A Holographic Model for Quantum Critical Responses

TL;DR

This work builds a self-consistent holographic model for quantum critical responses by introducing a bulk scalar 𝜙 dual to a relevant boundary operator 𝒪 with dimension Δ, coupled to Weyl curvature to generate a finite-temperature condensate ⟨𝒪⟩_T and to a bulk gauge field that yields current correlators. The finite-temperature conductivity σ(ω) is computed in a perturbative regime and shown to depend on Δ and the product α1α2, with σ(ω) closely matching Katz et al.'s simple ansatz across a wide parameter range, and with a high-frequency expansion that can be rederived from the boundary JJ OPE, giving a concrete C_{JJO} coefficient. The analysis connects holographic results to Wilson-Fisher CFT data, showing how the OPE controls the T/ω corrections and how large-N factorization yields higher-order tail terms, including potential multicritical scenarios when Δ<3/2 via composite operators. The framework provides a physically constrained platform to analytically continue QMC data, study detuning away from the QCP, and lay groundwork for exploring the phase diagram near quantum criticality in 2+1 dimensions and beyond.

Abstract

We analyze the dynamical response functions of strongly interacting quantum critical states described by conformal field theories (CFTs). We construct a self-consistent holographic model that incorporates the relevant scalar operator driving the quantum critical phase transition. Focusing on the finite temperature dynamical conductivity $σ(ω,T)$, we study its dependence on our model parameters, notably the scaling dimension of the relevant operator. It is found that the conductivity is well-approximated by a simple ansatz proposed by Katz et al [1] for a wide range of parameters. We further dissect the conductivity at large frequencies $ω>> T$ using the operator product expansion, and show how it reveals the spectrum of our model CFT. Our results provide a physically-constrained framework to study the analytic continuation of quantum Monte Carlo data, as we illustrate using the O(2) Wilson-Fisher CFT. Finally, we comment on the variation of the conductivity as we tune away from the quantum critical point, setting the stage for a comprehensive analysis of the phase diagram near the transition.

Figures (10)

• Figure 1: Phase diagram near a quantum critical point (QCP). The physics in the shaded region ("fan") is dominated by the thermally excited theory of the QCP. The transition is driven by a relevant operator with coupling $\lambda$ and scaling dimension $\Delta=d-1/\nu$, where $d$ is the spacetime dimension, and $\nu$ the "correlation length" critical exponent. This paper mainly focuses on the $\lambda\!=\!0$ line (dotted); for detuning effects see section \ref{['non critical']} and fig. \ref{['critical cond']}.
• Figure 2: A demonstration of the holographic model: real part of the conductivity as a function of frequency for various values of the scaling dimension of the scalar operator $\Delta$ with $\alpha_1 \alpha_2=0.1$ (left), and for various choices of $\alpha_1 \alpha_2$ with $\Delta=1.5$ (right). Note that $\alpha_1,\,\alpha_2$ are proportional to the OPE coefficients $C_{TTO},\,C_{JJO}$, respectively, of the boundary CFT (see Table \ref{['table']}).
• Figure 3: The scalar profile $\phi(u)$ for $\Delta=1.5$ (left) and $\Delta = 4$ (right). The solid black line is the exact solution, while the dashed blue line is the power-law profile $\phi(u) = \phi_{1}\, u^\Delta$, as used in katz. To compare the two profiles, $\phi_{1}$ is fixed to 1 so that the two profiles match to leading order as $u\to0$.
• Figure 4: On the left, we have the diffusion as a function of scaling dimension $\Delta$ with $\alpha_1\alpha_2 = 0.1$. The solid black line is the diffusion constant for the holographic model while for comparison, the dashed blue line is the diffusion calculated using the power-law profile $\phi(u) =\phi_{1} \,u^\Delta$ (and with the coefficient $\phi_{1}$ chosen to match to holographic solution for each $\Delta$). On the right we have the DC conductivity $\sigma_0 / \sigma_\infty$ as a function of the scaling dimension $\Delta$ with $\alpha_1 \alpha_2 = 0.1$. The solid black line is for the holographic model while the dashed blue line is found using $\phi(u) =\phi_{1}\, u^\Delta$.
• Figure 5: Plots of the conductivity for Euclidean (left) and real (right) frequencies for $\Delta=1.5$ with $\phi_{1} \alpha_2$ fit to the quantum Monte Carlo data for the O(2) Wilson-Fisher CFT katznatphys (see also chen). The solid black line represents the conductivity using the scalar profile given in eq. (\ref{['solution']}) with $\phi_{1}\alpha_2 = 0.589$, while the dashed blue line represents the value for the conductivity using the simple power-law profile $\phi(u) = \phi_{1}\,u^\Delta$ with $\phi_{1}\alpha_2 = 0.611$.