Nilpotent orbits in bad characteristic and the Springer correspondence

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 109-112).Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p, g the Lie algebra of G and g* the dual vector space of g. This thesis is concerned with nilpotent orbits in g and g* and the Springer correspondence for g and g* when p is a bad prime. Denote W the set of isomorphism classes of irreducible representations of the Weyl group W of G. Fix a prime number 1 7 p. We denote ... the set of all pairs (c, F), where c is a nilpotent G-orbit in g (resp. g*) and F is an irreducible G-equivariant Q1-local system on c (up to isomorphism). In chapter 1, we study the Springer correspondence for g when G is of type B, C or D (p = 2). The correspondence is a bijective map from W to 2t.. In particular, we classify nilpotent G-orbits in g (type B, D) over finite fields of characteristic 2. In chapter 2, we study the Springer correspondence for g* when G is of type B, C or D (p = 2). The correspondence is a bijective map from ... . In particular, we classify nilpotent G-orbits in g* over algebraically closed and finite fields of characteristic 2. In chapter 3, we give a combinatorial description of the Springer correspondence constructed in chapter 1 and chapter 2 for 8 and g*. In chapter 4, we study the nilpotent orbits in 8* and the Weyl group representations that correspond to the pairs ... under Springer correspondence when G is of an exceptional type. Chapters 1, 2 and 3 are based on the papers [X1, X2, X3]. Chapter 4 is based on some unpublished work.by Ting Xue.Ph.D

Pairing Symmetry in Iron-Pnictide Superconductor KFe2_2As2_2

The pairing symmetry is one of the major issues in the study of iron-based superconductors. We adopt a low-energy effective kinetic model based on the first-principles band structure calculations combined with the $J_1$-$J_2$ model for KFe$_2$As$_2$, the phase diagram of pairing symmetries is constructed. Putting the values of $J_1$ and $J_2$ of the $J_1$-$J_2$ model obtained by the first-principles calculations into this phase diagram, we find that the pairing symmetry for KFe$_2$As$_2$ is a nodal $d_{xy}$-wave in the folded Brillouin zone with two iron atoms per unit cell. This is in good agreement with experiments observed a nodal order parameter.Comment: 5 pages, 4 figures (The pairing symmetry is dependent on choosing an effective tight-binding model. In the publication version, we adopt a ten-orbital model by using the maximally localized Wannier functions based on the first-principles band structure calculations, and give an s-wave pairing for KFe$_2$As$_2$

Probing the topology in band insulators

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 131-133).Topological Insulator is a newly found state of matter. Unlike phases described by the traditional Landau theory of symmetry breaking, the topological phases do not break symmetry, and it is not obvious in which measurable quantity will the topological index manifest itself. In this thesis, our main goal is to understand how topological classification produces measurable consequences in periodic insulators. We first warm up by investigating the charge conjugation invariant insulator in one spatial dimension. We show there are two topological distinct classes and derive an integral formula for the topological index that distinguishes between them. We then show that the topological index appear as a Berry's phase when one adiabatically turns on a electric field. We then study the effective theory induced by this Berry's phase and show that there are measurable consequences. We then generalize the discussion to three spatial dimensions. It is hard to capture the topological terms in the effective theory by conventional perturbation methods. We then introduce a new formalism to calculate properties produced by those topological terms such as the polarization and the magnetization, in a unified way. The formalism is based on a perturbative expansion of the Green's functions in powers of a uniform field strength, instead of the potential. In particular, this formalism allows us to capture the effective action describing the three dimensional topological insulator defined under time reversal symmetry, which previously can only be calculated via pumping. Finally, we discuss measurable consequences from the effective theory, in various different boundary settings. Among the properties we have calculated, we find we can identify part of them as of bulk nature, and some other part of them more as an effect associated with boundaries. For the part that are associated with boundaries, the Maxwell relation in the bulk can be violated. For example, the isotropic orbital magneto-polarizability and the orbital electric-susceptibility are different with periodic boundary conditions. However, they become identical whenever there is a boundary.by Kuang-Ting Chen.Ph.D

Quantum trajectories for Brownian motion

We present the stochastic Schroedinger equation for the dynamics of a quantum particle coupled to a high temperature environment and apply it the dynamics of a driven, damped, nonlinear quantum oscillator. Apart from an initial slip on the environmental memory time scale, in the mean, our result recovers the solution of the known non-Lindblad quantum Brownian motion master equation. A remarkable feature of our approach is its localization property: individual quantum trajectories remain localized wave packets for all times, even for the classically chaotic system considered here, the localization being stronger the smaller $\hbar$.Comment: 4 pages, 3 eps figure

Gap symmetry and structure of Fe-based superconductors

The recently discovered Fe-pnictide and chalcogenide superconductors display low-temperature properties suggesting superconducting gap structures which appear to vary substantially from family to family, and even within families as a function of doping or pressure. We propose that this apparent nonuniversality can actually be understood by considering the predictions of spin fluctuation theory and accounting for the peculiar electronic structure of these systems, coupled with the likely 'sign-changing s-wave' (s\pm) symmetry. We review theoretical aspects, materials properties and experimental evidence relevant to this suggestion, and discuss which further measurements would be useful to settle these issues.Comment: 86 pages, revie

Measurement of the Spectroscopy of Orbitally Excited B Mesons at LEP

We measure the masses, decay widths and relative production rate of orbitally excited B mesons using 1.25 million hadronic Z decays recorded by the L3 detector. B-meson candidates are inclusively reconstructed and combined with charged pions produced at the primary event vertex. An excess of events above the expected background in the B\pi mass spectrum in the region 5.6-5.8 GeV is interpreted as resulting from the decay B_u,d^** -> B^(*)\pi, where B_u,d^** denotes a mixture of l=1 B-meson states containing a u or a d quark. A fit to the mass spectrum yields the masses and decay widths of the B_1^* and B_2^* spin states, as well as the branching fraction for the combination of l=1 states. In addition, evidence is presented for the existence of an excited B-meson state or mixture of states in the region 5.9-6.0 GeV

Search for Charged Higgs Bosons in e+e- Collisions at \sqrt{s} = 189 GeV

A search for pair-produced charged Higgs bosons is performed with the L3 detector at LEP using data collected at a centre-of-mass energy of 188.6 GeV, corresponding to an integrated luminosity of 176.4 pb^-1. Higgs decays into a charm and a strange quark or into a tau lepton and its associated neutrino are considered. The observed events are consistent with the expectations from Standard Model background processes. A lower limit of 65.5 GeV on the charged Higgs mass is derived at 95 % confidence level, independent of the decay branching ratio Br(H^{+/-} -> tau nu)

Second Generation Leptoquark Search in p\bar{p} Collisions at s\sqrt{s} = 1.8 TeV

We report on a search for second generation leptoquarks with the D\O\ detector at the Fermilab Tevatron $p\bar{p}$ collider at $\sqrt{s}$ = 1.8 TeV. This search is based on 12.7 pb$^{-1}$ of data. Second generation leptoquarks are assumed to be produced in pairs and to decay into a muon and quark with branching ratio $\beta$ or to neutrino and quark with branching ratio $(1-\beta)$. We obtain cross section times branching ratio limits as a function of leptoquark mass and set a lower limit on the leptoquark mass of 111 GeV/c$^{2}$ for $\beta = 1$ and 89 GeV/c$^{2}$ for $\beta = 0.5$ at the 95%\ confidence level.Comment: 18 pages, FERMILAB-PUB-95/185-