Elements of Phase Transitions and Critical Phenomena

Abstract

Phase transitions and critical phenomena have consistently been among the principal subjects of active studies in statistical physics. The simple act of transforming one state of matter or phase into another, for instance by changing the temperature, has always captivated the curious mind. This book provides an introductory account on the theory of phase transitions and critical phenomena, a subject now recognized to be indispensable for students and researchers from many fields of physics and related disciplines. The first five chapters are very basic and quintessential, and cover standard topics such as mean-field theories, the renormalization group and scaling, universality, and statistical field theory methods. The remaining chapters develop more advanced concepts, including conformal field theory, the Kosterlitz-Thouless transition, the effects of randomness, percolation, exactly solvable models, series expansions, duality transformations, and numerical techniques. Moreover, a comprehensive series of appendices expand and clarify several issues not developed in the main text. The important role played by symmetry and topology in understanding the competition between phases and the resulting emergent collective behaviour, giving rise to rigidity and soft elementary excitations, is stressed throughout the book. Serious attempts have been directed toward a self-contained modular approach so that the reader does not have to refer to other sources for supplementary information. Accordingly, most of the concepts and calculations are described in detail, sometimes with additional/auxiliary descriptions given in appendices and exercises. The latter are presented as the topics develop with solutions found at the end of the book, thus giving the text a self-learning character.

Duality analysis on random planar lattices

“…The limit of q → 1 gives the bond-percolation thresholds on the random planar lattice as in the literature [15]. The leading order in ǫ with q = 1 + ǫ gives…”

“…Then we must take the summation over several internal spins and evaluate the configurational average for all bonds on the cluster. The complexity of the former computation can be reduced by the special technique as the numerical transfer matrix [15,27] Appendix C: Relation between spin glass model and quantum error correction…”

“…Its fixed point condition exp(β c J) = (exp(β c J) + q − 1)/(exp(β c J) − 1) gives the critical temperature. We can evaluate the exact value of the internal energy and the rigorous bound for the specific heat at the critical point from the relation of the partition function (2) through the appropriate derivatives as well as its location [15].…”

“…Usually, the duality analyses are limited to the case on the self-dual lattice. For non-self-dual lattices, such as the triangular and hexagonal lattices, the duality analysis gives only the relationship between critical points for mutually dual pair [2, 14,15].…”

“…3 and 4, (p mix , exp(−β c J)) → (1 − p mix , (exp(−β c J) − 1)/(1+(q−1) exp(−β c J)), is satisfied for any value of p mix and q. We can estimate the bond-percolation threshold on the random planar lattice through p th c = exp(−β c J) − 1 on the random planar lattice via a technical limit of q → 1 [15]. The obtained results are also listed in Table I.…”