Dynamical response near quantum critical points

TL;DR

This work studies high-frequency response functions in the vicinity of quantum critical points by allowing for both detuning from the critical coupling and finite temperature, and considers general dimensions and dynamical exponents to lead to a unified understanding of sum rules.

Abstract

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 conformal 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.

Figures (4)

• Figure 1: Phase diagram near a canonical quantum critical point. $g$ is the non-thermal coupling that needs to be tuned. The dotted lines roughly delimit the quantum critical "fan".
• Figure 2: Log-log plot of the asymptotic behavior of $\sigma(\mathrm i\Omega_n)$ at imaginary frequencies, in the disordered phase of the $\mathrm{O}(2)$ model, computed using QMC in the limit $T\!\to\!0$. Each set of colored dots represents a different detuning $g$. $m\!\propto\! g^\nu$ is the single particle gap. The line is the field theory prediction (\ref{['eq:asympt']}) at large $\Omega_n$, with $\nu=0.67$.
• Figure 3: Integration contour in the complex $z=\omega^2$ plane; the radius of the circle is taken to be arbitrarily large. $F(z)$, defined in Eq.(\ref{['eq:F']}), is analytic everywhere except on the non-negative real axis, as indicated by the thick gray line.
• Figure 4: (a) Conductivity at Matsubara frequencies, $\sigma(\mathrm i\Omega_n)$ in the disordered phase $\delta u>0$. (b) Universal scaling function $f_+(\mathrm i\Omega_n/m)$ obtained from Eq.(\ref{['eq:non_uniscale']}) after subtraction the non universal high frequency cutoff corrections to scaling using $\Omega_c/m=100$. In both panels different curves correspond to difference detuning parameters $\delta u$