SciPost Submission Page
by Daniel Areán, Karl Landsteiner, Ignacio Salazar Landea
This is not the current version.
|As Contributors:||Karl Landsteiner|
|Arxiv Link:||https://arxiv.org/abs/1912.06647v1 (pdf)|
|Date submitted:||2020-03-27 01:00|
|Submitted by:||Landsteiner, Karl|
|Submitted to:||SciPost Physics Core|
Quantum theory can be formulated with certain non-Hermitian Hamiltonians. An anti-linear involution, denoted by PT, is a symmetry of such Hamiltonians. In the PT-symmetric regime the non-Hermitian Hamiltonian is related to a Hermitian one by a Hermitian similarity transformation. We extend the concept of non-Hermitian quantum theory to gauge-gravity duality. Non-Hermiticity is introduced via boundary conditions in asymptotically AdS spacetimes. At zero temperature the PT phase transition is identified as the point at which the solutions cease to be real. Surprisingly for solutions containing black holes real solutions can be found well outside the quasi-Hermitian regime. These backgrounds are however unstable to fluctuations which establishes the persistence of the holographic dual of the PT phase transition at finite temperature.
Submission & Refereeing History
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Anonymous Report 3 on 2020-5-17 Invited Report
- Cite as: Anonymous, Report on arXiv:1912.06647v1, delivered 2020-05-17, doi: 10.21468/SciPost.Report.1694
A simple idea presented in a short paper that is to the point.
It is not clear where future applications or developments may lead for PT symmetry in holography.
It is not clear where future applications or developments may lead for PT symmetry in holography.
This paper revolves around the idea of so-called PT symmetric quantum systems. The paper proposes a first realisation of such a system in holography by employing a bulk dual setup well-known from the study of holographic superconductors, albeit with differing boundary conditions.
To the best of my knowledge PT symmetric quantum systems have not previously been considered in holography and, as such I have enjoyed reading this work.
I would like to ask the authors to address the following questions or suggestions:
1. I have found the way the holographic example of a PT symmetric quantum system is introduced to be a little unclear.
- The relationship between the simple QM example given in the introduction and the actual holographic model is not clear. One might invoke, for example, the crucial role of the SU(2) symmetry in the QM system, which is absent in the holographic setup. Could the authors clarify the relationship between the two models further?
- In the paragraph before the section “holography” the authors seem to insist on the fact that the Lie group that is used be simple. This does not appear obvious from the examples before and after.
- Neither the parameter $\tilde M$ nor the parameter $M$ are defined, nor is their mutual relationship. Are they both real?
- It would appear to me that the crux of their setup lies in the analytic continuation of the boundary conditions, and hence the bulk solutions, such that $\bar\phi$ is no longer the complex conjugate of $\phi$. If this understanding of mine is correct, it might be nice to state this somewhere explicitly, it may even obviate the need to transform between various parameters $\tilde M$, $M$ and $x$, which all encode in some form or another this boundary condition.
2. Some further questions aimed at clarification:
- The simple $T=0$ solutions of domain-wall type are strongly tied to the form of the bulk action they chose, which is not fully dictated by their symmetry considerations. What is the motivation for the action (7) in particular? Could the authors comment on whether they expect similar physics for other choices of the potential?
- After Eq. (12) the authors invoke the equivalence of non-Hermitian and Hermitian theories in the PT-symmetric regime, citing Ref. . However, this point had already been discussed before, and backed up by Refs. [3-5]. Is there something qualitatively different that I am missing?
3. Some suggestions / comments:
- The authors leave the reader with the puzzling fact that it seems like the finite-temperature solutions they have found can be extended beyond the expected range of $x<1$, but appear to have an instability beyond $x=1$. It would be useful to address such puzzles in a top-down setting (and the authors have helpfully cited some such examples). In those settings one can hope to be more precise on the exact nature of the dual (i.e. boundary theory), and how the PT symmetry and its breaking are realised.
- I would have enjoyed seeing a bit more on motivations as to why the authors are constructing PT symmetric systems in holography. They mention some relations to Goldstone mode physics in the conclusions, however it is not clear to me what a holographic model would bring to this study. Perhaps the authors could clarify this, and at the same time elaborate a little more on the more general perspective the wish to take on PT symmetry in holography.
Anonymous Report 2 on 2020-4-20 Invited Report
- Cite as: Anonymous, Report on arXiv:1912.06647v1, delivered 2020-04-20, doi: 10.21468/SciPost.Report.1630
I want to thaks the authors for their explanations, these are my comments to their reply:
1) I guess my previous comment was not clear enough, I think readers would benefit if a discussion about PT in the holographic model (along the lines in the author's reply) is introduced in the paper. Regarding the reply, in the end I did not understand what the action of T on $(a,b)$ should be when they take complex values. From their comments it looked like they should be invariant, but that would not be consistent with complex conjugation of the metric when this last one takes complex values, as all of them are real in the ordinary setup.
2) Equation (19) does not involve metric fluctuations and when x=1, the only solution for the gauge field is trivial. So it would look like the fluctuation at that point is purely coming from the scalar field and neither the metric nor the gauge field are involved. So I still do not understand the comment about the diffussion mode, or Figure 3 does not represent the solutions of (19)?
3) The first order transition could happen before the system becomes locally unstable, in this case for|x|<1. It is not mandatory that there are two cycles, the relevant topology change concerns only the Euclidean time direction. One may consider a Hawking-Page transition for a planar black hole, for instance the AdS black brane versus the thermal AdS solution, with the thermal AdS solution being the same as the zero temperature solution with a compactified Euclidean time direction, a similar situation could be possible here with the zero temperature solution (15). Whether there is a transition or not would depend on the values of the free energies for each solution, in the AdS case the black brane always has lower free energy, but maybe this case is different.
4) The $q$ factor in In equation (19) I was referring to would be in the term that mixes the background scalar $\psi$ with the gauge field $a_t$ in the first line of the first equation, not the one the authors point out.
Anonymous Report 1 on 2020-4-10 Invited Report
- Cite as: Anonymous, Report on arXiv:1912.06647v1, delivered 2020-04-10, doi: 10.21468/SciPost.Report.1619
This paper presents a possible gravity dual realization of a non- Hermitian, ''PT symmetric'' strongly coupled quantum field theory. Non-Hermitian PT symmetric theories have been studied in the context of dissipative and open systems, as well as formal developments of quantum mechanics.
By analogy with a simple quantum mechanical system it is proposed that a holographic non-Hermitian model can be obtained from a Hermitian model by a complexified symmetry transformation. The concrete model the authors present is a complex scalar field coupled to gravity and a Maxwell field, with non-Hermitian boundary conditions for the scalar, which would map to non-Hermitian couplings in the field theory dual. Among the possible boundary conditions, the authors identify a ''quasi-Hermitian'' subset that can be obtained from Hermitian conditions through a complexified global $U(1)$ rotation acting on the complex scalar. They construct zero and finite temperature solutions of the gravity model and find that stable real solutions only exist for quasi-Hermitian boundary conditions.
To my knowledge, there have been no previous works attempting to describe non-Hermitian systems in the context of gauge/gravity duality, this is a new an interesting application and certainly deserves to be published. There is maybe some room for improvement, some minor comments/questions I have are the following:
1) The authors identify the quasi-Hermitian regime with a PT-symmetric regime, and otherwise they mention that PT would be broken. It would be worthwhile to specify exactly what PT transformations are in the holographic model and how they are broken in one case but not the other.
2) In the paragraph below Eq. (20) it looks like the authors are saying that at $x=1$ the lowest QNM they are following becomes the hydrodynamic mode for charge diffusion. This sounds a bit strange, from (19) it looks like at at $x=1$ the fluctuation for the gauge field should be set to zero and the mode is purely a fluctuation of the scalar, that will be decoupled from the metric and the gauge field (the charge of the scalar is effectively zero).
3) For $x>1$ the authors find that real finite temperature solutions become unstable and were not able to find other numerical solutions. I was wondering if a transition of Hawking-Page type would be possible to the thermal version of the zero temperature solutions they found previously. By thermal version I mean that in the analytic continuation to Euclidean signature they would have a compact time direction of finite length, corresponding to the inverse temperature.
4) Some possible typos: below equation (15), in the paragraph starting ''The IR boundary conditions (16)..'' do the authors actually mean the previous equation, (15)?
In equation (19) it looks like there is a factor $q$ missing in the first equation in the term proportional to the gauge field.