Ground states of holographic superconductors

TL;DR

This work analyzes the zero-temperature ground states of the Abelian Higgs model in AdS_4 under finite chemical potential, focusing on whether the infrared (IR) physics flows to emergent conformal symmetry or Lifshitz-like scaling. By studying IR asymptotics around candidate AdS_4_IR fixed points and constructing Lifshitz domain walls, the authors derive precise criteria: if the dimension Δ_Φ of the current dual to the time component satisfies Δ_Φ > 3 (equivalently q L_IR ψ_IR > 1), a conformal IR is possible via charged domain walls; otherwise, Lifshitz scaling governs the IR, with the positive-mass quadratic potential yielding a continuous Lifshitz family with explicit relations q^2 = z m^2/[2(z-1)], ψ_0, and L_0. For the W-shaped quartic potential, Lifshitz IR behavior arises in a broader parameter regime, and a detailed phase diagram shows when AdS_4-to-AdS_4 versus AdS_4-to-Lifshitz flows occur, including oscillatory approaches to Lifshitz fixed points. The findings illuminate how a conserved current acquiring an anomalous dimension through condensation can drive diverse IR dynamics in holographic superconductors, and they raise open questions about stability and ground-state energetics that warrant further numerical and analytical work.

Abstract

We investigate the ground states of the Abelian Higgs model in AdS_4 with various choices of parameters, and with no deformations in the ultraviolet other than a chemical potential for the electric charge under the Abelian gauge field. For W-shaped potentials with symmetry-breaking minima, an analysis of infrared asymptotics suggests that the ground state has emergent conformal symmetry in the infrared when the charge of the complex scalar is large enough. But when this charge is too small, the likeliest ground state has Lifshitz-like scaling in the infrared. For positive mass quadratic potentials, Lifshitz-like scaling is the only possible infrared behavior for constant nonzero values of the scalar. The approach to Lifshitz-like scaling is shown in many cases to be oscillatory.

Figures (7)

• Figure 1: (COLOR ONLINE) The number of irrelevant perturbations to the Lifshitz solution for a quadratic potential, as a function of $z>1$ and $m^2 L^2 > 0$. The two powers $\beta_\psi(1,1)$ and $\beta_\psi(1,-1)$ that characterize infrared perturbations away from this solution fall into one of the four categories described in the text. The four categories meet at the point $(z, m^2L^2)=(2, 0)$. Point $A$ corresponds to an example flow discussed in the text and displayed in figure \ref{['ColdBH']}.
• Figure 2: (COLOR ONLINE) The blue curves are $\psi$ and $gL_0/r$ for a very cold superconducting black hole based on the positive mass quadratic potential with $qL = 3$ and $m^2 L^2 = 1$. The temperature of this black hole is $T/\mu\approx 2.356\times 10^{-14}$, twelve orders of magnitude lower than the highest temperature at which the Abelian gauge symmetry is broken by $\psi$, $T_c/\mu\approx 0.0864$. The dotted red curves represent near-horizon fits to zero-temperature ansatzes that describe perturbations away from an infrared Lifshitz fixed points.
• Figure 3: (COLOR ONLINE) The behavior of $\Delta_{\Phi}$ as a function of $z$ for the quartic potential (\ref{['VChoice']}). The vertical purple line reminds us that for $\Delta_{\Phi} > 3$, domain walls with emergent conformal symmetry are allowed. The various curves show how $\Delta_{\Phi}$ behaves as a function of $z$ in backgrounds with Lifshitz scaling. Each curve corresponds to a definite value of the rescaled quartic coupling $\tilde{u}$. For $\tilde{u} < 1$, $z_m$ is the value of $z$ where $\Delta_{\Phi}$ is maximized.
• Figure 4: (COLOR ONLINE) When there is a Lifshitz solution at a given point $(z, \Delta_{\Phi})$, the two powers $\beta_{\psi}(1,1)$ and $\beta_{\psi}(1,-1)$ that characterize perturbations away from this solution in the infrared fall into one of the five categories described in the text and summarized briefly in the legend. In the plot above, we have taken $m^2=-2$ and $L=1$. The detail on the right shows that the five categories meet at the point $(z, \Delta_{\Phi})\approx (1.715, 3.061)$.
• Figure 5: (COLOR ONLINE) A flow between two conformal fixed points.