Quantum critical phenomena

This paper proposes an approach to the study of critical phenomena in quantum-mechanical systems at zero or low temperatures, where classical free-energy functionals of the Landau-Ginzburg-Wilson sort are not valid. The functional integral transformations first proposed by Stratonovich and Hubbard allow one to construct a quantum-mechanical generalization of the Landau-Ginzburg-Wilson functional in which the order-parameter field depends on (imaginary) time as well as space. Since the time variable lies in the finite interval [$0,-i\beta$], where $\beta$ is the inverse temperature, the resulting description of a $d$-dimensional system shares some features with that of a ($d+1\mathrm{}$)-dimensional classical system which has finite extent in one dimension. However, the analogy is not complete, in general, since time and space do not necessarily enter the generalized free-energy functional in the same way. The Wilson renormalization group is used here to investigate the critical behavior of several systems for which these generalized functionals can be constructed simply. Of these, the itinerant ferromagnet is studied in greater detail. The principal results of this investigation are (i) at zero temperature, in situations where the ordering is brought about by changing a coupling constant, the dimensionality which separates classical from nonclassical critical-exponent behavior is not 4, as is usually the case in classical statistics, but $4-z$ dimensions, where $z$ depends on the way the frequency enters the generalized free-energy functional. When it does so in the same way that the wave vector does, as happens in the case of interacting magnetic excitons, the effective dimensionality is simply increased by 1; $z=1\mathrm{}$. It need not appear in this fashion, however, and in the examples of itinerant antiferromagnetism and clean and dirty itinerant ferromagnetism, one finds $z=2, 3, \mathrm{and} 4$, respectively. (ii) At finite temperatures, one finds that a classical statistical-mechanical description holds (and nonclassical exponents, for $d<\mathrm{}4\mathrm{}$) very close to the critical value of the coupling $U_c$, when $\frac{(U-U_c)}{U_c}\ll {\left(\frac{T}{U_c}\right)}^{\frac{2}{z}}$. $\frac{z}{2}$ is therefore the quantum-to-classical crossover exponent.