3,276 research outputs found
Deformation rings and parabolic induction
We study deformations of smooth mod representations (and their duals) of
a -adic reductive group . Under some mild genericity condition, we prove
that parabolic induction with respect to a parabolic subgroup defines an
isomorphism between the universal deformation rings of a supersingular
representation of and of its parabolic induction
. As a consequence, we show that every Banach lift of is
induced from a unique Banach lift of .Comment: 28 pages, minor changes, final versio
A simpler approach to obtaining an O(1/t) convergence rate for the projected stochastic subgradient method
In this note, we present a new averaging technique for the projected
stochastic subgradient method. By using a weighted average with a weight of t+1
for each iterate w_t at iteration t, we obtain the convergence rate of O(1/t)
with both an easy proof and an easy implementation. The new scheme is compared
empirically to existing techniques, with similar performance behavior.Comment: 8 pages, 6 figures. Changes with previous version: Added reference to
concurrently submitted work arXiv:1212.1824v1; clarifications added; typos
corrected; title changed to 'subgradient method' as 'subgradient descent' is
misnome
Block-Coordinate Frank-Wolfe Optimization for Structural SVMs
We propose a randomized block-coordinate variant of the classic Frank-Wolfe
algorithm for convex optimization with block-separable constraints. Despite its
lower iteration cost, we show that it achieves a similar convergence rate in
duality gap as the full Frank-Wolfe algorithm. We also show that, when applied
to the dual structural support vector machine (SVM) objective, this yields an
online algorithm that has the same low iteration complexity as primal
stochastic subgradient methods. However, unlike stochastic subgradient methods,
the block-coordinate Frank-Wolfe algorithm allows us to compute the optimal
step-size and yields a computable duality gap guarantee. Our experiments
indicate that this simple algorithm outperforms competing structural SVM
solvers.Comment: Appears in Proceedings of the 30th International Conference on
Machine Learning (ICML 2013). 9 pages main text + 22 pages appendix. Changes
from v3 to v4: 1) Re-organized appendix; improved & clarified duality gap
proofs; re-drew all plots; 2) Changed convention for Cf definition; 3) Added
weighted averaging experiments + convergence results; 4) Clarified main text
and relationship with appendi
Emergent Fermions and Anyons in the Kitaev Model
We study the gapped phase of the Kitaev model on the honeycomb lattice using
perturbative continuous unitary transformations. The effective low-energy
Hamiltonian is found to be an extended toric code with interacting anyons.
High-energy excitations are emerging free fermions which are composed of
hardcore bosons with an attached string of spin operators. The excitation
spectrum is mapped onto that of a single particle hopping on a square lattice
in a magnetic field. We also illustrate how to compute correlation functions in
this framework. The present approach yields analytical perturbative results in
the thermodynamical limit without using the Majorana or the Jordan-Wigner
fermionization initially proposed to solve this problem.Comment: 4 pages, 5 figures, published versio
Perturbative study of the Kitaev model with spontaneous time-reversal symmetry breaking
We analyze the Kitaev model on the triangle-honeycomb lattice whose ground
state has recently been shown to be a chiral spin liquid. We consider two
perturbative expansions: the isolated-dimer limit containing Abelian anyons and
the isolated-triangle limit. In the former case, we derive the low-energy
effective theory and discuss the role played by multi-plaquette interactions.
In this phase, we also compute the spin-spin correlation functions for any
vortex configuration. In the isolated-triangle limit, we show that the
effective theory is, at lowest nontrivial order, the Kitaev honeycomb model at
the isotropic point. We also compute the next-order correction which opens a
gap and yields non-Abelian anyons.Comment: 7 pages, 4 figures, published versio
Solids and supersolids of three-body interacting polar molecules in an optical lattice
We study the physics of cold polar molecules loaded into an optical lattice
in the regime of strong three-body interactions, as put forward recently by
B\"uchler [Nature Phys. 3, 726 (2007)]. To this end quantum Monte Carlo
simulations, exact diagonalization and a semiclassical approach are used to
explore hardcore bosons on the two-dimensional square lattice which interact
solely by long ranged three-body terms. The resulting phase diagram shows a
sequence of solid and supersolid phases. Our findings are directly relevant for
future experimental implementations and open a new route towards the discovery
of a lattice supersolid phase in experiment.Comment: 4+ pages, 4 figures, published versio
Bound states in two-dimensional spin systems near the Ising limit: A quantum finite-lattice study
We analyze the properties of low-energy bound states in the transverse-field
Ising model and in the XXZ model on the square lattice. To this end, we develop
an optimized implementation of perturbative continuous unitary transformations.
The Ising model is studied in the small-field limit which is found to be a
special case of the toric code model in a magnetic field. To analyze the XXZ
model, we perform a perturbative expansion about the Ising limit in order to
discuss the fate of the elementary magnon excitations when approaching the
Heisenberg point.Comment: 21 pages, 18 figures, published versio
Magnetization plateaux in an extended Shastry-Sutherland model
We study an extended two-dimensional Shastry-Sutherland model in a magnetic
field where besides the usual Heisenberg exchanges of the Shastry-Sutherland
model two additional SU(2) invariant couplings are included. Perturbative
continous unitary transformations are used to determine the leading order
effects of the additional couplings on the pure hopping and on the long-range
interactions between the triplons which are the most relevant terms for small
magnetization. We then compare the energy of various magnetization plateaux in
the classical limit and we discuss the implications for the two-dimensional
quantum magnet SrCu(BO).Comment: 8 pages, Proceedings of the HFM2008 Conferenc
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