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Quantum Critical Matter and Phase Transitions Chapter 280 297 These scaling relations highly constrain the possible values the critical exponents can take. In fact, only the symmetry of the order parameter and the dimensionality of the system are relevant in order to determine the critical exponents of a transition. This is the concept of universality: every CPT belongs to a universality class (Ising, XY, Heisenberg, etc.) of transitions with the same critical exponents, even though the underlying microscopic physics can be completely different. The beautiful principle of universality has made the study of CPTs among the most popular in modern science. As early as 1971, Stanley published the first book on phase transitions and critical phenomena (Stanley, 1971), followed by Ma (1976), Goldenfeld (1992), Yeomans (1992), and Herbut (2010). Into the 2000s, a series of multivolume monographs have appeared, edited by Domb and Lebowitz (2001). At the pres- ent CPTs are well understood, from both experimental as well as theoretical points of view (Nishimori and Ortiz, 2010). Whenever a new magnetic material is fabricated, one just needs to verify if a CPT occurs and check its universality class. Once that is known, all is known about that particular phase transition. 2.2 Quantum Critical Theory A QPT is a second order phase transition that occurs at zero temperature, as a function of pressure, magnetic field or any tuning parameter g other than temperature. A standard way of obtaining such a transition is by suppressing the critical temperature of an ordered state—such as a ferromagnet or antiferromagnet—to zero temperature by, for example, pressure. When the pressure exceeds the critical pressure, a quantum disordered regime appears. Exactly at the critical value of the tuning parameter g1⁄4gc, the system develops many unconventional properties. The relevance of QPTs is that for nonzero temperatures close to the critical value gc the system is still domi- nated by quantum critical excitations. Thus can arise unconventional specific heat, resistivity, magnetic susceptibility, and so on. This “quantum critical regime,” as shown in the typical phase diagram of Fig. 2, is thought to be the key toward understanding many abnormal properties of materials. In this section, we first provide the traditional theory of how such a quantum critical regime appears. As for all phase transitions, the key lies in the concept of scaling functions and the corresponding exponents close to the transition. In fact, there is an intricate link between QPTs and thermal phase transitions, which becomes clear by considering the example of the Ising model. Critical exponents for any given model can be computed using the meth- ods of statistical physics. The starting point is to define the microscopic Hamiltonian H relevant for the system under study. For the Ising model, a model of interacting spins that can describe (anti)ferromagnetic phase transi- tions, the Hamiltonian is X H 1⁄4 J S zi S zj , (5) hijiPDF Image | HANDBOOK ON THE PHYSICS AND CHEMISTRY OF RARE EARTHS
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