Universal anomalous dimensions at large spin and large twist

In this paper we consider anomalous dimensions of double trace operators at large spin (l) and large twist (tau) in CFTs in arbitrary dimensions (d >= 3). Using analytic conformal bootstrap methods, we show that the anomalous dimensions are universal in the limit l >> tau >> 1. In the course of the derivation, we extract an approximate closed form expression for the conformal blocks arising in the four point function of identical scalars in any dimension. We compare our results with two different calculations in holography and find perfect agreement.

PERIODIC CONTROLS IN AN OSCILLATING DOMAIN: CONTROLS VIA UNFOLDING AND HOMOGENIZATION

An optimal control problem in a two-dimensional domain with a rapidly oscillating boundary is considered. The main features of this article are on two points, namely, we consider periodic controls in the thin periodic slabs of period epsilon > 0, a small parameter, and height O(1) in the oscillatory part, and the controls are characterized using unfolding operators. We then do a homogenization analysis of the optimal control problems as epsilon -> 0 with L-2 as well as Dirichlet (gradient-type) cost functionals.

Ancilla-assisted measurements on quantum ensembles: general protocols and applications in NMR quantum information processing

Quantum ensembles form easily accessible architectures for studying various phenomena in quantum physics, quantum information science and spectroscopy. Here we review some recent protocols for measurements in quantum ensembles by utilizing ancillary systems. We also illustrate these protocols experimentally via nuclear magnetic resonance techniques. In particular, we shall review noninvasive measurements, extracting expectation values of various operators, characterizations of quantum states and quantum processes, and finally quantum noise engineering.

Analytic bootstrap at large spin

We use analytic conformal bootstrap methods to determine the anomalous dimensions and OPE coefficients for large spin operators in general conformal field theories in four dimensions containing a scalar operator of conformal dimension Delta(phi). It is known that such theories will contain an in finite sequence of large spin operators with twists approaching 2 Delta(phi) + 2n for each integer n. By considering the case where such operators are separated by a twist gap from other operators at large spin, we analytically determine the n, Delta(phi) dependence of the anomalous dimensions. We find that for all n, the anomalous dimensions are negative for Delta(phi) satisfying the unitarity bound. We further compute the first subleading correction at large spin and show that it becomes universal for large twist. In the limit when n is large, we find exact agreement with the AdS/CFT prediction corresponding to the Eikonal limit of a 2-2 scattering with dominant graviton exchange.

Trace Formulae in Operator Theory

In this article, we survey several kinds of trace formulas that one encounters in the theory of single and multi-variable operators. We give some sketches of the proofs, often based on the principle of finite-dimensional approximations to the objects at hand in the formulas.

Action of Hopf on the twisted modular Hecke operator

Let Gamma subset of SL2(Z) be a principal congruence subgroup. For each sigma is an element of SL2(Z), we introduce the collection A(sigma)(Gamma) of modular Hecke operators twisted by sigma. Then, A(sigma)(Gamma) is a right A(Gamma)-module, where A(Gamma) is the modular Hecke algebra introduced by Connes and Moscovici. Using the action of a Hopf algebra h(0) on A(sigma)(Gamma), we define reduced Rankin-Cohen brackets on A(sigma)(Gamma). Moreover A(sigma)(Gamma) carries an action of H 1, where H 1 is the Hopf algebra of foliations of codimension 1. Finally, we consider operators between the levels A(sigma)(Gamma), sigma is an element of SL2(Z). We show that the action of these operators can be expressed in terms of a Hopf algebra h(Z).

Efficient density matrix renormalization group algorithm to study Y junctions with integer and half-integer spin

An efficient density matrix renormalization group (DMRG) algorithm is presented and applied to Y junctions, systems with three arms of n sites that meet at a central site. The accuracy is comparable to DMRG of chains. As in chains, new sites are always bonded to the most recently added sites and the superblock Hamiltonian contains only new or once renormalized operators. Junctions of up to N = 3n + 1 approximate to 500 sites are studied with antiferromagnetic (AF) Heisenberg exchange J between nearest-neighbor spins S or electron transfer t between nearest neighbors in half-filled Hubbard models. Exchange or electron transfer is exclusively between sites in two sublattices with N-A not equal N-B. The ground state (GS) and spin densities rho(r) = < S-r(z)> at site r are quite different for junctions with S = 1/2, 1, 3/2, and 2. The GS has finite total spin S-G = 2S(S) for even (odd) N and for M-G = S-G in the S-G spin manifold, rho(r) > 0(< 0) at sites of the larger (smaller) sublattice. S = 1/2 junctions have delocalized states and decreasing spin densities with increasing N. S = 1 junctions have four localized S-z = 1/2 states at the end of each arm and centered on the junction, consistent with localized states in S = 1 chains with finite Haldane gap. The GS of S = 3/2 or 2 junctions of up to 500 spins is a spin density wave with increased amplitude at the ends of arms or near the junction. Quantum fluctuations completely suppress AF order in S = 1/2 or 1 junctions, as well as in half-filled Hubbard junctions, but reduce rather than suppress AF order in S = 3/2 or 2 junctions.

MIXED NORM ESTIMATES FOR THE RIESZ TRANSFORMS ON SU (2)

In this paper we prove mixed norm estimates for Riesz transforms on the group SU(2). From these results vector valued inequalities for sequences of Riesz transforms associated to Jacobi differential operators of different types are deduced.

Integrals of motion for one-dimensional Anderson localized systems

Anderson localization is known to be inevitable in one-dimension for generic disordered models. Since localization leads to Poissonian energy level statistics, we ask if localized systems possess `additional' integrals of motion as well, so as to enhance the analogy with quantum integrable systems. We answer this in the affirmative in the present work. We construct a set of nontrivial integrals of motion for Anderson localized models, in terms of the original creation and annihilation operators. These are found as a power series in the hopping parameter. The recently found Type-1 Hamiltonians, which are known to be quantum integrable in a precise sense, motivate our construction. We note that these models can be viewed as disordered electron models with infinite-range hopping, where a similar series truncates at the linear order. We show that despite the infinite range hopping, all states but one are localized. We also study the conservation laws for the disorder free Aubry-Andre model, where the states are either localized or extended, depending on the strength of a coupling constant. We formulate a specific procedure for averaging over disorder, in order to examine the convergence of the power series. Using this procedure in the Aubry-Andre model, we show that integrals of motion given by our construction are well-defined in localized phase, but not so in the extended phase. Finally, we also obtain the integrals of motion for a model with interactions to lowest order in the interaction.

Homogeneous Hermitian holomorphic vector bundles and the Cowen-Douglas class over bounded symmetric domains

It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic group on finite-dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. We write down an equivariant constant coefficient differential operator that intertwines the bundle with the direct sum of its irreducible factors. As an application, we show that in the case of the closed unit ball in C-n all homogeneous n-tuples of Cowen-Douglas operators are similar to direct sums of certain basic n-tuples. (c) 2015 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.