Normalizable fermion modes in a holographic superconductor

TL;DR

The paper analyzes charged bulk fermions in a zero-temperature AdS_4 domain-wall superconducting background and demonstrates that normal modes form continuous bands within a compact 'preferred wedge' in the (ω,k) plane, as predicted by a semiclassical Bohr–Sommerfeld construction. Numerical solutions of the Dirac equation confirm multiple bands, with massless fermions typically yielding ungapped bands that cross ω=0, while certain parameter choices produce a gapped band; bands approximately satisfy (ω+qΦ_UV)^2/v_UV^2 − k^2 ≈ m_eff^2. This work highlights a qualitative difference from RNAdS, where isolated ω=0 modes appear, and discusses implications for Fermi surfaces in the dual field theory and possible extensions to embedding in maximal supergravity and higher-dimensional AdS spacetimes. The results provide a framework for understanding bulk fermionic spectra in holographic superconductors and potential connections to zero-temperature condensed-matter phenomena.

Abstract

We consider fermions in a zero-temperature superconducting anti-de Sitter domain wall solution and find continuous bands of normal modes. These bands can be either partially filled or totally empty and gapped. We present a semi-classical argument which approximately captures the main features of the normal mode spectrum.

Figures (5)

• Figure 1: The metric, gauge field and scalar field for the domain wall solution in M-theory.
• Figure 2: The green wedge is where (\ref{['SummaryRelation']}) holds. This is the region where our heuristic geometric optics arguments indicate that one might find fermion normal modes. The grey and black curves are an approximate depiction of the hyperbolas (\ref{['HyperboloidEstimate']}), representing an approximate WKB treatment of where normal modes lie. Only the black parts of the curve correspond to actual normal modes; the grey parts are where normal modes might have been if the green wedge had been larger.
• Figure 3: Fermion normal modes in the $AdS_4$ domain wall for $m=0$ and $q L_{\rm UV}=10$. The black lines mark the boundary of the "preferred wedge" \ref{['SummaryRelation']}. The red lines correspond to normal modes where $u^+_1$ and $u^-_2$ are nonzero (poles in $G_{22}$) while the blue lines correspond to normal modes where the $u^+_2$ and $u^-_1$ are nonzero (poles in $G_{11}$). The gray dot-dashed lines are one parameter fits and the black dashed lines are three parameter fits to hyperbola. The red and blue dots mark the location of the normal modes shown in Fig. \ref{['fig:twomodes']}.
• Figure 4: The wave functions for two fermion normal modes of the $AdS_4$ domain wall for $m=0$ and $q L_{\rm UV}=10$. We note that in the $\kappa_{\rm IR}>0$ region, we can always choose the $u^+_a$ to be purely real, in which case it follows that the $u^-_a$ are purely imaginary. The plot on the left shows the real part of $u^+_1$ (blue) and the imaginary part of $u^-_2$ (green) corresponding to the blue dot in Fig. \ref{['fig:normalmodesqL10']}. The plot on the right shows the real part of $u^+_2$ (red) and the imaginary part of $u^-_1$ (cyan) corresponding to the red dot in Fig. \ref{['fig:normalmodesqL10']}.
• Figure 5: Fermion normal modes of the $AdS_4$ domain wall for $m L_{\rm IR}=1$ and $q L_{\rm UV}=3/2$. The black lines mark the boundary of the "preferred wedge" \ref{['SummaryRelation']}. The blue line corresponds to normal modes where the $u^+_2$ and $u^-_1$ are nonzero (poles in $G_{11}$). Notice it never intersects the $\omega=0$ line (red). So this is a gapped band.