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298 Handbook on the Physics and Chemistry of Rare Earths FIG. 2 The typical phase diagram of a quantum phase transition. The horizontal axis represents a tuning parameter g, which can be pressure, magnetic field, or particle density; the vertical axis represents temperature. The high g regime to the right represents the “quantum disordered” phase: a paramagnet or just a regular Fermi liquid. At low g there is an ordered phase, for example, a (anti) ferromagnet. The quantum critical point resides exactly at T 1⁄4 0 where the system goes from the ordered to the disordered phase. At this critical value of g1⁄4gc, the nonzero temperature regime is denoted as “quantum critical.” In this part of the phase diagram, many properties such as the specific heat or the resistivity are unconventional due to the vicinity of the quantum critical point. Such unconventional behavior is usually referred to as “non-Fermi liquid” behavior. where Szi is the z-component of a classical spin at a lattice site i. For J > 0, this model has a ferromagnetic ground state and we therefore expect that below some temperature the system develops spontaneous magnetization. P bEn The Ising model is classical, so that the partition function Z 1⁄4 ne , where the sum runs over all possible spin states with energy En. Any macro- scopic quantities that we are interested in, such as magnetization or specific heat, can be derived from the free energy F 1⁄4kBTlogZ. In some cases, the partition function and, hence, the free energy can be computed exactly, in most cases, however, we need to resort to an approximation scheme. The most successful theory is Landau mean field theory: here we assume that spins interact only with the average field of all other spins, given by m(x). A free energy functional can then be written as a function of the magnetiza- tion m(x) and temperature T. In general, for a d-dimensional system, we have Z Fðm,TÞ1⁄4 ddx cðrmðxÞÞ2 +a2ðTÞmðxÞ2 +a4mðxÞ4 +⋯ (6) where a2, a4, and c are some parameters. For a CPT, Landau showed, we must have a2 $ t with t the reduced temperature, a4 and c are positive definite. The magnetization can be found by minimizing Fðm, TÞ with respect to the order parameter m(x). From just these very general considerations, we can compute critical expo- nents. For example, minimizing the free energy with respect to a homoge- neous m gives us the order parameter exponent b 1⁄4 1/2. The second derivative CV 1⁄4@2F yields the specific heat exponent a1⁄40. Explicitly writing @T2PDF Image | HANDBOOK ON THE PHYSICS AND CHEMISTRY OF RARE EARTHS
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