Superconductivity and Abelian Chiral Anomalies

Motivated by the geometric character of spin Hall conductance, the topological invariants of generic superconductivity are discussed based on the Bogoliuvov-de Gennes equation on lattices. They are given by the Chern numbers of degenerate condensate bands for unitary order, which are realizations of Abelian chiral anomalies for non-Abelian connections. The three types of Chern numbers for the $x,y$ and $z$-directions are given by covering degrees of some doubled surfaces around the Dirac monopoles. For nonunitary states, several topological invariants are defined by analyzing the so-called $q$-helicity. Topological origins of the nodal structures of superconducting gaps are also discussed.Comment: An example with a figure and discussions are supplemente

Berry phase and quantized Hall effect in three-dimension

We consider Bloch electrons in the electromagnetic field and argue the relation between the Berry phase and the quantized Hall conductivity in three-dimension. The Berry phase we consider here is induced by the adiabatic change of the time-dependent vector potential. The relation has been shown in two-dimensional systems, and we generalize the relation in three-dimensional systems.Comment: corrected some typos. Accepted for publication in J. Phys. Soc. Jp

Exact Results on Superconductivity due to Interband Coupling

We present a family of exactly solvable models at arbitrary filling in any dimensions which exhibit novel superconductivity with interband pairing. By the use of the hidden $SU(2)$ algebra the Hamiltonians were diagonalized explicitly. The zero-temperature phase diagrams and the thermodynamic properties are discussed. Several new properties are revealed which are different from those of the BCS-type superconductor

Quantum Group, Bethe Ansatz and Bloch Electrons in a Magnetic Field

The wave functions for two dimensional Bloch electrons in a uniform magnetic field at the mid-band points are studied with the help of the algebraic structure of the quantum group $U_q(sl_2)$. A linear combination of its generators gives the Hamiltonian. We obtain analytical and numerical solutions for the wave functions by solving the Bethe Ansatz equations, proposed by Wiegmann and Zabrodin on the basis of above observation. The semi-classical case with the flux per plaquette $\phi=1/Q$ is analyzed in detail, by exploring a structure of the Bethe Ansatz equations. We also reveal the multifractal structure of the Bethe Ansatz solutions and corresponding wave functions when $\phi$ is irrational, such as the golden or silver mean.Comment: 30 pages, 11 GIF figures(use xv, or WWW browser

Quantum Hall Effect in Three-dimensional Field-Induced Spin Density Wave Phases with a Tilted Magnetic Field

The quantum Hall effect in the three-dimensional anisotropic tight-binding electrons is investigated in the field-induced spin density wave phases with a magnetic field tilted to any direction. The Hall conductivity, $\sigma_{xy}$ and $\sigma_{xz}$, are shown to be quantized as a function of the wave vector of FISDW, while $\sigma_{yz}$ stays zero, where $x$ is the most conducting direction and $y$ and $z$ are perpendicular to $x$.Comment: 18 pages, REVTeX 3.0, 1 figure is available upon request, to be published in Physical Review

Conductivity of 2D lattice electrons in an incommensurate magnetic field

We consider conductivities of two-dimensional lattice electrons in a magnetic field. We focus on systems where the flux per plaquette $\phi$ is irrational (incommensurate flux). To realize the system with the incommensurate flux, we consider a series of systems with commensurate fluxes which converge to the irrational value. We have calculated a real part of the longitudinal conductivity $\sigma_{xx}(\omega)$. Using a scaling analysis, we have found $\Re\sigma_{xx}(\omega)$ behaves as $1/\omega ^{\gamma}$ \,$(\gamma =0.55)$ when $\phi =\tau,(\tau =\frac{\sqrt{5}-1}{2})$ and the Fermi energy is near zero. This behavior is closely related to the known scaling behavior of the spectrum.Comment: 16 pages, postscript files are available on reques

Adiabatic connection between the RVB State and the ground state of the half filled periodic Anderson model

A one-parameter family of models that interpolates between the periodic Anderson model with infinite repulsion at half-filling and a model whose ground state is exactly the Resonating-Valence-Bond state is studied. It is shown numerically that the excitation gap does not collapse. Therefore the ground states of the two models are adiabatically connected.Comment: 6 pages, 3 figures Revte

A General Theorem Relating the Bulk Topological Number to Edge States in Two-dimensional Insulators

We prove a general theorem on the relation between the bulk topological quantum number and the edge states in two dimensional insulators. It is shown that whenever there is a topological order in bulk, characterized by a non-vanishing Chern number, even if it is defined for a non-conserved quantity such as spin in the case of the spin Hall effect, one can always infer the existence of gapless edge states under certain twisted boundary conditions that allow tunneling between edges. This relation is robust against disorder and interactions, and it provides a unified topological classification of both the quantum (charge) Hall effect and the quantum spin Hall effect. In addition, it reconciles the apparent conflict between the stability of bulk topological order and the instability of gapless edge states in systems with open boundaries (as known happening in the spin Hall case). The consequences of time reversal invariance for bulk topological order and edge state dynamics are further studied in the present framework.Comment: A mistake corrected in reference

Numerical study of the hidden antiferromagnetic order in the Haldane phase

We investigate the string correlation functions proposed by den Nijs and Rommelse for S=1 Heisenberg antiferromagnets in one dimension. The Hamiltonian with DΣi(Siz)2 term is diagonalized by the Lanczos method to obtain the ground state. We calculate both the usual spin-spin correlation functions and the string correlation functions not only in the z direction (quantized direction) but also in the x direction to investigate the Z2×Z2 symmetry breaking recently proposed by Kennedy and Tasaki. We find that the long-range string correlation, which is argued to exist in the Haldane disordered phase, in fact, exists at the Heisenberg point D=0 by a finite-size analysis. We can show explicitly that the string correlation in the x direction signifies the difference between the Haldane phase and the Néel phase, which appears for the D<0, ‖D‖≳1 case. In the large-D (D≳1) phase, all spin-spin correlations are of short range as expected. There is a significant enhancement in the usual and string correlations in the x direction at the boundary between the Haldane phase and the large-D phase