Exact holographic superfluid at the Bogomolny point: transport, large charge, and R\'enyi entropies

Siqing Yu

REVIEW

Exact holographic superfluid at the Bogomolny point: transport, large charge, and R\'enyi entropies

Siqing Yu

High Energy Physics - Theory · Submitted 29 Oct 2026

Pith's verdict: New closed-form solution of the probe-limit holographic superfluid at the Bogomolny point, with strong numerical cross-checks; the global lowest-state claim is a real, honestly-flagged gap, and section 5.2 has a factor-of-two error in the charge-two dimension.

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arXiv ID2610.40371 (version 3)

Pith IDTWZLS67A, Pith's permanent record of this paper and its review

CitationSiqing Yu. "Exact holographic superfluid at the Bogomolny point: transport, large charge, and R\'enyi entropies." arXiv:2610.40371 (2026). https://pith.science/paper/TWZLS67A

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Bottom line: this is a strong paper and it deserves referees. The reader's ACCEPT is right, and I verified the one concrete consistency concern in the stress-test note: it lands.

What's new: eq. (2.10) is a closed-form condensed phase of the probe-limit Abelian Higgs model in AdS5 at m^2L^2 = -4 and lambda = e^2/2, on every static uncharged maximally symmetric slicing. The closure argument is clean and shows both tunings are necessary. The stability theorem in section 3.3 does not need the exact form, and the two results the paper flags as independent of it — the positivity proof and the horizon formula for the attenuation — hold up. The transport results are checked against quasinormal modes to about 10^-7, the two independent codes agree to about 10^-10 on the sphere spectrum, and Herzog's near-critical series is resummed and reproduced. That is real evidence, not decoration.

Soft spots, in order of softness.

The global lowest-state claim is the load-bearing step that turns Delta(Q) into CFT data, and it is argued rather than proven: linear stability is local, the nodal-branch scan stops at three nodes, and inhomogeneous and spinning branches are unconstructed. The author flags this honestly ("arguably the lowest-energy one", "we expect the same ordering"), so it is a bounded, named gap; still, a referee should push on it.

There is a genuine small inconsistency in section 5.2. Eqs. (5.3) and (5.5) both give Delta(Q=2) = 4 + (1+lambda/e^2)/(6N), but the text states U = (1+lambda/e^2)/(12N) and "dimension 4+U" — a factor of two. The exact formula is unaffected, but the identification with the double-trace anomalous dimension and the claimed match with ref. [117] need to be rechecked and corrected.

Minor: no public code or commit hash, only "available on reasonable request". Everything is probe-limit, and the low-temperature regime T^4 >> 4 kappa5^2 Tc^4/(e^2 L^2) is stated explicitly as a condition. The Rényi section is the most conditional; it depends on the normalizable-mode discussion and on the interpretation of the homogeneous saddle, both scoped carefully by the author.

For whom: anyone working on holographic superfluids, the large-charge expansion, or charged Rényi entropies. I would bring it to reading group and cite it. My recommendation: accept and send to a referee who can check the sphere gap and the double-trace identification, asking for the factor-of-two fix and code release. Neither issue is fatal; the central result is solid.

Referee report

Significance

If the results hold, this is a valuable exact benchmark for a standard holographic superfluid model. The closed-form family is verified symbolically, the transport predictions are checked against independently computed quasinormal modes with relative agreement near 10^-7, the stability proof is general and clean, and the large-charge and Renyi-entropy results connect the exact solution to established effective-theory structures. The paper is unusually careful: the main numerical results are reproduced with two independent codes, the thermodynamic identities are checked against the renormalized action, and the limitations of the probe limit and of the global-minimality claim are stated. The residual issues are local consistency problems in the presentation rather than defects in the central derivations, with one important conditionality caveat: the identification of the soliton as the lowest state of charge Q is supported by local stability and a nodal-branch scan, but not proven globally.

Major objections (3)

[Section 2.2, Eq. (2.11)] The displayed charge density and condensate are inconsistent with the rest of the paper. On the planar brane the boundary expansion of A_t in (2.10) gives n = L mu^3/(2 e^2), and the first law together with Eq. (5.3) requires the sphere version n = L/(2 e^2) mu (mu^2 - 4/R^2). The formula printed as n = (L/e^2) mu^2 (mu^2 - 4 kappa/R^2) has an extra factor of mu and a missing factor of 1/2. Similarly, the condensate printed as L^{1/2} e p/sqrt(2) should be L^{1/2} p/(sqrt(2) e), as follows from e L rho = p u^2/(1 + mu^2 u^2/4) and the boundary expansion (2.5). This is a typographical error rather than a used result, but it must be corrected because Eq. (2.11) is the main output of the ansatz and is cited in Section 3.1 as the thermodynamic charge density.
[Section 5.2, small-charge paragraph] There is a factor-of-two inconsistency in the identification of U. From Eq. (5.3), the small-charge expansion is Delta/N = 2 q + q^2/16 + ... at the Bogomolny point, which gives Delta(Q=2) = 4 + 1/(4N). In the general-coupling form (5.5), Delta(Q=2) = 4 + (1 + lambda/e^2)/(6N). The text defines U = (1 + lambda/e^2)/(12N) and states that the charge-two operator has dimension 4 + U. The correct statement from the exact expansion is 4 + 2U. The sentence 'whose dimension is 4 + U' should be changed accordingly, or the definition of U should be adjusted so that the pair interaction U Q(Q-1)/2 matches the exact Delta(2).
[Sections 5.2 and 7] The conversion of Eq. (5.3) into CFT data rests on the unproved claim that the nodeless static spherical soliton is the lowest state of charge Q. The authors acknowledge this gap: local linear stability is a local-minimum statement, and the nodal scan covers static spherical branches with up to three nodes, while inhomogeneous branches, higher-nodal branches, and spinning states are not constructed. I do not regard this as a derivation error, and the paper already uses the word 'arguably' in the abstract. However, given that Delta(Q) is advertised as the lowest scaling dimension, the statement of Eq. (5.3) should explicitly say that it gives the dimension of the nodeless static spherical soliton within the probe sector, and that global minimality among all charge-Q states is a conjecture supported by local stability and the scanned branches. This conditionality should also be repeated at the abstract or introduction level, not only in the body. The authors acknowledge this: “Nodal branches.The energy argument shows that the soliton is a local minimum of the energy at fixed charge; it does not exclude solutions of lower energy.”

Minor objections (3)

[Section 4.2, text after Eq. (4.15)] The sentence containing 'Itshouldnotbecompared' is missing spaces; the wording should be cleaned up.
[Section 5.2, Eq. (5.4)] The statement that the second term is the interaction U Q(Q-1)/2 uses the large-Q approximation 1/2 U Q^2, but the same sentence then applies U to the charge-two operator. Since the exact Delta(2) contains 2U rather than U, the pair-interaction language should be reconciled with the charge-two statement in the same paragraph.
[Section 4.3, Eq. (4.20)] The low-temperature validity condition T^4 >> 4 kappa_5^2 T_c^4/(e^2 L^2) is stated in Section 7 and in the text near Eq. (4.20), but it would be helpful to repeat it explicitly at Eq. (4.20) itself, since that equation is one of the paper's headline transport results.

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The paper

The paper claims

At the Bogomolny point the holographic superfluid is solved in closed form on every static AdS5 background, yielding exact transport, large-charge, and Rényi-entropy predictions.

The reading

This paper claims that the standard holographic model of a strongly coupled superfluid — the Abelian Higgs model in five-dimensional anti-de Sitter space, treated in the probe limit where the matter does not bend the geometry — has a condensed phase in closed form, provided the scalar mass squared sits at the Breitenlohner–Freedman (BF) bound, the lowest value AdS stability permits, and the quartic self-coupling takes the Bogomolny value $\lambda = e^2/2$, where the vector and scalar fluctuation masses coincide. The solution (2.10) exists on every static, uncharged AdS$_5$ background whose spatial sections are maximally symmetric — planes, spheres, hyperbolic spaces, and axion-deformed branes — and each slicing answers a different question about the dual conformal field theory: the superfluid at all temperatures below $T_c$ on the planar brane, the lowest states of large charge on the sphere, and a homogeneous hairy phase for the black holes that compute charged Rényi entropies of a ball on hyperbolic space. The paper also proves a result that does not depend on the closed form: any nodeless probe-limit condensate in the standard quantization, with nonnegative quartic coupling and no superflow, has no exponentially growing linear mode. If these claims are right, a model previously treated numerically becomes a source of exact statements — the Gorter–Casimir two-fluid law holds exactly, the sound speed and attenuation are closed-form functions of temperature, the scaling dimension $\Delta(Q)$ at charge $Q$ is known for all $Q$, and the Rényi entropy phase sets in at index exactly one — giving benchmarks and sharp structure that numerics could only approximate.

What carries the argument

The load-bearing object is the closed-form member of the probe-limit Abelian Higgs sector, eq. (2.10), pinned down by two tunings: the scalar mass at the BF bound $m^2L^2 = -4$ and the quartic coupling at the Bogomolny value $\lambda = e^2/2$, where the vector and amplitude fluctuations have equal masses at the minimum of the potential and the potential becomes quadratic in the moment map $|\Psi|^2$ (the D-term form obtained by eliminating the auxiliary field of a U(1) vector multiplet). The radial problem then collapses to first-order equations, which at zero temperature reduce to a static sine-Gordon kink in the logarithm of the radial coordinate, $\tan(\vartheta/2) = \mu u/2$, and at all temperatures to the flow (2.21) with the constraint $c^2+\Phi^2 = 1$, whose residuals are proportional to the constraint so that every solution of the flow solves the field equations. The stability claim is carried by a separate mechanism: writing the scalar fluctuation as $\tau = \bar\rho w$ converts the indefinite quadratic form of the perturbation energy into a manifestly nonnegative expression $E_2 \ge 0$ (eq. 3.15), so a growing mode would have to be a zero-energy trivial mode.

Extended reading notes

Core claim

The central discovery is the exact family (2.10): with $m^2L^2 = -4$ and $\lambda = e^2/2$, the static condensed phase of the probe-limit Abelian Higgs model has $eL\bar\rho = p\,u^2/(1+\mu^2u^2/4)$ and $\bar B_t = \mu f(u)/(1+\mu^2u^2/4)$, where $f$ is the blackening function of the background, $p^2 = \tfrac12(\mu^2-\mu_c^2)(\mu^2+4/u_h^2)$, and $\mu_c^2 = 4(1/u_h^2+\kappa/R^2)$. Because the derivation uses only the facts that $f(0)=1$ and $f$ is quadratic in $u^2$, the same formula covers the planar black brane, global AdS$_5$ and its black holes, the hyperbolic black holes, and axion-deformed branes; the free energy of condensation has the mean-field form $-\frac{L}{8e^2}(\mu^2-\mu_c^2)^2$ on every slicing. At zero temperature the solution is a domain wall ending at an infrared fixed point where the broken-symmetry current and order parameter have dimensions $5$ and $2+2\sqrt3$; the coincidence of vector and amplitude masses there ($8/L^2$), a consequence of the Bogomolny coupling, is what makes the closed form close. The stability theorem, proved by rewriting the perturbation as $\tau = \bar\rho w$ and showing the perturbation energy is nonnegative, applies to any nodeless condensate with $\lambda\ge 0$ and no superflow, not just to the exact family. From these ingredients the paper derives exact transport on the plane ($n_s/n = 1-(T/T_c)^4$, $c_s^2 = \frac13[1-(T/T_c)^4]$, $\Gamma = 5T^5/(36\pi T_c^6)$), the closed-form dimension $\Delta(Q)$ on the sphere with its phonons and additional towers of states, and the closed-form Rényi-entropy contribution (6.4) on hyperbolic space, whose phase sets in at Rényi index exactly one at zero chemical potential.

Load-bearing premise

The load-bearing premise is the probe limit, $\kappa_5/(eL) \ll 1$, in which the superfluid fields do not backreact on the fixed AdS$_5$ metric; the paper states that where backreaction is not negligible, in particular for the sound formulas unless $T^4 \gg 4\kappa_5^2 T_c^4/(e^2L^2)$, the closed form, the infrared fixed-point dimensions, and the transport predictions all change, and the identification of the global soliton as the lowest state of charge $Q$ is argued only against static, spherically symmetric branches with up to three nodes.

Editorial extensions

If this is right

On the planar brane the superfluid is exactly two-fluid at every temperature below $T_c$: the superfluid fraction is $n_s/n = 1-(T/T_c)^4$, the normal component's DC conductivity is its normal-phase value times the normal fraction, and the sound speed $c_s^2 = \frac13[1-(T/T_c)^4]$, attenuation $\Gamma = 5T^5/(36\pi T_c^6)$, and third bulk viscosity all follow in closed form.
In global AdS$_5$, the dimension $\Delta(Q)$ of the lowest state of charge $Q$ is known in closed form for all $Q$ at leading order in $1/N$; it is convex and matches the superfluid effective theory with pressure $P = c_1(X-\frac23 R)^2$, and its large-charge expansion converges above a charge of order $N$ rather than being merely asymptotic.
The spectrum above the large-charge ground state contains, alongside the phonons, towers of states whose gaps move monotonically from the free-field values at $Q=0$ to the infrared fixed-point values (current dimension $5$, order-parameter dimension $2+2\sqrt3$) as $Q\to\infty$; these light states lie within the regime of validity of the superfluid effective theory but are not described by it.
On the hyperbolic slicing, the homogeneous hairy phase of the charged hyperbolic black holes contributes to the charged Rényi entropies of a ball in closed form (eq. 6.4); the phase sets in at Rényi index exactly one at zero chemical potential for any positive quartic coupling, and with normalizable modes only the normal phase is stable below $n_* \approx 6.58$.

Reading between the lines

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

The exact family gives a natural seed for a backreaction expansion: since the paper notes the backreaction is of relative order $N/C_T$, one could compute corrections to $\Delta(Q)$ and to the Wilson coefficients order by order around the closed form, turning the probe-limit benchmark into a systematic strong-coupling large-charge expansion.
The mechanism that makes the closed form close — equal vector and amplitude masses at an infrared fixed point, realized by a D-term potential — suggests a recipe for finding closed-form holographic superfluids elsewhere; the paper itself notes a top-down truncation has the same infrared spectrum, so this equal-mass structure may be more common than a closed form.
The charged Rényi-entropy result (6.4) is, by continuation to imaginary chemical potential, the input needed for symmetry-resolved entanglement of a ball in this holographic CFT; the paper leaves that continuation across the phase boundary undone, so performing it is a direct testable extension.
The convergence of the large-charge expansion above $Q \sim N$ is traced to a mean-field form shared with the weak-coupling semiclassics, so one may conjecture that convergence (rather than mere asymptoticity) of the leading large-charge data is generic to mean-field effective descriptions of the conformal superfluid, not a special feature of AdS.

Original abstract

The probe-limit Abelian Higgs model in AdS$_5$ is a standard holographic model of a strongly coupled superfluid, usually solved numerically. With the scalar mass at the Breitenlohner-Freedman bound and the quartic coupling at the Bogomolny value, its condensed phase has a closed form on every static, uncharged AdS$_5$ background with maximally symmetric spatial sections. We prove that it is the global minimum of the free energy at fixed charge, and linearly stable, as is any nodeless condensate of the model in standard quantization with nonnegative quartic coupling and no superflow. On the planar black brane it is the superfluid at all temperatures below $T_c$: the superfluid fraction follows the Gorter-Casimir two-fluid law exactly, and the speed and attenuation of sound are in closed form. In global AdS$_5$ it is the lowest state of charge $Q$ on $\mathbb{R}\times S^3$ within the model, with its scaling dimension in closed form at all $Q$, classically in the bulk. Its excitations include towers, from an infrared fixed point of the probe sector, that the superfluid effective theory does not describe. On the hyperbolic black holes that compute charged R\'enyi entropies of a ball, it is a homogeneous hairy phase whose condensate does not fall off toward the entangling surface. With a pinning field there, the phase sets in at R\'enyi index exactly one at zero chemical potential, as found for a neutral scalar. Well below a bound on the quartic coupling, the static, homogeneous solutions computed at other couplings and masses keep an infrared fixed point with towers, but not the exact two-fluid law.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-30: 1 request, 7,812 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.50 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.54 · compares with prior results 0.67 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The Abelian Higgs model in AdS$_5$, in the probe limit, is a standard holographic model of a strongly coupled superfluid, usually solved numerically. When the scalar mass saturates the Breitenlohner-Freedman bound and the quartic coupling takes the Bogomolny value, its condensed phase can be found in closed form on every static, uncharged AdS$_5$ background with maximally symmetric spatial sections. We prove that it is linearly stable, as is any nodeless probe-limit condensate in the standard quantization with a nonnegative quartic coupling and no superflow. On each slicing it answers a different question about the dual conformal field theory. On the planar black brane it is the superfluid at all temperatures below $T_c$: the superfluid fraction follows the Gorter-Casimir two-fluid law exactly, and the speed and the attenuation of sound are found in closed form. In global AdS$_5$ it describes a state of charge $Q$ on $\mathbb{R}\times S^3$, arguably the lowest-energy one, and gives its scaling dimension in closed form at all $Q$, classically in the bulk. Its excitations include towers that the superfluid effective theory does not describe; they come from an infrared fixed point of the probe sector. The hyperbolic slicing gives in closed form a homogeneous hairy phase of the black holes that compute the charged R\'enyi entropies of a ball. Its condensate does not vanish at the entangling surface. When sourced by operators inserted there, the phase sets in at R\'enyi index exactly one at zero chemical potential, as found for a neutral scalar.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The Abelian Higgs model in AdS$_5$, in the probe limit, is a standard holographic model of a strongly coupled superfluid, usually solved numerically.	background 1.00	0.05	0.04	asserted 1.00
s1	When the scalar mass saturates the Breitenlohner-Freedman bound and the quartic coupling takes the Bogomolny value, its condensed phase can be found in closed form on every static, uncharged AdS$_5$ background with maximally symmetric spatial sections.	result 0.65	0.43	0.46	asserted 0.99
s2	We prove that it is linearly stable, as is any nodeless probe-limit condensate in the standard quantization with a nonnegative quartic coupling and no superflow.	result 0.91	0.85	0.17	asserted 1.00
s3	On each slicing it answers a different question about the dual conformal field theory.	interpretation 0.93	0.37	0.08	asserted 0.99
s4	On the planar black brane it is the superfluid at all temperatures below $T_c$: the superfluid fraction follows the Gorter-Casimir two-fluid law exactly, and the speed and the attenuation of sound are found in closed form.	result 0.99	0.87	0.21	asserted 1.00
s5	In global AdS$_5$ it describes a state of charge $Q$ on $\mathbb{R}\times S^3$, arguably the lowest-energy one, and gives its scaling dimension in closed form at all $Q$, classically in the bulk.	result 0.93	0.84	0.13	asserted 0.73
s6	Its excitations include towers that the superfluid effective theory does not describe; they come from an infrared fixed point of the probe sector.	result 0.79	0.73	0.08	asserted 1.00
s7	The hyperbolic slicing gives in closed form a homogeneous hairy phase of the black holes that compute the charged R\'enyi entropies of a ball.	result 0.94	0.87	0.16	asserted 1.00
s8	Its condensate does not vanish at the entangling surface.	result 1.00	0.79	0.07	asserted 1.00
s9	When sourced by operators inserted there, the phase sets in at R\'enyi index exactly one at zero chemical potential, as found for a neutral scalar.	result 0.99	0.61	0.07	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The planar member of the family at fixed chemical potential

**Figure 2.** The soliton in global AdS5, eq. (2.10) with κ = 1 and uh → ∞: (a) the scalar, in units of the minimum ρ∗ = 2√ 2/eL of the potential, and (b) the gauge potential. 2.3 Structure The Bogomolny coupling. The potential V has its minimum at ρ 2 ∗ = −m2/λ. Around a homogeneous configuration ρ = ρ∗ the vector and the amplitude fluctuation have the masses m2 B = e 2 ρ 2 ∗ = − e 2m2 λ , m2 ρ = m2 + 3λρ2 ∗ = −2m2 , (2.15) which coincide precisely when λ = e 2/2, the Bogomolny coupling discussed in the introduction. At this coupling V = (e 2/2)(|Ψ| 2 + m2/e2 ) 2 up to a constan

**Figure 3.** Regular part of the optical conductivity on the exact background at fixed chemical

**Figure 4.** Longitudinal quasinormal modes on the exact background (dots). (a) Sound speed,

**Figure 5.** (a) The dimension per unit charge of the lowest state of charge

**Figure 6.** Gaps of the transverse vector modes above the lowest state of charge

**Figure 7.** Frequencies of the scalar-channel modes above the lowest state of charge

**Figure 8.** The phonon correction. (a) Zero-temperature sound on the plane:

**Figure 9.** (a) The contribution of the probe sector to the charged Rényi entropies of a ball,

Figures reproduced from arXiv:2610.40371 by the authors.

References

Discussion

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