<br /> developped by Pesch and Dahm. Both methods give a consistent result. Near the<br /> upper critical field H$_{c2}$, we observe a larger zero-energy density of<br /> states(ZEDOS) when the magnetic field is applied in the nodal direction, while<br /> much below H$_{c2}$, larger ZEDOS is observed under a field in the anti-nodal<br /> direction. We give a natural interpretation to this crossover behavior in terms<br /> of momentum distribution of low-energy quasiparticles. We examine the recent<br /> field angle variation experiments of thermal conductivity and specific heat.<br /> Comparison with our results suggest that special care should be taken to derive<br /> the position of line nodes from the experimental data. Combining the<br /> experimental data of the specific heat and our analyses, we conclude that<br /> Sr$_2$RuO$_4$ has vertical line nodes in the direction of the a-axis and the<br /> b-axis.