In the previous chapters, we discussed mean-field approaches such as the Curie–Weiss and Bragg–Williams theories, which provide a microscopic justification for macroscopic order parameters and spontaneous symmetry breaking. However, these approaches were based on specific microscopic Hamiltonians. Lev Davidovich Landau introduced a far more general and elegant framework: one that does not depend on microscopic details but on symmetry principles and analyticity of the free energy near the transition.
In the previous chapters, we discussed mean-field approaches such as the Curie–Weiss and Bragg–Williams theories, which provide a microscopic justification for macroscopic order parameters and spontaneous symmetry breaking. However, these approaches were based on specific microscopic Hamiltonians. Lev Davidovich Landau introduced a far more general and elegant framework: one that does not depend on microscopic details but on symmetry principles and analyticity of the free energy near the transition.
This and the four next chapters are devoted to the study of several aspects of the theory of phase transition. This has been for more than one century, and still remains, a subject of intense experimental and theoretical activity, with important breakthroughs having impacts on different fields in physics but also on other disciplines such as mathematics, economics, sociology, biology In the first section of this chapter, the transition between the ferromagnetic and paramagnetic phases of the Ising system will be studied in the framework of a mean field approximation. The problem will first be addressed by means of the intuitive Weiss molecular field approximation ( Section 9.1.2 ) and then by means of the more detailed Bragg-Williams approximation ( Section 9.1.3 ) which will serve as a base to develop, in Section 9.2 , a general theory of continuous phase transitions: the Landau mean field theory for which the concept of order parameter will first be introduced. Other types of phase transitions will then be studied using the mean field approximation. Phase transitions in liquid crystals will be studied in Section 9.3 . These are systems with a tensor order parameter which exhibit a rich phenomenology. Finally, in Section 9.4 , a transition of a fairly more elaborate nature than a simple first or second order phase transition will be discussed.
In the previous chapter, we developed the Landau theory of phase transitions, emphasizing how macroscopic order parameters and symmetry considerations determine the general form of the free energy near criticality. There, spatial variations of the order parameter were neglected, corresponding to the mean-field approximation . However, as we have seen, close to the critical temperature T c the correlation length ξ diverges, and fluctuations occur over arbitrarily large scales. In this regime, the assumption of spatial uniformity breaks down, and a local field description becomes necessary.
Abstract In this chapter, the singular changes accompanying a phase transition requires the existence of macroscopic correlations. As a consequence, phase transitions mark a stark departure from the noninteracting and linear responding systems we have considered thus far. Understanding phase transi- tions requires that we study strongly interacting models, pushed far into regimes of nonlinear response. This complication renders detailed studies largely intractable analytically. However, some simplifications exist that allow for a general theory of phase transitions to be developed. These are the notions of universality that accompany phase transition phenomenology, and the ability to classify phase transi- tions simply using ideas of spontaneous symmetry breaking. The former allows us to postulate simple models bereft for molecular detail and yet still provide quantitative predictions for observable behavior. The latter allows us to predict what phases can exist, and understand why phase transitions occur.
Among the many applications of statistical mechanics, some of the most intriguing and challenging theoretical problems arise in connection with phase transitions. These are abrupt changes of state such as occur, for example, when a liquid is transformed into a vapour, a ferromagnet loses its magnetization upon heating to its Curie temperature, or at the onset at sufficiently low temperatures of superfluidity or superconductivity. It is within the theory of phase transitions, too, that the mathematical relationships between statistical mechanics and relativistic field theories are most powerful. Indeed, the idea of spontaneous symmetry breaking, which lies at the heart of the theory of phase transitions, is the crucial ingredient that turns the gauge theories of chapter 8 into a real working model of the fundamental forces of nature, to be discussed in the next chapter.
Among the many applications of statistical mechanics, some of the most intriguing and challenging theoretical problems arise in connection with phase transitions. These are abrupt changes of state such as occur, for example, when a liquid is transformed into a vapour, a ferromagnet loses its magnetization upon heating to its Curie temperature, or at the onset at suciently low temperatures of super uidity or superconductivity. It is within the theory of phase transitions, too, that the mathematical relationships between statistical mechanics and relativistic eld theories are most powerful. Indeed, the idea of spontaneous symmetry breaking, which lies at the heart of the theory of phase transitions, is the crucial ingredient that turns the gauge theories of Chapter 8 into a real working model of the fundamental forces of nature, to be discussed in the next chapter.
While analyzing second order thermodynamical phase transitions, Lev Landau (the famous Russian physicist) introduced a very vital concept, the concept of an "order parameter". This not only amalgamated the previous fragmentary theoretical understanding of phase transitions (an arsenal of mean-field theories) but also it put forward the important theory of "spontaneous symmetry breaking". Today, order parameter concept is a paradigm both in condensed matter physics and in high energy physics, and Landau theory is a pinnacle of all mean-field theories. Mean field theories are good qualitative descriptors of the phase transition behavior. But all mean-field theories (including Landau's theory) fail at the critical point (the problem of large correlation length). The problems with large correlation length in quantum many-body systems are the hardest problems known in theoretical physics (both in condensed matter and in particle physics). It was Ken Wilson's physical insights and his powerful mathematical skills that opened a way to the solution of such hard problems. This manuscript is a perspective on these issues. Starting with simple examples of phase transitions (like ice/water; diamond/graphite etc.) we address the following important questions: Why does non-analyticity (sharp phase transitions) arise when thermodynamical functions (i.e., free energies etc) are good analytic functions? How does Landau's program unify all the previous mean-field theories? Why do all the mean-field theories fail near the critical point? How does Wilson's program go beyond all the mean-field theories? What is the origin emergence and universality?