The Kovacs effect in the one-dimensional Ising model: a linear response analysis

We analyze the so-called Kovacs effect in the one-dimensional Ising model with Glauber dynamics. We consider small enough temperature jumps, for which a linear response theory has been recently derived. Within this theory, the Kovacs hump is directly related to the monotonic relaxation function of the energy. The analytical results are compared with extensive Monte Carlo simulations, and an excellent agreement is found. Remarkably, the position of the maximum in the Kovacs hump depends on the fact that the true asymptotic behavior of the relaxation function is different from the stretched exponential describing the relevant part of the relaxation at low temperatures.Comment: accepted for publication in Phys. Rev.

Two charges on plane in a magnetic field I. "Quasi-equal" charges and neutral quantum system at rest cases

Low-lying bound states for the problem of two Coulomb charges of finite masses on a plane subject to a constant magnetic field $B$ perpendicular to the plane are considered. Major emphasis is given to two systems: two charges with the equal charge-to-mass ratio (quasi-equal charges) and neutral systems with concrete results for the Hydrogen atom and two electrons (quantum dot). It is shown that for these two cases, but when a neutral system is at rest (the center-of-mass momentum is zero), some outstanding properties occur: in double polar coordinates in CMS $(R, \phi)$ and relative $(\rho, \varphi)$ coordinate systems (i) the eigenfunctions are factorizable, all factors except for $\rho$-dependent are found analytically, they have definite relative angular momentum, (ii) dynamics in $\rho$-direction is the same for both systems being described by a funnel-type potential; (iii) at some discrete values of dimensionless magnetic fields $b \leq 1$ the system becomes {\it quasi-exactly-solvable} and a finite number of eigenfunctions in $\rho$ are polynomials. The variational method is employed. Trial functions are based on combining for the phase of a wavefunction (a) the WKB expansion at large distances, (b) the perturbation theory at small distances (c) with a form of the known analytically (quasi-exactly-solvable) eigenfunctions. For the lowest states with relative magnetic quantum numbers $s=0,1,2$ this approximation gives not less than 7 s.d., 8 s.d., 9 s.d., respectively, for the total energy $E(B)$ for magnetic fields $0.049\, \text{a.u.} < B < 2000\, \text{a.u.}$ (Hydrogen atom) and $0.025\, \text{a.u.}\eqslantless B \eqslantless 1000\, \text{a.u.}$ (two electrons).Comment: 38 pages, 8 figures, 11 table

Two charges on plane in a magnetic field: II. Moving neutral quantum system across a magnetic field

The moving neutral system of two Coulomb charges on a plane subject to a constant magnetic field $B$ perpendicular to the plane is considered. It is shown that the composite system of finite total mass is bound for any center-of-mass momentum $P$ and magnetic field strength; the energy of the ground state is calculated accurately using a variational approach. Their accuracy is cross-checked in a Lagrange-mesh method for $B=1$ a.u. and in a perturbation theory at small $B$ and $P$. The constructed trial function has the property of being a uniform approximation of the exact eigenfunction. For a Hydrogen atom and a Positronium a double perturbation theory in $B$ and $P$ is developed and the first corrections are found algebraically. A phenomenon of a sharp change of energy behavior for a certain center-of-mass momentum and a fixed magnetic field is indicated.Comment: 24 pages, 5 figures, 13 tables (6 in main body and 7 moved in a supplementary material), several clarifying sentences and 3 extra references added, Chapter II rectified, typos fixed, Annals of Physics (to be published

Conformal Laplace superintegrable systems in 2D: polynomial invariant subspaces

2nd-order conformal superintegrable systems in $n$ dimensions are Laplace equations on a manifold with an added scalar potential and $2n - 1$ independent 2nd order conformal symmetry operators. They encode all the information about Helmholtz (eigenvalue) superintegrable systems in an efficient manner: there is a 1-1 correspondence between Laplace superintegrable systems and Stackel equivalence classes of Helmholtz superintegrable systems. In this paper we focus on superintegrable systems in two dimensions, $n = 2$, where there are 44 Helmholtz systems, corresponding to 12 Laplace systems. For each Laplace equation we determine the possible 2-variate polynomial subspaces that are invariant under the action of the Laplace operator, thus leading to families of polynomial eigenfunctions. We also study the behavior of the polynomial invariant subspaces under a Stackel transform. The principal new results are the details of the polynomial variables and the conditions on parameters of the potential corresponding to polynomial solutions. The hidden gl_3-algebraic structure is exhibited for the exact and quasi-exact systems. For physically meaningful solutions, the orthogonality properties and normalizability of the polynomials are presented as well. Finally, for all Helmholtz superintegrable solvable systems we give a unified construction of 1D and 2D quasi-exactly solvable potentials possessing polynomial solutions, and a construction of new 2D PT-symmetric potentials is established.Comment: 28 page

Three-loop Correction to the Instanton Density. II. The Sine-Gordon potential

In this second paper on quantum fluctuations near the classical instanton configurations, see {\em Phys. Rev. D \bf 92}, 025046 (2015) and arXiv:1501.03993, we focus on another well studied quantum-mechanical problem, the one-dimensional Sine-Gordon potential (the Mathieu potential). Using only the tools from quantum field theory, the Feynman diagrams in the instanton background, we calculate the tunneling amplitude (the instanton density) to the three-loop order. The result confirms (to seven significant figures) the one recently recalculated by G. V. Dunne and M. \"{U}nsal, {\it Phys. Rev. \bf D 89}, 105009 (2014) from the resurgence perspective. As in the double well potential case, we found that the largest contribution is given by the diagrams originating from the Jacobian. We again observe that in the three-loop case individual Feynman diagrams contain irrational contributions, while their sum does not.Comment: 14 pages, 3 figures, 1 table, already published at Phys.Rev.D: a post published version with tadpole diagrams redrawn to mark clearly the Jacobian source, Note Added about significance of a tadpole diagram in 3-,4-,5-loops contribution

Fluctuations in quantum mechanics and field theories from a new version of semiclassical theory. II

This is the second paper on semiclassical approach based on the density matrix given by the Euclidean time path integral with fixed coinciding endpoints. The classical path, interpolating between this point and the classical vacuum, called "flucton", plus systematic one- and two-loop corrections, has been calculated in the first paper \cite{Escobar-Ruiz:2016aqv} for double-well potential and now extended for a number of quantum-mechanical problems (anharmonic oscillator, sine-Gordon potential). The method is based on systematic expansion in Feynman diagrams and thus can be extended to QFTs. We show that the loop expansion in QM reminds the leading log-approximations in QFT. In this sequel we present complete set of results obtained using this method in unified way. Alternatively, starting from the Schr\"{o}dinger equation we derive a {\it generalized} Bloch equation which semiclassical-like, iterative solution generates the loop expansion. We re-derive two loop expansions for all three above potentials and now extend it to three loops, which has not yet been done via Feynman diagrams. All results for both methods are fully consistent with each other. Asymmetric (tilted) double-well potential (non-degenerate minima) is also studied using the second method

Studying the Triple Higgs Self-Coupling Via e+e- --> b bar b HH, t bar t HH at Future Linear e+e- Colliders

We study the triple Higgs self-coupling at future $e^{+}e^{-}$ colliders energies, with the reactions $e^{+}e^{-}\to b \bar b HH$ and $e^{+}e^{-}\to t \bar t HH$. We evaluate the total cross section of $b\bar bHH$, $t\bar tHH$ and calculate the total number of events considering the complete set of Feynman diagrams at tree-level. The sensitivity of the triple Higgs coupling is considered in the Higgs mass range 110-190 $GeV$, for the energy which is expected to be available at a possible Next Linear $e^{+}e^{-}$ Collider with a center-of-mass energy $800, 1000, 1500$ $GeV$ and luminosity 1000 $fb^{-1}$.Comment: 15 pages, 10 figure

Neutral Higgs Boson Pair-Production and Trilinear Self-Couplings in the MSSM at ILC and CLIC Energies

We study pair-production as well as the triple self-couplings of the neutral Higgs bosons of the Minimal Supersymmetric Standard Model (MSSM) at the Future International Linear $e^{+}e^{-}$ Collider (ILC) and Compact Linear Collider (CLIC). The analysis is based on the reactions $e^{+}e^{-}\to b \bar b h_ih_i, t \bar t h_ih_i$ with $h_i=h, H, A$. We evaluate the total cross-section for both $b\bar bh_ih_i$, $t\bar th_ih_i$ and calculate the total number of events considering the complete set of Feynman diagrams at tree-level. We vary the triple couplings $\kappa\lambda_{hhh}$, $\kappa\lambda_{Hhh}$, $\kappa\lambda_{hAA}$, $\kappa\lambda_{HAA}$, $\kappa\lambda_{hHH}$ and $\kappa\lambda_{HHH}$ within the range $\kappa=-1$ and +2. The numerical computation is done for the energies expected at the ILC with a center-of-mass energy 500, 1000, 1600 $GeV$ and a luminosity 1000 $fb^{-1}$. The channels $e^{+}e^{-}\to b \bar b h_ih_i$ and $e^{+}e^{-}\to t \bar t h_ih_i$ are also discussed to a center-of-mass energy of 3 $TeV$ and luminosities of 1000 $fb^{-1}$ and 5000 $fb^{-1}$.Comment: 26 pages, 11 figure

Chaotic Dynamics of Comet 1P/Halley; Lyapunov Exponent and Survival Time Expectancy

The orbital elements of comet Halley are known to a very high precision, suggesting that the calculation of its future dynamical evolution is straightforward. In this paper we seek to characterize the chaotic nature of the present day orbit of comet Halley and to quantify the timescale over which its motion can be predicted confidently. In addition, we attempt to determine the timescale over which its present day orbit will remain stable. Numerical simulations of the dynamics of test particles in orbits similar to that of comet Halley are carried out with the Mercury 6.2 code. On the basis of these we construct survival time maps to assess the absolute stability of Halley's orbit, frequency analysis maps, to study the variability of the orbit and we calculate the Lyapunov exponent for the orbit for variations in initial conditions at the level of the present day uncertainties in our knowledge of its orbital parameters. On the basis of our calculations of the Lyapunov exponent for comet Halley, the chaotic nature of its motion is demonstrated. The e-folding timescale for the divergence of initially very similar orbits is approximately 70 years. The sensitivity of the dynamics on initial conditions is also evident in the self-similarity character of the survival time and frequency analysis maps in the vicinity of Halley's orbit, which indicates that, on average, it is unstable on a timescale of hundreds of thousands of years. The chaotic nature of Halley's present day orbit implies that a precise determination of its motion, at the level of the present day observational uncertainty, is difficult to predict on a timescale of approximately 100 years. Furthermore, we also find that the ejection of Halley from the solar system or its collision with another body could occur on a timescale as short as 10,000 years.Comment: 11 pages, 10 figures, accepted for publication in MNRA

Three-loop Correction to the Instanton Density. I. The Quartic Double Well Potential

This paper deals with quantum fluctuations near the classical instanton configuration. Feynman diagrams in the instanton background are used for the calculation of the tunneling amplitude (the instanton density) in the three-loop order for quartic double-well potential. The result for the three-loop contribution coincides in six significant figures with one given long ago by J.~Zinn-Justin. Unlike the two-loop contribution where all involved Feynman integrals are rational numbers, in the three-loop case Feynman diagrams can contain irrational contributions.Comment: 15 pages, 3 figures, 1 table, already published at Phys.Rev.D92 (2015) 025046, 089902(erratum), a post published version with tadpole diagrams redrawn to clearly mark the Jacobian source and misprint on 3-loop tadpole contribution on p.8 fixe