Critical phenomena and renormalization-group theory

The distribution function P L ( s ) of the local order parameters in finite blocks of linear dimension L is studied for Ising lattices of dimensionality d = 2,3 and 4. Apart from the case where the block is a subsystem of an infinite lattice, also the distribution in finite systems with free [ P L ( f ) ( s )] and periodic [ P L ( p ) ( s )] boundary conditions is treated. Above the critical point T c , these distributions tend for large L towards the same gaussian distribution centered around zero block magnetization, while below T c these distributions tend towards two gaussians centered at ± M , where M is the spontaneous magnetization appearing in the infinite systems. However, below T c the wings of the distribution at small | s | are distinctly nongaussian, reflecting two-phase coexistence. Hence the distribution functions can be used to obtain the interface tension between ordered phases. At criticality, the distribution functions tend for large L towards scaled universal forms, though dependent on the boundary conditions. These scaling functions are estimated from Monte Carlo simulations. For subsystem-blocks, good agreement with previous renormalization group work of Bruce is obtained. As an application, it is shown that Monte Carlo studies of critical phenomena can be improved in several ways using these distribution functions: ( i ) standard estimates of order parameter, susceptibility, interface tension are improved ( ii ) T c can be estimated independent of critical exponent estimates ( iii ) A Monte Carlo “renormalization group” similar to Nightingale's phenomenological renormalization is proposed, which yields fairly accurate exponent estimates with rather moderate effort ( iv ) Information on coarse-grained hamiltonians can be gained, which is particularly interesting if the method is extended to more general Hamiltonians.

Finite size scaling analysis of Ising model block distribution functions

An introductory review of various concepts about first-order phase transitions is given. Rules for classification of phase transitions as second or first order are discussed, as well as exceptions to these rules. Attention is drawn to the rounding of first-order transitions due to finite-size or quenched impurities. Computational methods to calculate phase diagrams for simple model Hamiltonians are also described. Particular emphasis is laid on metastable states near first-order phase transitions, on the 'stability limits' of such states (e.g. the 'spinodal curve' of the gas-liquid transition) and on the dynamic mechanisms by which metastable states decay (nucleation and growth of droplets of a new phase, etc.).

https://iopscience.iop.org/article/10.1088/0034-4885/50/7/001/pdf

Theory of first-order phase transitions

This paper is a self-contained survey of most of the results known about oriented percolation. The table of contents below should give an idea of the topics which will be covered. A more detailed account can be found in the first section.

/pdf/oriented-percolation-in-two-dimensions-552o196dgr.pdf

Oriented Percolation in Two Dimensions

Finite Size Effects in Phase Transitions

A renormalization group transformation is introduced with the help of which critical properties of infinite systems can be related to finite systems. As a numerical example the method is applied to the two-dimensional Ising model. The critical point and critical point exponent are computed in addition to the amplitude of the logarithmic singularity in the specific heat.

M.P Nightingale

Papers

Scaling Theory and Finite Systems