Dimensional Reduction in Quantum Critical Systems

Abstract

This chapter discusses dimensional reduction in 3+1D antiferromagnets. The observable of interest is the order parameter (staggered magnetisation). At zero temperature, the order parameter is well described by a 3+1D theory. In the near vicinity of the Neel temperature (\(T_N\)), the system is expected to crossover into a regime whereby the order parameter is described by a theory in one lower dimension. This is ‘dimensional reduction’. A recent analysis of the order paramter versus temperature, performed on the data for 3+1D antiferromagnet TlCuCl\(_3\) [1], has identified the crossover boundaries to the dimensionally reduced regime. It was found that on approach to \(T_N\) a relatively sharp crossover occurs, whereby the order parameter near \(T_N\) displays scaling in accord with the 3D theory. In this chapter we analyse the same scenario. Starting from a \(T=0\), 3+1D quantum field theory, and perturbatively including temperature, we quantitatively describe the experimental data on TlCuCl\(_3\) across the range \(0\le T\le T_N\). Our theory cannot reproduce the 3D scaling exponents in the limit \(T\rightarrow T_N\), yet it fully accounts for the observed sharp crossover behaviour in TlCuCl\(_3\). Ultimately, we expect observables to scale with exponents of a lower dimensional theory, however, our results provide a new perspective on the enigmatic problem of dimensional reduction.