Unconstrained Variables and Equivalence Relations for Lattice Gauge Theories

We write the partition function for a lattice gauge theory, with compact gauge group, exactly in terms of unconstrained variables and show that, in the mean field approximation, the dynamics of pure gauge theories, invariant under compact, continuous,groups of rank 1 is the same for all. We explicitly obtain the equivalence for the case of SU(2) and U(1) and show that it obtains, also, if we consider saddle point configurations that are not,necessarily, uniform, but only proportional to the identity for both groups. This implies that the phase diagrams of the (an)isotropic SU(2) theory and the (an)isotropic U(1) theory in any dimension are identical, within this approximation, up to a re-evaluation of the numerical values of the coupling constants at the transitions. Only nonuniform field configurations, that, also, belong to higher dimensional representations for Yang--Mills fields, will be able to p robe the difference between them. We also show under what conditions the global symmetry of an anisotropic term in the lattice action can be promoted to a gauge symmetry of the theory on layers and point out how deconstruction and flux compactification scenaria may thus be studied on the lattice.Comment: 14 pages, LateX2e. Expanded presentation of equivalence relation. Added discussion on how the global symmetry of the anisotropic term can be promoted to a gauge symmetry on a laye

Anomaly cancellation for anisotropic lattice fields with extra dimensions

The current flow from the bulk is due to the anomaly on the brane-but the absence of current flow is not, necessarily, due to anomaly cancellation, but to the absence of the chiral zero modes themselves, due to the existence of the layered phase. This can be understood in terms of the difference between the Chern-Simons terms in three and five dimensions. Thus the anomaly cancellation in four dimensions, which is essential for shielding the boundary from quantum effects within the bulk, makes sense only along the transition line between the layered and the Coulomb phase, which, in turn, requires the presence of a compact U(1) factor for the gauge group.Comment: 6 pages, 4 figures, LaTeX2e, uses PoS. Contribution to The XXVII International Symposium on Lattice Field Theory - LAT2009, July 26-31 2009,Peking University, Beijing, Chin

Second Order Phase Transition in Anisotropic Lattice Gauge Theories with Extra Dimensions

Field theories with extra dimensions live in a limbo. While their classical solutions have been the subject of considerable study, their quantum aspects are difficult to control. A special class of such theories are anisotropic gauge theories. The anisotropy was originally introduced to localize chiral fermions. Their continuum limit is of practical interest and it will be shown that the anisotropy of the gauge couplings plays a crucial role in opening the phase diagram of the theory to a new phase, that is separated from the others by a second order phase transition. The mechanism behind this is generic for a certain class of models, that can be studied with lattice techniques. This leads to new perspectives for the study of quantum effects of extra dimensions.Comment: 7 pages, 1 figure. Uses PoS.cls. Contribution to The XXVIII International Symposium on Lattice Filed Theory, June 14-19,2010,Villasimius, Sardinia Ital

A functional calculus for the magnetization dynamics

A functional calculus approach is applied to the derivation of evolution equations for the moments of the magnetization dynamics of systems subject to stochastic fields. It allows us to derive a general framework for obtaining the master equation for the stochastic magnetization dynamics, that is applied to both, Markovian and non-Markovian dynamics. The formalism is applied for studying different kinds of interactions, that are of practical relevance and hierarchies of evolution equations for the moments of the distribution of the magnetization are obtained. In each case, assumptions are spelled out, in order to close the hierarchies. These closure assumptions are tested by extensive numerical studies, that probe the validity of Gaussian or non--Gaussian closure Ans\"atze.Comment: 17 pages, 5 figure

Chaotic Information Processing by Extremal Black Holes

We review an explicit regularization of the AdS$_2$/CFT$_1$ correspondence, that preserves all isometries of bulk and boundary degrees of freedom. This scheme is useful to characterize the space of the unitary evolution operators that describe the dynamics of the microstates of extremal black holes in four spacetime dimensions. Using techniques from algebraic number theory to evaluate the transition amplitudes, we remark that the regularization scheme expresses the fast quantum computation capability of black holes as well as its chaotic nature.Comment: 8 pages, 2 JPEG figues. Contribution to the VII Black Holes Workshop, Aveiro PT, Decemeber 201

Noisy SUSY

We review the idea, put forward in 1982, by Parisi and Sourlas, that the bath of fluctuations, with which a physical system is in equilibrium, can be resolved by the superpartners of the degrees of freedom, defined by the classical action. This implies, in particular, that fermions can be described in terms of their superpartners, using the Nicolai map. We focus on the question, whether the fluctuations of scalar fields can, in fact, produce the absolute value of the stochastic determinant itself, whose contribution to the action can be identified with the fermionic degrees of freedom and present evidence supporting this idea in two spacetime dimensions. The same idea leads to a new formulation of supersymmetric QED. We also review the obstacles for extending this approach to Yang-Mills theories and report on progress for evading the obstructions for obtaining interacting theories in three and four spacetime dimensions. This implies, in particular, that it is possible to describe the effects of fermions in numerical simulations, through their superpartners.Comment: 13 pages, 2 figures. Written contribution to the Corfu Summer Institute 2022 "School and Workshops on Elementary Particle Physics and Gravity", 28 August - 1 October, 202

Non-Markovian magnetization dynamics for uniaxial nanomagnets

A stochastic approach for the description of the time evolution of the magnetization of nanomagnets is proposed, that interpolates between the Landau-Lifshitz-Gilbert and the Landau-Lifshitz-Bloch approximations, by varying the strength of the noise. Its finite autocorrelation time, i.e. when it may be described as colored, rather than white, is, also, taken into account and the consequences, on the scale of the response of the magnetization are investigated. It is shown that the hierarchy for the moments of the magnetization can be closed, by introducing a suitable truncation scheme, whose validity is tested by direct numerical solution of the moment equations and compared to the averages obtained from a numerical solution of the corresponding colored stochastic Langevin equation. This comparison is performed on magnetic systems subject to both an external uniform magnetic field and an internal one-site uniaxial anisotropy.Comment: 4 pages, 3 figure

Quantum Magnets and Matrix Lorenz Systems

The Landau--Lifshitz--Gilbert equations for the evolution of the magnetization, in presence of an external torque, can be cast in the form of the Lorenz equations and, thus, can describe chaotic fluctuations. To study quantum effects, we describe the magnetization by matrices, that take values in a Lie algebra. The finite dimensionality of the representation encodes the quantum fluctuations, while the non-linear nature of the equations can describe chaotic fluctuations. We identify a criterion, for the appearance of such non-linear terms. This depends on whether an invariant, symmetric tensor of the algebra can vanish or not. This proposal is studied in detail for the fundamental representation of $\mathfrak{u}(2)=\mathfrak{u}(1)\times\mathfrak{su}(2)$. We find a knotted structure for the attractor, a bimodal distribution for the largest Lyapunov exponent and that the dynamics takes place within the Cartan subalgebra, that does not contain only the identity matrix, thereby can describe the quantum fluctuations.Comment: 5 pages, 3 figures. Uses jpconf style. Presented at the ICM-SQUARE 4 conference, Madrid, August 2014. The topic is a special case of the content of 1404.7774, currently under revisio

The quantum cat map on the modular discretization of extremal black hole horizons

Based on our recent work on the discretization of the radial AdS$_2$ geometry of extremal BH horizons,we present a toy model for the chaotic unitary evolution of infalling single particle wave packets. We construct explicitly the eigenstates and eigenvalues for the single particle dynamics for an observer falling into the BH horizon, with time evolution operator the quantum Arnol'd cat map (QACM). Using these results we investigate the validity of the eigenstate thermalization hypothesis (ETH), as well as that of the fast scrambling time bound (STB). We find that the QACM, while possessing a linear spectrum, has eigenstates, which are random and satisfy the assumptions of the ETH. We also find that the thermalization of infalling wave packets in this particular model is exponentially fast, thereby saturating the STB, under the constraint that the finite dimension of the single--particle Hilbert space takes values in the set of Fibonacci integers.Comment: 28 pages LaTeX2e, 8 jpeg figures. Clarified certain issues pertaining to the relation between mixing time and scrambling time; enhanced discussion of the Eigenstate Thermalization Hypothesis; revised figures and updated references. Typos correcte