Real root finding for equivariant semi-algebraic systems

Let $R$ be a real closed field. We consider basic semi-algebraic sets defined by $n$-variate equations/inequalities of $s$ symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by $2d < n$. Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most $2d-1$ distinct coordinates. Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by $s$ polynomials of degree $d$ in time $(sn)^{O(d)}$. This improves the state-of-the-art which is exponential in $n$. When the variables $x_1, \ldots, x_n$ are quantified and the coefficients of the input system depend on parameters $y_1, \ldots, y_t$, one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time $(sn)^{O(dt)}$

Relation between concurrence and Berry phase of an entangled state of two spin 1/2 particles

We have studied here the influence of the Berry phase generated due to a cyclic evolution of an entangled state of two spin 1/2 particles. It is shown that the measure of formation of entanglement is related to the cyclic geometric phase of the individual spins. \\Comment: 6 pages. Accepted in Europhys. Letters (likely to be published in vol 73, pp1-6 (2006)

Multi-parameter deformed and nonstandard Y(glM)Y(gl_M) Yangian symmetry in integrable variants of Haldane-Shastry spin chain

By using `anyon like' representations of permutation algebra, which pick up nontrivial phase factors while interchanging the spins of two lattice sites, we construct some integrable variants of Haldane-Shastry (HS) spin chain. Lax equations for these spin chains allow us to find out the related conserved quantities. However, it turns out that such spin chains also possess a few additional conserved quantities which are apparently not derivable from the Lax equations. Identifying these additional conserved quantities, and the usual ones related to Lax equations, with different modes of a monodromy matrix, it is shown that the above mentioned HS like spin chains exhibit multi-parameter deformed and `nonstandard' variants of $Y(gl_M)$ Yangian symmetry.Comment: 18 pages, latex, no figure

Scaling and universality in coupled driven diffusive models

Inspired by the physics of magnetohydrodynamics (MHD) a simplified coupled Burgers-like model in one dimension (1d), a generalization of the Burgers model to coupled degrees of freedom, is proposed to describe 1dMHD. In addition to MHD, this model serves as a 1d reduced model for driven binary fluid mixtures. Here we have performed a comprehensive study of the universal properties of the generalized d-dimensional version of the reduced model. We employ both analytical and numerical approaches. In particular, we determine the scaling exponents and the amplitude-ratios of the relevant two-point time-dependent correlation functions in the model. We demonstrate that these quantities vary continuously with the amplitude of the noise cross-correlation. Further our numerical studies corroborate the continuous dependence of long wavelength and long time-scale physics of the model on the amplitude of the noise cross-correlations, as found in our analytical studies. We construct and simulate lattice-gas models of coupled degrees of freedom in 1d, belonging to the universality class of our coupled Burgers-like model, which display similar behavior. We use a variety of numerical (Monte-Carlo and Pseudospectral methods) and analytical (Dynamic Renormalization Group, Self-Consistent Mode-Coupling Theory and Functional Renormalization Group) approaches for our work. The results from our different approaches complement one another. Possible realizations of our results in various nonequilibrium models are discussed.Comment: To appear in JSTAT (2009); 52 pages in JSTAT format. Some figure files have been replace

Axisymmetric Soil-Structure Interaction by Substructure Approach

The effect of soil-structure interaction is incorporated into an existing finite element computer program for axisymmetric shells and plates using substructure approach and energy transmitting boundaries. The results of numerical experimentation for a tall chimney and a typical hyperboloidal cooling tower are presented

High TcT_c Superconductivity, Skyrmions and the Berry Phase

It is here pointed out that the antiferromagnetic spin fluctuation may be associated with a gauge field which gives rise to the antiferromagnetic ground state chirality. This is associated with the chiral anomaly and Berry phase when we consider the two dimensional spin system on the surface of a 3D sphere with a monopole at the centre. This realizes the RVB state where spinons and holons can be understood as chargeless spins and spinless holes attached with magnetic flux. The attachment of the magnetic flux of the charge carrier suggest, that this may be viewed as a skyrmion. The interaction of a massless fermion representing a neutral spin with a gauge field along with the interaction of a spinless hole with the gauge field enhances the antiferromagnetic correlation along with the pseudogap at the underdoped region. As the doping increases the antiferromagnetic long range order disappears for the critical doping parameter $\delta_{sc}$. In this framework, the superconducting pairing may be viewed as caused by skyrmion-skyrmion bound states.Comment: 10 pages, accepted in Phys. Rev.

Berry phase and fidelity susceptibility of the three-qubit Lipkin-Meshkov-Glick ground state

Berry phases and quantum fidelities for interacting spins have attracted considerable attention, in particular in relation to entanglement properties of spin systems and quantum phase transitions. These efforts mainly focus either on spin pairs or the thermodynamic infinite spin limit, while studies of the multipartite case of a finite number of spins are rare. Here, we analyze Berry phases and quantum fidelities of the energetic ground state of a Lipkin-Meshkov-Glick (LMG) model consisting of three spin-1/2 particles (qubits). We find explicit expressions for the Berry phase and fidelity susceptibility of the full system as well as the mixed state Berry phase and partial-state fidelity susceptibility of its one- and two-qubit subsystems. We demonstrate a realization of a nontrivial magnetic monopole structure associated with local, coordinated rotations of the three-qubit system around the external magnetic field.Comment: The title of the paper has been changed in this versio