Towards the Lattice Effects on the Holographic Superconductor

TL;DR

This work investigates how a lattice background affects the zero-frequency response in holographic superconductors by analyzing a toy bulk model with a massive U(1) gauge field. Introducing a periodic lattice potential $V(u,x)=rac{1}{L^2}ig(V_0(u)+\epsilon\delta V(u)\cos qx\big)$ in the probe limit, the authors compute the AC conductivity and perturbative corrections $\sigma^{(0)}$, $\sigma^{(1)}$, and $\overline{\sigma}^{(2)}$; they find a robust zero-frequency delta function peak in Im[$\sigma$] driven by translational invariance, whose weight is reduced by lattice effects and decreases with lattice wavenumber $q$ (e.g., $\tilde{C}(q)$ grows with $q$ for the first-order term, while $C^{(2)}(q)<0$ and decreases with $q$ for the averaged second-order term). The results suggest the superfluid component remains under lattice perturbations in this toy setting, though backreaction and dynamical scalar effects are expected to modify the quantitative behavior. The study highlights the interplay between translational symmetry breaking and U(1) breaking in holographic contexts and points to future work including gravity backreaction and full scalar dynamics to assess nonperturbative lattice effects.

Abstract

We study the lattice effects on the simple holographic toy model; massive U(1) gauge theory for the bulk action. The mass term is for the U(1) gauge symmetry breaking in the bulk. Without the lattice, the AC conductivity of this model shows similar results to the holographic superconductor with the energy gap. On this model, we introduce the lattice effects, which induce the periodic potential and break the translational invariance of the boundary field theory. Without the lattice, due to the translational invariance and the mass term, there is a delta function peak at zero frequency on the AC conductivity. We study how this delta function peak is influenced by the lattice effects, which we introduce perturbatively. In the probe limit, we evaluate the perturbative corrections to the conductivities at very small frequency limit. We find that the delta function peak remains, even after the lattice effects are introduced, although its weight reduces perturbatively. We also study the lattice wavenumber dependence of this weight. Our result suggests that in the U(1) symmetry breaking phase, the delta function peak is stable against the lattice effects at least perturbatively.

Figures (24)

• Figure 1: Im[$\sigma^{(0)}$] is plotted for various $\omega$ (red) in the case (i). The best fitting curve (blue) is Im$[\sigma^{(0)}(\omega)]\simeq-9.4\times 10^{-4}+3.1/\omega$.
• Figure 2: Re[$\sigma^{(0)}$] is plotted for various $\omega$ in the case (i). There is a delta function peak at $\omega = 0$.
• Figure 3: Im[$\sigma^{(0)}$] is plotted for various $\omega$ (red) in the case (ii). The best fitting curve (blue) is Im$[\sigma^{(0)}(\omega)]\simeq-8.4\times 10^{-5}+3.25/\omega$.
• Figure 4: Re[$\sigma^{(0)}$] is plotted for various $\omega$ in the case (ii). There is a delta function peak at $\omega = 0$.
• Figure 5: (color online) $\hbox{Im}[\sigma(\omega)^{(1)}]$ in the case (i) for $q=2$. The best fitting curve is $\sigma^{(1)}\simeq 9.2\times 10^{-5}+0.41/\omega$