We study soliton and black hole solutions with scalar hair in AdS in a theory with a Maxwell field and a charged scalar field with double trace boundary conditions, which can trigger the dual boundary theory to become a superconductor. We investigate the phase diagram as a function of the temperature $\textit{T}$ and of the double trace coupling $\textit{κ}$ and we find a rich pattern of phase transitions, which can be of the Hawking-Page kind or can be due to the condensation of the order parameter. We also find a transition between vortex solutions and the zero temperature limit of the black hole for a critical value of the double trace coupling $\textit{κ}$. The Little-Park periodicity is realized for the dual of the black hole solution with hair as a shift in the winding number and in the gauge field.

Italy

Roberto Auzzi

Stefano Bolognesi

Giuseppe Nardelli

Chile

Gianni Tallarita

Japan

Nicolò Zenoni

Solitonic vortices and black holes with vortex hair in AdS

In the framework of the static patch approach to de Sitter holography introduced in [L. Susskind, J. Hologr. Appl. Phys. 1, 1 (2021)], the growth of holographic complexity has a hyperfast behavior, which leads to a divergence in a finite time. This is very different from the anti–de Sitter (AdS) spacetime, where instead the complexity rate asymptotically reaches a constant value. We study holographic volume complexity in a class of asymptotically AdS geometries which include de Sitter bubbles in their interior. With the exception of the static bubble case, the complexity obtained from the volume of the smooth extremal surfaces which are anchored just to the AdS boundary has a similar behavior to the AdS case, because it asymptotically grows linearly with time. The static bubble configuration has a zero complexity rate and corresponds to a discontinuous behavior, which resembles a first order phase transition. If instead we consider extremal surfaces which are anchored at both the AdS boundary and the de Sitter stretched horizon, we find that complexity growth is hyperfast, as in the de Sitter case.

Gabriel Pedde Ungureanu

Volume complexity of dS bubbles

In the presence of a scalar hair perturbation, the Cauchy horizon of a Reissner-Nordström black hole disappears and is replaced by the rapid collapse of the Einstein-Rosen bridge, which leads to a Kasner singularity [1, 2]. We study the time-dependence of holographic complexity, both for the volume and for the action proposals, in a class of models with hairy black holes. Volume complexity can only probe a portion of the black hole interior that remains far away from the Kasner singularity. We provide numerical evidence that the Lloyd bound is satisfied by the volume complexity rate in all the parameter space that we explored. Action complexity can instead probe a portion of the spacetime closer to the singularity. In particular, the complexity rate diverges at the critical time t  c  for which the Wheeler-DeWitt patch touches the singularity. After the critical time the action complexity rate approaches a constant. We find that the Kasner exponent does not directly affect the details of the divergence of the complexity rate at t = t  c  and the late-time behaviour of the complexity. The Lloyd bound is violated by action complexity at finite time, because the complexity rate diverges at t = t  c . We find that the Lloyd bound is satisfied by the asymptotic action complexity rate in all the parameter space that we investigated.

Israel

France

Eliezer Rabinovici

Fidel Schaposnik Massolo

On the time dependence of holographic complexity for charged AdS black holes with scalar hair

The holographic complexity of formation for the AdS3 2-sided Randall-Sundrum model and the AdS3/BCFT2 models is logarithmically divergent according to the volume conjecture, while it is finite using the action proposal. One might be tempted to conclude that the UV divergences of the volume and action conjectures are always different for defects and boundaries in two-dimensional conformal field theories. We show that this is not the case. In fact, in Janus AdS3 we find that both volume and action proposals provide the same kind of logarithmic divergences.

Stefano Baiguera

Denmark

Sara Bonansea

Action complexity in the presence of defects and boundaries

An analytic static monopole solution is found in global AdS4, in the limit of small backreaction. This solution is mapped in Poincaré patch to a falling monopole configuration, which is dual to a local quench triggered by the injection of a condensate. Choosing boundary conditions which are dual to a time-independent Hamiltonian, we find the same functional form of the energy-momentum tensor as the one of a quench dual to a falling black hole. On the contrary, the details of the spread of entanglement entropy are very different from the falling black hole case, where the quench induces always a higher entropy compared to the vacuum, i.e. ∆S &gt; 0. In the propagation of entanglement entropy for the monopole quench, there is instead a competition between a negative contribution to ∆S due to the scalar condensate and a positive one carried by the freely propagating quasiparticles generated by the energy injection.

Belgium

Stefania Caggioli

Maria Martinelli

A falling magnetic monopole as a holographic local quench

We investigate the complexity=volume proposal in the case of Janus AdS3 geometries, both at zero and finite temperature. The leading contribution coming from the Janus interface is a logarithmic divergence, whose coefficient is a function of the dilaton excursion. In the presence of the defect, complexity is no longer topological and becomes temperature-dependent. We also study the time evolution of the extremal volume for the time-dependent Janus BTZ black hole. This background is not dual to an interface but to a pair of entangled CFTs with different values of the couplings. At late times, when the equilibrium is restored, the couplings of the CFTs do not influence the complexity rate. On the contrary, the complexity rate for the out-of-equilibrium system is always smaller compared to the pure BTZ black hole background.

Kristian Toccacelo

Volume complexity for Janus AdS3 geometries

Computational complexity is a quantum information concept that recently has found applications in the holographic understanding of the black hole interior. We consider quantum computational complexity for <math><mi>n</mi></math> qubits using Nielsen’s geometrical approach. In the definition of complexity there is a big amount of arbitrariness due to the choice of the penalty factors, which parametrizes the cost of the elementary computational gates. In order to reproduce desired features in holography, such as ergodicity and exponential maximal complexity for large number of qubits <math><mi>n</mi></math>, negative curvatures are required. With the simplest choice of penalties, this is achieved at the price of singular sectional curvatures in the large <math><mi>n</mi></math> limit. We investigate a choice of penalties in which we can obtain negative curvatures in a smooth way. We also analyze the relation between operator and state complexities, framing the discussion with the language of Riemannian submersions. This provides a direct relation between geodesics and curvatures in the unitaries and the states spaces, which we also exploit to give a closed-form expression for the metric on the states in terms of the one for the operators. Finally, we study conjugate points for a large number of qubits in the unitary space and we provide a strong indication that maximal complexity scales exponentially with the number of qubits in a certain regime of the penalties space.

United States

G. Bruno De Luca

United Kingdom

Andrea Legramandi

Geometry of quantum complexity

We analytically compute subsystem action complexity for a segment in the BTZ black hole background up to the finite term, and we find that it is equal to the sum of a linearly divergent term proportional to the size of the subregion and of a term proportional to the entanglement entropy. This elegant structure does not survive to more complicated geometries: in the case of a two segments subregion in AdS3, complexity has additional finite contributions. We give analytic results for the mutual action complexity of a two segments subregion.

India

Pratim Roy

On subregion action complexity in AdS3 and in the BTZ black hole

We investigate vortex lattice solutions in a holographic superconductor model in asymptotically AdS4 spacetime which includes the gravitational backreaction of the vortex. The circular cell approximation, which is known to give a good result for several physical quantities in the Ginzburg-Landau model, is used. The critical magnetic fields and the magnetization curve are computed. The vortex lattice profiles are compared to expectations from the Abrikosov solution in the regime nearby the upper critical magnetic field H2c for which superconductivity is lost.

The holographic vortex lattice using the circular cell method

We compute the ultraviolet divergences of holographic subregion complexity for the left and right factors of the thermofield double state in warped AdS3 black holes, both for the action and the volume conjectures. Besides the linear divergences, which are also present in the BTZ black hole, additional logarithmic divergences appear. For the action conjecture, these log divergences are not affected by the arbitrarity in the length scale associated with the counterterm needed to ensure reparameterization invariance. We find that the subregion action complexity obeys the superadditivity property for the thermofield double in warped AdS3, independently from the action counterterm coefficient. We study the temperature dependence of subregion complexity at constant angular momentum and we find that it is correlated with the sign of the specific heat.