Phase Transitions and Critical Points

Joydeep Chakravarty, Diksha Jain

Abstract In this work, we define and calculate critical exponents associated with higher order thermodynamic phase transitions. Such phase transitions can be classified into two classes: with or without a local order parameter. For phase transitions involving a local order parameter, we write down the Landau theory and calculate critical exponents using the saddle point approximation. Further, we investigate fluctuations about the saddle point and demarcate when such fluctuations dominate over saddle point calculations by introducing the generalized Ginzburg criteria. We use Wilsonian renormalization group (RG) to derive scaling forms for observables near criticality and obtain scaling relations between the critical exponents. Afterwards, we find out fixed points of the RG flow using the one-loop beta function and calculate critical exponents about the fixed points for third and fourth order phase transitions.

James G. Glimm, Arthur Jaffe

Jan Tobochnik

This Resource Letter provides a guide to the literature on critical point phenomena and phase transitions. Journal articles and books are cited for the following topics: continuous phase transitions, critical points, model systems with special attention to the Ising model and the Kosterlitz–Thouless transition, mean-field theory, computer simulation methods, scaling, universality, and the renormalization group.

David A. Lavis, Roman Frigg

John F. Nagle

Ameneh Kargarian, Maryam Ghapanvari

In this paper‎, ‎we study the nature of the dynamics in second-order Quantum Phase Transition (QPT) between vibrational ( ) and -unstable ( ) nuclear shapes‎. ‎Using a transitional Hamiltonian according to an affine SU(1,1) algebra in combination with a coherent state formalism, Shape Phase Transitions (SPT) in odd-nuclei in the framework of the Interacting Boson Fermion Model (IBFM) are investigated‎. ‎Classical analysis reveals a change in the system along with the transition in a critical point‎. ‎The role of a fermion with angular momentum j at the critical point on quantum phase transitions in bosonic systems is investigated via a semi-classical approach‎. ‎The effect of the coupling of the odd particle to an even-even boson core is discussed along with the shape transition and‎, ‎in particular‎, ‎at the critical point‎. Our study confirms the importance of the odd nuclei as necessary signatures to characterize the occurrence of the phase transition and determine the critical point's precise position‎.