Universal relaxational dynamics near two-dimensional quantum critical points

We describe the nonzero temperature $\mathrm{}\left(T\right)\mathrm{}$, low frequency $\left(\omega \right)$ dynamics of the order parameter near quantum critical points in two spatial dimensions $\mathrm{}\left(d\right)\mathrm{}$, with a special focus on the regime $ħ\omega \ll k_{B}T$. For the case of a “relativistic,” $\mathrm{O}\mathrm{}\left(n\right)\mathrm{}$-symmetric, bosonic quantum field theory we show that, for small $\epsilon =3\mathrm{}-d$, the dynamics is described by an effective classical model of waves with a quartic interaction. We provide analytical and numerical analyses of the classical wave model directly in $d=2\mathrm{}$. We describe the crossover from the finite frequency, “amplitude fluctuation,” gapped quasiparticle mode in the quantum paramagnet (or Mott insulator), to the zero frequency “phase” $(n>~2)$ or “domain wall” $\mathrm{}(n=1\mathrm{})\mathrm{}$ relaxation mode near the ordered state. For static properties, we show how a surprising, duality-like transformation allows an exact treatment of the strong-coupling limit for all n. For $n=2\mathrm{}$, we compute the universal T dependence of the superfluid density below the Kosterlitz-Thouless temperature, and discuss implications for the high temperature superconductors. For $n=3\mathrm{}$, our computations of the dynamic structure factor relate to neutron scattering experiments on ${\mathrm{La}}_{1.85}{\mathrm{Sr}}_{0.15}{\mathrm{Cu}\mathrm{}\mathrm{O}}_4$, and to light scattering experiments on double layer quantum Hall systems. We expect that closely related effective classical wave models will apply also to other quantum critical points in $d=2\mathrm{}$. Although computations in appendixes do rely upon technical results on the $\epsilon$-expansion of quantum critical points obtained in earlier papers, the physical discussion in the body of the paper is self-contained, and can be read without consulting these earlier works.