Non-analyticity in Holographic Complexity near Critical points

The region near a critical point is studied using holographic models of second-order phase transitions. In a previous paper, we argued that the quantum circuit complexity of the vacuum ($C_0$) is the largest at the critical point. When deforming away from the critical point by a term $\int d^d x \, \tau \, O_\Delta$ the complexity $C(\tau)$ has a piece non-analytic in $\tau$, namely $C_0 -C(\tau) \sim |\tau-\tau_c|^{\nu(d-1)} + \mathrm{analytic} $. Here, as usual, $\nu=\frac{1}{d-\Delta}$ and $\xi$ is the correlation length $\xi\sim |\tau-\tau_c|^{-\nu}$ and there are possible logarithmic corrections to this expression. That was derived using numerical results for the Bose-Hubbard model and general scaling considerations. In this paper, we show that the same is valid in the case of holographic complexity providing evidence that the results are universal, and at the same time providing evidence for holographic computations of complexity.