Universal relaxational dynamics near two-dimensional quantum critical points

We describe the nonzero temperature $(T)$, low frequency (ω) dynamics of the order parameter near quantum critical points in two spatial dimensions $(d)$, with a special focus on the regime ωkBT. For the case of a ``relativistic,'' O(n)-symmetric, bosonic quantum field theory we show that, for small ε=3-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 $d=2$. 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'' $(n=1)$ 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 $n=2$, we compute the universal T dependence of the superfluid density below the Kosterlitz-Thouless temperature, and discuss implications for the high temperature superconductors. For $n=3$, our computations of the dynamic structure factor relate to neutron scattering experiments on La1.85Sr0.15CuO₄, 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 $d=2$. Although computations in appendixes do rely upon technical results on the ε-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.