Spin of the ground state and the flux phase problem on the ring

As a continuation of our previous work, we derive the optimal flux phase which minimizes the ground state energy in the one-dimensional many particle systems, when the number of particles is odd in the absence of on-site interaction and external potential. Moreover, we study the relationship between the flux on the ring and the spin of the ground state through which we derive some information on the sum of the lowest eigenvalues of one-particle Hamiltonians

Nonlinear Criterion for the Stability of Molecular Clouds

Dynamically significant magnetic fields are routinely observed in molecular clouds, with mass-to-flux ratio lambda = (2 pi sqrt{G}) (Sigma/B) ~ 1 (here Sigma is the total column density and B is the field strength). It is widely believed that ``subcritical'' clouds with lambda < 1 cannot collapse, based on virial arguments by Mestel and Spitzer and a linear stability analysis by Nakano and Nakamura. Here we confirm, using high resolution numerical models that begin with a strongly supersonic velocity dispersion, that this criterion is a fully nonlinear stability condition. All the high-resolution models with lambda <= 0.95 form ``Spitzer sheets'' but collapse no further. All models with lambda >= 1.02 collapse to the maximum numerically resolvable density. We also investigate other factors determining the collapse time for supercritical models. We show that there is a strong stochastic element in the collapse time: models that differ only in details of their initial conditions can have collapse times that vary by as much as a factor of 3. The collapse time cannot be determined from just the velocity dispersion; it depends also on its distribution. Finally, we discuss the astrophysical implications of our results.Comment: 11 pages, 5 figure

Endotrivial Modules for the General Linear Group in a Nondefining Characteristic

Suppose that $G$ is a finite group such that $\operatorname{SL}(n,q)\subseteq G \subseteq \operatorname{GL}(n,q)$, and that $Z$ is a central subgroup of $G$. Let $T(G/Z)$ be the abelian group of equivalence classes of endotrivial $k(G/Z)$-modules, where $k$ is an algebraically closed field of characteristic~$p$ not dividing $q$. We show that the torsion free rank of $T(G/Z)$ is at most one, and we determine $T(G/Z)$ in the case that the Sylow $p$-subgroup of $G$ is abelian and nontrivial. The proofs for the torsion subgroup of $T(G/Z)$ use the theory of Young modules for $\operatorname{GL}(n,q)$ and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules

Drude Weight of the Two-Dimensional Hubbard Model -- Reexamination of Finite-Size Effect in Exact Diagonalization Study --

The Drude weight of the Hubbard model on the two-dimensional square lattice is studied by the exact diagonalizations applied to clusters up to 20 sites. We carefully examine finite-size effects by consideration of the appropriate shapes of clusters and the appropriate boundary condition beyond the imitation of employing only the simple periodic boundary condition. We successfully capture the behavior of the Drude weight that is proportional to the squared hole doping concentration. Our present result gives a consistent understanding of the transition between the Mott insulator and doped metals. We also find, in the frequency dependence of the optical conductivity, that the mid-gap incoherent part emerges more quickly than the coherent part and rather insensitive to the doping concentration in accordance with the scaling of the Drude weight.Comment: 9 pages with 10 figures and 1 table. accepted in J. Phys. Soc. Jp

Ring Formation in Magnetically Subcritical Clouds and Multiple Star Formation

We study numerically the ambipolar diffusion-driven evolution of non-rotating, magnetically subcritical, disk-like molecular clouds, assuming axisymmetry. Previous similar studies have concentrated on the formation of single magnetically supercritical cores at the cloud center, which collapse to form isolated stars. We show that, for a cloud with many Jeans masses and a relatively flat mass distribution near the center, a magnetically supercritical ring is produced instead. The supercritical ring contains a mass well above the Jeans limit. It is expected to break up, through both gravitational and possibly magnetic interchange instabilities, into a number of supercritical dense cores, whose dynamic collapse may give rise to a burst of star formation. Non-axisymmetric calculations are needed to follow in detail the expected ring fragmentation into multiple cores and the subsequent core evolution. Implications of our results on multiple star formation in general and the northwestern cluster of protostars in the Serpens molecular cloud core in particular are discussed.Comment: 25 pages, 4 figures, to appear in Ap

Magnetically Regulated Star Formation in 3D: The Case of Taurus Molecular Cloud Complex

We carry out three-dimensional MHD simulations of star formation in turbulent, magnetized clouds, including ambipolar diffusion and feedback from protostellar outflows. The calculations focus on relatively diffuse clouds threaded by a strong magnetic field capable of resisting severe tangling by turbulent motions and retarding global gravitational contraction in the cross-field direction. They are motivated by observations of the Taurus molecular cloud complex (and, to a lesser extent, Pipe Nebula), which shows an ordered large-scale magnetic field, as well as elongated condensations that are generally perpendicular to the large-scale field. We find that stars form in earnest in such clouds when enough material has settled gravitationally along the field lines that the mass-to-flux ratios of the condensations approach the critical value. Only a small fraction (of order 1% or less) of the nearly magnetically-critical, condensed material is turned into stars per local free-fall time, however. The slow star formation takes place in condensations that are moderately supersonic; it is regulated primarily by magnetic fields, rather than turbulence. The quiescent condensations are surrounded by diffuse halos that are much more turbulent, as observed in the Taurus complex. Strong support for magnetic regulation of star formation in this complex comes from the extremely slow conversion of the already condensed, relatively quiescent C$^{18}$O gas into stars, at a rate two orders of magnitude below the maximum, free-fall value. We analyze the properties of dense cores, including their mass spectrum, which resembles the stellar initial mass function.Comment: submitted to Ap