Confidence regions for high quantiles of a heavy tailed distribution

Estimating high quantiles plays an important role in the context of risk management. This involves extrapolation of an unknown distribution function. In this paper we propose three methods, namely, the normal approximation method, the likelihood ratio method and the data tilting method, to construct confidence regions for high quantiles of a heavy tailed distribution. A simulation study prefers the data tilting method.Comment: Published at http://dx.doi.org/10.1214/009053606000000416 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Detecting crystal symmetry fractionalization from the ground state: Application to Z2\mathbb Z_2 spin liquids on the kagome lattice

In quantum spin liquid states, the fractionalized spinon excitations can carry fractional crystal symmetry quantum numbers, and this symmetry fractionalization distinguishes different topologically ordered spin liquid states. In this work we propose a simple way to detect signatures of such crystal symmetry fractionalizations from the crystal symmetry representations of the ground state wave function. We demonstrate our method on projected $\mathbb Z_2$ spin liquid wave functions on the kagome lattice, and show that it can be used to classify generic wave functions. Particularly our method can be used to distinguish several proposed candidates of $\mathbb Z_2$ spin liquid states on the kagome lattice.Comment: main text: 6 pages, 1 figure. supplemental material: 8 pages, 2 figures. Added a few references and the journal referenc

Anomalous Crystal Symmetry Fractionalization on the Surface of Topological Crystalline Insulators

The surface of a three-dimensional topological electron system often hosts symmetry-protected gapless surface states. With the effect of electron interactions, these surface states can be gapped out without symmetry breaking by a surface topological order, in which the anyon excitations carry anomalous symmetry fractionalization that cannot be realized in a genuine two-dimensional system. We show that for a mirror-symmetry-protected topological crystalline insulator with mirror Chern number $n=4$, its surface can be gapped out by an anomalous $\mathbb Z_2$ topological order, where all anyons carry mirror-symmetry fractionalization $M^2=-1$. The identification of such anomalous crystalline symmetry fractionalization implies that in a two-dimensional $\mathbb Z_2$ spin liquid the vison excitation cannot carry $M^2=-1$ if the spinon carries $M^2=-1$ or a half-integer spin.Comment: 6+8 pages, 2 figures. v2: added a new section in the supplemental material, the journal reference and some other change