Understanding the real time dynamics of quantum systems without quasiparticles constitutes an important yet challenging problem. We study the superfluid-insulator quantum-critical point of bosons on a two-dimensional lattice, a system whose excitations cannot be described in a quasiparticle basis. We present detailed quantum Monte Carlo results for two separate lattice realizations: their low-frequency conductivities are found to have the same universal dependence on imaginary frequency and temperature. We then use the structure of the real time dynamics of conformal field theories described by the holographic gauge/gravity duality to make progress on the dicult problem of analytically continuing the Monte Carlo data to real time. Our method yields quantitative and experimentally testable results on the frequency-dependent conductivity near the quantum critical point, and on the spectrum of quasinormal modes in the vicinity of the superfluid-insulator quantum phase transition. Extensions to other observables and universality classes are discussed.
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 obtain exact, nonperturbative results for the zero- and finite-temperature dynamical responses at the semimetal-superconductor quantum critical point (QCP) of two-dimensional Dirac fermions, which can occur on the surface of a topological insulator. This strongly interacting QCP is described by a conformal field theory with emergent
N=2 supersymmetry, which imposes strong constraints. At zero temperature, we obtain closed-form results for the optical conductivity, dynamical shear viscosity, and corner entanglement entropy. We also make exact statements for the optical conductivity at finite temperature regarding large-frequency asymptotics and sum rules. Finally, we discuss experimental considerations for the observation of these signatures in topological insulators.
Following on from our previous work [M. J. Bhaseen et al., Phys. Rev. Lett. 98, 166801 (2007)] we examine the finite temperature magnetothermoelectric response in the vicinity of a quantum critical point (QCP). We begin with general scaling considerations relevant to an arbitrary QCP, either with or without Lorentz invariance, and in arbitrary dimension. In view of the broad connections to high-temperature superconductivity and cold atomic gases, we focus on the quantum critical fluctuations of the relativistic Landau-Ginzburg theory. This paradigmatic model arises in many contexts and describes the (particle-hole symmetric) superfluid-Mott insulator quantum phase transition in the Bose-Hubbard model. The application of a magnetic field opens up a wide range of physical observables, and we present a detailed overview of the charge and thermal transport and thermodynamic response. We combine several different approaches including the epsilon expansion and associated quantum Boltzmann equation, entropy drift, and arguments based on Lorentz invariance. The results differ markedly from the zero-field case, and we include an extended discussion of the finite thermal conductivity which emerges in the presence of a magnetic field. We derive an integral equation that governs its response and explore the crossover upon changing the magnetic field. This equation may be interpreted as a projection equation in the low-field limit, and clearly highlights the important role of collision invariants (or zero modes) in the hydrodynamic regime. Using an epsilon expansion around three dimensions, our analytic and numerical results interpolate between our previously published value and the exact limit of two-dimensional relativistic magnetohydrodynamics.
We study the Higgs excitation in the Goldstone phase of the relativistic O(
N) model in two spatial dimensions at zero temperature. The response functions of the order parameter, and its magnitude squared, become universal functions of frequency in the vicinity of the quantum critical point described by the Wilson-Fisher fixed point, and we compute them to next-to-leading order in
1/N. The Higgs particle has an infrared singular decay to gapless Goldstone excitations, and its response functions are characterized by a pole in the lower half of the complex frequency plane. The pole acquires a nonzero real part only at next-to-leading order in
1/N, demonstrating that the Higgs excitation has an oscillatory component even in the scaling limit. Both the real and imaginary parts of the pole position vanish with the correlation length exponent $\ensuremath{\nu}$ upon approaching the critical point. We present evidence that the spectral density of the O(
N)-invariant amplitude-squared of the order parameter has a peak at a nonzero frequency in the scaling limit. We connect our results to recent experimental studies of the superfluid-insulator quantum phase transition of ultracold bosonic atoms in optical lattices.
We compute the zero-temperature conductivity in the two-dimensional quantum
O(N) model using a nonperturbative functional renormalization-group approach. At the quantum critical point we find a universal conductivity ${\ensuremath{\sigma}}^{*}/{\ensuremath{\sigma}}_{Q}^{}$ (with ${\ensuremath{\sigma}}_{Q}^{}={q}^{2}/h$ the quantum of conductance and
q the charge) in reasonable quantitative agreement with quantum Monte Carlo simulations and conformal bootstrap results. In the ordered phase the conductivity tensor is defined, when $N\ensuremath{\ge}3$, by two independent elements, ${\ensuremath{\sigma}}_{\mathrm{A}}(\ensuremath{\omega})$ and ${\ensuremath{\sigma}}_{\mathrm{B}}(\ensuremath{\omega})$, respectively associated with
SO(N) rotations which do and do not change the direction of the order parameter. Whereas ${\ensuremath{\sigma}}_{\mathrm{A}}(\ensuremath{\omega}\ensuremath{\rightarrow}0)$ corresponds to the response of a superfluid (or perfect inductance), the numerical solution of the flow equations shows that ${lim}_{\ensuremath{\omega}\ensuremath{\rightarrow}0}{\ensuremath{\sigma}}_{\mathrm{B}}(\ensuremath{\omega})/{\ensuremath{\sigma}}_{Q}^{}={\ensuremath{\sigma}}_{\mathrm{B}}^{*}/{\ensuremath{\sigma}}_{Q}^{}$ is a superuniversal (i.e.,
N-independent) constant. These numerical results, as well as the known exact value ${\ensuremath{\sigma}}_{\mathrm{B}}^{*}/{\ensuremath{\sigma}}_{Q}^{}=\ensuremath{\pi}/8$ in the large-
N limit, allow us to conjecture that ${\ensuremath{\sigma}}_{\mathrm{B}}^{*}/{\ensuremath{\sigma}}_{Q}^{}=\ensuremath{\pi}/8$ holds for all values of
N, a result that can be understood as a consequence of gauge invariance and asymptotic freedom of the Goldstone bosons in the low-energy limit.