The truncated quantum mechanical n-vector model in one dimension is studied by means of the phenomenological renormalisation group (PRG) and the block renormalisation group (BRG) for general values of n. Particular emphasis is put on the extrapolation of the finite size results using the finite size scaling hypothesis. Accurate estimates of the critical properties can be made for 0<or approximately=n<or approximately=2. The two methods used are critically compared.
The truncated quantum mechanical n-vector model in one dimension is studied by means of the phenomenological renormalisation group (PRG) and the block renormalisation group (BRG) for general values of n. Particular emphasis is put on the extrapolation of the finite size results using the finite size scaling hypothesis. Accurate estimates of the critical properties can be made for 0<or approximately=n<or approximately=2. The two methods used are critically compared.
The thesis consists of five major parts or chapters, the first of which is an introduction and review relating the field of investigation to problems in quantum field theory and statistical mechanics.Although the thesis generally deals with eigenvalue bounding techniques, these methods are applied with a view to calculating the critical properties of infinite systems [7].The approach advocated in the thesis intends the use of approximate methods to calculate the thermodynamics of large, but finite lattices.Finite size scaling [3] can then be used to estimate the bulk system thermodynamics, especially critical phenomena, from the approximate large lattice data.The approximations used include projected mean fields [5], for both ground and excited states, correlated product wave functions [6], which are extended to kink states, and blocking approximations [2].Part of this work is motivated by the promise these wave functions have in their application to variationally guided Monte Carlo methods for finite lattices [4], Also, it is shown that the critical indices predicted by the finite size rescaling formalism, are left invariant by the application of an approximate (blocking) renormalisation group to the finite lattices involved.Moreover, this invariance of the finite size rescaling equations is used to define a sequence transformation suitable for improving
Some renormalization group approaches have been proposed during the last few years which are close in spirit to the Nightingale phenomenological procedure. In essence, by exploiting the finite size scaling hypothesis, the approximate critical behavior of model on infinite lattice is obtained through the exact computation of some thermal quantities of the model on finite clusters. In this work some of these methods are reviewed, namely the mean field renormalization group, the effective field renormalization group and the finite size scaling renormalization group procedures. Although special emphasis is given to the mean field renormalization group (since it has been, up to now, much more used to study a wide variety of different systems) a discussion of their potentialities and interrelations to other methods is also presented.
Introduction The real space renormalisation group has a wide application in the study of classical critical phenomena. For quantum systems, several versions of the real space renormalisation group have been proposed (Jullien, 1981; Pfeuty et al ., 1982; dos Santos, 1982). The real space RG is the most direct way of implementing the scaling ideas presented in Chapter 1. The operation of elimination of degrees of freedom as the system is reduced to blocks of smaller sizes under a change of length scale is directly and intuitively represented as mathematical equations. In particular, the block method has been used to study different problems of strongly correlated systems (Pfeuty et al ., 1982) and spin chains (White, 1992, 1993) with different degrees of sophistication. We show here in detail a simple form of this technique applied to the case of the one-dimensional Ising model in a transverse field. The purpose is to illustrate the ideas introduced earlier in Chapter 1 with a concrete example where a quantum critical point is associated with an unstable fixed point. Also, we show how the stable attractors or fixed points can characterise the different phases and how the critical exponents can be obtained. Although the actual values for the quantum critical point and critical exponents that we obtain using a small block are not exact, a procedure is outlined which can be used to improve these results. In this sense, the block method presented here represents a controlled approximation, which can be systematically improved to yield in the limit of very large blocks essentially the correct results. When compared to the k -space version of the RG (to be presented in the next two chapters), the real space version turns out to be in general more difficult to control, in the sense of producing a systematic improvement of the results. On the other hand, this method yields without much effort the attractive fixed points of the different phases, allowing for their characterisation. This turns out to be useful since the nature of the phase transition, whether first or second order, can be determined by exponents calculated in the attractor of this phase (Nienhuis and Naunberg, 1975).
The phenomenological renormalisation group transformation, introduced by Nightingale (1976), is extended to a form suitable for the study of a quantum mechanical system at T=0. The spin- 1 / 2 Ising model in a transverse field Gamma , in one and two dimensions, is considered. The new scaling form, called phenomenological time scaling, is derived by exploiting the connection between a d dimensional quantum mechanical system at T=0 and a (d+1) dimensional classical system in the anisotropic limit of weakly coupled chains. Estimates are made of the critical field Gamma c and the exponent nu associated with the vanishing of the first excitation energy at T=0. These results also provide some information about the thermal critical behaviour of the anisotropic three-dimensional Ising model.
Abstract A number of numerical calculations, like Monte Carlo or transfer matrix calculations, are performed with systems in which the size in several or all dimensions is finite. To extrapolate the results to the infinite system, it is thus necessary to understand how the infinite size limit is reached. In particular in a system in which the forces are short range, no phase transition can occur in a finite volume, or in a geometry in which the size is infinite only in one dimension. This indicates that the infinite-size extrapolation is somewhat non-trivial. In this chapter, the problem is analysed in the case of second-order phase transitions, in the framework of the N-vector model. The existence of a finite-size scaling is established, extending renormalization group (RG) arguments to this new situation. Then, finite volume geometry and cylindrical geometry, in which the size is finite in all dimensions except one, are distinguished. It is explained how to adapt the methods used in the case of infinite systems to calculate the new universal quantities appearing in finite-size effects, for example, in d = 4−ϵ or d = 2+ϵ dimensions. Special properties of the commonly used periodic boundary conditions are emphasized.
The critical behaviour of the spin- Ising model with a transverse field at zero temperature in triangular and square lattices is investigated through a finite-size rescaling transformation. This technique enables the critical properties of infinite systems to be calculated from the scaling properties of systems of finite size, in a generalisation of the so-called phenomenological renormalisation group. The critical field and correlation length exponent calculated using systems of fairly small sizes have reasonable accuracy, but calculations on larger clusters are limited by computer time and storage.