The subject of this paper is to study the critical N-vector model in 4-\epsilon dimensions in one-loop order. We analyze the spectrum of anomalous dimensions of composite operators with an arbitrary number of fields and gradients. For composite operators with three elementary fields and gradients we work out the complete spectrum of anomalous dimensions, thus extending the old solution of Wilson and Kogut for two fields and gradients. In the general case we prove some properties of the spectrum, in particular a lower limit 0+O(\epsilon^2). Thus one-loop contributions generally improve the stability of the nontrivial fixed point in contrast to some 2+\epsilon expansions. Furthermore we explicitly find conformal invariance at the nontrivial fixed point.
In a recent publication we have investigated the spectrum of anomalous dimensions for arbitrary composite operators in the critical N-vector model in 4-\epsilon dimensions. We could establish properties like upper and lower bounds for the anomalous dimensions in one-loop order. In this paper we extend these results and explicitly derive parts of the one-loop spectrum of anomalous dimensions. This analysis becomes possible by an explicit representation of the conformal symmetry group on the operator algebra. Still the structure of the spectrum of anomalous dimensions is quite complicated and does generally not resemble the algebraic structures familiar from two-dimensional conformal field theories.
The spectrum of critical exponents of the N-vector model in 4-\epsilon dimensions is investigated to the second order in \epsilon. A generic class of one-loop degeneracies that has been reported in a previous work is lifted in two-loop order. One- and two-loop results lead to the conjecture that the spectrum possesses a remarkable hierarchical structure: The naive sum of any two anomalous dimensions generates a limit point in the spectrum, an anomalous dimension plus a limit point generates a limit point of limit points and so on. An infinite hierarchy of such limit points can be observed in the spectrum.
It has been shown by several authors that a certain class of composite operators with many fields and gradients endangers the stability of nontrivial fixed points in 2+eps expansions for various models. This problem is so far unresolved. We investigate it in the N-vector model in an 1/N expansion. By establishing an asymptotic naive addition law for anomalous dimensions we demonstrate that the first orders in the 2+eps expansion can lead to erroneous interpretations for high-gradient operators. While this makes us cautious against over-interpreting such expansions (either 2+eps or 1/N), the stability problem in the N-vector model persists also in first order in 1/N below three dimensions.
We formulate the geometrical string which has been proposed in earlier articles on the euclidean lattice. There are two essentially distinct cases which correspond to non-self-avoiding surfaces and to soft-self-avoiding ones. For the last case it is possible to find such spin systems with local interaction which reproduce the same surface dynamics. In the three-dimensional case this spin system is a usual Ising ferromagnet with additional diagonal antiferromagnetic interaction and with specially adjusted coupling constants. In the four-dimensional case the spin-system coincides with the gauge Ising system with an additional double-plaquette interaction and also with specially tuned coupling constants. We extend this construction to random walks and random hypersurfaces (membrane and p-branes) of high dimensionality. We compare these spin systems with the eight-vertex model and BNNNI models.
We are able to perform the duality transformation of the spin system which was found before as a lattice realization of the string with linear action. In four and higher dimensions this spin system can be described in terms of a two-plaquette gauge hamiltonian. The duality transformation is constructed in geometrical and algebraic language. The dual hamiltonian represents a new type of spin system with local gauge invariance. At each vertex \xi there are d(d-1)/2 Ising spins \Lambda_{\ny,my}=\Lambda_{\my,\ny}, \my \neq \ny = 1,...,d and one Ising spin \Gamma on every link (\xi,\xi+e_{\my}). For the frozen spin \Gamma=1 the dual hamiltonian factorizes into d(d-1)/2 two-dimensional Ising ferromagnets and into antiferromagnets in the case \Gamma=-1. For fluctuating \Gamma it is a sort of spin-glass system with local gauge invariance. The generalization to p-membranes is given.
We prove the existence of an ordered low-temperature phase in a model of soft-self-avoiding closed random surfaces on a cubic lattice by a suitable extension of Peierls contour method. The statistical weight of each surface configuration depends only on the mean extrinsic curvature and on an interaction term arising when two surfaces touch each other along some contour. The model was introduced by F.J. Wegner and G.K. Savvidy as a lattice version of the gonihedric string, which is an action for triangulated random surfaces.