Constraining quantum critical dynamics: 2+1D Ising model and beyond

TL;DR

This work analyzes the large frequency or momentum asymptotics of observables, which are used to derive powerful sum rules and inequalities in the linear-response dynamics of conformal QC systems at finite temperature, in spatial dimensions above 1.

Abstract

Quantum critical (QC) phase transitions generally lead to the absence of quasiparticles. The resulting correlated quantum fluid, when thermally excited, displays rich universal dynamics. We establish non-perturbative constraints on the linear-response dynamics of conformal QC systems at finite temperature, in spatial dimensions above one. Specifically, we analyze the large frequency/momentum asymptotics of observables, which we use to derive powerful sum rules and inequalities. The general results are applied to the O(N) Wilson-Fisher fixed point, describing the QC Ising model when N = 1. We focus on the order parameter and scalar susceptibilities, and the dynamical shear viscosity. Connections to simulations, experiments and gauge theories are made.

Figures (3)

• Figure 1: a) Phase diagram near a quantum critical point (QCP). b) Asymptotic behavior of the Euclidean susceptibility associated with an operator $\mathcal{O}$ of scaling dimension $\Delta$: $\chi^{\rm E}(i\omega_n)=\langle \mathcal{O}(-\omega_n)\mathcal{O}(\omega_n)\rangle_T$. c) Schematic operator product expansion (OPE) determining the asymptotics of $\chi$. "desc." denotes the descendants of the primary $\mathcal{O}_n$ (dimension $\Delta_n$).
• Figure 2: Thermal mass $m_T/T=\Theta_d$, as well as expansion coefficients $a_0,a_g$ appearing in $\chi_s(|k|\gg T)$ as a function of the spatial dimension $d$. Interestingly, $a_g$ vanishes in $2+1$D.
• Figure 3: a) Scalar response of the O$(N)$ model ($N=\infty$) in various spacetime dimensions $D$. The dashed vertical arrow represents a delta function $\delta(\omega)$. b) Weight of this delta function versus $D$.