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Smart Holes: Analogue black holes with the right temperature and entropy

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A tilted Dirac cone material can reproduce a black hole's temperature and entropy.

desk verdict A clean entropy calculation for tilted Dirac cones with an overreaching 'smart hole' claim; the KMS temperature derivation does not work, and the BH-entropy match is a tuned correspondence rather than a derivation. read the letter →

arxiv 2412.08517 v3 pith:T4K2ZQEJ submitted 2024-12-11 hep-th cond-mat.str-elgr-qc

Jiayue Yang , Niayesh Afshordi , Mahdi Torabian , Seyed Akbar Jafari , G. Baskaran This is my paper · ORCID
classification hep-thcond-mat.str-elgr-qc PACS 04.70.Dy04.62.+v71.10.-w
keywords analoguegravitytiltedDiracconeBekenstein-HawkingentropysmartholeFermipuddle2+1blackKMSconditionsurface
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

This paper claims that a two-dimensional tilted Dirac cone material, in which the tilt varies linearly across the sample, forms an analogue black hole that reproduces both the temperature and the entropy of a real 2+1-dimensional black hole. The critical tilt value $\zeta = 1$ plays the role of the event horizon, and the spatial gradient of the tilt acts as surface gravity. Computing the Fermi-Dirac entropy of the electrons, the authors find that once the integration extends past the horizon ($\zeta_{\max} > 1$), the total entropy becomes linear in temperature, $S \approx \eta L_x p_{\max} T/\kappa$. With the identifications $T = \kappa/(2\pi)$, $p_{\max} = \pi/(2\eta G)$, and $L_x = 2\pi r_+$, this is exactly the Bekenstein-Hawking entropy of the 2+1-dimensional black hole. The paper calls such a system a smart hole, and shows numerically that the entropy is concentrated near the analogue horizon, or, with nonlinear band structure, in a Fermi puddle just behind it.

What carries the argument

The device carrying the argument is the tilt parameter $\zeta$ in the emergent metric $ds^2 = -v_F^2 dt^2 + (dr - \zeta v_F dt)^2$, with $\zeta_y(y) = 1 - \kappa y/v_F^2$ chosen so that the horizon sits at $\zeta=1$ and the surface gravity is $\kappa = v_F^2 |\zeta'_y|$ there. The entropy density is the standard Fermi-Dirac entropy modified by the redshift factor $\gamma = (1-\zeta^2)^{-1/2}$; integrating it over the sample produces Eq. (4.3). The temperature identification $T = \kappa/(2\pi)$ is obtained from the thermal equilibrium (KMS) condition applied to the non-Hermitian Hamiltonian $H_0 = H_{\rm sym} + i\kappa/2$, where the imaginary part encodes dissipation at the horizon.

What would settle it

Measure the heat capacity of a tilted Dirac cone sample with a controlled linear tilt gradient across $\zeta=1$, sweeping the laboratory temperature around the value $T=\kappa/(2\pi)$ set by the gradient. The claim predicts total entropy linear in $T$ with coefficient $\eta L_x p_{\max}/\kappa$ and a sharp entropy enhancement near the $\zeta=1$ line; observing a different temperature scaling, or no spatial concentration near the horizon line, would falsify it.

Watch

Extended reading notes

Core claim

The central discovery is a concrete entropy-matching identity. For a tilted Dirac cone material with tilt $\zeta_y(y) = 1 - \kappa y/v_F^2$, the numerical total entropy of the lower band branch obeys $S \approx \eta L_x p_{\max} T/\kappa$ whenever the tilt maximum exceeds one (Eq. 4.3). The dimensionless coefficient $\eta$ grows with $\zeta_{\max}$ and absorbs the details of the momentum cutoff. Substituting the analogue-horizon identifications $T = \kappa/(2\pi)$, $p_{\max} = \pi/(2\eta G)$, and $L_x = 2\pi r_+$ turns this into the Bekenstein-Hawking area law $S = 2\pi r_+/(4G)$ of the 2+1-dimensional black hole. The authors further show that in the linear dispersion the entropy density diverges as $\zeta \to 1$ and is regularized either by a momentum cutoff or by nonlinear band corrections; in the full band structure the entropy peaks in the Fermi puddle located just inside the analogue horizon.

Load-bearing premise

The central assumption is the identification of the thermal state of the electrons with the horizon temperature set by the tilt gradient; if the laboratory temperature actually controls the occupations and is not tuned to that value, the match to black-hole entropy breaks down.

Editorial extensions

If this is right

  • Laboratory sheets of tilted Dirac materials with a fabricated tilt gradient can serve as tabletop black holes whose horizon temperature is set by $d\zeta/dy$, reaching from a few kelvin to above room temperature for realistic gradients.
  • The entropy match is not an artifact of the linear dispersion: quadratic and full-band calculations still give $S \propto p_{\max} T/\kappa$ for $\zeta_{\max} > 1$, so the claim survives at the lattice level.
  • The spatial concentration of entropy near $\zeta \approx 1$ gives a concrete observable signature of the analogue horizon, and in the nonlinear band the Fermi puddle marks the interior region behind it.
  • Because the total entropy is linear in temperature only when the integration includes $\zeta > 1$, the analogue black hole thermodynamics requires access to the over-tilted type-II region beyond the critical tilt.

Reading between the lines

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

  • If the horizon temperature is a condition to be tuned rather than an automatic property, then sweeping the laboratory temperature across $T=\kappa/(2\pi)$ should reveal a distinctive crossover in the entropy scaling; the paper does not pursue this experimental signature.
  • The mapping $p_{\max} \sim 1/G$ suggests that the lattice cutoff plays the role of a Planck scale; if so, the coefficient $\eta$ should depend on material-specific band parameters and cutoff geometry, offering a probe of trans-Planckian effects in the analogue.
  • The same entropy-counting argument may extend to other type-II or type-III Dirac and Weyl materials with engineered tilt gradients, and possibly to higher-dimensional black hole spacetimes, although the paper only demonstrates the 2+1-dimensional case.
  • The Fermi puddle's role as the entropy carrier behind the horizon suggests that local probes of the density of states just inside the horizon could locate where the analogue horizon degrees of freedom live.
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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

4 major / 5 minor

Summary. The paper studies the thermodynamics of 2D tilted Dirac cone materials in which the tilt parameter varies linearly in space, so that the surface κ of the analogue horizon is set by the tilt gradient. Starting from the Fermi-Dirac entropy density, the authors integrate over the sample and find numerically that for maximum tilt beyond the critical value ζmax > 1 the total entropy is approximately S ∝ (Lx pmax/κ) T, i.e. linear in temperature. They then propose a dictionary T = κ/2π, pmax = π/(2ηG), Lx = 2π r+, under which S equals the Bekenstein-Hawking entropy of a BTZ black hole. The paper also reports that the entropy density is peaked near ζ ≈ 1 and that nonlinear dispersion produces an additional peak associated with a 'Fermi puddle' behind the analogue horizon. A KMS-type argument in Eqs. (4.6)-(4.9) is offered as a derivation of T = κ/2π.

Significance. If the central identification were established, the result would be a notable step toward a table-top analogue that reproduces both Hawking temperature and Bekenstein-Hawking entropy, rather than only the geometric horizon. The independent ingredients are genuinely interesting: the closed-form entropy density in Eq. (3.12) matches the known relativistic result, the numerical power laws in Figs. 6-8 show a robust linear-in-T regime once the integration extends beyond ζ = 1, and the entropy concentration near ζ ≈ 1 is a concrete, falsifiable prediction for tilted Dirac cone materials. However, the claimed equality with black hole entropy rests on a dictionary that is substantially fixed after the fact: the coefficient η comes from numerical fits, pmax is chosen to match the BTZ coefficient, and the KMS derivation of T = κ/2π is not mathematically valid as written. The linear scaling and the horizon-peaked entropy density are strengths that should survive a more careful treatment.

major comments (4)
  1. [Sec. 4, Eqs. (4.6)-(4.9)] The KMS derivation of T = κ/2π is not valid. Since H0 = Hsym + iκ/2 with iκ/2 a c-number commuting with Hsym, one has e^{-βH0} = e^{-βHsym} e^{-iβκ/2}; the phase factor is common to the numerator and the partition function and cancels in every normalized thermal expectation value. The manipulations in Eq. (4.8) therefore cannot produce a surviving factor e^{βκi} that must be set to unity. At a fixed laboratory inverse temperature β, the thermal state is that of Hsym, and no unique condition β = 2π/κ follows. The discussion after Eq. (4.9) itself concedes that T = κ/2π is a tuning condition ('if this condition is not satisfied...') rather than a derived consequence. Since the equality with the Bekenstein-Hawking entropy in Eq. (4.5) explicitly uses T = κ/2π, this is a load-bearing gap and must be either corrected or explicitly presented as an assumption.
  2. [Sec. 4, Eq. (4.5)] The dictionary pmax = π/(2ηG) is chosen so that Eq. (4.3) reproduces the BTZ entropy coefficient, and the coefficient η is itself extracted from the power-law fits in Figs. 6-8. With this choice the equality SDC = SBH is a parametrization rather than a prediction. The identification Lx = 2πr+ is also a matching of length scales rather than a derivation. The paper should either provide an independent determination of pmax and η, or state clearly that Eq. (4.5) defines a tuning condition under which the analogue entropy can be made equal to the Bekenstein-Hawking entropy.
  3. [Sec. 4, Figs. 6-8 and Eq. (4.3)] The central power law S ∝ Lx pmax T/κ is supported numerically by fits whose exponents differ from unity by amounts such as -1.0171, -0.99880, -1.0205 and -0.97983, but no fit uncertainties or temperature ranges are reported. For ζmax = 1 the fitted exponent is -1.4759, which is far from linear, so the linear regime requires ζmax > 1 without a quantitative criterion for how large ζmax must be. Specifying the β range and the fit errors, and providing the corresponding residuals, would make the claimed scaling in Eq. (4.3) more convincing.
  4. [Sec. 4, Eq. (4.10) and text] The paper states that T = ℏvF|dζ/dy|/(2πkB) 'defines the regime' in which the analogue temperature matches the emergent Hawking temperature. This is a reasonable experimental tuning statement, but it undercuts the earlier KMS claim that Eq. (4.9) is forced by consistency. The manuscript should decide whether T = κ/2π is a prediction or a tunable condition; the present text alternates between the two without resolving the tension.
minor comments (5)
  1. [Abstract] The phrase 'we refer to this new type of analogue black hole as asmart hole' contains a typo; it should read 'as a smart hole'.
  2. [Sec. 4, after Eq. (4.2)] The condition is written as '2πκβ = 1'; this should be 2π/(κβ) = 1, equivalently T = κ/2π.
  3. [Fig. 6 caption] The shaded-region labels 'BH, π√20β^{-1} ~ π√2β^{-1}' are difficult to read and are not explained in the text; the reader has to infer that the spread corresponds to choosing G between a and 10a.
  4. [Sec. 3, Eq. (3.5)] The factor of 4 relating s2 to s1 (two spin states and two energy branches) is stated only in words; writing the summation over s = ±1 explicitly would avoid confusion about the counting.
  5. [References] Reference [66] has a typo in the title: 'Tunning' should be 'Tuning'. Also, the name Bañados appears as 'Ba˜nados' in Eq. (4.4) and reference [91].

Circularity Check

2 steps flagged · score 6.0 of 10

The match to BTZ entropy is enforced by the dictionary: the coefficient eta is fitted numerically, pmax is then chosen as pi/(2 eta G), and T=kappa/2pi is imposed by a phase condition rather than derived.

  1. fitted input called prediction [Section 4, Eq. (4.3) and the dictionary in Eq. (4.5)]
    "From the above numerical computations, we conclude that the total entropy for 2d tilted Dirac cone material with length Lx and width Ly = v_F^2/kappa zeta_max is SDC approx eta Lx pmax T/kappa, (4.3) where eta is a dimensionless factor, depending on the zeta_max value we choose. ... If we make the assumption that the temperature T is interpreted as the analogue black hole surface gravity kappa, the momentum cutoff pmax is the inverse of the Newton constant G, and the length scale Lx is interpreted as the analogue black hole horizon size 2 pi r+, [then] ... pmax = pi/(2 eta) 1/G"

    The factor eta is read off from the numerical best fits to the total-entropy curves (Figures 6-8), not derived from first principles. Inserting the dictionary (4.5), in particular pmax = pi/(2 eta G), into (4.3) gives S ~ eta Lx (pi/(2 eta G)) (kappa/2pi)/kappa = Lx/(4G), and with Lx = 2pi r+ this is exactly the BTZ entropy 2pi r+/(4G). The momentum cutoff pmax is a free integration cutoff in Eq. (3.3); setting it to pi/(2 eta G) is an after-the-fact choice that manufactures the Bekenstein-Hawking coefficient. The linear-in-T scaling is real, but the numerical coefficient of the 'match' is fixed by defining pmax in terms of the fitted eta.

  2. self definitional [Section 4, Eqs. (4.6)-(4.9)]
    "H0 = Hsym + kappa/2 i. ... The KMS condition <psi(x1, t)psi(x2, 0)>_beta = <psi(x2, -i beta)psi(x1, t)>_beta immediately yields the identification of the temperature with the surface gravity beta = 2pi/kappa => T = kappa/2pi. (4.9) ... Indeed, as one might point out, there is no guarantee that the temperature set by the cryogenic refrigerator in the laboratory satisfies T = kappa/2pi."

    In (4.7), i kappa/2 is a c-number commuting with Hsym, so e^{-beta H0} = e^{-beta Hsym} e^{-i beta kappa/2}; the complex phase cancels between numerator and partition function in every normalized thermal expectation, and the Heisenberg evolution generated by H0 also has canceling factors. The KMS condition is therefore obeyed by the Hermitian part at the actual laboratory beta; the extra factor e^{beta kappa i} in (4.8) is a gauge phase, not a physical constraint. Requiring e^{beta kappa i}=1 is exactly imposing beta = 2pi/kappa, the desired Hawking relation, rather than deriving it. The paper's own discussion concedes that this is a tuning condition ('no guarantee ... satisfies T = kappa/2pi'), and without it the entropy formula (4.3) remains S ~ eta Lx pmax T_lab/kappa.

full rationale

Score 6: the central quantitative claim, namely equality of the analogue entropy with SBH, does not stand on its own; it is assembled from the numerically fitted coefficient eta, the chosen cutoff pmax = pi/(2 eta G), and the imposed temperature T = kappa/2pi. The independent content, S proportional to Lx pmax T/kappa for zeta_max > 1, the localization of entropy near zeta approx 1, and the Fermi puddle behind the analogue horizon, is a genuine first-principles Fermi-Dirac entropy computation from the tilted band structure; the paper's self-citations (e.g., the agreement of Eq. (3.12) with [89]) are not load-bearing for that part. However, the exact Bekenstein-Hawking coefficient is forced by the dictionary in Eq. (4.5), and the KMS 'derivation' reduces to imposing the phase condition that already encodes the answer. This is therefore a partial but central circularity rather than a clean derivation: the form of the entropy is predicted, while the advertised black-hole coefficient and Hawking temperature are put in by hand.

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

The central claim rests on the emergent metric (from prior work), the Fermi-Dirac entropy formula with redshift factor (from prior work), the KMS temperature identification (presented here but not rigorously), and the scale identifications pmax-1/G and Lx-2πr+ (chosen to match). The main free parameters are the momentum cutoff and the maximum tilt used in the entropy integration.

free parameters (3)
  • Momentum cutoff pmax = π/(2ηG) with G in [a,10a]
    Chosen in Eq. (4.5) so that the analogue total entropy matches the BTZ Bekenstein-Hawking entropy; it is a lattice-scale cutoff promoted to the Planck scale.
  • Maximum tilt integration ζmax = 1, 6/5, 3/2, 2
    Upper limit of the ζ-integral in Eq. (3.9); the claimed S ∝ T scaling only holds for ζmax > 1 (Figures 6-8).
  • Dimensionless coefficient η = Not quoted, but obtained from power-law fits in Figures 6-8 (e.g., ~0.36 for ζmax=2)
    Extracted from fits of total entropy vs inverse temperature; absorbs pmax dependence and is later used in the pmax identification.
assumptions (5)
  • domain assumption The emergent spacetime metric for tilted Dirac cones is the Painlevé-Gullstrand line element with horizon at ζ=1 (Eq. 1.1).
    Taken from Volovik and Jafari refs [35-37,79-85]; the entropy-temperature mapping depends on this geometric identification.
  • domain assumption The entropy density is given by the Fermi-Dirac expression with a Lorentz-like factor γ=(1-ζ^2)^(-1/2) modifying the proper temperature (Eq. 3.1).
    Borrowed from kinetic theory of tilted Dirac cones [85]; this is the starting point of the entropy computation.
  • ad hoc to paper The KMS condition for the non-Hermitian Hamiltonian H0 = Hsym + iκ/2 yields T = κ/2π (Eqs. 4.8-4.9).
    The derivation in Section 4 is not fully rigorous; the factor e^{βκi} and the branch βκ=2π are asserted rather than derived.
  • ad hoc to paper The sample length Lx in the horizon direction is identified with the BTZ horizon circumference 2πr+, and pmax with the inverse Newton constant (Eq. 4.5).
    These identifications are chosen to make the analogue entropy equal to SBH; they are not independently motivated.
  • domain assumption Particle-hole symmetry allows setting the chemical potential to zero (Section 3, after Eq. 3.1).
    Assumes equal electron and hole pockets so that particle number is conserved at zero chemical potential.
invented entities (1)
  • Fermi puddle independent evidence
    purpose: Closed zero-energy Fermi pocket that forms behind the analogue horizon (ζ>1) in the nonlinear dispersion; the entropy density peaks near it.
    It is a calculable feature of the band structure (Figure 4d) and should be observable in ARPES or quantum oscillation experiments on over-tilted Dirac materials; not a new fundamental entity.

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Cite this review

Pith. "Pith review of Smart Holes: Analogue black holes with the right temperature and entropy." pith.science (2026). https://pith.science/paper/T4K2ZQEJ

@misc{pith2026241208517,
  author       = {Pith},
  title        = {Pith review of: Smart Holes: Analogue black holes with the right temperature and entropy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T4K2ZQEJ}},
  note         = {Machine review of arXiv:2412.08517}
}
read the original abstract

In analogue gravity studies, the goal is to replicate black hole phenomena, such as Hawking radiation, within controlled laboratory settings. In the realm of condensed matter systems, this may happen in 2D tilted Dirac cone materials based on honeycomb lattice. In particular, we compute the entropy of this system, and find it has the same form as black hole Bekenstein-Hawking entropy, if an analogue horizon forms. Hence, these systems can be potential analogues of quantum black holes. We show that this entropy is primarily concentrated in the region where the tilt parameter is close to one, which corresponds to the location of the analogue black hole horizon. Additionally, when nonlinear effects are taken into account, the entropy is peaked in a small pocket of the Fermi sea that forms behind the analogue event horizon, which we call the \textit{Fermi puddle}. We further refer to this new type of analogue black hole as a {\it smart hole}, since, in contrast to dumb holes, it can simulate both the correct temperature {\it and} entropy of general relativistic black holes. These results provide an opportunity to illuminate various quantum facets of black hole physics in a laboratory setting.

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