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Field theory in a finite geometry: Finite size scaling

Jean Zinn‐Justin

Published Jun 19, 2019• 0 citations
Publisher

Abstract

Abstract Computer simulations of critical statistical systems or quantum field theory models are performed with systems where sizes are finite. In transfer matrix calculations, all sizes but one are also finite. In systems where the correlation length is large, it is thus important to understand how the infinite size limit is reached. This problem is investigated in Chapter 19. RG equations allow proving the properties of universality and of finite size scaling. When the correlation length is larger than the linear system size, a phenomenon of dimensional reduction is observed. With periodic boundary conditions, fields have a zero mode. A local expansion generates an effective field theory for the zero mode.

DOI
10.1093/oso/9780198787754.003.0019

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Calculations on finite (or partially finite) systems, such as Monte-Carlo simulations or the diagonalization of a finite transfer matrix, are often extrapolated to the thermodynamic limit by the use of scaling assumptions involving the finite sizes L of the system. It is shown here that these finite size scaling laws follow from the fact that : i) in the renormalization group flow equations the length L is not renormalized ; ii) there is no singularity at the fixed point. This second property fails in four (and above four) dimensions and finite size scaling does not hold there. An illustration of these points is presented, with detailed calculations of the large N limit of the N- vector model. As a consequence, the ε-expansion of finite size scaling properties is singular.
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.
A d-dimensional finite quantum model system confined to a general hypercubical geometry with linear spatial size L and ``temporal size'' 1/T (T - temperature of the system) is considered in the spherical approximation under periodic boundary conditions. Because of its close relation with the system of quantum rotors it represents an effective model for studying the low-temperature behaviour of quantum Heisenberg antiferromagnets. Close to the zero-temperature quantum critical point the ideas of finite-size scaling are used for studying the critical behaviour of the model. For a film geometry in different space dimensions $\halfσ
We address the problem of the definition of the finite-volume correlation length. First, we study the large-N limit of the N-vector model, and we show the existence of several constraints on the definition if regularity of the finite-size scaling functions and correct anomalous behaviour above the upper critical dimension are required. Then, we study in detail a model in which the zero mode is prohibited. Such. a model is a generalization of the fixed-magnetization Ising model which is equivalent to the lattice gas. Also in this case, we find that the finite-volume correlation length must satisfy appropriate constraints in order to obtain regular finite-size scaling functions, and, above the upper critical dimension, an anomalous scaling behaviour. The large-N results are confirmed by a one-loop calculation in the lattice phi (4) theory.