[Paper Review] Analytic quantum critical points from holography

Jie Ren|arXiv (Cornell University)|Oct 9, 2012

Black Holes and Theoretical Physics3 citations

TL;DR

This paper presents exact analytic solutions for the Klein-Gordon equation of a scalar field in the extremal Reissner-Nordström-AdS₅ black hole background, enabling the holographic extraction of Green’s functions near zero frequency. It classifies quantum critical points into hybridized, bifurcating, and mixed types based on instabilities from zero modes or imaginary IR scaling exponents, with phase diagrams analytically derived without numerical tuning.

ABSTRACT

We find exact, analytic solutions of the Klein-Gordon equation for a scalar field in the background of the extremal Reissner-Nordstrom-AdS_5 black hole. The Green's function near a quantum critical point for a strongly coupled system can be extracted holographically from an exact solution for the scalar at zero frequency (ω), but arbitrary momentum (k), mass, and charge. By examining the Green's function near ω=0, there are two types of instability: the first one is triggered by a zero mode, and gives a hybridized critical point; the second one is triggered by the instability of the IR geometry, and gives a bifurcating critical point. The two types of instability can happen at the same time, and give a mixed critical point. Without tuning an extra parameter, only the second type of instability can happen at k=0. At the critical point with the superfluid velocity, the scalar can develop either type of instability, depending on the parameters. The zero mode can also be obtained by tuning a double trace deformation. The phase diagrams can be analytically drawn.

Motivation & Objective

• To provide exact analytic solutions for the Klein-Gordon equation at zero frequency in the extremal Reissner-Nordström-AdS₅ black hole background.
• To classify quantum critical points in strongly coupled systems using holographic duality, distinguishing between hybridized, bifurcating, and mixed critical behaviors.
• To derive phase diagrams analytically for quantum critical points without relying on numerical approximations.
• To examine the role of superfluid velocity and double trace deformations in tuning critical behavior and zero mode emergence.

Proposed method

• Solve the Klein-Gordon equation for a scalar field with mass $ m $, charge $ q $, and arbitrary momentum $ \mathbf{k} $ at $ \omega = 0 $ in the extremal RN-AdS₅ geometry.
• Use the standard and alternative quantization schemes to extract the boundary Green’s function from the bulk scalar solution.
• Identify instabilities via the IR scaling exponent $ \nu_k = \frac{1}{2\sqrt{3}}\sqrt{m^2 + k^2 - 2q^2 + 3} $, where imaginary $ \nu_k $ signals IR geometry instability.
• Determine zero modes from the condition $ \nu_k = \frac{q}{\sqrt{3}} - n_\pm - \frac{\Delta_\pm - 1}{2} $, with $ n_\pm \in \mathbb{N}_0 $, indicating UV-driven hybridized instabilities.
• Construct the full Green’s function by combining UV data (from $ \omega = 0 $ solution) and IR data (from AdS₂ near-horizon limit).
• Use Whittaker functions to express the IR Green’s function $ \mathcal{G}_k(\omega) $, with asymptotic behavior $ \sim (-2i\omega)^{2\nu_k} $, enabling analysis near critical points.

Experimental results

Research questions

• RQ1 What types of quantum critical points emerge from holographic scalar field instabilities in extremal RN-AdS₅, and how are they classified?
• RQ2 How do zero modes and IR geometry instabilities coexist or compete in determining the nature of the critical point?
• RQ3 Can phase diagrams for quantum critical points be analytically constructed without numerical input, particularly at $ \mathbf{k} = 0 $ and $ \omega = 0 $?
• RQ4 What role does the superfluid velocity play in accessing different types of critical points, including mixed or marginal ones?
• RQ5 How does tuning a double trace deformation affect the emergence of zero modes and the resulting critical behavior?

Key findings

• Exact analytic solutions for the Klein-Gordon equation at $ \omega = 0 $ are obtained in extremal RN-AdS₅, enabling full holographic computation of the Green’s function without numerical approximation.
• Two distinct instability mechanisms are identified: (1) zero-mode-driven hybridized critical points (requiring large $ q $), and (2) IR geometry instability via imaginary $ \nu_k $, leading to bifurcating critical points.
• At $ \mathbf{k} = 0 $, only the bifurcating critical point occurs without tuning an extra parameter, while tuning the superfluid velocity allows access to all three types: hybridized, bifurcating, and mixed.
• Mixed critical points, including marginal ones, arise when both zero-mode and IR instabilities coexist, with the latter dominating at $ \mathbf{k} = 0 $.
• The Green’s function at $ \omega = 0 $ is finite when $ \Delta_+ $ is not an integer, but diverges when $ \Delta_+ \in \mathbb{Z}_{\geq 2} $, requiring renormalization via $ \delta \to 0 $ and UV cutoff $ L_{\text{UV}} $, with consistency condition $ |\delta \ln(L_{\text{UV}}/L)| \ll 1 $.
• The IR Green’s function $ \mathcal{G}_k(\omega) $ is expressed in terms of Whittaker functions and gamma functions, with $ \mathcal{G}_{k=0}^{(T)}(\omega) \sim -1 + 2\nu \mathcal{G}_0^{(T)}(\omega) $ near $ \nu \to 0 $, confirming the bifurcation behavior at criticality.