SciPost Submission Page
Diving into a holographic superconductor
by Sean A. Hartnoll, Gary T. Horowitz, Jorrit Kruthoff, Jorge E. Santos
This is not the current version.
Submission summary
As Contributors:  Jorrit Kruthoff 
Arxiv Link:  https://arxiv.org/abs/2008.12786v2 (pdf) 
Date submitted:  20200928 05:03 
Submitted by:  Kruthoff, Jorrit 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Charged black holes in antide Sitter space become unstable to forming charged scalar hair at low temperatures $T < T_\text{c}$. This phenomenon is a holographic realization of superconductivity. We look inside the horizon of these holographic superconductors and find intricate dynamical behavior. The spacetime ends at a spacelike Kasner singularity, and there is no Cauchy horizon. Before reaching the singularity, there are several intermediate regimes which we study both analytically and numerically. These include strong Josephson oscillations in the condensate and possible 'Kasner inversions' in which after many efolds of expansion, the EinsteinRosen bridge contracts towards the singularity. Due to the Josephson oscillations, the number of Kasner inversions depends very sensitively on $T$, and diverges at a discrete set of temperatures $\{T_n\}$ that accumulate at $T_c$. Near these $T_n$, the final Kasner exponent exhibits fractallike behavior.
Current status:
Submission & Refereeing History
You are currently on this page
Reports on this Submission
Anonymous Report 3 on 20201027 Invited Report
Strengths
see report
Weaknesses
see report
Report
This manuscript studies the interior of planar AdS black hole after the complex scalar field begins to condensate. It finds there is no inner horizon. It also observes the Josephson oscillations and the Kasner behavior of singularity according to numerical computations and semianalytical analyses.
Basically, I think results in this manuscript are interesting and should be published but I feel there are still a few of aspects that should be improved. I call the analytical arguments of this manuscript to be`` semianalytical analyses’’ because the authors made a few of addition assumptions without enough arguments or proofs. The authors used "posteriori’’ to support the validity of these assumptions. However, I feel this has suspicion of circularity.
In the Sec.4.1, authors claimed that the mass terms of scalar field and charged terms of Maxwell term are neglectable when $T$ is closed to $T_c$, then they find that Eqs (6)(9) are simplified into Eqs.(12)(15). It is clear that the mass term in Eq (9) is neglectable as scalar field is small. However, at least for me, it is completely difficult to understand why we can do that in other three equations.
For example, the Eq. (7) is only a linear equation of $\phi$, I do not understand why the mass can be dropped. Particularly, from Eq. (6) it seems that $\Phi/f$ approaches to finite when we come close to the wouldbe inner horizon. Thus, the $\Phi^2/f$ term will approach to zero. If these is true, we would obtain a conclusion that, in Eq. (13), the $\Phi$ term should be dropped but the mass term should be kept.
The similar puzzlement also appears when the authors obtained Eqs. (28) and (29).
There is also another place which confuses me. In Sec. 4.5, authors use equation (12) (15) to obtain Eq. (28) and (29). Then they discussed lower temperature case, i.e. $T\ll T_c$. But in Sec. 4.1, authors said “At temperatures close to Tc…. At that point there are…”. If my understanding is correct, the equations (12)(15) is obtained in the case when $T$ is closed to $T_c$. Why can they be used when $T$ is much smaller than $T_c$？
In addition, between Eqs. (35) and (37), I found, if I do not make mistake,
$p_x=2/(3+\alpha^2)$ and $p_\phi=2\alpha/(3+\alpha^2)$
Then $p_\phi^2+p_t^2+2p_x^2\neq1$.
A typo may be in the part between Eqs. (35) and (37).
To summary, I suggest the authors try to give more arguments and explanations on their assumptions of analytical analyses. I would be happy to recommend the publication after above issues are responded suitably in their revised version.
Anonymous Report 2 on 20201024 Invited Report
Report
The manuscript “Diving into a holographic superconductor” by S. Hartnoll, et.al. studies the geometry of black hole interior for classical AdS EinsteinMaxwellcharged scalar gravity at low temperatures $T<T_c$ when a nonzero scalar field is favored. Using both analytic analysis and numerical calculations they found that no inner horizon exists and along the timelike coordinate inside the black hole, the geometry would pass through different intermediate regimes, including “collapse of EinsteinRosen bridge”, “Josephson oscillations”, the Kasner universe and finally reaches a spacelike Kasner singularity.
The geometry inside the black hole horizon is currently being paid attention to, which might be related to quantum information of quantum fields outside the black hole horizon. This work provides an intuitive example of the black hole interior in the classical regime. The physics described in the manuscript is clear and the picture is easy to follow by the readers. The analytic analysis is further confirmed by numerics which should be sound and trustable. I am only a little confused by certain analytic arguments about what terms could be ignored and what terms should be important. I list several of the confusing arguments as examples in the following.
1) In the second paragraph of section 4.2, $\delta z$ dependence in functions of $f$, $\chi$, etc. is kept but in the equations is not kept, is it possible to have a simple explanation why $\delta z$ dependence in the parameters of the equations could be ignored and why $\delta z$ dependence in the functions have to be kept?
2) Under (17), what is the value of $g_{tt}$ at $z=z_0$?
3) On the top of page 10, it is argued that $\Phi’$ could be ignored because $\delta \Phi \to 0$ as $\delta z \to 0$, however we need $\delta \phi/\delta z\to 0$ for $\Phi’$ to be ignored.
4) In the first paragraph of section 4.2, why is this called the “collapse of the EinsteinRosen bridge”?
I do not doubt that these arguments should be correct as they have been further confirmed by numerics. Thus I suggest the authors elaborate a little more on the explanations for these arguments and make them more clear and detailed. I recommend the manuscript to be published after these modifications to the text.
Anonymous Report 1 on 2020102 Invited Report
Strengths
See report
Weaknesses
See report
Report
This paper fills an important gap in the holographic realization of superconductivity, by systematically studying the dynamics of the relevant fields in the dual black hole interior. The authors show that the charged scalar field eliminates the inner Cauchy horizon. Hence the singularity inside the black hole is spacelike and a BKLlike analysis applies.
The analysis of the EinsteinMaxwellscalar system near a spacelike singularity by BKL techniques has a long history, and the dynamical features near the spacelike singularity uncovered by the authors confirms to some extent old results (see note below). But the holographic understanding is certainly original and extremely interesting, so that I recommend publication.
I would just suggest that the authors read and perhaps take into account the information in the note below. Whatever they choose to do, however, I think that the paper can be accepted and I do not need to see the manuscript again.
Note: Rigorous analytical results on the “subcritical” EinsteinMaxwellscalar system can be found in Commun.Math.Phys. 218 (2001) 479511 (grqc/0001047) and Annales Henri Poincare 3 (2002) 10491111 (grqc/0202069). Information on the generic approach to a final Kasner regime was given earlier in Phys. Lett. B175 ( 1986) 129132 (being “generic”, the approach does not exclude an infinite number of oscillations (“mixmaster type”) for conditions of “zero measure”).
Requested changes
See report