Scaling and Renormalization in Statistical Physics

Abstract

This text provides a thoroughly modern graduate-level introduction to the theory of critical behaviour. Beginning with a brief review of phase transitions in simple systems and of mean field theory, the text then goes on to introduce the core ideas of the renormalization group. Following chapters cover phase diagrams, fixed points, cross-over behaviour, finite-size scaling, perturbative renormalization methods, low-dimensional systems, surface critical behaviour, random systems, percolation, polymer statistics, critical dynamics and conformal symmetry. The book closes with an appendix on Gaussian integration, a selected bibliography, and a detailed index. Many problems are included. The emphasis throughout is on providing an elementary and intuitive approach. In particular, the perturbative method introduced leads, among other applications, to a simple derivation of the epsilon expansion in which all the actual calculations (at least to lowest order) reduce to simple counting, avoiding the need for Feynman diagrams.

Finite size effects in ferromagnetic spin chains and quantum corrections to classical strings

“…In [41] the leading quantum correction to their classical energy was calculated, finding that the first term in It is interesting to compare expression (4.9) with the finite size corrections to the ground state energy E(L) of a critical Hamiltonian H defined in a strip of width L, which is given by E 0 (L) = −πc/(6L) where c is the Virasoro central charge [54]. A well known example is the antiferromagnetic Heisenberg Hamiltonian, where c = 1, which is at the heart of the bosonization approach to this spin chain [55].…”

Minimalistic real-space renormalization of Ising and Potts Models in two dimensions

“…Any fixed point R(K * ) = K * therefore corresponds to either a divergent or a vanishing correlation length. It is well known that the eigenvalues of the linear stability matrix at that fixed point are related to the critical exponents characterising the corresponding continuous phase transition [10][11][12]. The aim of the present work is to demonstrate that very good estimates of those critical exponents, some unexpected, can be produced on the basis of very small systems.…”

“…To formalize the block spin procedure, we introduce the function [11] T(S; s) = 1 if S is a valid map of s 0 otherwise ,…”

“…These exponents characterize the scale invariance in the neighborhood of the fixed point. As discussed extensively in the textbooks [11], they are related to those eigenvalues λ x of the Jacobian…”

“…Firstly the original lattice is divided into disjoint blocks of b × b spins. These blocks are then replaced by a single spin using a majority rule [11] (any ambiguity is resolved by replacing a block by the top left spin) and finally, the lattice is rescaled as to restore the distance between nearest neighbors to that on the original lattice. During this process, the number of degrees of freedom decreases by a factor of b d , while their density remains unchanged.…”