Generalized Volume-Complexity For Two-Sided Hyperscaling Violating Black Branes

In this paper, we investigate generalized volume-complexity $\mathcal{C}_{\rm gen}$ for a two-sided uncharged HV black brane in $d+2$ dimensions. This quantity which was recently introduced in [arXiv:2111.02429], is an extension of volume in the Complexity=Volume (CV) proposal, by adding higher curvature corrections with a coupling constant $\lambda$ to the volume functional. We numerically calculate the growth rate of $\mathcal{C}_{\rm gen}$ for different values of the hyperscaling violation exponent $\theta$ and dynamical exponent $z$. It is observed that $\mathcal{C}_{\rm gen}$ always grows linearly at late times provided that we choose $\lambda$ properly. Moreover, it approaches its late time value from below. For the case $\lambda=0$, we find an analytic expression for the late time growth rate for arbitrary values of $\theta$ and $z$. However, for $\lambda \neq 0$, the late time growth rate can only be calculated analytically for some specific values of $\theta$ and $z$. We also examine the dependence of the growth rate on $d$, $\theta$, $z$ and $\lambda$. Furthermore, we calculate the complexity of formation obtained from volume-complexity and show that it is not UV divergent. We also examine its dependence on the thermal entropy and temperature of the black brane. At the end, we also numerically calculate the growth rate of $\mathcal{C}_{\rm gen}$ for the case where the higher curvature corrections are a linear combination of the Ricci scalar, square of the Ricci tensor and square of the Riemann tensor. We show that for appropriate values of the coupling constants, the late time growth rate is again linear.