Circuit complexity near critical points

We consider the Bose-Hubbard model in two and three spatial dimensions numerically compute quantum circuit complexity of ground state Mott insulator superfluid phases using a mean field approximation with additional quadratic fluctuations. After mapping to qubit system, result is given by associated Bogoliubov transformation applied reference taken be state. In particular, has peaks at $O(2)$ critical points where system can described relativistic theory. Given that we use gaussian approximation, near criticality numerical results agree free theory calculation. To go beyond general scaling arguments imply that, as approach point $t\rightarrow t_c$, there non-analytic behavior $c_2(t)$ form $|c_2(t) - c_2(t_c)| \sim |t-t_c|^{\nu d}$, up possible logarithmic corrections. Here $d$ number $\nu$ usual exponent for correlation length $\xi\sim|t-t_c|^{-\nu}$. As check, $d=2$ this agrees computation if $\nu=\frac{1}{2}$. Finally, AdS/CFT methods, study higher dimensional examples confirm argument non-gaussian strongly interacting theories have gravity dual.