Generalized scaling function from light-cone gauge AdS_5 x S^5 superstring

We revisit the computation of the 2-loop correction to the energy of a folded spinning string in AdS_5 with an angular momentum J in S^5 in the scaling limit log S, J >>1 with J / log S fixed. This correction gives the third term in the strong-coupling expansion of the generalized scaling function. The computation, using the AdS light-cone gauge approach developed in our previous paper, is done by expanding the AdS_5 x S^5 superstring partition function near the generalized null cusp world surface associated to the spinning string solution. The result corrects and extends the previous conformal gauge result of arXiv:0712.2479 and is found to be in complete agreement with the corresponding terms in the generalized scaling function as obtained from the asymptotic Bethe ansatz in arXiv:0805.4615 (and also partially from the quantum O(6) model and the Bethe ansatz data in arXiv:0809.4952). This provides a highly nontrivial strong coupling comparison of the Bethe ansatz proposal with the quantum AdS_5 x S^5 superstring theory, which goes beyond the leading semiclassical term effectively controlled by the underlying algebraic curve. The 2-loop computation we perform involves all the structures in the AdS light-cone gauge superstring action of hep-th/0009171 and thus tests its ultraviolet finiteness and, through the agreement with the Bethe ansatz, its quantum integrability. We do most of the computations for a generalized spinning string solution or the corresponding null cusp surface that involves both the orbital momentum and the winding in a large circle of S^5.Comment: 50 pages, late

Generalized Unitary Coupled Cluster Wavefunctions for Quantum Computation

We introduce a unitary coupled-cluster (UCC) ansatz termed $k$-UpCCGSD that is based on a family of sparse generalized doubles (D) operators which provides an affordable and systematically improvable unitary coupled-cluster wavefunction suitable for implementation on a near-term quantum computer. $k$-UpCCGSD employs $k$ products of the exponential of pair coupled-cluster double excitation operators (pCCD), together with generalized single (S) excitation operators. We compare its performance in both efficiency of implementation and accuracy with that of the generalized UCC ansatz employing the full generalized SD excitation operators (UCCGSD), as well as with the standard ansatz employing only SD excitations (UCCSD). $k$-UpCCGSD is found to show the best scaling for quantum computing applications, requiring a circuit depth of $\mathcal O(kN)$, compared with $\mathcal O(N^3)$ for UCCGSD and $\mathcal O((N-\eta)^2 \eta)$ for UCCSD where $N$ is the number of spin orbitals and $\eta$ is the number of electrons. We analyzed the accuracy of these three ans\"atze by making classical benchmark calculations on the ground state and the first excited state of H$_4$ (STO-3G, 6-31G), H$_2$O (STO-3G), and N$_2$ (STO-3G), making additional comparisons to conventional coupled cluster methods. The results for ground states show that $k$-UpCCGSD offers a good tradeoff between accuracy and cost, achieving chemical accuracy for lower cost of implementation on quantum computers than both UCCGSD and UCCSD. Excited states are calculated with an orthogonally constrained variational quantum eigensolver approach. This is seen to generally yield less accurate energies than for the corresponding ground states. We demonstrate that using a specialized multi-determinantal reference state constructed from classical linear response calculations allows these excited state energetics to be improved

String Method for Generalized Gradient Flows: Computation of Rare Events in Reversible Stochastic Processes

Rare transitions in stochastic processes can often be rigorously described via an underlying large deviation principle. Recent breakthroughs in the classification of reversible stochastic processes as gradient flows have led to a connection of large deviation principles to a generalized gradient structure. Here, we show that, as a consequence, metastable transitions in these reversible processes can be interpreted as heteroclinic orbits of the generalized gradient flow. This in turn suggests a numerical algorithm to compute the transition trajectories in configuration space efficiently, based on the string method traditionally restricted only to gradient diffusions

Generalized superconductors from the coupling of a scalar field to the Einstein tensor and their refractive index in massive gravity

We construct the generalized superconductors from the coupling of a scalar field to the Einstein tensor in the massive gravity and investigate their negative refraction in the probe limit. We observe that the larger graviton mass and Einstein tensor coupling parameters both hinder the formation of the condensation, but the larger graviton mass or smaller coupling parameter makes it easier for the emergence of the Cave of Winds. Furthermore, we see that the larger graviton mass but smaller coupling parameter make the range of frequencies or the range of temperatures larger for which a negative Depine-Lakhtakia index occurs, which indicates that the graviton mass and Einstein tensor have completely different effects on the negative refraction. In addition, we find that the larger graviton mass and coupling parameters both can reduce the dissipation and improve the propagation in the holographic setup.Comment: 20 pages, 12 figure

An overview of the proper generalized decomposition with applications in computational rheology

We review the foundations and applications of the proper generalized decomposition (PGD), a powerful model reduction technique that computes a priori by means of successive enrichment a separated representation of the unknown field. The computational complexity of the PGD scales linearly with the dimension of the space wherein the model is defined, which is in marked contrast with the exponential scaling of standard grid-based methods. First introduced in the context of computational rheology by Ammar et al. [3] and [4], the PGD has since been further developed and applied in a variety of applications ranging from the solution of the Schrödinger equation of quantum mechanics to the analysis of laminate composites. In this paper, we illustrate the use of the PGD in four problem categories related to computational rheology: (i) the direct solution of the Fokker-Planck equation for complex fluids in configuration spaces of high dimension, (ii) the development of very efficient non-incremental algorithms for transient problems, (iii) the fully three-dimensional solution of problems defined in degenerate plate or shell-like domains often encountered in polymer processing or composites manufacturing, and finally (iv) the solution of multidimensional parametric models obtained by introducing various sources of problem variability as additional coordinates

Biscale Chaos in Propagating Fronts

The propagating chemical fronts found in cubic autocatalytic reaction-diffusion processes are studied. Simulations of the reaction-diffusion equation near to and far from the onset of the front instability are performed and the structure and dynamics of chemical fronts are studied. Qualitatively different front dynamics are observed in these two regimes. Close to onset the front dynamics can be characterized by a single length scale and described by the Kuramoto-Sivashinsky equation. Far from onset the front dynamics exhibits two characteristic lengths and cannot be modeled by this amplitude equation. An amplitude equation is proposed for this biscale chaos. The reduction of the cubic autocatalysis reaction-diffusion equation to the Kuramoto-Sivashinsky equation is explicitly carried out. The critical diffusion ratio delta, where the planar front loses its stability to transverse perturbations, is determined and found to be delta=2.300.Comment: Typeset using RevTeX, fig.1 and fig.4 are not available, mpeg simulations are at http://www.chem.utoronto.ca/staff/REK/Videos/front/front.htm