M.C.R.G. Study of Fixed-connectivity Surfaces

We apply Monte Carlo Renormalization group to the crumpling transition in random surface models of fixed connectivity. This transition is notoriously difficult to treat numerically. We employ here a Fourier accelerated Langevin algorithm in conjunction with a novel blocking procedure in momentum space which has proven extremely successful in $\lambda\phi^4$. We perform two successive renormalizations in lattices with up to $64^2$ sites. We obtain a result for the critical exponent $\nu$ in general agreement with previous estimates and similar error bars, but with much less computational effort. We also measure with great accuracy $\eta$. As a by-product we are able to determine the fractal dimension $d_H$ of random surfaces at the crumpling transition.Comment: 35 pages,Latex file, 6 Postscript figures uuencoded,uses psfig.sty 2 misspelled references corrected and one added. Paper unchange

Spin Glasses on Thin Graphs

In a recent paper we found strong evidence from simulations that the Isingantiferromagnet on ``thin'' random graphs - Feynman diagrams - displayed amean-field spin glass transition. The intrinsic interest of considering such random graphs is that they give mean field results without long range interactions or the drawbacks, arising from boundary problems, of the Bethe lattice. In this paper we reprise the saddle point calculations for the Ising and Potts ferromagnet, antiferromagnet and spin glass on Feynman diagrams. We use standard results from bifurcation theory that enable us to treat an arbitrary number of replicas and any quenched bond distribution. We note the agreement between the ferromagnetic and spin glass transition temperatures thus calculated and those derived by analogy with the Bethe lattice, or in previous replica calculations. We then investigate numerically spin glasses with a plus or minus J bond distribution for the Ising and Q=3,4,10,50 state Potts models, paying particular attention to the independence of the spin glass transition from the fraction of positive and negative bonds in the Ising case and the qualitative form of the overlap distribution in all the models. The parallels with infinite range spin glass models in both the analytical calculations and simulations are pointed out.Comment: 13 pages of LaTex and 11 postscript figures bundled together with uufiles. Discussion of first order transitions for three or more replicas included and similarity to Ising replica magnet pointed out. Some additional reference

Field Theoretic Calculation of the Universal Amplitude Ratio of Correlation Lengths in 3D-Ising Systems

In three-dimensional systems of the Ising universality class the ratio of correlation length amplitudes for the high- and low-temperature phases is a universal quantity. Its field theoretic determination apart from the $\epsilon$-expansion represents a gap in the existing literature. In this article we present a method, which allows to calculate this ratio by renormalized perturbation theory in the phases with unbroken and broken symmetry of a one-component $\phi^4$-theory in fixed dimensions $D=3$. The results can be expressed as power series in the renormalized coupling constant of either of the two phases, and with the knowledge of their fixed point values numerical estimates are obtainable. These are given for the case of a two-loop calculation.Comment: 14 pages, MS-TPI-94-0

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.Comment: 9 pages, FUB-HEP 9/92, HLRZ Preprint 56/92, August 199

High-Temperature Series Analyses of the Classical Heisenberg and XY Model

Although there is now a good measure of agreement between Monte Carlo and high-temperature series expansion estimates for Ising ($n=1$) models, published results for the critical temperature from series expansions up to 12{\em th} order for the three-dimensional classical Heisenberg ($n=3$) and XY ($n=2$) model do not agree very well with recent high-precision Monte Carlo estimates. In order to clarify this discrepancy we have analyzed extended high-temperature series expansions of the susceptibility, the second correlation moment, and the second field derivative of the susceptibility, which have been derived a few years ago by L\"uscher and Weisz for general $O(n)$ vector spin models on $D$-dimensional hypercubic lattices up to 14{\em th} order in $K \equiv J/k_B T$. By analyzing these series expansions in three dimensions with two different methods that allow for confluent correction terms, we obtain good agreement with the standard field theory exponent estimates and with the critical temperature estimates from the new high-precision MC simulations. Furthermore, for the Heisenberg model we reanalyze existing series for the susceptibility on the BCC lattice up to 11{\em th} order and on the FCC lattice up to 12{\em th} order.Comment: 15 pages, Latex, 2 PS figures not included. FUB-HEP 18/92 and HLRZ 76/9

Accurate Estimates of 3D Ising Critical Exponents Using the Coherent-Anomaly Method

An analysis of the critical behavior of the three-dimensional Ising model using the coherent-anomaly method (CAM) is presented. Various sources of errors in CAM estimates of critical exponents are discussed, and an improved scheme for the CAM data analysis is tested. Using a set of mean-field type approximations based on the variational series expansion approach, accuracy comparable to the most precise conventional methods has been achieved. Our results for the critical exponents are given by \alpha=\afin, \beta=\bfin, \gamma=\gfin and \delta=\dfin.Comment: 16 pages, latex, 1 postscript figur

Polchinski equation, reparameterization invariance and the derivative expansion

The connection between the anomalous dimension and some invariance properties of the fixed point actions within exact RG is explored. As an application, Polchinski equation at next-to-leading order in the derivative expansion is studied. For the Wilson fixed point of the one-component scalar theory in three dimensions we obtain the critical exponents \eta=0.042, \nu=0.622 and \omega=0.754.Comment: 28 pages, LaTeX with psfig, 12 encapsulated PostScript figures. A number wrongly quoted in the abstract correcte

Three-State Anti-ferromagnetic Potts Model in Three Dimensions: Universality and Critical Amplitudes

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 $\gamma /\nu=1.973(9)$ and $\nu=0.664(4)$.Comment: 18pages + 3figures, KL-TH-94/5 , CERN-TH.7183/9

Critical Behaviour of the 3D XY-Model: A Monte Carlo Study

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 $\gamma /\nu=1.976(6)$ and $\nu=0.662(7)$.Comment: 24 pages (4 PS-Figures Not included, Revtex 3.O file), report No.: CERN-TH.6885/93, KL-TH-93/1

O(N) models within the local potential approximation

Using Wegner-Houghton equation, within the Local Potential Approximation, we study critical properties of O(N) vector models. Fixed Points, together with their critical exponents and eigenoperators, are obtained for a large set of values of N, including N=0 and N\to\infty. Polchinski equation is also treated. The peculiarities of the large N limit, where a line of Fixed Points at d=2+2/n is present, are studied in detail. A derivation of the equation is presented together with its projection to zero modes.Comment: 27 pages, LaTeX with psfig, 7 PostScript figures. One reference corrected and one added with respect to the journal versio