38457740
Finding all the stationary points of a potential-energy landscape via numerical polynomial-homotopy-continuation method.
The stationary points (SPs) of a potential-energy landscape play a crucial role in understanding many of the physical or chemical properties of a given system. However, unless they are found analytically, no efficient method is available to obtain all the SPs of a given potential. We present a method, called the numerical polynomial-homotopy-continuation method, which numerically finds all the SPs, and is embarrassingly parallelizable. The method requires the nonlinearity of the potential to be polynomial-like, which is the case for almost all of the potentials arising in physical and chemical systems. We also certify the numerically obtained SPs so that they are independent of the numerical tolerance used during the computation. It is then straightforward to separate out the local and global minima. As a first application, we take the XY model with power-law interaction, which is shown to have a polynomial-like nonlinearity, and we apply the method.
2452577
Lattice Landau gauge via Stereographic Projection
Centre for the Subatomic Structure of Matter, School of Chemistry and Physics,The University of Adelaide, SA 5005, AustraliaE-mail: lorenz.smekal@adelaide.edu.au,alexander.jorkowski@student.adelaide.edu.au,dhagash.mehta@adelaide.edu.au, andre.sternbeck@adelaide.edu.auThe complete cancellation of Gribov copies and the Neuberger 0=0 problem of lattice BRST canbe avoided in modiﬁed lattice Landau gauge. In compact U(1), where the problem is a latticeartifact, there remain to be Gribov copies but their number is exponentially reduced. Moreover,there is no cancellation of copies there as the sign of the Faddeev-Popov determinant is posi-tive. Applied to the maximal Abelian subgroup this avoids the perfect cancellation amongst theremaining Gribov copies for SU(N) also. In addition, based on a deﬁnition of gauge ﬁelds onthe lattice as stereographically-projected link variables, it provides a framework for gauge ﬁxedMonte-Carlo simulations. This will include all Gribov copies in the spirit of BRST. Their averageis not zero, as demonstrated explicitly in simple models. This might resolve present discrepanciesbetween gauge-ﬁxed lattice and continuum studies of QCD Green’s functions.8th Conference Quark Conﬁnement and the Hadron SpectrumSeptember 1-6, 2008Mainz, Germany
118852174
Lattice Landau Gauge and Algebraic Geometry
Finding the global minimum of a multivariate function efficiently is a fundamental yet difficult problem in many branches of theoretical physics and chemistry. However, we observe that there are many physical systems for which the extremizing equations have polynomial-like non-linearity. This allows the use of Algebraic Geometry techniques to solve these equations completely. The global minimum can then straightforwardly be found by the second derivative test. As a warm-up example, here we study lattice Landau gauge for compact U(1) and propose two methods to solve the corresponding gauge-fixing equations. In a first step, we obtain all Gribov copies on one and two dimensional lattices. For simple 3x3 systems their number can already be of the order of thousands. We anticipate that the computational and numerical algebraic geometry methods employed have far-reaching implications beyond the simple but illustrating examples discussed here.
118601555
't Hooft-Polyakov monopoles in lattice SU(N)+adjoint Higgs theory
We investigate twisted C-periodic boundary conditions in SU(N) gauge field theory with an adjoint Higgs field. We show that with a suitable twist for even N one can impose a nonzero magnetic charge relative to residual U(1) gauge groups in the broken phase, thereby creating a 't Hooft-Polyakov magnetic monopole. This makes it possible to use lattice Monte Carlo simulations to study the properties of these monopoles in the quantum theory.
216050660
Supersymmetric lattice fermions on the triangular lattice: superfrustration and criticality
We study a model for itinerant, strongly interacting fermions where a judicious tuning of the interactions leads to a supersymmetric Hamiltonian. On the triangular lattice this model is known to exhibit a property called superfrustration, which is characterized by an extensive ground state entropy. Using a combination of numerical and analytical methods we study various ladder geometries obtained by imposing doubly periodic boundary conditions on the triangular lattice. We compare our results to various bounds on the ground state degeneracy obtained in the literature. For all systems we find that the number of ground states grows exponentially with system size. For two of the models that we study we obtain the exact number of ground states by solving the cohomology problem. For one of these, we find that via a sequence of mappings the entire spectrum can be understood. It exhibits a gapped phase at 1/4 filling and a gapless phase at 1/6 filling and phase separation at intermediate fillings. The gapless phase separates into an exponential number of sectors, where the continuum limit of each sector is described by a superconformal field theory.