Direct and indirect loop equations in lattice Yang-Mills theory

We apply a quantum teleportation protocol based on the Hayden-Preskill thought experiment to quantify how scrambling a given quantum evolution is. It has an advantage over the direct measurement of out-of-time ordered correlators when used to diagnose the information scrambling in the presence of decoherence effects stemming from a noisy quantum device. We demonstrate the protocol by applying it to two physical systems: Ising spin chain and SU(2) lattice Yang-Mills theory. To this end, we numerically simulate the time evolution of the two theories in the Hamiltonian formalism. The lattice Yang-Mills theory is implemented with a suitable truncation of Hilbert space on the basis of the Kogut-Susskind formalism. On a two-leg ladder geometry and with the lowest nontrivial spin representations, it can be mapped to a spin chain, which we call it Yang-Mills-Ising model and is also directly applicable to future digital quantum simulations. We find that the Yang-Mills-Ising model shows the signal of information scrambling at late times.

In this paper I analyse lattice Yang-Mills theories with continuous time. After a short discussion of more conceptual questions, such as the existence of a Hamilton operator in the infinite volume limit, I study the phase diagram. The existence of a strong coupling/low temperature confinement phase (which was not proven up to now) is established for arbitrary compact groups, continuous or discrete. For discrete compact groups the deconfinement region decomposes into (at least) two phases, which are distinguished by the behaviour of spatial Wilson loops: a deconfinement phase where spatial Wilson loops still show area law behaviour, and a “freezing” phase with perimeter law behaviour for spatial Wilson loops. The methods to prove these results rely on cluster expansion methods, combined with renormalisation ideas.

We study the shape of the flux tube in lattice Yang-Mills theories and in particular its intrinsic width. In the framework of the Effective String Theory description of the confining flux tube this intrinsic width has no measurable effects on the inter-quark static potential, but it can be precisely detected looking at the profile of the flux tube. We address this problem with a set of high precision simulations in the (2+1) dimensional $\mathrm{SU}(2)$ model. We find two different behaviours as a function of the temperature. In the low temperature regime ($T \ll T_c$) we find a good agreement with an expression inspired by the dual superconductive model of confinement. In the high temperature regime ($T \lesssim T_c$) our data agree with a model based on the Svetitsky-Yaffe mapping. All our data in this regime can be described in terms of only one length scale, the intrinsic width, which turns out to be the same scale appearing in the confining inter-quark static potential.

Since the inception of lattice QCD, a natural definition for the Yang-Mills instanton on lattice has been long sought for. In a recent work [1], one of authors showed the natural solution has to be organized in terms of bundle gerbes in higher homotopy theory / higher category theory, and introduced the principles for such a categorical construction. To pave the way towards actual numerical implementation in the near future, nonetheless, an explicit construction is necessary. In this paper we provide such an explicit construction for SU(2) gauge theory, with technical aspects inspired by Lüscher’s 1982 geometrical construction [2]. We will see how the latter is in a suitable sense a saddle point approximation to the full categorical construction. The generalization to SU(N) will be discussed. The construction also allows for a natural definition of lattice Chern-Simons-Yang-Mills theory in three spacetime dimensions.

We discuss finite temperature lattice Yang-Mills theory with special attention to the confinement problem. The relationship between the confinement criteria of Wilson, Polyakov, and 't Hooft is clarified by establishing a string of inequalities between the corresponding string tensions. The close connection between finite temperature Yang-Mills models and spin models is exploited to obtain new and rather sharp upper bounds for the critical coupling constant above which there is confinement. This same analogy also allows us to establish infrared bounds for the gauge models that yield a lower bound for this critical coupling and thereby show the existence of a weak coupling regime without confinement at nonzero temperature in three or more space dimensions. Finally we discuss extension of our results to other forms of the lattice action, the Hamiltonian lattice models of Kogut and Susskind and 't Hooft'sN → ∞ limit.

Loop equation and Wilson line correlators in non-commutative gauge theories

Averaging of the system of equations of elasticity theory in a thin periodically inhomogeneous layer

We present an efficient numerical method for computing electromagnetic (EM) scattering of arbitrary three‐dimensional (3-D) local inhomogeneities buried in a uniform or two‐layered earth. In this scheme the inhomogeneity is enclosed by a volume whose conductivity is discretized by a finite‐element mesh and whose boundary is only a slight distance away from the inhomogeneity. The scheme uses two sets of independent equations. The first is a set of finite‐element equations derived from a variational integral, and the second is a mathematical expression for the fields at the boundany in terms of electric fields inside the boundary. The Green’s function is used to derive the second set of equations. An iterative algorithm has been developed to solve these two sets of equations. The solutions are the electric fields at nodes inside the finite‐element mesh. The scattered fields anywhere may then be obtained by performing volume integrations over the inhomogeneous region. The scheme is used for modeling 3-D inhomogeneities with plane‐wave and magnetic dipole sources. The results agree with earlier model analyses using the finite‐element technique.

Consistent equations for turbulent open-channel flows on a smooth bottom are derived using a turbulence model of mixing length and an asymptotic expansion in two layers. A shallow-water scaling is used in an upper – or external – layer and a viscous scaling is used in a thin viscous – or internal – layer close to the bottom wall. A matching procedure is used to connect both expansions in an overlap domain. Depth-averaged equations are then obtained in the approximation of weakly sheared flows which is rigorously justified. We show that the Saint-Venant equations with a negligible deviation from a flat velocity profile and with a friction law are a consistent set of equations at a certain level of approximation. The obtained friction law is of the Kármán–Prandtl type and successfully compared to relevant experiments of the literature. At a higher precision level, a consistent three-equation model is obtained with the mathematical structure of the Euler equations of compressible fluids with relaxation source terms. This new set of equations includes shearing effects and adds corrective terms to the Saint-Venant model. At this level of approximation, energy and momentum resistances are clearly distinguished. Several applications of this new model that pertains to the hydraulics of open-channel flows are presented including the computation of backwater curves and the numerical resolution of the growing and breaking of roll waves.

We review the Pinch Technique Background Field Method (PT-BFM) framework for formulating and solving the Schwinger-Dyson equations of Yang-Mills theories. In particular, we show how within this framework it is possible to write a new set of Schwinger-Dyson equations that (i) accommodate the dynamical gluon mass generation through Schwinger’s mechanism, and (ii) have much better truncation properties than the the conventional equations. The resulting solutions show (in the Landau gauge) an infra-red saturating gluon propagator and ghost dressing function, in agreement with all lattice studies to date for both SU(2) and SU(3) gauge groups as well as 3 and 4 space-time dimensions. We also briefly discuss how a massive gluon enables selfconsistently confinement through the condensation of thick vortices, and study other infra-red characteristic quantities such as the Kugo-Ojima function and the effective charge.

Retrieval of integrated water vapor content (WVC) from remote sensing data is often ill-posed because of insufficient observational information. There are many factors that cause WVC changes, which yield instability in the accuracy of many traditional algorithms. To overcome this problem, we developed a novel fully-coupled paradigm for the robust retrieval of WVC from thermal infrared remote sensing data. Through the derivation of the physical radiative transfer equation, we determined two conditions that need to be satisfied for the deep learning retrieval paradigm of WVC. The first condition is that the input parameters and output parameters of the deep learning need to be able to build a complete set of solvable equations in theory. The second condition is that, if there is a strong relationship between input parameters and output parameters, it can be directly retrieved. If it is a weak relationship, we need to use prior knowledge to improve the portability and accuracy of the algorithm. The training and test data of deep learning is composed of representative solutions of physical methods and solutions of statistical methods. The representative solutions of the physical methods were obtained from the physical forward model, and the statistical solutions were obtained from multi-source data which can compensate for the defect that the physical model cannot simulate mixed pixels. MODIS L1B data was used for case analysis of paradigm retrieval, and the analysis indicated that four thermal infrared bands were usually needed as the input parameters of deep learning and the integrated water vapor content as the output parameter. When land surface temperature and emissivity were taken as prior knowledge, the root-mean-square error (RMSE) of the retrieved WVC was 0.07 g/cm2. The optimal accuracy RMSE was 0.27 g/cm2. When there was a strong correlation between input parameters and output parameters, i.e., if there were two bands that were very sensitive to WVC in the band combination, high-precision retrieval could also be achieved without prior knowledge. All the analyses show that the paradigm of deep learning coupling physics and statistics can accurately retrieve WVC, which is a significant improvement on the traditional method and solves the problem of lack of physical interpretation of deep learning.

A hierarchy of approximate elastodynamic plate theories is deduced from the three-dimensional equations of elasticity. The derivation is based upon asymptotic expansions in a parameter viewed as the ratio of plate half thickness to a dominant signal wavelength of motion. Approximate theories of any desired order of accuracy are contained in a single compact set of partial differential equations with the order of approximation determined by the order of truncation of the differential operators appearing in the equations. As examples, particular cases of free extensional and forced flexural vibrations are studied. Lower-order theories for these cases are compared to some existing approximate theories of the same order.

: The Biot theory of propagation in a porous medium provides a mathematical framework for studying acoustic interaction with the seafloor. The theory considers the two-phase porous nature of marine sediments in contrast to the classical models of wave propagation in the seafloor that consider marine sediments as an extended single-phase fluid or solid. A boundary value problem is set up and solved for a line source in a fluid medium above a poro- viscoelastic halfspace. Expressions for the reflected and transmitted field are given in integral form and asymptotic expansions in the high-frequency, far- field limit. A set of simultaneous equations is solved to give plane wave reflection and transmission (Type I, Type II and shear wave) coefficients. These equations also yield the Scholte Pseudo-Scholte, and pseudo-Rayleigh wave phase velocities and attenuations. The plane wave coefficients and the surface wave velocities and attenuations are compared with commensurate quantities from the single-phase theories. Recent experiments of high frequency transmission through a water-sand interface have indicated anomalously high transmitted energy at angles near the critical angle. The Type II wave (predicted by Biot theory but not the single-phase theories) was suspected as a possible reason for the anomalies. Calculations indicate that the contributions from the Type II wave are negligible and thus not the cause of the anomalies. Much of the anomalous behavior appeared to arise simply as a result of an incorrect compressional wave attenuation in the geoacoustic model.

A spherical α2-dynamo is presented as an expansion in the free decay modes of the magnetic field. In the limit of vanishing viscosity the momentum equation yields various asymptotic expansions for the flow, depending on the precise form of the dissipation and boundary conditions applied. A new form for the dissipation is introduced that greatly simplifies this asymptotic expansion. When these expansions are substituted back into the induction equation, a set of modal amplitude equations is derived, and solved for various distributions of the α-effect. For all choices of α the solutions approach the Taylor state, but the manner in which this occurs can vary, as previously found by Soward and Jones (1983). Furthermore, as hypothesized by Malkus and Proctor (1975), but not previously demonstrated, the post-Taylor equilibration is indeed independent of the viscosity in the asymptotic limit, and depending on the choice of a may be either steady-state or oscillatory.

Introduction to Molecular Dynamics Simulations: Applications in Hard and Soft Condensed Matter Physics