1,720,969 research outputs found
Conformal-invariance, Finite-size Scaling and Surface Magnetic Exponent of the Potts-model In 2 Dimensions
No full textThe analysis of renormalisation group equations for the q-state Potts model in two dimensions is generalised to include the presence of a surface field Hs besides the bulk one, HB. Combined with conformal invariance results, this allows prediction of a surface susceptibility behaviour delta 2/f delta Hs delta HB approximately L-1/8(1-C'lnL)15/16 for a q=4 system of size L at the critical temperature. This prediction is checked by a Monte Carlo based finite-size scaling analysis, which also nicely reproduces the exact and conjectured magnetic surface exponents for q=2 and 3, respectively. For the Baxter-Wu model similar methods give results consistent with the q=4 behaviour without logarithmic corrections (c'=0)
Scaling and Fractal Dimension of Ising Clusters At the D=2 Critical-point
No full textUsing formal arguments based on conformal invariance and on the connection between correlated-site percolation and the q-state Potts model with vacancies, we show that the exponents describing Ising clusters at Onsager’s critical point are those of the tricritical q=1 Potts model. This implies, in particular, a fractal dimension d ̄=(187/96 and a percolative susceptibility exponent γ=(91/48, in good agreement with existing numerical estimates. This d ̄ is also clearly supported by a new very accurate Monte Carlo finite-size scaling determination. We also conjecture an exponent yJ=(13/24 controlling the crossover between clusters and droplets
Surface Critical-behavior of Some 2-dimensional Lattice Models
No full textA review is given of recent work on the ordinary surface critical behaviour of systems in two dimensions. Several models of interest in statistical mechanics are considered: Potts model, percolation, Ising clusters, ZN-model, O(n) model and polymers. Numerical results for surface exponents, obtained by suitable finite size scaling extrapolations, are discussed in the light of recent advances based on the conformal invariance approach.
Surface exponents are often seen as important tests of conformal invariance predictions. In other cases these exponents provide important information for a location of the problem within the classification schemes offered by the conformal approach, and a determination of its universality class. A relevant example of the first aspect is the study of the q-state Potts model with q near 4, for which an analytical study of logarithmic scaling corrections is needed to achieve a successful test. The latter point of view applies, e.g., to the more controversial cases of polymers at the theta point and critical Ising clusters. Emphasis is put on the importance of an integrated study of both bulk and surface properties. Relevant issues, like the possible existence of analytical expressions for the indices in particular model families, or of general relationships between bulk and surface exponents, are critically discussed.
The new problem of critical behaviour at fractal boundaries is also considered for random (RW) and self-avoiding walks (SAW). From the numerical analysis of this problem remarkable universalities of the surface exponents seem to emerge, which, in the case of SAW’s, are still far from being understood
Thermodynamics of the Hubbard Chain - Temperature Extension of Ground-state Renormalization For Fermions
No full textA recently proposed ground-state renormalisation-group approach to the Hubbard model is generalised to finite temperatures. This new method provides a global and consistent description of the thermodynamics of the model at all temperatures. Application to the Hubbard chain with a half-filled band gives values for various thermodynamic functions which compare remarkably well with existing exact or numerical data, over all temperature ranges. The difficulties posed by Fermi-Dirac statistics in the set-up of the transformation are discussed to some extent and it is shown that these do not prevent one, at least in principle, from extending the approach to higher dimensions
Nonuniversality In the Collapse of 2-dimensional Branched Polymers
No full textIn this paper we study the complete phase diagram of a model of interacting branched polymers. The model we consider is a lattice animal one, where the collapse transition can be driven both by a contact fugacity between two occupied nearest neighbours and by a fugacity related to each occupied edge. Using a Potts model formulation of the problem we conjecture the existence of two different universality classes for the theta transitions (with thermal exponents, nu and phi, equal to (1/2, 2/3) and (8/15, 8/15)), separated by a higher-order percolation point. We also present convincing numerical evidence for these exponent values using a transfer-matrix approach. We discuss the possibility of a collapse-collapse transition and we predict the behaviour of our model when an adsorbing surface is included
A Unified View On Various Iterative Approaches To the Ground-state and Finite Temperature Properties of Quantum Spin Systems
No full textWe show that several iterative techniques, which were proposed for calculating ground-state properties of quantum models, can all be derived as the zero-temperature limit of a real-space quantum renormalization method. As a consequence, all these techniques are systematically improvable by a perturbative approach, and may be extended in a natural way to nonzero temperatures
Self-avoiding Walks In the Presence of Strongly Correlated, Annealed Vacancies
No full textTwo models coupling self-avoiding walks (SAWs) to Ising vacancies are studied. When the walk is performed on the hexagonal lattice and the vacancies are on the dual, an exact description of the behavior of the walk as a function of the Ising coupling is given. The walk collapses from SAW to FTHETA’-point behavior at the Ising critical point where its fractal dimension equals 11/8, as for the hull of Ising clusters. A walk on the square lattice, with vacancies either on the same or on the dual lattice, has SAW behavior even at Ising criticality. This is shown numerically and, within the conformal-invariance approach, poses a problem for which a conjectural solution is proposed
Boundary Critical-behavior of D = 2 Self-avoiding Walks On Correlated and Uncorrelated Vacancies
No full textIn this paper we present exact results for the critical exponents of interacting self-avoiding walks with ends at a linear boundary. Effective interactions are mediated by vacancies, correlated and uncorrelated, on the dual lattice. By choosing different boundary conditions, several ordinary and special regimes can be described in terms of clusters geometry and of critical and low-temperature properties of the O(n = 1) model. In particular, the problem of boundary exponents at the THETA-point is fully solved, and implications for THETA-point universality are discussed. The surface crossover exponent at the special transition of noninteracting self-avoiding walks is also interpreted in terms of percolation dimensions
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