Phase transitions and critical phenomena: Universality and global multicritical phase diagrams from position-space renormalization-group studies

The position-space renormalization-group (PSRG) approach has given impressive results in studies of phase transitions and critical phenomena. Most work has focussed on the local critical properties of rather simple models. After presenting background on phase transitions and reviewing the PSRG procedure in general, we extend this approach to more complicated systems anq global multicritical behavior by two specific studies. First, we introduce the concept of restructuring: A single restructuring transformation maps a given system onto another one with different, hopefully more manageable structure; this prefaces the multiple rescaling transformations of a PSRG treatment. Structurally different systems can thus eventually map onto the same fixed point and manifestly fall into the same universality class. The scope of application of a given PSRG calculation is enlarged. We demonstrate the restructuring concept by constructing a transformation which maps Ising models of arbitrary spin s onto a spin-1/2 Ising mOdel with further-neighbor and many-site interactions. We calculate the critical interactions (i.e., the inverse critical temperatures) of the spin-s Ising models on the triangular lattice, using this spinrestructuring transformation and the results of Niemeijer and van Leeuwen for the critical subspace of the corresponding spin-l/2 Ising model. Our evaluations agree very well with series data. Second, the unified global treatment of a phase diagram with a rich variety of transitions is illustrated: The spin-l Ising (Blume-Emery-Griffiths-Potts) model on the square lattice with nearest-neighbor ferromagnetic exchange interactions (both bilinear (J) and biquadratic (K» and crystal field interaction (A) is studied via a PSRG transformation. By building symmetries into a truncated transformation, we are able to treat a complex problem with a simple transformation. The resulting phase diagram in J, K, A space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end-points. These three lines join at a special tricritical point corresponding to the transition of the 3-state Potts model. The overall phase diagram is qualitatively similar to that obtained with the meanfield approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (H) and another odd interaction (L) is performed. A one-adjusted parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Physics

Year dc:date

2011

Author dc:creator

• Berker, Ahmet Nihat

Contributors dc:contributor

• Wortis, M.

• phase transitions
• critical phenomena
• position-space renormalization-group (PSRG)
• local critical properties

Statement dc:rights

• 1977 Ahmet Nihat Berker

Language dc:language

en

Identifier

252674

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/25633