Quantum Complexity of Nonlocal Field Theories

Entanglement entropy for nonlocal field theories displays a universal ``volume law" scaling \cite{Barbon:2008ut, Karczmarek:2013xxa, Shiba:2013jja, Pang:2014tpa} as opposed to the ``area law" scaling for local field theories. The aim of this work is to determine whether complexity displays any such an universal scaling laws. The field theories considered here are obtained by deforming $\mathcal{N}=4$ SYM theory by higher dimension operators introducing nonlocality, namely a dipole deformation and noncommutativity (NCSYM) by turning on world volume Kalb-Ramond $B$ field. The dual gravity backgrounds have a running dilaton, in addition to the $B$-field background, which alter AdS asymptotics. Our results capture nonlocality in the hyperscaling behavior for complexity. We also compute the subregion complexity which display phase transitions in the nonlocal field theories with the transition point being the same as that for the phase transition of entanglement entropy \cite{Karczmarek:2013xxa}. These new results dovetail nicely with our findings from our previous works \cite{Chakraborty:2020fpt, Katoch:2022hdf, Bhattacharyya:2022ren} on other lower dimensional nonlocal field theories such as little string theories (LSTs) and warped conformal field theories (WCFTs).