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.
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
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.