Strongly correlated quantum critical points with conformal symmetry in two spatial dimensions describe many phase transitions in condensed matter systems, as for example that from superfluid to insulator. Combining field theory methods, Monte Carlo simulations, and the holographic duality from string theory, the authors use an operator product expansion to derive non-perturbative results for these systems, including the leading terms in the frequency dependence of the nonzero temperature conductivity and sum rules obeyed by the conductivity. These methods and results are applicable to a wide class of conformal field theories for condensed matter systems.
We study charge transport of quantum critical points described by conformal field theories in 2+1 space-time dimensions. The transport is described by an effective field theory on an asymptotically anti-de Sitter space-time, expanded to fourth order in spatial and temporal gradients. The presence of a horizon at nonzero temperatures implies that this theory has quasinormal modes with complex frequencies. The quasinormal modes determine the poles and zeros of the conductivity in the complex frequency plane, and so fully determine its behavior on the real frequency axis, at frequencies both smaller and larger than the absolute temperature. We describe the role of particle-vortex or S duality on the conductivity, specifically how it maps poles to zeros and vice versa. These analyses motivate two sum rules obeyed by the quantum critical conductivity: the holographic computations are the first to satisfy both sum rules, while earlier Boltzmann-theory computations satisfy only one of them. Finally, we compare our results with the analytic structure of the
O(N) model in the large-
N limit, and other CFTs.
We compute the universal conductivity of the (2+1)-dimensional XY universality class, which is realized for a superfluid-to-Mott insulator quantum phase transition at constant density. Based on large-scale Monte Carlo simulations of the classical (2+1)-dimensional J-current model and the two-dimensional Bose-Hubbard model, we can precisely determine the conductivity on the quantum critical plateau, σ(∞) = 0.359(4)σQ with σQ the conductivity quantum. The universal conductivity curve is the standard example with the lowest number of components where the bottoms-up AdS/CFT correspondence from string theory can be tested and made to use [R. C. Myers, S. Sachdev, and A. Singh, Phys. Rev. D 83, 066017 (2011)]. For the first time, the shape of the σ(iω(n)) - σ(∞) function in the Matsubara representation is accurate enough for a conclusive comparison and establishes the particlelike nature of charge transport. We find that the holographic gauge-gravity duality theory for transport properties can be made compatible with the data if temperature of the horizon of the black brane is different from the temperature of the conformal field theory. The requirements for measuring the universal conductivity in a cold gas experiment are also determined by our calculation.
Relativistic O(
N) field theories are studied near the quantum-critical point in two space dimensions. We compute dynamical correlations by large-scale Monte Carlo simulations and numerical analytic continuation. In the ordered side, the scalar spectral function exhibits a universal peak at the Higgs mass. For
N=3 and 4, we confirm its ${\ensuremath{\omega}}^{3}$ rise at low frequency. On the disordered side, the spectral function exhibits a sharp gap. For
N=2, the dynamical conductivity rises above a threshold at the Higgs mass (density gap), in the superfluid (Mott insulator) phase. For charged bosons (Josephson arrays), the power-law rise above the Higgs mass increases from two to four. Approximate charge-vortex duality is reflected in the ratio of imaginary conductivities on either side of the transition. We determine the critical conductivity to be ${\ensuremath{\sigma}}_{\mathrm{c}}^{*}=0.3(\ifmmode\pm\else\textpm\fi{}0.1)\ifmmode\times\else\texttimes\fi{}4{e}^{2}/h$ and describe a generalization of the worm algorithm to
N>2. We use a singular value decomposition error analysis for the numerical analytic continuation.
We study high-frequency response functions, notably the optical conductivity, in the vicinity of quantum critical points (QCPs) by allowing for both detuning from the critical coupling and finite temperature. We consider general dimensions and dynamical exponents. This leads to a unified understanding of sum rules. In systems with emergent Lorentz invariance, powerful methods from quantum field theory allow us to fix the high-frequency response in terms of universal coefficients. We test our predictions analytically in the large-N O(N) model and using the gauge-gravity duality and numerically via quantum Monte Carlo simulations on a lattice model hosting the interacting superfluid-insulator QCP. In superfluid phases, interacting Goldstone bosons qualitatively change the high-frequency optical conductivity and the corresponding sum rule.
We describe the nonzero temperature
(T), low frequency $(\ensuremath{\omega})$ dynamics of the order parameter near quantum critical points in two spatial dimensions
(d), with a special focus on the regime $\ensuremath{\Elzxh}\ensuremath{\omega}\ensuremath{\ll}{k}_{B}T$. For the case of a ``relativistic,''
O(n)-symmetric, bosonic quantum field theory we show that, for small $\ensuremath{\epsilon}=3\ensuremath{-}d$, 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 $\ensuremath{\epsilon}$-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.