arXiv subjects

Christian Holm

Publications and source records attributed to Christian Holm.

136 records · Page 8Linked to original sources

arXiv1996

Fixed versus random triangulations in 2D simplicial Regge calculus

We study 2D quantum gravity on spherical topologies using the Regge calculus approach with the $dl/l$ measure. Instead of a fixed non-regular triangulation which has been used before, we study for each system size four different random triangulations, which are obtained according to the standard Voronoi-Delaunay procedure. We compare both approaches quantitatively and show that the difference in the expectation value of $R^2$ between the fixed and the random triangulation depends on the lattice size and the surface area $A$. We also try again to measure the string susceptibility exponents through a finite-size scaling Ansatz in the expectation value of an added $R^2$ interaction term in an approach where $A$ is held fixed. The string susceptibility exponent $γ_{str}'$ is shown to agree with theoretical predictions for the sphere, whereas the estimate for $γ_{str}$ appears to be too negative.

hep-lat↗

arXiv1996

Critical Exponents of the Classical Heisenberg Ferromagnet

In a recent letter, R.G. Brown and M. Ciftan (Phys. Rev. Lett. 76, 1352, 1996) reported high precision Monte Carlo (MC) estimates of the static critical exponents of the classical 3D Heisenberg model, which stand in sharp contrast to values obtained by four independent approaches, namely by other recent high statistics MC simulations, high-temperature series analyses, field theoretical methods, and experimental studies. In reply to the above cited work we submitted this paper as a comment to Phys. Rev. Lett.

hep-lat↗

arXiv1995

Ising Spins on a Gravitating Sphere

We investigated numerically an Ising model coupled to two-dimensional Euclidean gravity with spherical topology, using Regge calculus with the $dl/l$ path-integral measure to discretize the gravitational interaction. Previous studies of this system with toroidal topology have shown that the critical behavior of the Ising model remains in the flat-space Onsager universality class, contrary to the predictions of conformal field theory and matrix models. Implementing the spherical topology as triangulated surfaces of three-dimensional cubes, we find again strong evidence that the critical exponents of the Ising transition are consistent with the Onsager values, and that KPZ exponents are definitely excluded.

hep-lat↗

arXiv1995

Measuring the string susceptibility in 2D simplicial quantum gravity using the Regge approach

We use Monte Carlo simulations to study pure 2D Euclidean quantum gravity with $R^2$-interaction on spherical topologies, employing Regge's formulation. We attempt to measure the string susceptibility exponent $γ_{\rm str}$ by using a finite-size scaling Ansatz in the expectation value of $R^2$, as has been done in a previous study by Bock and Vink ( hep-lat/9406018 ). By considerably extending the range and statistics of their study we find that this Ansatz is plagued by large systematic errors. The $R^2$ specific string susceptibility exponent $\GS'$ is found to agree with theoretical predictions, but its determination also is subject to large systematic errors and the presence of finite-size scaling corrections. To circumvent this obstacle we suggest a new scaling Ansatz which in principle should be able to predict both, $\GS$ and $\GS'$. First results indicate that this requires large system sizes to reduce the uncertainties in the finite-size scaling Ansätze. Nevertheless, our investigation shows that within the achievable accuracy the numerical estimates are still compatible with analytic predictions, contrary to the recent claim by Bock and Vink.

hep-lat↗

arXiv1995

New results on $γ_{\rm str}$ in 2D quantum gravity using Regge calculus

We study 2D quantum gravity on spherical topologies using the Regge calculus approach. Our goal is to shed new light upon the validity of the Regge approach to quantum gravity, which has recently been questioned in the literature. We incorporate an $R^2$ interaction term and investigate its effect on the value of the string susceptibility exponent $γ_{\rm str}$ using two different finite-size scaling Ansätze. Our results suggest severe shortcomings of the methods used so far to determine $\GS$ and show a possible cure of the problems. To have better control over the influence of irregular vertices, we choose besides the almost regular triangulation of the sphere as the surface of a cube a random triangulation according to the Voronoi-Delaunay prescription.

hep-lat↗

arXiv1995

Measure dependence of 2D simplicial quantum gravity

We study pure 2D Euclidean quantum gravity with $R^2$ interaction on spherical lattices, employing Regge's formulation. We attempt to measure the string susceptibility exponent $γ_{\rm str}$ by using a finite-size scaling Ansatz in the expectation value of $R^2$. To check on effects of the path integral measure we investigate two scale invariant measures, the "computer" measure $dl/l$ and the Misner measure $dl/\sqrt A$.

hep-lat↗

arXiv1995

The Ising transition in 2D simplicial quantum gravity - can Regge calculus be right?

We report a high statistics simulation of Ising spins coupled to 2D quantum gravity in the Regge calculus approach using triangulated tori with up to $512^2$ vertices. For the constant area ensemble and the $dl/l$ functional measure we definitively can exclude the critical exponents of the Ising phase transition as predicted for dynamically triangulated surfaces. We rather find clear evidence that the critical exponents agree with the Onsager values for static regular lattices, independent of the coupling strength of an $R^2$ interaction term. For exploratory simulations using the lattice version of the Misner measure the situation is less clear.

hep-lat↗

arXiv1993

Monte Carlo Study of Topological Defects in the 3D Heisenberg Model

We use single-cluster Monte Carlo simulations to study the role of topological defects in the three-dimensional classical Heisenberg model on simple cubic lattices of size up to $80^3$. By applying reweighting techniques to time series generated in the vicinity of the approximate infinite volume transition point $K_c$, we obtain clear evidence that the temperature derivative of the average defect density $d\langle n \rangle/dT$ behaves qualitatively like the specific heat, i.e., both observables are finite in the infinite volume limit. This is in contrast to results by Lau and Dasgupta [{\em Phys. Rev.\/} {\bf B39} (1989) 7212] who extrapolated a divergent behavior of $d\langle n \rangle/dT$ at $K_c$ from simulations on lattices of size up to $16^3$. We obtain weak evidence that $d\langle n \rangle/dT$ scales with the same critical exponent as the specific heat.As a byproduct of our simulations, we obtain a very accurate estimate for the ratio $α/ν$ of the specific-heat exponent with the correlation-length exponent from a finite-size scaling analysis of the energy.

hep-lat↗

arXiv1993

Critical Exponents of the Classical 3D Heisenberg Model: A Single-Cluster Monte Carlo Study

We have simulated the three-dimensional Heisenberg model on simple cubic lattices, using the single-cluster Monte Carlo update algorithm. The expected pronounced reduction of critical slowing down at the phase transition is verified. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors. In one set of simulations we employ the usual finite-size scaling methods to compute the critical exponents $ν,α,β,γ, η$ from a few measurements in the vicinity of the critical point, making extensive use of histogram reweighting and optimization techniques. In another set of simulations we report measurements of improved estimators for the spatial correlation length and the susceptibility in the high-temperature phase, obtained on lattices with up to $100^3$ spins. This enables us to compute independent estimates of $ν$ and $γ$ from power-law fits of their critical divergencies.

hep-lat↗

arXiv1992

Finite-Size Scaling Study of the Three-Dimensional Classical Heisenberg Model

We use the single-cluster Monte Carlo update algorithm to simulate the three-dimensional classical Heisenberg model in the critical region on simple cubic lattices of size $L^3$ with $L=12, 16, 20, 24, 32, 40$, and $48$. By means of finite-size scaling analyses we compute high-precision estimates of the critical temperature and the critical exponents, using extensively histogram reweighting and optimization techniques. Measurements of the autocorrelation time show the expected reduction of critical slowing down at the phase transition. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors in finite-size scaling analyses.

hep-lat↗