Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian

Abstract

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.

Synchronization in complex networks of phase oscillators: A survey

“…For arbitrary network topologies and weights the equilibrium and potential energy landscape of the oscillator network (1) has been studied by different communities, see (Tavora and Smith, 1972a;Korsak, 1972;Araposthatis et al, 1981;Baillieul and Byrnes, 1982;Mehta and Kastner, 2011). We particularly recommend the article (Araposthatis et al, 1981), where various surprising and counter-intuitive examples are reported.…”

“…Likewise, for the transient analysis, the ∞ -type contraction Lyapunov function (32) is a powerful analysis concepts for complete graphs and still needs to be extended to arbitrary connected graphs. Regarding the potential and equilibrium landscape, a few interesting and still unresolved conjectures can be found in Tavora and Smith (1972a); Araposthatis et al (1981); Baillieul and Byrnes (1982); Mehta and Kastner (2011);Korsak (1972) and pertain to the number of (stable) equilibria and topological properties of the equilibrium set. Finally, the complex networks, nonlinear dynamics, and statistical physics communities found various interesting scaling laws in their statistical and numerical analyses of random graph models, such as conditions depending on the spectral ratio λ 2 /λ n of the Laplacian eigenvalues, interesting results for correlations between the degree deg i and the natural frequency ω i , and degree-dependent synchronization conditions (Nishikawa et al, 2003;Moreno and Pacheco, 2004;Restrepo et al, 2005;Boccaletti et al, 2006;Gómez-Gardeñes et al, 2007;Arenas et al, 2008;Kalloniatis, 2010;Skardal et al, 2013).…”

Potential energy landscapes for the 2D XY model: Minima, transition states, and pathways

“…Given the wide-ranging applications of this model, which include superconductivity, superfluidity, liquid crystals, Josephson junctions, and the fundamental importance of this Hamiltonian in lattice gauge theory [8][9][10] …”

“…H also corresponds to the lattice Landau gauge functional for a compact U(1) lattice gauge theory [8][9][10], and to the nearest-neighbor Kuramoto model with homogeneous frequency, where the stationary points constitute special configurations in phase space from a non-linear dynamical systems viewpoint [11].…”

Certification and the potential energy landscape

“…A variety of techniques have been deveoped within the framework of potential energy landscape theory [1,2], with applications to many-body systems as diverse as metallic clusters, biomolecules and their folding transitions, and glass formers. Except for rare examples, like the one-dimensional XY model [3], it is not usually possible to obtain the SPs analytically because solving the nonlinear stationary equations can be an extremely difficult task. Hence, one has to rely upon numerical methods.…”