arXiv subjects
Publications and source records attributed to M. Hasenbusch.
We compare predictions of the Capillary Wave Model beyond its Gaussian approximation with Monte Carlo results for the energy gap and the surface energy of the 3D Ising model in the scaling region. Our study reveals that the finite size effects of these quantities are well described by the Capillary Wave Model, expanded to two--loop order ( one order beyond the Gaussian approximation).
We describe a new method to compute renormalized coupling constants in a Monte Carlo renormalization group calculation. The method can be used for a general class of models, e.g., lattice spin or gauge models. The basic idea is to simulate a joint system of block spins and canonical demons. In contrast to the Microcanonical Renormalization Group invented by Creutz et al. our method does not suffer from systematical errors stemming from a simultaneous use of two different ensembles. We present numerical results for the $O(3)$ nonlinear $σ$-model.
We present the results of a Monte Carlo study of the three-dimensional XY model and the three-dimensional antiferromagnetic three-state Potts model. In both cases we compute the difference in the free energies of a system with periodic and a system with antiperiodic boundary conditions in the neighbourhood of the critical coupling. From the finite-size scaling behaviour of this quantity we extract values for the critical temperature and the critical exponent nu that are compatible with recent high statistics Monte Carlo studies of the models. The results for the free energy difference at the critical temperature and for the exponent nu confirm that both models belong to the same universality class.
We confirm the Kosterlitz-Thouless scenario of the roughening transition for three different Solid-On-Solid models: the Discrete Gaussian model, the Absolute-Value-Solid-On-Solid model and the dual transform of the XY model with standard (cosine) action. The method is based on a matching of the renormalization group flow of the candidate models with the flow of a bona fide KT model, the exactly solvable BCSOS model. The Monte Carlo simulations are performed using efficient cluster algorithms. We obtain high precision estimates for the critical couplings and other non-universal quantities. For the XY model with cosine action our critical coupling estimate is $β_R^{XY}=1.1197(5)$. For the roughening coupling of the Discrete Gaussian and the Absolute-Value-Solid-On-Solid model we find $K_R^{DG}=0.6645(6)$ and $K_R^{ASOS}=0.8061(3)$, respectively.
Deconfinement and screening of higher-representation sources in finite-temperature $SU(2)$ lattice gauge theory is investigated by both analytical and numerical means. The effective Polyakov-line action at strong coupling is simulated by an efficient cluster-updating Monte Carlo algorithm for the case of $d\!=\!4$ dimensions. The results compare very favourably with an improved mean-field solution. The limit $d\!\to\!\infty$ of the $SU(2)$ theory is shown to be highly singular as far as critical behaviour is concerned. In that limit the leading amplitudes of higher representation Polyakov lines vanish at strong coupling, and subleading exponents become dominant. Each of the higher-representation sources then effectively carry with them their own critical exponents.
We present the results of a Monte Carlo study of the three-dimensional anti-ferromagnetic 3-state Potts model. We compute various cumulants in the neighbourhood of the critical coupling. The comparison of the results with a recent high statistics study of the 3D XY model strongly supports the hypothesis that both models belong to the same universality class. From our numerical data for the anti-ferromagnetic 3-state Potts model we obtain for the critical coupling $\coup_c=0.81563(3)$, and for the static critical exponents $γ/ν=1.973(9)$ and $ν=0.664(4)$.
We propose an effective transfer-matrix method that allows a measurement of tunnelling correlation lengths that are orders of magnitude larger than the lattice extension. Combining this method with a particularly efficient implementation of the multimagnetical algorithm we were able to determine the interface tension of the 3D Ising model close to criticality with a relative error of less than 1 per cent.
We describe a Monte Carlo procedure for the simulation of dynamically triangulate random surfaces with a boundary (topology of a disk). The algorithm keeps the total number of triangles fixed, while the length of the boundary is allowed to fluctuate. The algorithm works in the presence of matter fields. We here present results for the pure gravity case. The algorithm reproduces the theoretical expectations.
We compare Monte Carlo results for the interface tension and interface energy of the 3-dimensional Ising model with Padé and inhomogeneous differential approximants of the low temperature series that was recently extended by Arisue to $17^{\rm th}$ order in $u=\exp(-4β)$. The series is expected to suffer from the roughening singularity at $u\approx 0.196$. The comparison with the Monte Carlo data shows that the Padé and inhomogeneous differential approximants fail to improve the truncated series result of the interface tension and the interface energy in the region around the roughening transition. The Monte Carlo data show that the specific heat displays a peak in the smooth phase. Neither the truncated series nor the Padé approximants find this peak. We also compare Monte Carlo data for the energy of the ASOS model with the corresponding low temperature series that we extended to order $u^{12}$.
We investigate the surface width $W$ of solid-on-solid surfaces in the vicinity of the roughening temperature $T_r$. Above $T_r$, $W^2$ is expected to diverge with the system size $L$ like $\ln L$. However, close to $T_r$ a clean $\ln{L}$ behavior can only be seen on extremely large lattices. Starting from the Kosterlitz-Thouless renormalization group, we derive an improved formula that describes the small $L$ behavior on both sides of $T_r$. For the Discrete Gaussian model, we used the valleys-to-mountains-reflections cluster algorithm in order to simulate the fluctuating solid-on-solid surface. The base plane above which the surface is defined is an $L \times L$ square lattice. In the simulation we took $8\leq L\leq 256$. The improved formula fits the numerical results very well. {}From the analysis, we estimate the roughening temperature to be $T_r = 0.755(3)$.
We present the results of a study of the three-dimensional $XY$-model on a simple cubic lattice using the single cluster updating algorithm combined with improved estimators. We have measured the susceptibility and the correlation length for various couplings in the high temperature phase on lattices of size up to $L=112$. At the transition temperature we studied the fourth-order cumulant and other cumulant-like quantities on lattices of size up to $L=64$. From our numerical data we obtain for the critical coupling $\coup_c=0.45420(2)$, and for the static critical exponents $γ/ν=1.976(6)$ and $ν=0.662(7)$.
We study the use of effective transfer matrices for the numerical computation of masses (or correlation lengths) in lattice spin models. The effective transfer matrix has a strongly reduced number of components. Its definition is motivated by a renormalization group transformation of the full model onto a 1-dimensional spin model. The matrix elements of the effective transfer matrix can be determined by Monte Carlo simulation. We show that the mass gap can be recovered exactly from the spectrum of the effective transfer matrix. As a first step towards application we performed a Monte Carlo study for the 2-dimensional Ising model. For the simulations in the broken phase we employed a multimagnetical demon algorithm. The results for the tunnelling correlation length are particularly encouraging.
We verify the Kosterlitz Thouless scenario for three different SOS (solid-on-solid) models, including the dual transforms of XY-models with Villain and with cosine action. The method is based on a matching of the renormalization group (RG) flow of the candidate models with the flow of a bona fide KT model, the exactly solvable BCSOS model. We obtain high precision estimates for the critical couplings and other non-universal quantities.
We adapt the VMR (valleys-to-mountains reflections) algorithm, originally devised by us for simulations of SOS models, to the BCSOS model. It is the first time that a cluster algorithm is used for a model with constraints. The performance of this new algorithm is studied in detail in both phases of the model, including a finite size scaling analysis of the autocorrelations.