We study the Sakai-Sugimoto model of holographic QCD at zero temperature and finite chemical potential. We find that as the baryon chemical potential is increased above a critical value, there is a phase transition to a nuclear matter phase characterized by a condensate of instantons on the probe D-branes in the string theory dual. As a result of electrostatic interactions between the instantons, this condensate expands towards the UV when the chemical potential is increased, giving a holographic version of the expansion of the Fermi surface. We argue based on properties of instantons that the nuclear matter phase is necessarily inhomogeneous to arbitrarily high density. This suggests an explanation of the "chiral density wave" instability of the quark Fermi surface in large N c QCD at asymptotically large chemical potential. We study properties of the nuclear matter phase as a function of chemical potential beyond the transition and argue in particular that the model can be used to make a semi-quantitative prediction of the binding energy per nucleon for nuclear matter in ordinary QCD.

In [1] a holographic black hole solution is discussed which exhibits a superconductor like transition. In the superconducting phase the black holes show infinite DC conductivity. This gives rise to the possibility of deforming the solutions by turning on a time independent current (supercurrent), without any electric field. This type of deformation does not exist for normal (non-superconducting) black holes, due to the no-hair theorems. In this paper we have studied such a supercurrent solution and the associated phase diagram. Interestingly, we have found a "special point" (critical point) in the phase diagram where the second order superconducting phase transition becomes first order. Supercurrent in superconducting materials is a well studied phenomenon in condensed matter systems. We have found some qualitative agreement with known results.

We show that a D3/D7 system (in the limit of zero quark mass) at finite isospin chemical potential goes through a superconductor (superfluid) like phase transition. This is similar to a flavored superfluid phase studied in the QCD literature, where mesonic operators condense. We have studied the frequency dependent conductivity of the condensate and found a delta function peak in the zero frequency limit. This is an example of superconductivity in a string theory context. Consequently we have found a superfluid/supercurrent type solution and studied the associated phase diagram. The superconducting transition changes from second order to first order at a critical superfluid velocity. We have studied various properties of the superconducting system like superfluid density, energy gap, second sound etc. We investigate the possibility of the isospin chemical potential modifying the embedding of the flavor branes by checking whether the transverse scalars also condense at low temperatures. This however does not seem to be the case.

In this work we discuss the zero temperature limit of a "p-wave" holographic superconductor. The bulk description consists of a non-Abelian SU (2) gauge fields minimally coupled to gravity. We numerically construct the zero temperature solution which is the gravity dual of the superconducting ground state of the "p-wave" holographic superconductors. The solution is a smooth soliton with zero horizon size and shows an emergent conformal symmetry in the IR. We found the expected superconducting behavior. Using the near horizon analysis we show that the system has a "hard gap" for the relevant gauge field fluctuations. At zero temperature the real part of the conductivity is zero for an excitation frequency less than the gap frequency. This is in contrast with what has been observed in similar scalargravity-gauge systems (holographic superconductors). We also discuss the low but finite temperature behavior of our solution.