Exact microscopic wave function for a topological quantum membrane

The higher dimensional quantum Hall liquid constructed recently supports stable topological membrane excitations. Here we introduce a microscopic interacting Hamiltonian and present its exact ground state wave function. We show that this microscopic ground state wave function describes a topological quantum membrane. We also construct variational wave functions for excited states using the non-commutative algebra on the four sphere. Our approach introduces a non-perturbative method to quantize topological membranes

SO(5) Quantum Nonlinear sigma Model Theory of the High Tc Superconductivity

We show that the complex phase diagram of high $T_c$ superconductors can be deduced from a simple symmetry principle, a $SO(5)$ symmetry which unifies antiferromagnetism with $d$ wave superconductivity. We derive the approximate $SO(5)$ symmetry from the microscopic Hamiltonian and show furthermore that this symmetry becomes exact under the renormalization group flow towards a bicritical point. With the help of this symmetry, we construct a $SO(5)$ quantum nonlinear $\sigma$ model to describe the effective low energy degrees of freedom of the high $T_c$ superconductors, and use it to deduce the phase diagram and the nature of the low lying collective excitations of the system. We argue that this model naturally explains the basic phenomenology of the high $T_c$ superconductors from the insulating to the underdoped and the optimally doped region.Comment: 36 pages, 1 Postscript figur

To see a world in a grain of sand

Throughout John Wheeler's career, he wrestled with big issues like the fundamental length, the black hole and the unification of quantum mechanics and relativity. In this essay, I argue that solid state physics -- historically the study of silicon, semiconductors and sand grains -- can give surprisingly deep insights into the big questions of the world.Comment: A chapter for John Wheeler's 90's birthday festschrif

A Progress Report on the SO(5) Theory of High T_c Superconductivity

In this talk I give a brief update on the recent progress in the SO(5) theory of high T_c superconductivity (Science, 275: 1089,1997). Reviewed topics include SO(5) ladders, the unification of BCS and SDW quasi-particles in the SO(5) theory and the microscopic origin of the condensation energy.Comment: To be published by Springer Verlag in the Proceedings of the Grand Finale Taniguchi Symposium on "The Physics and Chemistry of Transition Metal Oxides

Positivity of heights of codimension 2 cycles over function field of characteristic 0

In this note, we show how the classical Hodge index theorem implies the Hodge index conjecture of Beilinson for height pairing of homologically trivial codimension two cycles over function field of characteristic 0. Such an index conjecture has been used in our paper on Gross-Schoen cycles to deduce the Bogomolov conjecture and a lower bound for Hodge class (or Faltings height) from some conjectures about metrized graphs which have just been recently proved by Zubeyir Cinkir

Topological insulators from the Perspective of first-principles calculations

Topological insulators are new quantum states with helical gapless edge or surface states inside the bulk band gap.These topological surface states are robust against the weak time-reversal invariant perturbations, such as lattice distortions and non-magnetic impurities. Recently a variety of topological insulators have been predicted by theories, and observed by experiments. First-principles calculations have been widely used to predict topological insulators with great success. In this review, we summarize the current progress in this field from the perspective of first-principles calculations. First of all, the basic concepts of topological insulators and the frequently-used techniques within first-principles calculations are briefly introduced. Secondly, we summarize general methodologies to search for new topological insulators. In the last part, based on the band inversion picture first introduced in the context of HgTe, we classify topological insulators into three types with s-p, p-p and d-f, and discuss some representative examples for each type.Comment: 10 pages, 7 figure

Some New Questions on Point-countable Covers and Sequence-covering Mappings

In this survey, 37 questions on point-countable covers and sequence-covering mappings are listed, in which some of these questions have been answered. These questions are mainly related to the theory of generalized metric spaces, involving point-countable covers, sequence-covering mappings, images of metric spaces and hereditarily closure-preserving families.Comment: 17 page

Tuning order in cuprate superconductors

This article presents our perspective on STM measurements by Hoffman et al. (cond-mat/0201348) of the vortex lattice in BSCCO. We discuss implications of these measurements for various theories of the cuprate superconductors.Comment: 7 pages and 2 figure

Weyl Semimetal and Topological Phase Transition in Five Dimensions

We study two Weyl semimetal generalizations in five dimensions (5d) which have Yang monopoles and linked Weyl surfaces in the Brillouin zone, respectively, and carry the second Chern number as a topological number. In particular, we show a Yang monopole naturally reduces to a Hopf link of two Weyl surfaces when the $\mathbf{TP}$ (time-reversal combined with space-inversion) symmetry is broken. We then examine the phase transition between insulators with different topological numbers in 5d. In analogy to the 3d case, 5d Weyl semimetals emerge as intermediate phases during the topological phase transition.Comment: 6 pages, 4 figure

Quantum Anomalous Hall Effect in Magnetic Topological Insulator GdBiTe3

The quantum anomalous Hall (QAH) state is a two-dimensional bulk insulator with a non-zero Chern number in absence of external magnetic fields. Protected gapless chiral edge states enable dissipationless current transport in electronic devices. Doping topological insulators with random magnetic impurities could realize the QAH state, but magnetic order is difficult to establish experimentally in the bulk insulating limit. Here we predict that the single quintuple layer of GdBiTe3 film could be a stoichiometric QAH insulator based on ab-initio calculations, which explicitly demonstrate ferromagnetic order and chiral edge states inside the bulk gap. We further investigate the topological quantum phase transition by tuning the lattice constant and interactions. A simple low-energy effective model is presented to capture the salient physical feature of this topological material.Comment: 18pages, 4figure