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300 Handbook on the Physics and Chemistry of Rare Earths called imaginary time t 2 [0, b). At zero temperature, the free energy for the d-dimensional quantum Ising model becomes Z no Fðm, gÞ1⁄4 ddxdt cð@mmðx, tÞÞ2 +a2ðgÞmðx, tÞ2 +a4mðx, tÞ4 +⋯ , (10) so that the tuning parameter is now g as opposed to T for the classical phase transition. The parameter a2$(ggc) changes sign when the transverse field is increased. At the transition, not only spatial correlation length diverges, but also the imaginary time correlations xt. They are related by the dynamical critical exponent z, xt $xz: (11) It is easy to see that for the Ising model, the dynamical critical exponent is z1⁄41, since x and t are treated on the same footing. All other critical exponents can be derived from the critical exponents of the (d+1)-dimensional classical Ising model. We have m$(ggc)b for the order parameter, and similar expressions for the susceptibility exponents and the correlation length exponents. All of those are listed in Table 1. Note that there is not an analogue of the specific heat exponent at zero temperature. In conclusion, we note that both classical and quantum field theories are described by a free energy functional that depends on the order parameter. The quantum transverse field Ising model in d dimensions is equivalent to the classical Ising model in (d+1) dimensions. This allows us to find the crit- ical exponents associated by the QPT in the quantum Ising model from the classical exponents. In Section 2.4.1, we will discuss Li(HoY)F4, a rare earth material that displays characteristics of a quantum Ising phase transition. 2.3 HM Theories of Itinerant Magnets In many rare earth materials, the magnetism is not caused by localized spins as described by the Ising model. Instead, it consists of conducting (itinerant) electrons that undergo a spin-density wave (SDW) transition or a supercon- ducting transition. The theory of QPTs in the presence of interacting electrons was pioneered by Hertz (1976), Millis (1993), and Moriya (1985). Their theory starts with interacting electrons, for which one can write down the partition function at inverse temperature b, defined by Eq. (7). These interacting electrons have an instability toward the formation of some kind of magnetic order. The magnitude of the magnetic order is parametrized by the order parameter field C(x, t) where x is the spatial coordinate and t the imaginary time—the extra dimension as a consequence of quantum physics. The key assumption in HM theories is that the low-energy physical description of this system can do without referring to electrons. All electronic degrees of freedom can be integrated out and we are left with an effective freePDF Image | HANDBOOK ON THE PHYSICS AND CHEMISTRY OF RARE EARTHS
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