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Non-Hermitian Holographic Flows to Little Rip Cosmologies

T0 review · 2 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read The paper claims that inside a holographic black hole dual to a non-Hermitian PT-symmetric QFT, the deep interior is an isotropic Little Rip cosmology, not a Kasner universe.

desk verdict A genuinely new non-Kasner Little Rip interior in a non-Hermitian holographic model, with strong numerics and one unproven branch choice carrying the central claim. read the letter →

arxiv 2607.14284 v1 pith:AHNAHDSS submitted 2026-07-15 hep-th gr-qcquant-ph

Daniel Arean , David Garcia-Fariña , Juan F. Pedraza This is my paper · ORCID
classification hep-thgr-qcquant-ph PACS 04.70.-s11.25.Tq98.80.-k
keywords non-HermitianholographyPTsymmetryblackholeinteriorKasnersingularityLittleRipcosmologynullenergyconditionFLRWholographiccorrelators
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that inside a holographic black hole dual to a non-Hermitian PT-symmetric quantum field theory, the deep interior evades the standard Kasner picture. In the PT-restored phase, where the null energy condition is violated, the geometry approaches an isotropic FLRW cosmology with scale factor a(τ)=exp(exp(ατ)) with α=4√(2v/3) — a Little Rip that expands super-exponentially and blows up only at infinite proper time. The authors show that this interior leaves a distinguishing imprint on two-sided heavy-operator correlators: the renormalized geodesic length falls as an inverse logarithm rather than a power law, so it can be told apart from a Kasner interior. If correct, this is the first concrete holographic realization of a Little Rip cosmology and a step toward a Little Rip/CFT correspondence.

What carries the argument

Central mechanism: an analytic continuation across the non-analytic surface z=z_σ. Using u=√(z_σ-z), the metric is smooth at u=0; the paper proposes to continue to u<0 (flipping the square-root sign), producing a real new patch. In that patch the near-singularity solution has f∝(log z̃)^2 and ψ∝√(-log z̃), which in comoving time gives the scale factor a(τ)=exp(exp(ατ)). The observable probe is the renormalized length of spacelike two-sided heavy-operator geodesics, whose large-energy limit is governed by the near-singularity region and yields L_ren(E)~ (π/(2α)) log(E/√g0) — a non-algebraic decay slower than any power law.

What would settle it

Numerically evolve the full bulk equations through z=z_σ along the complex branch and compute the resulting two-sided heavy-operator correlator; if the renormalized geodesic length decays as a power law in E rather than as 1/log(E), the real-patch Little Rip is not the physical interior.

Watch

Extended reading notes

Core claim

The central discovery is a new black hole interior in holography: in the real, PT-restored phase (1<|η|<η_c, v>0), the metric becomes non-analytic at a surface z=z_σ but can be continued through it to a real second patch. In that patch, near the endpoint z̃→0, the geometry is ds^2=-dτ^2+exp(2exp(ατ))(g0 dt^2+dx^2), α=4√(2v/3): a spatially flat FLRW universe with super-accelerated expansion. This is a Little Rip: curvature invariants diverge only as τ→∞, so the spacetime is geodesically complete. The paper derives the near-singularity analytic forms (g→-g0, f∝-(log z̃)^2, ψ∝√(-log z̃)) and the effective-fluid equation of state p=-ϵ-2α√(ϵ/3), and shows the same phenomenon with a(τ)=exp(ατ^2) f

Load-bearing premise

The physical continuation across the non-analytic surface z=z_σ is assumed by hand: the paper chooses the real branch (flipping the square-root sign) over the complex one, motivated by the TFD interpretation, but this choice is not derived from the equations of motion; if the complex branch were the physical one, the Little Rip geometry would not be the endpoint.

Editorial extensions

If this is right

  • In the PT-restored phase of this model, black hole interiors can be isotropic and super-accelerated rather than Kasner, so the BKL/Kasner paradigm is not universal once the null energy condition is violated.
  • The Little Rip interior is geodesically complete: the curvature blow-up is only asymptotic, occurring at infinite proper time.
  • Two-sided heavy-operator correlators distinguish the Little Rip from a Kasner interior: the renormalized geodesic length decays as ~1/log(E) rather than as a power of E.
  • The construction gives a concrete holographic realization of a Little Rip cosmology hidden behind a horizon, laying groundwork for a Little Rip/CFT correspondence.
  • For a purely quadratic scalar potential the same non-Kasner Little Rip behavior appears with scale factor a(τ)=exp(ατ^2), showing the asymptotics depend on the scalar potential.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If this interior is generic for holographic models with controlled NEC violation, boundary correlators could serve as a laboratory for probing eternal expansion regimes that are otherwise inaccessible.
  • The potential dependence of the scale factor suggests a wider family of 'generalized Little Rip' interiors (e.g., a(τ)~exp(τ^β)) could be engineered, each with its own boundary signature; computing the corresponding geodesic lengths would be a direct extension.
  • The existence of a real, geodesically complete interior in a theory that violates the NEC may inform debates about cosmic censorship and singularity resolution in asymptotically AdS spacetimes, though the paper does not address black hole formation.
  • The inverse-log decay of L_ren(E) is sharp enough that a lattice or tensor-network simulation of the boundary non-Hermitian theory might test the correspondence numerically.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies black hole interiors in the non-Hermitian PT-symmetric holographic model of [40]. In the PT-restored phase (1<|eta|<eta_c, v>0), the numerically constructed interior reaches a regular surface z=z_sigma where the metric fields have a half-integer expansion in sqrt(z_sigma-z). The authors propose to continue through this surface into a second real patch (u<0), and find that as the new radial coordinate tends to zero the geometry becomes a spatially flat FLRW universe with scale factor a(tau)=exp(exp(alpha tau)), alpha=4 sqrt(2v/3). They interpret this as a Little Rip cosmology, supported by NEC-violating matter, with curvature blow-up only at tau->infinity and hence geodesic completeness. They further compute two-sided heavy-operator geodesics and claim that the singularity-induced renormalized length behaves as L_sing(E) ~ pi/(2 alpha log(E/sqrt(g0))), in contrast to the power-law behavior of Kasner interiors. Appendix B repeats the analysis for v=0, where a(tau)=exp(alpha tau^2). The paper includes extensive numerical checks of the near-singularity expansions down to extremely small scales.

Significance. If the proposed continuation is accepted, the paper's central claim is significant: it provides a concrete holographic model in which the deep interior does not follow the BKL/Kasner paradigm, despite being a real, Lorentzian, black-hole interior. The near-singularity expansions (5.1)-(5.4) and (B.1)-(B.4) are explicit and match the numerics over an enormous dynamic range, with the leading coefficients fixed by m^2, v, and eta; this is a strong internal check. The identification of a Little Rip endpoint and the derivation of a non-algebraic geodesic signature are also conceptually novel. However, the result is conditional on a branch choice made in section 4.2, and the printed derivation of the inverse-log geodesic behavior contains inconsistencies. These issues must be resolved before the main claims can be regarded as firmly established.

major comments (2)
  1. [§4.2, Eq. (4.6)] The physical continuation into the u<0 patch is introduced with the words 'we propose', not derived from the equations of motion or from a holographic selection rule. The surface u=0 is regular, but the initial data at u=0 do not by themselves single out the real u<0 branch among possible real or complex extensions. The arguments based on real geodesics and on the TFD/Schwinger-Keldysh interpretation are consistency conditions, not a uniqueness proof. Since the Little Rip geometry (5.5) and the geodesic signature (5.16) are properties of the u<0 patch, the paper's headline claim is conditional on this branch choice. Please either (i) derive the branch from the holographic dictionary, e.g. from the real-time/Schwinger-Keldysh contour or from the requirement of a real maximal analytic extension, or (ii) explicitly state the result as conditional and analyze the alternative complex continua
  2. [§5.3, Eqs. (5.15)-(5.16)] As printed, the derivation of L_sing(E) is not reproducible. After the rescaling u=tilde z/tilde z_*, the integrand obtained from (5.15) contains an additional factor of 1/u relative to what is written (the combination of the Jacobian and the 1/tilde z in the denominator gives 1/u^2 before the square-root is rewritten). Moreover, the limit tilde z_*->0 sends the lower integration limit tilde z_LR/tilde z_* to infinity, so the factorization of 1/log tilde z_* over a finite u-integral is not justified as stated. The sign/branch of the square root inside the horizon also needs to be specified. Please re-derive the large-E limit carefully; the claimed 1/log E signature depends on this calculation.
minor comments (5)
  1. [§1, §6] The statements that 'the renormalized geodesic length ... vanishes slower than any power law' should be clarified: the full renormalized length contains a universal -2 log E UV piece. The non-algebraic behavior is the singularity-induced correction L_sing(E), not the total L_ren(E).
  2. [Throughout] The abbreviation 'FLR W' appears many times; it should be 'FLRW'. In §4.2, 'casted' should be 'cast'.
  3. [§5.3, Eq. (5.16)] The notation in the intermediate step 'integral from 1 to infinity' with sqrt(1/u^2-1) is ambiguous because the integrand is imaginary on that domain. Please specify the contour or the intended real integral, e.g. after a substitution w=1/u.
  4. [Figures 4 and 6] The horizontal axes labeled 'z / z_tilde' are confusing since z and tilde z are coordinates on different patches. Please label them more explicitly, e.g. z/z_sigma and tilde z/z_sigma.
  5. [§4.2, Eq. (4.4)] It would help to state explicitly that g0>0 in the sign convention used, and to note that the sign flip in (4.6) is what makes the metric real in the second patch.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Little Rip interior and its geodesic signature are derived from the EOMs; the u<0 branch choice is an acknowledged assumption, not a circular reduction.

full rationale

The paper's derivation chain is self-contained. The deep-interior metric (5.5) follows from the asymptotic solution (5.1)-(5.3), which the paper states is admitted by the model and verifies numerically; the leading coefficients are fixed by m^2, v, eta, with the only free constant g0 absorbable into a time redefinition. The FLRW form and scale factor a(tau)=exp(exp(alpha tau)) are coordinate re-expressions of that solution, not inputs. The heavy-operator geodesic length L_sing ~ pi/(2 alpha log(E/sqrt(g0))) is then computed from the same geometry by the standard integral (5.15)-(5.16); it is a consequence, not a fitted parameter. The only underived element is the choice to continue through z=z_sigma to the real u<0 patch: the paper explicitly says "we propose that the physical continuation is instead to define a new patch to cover the region u<0" (Section 4.2). This is an acknowledged selection rule / physical assumption and a potential correctness concern, but it is not circular in the sense of equating the prediction to its input: the Little Rip is not used to define the branch, and the branch is not fitted to the claimed observable. Self-citations such as [40] supply the model and phase diagram as prior external work; they are not invoked as a uniqueness theorem and do not force the interior result. Therefore no circular step is exhibited.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central claim rests on: (1) the validity of the non-Hermitian holographic dictionary from [40]; (2) the phase diagram from [42]; (3) the ad hoc choice of real analytic continuation at z_sigma; (4) the ansatz (5.1)-(5.4) checked numerically. Free parameters are limited to numerically determined critical values (eta_c, eta_s) and g0 which is absorbed. No new entities are introduced.

free parameters (3)
  • eta_s (critical eta for end of Kasner behavior in subdominant branch) = approx 1.47 for v=1, T/M=0.5
    Determined numerically from the divergence of Kasner coefficients in figure 3; not derived analytically. This parameter decides which branch of the PT-restored phase has Kasner vs Little Rip interiors.
  • eta_c (critical value where real solutions cease to exist) = approx 1.49 for v=1, T/M=0.5; approx 2.43 for T/M=1
    Quoted from [42] and used to define the PT-broken phase boundary. Numerically determined; no analytic formula given. The branch structure of the paper depends on it.
  • g0 (constant value of -g in the near-singularity regime) = not given numerically; absorbed into time redefinition
    The paper states this is the only free coefficient at leading order and that it can be reabsorbed into the definition of time. It enters the final geodesic-time-shift formula (5.18) as sqrt(g0)*Delta t, so it affects the precise form of the correlator signature, though not the qualitative log behavior.
assumptions (5)
  • domain assumption The standard Gubser-Rocha/EM-scalar holographic model (3.6) is a valid dual to a non-Hermitian PT-symmetric QFT when sources are complexified, as established in [40].
    The entire paper builds on the claim that (3.6) with boundary conditions (3.10)-(3.11) describes the deformation (3.1). This is taken from prior work by overlapping authors; no independent derivation is given.
  • domain assumption Existence and stability classification of the PT-restored branches is taken from [40, 42] (two branches, dominance, critical eta_c).
    Section 3.1 summarizes phase diagram from [42] without re-deriving it. The interior analysis depends on the branch structure: the dominant branch and the subdominant branch for eta > eta_s exhibit the Little Rip.
  • standard math The two-sided heavy-operator correlator is well approximated by e^{-Delta L_ren} in the geodesic approximation for large Delta (eq. (5.11)).
    Standard AdS/CFT geodesic approximation from [87, 88], accepted in the field. Not new.
  • ad hoc to paper The 'physical continuation' across z=z_sigma is the real branch defined by (4.6), i.e., exchanging sqrt(z_sigma - z) for -sqrt(z_sigma - tilde z).
    This is the load-bearing modeling choice discussed in the weakest_assumption field. It is motivated by preserving reality and TFD/Schwinger-Keldysh interpretation, but not derived from Einstein's equations or initial data. §4.2: 'we propose that the physical continuation is instead to define a new patch...'
  • domain assumption At large z (deep interior), the scalar field is real and diverges as sqrt(log) in the PT-restored phase, and the leading behavior is governed by the approximate equations used to derive (5.1)-(5.4).
    This is checked numerically, but the analytic solution (5.1)-(5.3) is an ansatz motivated by the numerics: 'we characterize the singularity by noting that our model admits the following approximate solution...'. It is a justified assumption, but it is an input to the analytic derivation rather than a derivation from first principles alone. It does not include the target result as an input, so the circularity burden is modest.
invented entities (1)
  • No new particles, forces, or dimensions are invented.
    purpose: N/A
    The paper uses only the existing fields of the Gubser-Rocha model and the standard PT-symmetric non-Hermitian deformation. The Little Rip is a geometric phase of the interior, not an entity. Therefore no graviton-problem-style invented entities.

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Pith. "Pith review of Non-Hermitian Holographic Flows to Little Rip Cosmologies." pith.science (2026). https://pith.science/paper/AHNAHDSS

@misc{pith2026260714284,
  author       = {Pith},
  title        = {Pith review of: Non-Hermitian Holographic Flows to Little Rip Cosmologies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHNAHDSS}},
  note         = {Machine review of arXiv:2607.14284}
}
abstract

Spacelike singularities supported by matter satisfying the null energy condition (NEC) are expected to fall within the Belinski--Khalatnikov--Lifshitz (BKL) paradigm. We show that controlled violations of the NEC in holography can lead to different black hole interiors. In a holographic model dual to a strongly coupled non-Hermitian $\mathcal{PT}$-symmetric QFT, we uncover a novel non-Kasner regime within its real, $\mathcal{PT}$-restored phase. The deep-interior geometry describes an isotropic FLRW cosmology undergoing super-accelerated expansion and approaching a Little Rip. This regime leaves a characteristic imprint on two-sided heavy-operator correlators, allowing it to be distinguished from a standard Kasner interior. Our construction provides a concrete holographic realization of a Little Rip cosmology and lays the groundwork for a Little Rip/CFT correspondence, in which such cosmological regimes can be explored through observables in a non-Hermitian quantum field theory.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Critical and multicritical Kasner scaling in holographic phase transitions

    hep-th 2026-08 conditional novelty 7.0 of 10

    The deviation of the first internal Kasner exponent from the Schwarzschild value scales as the square of the condensate, giving temperature exponents 1/r at r-th order multicritical points.

Reference graph

Works this paper leans on

90 extracted references · 79 linked inside Pith · cited by 1 Pith paper

  1. [40]

    Are´ an, K

    D. Are´ an, K. Landsteiner and I. Salazar Landea,Non-hermitian holography,SciPost Phys.9(2020) 032, [1912.06647]

    arXiv 2020

  2. [1]

    C. W. Misner,Mixmaster Universe,Phys. Rev. Lett.22(May, 1969) 1071–1074

    1969

  3. [2]

    V. A. Belinski, I. M. Khalatnikov and E. M. Lifshitz,Oscillatory approach to a singular point in the relativistic cosmology,Adv. Phys.19(1970) 525–573

    1970

  4. [3]

    Lifshitz, I

    E. Lifshitz, I. Lifshitz and I. Khalatnikov,Asymptotic analysis of oscillatory mode of approach to a singularity in homogeneous cosmological models,Sov. Phys. JETP32 (1971) 173

    1971

  5. [4]

    Oling and J

    G. Oling and J. F. Pedraza,Mixmasters in Wonderland: Chaotic dynamics from Carroll limits of gravity,SciPost Phys. Core8(2025) 025, [2409.05836]

    arXiv 2025

  6. [5]

    Damour and M

    T. Damour and M. Henneaux,Chaos in superstring cosmology,Phys. Rev. Lett.85 (2000) 920–923, [hep-th/0003139]

    arXiv 2000

  7. [6]

    Damour and M

    T. Damour and M. Henneaux,E(10), BE(10) and arithmetical chaos in superstring cosmology,Phys. Rev. Lett.86(2001) 4749–4752, [hep-th/0012172]

    arXiv 2001

  8. [7]

    Damour, M

    T. Damour, M. Henneaux and H. Nicolai,Cosmological billiards,Class. Quant. Grav.20(2003) R145–R200, [hep-th/0212256]. – 28 –

    arXiv 2003

Show all 90 references
  1. [8]

    Henneaux, D

    M. Henneaux, D. Persson and P. Spindel,Spacelike Singularities and Hidden Symmetries of Gravity,Living Rev. Rel.11(2008) 1, [0710.1818]

    2008 arXiv

  2. [9]

    Andersson and A

    L. Andersson and A. D. Rendall,Quiescent Cosmological Singularities,Commun. Math. Phys.218(2001) 479–511, [gr-qc/0001047]

    2001 arXiv

  3. [10]

    Damour, M

    T. Damour, M. Henneaux, A. D. Rendall and M. Weaver,Kasner-Like Behaviour for Subcritical Einstein-Matter Systems,Annales Henri Poincare3(2002) 1049–1111, [gr-qc/0202069]

    2002 arXiv

  4. [11]

    Fidkowski, V

    L. Fidkowski, V. E. Hubeny, M. Kleban and S. Shenker,The Black Hole Singularity in AdS/CFT,JHEP02(2004) 014, [hep-th/0306170]

    2004 arXiv

  5. [12]

    Festuccia and H

    G. Festuccia and H. Liu,Excursions beyond the horizon: Black hole singularities in yang–mills theories. i,JHEP04(2006) 044, [hep-th/0506202]

    2006 arXiv

  6. [13]

    Jørstad, R

    E. Jørstad, R. C. Myers and S.-M. Ruan,Complexity=anything: Singularity probes, JHEP07(2023) 223, [2304.05453]

    2023 arXiv

  7. [14]

    Are´ an, H.-S

    D. Are´ an, H.-S. Jeong, J. F. Pedraza and L.-C. Qu,Kasner Interiors from Analytic Hairy Black Holes,JHEP11(2024) 138, [2407.18430]

    2024 arXiv

  8. [15]

    Caceres, A

    E. Caceres, A. Kundu, A. K. Patra and S. Shashi,Trans-IR flows to black hole singularities,Phys. Rev. D106(2022) 046005, [2201.06579]

    2022 arXiv

  9. [16]

    Frenkel, S

    A. Frenkel, S. A. Hartnoll, J. Kruthoff and Z. D. Shi,Holographic flows from CFT to the Kasner universe,JHEP08(2020) 003, [2004.01192]

    2020 arXiv

  10. [17]

    S. A. Hartnoll, G. T. Horowitz, J. Kruthoff and J. E. Santos,Gravitational duals to the grand canonical ensemble abhor Cauchy horizons,JHEP10(2020) 102, [2006.10056]

    2020 arXiv

  11. [18]

    S. A. Hartnoll, G. T. Horowitz, J. Kruthoff and J. E. Santos,Diving into a holographic superconductor,SciPost Phys.10(2021) 009, [2008.12786]

    2021 arXiv

  12. [19]

    Sword and D

    L. Sword and D. Vegh,Kasner geometries inside holographic superconductors,JHEP 04(2022) 135, [2112.14177]

    2022 arXiv

  13. [20]

    Sword and D

    L. Sword and D. Vegh,What lies beyond the horizon of a holographic p-wave superconductor,JHEP12(2022) 045, [2210.01046]

    2022 arXiv

  14. [21]

    Y.-Q. Wang, Y. Song, Q. Xiang, S.-W. Wei, T. Zhu and Y.-X. Liu,Holographic flows with scalar self-interaction toward the Kasner universe,2009.06277

    2009 arXiv

  15. [22]

    S. A. H. Mansoori, L. Li, M. Rafiee and M. Baggioli,What’s inside a hairy black hole in massive gravity?,JHEP10(2021) 098, [2108.01471]

    2021 arXiv

  16. [23]

    Liu, H.-D

    Y. Liu, H.-D. Lyu and A. Raju,Black hole singularities across phase transitions, JHEP10(2021) 140, [2108.04554]

    2021 arXiv

  17. [24]

    Das and A

    S. Das and A. Kundu,RG flows and thermofield-double states in holography,JHEP 04(2022) 167, [2112.11675]

    2022 arXiv

  18. [25]

    Y.-S. An, L. Li, F.-G. Yang and R.-Q. Yang,Interior structure and complexity – 29 – growth rate of holographic superconductor from M-theory,JHEP08(2022) 133, [2205.02442]

    2022 arXiv

  19. [26]

    Auzzi, S

    R. Auzzi, S. Bolognesi, E. Rabinovici, F. I. Schaposnik Massolo and G. Tallarita,On the time dependence of holographic complexity for charged AdS black holes with scalar hair,JHEP08(2022) 235, [2205.03365]

    2022 arXiv

  20. [27]

    Mirjalali, S

    M. Mirjalali, S. A. Hosseini Mansoori, L. Shahkarami and M. Rafiee,Probing inside a charged hairy black hole in massive gravity,JHEP09(2022) 222, [2206.02128]

    2022 arXiv

  21. [28]

    S. A. Hartnoll,Wheeler-DeWitt states of the AdS-Schwarzschild interior,JHEP01 (2023) 066, [2208.04348]

    2023 arXiv

  22. [29]

    S. A. Hartnoll and N. Neogi,AdS black holes with a bouncing interior,SciPost Phys. 14(2023) 074, [2209.12999]

    2023 arXiv

  23. [30]

    Caceres and S

    E. Caceres and S. Shashi,Anisotropic flows into black holes,JHEP01(2023) 007, [2209.06818]

    2023 arXiv

  24. [31]

    Liu and H.-D

    Y. Liu and H.-D. Lyu,Interior of helical black holes,JHEP09(2022) 071, [2205.14803]

    2022 arXiv

  25. [32]

    L.-L. Gao, Y. Liu and H.-D. Lyu,Black hole interiors in holographic topological semimetals,JHEP03(2023) 034, [2301.01468]

    2023 arXiv

  26. [33]

    Caceres, S

    E. Caceres, S. Shashi and H.-Y. Sun,Imprints of phase transitions on Kasner singularities,Phys. Rev. D109(2024) 126018, [2305.11177]

    2024 arXiv

  27. [34]

    M. J. Blacker and S. Ning,Wheeler DeWitt states of a charged AdS 4 black hole, JHEP12(2023) 002, [2308.00040]

    2023 arXiv

  28. [35]

    C´ aceres, A

    E. C´ aceres, A. K. Patra and J. F. Pedraza,Shock waves, black hole interiors and holographic RG flows,JHEP07(2024) 052, [2311.12940]

    2024 arXiv

  29. [36]

    Carballo, A

    J. Carballo, A. K. Patra and J. F. Pedraza,Diving inside holographic metals,JHEP 05(2025) 072, [2408.07748]

    2025 arXiv

  30. [37]

    C´ aceres,´A

    E. C´ aceres,´A. J. Murcia, A. K. Patra and J. F. Pedraza,Kasner eons with matter: holographic excursions to the black hole singularity,JHEP12(2024) 077, [2408.14535]

    2024 arXiv

  31. [38]

    De Clerck, S

    M. De Clerck, S. A. Hartnoll and J. E. Santos,Mixmaster chaos in an AdS black hole interior,JHEP07(2024) 202, [2312.11622]

    2024 arXiv

  32. [39]

    C´ aceres,´A

    E. C´ aceres,´A. J. Murcia, A. K. Patra and J. F. Pedraza,Higher-dimensional BKL dynamics in AdS black holes,2603.06560

  33. [41]

    Morales-Tejera and K

    S. Morales-Tejera and K. Landsteiner,Non-Hermitian quantum quenches in holography,SciPost Phys.14(2023) 030, [2203.02524]

    2023 arXiv

  34. [42]

    Z.-Y. Xian, D. Rodr ´ ıguez Fern´ andez, Z. Chen, Y. Liu and R. Meyer,Electric – 30 – conductivity in non-Hermitian holography,SciPost Phys.16(2024) 004, [2304.11183]

    2024 arXiv

  35. [43]

    Arean, D

    D. Arean, D. Garcia-Fari˜ na and K. Landsteiner,Strongly Coupled PT-Symmetric Models in Holography,Entropy27(2025) 13, [2411.18471]

    2025 arXiv

  36. [44]

    Arean and D

    D. Arean and D. Garcia-Fari˜ na,Holographic non-Hermitian lattices and junctions and their RG flows,JHEP07(2025) 276, [2410.13584]

    2025 arXiv

  37. [45]

    J. M. Begines, A. F. Faedo, C. Hoyos and D. Musso,Supergravity flows, wormholes and their pseudo-Hermitian holographic duals,2605.16168

    arXiv

  38. [46]

    P. H. Frampton, K. J. Ludwick and R. J. Scherrer,The Little Rip,Phys. Rev. D84 (2011) 063003, [1106.4996]

    2011 arXiv

  39. [47]

    C. M. Bender and S. Boettcher,Real spectra in nonHermitian Hamiltonians having PT symmetry,Phys. Rev. Lett.80(1998) 5243–5246, [physics/9712001]

    1998 arXiv

  40. [48]

    C. M. Bender,Introduction to PT-Symmetric Quantum Theory,Contemp. Phys.46 (2005) 277–292, [quant-ph/0501052]

    2005 arXiv

  41. [49]

    P. D. Mannheim,Antilinearity Rather than Hermiticity as a Guiding Principle for Quantum Theory,J. Phys. A51(2018) 315302, [1512.04915]

    2018 arXiv

  42. [50]

    Fring,An Introduction to PT-Symmetric Quantum Mechanics-Time-Dependent Systems,J

    A. Fring,An Introduction to PT-Symmetric Quantum Mechanics-Time-Dependent Systems,J. Phys. Conf. Ser.2448(2023) 012002, [2201.05140]

    2023 arXiv

  43. [51]

    Mostafazadeh,PseudoHermiticity versus PT symmetry

    A. Mostafazadeh,PseudoHermiticity versus PT symmetry. The necessary condition for the reality of the spectrum,J. Math. Phys.43(2002) 205–214, [math-ph/0107001]

    2002 arXiv

  44. [52]

    Mostafazadeh,PseudoHermiticity versus PT symmetry 2

    A. Mostafazadeh,PseudoHermiticity versus PT symmetry 2. A Complete characterization of nonHermitian Hamiltonians with a real spectrum,J. Math. Phys. 43(2002) 2814–2816, [math-ph/0110016]

    2002 arXiv

  45. [53]

    Mostafazadeh,PseudoHermiticity versus PT symmetry 3: Equivalence of pseudoHermiticity and the presence of antilinear symmetries,J

    A. Mostafazadeh,PseudoHermiticity versus PT symmetry 3: Equivalence of pseudoHermiticity and the presence of antilinear symmetries,J. Math. Phys.43 (2002) 3944–3951, [math-ph/0203005]

    2002 arXiv

  46. [54]

    C. M. Bender, D. C. Brody and H. F. Jones,Extension of PT symmetric quantum mechanics to quantum field theory with cubic interaction,Phys. Rev. D70(2004) 025001, [hep-th/0402183]

    2004 arXiv

  47. [55]

    Alexandre, J

    J. Alexandre, J. Ellis and P. Millington,Discrete spacetime symmetries, second quantization, and inner products in a non-Hermitian Dirac fermionic field theory, Phys. Rev. D106(2022) 065003, [2201.11061]

    2022 arXiv

  48. [56]

    Li and Q

    Y.-D. Li and Q. Wang,Isospectral local Hermitian theory for the PT-symmetric iϕ3 quantum field theory,Phys. Rev. D111(2025) 025016, [2412.10732]

    2025 arXiv

  49. [57]

    Wiersig,Enhancing the Sensitivity of Frequency and Energy Splitting Detection by Using Exceptional Points: Application to Microcavity Sensors for Single-Particle Detection,Phys

    J. Wiersig,Enhancing the Sensitivity of Frequency and Energy Splitting Detection by Using Exceptional Points: Application to Microcavity Sensors for Single-Particle Detection,Phys. Rev. Lett.112(2014) 203901. – 31 –

    2014

  50. [58]

    Chen, S ¸

    W. Chen, S ¸. Kaya ¨Ozdemir, G. Zhao, J. Wiersig and L. Yang,Exceptional points enhance sensing in an optical microcavity,Nature548(2017) 192–196

    2017

  51. [59]

    Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao and D. N. Christodoulides, Unidirectional Invisibility Induced byPT-Symmetric Periodic Structures,Phys. Rev. Lett.106(May, 2011) 213901

    2011

  52. [60]

    Feng, Y.-L

    L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida et al.,Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,Nature Materials12(2012) 108–113

    2012

  53. [61]

    Peng, S ¸

    B. Peng, S ¸. K. ¨Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long et al., Parity–time-symmetric whispering-gallery microcavities,Nature Phys.10(2014) 394–398, [1308.4564]

    2014 arXiv

  54. [62]

    K.¨Ozdemir, S

    S ¸. K.¨Ozdemir, S. Rotter, F. Nori and L. Yang,Parity–time symmetry and exceptional points in photonics,Nature Materials18(2019) 783–798

    2019

  55. [63]

    Ashida, Z

    Y. Ashida, Z. Gong and M. Ueda,Non-Hermitian physics,Adv. Phys.69(2021) 249–435, [2006.01837]

    2021 arXiv

  56. [64]

    K. Ding, C. Fang and G. Ma,Non-Hermitian topology and exceptional-point geometries,Nature Rev. Phys.4(2022) 745–760, [2204.11601]

    2022 arXiv

  57. [65]

    Gurarie,Logarithmic Operators in Conformal Field Theory,Nuclear Physics B 410(1993) 535–549, [hep-th/9303160]

    V. Gurarie,Logarithmic Operators in Conformal Field Theory,Nuclear Physics B 410(1993) 535–549, [hep-th/9303160]

    1993 arXiv

  58. [66]

    Cardy,Logarithmic Conformal Field Theories as Limits of Ordinary CFTs and Some Physical Applications,Journal of Physics A: Mathematical and Theoretical46 (2013) 494001, [1302.4279]

    J. Cardy,Logarithmic Conformal Field Theories as Limits of Ordinary CFTs and Some Physical Applications,Journal of Physics A: Mathematical and Theoretical46 (2013) 494001, [1302.4279]

    2013 arXiv

  59. [67]

    Grumiller and N

    D. Grumiller and N. Johansson,Gravity duals for logarithmic conformal field theories,J. Phys. Conf. Ser.222(2010) 012047, [1001.0002]

    2010 arXiv

  60. [68]

    Grumiller, W

    D. Grumiller, W. Riedler, J. Rosseel and T. Zojer,Holographic applications of logarithmic conformal field theories,J. Phys. A46(2013) 494002, [1302.0280]

    2013 arXiv

  61. [69]

    Io, F.-H

    I.-F. Io, F.-H. Huang and C.-T. Hsieh,Non-Hermitian free-fermion critical systems and logarithmic conformal field theory,2602.02649

    arXiv

  62. [70]

    C. M. Bender, D. C. Brody and H. F. Jones,Scalar Quantum Field Theory with a Complex Cubic Interaction,Physical Review Letters93(Dec., 2004)

    2004

  63. [71]

    Alexandre, J

    J. Alexandre, J. Ellis, P. Millington and D. Seynaeve,Spontaneous symmetry breaking and the Goldstone theorem in non-Hermitian field theories,Phys. Rev. D 98(2018) 045001, [1805.06380]

    2018 arXiv

  64. [72]

    P. D. Mannheim,Goldstone bosons and the Englert-Brout-Higgs mechanism in non-Hermitian theories,Phys. Rev. D99(2019) 045006, [1808.00437]

    2019 arXiv

  65. [73]

    Fring and T

    A. Fring and T. Taira,Goldstone bosons in different PT-regimes of non-Hermitian scalar quantum field theories,Nucl. Phys. B950(2020) 114834, [1906.05738]. – 32 –

    2020 arXiv

  66. [74]

    Alexandre, J

    J. Alexandre, J. Ellis, P. Millington and D. Seynaeve,Spontaneously Breaking Non-Abelian Gauge Symmetry in Non-Hermitian Field Theories,Phys. Rev. D101 (2020) 035008, [1910.03985]

    2020 arXiv

  67. [75]

    Fring and T

    A. Fring and T. Taira,Pseudo-Hermitian approach to Goldstone’s theorem in non-Abelian non-Hermitian quantum field theories,Phys. Rev. D101(2020) 045014, [1911.01405]

    2020 arXiv

  68. [76]

    Alexandre, J

    J. Alexandre, J. Ellis and P. Millington,PT-symmetric non-Hermitian quantum field theories with supersymmetry,Phys. Rev. D101(2020) 085015, [2001.11996]

    2020 arXiv

  69. [77]

    A. M. Begun, M. N. Chernodub and A. V. Molochkov,Phase diagram and vortex properties of a PT-symmetric non-Hermitian two-component superfluid,Phys. Rev. D104(2021) 056024, [2105.07453]

    2021 arXiv

  70. [78]

    R. A. Battye, S. J. Cotterill and P. Millington,Vortex stability in pseudo-Hermitian theories,2511.14300

  71. [79]

    J. M. Maldacena,Eternal black holes in anti-de Sitter,JHEP04(2003) 021, [hep-th/0106112]

    2003 arXiv

  72. [80]

    D. C. Brody,Biorthogonal quantum mechanics,J. Phys. A47(2013) 035305, [1308.2609]

    2013 arXiv

  73. [81]

    Mostafazadeh,Pseudo-Hermitian Representation of Quantum Mechanics,Int

    A. Mostafazadeh,Pseudo-Hermitian Representation of Quantum Mechanics,Int. J. Geom. Meth. Mod. Phys.7(2010) 1191–1306, [0810.5643]

    2010 arXiv

  74. [82]

    S. S. Gubser and F. D. Rocha,The gravity dual to a quantum critical point with spontaneous symmetry breaking,Phys. Rev. Lett.102(2009) 061601, [0807.1737]

    2009 arXiv

  75. [83]

    C. P. Herzog and D. T. Son,Schwinger-Keldysh propagators from AdS/CFT correspondence,JHEP03(2003) 046, [hep-th/0212072]

    2003 arXiv

  76. [84]

    Skenderis and B

    K. Skenderis and B. C. van Rees,Real-time gauge/gravity duality,Phys. Rev. Lett. 101(2008) 081601, [0805.0150]

    2008 arXiv

  77. [85]

    Skenderis and B

    K. Skenderis and B. C. van Rees,Real-time gauge/gravity duality: Prescription, Renormalization and Examples,JHEP05(2009) 085, [0812.2909]

    2009 arXiv

  78. [86]

    C. M. Bender,Making sense of non-Hermitian Hamiltonians,Rept. Prog. Phys.70 (2007) 947, [hep-th/0703096]

    2007 arXiv

  79. [87]

    Balasubramanian and S

    V. Balasubramanian and S. F. Ross,Holographic particle detection,Phys. Rev. D61 (2000) 044007, [hep-th/9906226]

    2000 arXiv

  80. [88]

    Louko, D

    J. Louko, D. Marolf and S. F. Ross,On geodesic propagators and black hole holography,Phys. Rev. D62(2000) 044041, [hep-th/0002111]

    2000 arXiv

  81. [89]

    Skenderis,Lecture notes on holographic renormalization,Class

    K. Skenderis,Lecture notes on holographic renormalization,Class. Quant. Grav.19 (2002) 5849–5876, [hep-th/0209067]

    2002 arXiv

  82. [90]

    S. A. Hartnoll, C. P. Herzog and G. T. Horowitz,Holographic Superconductors, JHEP12(2008) 015, [0810.1563]. – 33 –

    2008 arXiv

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