Inverse anisotropic catalysis and complexity

Abstract In this work, the effect of anisotropy on computational complexity is considered by CA proposal in holographic two-sided black brane dual of a strongly coupled gauge theory. It is shown that due to the confinement–deconfinement phase transition, there are two different behaviors: with an increase in anisotropy, there is an increase in the complexity growth rate in small anisotropy and a decrease in the complexity growth rate in large anisotropy. In the extreme case, very large anisotropy leads to the triviality of the complexity growth rate and the complexity itself, which means that in this case, getting the target state from the reference state is achieved with no effort or the identity of two states. Moreover, we suggest that $$\frac{1}{M}\frac{dC}{dt}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>M</mml:mi> </mml:mfrac> <mml:mfrac> <mml:mrow> <mml:mi>dC</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>dt</mml:mi> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:math> is a better representation of system degrees of freedom rather than the complexity growth rate $$\frac{dC}{dt}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mrow> <mml:mi>dC</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>dt</mml:mi> </mml:mrow> </mml:mfrac> </mml:math> and show that how it is related to inverse anisotropic catalysis. In addition, we consider the one-sided black brane dual to the quantum quench and show that increase in anisotropy comes with decrease in complexity regardless of the anisotropy value which is due to the fact that the system does not experience a phase transition.