Lattice Landau Gauge and Algebraic Geometry

Mehta, Dhagash; Sternbeck, André; Smekal, Lorenz von; Williams, Anthony G.

doi:10.22323/1.087.0025

Cited by 46 publications

(68 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is described in section 3. After restricting to the first Gribov region, the remainders of the residual gauge orbits still possess a large number of Gribov copies [69,[86][87][88][89]. This set will also be denoted as the residual gauge orbit in the following, to avoid the term residual of the residual gauge orbit.…”

Section: Proposals For Resolving the Gribov-singer Ambiguity 251 Gmentioning

confidence: 99%

“…This set will also be denoted as the residual gauge orbit in the following, to avoid the term residual of the residual gauge orbit. In fact, in an infinite volume this number is likely infinite, and in a finite volume V it appears to be a rapidly rising function of V [88][89][90], possibly even proportional to exp V [84].…”

Section: Proposals For Resolving the Gribov-singer Ambiguity 251 Gmentioning

confidence: 99%

“…It is also in general non-trivial how to find all Gribov copies, so that they can be counted. Thus, except for very special systems [88], only a lower limit can be posed on the number of Gribov copies. Examples for these are shown for two, three, and four space-time dimension on a finite lattice 12 in figure 1.…”

Section: Proposals For Resolving the Gribov-singer Ambiguity 251 Gmentioning

confidence: 99%

See 2 more Smart Citations

Gauge bosons at zero and finite temperature

2013

View full text Add to dashboard Cite

Gauge theories of the Yang-Mills type are the single most important building block of the standard model of particle physics and beyond. They are an integral part of the strong and weak interactions, and in their Abelian version of electromagnetism. Since Yang-Mills theories are gauge theories their elementary particles, the gauge bosons, cannot be described without fixing a gauge. Therefore, to obtain their properties a quantized and gauge-fixed setting is necessary.Beyond perturbation theory, gauge-fixing in non-Abelian gauge theories is obstructed by the Gribov-Singer ambiguity, which requires the introduction of non-local constraints. The construction and implementation of a method-independent gauge-fixing prescription to resolve this ambiguity is the single most important first step to describe gauge bosons beyond perturbation theory. Proposals for such a procedure, generalizing the perturbative Landau gauge, are described here. Their implementation are discussed for two example methods, lattice gauge theory and the quantum equations of motion.After gauge-fixing, it is possible to study gauge bosons in detail. The most direct access is provided by their correlation functions. The corresponding two-and three-point correlation functions are presented at all energy scales. These give access to the properties of the gauge bosons, like their absence from the asymptotic physical state space, particle-like properties at high energies, and the running coupling. Furthermore, auxiliary degrees of freedom are introduced during gauge-fixing, and their properties are discussed as well. These results are presented for two, three, and four dimensions, and for various gauge algebras.Finally, the modifications of the properties of gauge bosons at finite temperature are presented. Evidence is provided that these reflect the phase structure of Yang-Mills theory. However, it is found that the phase transition is not deconfining the gauge bosons, although the bulk thermodynamical behavior is of a Stefan-Boltzmann type. The resolution of this apparent contradiction is also presented. In addition, this resolution provides an explicit and constructive solution to the Linde problem.Thus, the technical and conceptual framework presented here can be taken as a basis how to determine correlation functions in Yang-Mills theory, therefore opening up the avenue to investigate theories of direct practical relevance. The status of this effort will be briefly described, alongside with connections to other approaches to Yang-Mills theory beyond perturbation theory.

show abstract

Section: Proposals For Resolving the Gribov-singer Ambiguity 251 Gmentioning

confidence: 99%

Section: Proposals For Resolving the Gribov-singer Ambiguity 251 Gmentioning

confidence: 99%

Section: Proposals For Resolving the Gribov-singer Ambiguity 251 Gmentioning

confidence: 99%

See 1 more Smart Citation

Gauge bosons at zero and finite temperature

2013

View full text Add to dashboard Cite

show abstract

“…, q s N ) with q s j ∈ {0, ± 6µ 2 /λ}. Knowledge of these solutions permits us to continue them to J > 0 by numerical continuation; see [13] for a description of the homotopy continuation method we have actually been using. Under certain conditions on the initial (decoupled) and final (coupled) potentials, this method is known to yield all stationary points of V .…”

mentioning

confidence: 99%

Phase Transitions Detached from Stationary Points of the Energy Landscape

Kastner

Mehta

2011

Phys. Rev. Lett.

View full text Add to dashboard Cite

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...

show abstract

“…More specifically, even if a numerical approximation is heuristically validated, it could turn out to be a nonsolution at higher precision, or Newton iterations may have unpredictable behavior, such as attracting cycles and chaos, when applied to points that are not in a basin of attraction [4][5][6][7][8] of some solution. We note that if the given system is a set of polynomial equations, then one can use numerical polynomial homotopy continuation [9][10][11][12][13][14][15][16][17][18][19][20][21][22] to compute all the isolated solutions (see e.g. [23][24][25] for some related approaches).…”

Section: Introductionmentioning

confidence: 99%

Certification and the potential energy landscape

Mehta

Hauenstein

Wales

2014

The Journal of Chemical Physics

View full text Add to dashboard Cite

Typically, there is no guarantee that a numerical approximation obtained using standard nonlinear equation solvers is indeed an actual solution, meaning that it lies in the quadratic convergence basin. Instead, it may lie only in the linear convergence basin, or even in a chaotic region, and hence not converge to the corresponding stationary point when further optimization is attempted. In some cases, these non-solutions could be misleading. Proving that a numerical approximation will quadratically converge to a stationary point is termed certification. In this report, we provide details of how Smale's α-theory can be used to certify numerically obtained stationary points of a potential energy landscape, providing a mathematical proof that the numerical approximation does indeed correspond to an actual stationary point, independent of the precision employed.

show abstract

Lattice Landau Gauge and Algebraic Geometry

Cited by 46 publications

References 28 publications

Gauge bosons at zero and finite temperature

Gauge bosons at zero and finite temperature

Phase Transitions Detached from Stationary Points of the Energy Landscape

Certification and the potential energy landscape

Contact Info

Product

Resources

About