Measuring the Hausdorff Dimension of Quantum Mechanical Paths

We measure the propagator length in imaginary time quantum mechanics by Monte Carlo simulation on a lattice and extract the Hausdorff dimension $d_{H}$. We find that all local potentials fall into the same universality class giving $d_{H}=2$ like the free motion. A velocity dependent action ($S \propto \int dt \mid \vec{v} \mid^{\alpha}$) in the path integral (e.g. electrons moving in solids, or Brueckner's theory of nuclear matter) yields $d_{H}=\frac{\alpha }{\alpha - 1}$ if $\alpha > 2$ and $d_{H}=2$ if $\alpha \leq 2$. We discuss the relevance of fractal pathes in solid state physics and in $QFT$, in particular for the Wilson loop in $QCD$.Comment: uuencoded and compressed shell archive file. 8 pages with 7 figure

Chern-Simons term in the 4-dimensional SU(2) Higgs Model

Using Seiberg's definition for the geometric charge in SU(2) lattice gauge theory, we have managed to apply it also to the Chern-Simons term. We checked the periodic structure and determined the Chern-Simons density on small lattices $L^4$ and $L^3 \times 2,\, 4$ with L=4,\, 6,\mbox{ and }8 near the critical region in the SU(2) Higgs model. The data indicate that tunneling is increased at high temperature.Comment: 7 pages plus 4 PS figure

A Lower Bound on TSR/mHT_{SR}/{m_{\rm H}} in the O(4) Model on Anisotropic Lattices

Results of an investigation of the $O(4)$ spin model at finite temperature using anisotropic lattices are presented. In both the large $N$ approximation and numerical simulations using the Wolff cluster algorithm we find that the ratio of the symmetry restoration temperature $T_{\rm SR}$ to the Higgs mass $m_{\rm H}$ is independent of the anisotropy $\xi$. From the numerical simulations we obtain a lower bound of $T_{\rm SR} / m_{\rm H} \simeq 0.58 \pm 0.02$ at a value for the Higgs mass $m_{\rm H}a_s \simeq 0.5$, which is lowered further by about $10\%$ at $m_{\rm H}a_s \simeq 1$. Requiring certain timelike correlation functions to coincide with their spacelike counterparts, quantum and scaling corrections to the anisotropy are determined and are found to be small, i.e., the anisotropy is found to be close to the ratio of spacelike and timelike lattice spacings.Comment: 16 pages with 4 ps figures included. LaTeX file. BI-TP 92/27, FSU-SCRI-92-101, HLRZ-92-40, TIFR/TH/92-4

Vacuum Tunneling and Periodic Structure in Lattice Higgs Models

Using a geometric definition for the lattice Chern-Simons term in even dimensions, we have studied the distribution of Chern-Simons numbers for the 2d-U(1) and the 4d-SU(2) lattice Higgs models. The periodic structure of the distributions is preserved in our lattice formulation and has been examined in detail. In both cases the finite size effects visible in the distribution of Chern-Simons numbers are well accounted for by the Haar measure. Moreover, we find that $\langle N_{CS}^2 \rangle$ grows with the spatial volume. We also find numerical evidence that tunneling in 4d is increased at high temperature. (PS-File including Figures available via E-mail: [email protected])Comment: 25 pages, 5 figures, LaTeX (+ PicLaTeX) file, HLRZ-93-08 and BI-TP 93/0

Fluctuations of Chern-Simons numbers in the SU(2) Higgs model

Karsch F, Laursen ML, Neuhaus T, Plache B. Fluctuations of Chern-Simons numbers in the SU(2) Higgs model. International journal of modern physics, C. 1993;4(06):1195-1204.We investigate vacuum transitions in lattice higgs models at finite temperature. The 2 dimensional U(1) Higgs model is used as a toy model. In the 4 dimensional SU(2) Higgs model the region of the phase transition and temperatures above it are considered. The couplings (beta, kappa, lambda) = (2.25, 0.27, 0.5) and (8.0, 0.12996, 0.0017235) correspond to masses in lattice units (a(sigma) m(H), a(sigma) m(W)) of (0.02, 0.05) and (0.2, 0.2), respectively. The algorithm is described and a parallelized version is proposed. Taking the influence of the finite lattice into account we discuss temperature effects. We compare our results with perturbative estimates and claim that they link low and high temperature approximations

A Study of symmetry restoration at finite temperature in the O(4) model using anisotropic lattices

Gavai RV, Heller UM, Karsch F, Plache B, Neuhaus T. A Study of symmetry restoration at finite temperature in the O(4) model using anisotropic lattices. Nucl.Phys.Proc.Suppl. 1992;26:539-541

Thermal Fluctuations of Chern-Simons Numbers in the Lattice SU(2) Higgs Model

We study the temperature dependence of the Chern-Simons number fluctuations in the SU(2) Higgs Model on Euclidean lattices with spatial sizes up to 20^3. Temperatures well above the Higgs phase transition $T_H$ are achieved on anisotropic lattices. Numerical results are compared to perturbative results on finite lattices as well as in continuum perturbation theory. We find qualitative agreement with perturbative estimates and see at high temperatures a tendency towards static configurations. Up to temperatures $T \simeq 2T_H$ we find indication that tunneling between vacuums with different Chern-Simons numbers are still exponentially suppressed. (Full PostScript file available via Email from [email protected] )Comment: 18 pages latex, (+piclatex+epsf (for 6 figures)), HLRZ-93-39 BI-TP 93/2