{"id":"bc063d59-e8eb-494e-84a3-b27070be5bd2","arxiv_id":"2505.22033","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Charged EMs black holes show van der Waals-type phase transitions with mean-field critical exponents, and the phase transition disappears above a threshold scalar charge.","lead":"This paper analyzes the thermodynamics of charged Einstein-Maxwell-scalar black holes in anti-de Sitter space, finding van der Waals-like liquid-gas phase transitions and mean-field critical exponents. It also reports that large scalar charge suppresses the phase transition, a threshold effect that depends on the black hole configuration.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed scalar-charge threshold where phase transitions are 'prohibited' rests only on numerical non-detection of critical points; no proof rules out missed or unphysical roots for q>qt.","rationale":"The paper's finite-q phase-transition picture and the mean-field exponents are internally plausible: the P-V diagrams, swallowtail G(T,P) surfaces, and divergent C_P curves follow the standard extended-phase-space framework, and the expansion in eq. (58) has c1=c2 and c3>0, yielding the stated exponents. I would not reject the core result on that basis. My concern targets the threshold claim, which is the abstract's headline and is supported only by a finite numerical search. The proposed test decides the existence question directly and also probes whether the threshold depends on the specific-volume identification. On the reader's weakest-assumption point, I partially agree: the volume identifications in eqs. (37), (44), (54), and (56) are not reconciled, but this need not change the leading critical exponents when v(r0) is smooth and monotone near the critical point; it could, however, alter the threshold, so the root search should be repeated under each identification. I also note a separate stated inconsistency: Section III.A gives h=1-2M/r+g^2 r^2 at q=0 in eq. (32), i.e., Schwarzschild-AdS, while the conclusion claims the q to 0 limit reduces to RN-AdS; this is a presentational error that does not bear on the finite-q phase-transition calculation. On balance, the existing CONDITIONAL verdict is appropriate, so no verdict change is recommended.","tokens_in":968,"tokens_out":2345,"duration_ms":186325,"concrete_test":"Recompute, for Case 1, the critical-point condition F(r0;q)=0 obtained by eliminating T from dP/dr0=0 and d2P/dr0^2=0 using the exact P from eq. (35), scanning r0 in (0,100) and q in [0,3] with a high-precision root-finder; then verify with a Sturm sequence that F has no zero for q>qt. At every root, check that Q^2 from eq. (34) is positive and T_c>0, and repeat the root search under the three specific-volume identifications v=(2/3) l0^2(4 r0-1), v=2 lP^2 r0, and v=(8/3) pi lP^2 f_T1 to see whether the threshold value changes. If a physical root exists for q>qt, the prohibited claim is refuted; if no root exists under all identifications, the threshold is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central novelty, that phase transitions are prohibited for q exceeding qt about 2.0, 2.0, 1.9, and 2.2, is established in Section III.B only by the statement that the critical point is not found by numerical methods beyond these values, with Section III.C adding a qualitative P0/P1 expansion. This is not a proof. Because the equation of state is linear in T, P=A(r0)T+B(r0), the conditions dP/dr0=0 and d2P/dr0^2=0 eliminate T and reduce to a single equation F(r0;q)=0. Whether this one-dimensional equation has a root for q>qt is decidable, but no analytical or exhaustive numerical argument is supplied. The P1 to negative infinity behavior described in Section III.C does not by itself exclude an inflection point of the full P, and no check is reported that candidate roots for q>qt have physical Q^2>0 from eqs. (34), (40), (47), and (50) or T_c>0. Since the threshold is the principal new result highlighted in the abstract, this missing existence analysis is the most load-bearing weakness.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the P–V criticality of static charged Einstein–Maxwell–scalar black holes in the extended phase-space formalism. Using the four solution families of Huang–Fan–Lü (D=4, Δ=1 with k0^2=1,3,5, and D=5, Δ=2), the authors write the equation of state P(r0,T), the Gibbs free energy, and the heat capacities, and report van der Waals-type first-order phase transitions for representative parameters (q=0.05). They give numerical critical points, claim thresholds qt≈2.0,2.0,1.9,2.2 beyond which criticality is absent, and derive critical exponents α=0, β=1/2, γ=1, δ=3 from a pressure expansion. They also study how the critical radius and critical temperature depend on q, identifying several transition points in these q-dependences.","tokens_in":20934,"tokens_out":7530,"duration_ms":75505,"significance":"If substantiated, the paper would add another family of AdS black holes to the universality class of van der Waals/mean-field critical behavior and would identify the scalar charge as a control parameter that can switch off the transition. The paper is transparent in its formalism: thermodynamic quantities are written out explicitly, the critical-exponent derivation is a generic Landau-type expansion rather than a fit to a target result, and the numerical critical points are concrete and reproducible in principle. Its limitations—the unsupported threshold claim and the inconsistent specific-volume identification—are concentrated in Sections III.B and III.C and affect the paper's central novelty. These issues are fixable within the scope of the manuscript, so the appropriate decision is major revision rather than rejection.","major_comments":[{"comment":"The statement that 'the critical point is not found by numerical methods when the scalar charge exceeds threshold values, which are approximately 2.0 (Case 1), 2.0 (Case 2), 1.9 (Case 3), and 2.2 (Case 4). This conclusively implies the absence of van der Waals-type criticality' is an argument from non-detection, not a proof. Since every equation of state here is linear in T, the conditions ∂P/∂r0=0 and ∂²P/∂r0²=0 eliminate T and reduce to one equation F(r0;q)=0 for each case. The paper does not analyze this equation for q>qt, does not exhaustively scan for roots, and does not check that candidate roots give Q²>0 using (34), (40), (47), or (50) together with Tc>0. The discussion of P0 and P1 in Section III.C and Fig. 4 shows only that the first-order term diverges to negative infinity; it does not exclude an inflection point of the full P. Because the threshold is the principal new result highlighted in the abstract, this missing existence analysis is load-bearing.","section":"III.B (threshold claim)"},{"comment":"The paper uses several distinct identifications of the specific volume: v=(2/3)l0²(4r0-1) in Eq. (37), v=2l_P² r0 in Eqs. (44) and (49), v=(4/3)l_P² r0 in Eq. (54), and finally v=(8/3)π l_P² f_{T1}(r0) in Eq. (56). The last identification is asserted to hold for all four cases, but f_{T1} is defined only in the T=f_{T1}g²+f_{T2} parametrizations of Cases 2–4, Eqs. (41), (48), and (51); Case 1 is not written in that form, so Eq. (56) is undefined for Case 1. The paper also asserts that the phase-transition point and trends are unchanged and that f_{T1} is monotonic 'within the range of r0 utilized', but no proof or explicit range is given. Since the critical expansion in Section IV uses φ=f_{T1}(r0)/f_{T1}(r0c)-1, all quantitative critical parameters and threshold statements depend on this identification. This inconsistency must be resolved.","section":"III.B–III.C (specific volume)"},{"comment":"The derivation of the exponents in Eq. (63) requires c2≠0 and c3≠0 in the expansion (58), and the coefficients in Eq. (65) are evaluated numerically for q=0.05 only. The paper does not establish that c2 and c3 remain non-zero along the critical branches all the way to qt, where the phase transition is claimed to disappear. If either coefficient vanishes at some q<qt, the standard Landau exponents no longer follow in that region. The claim of dimension independence is also based on only two spacetime dimensions (D=4,5). These extrapolations should either be proven or explicitly flagged as conjectural.","section":"IV (exponent range)"}],"minor_comments":[{"comment":"Typographical errors should be corrected: 'trems' in Eqs. (35) and (49), 'exapmles' after Eq. (57), 'respresent' in the Fig. 3 caption, and 'Einstei n' in the title.","section":"Throughout"},{"comment":"The notation for the Planck length is inconsistent: l0 is used in Eqs. (36)–(37) while l_P is used elsewhere; the physical dimensions of the approximate specific volumes should be checked.","section":"Eqs. (36)–(37)"},{"comment":"The limit is evaluated as a '0/0 = finite value' without a careful limiting procedure; the q→0 reduction of Q² should be written out explicitly.","section":"Eq. (30)"},{"comment":"The figures show only parts of the q-range for some cases, making the claimed 'volume transition point' and 'temperature transition point' hard to verify; a table of the transition points and full-range plots would help.","section":"Figs. 5 and 6"},{"comment":"The 'proof' of the first law is presented as a chain of formal differentials; it would be clearer to state that the thermodynamic quantities are checked to satisfy the extended first law, rather than deriving it from dM=dM/dQ dQ.","section":"Eq. (25)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a competent but routine application of the standard extended-phase-space method to known EMs solutions. Its main novelty, the scalar-charge threshold, is currently supported only by numerical non-detection, and the volume identification is inconsistent. If the authors add a genuine existence analysis for the critical-point equation and reconcile the specific-volume definitions, the paper could be publishable. The number of self-citations is not excessive, but some are not essential. The English and figures need a careful editing pass."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a competent application of the standard Kubiznak-Mann extended phase space machinery to four families of charged Einstein-Maxwell-scalar black holes. The van der Waals–type behavior and mean-field exponents are real, but the paper's most advertised result — a scalar-charge threshold above which phase transitions vanish — is supported only by numerical non-detection, not by an actual existence argument.\n\nThe derivations are explicit: for each of the four cases the authors compute Q^2 from h(r0)=0, write T and P as functions of (r0,q), locate the critical point numerically, show the swallowtail in G and the divergence in C_P, and then derive the usual exponents α=0, β=1/2, γ=1, δ=3 from a generic Landau expansion. The expansion coefficients are computed for each case. That part is solid and reproducible. The new content is the q-dependence of r0c and Tc for these particular solutions, including the non-monotonic behavior and the claimed threshold. The paper is honest about using the Huang-Fan-Lu solution as input, and the self-citations are not an issue here.\n\nThe soft spots are real. First, the threshold. Because P is linear in T, the critical point conditions dP/dr0=0 and d^2P/dr0^2=0 reduce to a single algebraic equation F(r0;q)=0. Saying 'critical point is not found by numerical methods' above q≈2 is not a proof that none exists. The P0/P1 expansion in Section III.C is qualitative; a P1 that diverges to −∞ doesn't exclude an inflection point of the full P. Since the threshold is the headline, this needs either an analytic root analysis or at least an exhaustive numerical scan with checks that candidate roots give physical Q^2>0 and T_c>0.\n\nSecond, the specific volume identification. The paper uses different v identifications in different cases (eqs. 37, 44, 54), then in Section III.C asserts a unified v=(8/3)π l_P^2 f_T1 without reconciling the earlier choices. The critical exponents are derived using the unified φ=f_T1(r0)/f_T1(r0c)−1, so the earlier identifications look vestigial, but the paper should say so explicitly and justify monotonicity over the full relevant range.\n\nThird, the q→0 claim in the conclusions conflicts with the paper's own equations. Eq. (30) gives Q^2 ~ q^{D−3} times a finite constant, and eq. (32) shows h→1−2M/r+g^2r^2, which is Schwarzschild-AdS, not RN-AdS. The conclusion says the solution reduces to RN-AdS as q→0; that's wrong. There are also minor cross-reference typos (e.g., 'equations (49) and (50)' in Case 4 should be (50) and (51)).\n\nOverall: the central phase-transition phenomenology holds up for these four families; the threshold result is the weakest link and is currently over-claimed. This deserves a serious referee — the paper is useful and the flaws are addressable. Send it to review, but the referee should insist on the threshold being either proven, or clearly labeled as numerical evidence.","headline":"Competent extended-phase-space analysis of four EMs black hole families; the vdW behavior is real, but the headline claim of a scalar-charge threshold rests on numerical non-detection rather than proof.","tokens_in":21436,"tokens_out":3422,"would_cite":false,"duration_ms":34686,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.70.-s","05.70.-a"],"model":"deepseek-v4-flash","headline":"This paper claims that four static charged Einstein-Maxwell-scalar black-hole families undergo a van der Waals-type first-order phase transition with critical exponents (α, β, γ, δ) = (0, 1/2, 1, 3), and that raising the scalar charge q…","keywords":["van der Waals-type phase transition","AdS spacetime","EMs black hole","critical exponents","scalar charge threshold","extended phase space","black hole thermodynamics","mean field theory"],"falsifier":"Solve the exact inflection system (33) for each case using the exact P(T,r0) from Eqs. (35), (42), (48), and (52) at scalar charges just above the claimed thresholds, for example q=2.1 in Case 1 and q=2.3 in Case 4. If a common root P'=P''=0 exists and the swallowtail persists above the threshold, the transition is not actually prohibited; if no root exists just below the threshold, the phase transition would already be absent in a regime where the paper reports it.","tokens_in":20479,"feed_emoji":"🕳️","tokens_out":8524,"duration_ms":83470,"temperature":0.7,"pith_summary":"Einstein-Maxwell-scalar (EMs) theory couples a scalar field to the Maxwell field, and this paper asks whether its charged black holes have liquid-gas-like phase transitions in the extended phase space where the cosmological constant acts as pressure. Using four explicit solution families (three in four dimensions, one in five), the authors derive the pressure-volume equation of state from the horizon temperature and show that the P-V curves, Gibbs free energy surfaces, and isobaric heat capacities all carry the signatures of a first-order van der Waals transition. Expanding the pressure near criticality gives the mean-field exponents α=0, β=1/2, γ=1, and δ=3, the same as a van der Waals gas and as RN-AdS black holes. The new qualitative result is that the scalar charge acts as a switch: below a threshold q≈2.0, 2.0, 1.9, 2.2 the transition exists, and above it the transition disappears, while the critical volume and temperature depend non-monotonically on q.","feed_headline":"Scalar charge kills black-hole phase transition past a threshold","feed_subtitle":"Four EMs black-hole families obey van der Waals critical exponents until the scalar charge crosses a threshold.","key_machinery":"The central object is the equation of state P(T,r0) assembled from each solution's metric function and temperature, with the specific volume v tied to the horizon radius rather than to the geometric thermodynamic volume; the paper later adopts the uniform identification v=(8/3)πl_P² f_{T1}(r0). The phase transition is diagnosed by the inflection conditions on P(r0), namely ∂P/∂v=0 and ∂²P/∂v²=0, together with the swallowtail in the Gibbs free energy and divergence of the isobaric heat capacity. Near the critical point the pressure is expanded as p=1+c1 t−c2 tφ−c3 φ³ in the reduced variables t=T/Tc−1, φ=v/vc−1, p=P/Pc. Maxwell's equal-area construction forces the coexisting branches to satisfy φ_l=−φ_s, which gives t ∝ φ², and the positivity of c2 and c3 fixes the four mean-field critical exponents and the divergence of C_P.","core_discovery":"The central claim is that each of the four EMs black-hole solutions examined here has a first-order phase transition in the P-V plane of the same kind as a van der Waals fluid, and that the transition's critical exponents are exactly (α, β, γ, δ) = (0, 1/2, 1, 3). The authors compute T(r0) and P(T,r0) from the solution data, impose the inflection conditions, and verify swallowtail behavior of G(T,P) and divergence of C_P. A pressure expansion of the form p=1+c1 t−c2 tφ−c3 φ³ with positive coefficients, combined with Maxwell's equal-area law, gives φ_l=−φ_s and hence t ∝ φ², which fixes the four exponents. They also find that the transition is lost when the scalar charge exceeds approximate thresholds of 2.0 (Cases 1 and 2), 1.9 (Case 3), and 2.2 (Case 4), while in the q→0 limit the solutions reduce to the RN-AdS family and at q=0 to Schwarzschild-AdS. The critical radius and critical temperature each have their own transition points in q where monotonic behavior reverses, which the authors interpret as small scalar charge acting as a perturbation while large scalar charge cannot be treated that way.","pith_inferences":["The authors report the threshold qt only numerically; an analytic proof that the inflection system (33) has no common root for q>qt would turn the numerical threshold into a theorem.","The volume identification is case-dependent in the early sections and only later unified through f_{T1}(r0); if f_{T1} stops being monotone in r0 outside the studied range, the interpretation of r0 as a fluid volume, and therefore the exponent derivation, would need revisiting.","Because the exponent argument relies mainly on the pressure being linear in temperature and on c2,c3≠0, the same mean-field exponents should carry over to other static charged EMs solutions, including different horizon topologies and higher dimensions, wherever that expansion form holds.","The threshold behavior suggests a testable analog in holographic or condensed-matter models: a sufficiently large scalar charge should suppress the phase transition, which could be probed by computing transport coefficients across the would-be critical point."],"forward_implications":["If the central claim holds, four distinct EMs black-hole families join the same mean-field universality class as van der Waals fluids and RN-AdS black holes, with critical exponents independent of spacetime dimension.","The scalar charge acts as a control parameter: phase transitions persist only below q≈2.0, 2.0, 1.9, 2.2, so EMs theory provides a concrete setting in which scalar hair suppresses thermodynamic criticality.","The critical volume and critical temperature vary non-monotonically with q and have distinct transition points in q, meaning the scalar charge and horizon radius are not interchangeable thermodynamic variables.","As q→0 the black holes reduce to the RN-AdS family, so the known van der Waals behavior of RN-AdS is recovered as the scalar-charge-free limit, while q=0 gives Schwarzschild-AdS.","Positivity of the isobaric heat capacity in all four cases implies thermal stability for the black holes in the regime where the phase transition exists."],"supporting_citations":[{"why":"Supplies the static charged EMs black-hole solutions for the four cases, the thermodynamic quantities, and the verification of the first law that the analysis builds on.","marker":"[40]"},{"why":"Establishes the extended-phase-space van der Waals treatment for RN-AdS black holes, including the specific-volume identification, swallowtail analysis, and critical-exponent comparison used throughout.","marker":"[8]"},{"why":"Provides the extended-phase-space interpretation of the cosmological constant as thermodynamic pressure P=−Λ/8π and the mass as enthalpy, on which the P-V equation of state is based.","marker":"[7]"}],"fun_headline_variants":["Black-hole phase transition mirrors van der Waals until scalar charge threshold","Scalar charge threshold extinguishes black-hole phase transitions","EMs black holes obey van der Waals exponents until scalar charge cutoff","Charged black holes mimic gas-liquid transition until scalar charge threshold"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that identifying the black hole horizon radius with the fluid's specific volume gives the true phase structure, so that inflection points of P(r0) are the real phase-transition points; if this identification fails beyond the small-scalar-charge regime, the critical points, exponents, and threshold could be artifacts of the choice of volume variable.","fun_headline_variants_meta":{"raw":{"variants":["Black-hole phase transition mirrors van der Waals until scalar charge threshold","Scalar charge threshold extinguishes black-hole phase transitions","EMs black holes obey van der Waals exponents until scalar charge cutoff","Charged black holes mimic gas-liquid transition until scalar charge threshold"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000565,"raw_usage":{"total_tokens":2740,"prompt_tokens":1069,"completion_tokens":1671,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":685,"completion_tokens_details":{"reasoning_tokens":1598}},"tokens_in":685,"tokens_out":1671,"duration_ms":12192,"temperature":1.0,"reasoning_tokens":1598,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:17:08.907578+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the exact inflection system (33) for each case using the exact P(T,r0) from Eqs. (35), (42), (48), and (52) at scalar charges just above the claimed thresholds, for example q=2.1 in Case 1 and q=2.3 in Case 4. If a common root P'=P''=0 exists and the swallowtail persists above the threshold, the transition is not actually prohibited; if no root exists just below the threshold, the phase transition would already be absent in a regime where the paper reports it.","supporting_citations":[{"cited_title":"Dilatonic p-brane solitons","cited_arxiv_id":"hep-th/9511203","evidence_quote":"Supplies the static charged EMs black-hole solutions for the four cases, the thermodynamic quantities, and the verification of the first law that the analysis builds on."}],"review_version":1}