Numerical homotopies to compute generic points on positive dimensional algebraic sets

Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to various polynomial systems, such as the cyclic n-roots problem

Dynamical mass generation by source inversion: calculating the mass gap of the chiral Gross-Neveu model

We probe the U(N) chiral Gross-Neveu model with a source-term J\l{\Psi}\Psi. We find an expression for the renormalization scheme and scale invariant source $\hat{J}$, as a function of the generated mass gap. The expansion of this function is organized in such a way that all scheme and scale dependence is reduced to one single parameter $d$. We obtain a non-perturbative mass gap as the solution of $\hat{J}=0$. A physical choice for $d$ gives good results for $N>2$. The self-consistent minimal sensitivity condition gives a slight improvement.Comment: 14 pages, 2 figure

An intrinsic homotopy for intersecting algebraic varieties

Recently we developed a diagonal homotopy method to compute a numerical representation of all positive dimensional components in the intersection of two irreducible algebraic sets. In this paper, we rewrite this diagonal homotopy in intrinsic coordinates, which reduces the number of variables, typically in half. This has the potential to save a significant amount of computation, especially in the iterative solving portion of the homotopy path tracker. There numerical experiments all show a speedup of about a factor two

Spontaneous electromagnetic superconductivity of vacuum induced by a strong magnetic field: QCD and electroweak theory

Both in electroweak theory and QCD, the vacuum in strong magnetic fields develops charged vector condensates once a critical value of the magnetic field is reached. Both ground states have a similar Abrikosov lattice structure and superconducting properties. It is the purpose of these proceedings to put the condensates and their superconducting properties side by side and obtain a global view on this type of condensates. Some peculiar aspects of the superfluidity and backreaction of the condensates are also discussed.Comment: 7 pages, 4 figures. Talk presented at QCD@Work 2012: International Workshop on QCD - Theory and Experiment, June 18-21, Lecce, Ital

Dynamical mass generation by source inversion: Calculating the mass gap of the Gross-Neveu model

We probe the U(N) Gross-Neveu model with a source-term $J\bar{\Psi}\Psi$. We find an expression for the renormalization scheme and scale invariant source $\hat{J}$, as a function of the generated mass gap. The expansion of this function is organized in such a way that all scheme and scale dependence is reduced to one single parameter d. We get a non-perturbative mass gap as the solution of $\hat{J}=0$. In one loop we find that any physical choice for d gives good results for high values of N. In two loops we can determine d self-consistently by the principle of minimal sensitivity and find remarkably accurate results for N>2.Comment: 13 pages, 3 figures, added referenc

Entanglement renormalization for quantum fields

It is shown how to construct renormalization group flows of quantum field theories in real space, as opposed to the usual Wilsonian approach in momentum space. This is achieved by generalizing the multiscale entanglement renormalization ansatz to continuum theories. The variational class of wavefunctions arising from this RG flow are translation invariant and exhibit an entropy-area law. We illustrate the construction for a free non-relativistic boson model, and argue that the full power of the construction should emerge in the case of interacting theories.Comment: 4 pages: completely revised. Focus on a single non-relativistic model for clarit

Four qubits can be entangled in nine different ways

We consider a single copy of a pure four-partite state of qubits and investigate its behaviour under the action of stochastic local quantum operations assisted by classical communication (SLOCC). This leads to a complete classification of all different classes of pure states of four-qubits. It is shown that there exist nine families of states corresponding to nine different ways of entangling four qubits. The states in the generic family give rise to GHZ-like entanglement. The other ones contain essentially 2- or 3-qubit entanglement distributed among the four parties. The concept of concurrence and 3-tangle is generalized to the case of mixed states of 4 qubits, giving rise to a seven parameter family of entanglement monotones. Finally, the SLOCC operations maximizing all these entanglement monotones are derived, yielding the optimal single copy distillation protocol