Abstract
We investigate the dependence of the conductivity and the entanglement entropy on the space-time dimensionality $d$ in two holographic superconductors: one dual to a quantum critical point with spontaneous symmetry breaking, and the other modeled by a charged scalar that condenses at a sufficiently low temperature in the presence of a Maxwell field. In both cases the gravity background is asymptotically Anti de Sitter (AdS). In the large $d$ limit we obtain explicit analytical results for the conductivity at zero temperature and the entanglement entropy by a $1/d$ expansion. We show that the entanglement entropy is always smaller in the broken phase. As dimensionality increases, the entanglement entropy decreases, the coherence peak in the conductivity becomes narrower and the ratio between the energy gap and the critical temperature decreases. These results suggest that the condensate interactions become weaker in high spatial dimensions.
Highlights
We investigate the dependence of the conductivity and the entanglement entropy on the space-time dimensionality d in two holographic superconductors: one dual to a quantum critical point with spontaneous symmetry breaking, and the other modeled by a charged scalar that condenses at a sufficiently low temperature in the presence of a Maxwell field
The entanglement entropy decreases, the coherence peak in the conductivity becomes narrower and the ratio between the energy gap and the critical temperature decreases. These results suggest that the condensate interactions become weaker in high spatial dimensions
As dimensionality increases the condensation of the scalar occurs always close to the horizon as the gravitational effects of the black hole are only important in this region
Summary
We compute numerically the electrical conductivity in the probe limit for 3, 4, 5, 7 and 9 dimensions of the dual boundary theory and for two scalar masses m2 = 0, d + 1. A tentative explanation of this behaviour in the gravity dual is that [15, 16] as the dimensionality increases the condensation of the scalar gradually occurs closer to the horizon which corresponds to the less strongly interacting limit of the dual field theory. The ratio decreases monotonically as d increases and it is likely to converge to a finite value in the d → ∞ limit still above the prediction ∼ 3.528 of the Bardeen-Cooper-Schrieffer (BCS) theory of weakly coupled superconductors.
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