We propose a modified lattice Landau gauge based on stereographically projecting the link variables on the circle S 1 → R for compact U(1) or the 3-sphere S 3 → R 3 for SU(2) before imposing the Landau gauge condition. This can reduce the number of Gribov copies exponentially and solves the Gribov problem in compact U(1) where it is a lattice artifact. Applied to the maximal Abelian subgroup this might be just enough to avoid the perfect cancellation amongst the Gribov copies in a lattice BRST formulation for SU(N), and thus to avoid the Neuberger 0/0 problem. The continuum limit of the Landau gauge remains unchanged.

Abstract. We investigate 2-colour QCD with 2 flavours of Wilson fermion at nonzero temperature T and quark chemical potential µ, with a pion mass of 700 MeV (mπ/mρ = 0.8). From temperature scans at fixed µ we find that the critical temperature for the superfluid to normal transition depends only very weakly on µ above the onset chemical potential, while the deconfinement crossover temperature is clearly decreasing with µ. We find indications of a region of superfluid but deconfined matter at high µ and intermediate T . The static quark potential determined from the Wilson loop is found to exhibit a 'string tension' that increases at large µ in the 'deconfined' region. The electric (longitudinal) gluon propagator in Landau gauge becomes strongly screened with increasing temperature and chemical potential. The magnetic (transverse) gluon shows little sensitivity to temperature, and exhibits a mild enhancement at intermediate µ before becoming suppressed at large µ.PACS. 11.15Ha Lattice gauge theory -12.38Aw Lattice QCD calculations -21.65Qr Quark matter -12.38Mh Quark-gluon plasma

We argue that a generic instability afflicts vacua that arise in theories whose moduli space has large dimension. Specifically, by studying theories with multiple scalar fields we provide numerical evidence that for a generic local minimum of the potential the usual semiclassical bubble nucleation rate, Γ = A e −B , increases rapidly as function of the number of fields in the theory. As a consequence, the fraction of vacua with tunneling rates low enough to maintain metastability appears to fall exponentially as a function of the moduli space dimension. We discuss possible implications for the landscape of string theory. Notably, if our results prove applicable to string theory, the landscape of metastable vacua may not contain sufficient diversity to offer a natural explanation of dark energy.

Machine learning techniques are being increasingly used as flexible non-linear fitting and prediction tools in the physical sciences. Fitting functions that exhibit multiple solutions as local minima can be analysed in terms of the corresponding machine learning landscape. Methods to explore and visualise molecular potential energy landscapes can be applied to these machine learning landscapes to gain new insight into the solution space involved in training and the nature of the corresponding predictions. In particular, we can define quantities analogous to molecular structure, thermodynamics, and kinetics, and relate these emergent properties to the structure of the underlying landscape. This Perspective aims to describe these analogies with examples from recent applications, and suggest avenues for new interdisciplinary research.

The stationary points of the potential energy function V are studied for the φ 4 model on a two-dimensional square lattice with nearest-neighbor interactions. On the basis of analytical and numerical results, we explore the relation of stationary points to the occurrence of thermodynamic phase transitions. We find that the phase transition potential energy of the φ 4 model does in general not coincide with the potential energy of any of the stationary points of V . This disproves earlier, allegedly rigorous, claims in the literature on necessary conditions for the existence of phase transitions. Moreover, we find evidence that the indices of stationary points scale extensively with the system size, and therefore the index density can be used to characterize features of the energy landscape in the infinite-system limit. We conclude that the finite-system stationary points provide one possible mechanism of how a phase transition can arise, but not the only one. The stationary points of the potential energy function or other classical energy functions can be employed to calculate or estimate physical quantities. Well-known examples include transition state theory or Kramers's reaction rate theory for the thermally activated escape from metastable states, where the barrier height (corresponding to the difference between potential energies at certain stationary points of the potential energy function) plays an essential role. More recently, a large variety of related techniques has become popular under the name of energy landscape methods [1], with applications to many-body systems as diverse as metallic clusters, or biomolecules and their folding transitions. While the mentioned applications focus mostly on the numerical investigation of finite systems, the analysis of stationary points has also proved useful for analytical studies of N -body systems in the thermodynamic limit. One field of research where such methods have been fruitfully applied is disordered systems undergoing a dynamical glass transition [2].Another line of research based on stationary points but focusing on equilibrium phase transitions in the thermodynamic limit N → ∞, dates back to about the same time [3]. This approach, originally formulated in terms of topology changes of configuration space submanifolds, can be rephrased in terms of stationary points of the potential energy function V , i.e. configuration space points q s satisfying ∇V (q s ) = 0. The underlying idea can be understood as follows [4]: Thermodynamic equilibrium properties are encoded in the thermodynamic limit value of the microcanonical configurational entropywhere Γ denotes configuration space and dx its volume measure, Σ v ⊂ Γ is the hypersurface of constantpotential energy V = N v, and dΣ stands for the (N − 1)-dimensional Hausdorff measure on Σ v . At a stationary point, we have ∇V = 0, the integrand on the righthand side of (1) diverges, and we may expect the stationary point to give an important contribution to the integral. Indeed, it has been shown that, for finite N...

We study the stationary points of what is known as the lattice Landau gauge fixing functional in one-dimensional compact U(1) lattice gauge theory, or as the Hamiltonian of the one-dimensional random phase XY model in statistical physics. An analytic solution of all stationary points is derived for lattices with an odd number of lattice sites and periodic boundary conditions. In the context of lattice gauge theory, these stationary points and their indices are used to compute the gauge fixing partition function, making reference in particular to the Neuberger problem. Interpreted as stationary points of the one-dimensional XY Hamiltonian, the solutions and their Hessian determinants allow us to evaluate a criterion which makes predictions on the existence of phase transitions and the corresponding critical energies in the thermodynamic limit.

In the modern formulation of lattice gauge-fixing, the gauge fixing condition is written in terms of the minima or stationary points (collectively called solutions) of a gauge-fixing functional. Due to the non-linearity of this functional, it usually has many solutions called Gribov copies. The dependence of the number of Gribov copies, n[U] on the different gauge orbits plays an important role in constructing the Faddeev-Popov procedure and hence in realising the BRST symmetry on the lattice. Here, we initiate a study of counting n[U] for different orbits using three complimentary methods: 1. analytical results in lower dimensions, and some lower bounds on n[U] in higher dimensions, 2. the numerical polynomial homotopy continuation method, which numerically finds all Gribov copies for a given orbit for small lattices, and 3. numerical minimisation ("brute force"), which finds many distinct Gribov copies, but not necessarily all. Because n for the coset SU(N_c)/U(1) of an SU(N_c) theory is orbit-independent, we concentrate on the residual compact U(1) case in this article and establish that n is orbit-dependent for the minimal lattice Landau gauge and orbit-independent for the absolute lattice Landau gauge. We also observe that contrary to a previous claim, n is not exponentially suppressed for the recently proposed stereographic lattice Landau gauge compared to the naive gauge in more than one dimension.Comment: 39 pages, 15 eps figures. Published version: minor changes onl

Abstract-The manuscript addresses the problem of finding all solutions of power flow equations or other similar nonlinear system of algebraic equations. This problem arises naturally in a number of power systems contexts, most importantly in the context of direct methods for transient stability analysis and voltage stability assessment. We introduce a novel form of homotopy continuation method called the numerical polynomial homotopy continuation (NPHC) method that is mathematically guaranteed to find all the solutions without ever encountering a bifurcation. The method is based on embedding the real form of power flow equation in complex space, and tracking the generally unphysical solutions with complex values of real and imaginary parts of the voltage. The solutions converge to physical real form in the end of the homotopy. The so-called γ-trick mathematically rigorously ensures that all the paths are wellbehaved along the paths, so unlike other continuation approaches, no special handling of bifurcations is necessary. The method is embarrassingly parallelizable and can be applied to reasonably large sized systems. We demonstrate the technique by analysis of several standard test cases up to the 14-bus system size. Finally, we discuss possible strategies for scaling the method to large size systems, and propose several applications for transient stability analysis and voltage stability assessment.