Topology in Ordered Phases

PREFACE.

CONTENTS.

TOP 2005 symposium.

Group Photo.

No abstract received.

An introductory talk on a purpose of the project "Topological Science and Technology" is given so as specialists in various fields can share a common perception.

The phenomenon of quantum number fractionalization is explained. The relevance of non-trivial phonon field topology is emphasized.

Geometric phases accompanying adiabatic processes in quantum systems can be utilized as unitary gates for quantum computation. Optimization of control of the adiabatic process naturally leads to the isoholonomic problem. The isoholonomic problem in a homogeneous fiber bundle is formulated and solved completely.

We report the discovery of a Möbius crystal of NbSe₃, conventionally grown as ribbons and whiskers. We also reveal their formation mechanisms of which two crucial components are the spherical selenium (Se) droplet, which a NbSe₃ fiber wraps around due to surface tension, and the monoclinic (P2₁/m) crystal symmetry inherent in NbSe₃, which induces a twist in the strip when bent. Our crystals provide a non-fictitious Möbius world governed by a non-trivial real-space topology.

The superconducting states on a Möbius strip are studied based on Ginzburg-Landau theory and Bogoliubov-de Gennes theory. It is shown that, in a Möbius strip made of an anisotropic superconductor, the Little-Parks oscillation, which occurs when an magnetic flux is threading a superconducting ring, is significantly modified. Especially, when the flux is close to a half-odd-integer times the flux quantum, a new type of states appears, which we call the "nodal state". In these states the superconducting gap has a node in the middle of the strip along the circumference. We discuss the stability and the electronic properties of these states in two-dimensional case, where the thickness of the strip is negligible. A possible extension of this analysis to the thicker strips is also addressed.

Structure analyses of topological crystals were done using intensive synchrotron radiation from SPring-8. Firstly, directions of the crystal axes were determined using a highly sensitive oscillation camera under vacuum. The atomic arrangement of topological crystals was then determined by the newly developed X-ray camera under vacuum. Small but systematic shrinkage of the a axis with thickness was observed in ring crystals.

We have investigated transport properties of charge density wave (CDW) rings. The realization of CDW rings by synthesizing of niobium triselenide ring crystals provides a new system for investigation of topological effects in macroscopic quantum state. DC nonlinear conductivity measurement and the AC conductivity measurement are useful methods to investigate CDW dynamics. To investigate the topological effects, we cut the ring samples and measured AC conductivity again. We found anomalies of conductivity in DC and AC measurements. These anomalies could not be explained a simple parallel circuit model. These results suggest that the topology of CDW rings was reflected in CDW dynamics.

We study magnetic response of superconducting order parameter of one-dimensional ring. We solve Bogoliubov–de Gennes equation numerically without losing self-consistency, and obtain energy level of quasiparticle. It is found that magnetic oscillation has half period of quantum flux Φ_e ≡ hc/|e|, however, the behavior is unexpected — absolute value of pair potential increases when magnetic flux approaches .

We study theoretically possible coexistence of CDW and superconducting (SC) orders in a ring-shaped crystal of NbSe₃. Since the transfer integrals of inner and outer chains may differ due to the bending of the crystal, the electronic states may be different from chain to chain. This may lead to a coexisting state of CDW and superconductivity, and we examine this possibility by applying the Bogoliubov de Gennes method to a simple tight-binding model.

We studied growth of topological crystal. We investigated reaction time dependence of the number of produced ring crystals of TaS₃. As a result, we found the number of ring crystals decreased after it reached a peak. We proposed a model to explain the result. It is necessaty to consider the number of droplet by chalcogen and decomposition rate. A result of numerical simulation using the model reproduced the experimental result qualitatively. We discussed possible mechanisms about the decomposition process. Effects of the solid to vapor phase transition, the solid to liquid phase transition and mechanical force were considered. In those processes, we thought the solid to liquid phase transition was most plausible from binary phase diagrams. Our result will lead to an optimum condition of mass production of the topological crystals.

We have investigated sliding of a charge density wave (CDW) on one-dimensional conductor NbS₃ at room temperature. We confirmed existence of the CDW by transmission electron diffraction. The Peierls vector was determined as q = (0, 1/4b*, 1/4c*) from satellites. Nonlinear conductor with threshold field of 13.3 V/cm was exhibited in NbS₃ crystals. Moreover, we made prototype of the CDW field effect transistor. The result is that source-to-drain voltage was modulated by applied gate voltage, as the larger constant dc current around a threshold value E_T flowed.

We propose a new general model explaining how the crystal becomes the Möbius strip geometry.

To solve curved shape crystal, novel X-ray camera with two-axis sample rotator was developed. With use of this sample rotator, the preferred-orientation of micro-crystalline was nearly suppressed. The effect of two-axis rotation was effectively corrected by selecting integrated area of the imaging plate(IP). Its application to the topological crystals was presented briefly.

Ultrafast relaxation dynamics in a quasi-one-dimensional semiconductor (NbSe₄)₃I was investigated with time-resolved optical spectroscopy. New low frequency phonon modes and a central peak were exposed. We have identified a critical behavior connected with an eletronicaly driven structural phase transition at T_c = 274K.

We have investigated the quasi-one-dimensional (quasi-1D) compound NbSe3 ring and whisker crystals by means of a time-resolved optical pump-probe measurement. Around a Peierls transition temperature at which the 3D CDW ordering is achieved by adjusting the coulomb correlation between individual 1D chains, the single particle relaxation show a remarkable difference between the crystals, suggesting a difference of the phase correlation depending on the crystal topology.

Folding transition of a single semiflexible polymer under poor solvent conditions can lead to a large variety of morphologies. We performed an analysis on these morphological changes by simple theoretical treatment and classified the different conformations by using the topological index called genus. In particular, we investigated the effects of stiffness and thickness of the chain on the toroidal conformation, a typical morphology of condensed semiflexible chains. Our results are in a semi-quantitative agreement with single-chain observation on long duplex DNA molecules.

We have studied topological crystals, metal trichalcogenides (MX₃) and metal dichalcogenides (MX₂). The topological crystals are defined by following property. Both ends of system are combined each together, such as ring shape or tube shape. The system loses translational symmetry and gain rotational symmetry. Such the properties are allowed by low dimensional structure of the systems. Therefore once material is chosen, its dimensionality restricts feasibility of shapes. We succeeded in synthesizing TaSe₂ ring crystal, quasi-topological crystal and nanoscale heterojunctions of TaSe₃-TaSe₂-TaSe₃ by the de-chalcogenide method.

The high pressure effect on the topological change of the Fermi surface in Bi is investigated using microscope pump and probe system. We have observed drastic changes of electronic response which is relaxation of photo-excited carriers at phase transition pressure. The electronic responses are strongly correlated with change in the electronic band structure, we propose the drastic changes of electronic responses can be topological change of the Fermi surface caused by the phase transition.

We discuss the behavior of the large change of photoluminescence (PL) spectrum at room temperature due to the change of the intermolecular arrangement of 4, 4' – bis[9 – dicarbazolyl] – 2,2' – biphenyl (CBP). In this study, we report on PL characteristics in various aggregated morphologies such as deposited film and single crystal, and discuss its optical properties.

We have investigated the spin dynamics in the V15 cluster at low temperature by measuring proton spin-lattice relaxation time (T₁) as a function of temperature (T) and external magnetic field (H). Both the H and T dependences of 1/T₁ are well explained by a model in terms of the spin-phonon interaction above H_C where the ground state of the cluster is S=3/2. On the other hand, the T₁ data can not be explained by the model in the low magnetic field region where the ground state is two S=1/2 doubly degenerate states. The temperature independent behavior of 1/T₁ is observed at low magnetic field. These results suggest that an existence of another contribution to 1/T₁ in the low magnetic field region, which might be originated from peculiarities of S=1/2 triangle configuration in the V15 cluster.

NbSe₂ nanotubes were studied by scanning tunneling microscopy (STM). Topographic images of NbSe₂ nanotubes with length of 300-200 nm were obtained at room temperature. The measured diameter of 2-20 nm suggests that the nanotubes are single-walled. The bundle structure and Y-junction were found similarly to carbon nanotubes.

This paper reports a novel methodology to produce metal hydrides and the product's remarkable structure. In this method, metal hydride was synthesized in gas-phase and then sublimated into solid. High purity metal hydride of MgH₂ was synthesized by this method, in which raw material of magnesium was heated and evaporated in hydrogen atmosphere and then sublimate as the product of MgH₂. The product had fibrous shape, which is quite differ to conventional products. The fibers had uniform diameter of 500 nm and length ranging from 10 to 100 μm but no branches or curving. Transmission Electron Microscopy revealed that the fibers were formed with single crystalline.

We have measured magnetic properties of stable icosahedral quasicrystals Zn-M-Sc (M = Fe, Co and Ag). These quasicrystals were formed in Zn-Sc based alloys and possess high structural perfection which was confirmed by X-ray powder diffraction and selected-area electron diffraction. The magnetic susceptibility of the Zn_74.5Ag_9.5Sc₁₆ quasicrystal shows an increase with a rise in temperature over 80 K, which is accounted for by a temperature dependence of the Pauli paramagnetism. The temperature dependence of the magnetic susceptibility of Zn₇₇Fe₇Sc₁₆ and Zn₇₈Co₆Sc₁₆ quasicrystals follow Curie-Weiss law. The Fe in the Zn₇₇Fe₇Sc₁₆ quasicrystal has considerably large magnetic moments; the effective number of Bohr magneton per a Fe atom is estimated as 3.7 from Curie constant obtained by Curie-Weiss fitting, and shows spin glass behavior with the freezing temperature 7.0 K. This is the first case that most of Fe atoms have large magnetic moments in the stable icosahedral quasicrystals. On the other hands, the Zn₇₈Co₆Sc₁₆ dose not show large magnetic moments, and then, the singularity of the Zn₇₇Fe₇Sc₁₆ quasicrystal was clarified in this study.

A non-axisymmetric structure of accretion disks around the neutron star in Be/X-ray binaries is studied, analyzing the results from 3D SPH simulations performed by Hayasaki & Okazaki (2004)¹. It is found that ram pressure due to the phase-dependent mass transfer from the Be-star disk excites a one-armed, trailing spiral structure in the accretion disk around the neutron star. The spiral wave has a transient nature; it is excited around the periastron, when the material is transferred from the Be disk, and gradually damped afterwards. The disk changes its topology from circular to eccentric with the development of the spiral wave, and then from eccentric to circular with the decay of the siral wave during one orbital period. We also find that the orbital phase-dependence of the mass-accretion rate is mainly caused by the inward propagation of the spiral wave excited on the accretion disks.

Most of strongly correlated electronic systems show various types of symmetry breaking at the origin of degenerate ground states. The degeneracy allows for special topologically non trivial perturbations, the most known forms being domain walls, lines of vortices or dislocations. Of special interest are classes of correlated systems, the so-called Electronic Crystals including Wigner crystals, charge and spin density waves, … . Many of these systems have a continuous degeneracy like in incommensurate charge density waves (CDW). Contrary to usual crystals, the number of sites is not fixed and can be readjusted to absorb transferred electrons. Locally, the addition of electrons to the "condensate" of the crystalline order goes via topologically excitations such as solitons. Experiments performed on the charge density wave compound NbSe₃ will be described in which it is shown that solitons play a central role in the current conversion between CDW current and normal current via the promotion of phase slip processes.

We report studies of o-TaS₃ nanocrystals, including sample preparation of nanocrystals, fabrication of electrodes attached to the nanocrystals, and transport properties. Single crystals of o-TaS₃ were synthesized by chemical transport method. The observed temperature dependence of the nanoscale o-TaS₃ crystal did not show clear Peierls transition. The resistance was well described in terms of one-dimensional (1D) variable-range-hopping conduction over the wide range of temperature (100 K to 4.2 K). These features suggest transport properties of the nanoscale o-TaS₃ crystals were governed by soliton nucleation even at the highest observed temperature. We found finite-size effect of soliton transport in nanoscale o-TaS₃ crystals. One kind of the samples exhibit the standard activation formula, exp[– E₀/E], whereas other kind of the samples show a modified form, exp[– (E₀/E)²], where E₀ is a constant, and E is an applied field. We interpreted it as the dimensional crossover of soliton transport, in the framework provided by Hatakenaka et al. The system dimension relevant to soliton transport was either 1D or 2D, which depends on the system size. The sample of the smallest effective cross section, determined by nominal bulk resistively, exhibits the 1D behavior, whereas the larger samples showed the 2D one. This result is consistent with the idea that electric transport phenomenon of the nanoscale o-TaS₃ crystals should be attributed to soliton nucleation.

After a brief introduction on nodal superconductors, we review the topological defects in triplet superconductors such as UPt₃, Sr₂RuO₄, etc. This is in part motivated by the surprising discovery of Ana Celia Mota and her colleagues that in some triplet superconductors the flux motion is completely impeded (the ideal pinning). Among topological defects the most prominent is Abrikosov's vortex with quantum flux . Abrikosov's vortex is universal and ubiquitous and seen in both conventional and unconventional superconductors by the Bitter decoration technique, small angle neutron scattering (SANS), scanning tunneling microscopy (STM), micromagnetometer and more recently by Lorentz electron micrograph. In order to interpret the experiment by Mota et al a variety of textures are proposed. In particular, in analogy to superfluid ³He-A the -soliton and -soliton play the prominent role. We review these notions and point out possible detection of these domain walls and half-quantum vortices in some triplet superconductors.

We analyze the low temperature specific heat and magnetization in the mixed state to extract signatures which are able to discriminate various pairing symmetries. Microscopic calculations based on quasi-classical and Bogoliubov-de Gennes formalisms are performed. We emphasize the importance of low-energy excitations induced around a vortex core to understand the vortex physics.

Using Small Angle Neutron Scattering we have been able to observe for the first time a well defined Vortex lattice (VL) structure both in the hole-doped LSCO and electron-doped NCCO superconductors. Our measurements on optimally doped LSCO reveal the existence of a magnetic field-induced phase transition from a hexagonal to a square coordination of the VL. Various scenarios to explain such phase transition are presented. In NCCO as well a clear square VL could be detected, which is unexpectedly kept down to the lowest measurable magnetic fields.

We investigated energy dissipation of QPs confined in the naturally prepared nano-scale topological defects; vortex core of high-temperature superconductors (HTSC), by using microwave surface impedance measurement techniques. We found a moderately clean nature of the core as a rather universal property of HTSC. This might be related to a novel dissipation mechanism proposed recently theoretically. We also found a sublinear flux flow resistivity in YNi₂B₂C as a function of magnetic field, which is believed to be common to anisotropic superconductors.

The X-ray diffraction measurement has been measured on CeAl₂ under pressure up to 30 GPa. Although there was no pressure induced structure transition in the pressure range, An anomaly was observed on the compression curve. Detailed comparison was made with a band structure calculations, and investigated that the Lifshitz nature of the topological transition is responsible for the anomaly in the compression curve.

In this paper, we provide a possible explanation for the unusual orientation of square vortex lattice in a high-T_c cuprate La_2-xSr_xCuO₄ (LSCO), which was recently demonstrated by the small angle neutron scattering technique to be oriented along the anti-nodal directions of d-wave superconducting order parameter. Such an orientation of the square vortex lattice might be consistent with "nodal superconductivity" that the nodal Fermi arc around (π/2, π/2) plays a crucial role in the coherence of superconductivity or the coherent part of energy gap, that is, the effective superconducting gap formed by coherent pairing develops only on the nodal Fermi arc and is suppressed on the anti-nodal Fermi surface.

In this work, we have examined the spatial variation of low-temperature (T ≪ T_c) tunneling spectra in a slightly underdoped Bi₂Sr₂CaCu₂O_8+δ (BSCCO) sample using the STM/STS technique, to clarify whether the nano-scale granular type superconductivity is intrinsic or not. We report that the superconductivity takes place rather homogeneously at least in the slightly underdoped sample.

BEC of alkali-metal atoms is different from conventional BEC of ⁴He in several aspects. One of the remarkable properties of this newly discovered BEC is its internal degrees of freedom called the hyperfine spin. This spin is easily controllable by an external magnetic field. We proposed to utilise the hyperfine spin to create a vortex in the condensate by manipulating the external magnetic field, which was subsequently demonstrated experimentally at MIT and Kyoto University. In this contribution, we show that this vortex formation is understood in terms of the Berry phase and is topological in nature. Finally it is shown that the gravitational field introduces a subtle difference between topological vortex formation with light Na atoms (MIT) and with heavy Rb atoms (Kyoto).

Superfiuid ⁴He confined in nano-porous media provides us with novel topological matters, in which the system topology can be controlled by size, dimensionality and interconnectivity of the porous structures. We study superfluidity and liquid-solid phase transition of ⁴He confined in a porous glass which has nanopores of 2.5 nm in diameter. The pressure-temperature phase diagram is quite unprecedented: The superfiuid transition temperature approaches 0 K at 3.4 MPa, and the freezing pressure shifts up about 1 MPa from the bulk one. The features indicate that the confined ⁴He undergoes a superfiuid-nonsuperfluid-solid quantum phase transition at 0 K. The nonsuperfluid phase may be a localized Bose-condensed state, in which global phase coherence is destroyed by strong correlation between ⁴He atoms or by random potential (i.e. the Bose glass).

We formulate a conserving gapless mean-field theory for Bose-Einstein condensates based on a Luttinger-Ward thermodynamic functional. It is applied to a weakly interacting uniform gas with density n and s-wave scattering length a to clarify its basic thermodynamic properties. It is found that the condensation here occurs as a first-order transition. The shift of the transition temperature from the ideal-gas result T₀ is positive and given to the leading order by ΔT_C = 2.33an^1/3T₀, in agreement with a couple of previous estimates.

A formula of the spin current in mesoscopic superconductors is derived from the mean-field theory of superconductivity. The pair potentials in spin-triplet superconductivity is characterized by vectors. The spatial fluctuations of generate the spin current in equilibrium. We apply the obtained formula to the spin current in Josephson junctions and isolated rings of p-wave superconductors.

We discuss unusual competition between antiferromagnetic order and nonmagnetic hidden order in the heavy-electron compound URu₂Si₂, on the basis of neutron scattering, ²⁹Si-NMR and μSR measurements performed under hydrostatic pressure applied up to ~ 1.5 GPa. The experimental results suggenst the presence of a bicritical point in the P-T phase diagram, around which the antiferromagnetic order might surve as topological defects of the underlying hidden order.

We develop an alternative method to solve the Eilenberger equations numerically for the vortex-lattice states of type-II superconductors. Using it, we clarify the magnetic-field and impurity-concentration dependences of the magnetization, the entropy, the Pauli paramagnetism, and the mixing of higher Landau levels in the pair potential for two-dimensional s- and d_x²-y²-wave superconductors with a cylindrical Fermi surface.

Nuclear spin-lattice relaxation in the alternating-spin chains and the trimeric interwining double-chains is studied by means of a modified spin-wave theory. We consider the second-order process, where a nuclear spin flip induces virtual spin waves which are then scattered thermally via the four-magnon exchange interaction, where a nuclear spin directly interacts with spin waves via the hyperfine interaction. We point out a possibility of the three-magnon relaxation process.

We investigate wave function statistics at the critical point of the Anderson transition in two-dimensional symplectic systems with different topologies. It is found that the distribution function of critical wave function amplitudes depends on the topology even in infinite systems, while the typical value of the correlation dimension does not change for alternating the system topology. Topological effects, however, survive even in very large systems.

We investigate numerically critical properties of one-dimensional (1D) electron systems in the presence of long-range correlated diagonal disorder. Calculated results for the localization length ξ of eigenstates indicate the existence of the metal-insulator transition in 1D systems and elucidate non-trivial behavior of ξ as a function of the disorder strength. Finite-size scaling analysis is employed to obtain a precise value of transition points and the critical exponent ν, which reveals that every ν disobeys the Harris criterion ν > 2/d.

Magnetization measurements of heavy fermion superconductor URu₂Si₂ have been performed under hydrostatic pressures. The superconducting transition temperature decreases linearly with pressure up to 0.8 GPa and disappears around 1.2 GPa. The hysteretic magnetization shows charasteristic behaviors at high pressures, suggesting that the pinning properties change due to the switch of the normal state from the hidden order phase to the antiferromagnetic phase.

No abstract received.

Optical vortices, or phase singularities, are places in a scalar optical wave field where the phase is not defined. Within any beam cross-section they are observed as points of darkness. However, in 3D, these singularities are lines of darkness embedded within the light. The topologies of these dark lines can be imaged by finding the phase singularities in successive cross-sections.

A theoretical study is given of a new type of optical vortex accompanying nonlinear birefringence that is induced by the Kerr effect. This is called the optical spin vortex. To treat this object we start with the two-component nonlinear Schrödinger equation. The vortex is inherent in the spin texture caused by an anisotropy of dielectric tensor, for which a role of spin is played by the Stokes vector (or pseudo-spin). By using the effective Lagrangian for the pseudo-spin field, we give an explicit form for the vortex solution. We also examine the evolution equation of the new vortex with respect to the propagation direction.

We have observed a coherent collective excitation of the charge density wave (CDW) of the quasi-one-dimensional metal, NbSe₃, by means of ultrafast pump-probe measurement. The temperature dependence of the transient signal reveals that the collective amplitude mode (AM) disappears around 10K below a Peierls transition temperature (T_p), at which the inverse of single-particle (SP) relaxation time τ_sp becomes smaller than the AM frequency. In addition, the time-frequency analysis of the AM exhibits a time-developed frequency change. These results suggest that the instantaneously excited SPs deform the CDW gap and generate the coherent AM motion reflecting gap formation within their relaxation.

Coherent collective excitations of charge density wave (CDW) in the quasi-one-dimensional metals TaS₃ were observed as transient reflectivity changes with ultrafast time resolved spectroscopy. We have also successfully grown large TaS₃ crystal that is several times larger than the typical one. Possibility of phase soliton excitation in commensurate CDW phase was pointed out. We identified the coherent oscillation as phase soliton based on its' temperature dependence and anisotropy.

Surface acoustic waves are imaged in real time on an isotropic plane surface and also on isotropic spherical surfaces using an ultrafast optical technique. The effect of the difference in surface topology on the wave propagation is discussed.

Generation of optical vortices with a simple reflective plate inducing multilevel spiral-phase distribution was demonstrated. Furthermore, a characterization technique of topological materials using optical vortices was proposed.

A novel optical scanning system for real time imaging of surface acoustic waves is developed that is well adapted to the investigation of topological structures. We demonstrate its operation by application to an opaque anisotropic crystal and to an embedded microscopic disc. The results indicate that the system allows the monitoring of ultrafast phenomena on opaque surfaces with micron-order lateral spatial resolution.

We study the evolution of polarization of a transverse electromagnetic wave in nonlinear birefringent media, by solving the equation of motion for the Stokes parameters. Nonlinear oscillations occurring under the simultaneous effect of the linear and non-linear birefringence can be solved exactly. In particular, the forced oscillations of the circular polarization when the linear birefringence is periodically modulated along the wave propagation, are similar to the nonlinear Rabi resonance in NMR, and the influence of light intensity on the shape of the polarization-resonance line is studied for the case of incident wave with the linear polarization.

Acoustic phonon modes localized in an embedded wire structure are derived in terms of a scalar and vector potentials. The phonon modes comprise longitudinal and transverse waves, having angular momentum along the wire axis. Particle motion of the phonon modes shows peristaltic or rotating barber's pole motion, depending on the angular momentum. The dispersion relation and the density of states are also derived.

A brief account is given on the possibilities of mesoscopic superconductivity in low-noise amplifier and detector applications. In particular, three devices will be described: 1) Bloch oscillating transistor (BOT), 2) Inductively-read superconducting Cooper pair transistor (L-SET), and 3) Quantum capacitive phase detector (C-SET). The BOT is a low-noise current amplifier while the L-SET and C-SET act as ultra-sensitive charge and phase detectors, respectively. The basic operating principles and the main characteristics of these devices will be reviewed and discussed.

Equation of motion of a planar domain wall in a nanoscale ferromagnetic wire under electric current is derived based on microscopic description. The wall is shown to be driven by two mechanisms, spin transfer and momentum transfer, both arising from the exchange interaction, as has been argued by Berger. Domain wall under spin torque (spin transfer) is shown to have intrinsic pinning arising from hard-axis anisotropy energy, and that extrinsic pinning does not affect the threshold current. A resonating oscillation of domain wall occurs under AC current, and this was recently used for a spectroscopy of a single "domain wall particle". Nucleation of domain walls by spin torque is discussed. Fundamental mechanism of spin transfer effect is understood in terms of spin Josephson effect.

The Josephson effect is theoretically studied in two types of SQUID consisting of s wave superconductor and Sr₂RuO₄. Results show various response of the critical Josephson current to applied magnetic fields depending on the types of SQUID and on the pairing symmetries. In the case of a p_x + ip_y wave symmetry, the critical current in a corner SQUID becomes an asymmetric function of magnetic fields near the critical temperatures. Our results well explain a recent experimental finding [Nelson et. al, Science 306, 1151 (2004)].

Quantum tunneling of a relativistic fluxon through a rectangular potential is discussed. In the relativistic cases in the absence of friction, differences from non-relativistic cases are revealed. In order to extend the study to relativistic quantum tunneling with friction, path integral approach is also developed.

We studied two and three-dimensional superconducting networks experimentally. We succeeded in fabrication of three-dimensional structures of nickel and indium by anodizing aluminum, laser irradiation, and electrophoretic deposition. We observed superconducting transition of the indium ring fabricated by the technique. We also fabricated suqure lattice and observed two periods of oscillation of superconducting transition temperature with magnetic field.

AUTHOR INDEX.