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Register a new userHolographic complexity of the disk subregion in (2+1)-dimensional gapped systems
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Using the volume of the space enclosed by the Ryu-Takayanagi (RT) surface, we study the complexity of the disk-shape subregion (with radius R) in various (2+1)-dimensional gapped systems with gravity dual. These systems include a class of toy models with singular IR and the bottom-up models for quantum chromodynamics and fractional quantum Hall effects. Two main results are: i) in the large-R expansion of the complexity, the R-linear term is always absent, similar to the absence of topological entanglement entropy; ii) when the entanglement entropy exhibits the classic `swallowtail phase transition, the complexity is sensitive but reacts differently.
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We analyze the holographic subregion complexity in a $3d$ black hole with the vector hair. This $3d$ black hole is dual to a $1+1$ dimensional $p$-wave superconductor. We probe the black hole by changing the size of the interval and by fixing $q$ or $T$. We show that the universal part is finite across the superconductor phase transition and has competitive behaviors different from the finite part of entanglement entropy. The behavior of the subregion complexity depends on the gravitational coupling constant divided by the gauge coupling constant. When this ratio is less than the critical value, the subregion complexity increases as temperature becomes low. This behavior is similar to the one of the holographic $1+1$ dimensional $s$-wave superconductor arXiv:1704.00557. When the ratio is larger than the critical value, the subregion complexity has a non-monotonic behavior as a function of $q$ or $T$. We also find a discontinuous jump of the subregion complexity as a function of the size of the interval. The subregion complexity has the maximum when it wraps the almost entire spatial circle. Due to competitive behaviors between normal and condensed phases, the universal term in the condensed phase becomes even smaller than that of the normal phase by probing the black hole horizon at a large interval. It implies that the formed condensate decreases the subregion complexity like the case of the entanglement entropy.
We investigate general features of the evolution of holographic subregion complexity (HSC) on Vaidya-AdS metric with a general form. The spacetime is dual to a sudden quench process in quantum system and HSC is a measure of the ``difference between two mixed states. Based on the subregion CV (Complexity equals Volume) conjecture and in the large size limit, we extract out three distinct stages during the evolution of HSC: the stage of linear growth at the early time, the stage of linear growth with a slightly small rate during the intermediate time and the stage of linear decrease at the late time. The growth rates of the first two stages are compared with the Lloyd bound. We find that with some choices of certain parameter, the Lloyd bound is always saturated at the early time, while at the intermediate stage, the growth rate is always less than the Lloyd bound. Moreover, the fact that the behavior of CV conjecture and its version of the subregion in Vaidya spacetime implies that they are different even in the large size limit.
We present the results of our computation of the subregion complexity and also compare it with the entanglement entropy of a $2+1$--dimensional holographic superconductor which has a fully backreacted gravity dual with a stable ground sate. We follow the complexity equals volume or the CV conjecture. We find that there is only a single divergence for a strip entangling surface and the complexity grows linearly with the large strip width. During the normal phase the complexity increases with decreasing temperature, but during the superconducting phase it behaves differently depending on the order of phase transition. We also show that the universal term is finite and the phase transition occurs at the same critical temperature as obtained previously from the free energy computation of the system. In one case, we observe multivaluedness in the complexity in the form of an S curve.
We numerically investigate the evolution of the holographic subregion complexity during a quench process in Einstein-Born-Infeld theory. Based on the subregion CV conjecture, we argue that the subregion complexity can be treated as a probe to explore the interior of the black hole. The effects of the nonlinear parameter and the charge on the evolution of the holographic subregion complexity are also investigated. When the charge is sufficiently large, it not only changes the evolution pattern of the subregion complexity, but also washes out the second stage featured by linear growth.
We construct renormalized holographic entanglement entropy (HEE) and subregion complexity (HSC) in the CV conjecture for asymptotically AdS$_4$ and AdS$_5$ geometries under relevant perturbations. Using the holographic renormalization method developed in the gauge/gravity duality, we obtain counter terms which are invariant under coordinate choices. We explicitly define different forms of renormalized HEE and HSC, according to conformal dimensions of relevant operators in the $d=3$ and $d=4$ dual field theories. We use a general embedding for arbitrary entangling subregions and showed that any choice of the coordinate system gives the same form of the counter terms, since they are written in terms of curvature invariants and scalar fields on the boundaries. We show an explicit example of our general procedure. Intriguingly, we find that a divergent term of the HSC in the asymptotically AdS$_5$ geometry under relevant perturbations with operators of conformal dimensions in the range $0< Delta < frac{1}{2},, {rm and} ,, frac{7}{2}< Delta < 4$ cannot be cancelled out by adding any coordinate invariant counter term. This implies that the HSCs in these ranges of the conformal dimensions are not renormalizable covariantly.
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