Holographic subregion complexity in a moving strongly coupled plasma

Mahsa Lezgi; Mohammad Ali-Akbari; Mohammad Mahdi Daryaei Goki; Vahid Esrafilian

REVIEW 5 major objections 5 minor 45 references

Holographic subregion complexity in a moving strongly coupled plasma

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

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Pith's one-line read In a moving strongly coupled plasma, holographic subregion complexity grows with temperature, velocity, and strip length, and diverges as the square of the Lorentz factor as velocity approaches light speed, in d=2,3,4.

desk verdict A genuinely new numerical map of subregion complexity in boosted plasmas, but the universal gamma^2 claim is fitted from l=1 data only and is broader than the evidence. read the letter →

arxiv 2412.14755 v1 pith:YWHHE4KO submitted 2024-12-19 hep-th

Mohammad Mahdi Daryaei Goki , Mohammad Ali-Akbari , Mahsa Lezgi , Vahid Esrafilian This is my paper · ORCID

classification hep-th

keywords holographicsubregioncomplexityboostedblackbranemovingstronglycoupledplasmaLorentzfactordivergenceHRTsurfacefinitedifferencemethodstripAdS/CFT

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 studies holographic subregion complexity in a strongly coupled plasma moving at constant velocity, using the holographic proposal that identifies the complexity of a mixed state with the volume of the codimension-one hypersurface enclosed by the HRT surface (the covariant extremal surface used in holographic entanglement entropy). For a strip-shaped boundary region aligned with the boost, the authors solve the resulting extremal-volume equation numerically by finite differences in boundary dimensions $d=2,3,4$, without restricting temperature or velocity. Their central finding is that holographic subregion complexity increases with temperature, velocity, and subregion length, and that as the velocity approaches the speed of light the complexity diverges as the square of the Lorentz factor, $\gamma^2$, for arbitrary temperature and subregion length. They conclude that this $\gamma^2$ divergence is a universal feature of strip subregions in a moving plasma across the three dimensions studied.

What carries the argument

The central object is the volume functional $V=\int dz\,dx\,\mathcal{V}$ for the region enclosed by the HRT surface in the boosted black brane background, with integrand (9) and the associated Euler-Lagrange equation (10). The load-bearing computational step is the finite-difference solution of this nonlinear PDE for $t(x,z)$, with boundary values taken from the HRT profile $(x_\pm(z), t_\pm(z))$; the iteration relaxes all grid nodes to a tolerance of order $10^{-6}$. The Lorentz factor enters through the metric function $g(z)=\gamma^2(z/z_h)^d$, so the $\gamma^2$ divergence of the complexity as $v\to1$ tracks the divergence of this metric coefficient. The subtracted quantities $\mathcal{C}$ and $\hat{\mathcal{C}}$ compare the complexity against the static-plasma and pure-AdS baselines, and the fitted form $a(T)\gamma^2+b(T)$ is what converts the numerical curves into the claimed universal scaling law.

What would settle it

Compute the same subtracted complexity at velocities closer to 1 (for example $v=0.999$) with a finer mesh and a convergence study, and extract the divergence exponent from a log-log plot of $\mathcal{C}$ versus $\gamma$ at fixed $T$ and $l$. If the exponent is not 2, or if the fitted function requires higher powers of $\gamma$, the claimed universal $\gamma^2$ divergence fails. An analytic derivation of the leading velocity dependence from the volume functional that yields a different Lorentz-factor power would also settle the question.

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Extended reading notes

Core claim

The paper establishes numerically that for a strip-like subregion of length $l$ and infinite width in a $d$-dimensional strongly coupled plasma moving with constant velocity $v$ parallel to the strip's short edge, the holographic subregion complexity $C_A = V_{\gamma_A}/(8\pi G_N)$ is an increasing function of temperature $T$, velocity $v$, and subregion length $l$ for $d=2,3,4$. Working with the boosted black brane metric and using the HRT surface (the covariant extremal surface used in holographic entanglement entropy) as boundary data, the authors solve the nonlinear partial differential equation (10) for the extremal volume slice $t(x,z)$ by finite differences. Subtracting either the static-plasma value or the pure-AdS value gives the quantities $\mathcal{C}$ and $\hat{\mathcal{C}}$ defined in (11), both of which are positive and increase with all three parameters. Fitting the velocity dependence shows that as $v\to 1$ the complexity diverges as $a(T)\gamma^2+b(T)$, with $a,b\ll C_0$, so the divergence is characterized by the Lorentz factor squared $\gamma^2$; the same behavior is found in $d=2,3,4$ for arbitrary temperature and subregion length. The paper reads this as evidence for a universal $\gamma^2$ divergence for strip subregions when the plasma moves parallel to the short edge.

Load-bearing premise

The load-bearing premise is that equation (10) is the correct equation for the surface whose volume defines the complexity, and that it has a unique smooth solution matching the HRT surface; the paper states this without derivation, checks only one sample solution, and gives no mesh-convergence study.

Editorial extensions

If this is right

Holographic subregion complexity of a strip region in a moving plasma is larger than in a static plasma at the same temperature, so specifying the mixed state requires more information once the plasma moves.
As the plasma velocity approaches the speed of light, the subtracted complexity diverges like $\gamma^2$ for fixed $T$ and $l$ in $d=2,3,4$, making the divergence a candidate universal signature of the boost.
For small subregion lengths ($T_l\gg T$) the complexity approaches the pure-AdS value, so short-distance probes are insensitive to both temperature and velocity.
Higher boundary dimension increases the absolute value of the complexity and makes the differences between velocities and temperatures more pronounced at larger subregion lengths.
At low temperature the velocity dependence is suppressed because the metric coefficients reduce to $f(z)\to1$ and $g(z)\to0$, recovering the static vacuum limit.

Reading between the lines

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

The $\gamma^2$ scaling may reflect a kinematic boost effect rather than a dynamical property of the plasma; one way to test this is to repeat the calculation with the strip at an oblique angle to the boost direction and check whether the Lorentz-factor power changes.
The same finite-difference pipeline could be applied to other mixed-state observables, such as the entanglement wedge cross-section or mutual information, to see whether they share the $\gamma^2$ tail; if they do, the divergence would be a general property of the boosted geometry rather than of the specific volume functional.
The fitted form $a(T)\gamma^2+b(T)$ with $b\ll C_0$ suggests that in the strict large-$\gamma$ regime the subtracted complexity is controlled by a single temperature-dependent coefficient $a(T)$; deriving $a(T)$ analytically would turn the numerical fit into a closed-form prediction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my read. The paper does something legitimately new: it computes holographic subregion complexity for a boosted black brane non-perturbatively across d=2,3,4, for a range of temperatures and velocities, and shows monotonic growth in T, v, and strip length. The fits are impressively tight (sub-percent relative errors), and the consistency with the perturbative gamma^2 result from [40] is a real point in its favor. The numerical work is not trivial, and the authors are honest about the numerical challenges near v=1 and about citing the earlier perturbative calculations.

The soft spots are real but not fatal. The headline claim—universal gamma^2 divergence for arbitrary T and l—is not backed by the reported fits. Every scaling fit uses l=1, and only one or two temperatures per dimension. The l-dependence plots are all at fixed v, so they test monotonicity, not the divergence exponent. The paper's own footnote admits that high temperatures cannot be probed near v=1, so the 'arbitrary T' part is also not demonstrated. Calling this universal is an extrapolation. I think the stress-test note is correct on this point.

Two technical gaps also bother me. Equation (10) is stated without derivation, and for a numerical paper that is a missing load-bearing piece: the reader cannot check that the PDE is the correct Euler-Lagrange equation. And there is no mesh-convergence study, no code or data release, just one example solution and an iteration-count plot. That is not enough to establish numerical reliability, especially when the central claim is a fitted scaling law with 'b' discarded as numerical error.

That said, the monotonic trends are very likely correct, and the gamma^2 scaling is externally supported by [40]. This is a solid numerical study overreaching in its conclusions, not a flawed one. The paper deserves a serious referee, but the referee should push for a derivation or at least a clear justification of (10), a mesh-convergence analysis, and fits at multiple l and T values before the universality claim is accepted. I would bring it to a reading group as an example of how numerical hysteresis can look convincing while missing the stated domain of validity.

Referee Report

5 major / 5 minor

Summary. The manuscript computes holographic subregion complexity (HSC) for a strip-like boundary region in a boosted black-brane background in d = 2, 3, 4. The authors numerically solve a nonlinear PDE for the bulk profile t(x,z), using the HRT surface from [42] as boundary data, and then evaluate the enclosed volume. They report three qualitative results: HSC increases with plasma temperature T, boost velocity v, and strip length l; and as v approaches 1, the subtracted complexity C-C0 diverges as the Lorentz factor squared, gamma^2, which they tentatively call universal across dimensions, temperatures, and subregion lengths. The fits use the form a(T)gamma^2+b(T) and have small reported relative errors.

Significance. If the gamma^2 divergence and the monotonicity results are correct, the paper would provide the first all-temperature, all-velocity numerical treatment of holographic subregion complexity in this stationary moving-plasma background, going beyond the perturbative calculations of [39,40]. The numerical computation of a genuinely nonlinear PDE with HRT boundary data is nontrivial, and the small fit errors and the consistency with the perturbative gamma^2 result of [40] are genuine strengths. However, the paper's headline universality claim currently rests on a narrow set of fits and on an unproven identification of the fitted constant b with numerical error, so the significance of the paper depends on how much of that claim can be supported by additional evidence or by a more restricted statement.

major comments (5)

[Numerical results, d=2 and d=3,4; Figs. 2 and 5] The central claim that the gamma^2 divergence holds for arbitrary subregion length l is not supported by the data: every velocity-scaling fit is performed at l=1. Figures 4, 7, and 8 vary l only at fixed v, which tests monotonicity but cannot determine the v->1 divergence exponent. To support the abstract and the conclusions, the authors should either extract the divergence exponent from data at several values of l (for example by fitting C as a function of gamma at each l and showing the coefficient scales consistently), or explicitly restrict the universality claim to l=1.
[Numerical results; Figs. 2, 5, 6 and Appendix A] The claim of universality for arbitrary temperature is similarly not demonstrated. In d=2 the fits use only T=1/(6 pi) and T=1/(60 pi); in d=3 and d=4 each scaling fit is at a single temperature T=1/(4 pi). Moreover, Appendix A concedes that velocities close to one cannot be probed at high temperatures, so the v->1 exponent is not checked across a meaningful temperature range. The authors should either restrict the claim to the temperatures actually probed or perform fits at additional temperatures and show the gamma^2 coefficient and exponent are stable.
[Holographic subregion complexity, Eq. (10)] Equation (10), the second-order nonlinear PDE that is the core of the paper's numerical method, is stated without derivation. Since the entire numerical computation and all subsequent claims depend on this equation, the authors should provide its derivation from the volume functional (8)-(9), or at least a clear reference where it is derived. They should also comment on the existence, uniqueness, and single-valuedness of the solution t(x,z) for the HRT boundary data, since the text asserts single-valuedness without discussion.
[Appendix A and Fig. 10] The convergence evidence is not a mesh-convergence study. Figure 10 shows the number of unrelaxed grid nodes as a function of iteration count for one example at one set of parameters; it does not show how the computed complexity C depends on the mesh sizes Delta x and Delta z, nor does it quantify discretization error. This matters because the authors attribute the fitted constant b to numerical precision. I ask for a systematic grid-convergence test for the subtracted complexity, for example repeating a representative case at cell sizes 10^-3, 10^-4, and 10^-5, and reporting the resulting values of C and of the fitted b. Availability of the code and data would also substantially strengthen reproducibility.
[Numerical results, d=2; Fig. 2 and text after it] The treatment of the constant b in the fit C_bar = a(T)gamma^2 + b(T) is not justified. The authors state that a and b are of the same order and that both are much smaller than C0, and they then conclude that b is due to numerical error. The comparison with C0 is irrelevant for this conclusion: b is comparable to a (for example, for T=1/(6 pi), a=0.9260 and b=-0.8898), so over the accessible velocity range the data are fit by a constant plus a gamma^2 term. The asymptotic v->1 exponent would still be gamma^2 if b is constant, but the stronger statement C proportional to a(T)gamma^2 requires b to vanish in the continuum limit. The authors should test this by increasing the numerical precision and by a mesh-convergence study, rather than inferring it from a and b being small compared with C0.

minor comments (5)

[Introduction] There are several typos and reference formatting issues: 'Hubney' should be 'Hubeny' in the first occurrence and in reference [3]; Appendix A contains 'differntial'; reference [14] contains 'Infromation'; several arXiv identifiers are missing their archive prefix, e.g. [35] should read arXiv:hep-ph/0607062.
[Equation (11) and surrounding text] The notation is potentially confusing because C is used both for the unsubtracted complexity in Eq. (4) and for the subtracted quantity in Eq. (11a). The authors should introduce distinct symbols for the two quantities and use them consistently in the figures and captions.
[Fig. 2, right panel] The quantity Cv is used for dC/dv but is not defined before the figure; please define it explicitly and, if possible, state the numerical differentiation method used to obtain the slope.
[Fig. 1 and Appendix A] The red and blue curves in Fig. 1 are described as the plus and minus branches but the caption does not say which color corresponds to which branch; in addition, the discretization details in Appendix A state the maximum cell size is O(10^-3) but do not specify how nx, nz, and the tolerance were chosen for each run.
[Conclusions] The conclusions repeat the universality claim for 'all temperatures and subregion lengths', but the preceding numerical section explicitly notes that high temperatures cannot be probed near v->1. The wording should be reconciled with the actual data coverage so that the claim does not overstate the numerical evidence.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the γ² divergence is a fitted output of the numerical PDE solution, not an input; the arbitrary-T/l universality claim is an extrapolation (a correctness concern), not a circular reduction.

full rationale

The derivation chain is: boosted black-brane metric (1)–(2); Alishahiha's subregion-complexity proposal (4); volume functional (8)–(9); extremal PDE (10), stated without derivation; HRT profiles from [42] as boundary data; finite-difference solution; then a fit C(v)=a(T)γ²+b(T) to the numerically computed C(v). The γ² law is therefore an empirical fit to the output observable, not an input that defines the result, so it does not reduce by construction. The abstract's universal claim over arbitrary T and l goes beyond the fitted data (l=1 in all fits; only two temperatures for d=2 and one for d=3,4), and footnote 1 concedes high-T near-v→1 is numerically inaccessible; this is unsupported extrapolation, not circularity. The independent perturbative result of [40] by different authors provides external support for γ², and no load-bearing self-citation or imported uniqueness theorem is present. Omitted derivation of (10), lack of mesh-convergence study, and no code/data are correctness/reproducibility risks, not circular steps.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the boosted-brane geometry, the CV/HRT subregion complexity proposal, and a numerically solved PDE that is not derived in the paper. The only free parameters are the fit coefficients used to extract the gamma^2 law. No new physical entities are introduced.

free parameters (2)

a(T) = d=2: 0.9260 (T=pi/6), 0.0121 (T=pi/60); d=3: 5239.6824 (T=pi/4); d=4: 3.8888e6 (T=pi/4)
Coefficient of gamma^2 in the fitted function a(T)gamma^2+b(T) that defines the claimed divergence.
b(T) = d=2: -0.8898 (T=pi/6), -0.00884 (T=pi/60); d=3: -5150.1775; d=4: -3.8663e6
Constant term in the fit, discarded as numerical error; its near-cancellation with a at v=0 implies C(0) is approximately zero.

assumptions (5)

domain assumption Boosted black brane metric (1) is the correct holographic dual of a strongly coupled plasma moving with constant velocity.
Used throughout as the background; relies on the standard AdS/CFT dictionary.
domain assumption Subregion complexity is given by the volume enclosed by the HRT surface, C_A = V/(8 pi G_N), from [19].
Adopted without critical comparison to alternative mixed-state complexity proposals.
ad hoc to paper The volume functional (8)-(9) and Euler-Lagrange equation (10) are correct.
Equation (10) is stated without derivation and is the core equation being solved numerically.
domain assumption The HRT surface profiles (x±(z), t±(z)) from [42] are correct for the boosted black brane.
Used as boundary condition for the volume PDE; any error propagates into HSC.
ad hoc to paper The fit ansatz a(T)gamma^2+b(T) captures the true velocity dependence, with b attributable to numerical precision.
This assumption converts a numerical fit into the claimed universal gamma^2 divergence.

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Cite this review

Pith. "Pith review of Holographic subregion complexity in a moving strongly coupled plasma." pith.science (2026). https://pith.science/paper/YWHHE4KO

@misc{pith2026241214755,
  author       = {Pith},
  title        = {Pith review of: Holographic subregion complexity in a moving strongly coupled plasma},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YWHHE4KO}},
  note         = {Machine review of arXiv:2412.14755}
}

read the original abstract

We study holographic subregion complexity in a moving strongly coupled plasma in dimensions d = 2, 3, 4, which is holographically dual to a boosted black brane metric in a higher dimensional geometry. The proposal we employ is the one that identifies the complexity of a mixed state by the volume of codimensional-one hypersurface enclosed by Hubeny-Rangamani-Takayanagi surface. Using the finite difference method, the numerical calculations reveal that temperature, velocity, and subregion length all have an increasing effect on holographic subregion complexity. For arbitrary values of temperature and subregion length, as velocity approaches its relativistic upper limit, holographic subregion complexity exhibits a divergence. This divergence behavior observed in d = 2, 3, 4 seems to demonstrate a universal behavior and is characterized by the Lorentz factor squared, {\gamma}2.

Figures

Figures reproduced from arXiv: 2412.14755 by the authors.

**Figure 2.** FIG. 2: Left: The behavior of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3: The behavior of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4: The behavior of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5: Left [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6: The behavior of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7: The behavior of [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 8.** Figure 8: FIG. 8: The behavior of [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: FIG. 9: A schematic representation of the finite difference mesh of the solution area. The curve schematically represents [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: FIG. 10: Left: An example solution in [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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