{"id":"d408834b-a438-417a-9326-c15a6833765a","arxiv_id":"2608.09056","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"All holographic multipartite entanglement quantities in gapless planar systems share a single large-distance scaling exponent fixed by the infrared geometry, ranging from decay to volume-law growth.","lead":"This paper derives a rule for how multipartite quantum entanglement grows or decays over long distances in strongly interacting gapless systems described by holography. It predicts that, depending on the infrared geometry, this long-range entanglement can decay, stay constant, grow subextensively, or even scale with the system volume.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Universal multipartite scaling rests on an omitted shape-invariance proof for RT networks; as written, Eq. (2.24) is only proven for single RT surfaces. Supply the network-level verification or restrict the claim.","rationale":"The reader's weakest assumption is exactly where the central claim is least secured: Eq. (2.23) is asserted for every segment of an arbitrary RT network, but only a single-arch check is indicated and then waived. This is a genuine completeness gap, not a disagreement with consensus. My own check suggests the gap is probably closable: under (2.19) the reduced area element for a graph x_i(r) picks up a common factor lambda^{1+alpha_V/D}, which would imply shape invariance, but the paper does not present this, and the junction and force-balance part of the argument is not automatic when D != -1. A wrong sign in Section 4 (1-alpha_i+alpha_r = z_i instead of -z_i) is a secondary but real correctness defect. These issues do not overturn the result; they make the paper conditional on supplying the missing proof and correcting the sign. I therefore keep the reader's CONDITIONAL verdict rather than escalating to rejection. The paper does have independent support in the dominated-convergence argument for the single-surface l-r_star relation and in prior numerical studies of topological semimetal holography, so the concern is about completeness of the general proof rather than about the plausibility of the scaling law.","tokens_in":23005,"tokens_out":32568,"duration_ms":320011,"concrete_test":"Perform the omitted verification analytically: substitute the power-law metric (2.13) and the rescaling (2.19) into the area functional for a general graph segment x_i(r), and show that every segment type (arch, vertical wall, and Steiner junction force-balance condition) acquires the same prefactor lambda^{1+alpha_V/(1-alpha_i+alpha_r)}. If the prefactor is common, re-derive Eq. (2.23) for all segment types and Eq. (2.24) follows. If a junction angle in the reduced (x_i,r) metric fails to be invariant for z_i != 1, the universality claim already fails for Lifshitz IR geometries with multipartite networks.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Eq. (2.24), and its proof depends on the assertion, immediately after Eq. (2.20), that the entire minimal-surface network is shape-invariant under the rescaling (2.19). The authors explicitly state that the verification is lengthy and omit it. This matters because Eq. (2.20) is only the single-arch segment check. A general network also contains vertical-wall EWCS segments and Steiner-type junctions, whose endpoints are not simply r_star and whose force-balance conditions must be preserved by the rescaling. If shape invariance fails for any segment type or junction, Eq. (2.23) does not apply to that segment, and the universal exponent in Eq. (2.24) is not established for multipartite quantities; only single RT-surface entropies would be covered. The gap is likely fillable, since the reduced area functional for a graph x_i(r) transforms covariantly under (2.19) with the common prefactor lambda^{1+alpha_V/(1-alpha_i+alpha_r)}, but that computation must be written down, including the junction and force-balance step, before the universality claim is proven. Relatedly, Section 4 states 1-alpha_i+alpha_r = z_i for anisotropic HV; the correct combination is -z_i, a sign error that should be corrected even though it does not affect the z_i=0 AdS3 x R2 conclusion.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the large-distance scaling of multipartite entanglement in holographic gapless systems. It proposes that, for strip configurations in a static asymptotically-AdS bulk whose IR metric behaves as g_ii ∝ r^{2α_i}, g_rr ∝ r^{2α_r}, every nonvanishing entanglement quantity built from a finite network of minimal surfaces scales as Q ∝ λ^{1+α_V/(1-α_i+α_r)} under a common rescaling of all subregion sizes. It then applies this formula to hyperscaling-violating IR geometries, predicting decay, logarithmic, subextensive, or volume-law long-distance multipartite entanglement depending on the effective dimension d_eff, and to an anisotropic AdS_3×R^2 IR geometry, where long-range entanglement survives only along the gapless direction. The paper includes a proof outline via dominated convergence and a shape-invariance argument, and illustrates the results with the tripartite information I_3 and with previous numerical observations.","tokens_in":23232,"tokens_out":13634,"duration_ms":142643,"significance":"The claimed result, if fully established, would be a strong and falsifiable statement: it would reduce the IR scaling of many distinct holographic entanglement measures to a single exponent fixed by the IR metric, and it would provide a concrete nonlocal diagnostic of the IR universality class. The derivation is mostly analytic and does not fit free parameters to data; the citations to previous numerical work are motivational rather than load-bearing. The classification of hyperscaling-violating and anisotropic phases is clear, and the conditions under which long-range multipartite entanglement is enhanced are explicit. The main weaknesses are that the proof of the network-level statement is incomplete at a load-bearing point, and there is a sign error in the anisotropic section; both appear fixable within the manuscript's scope.","major_comments":[{"comment":"The central universality claim rests on the assertion, immediately after Eq. (2.20), that the entire minimal-surface network is shape-invariant under the rescaling (2.19). This is the load-bearing step, and it is omitted. Eq. (2.20) only checks invariance of the x_i-span of a single arch-shaped RT segment; it does not verify the scaling of vertical-wall EWCS segments once their endpoints on the RT surfaces are included, nor the scaling of Steiner-type junction nodes and their force-balance conditions. If shape invariance fails for any segment type or junction, Eq. (2.23) applies only to single RT surfaces and Eq. (2.24) is not established for multipartite quantities. Please supply the full verification, including the endpoint and junction steps, or explicitly restrict the claim to single RT surfaces.","section":"Section 2.2, Eqs. (2.19)–(2.24)"},{"comment":"The text states that 'the scaling components combination 1−α_i+α_r = z_i' for the anisotropic hyperscaling-violating metric. From the metric (3.1), α_i = z_i − θ/d and α_r = −1 − θ/d, so the correct combination is 1 − α_i + α_r = −z_i. As written, substituting z_i into the denominator of (2.24) would produce the wrong exponent; the entries of Table 3 appear to use the correct sign. Please correct this statement and check all formulas in Section 4 for consistency with the sign.","section":"Section 4, around Table 3"}],"minor_comments":[{"comment":"The dominated-convergence steps rely on asserted monotonicity and integrability of the dominating functions; these are plausible but should be verified in a sentence or two so that the asymptotic derivation is fully self-contained.","section":"Section 2.2, Eqs. (2.14) and (2.17)"},{"comment":"The numerical statements about c_x, c_z, and I_3 are reported without plots, parameter values, or fitting details; please include the relevant data or a precise description of the computation, even if it is a summary of previous work.","section":"Section 4, after Eq. (4.5)"},{"comment":"There are several typos: 'we we prove' at the end of the Introduction, 'hologaphy' in Section 2.1, 'qunatity' in the note to Table 3, and 'effect spatial dimension' in Section 3.2; these should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of JHEP and the central idea is attractive. My main concern is the omitted network-level shape-invariance proof, which is load-bearing for the universality claim; if the authors supply that verification and fix the sign error in Section 4, the paper would be suitable for publication. I do not see grounds for rejection, since the gap appears fillable and the rest of the analysis is coherent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new claim is Eq. (2.24): any nonvanishing holographic multipartite quantity built from a finite RT network in a strip configuration scales with one universal IR exponent, fixed by the power-law bulk metric. That is a real step beyond the authors' earlier numerical observations for specific measures and beyond the known entropy scaling in HV geometries. The paper is well organized, the regime classification in Table 1 is useful, and the explicit I3 formula in Eq. (3.4) makes the general claim concrete. The anisotropic AdS3 x R2 example, with directional gapping, is a nice application.\n\nThe main soft spot is the proof of Eq. (2.24). After Eq. (2.20) the authors state that shape invariance of the whole minimal-surface network under the rescaling (2.19) is verified by a lengthy computation and omit it. This is not a cosmetic omission. Eq. (2.20) is an arch-segment check; a general network also has vertical-wall EWCS segments and Steiner-type junctions, whose endpoints and force-balance conditions need to preserve the rescaling. If that fails, Eq. (2.24) covers only single RT surfaces and the universality claim collapses. I suspect the gap is fillable--the reduced area functional should transform covariantly with the stated prefactor--but the computation must be written down. As is, the central theorem is conditional.\n\nTwo smaller issues. In Section 4, the text says 1 - alpha_i + alpha_r = z_i for the anisotropic HV metric. Direct substitution from Eq. (3.1) gives -z_i. The z_i = 0 conclusion is unchanged, but the sign should be fixed. And the anisotropic section reports fits for c_z proportional to l_z^2 and I_3 constants without showing the numerical data or the fitting procedure; those claims are hard to check as written.\n\nThe citations to the authors' previous numerical work are motivational, not load-bearing, so the circularity burden is low. The dominated convergence argument is plausible; I did not find a problem there.\n\nWho is this for: people working on holographic entanglement measures, IR geometries, or long-range multipartite entanglement. It deserves a serious referee. I'd send it to review with the request that the omitted shape-invariance proof be supplied (or the claim restricted to single RT surfaces until it is), and that the sign and numerical presentation be cleaned up.","headline":"A genuine scaling-universality claim for multipartite RT quantities, but the proof rests on an omitted shape-invariance check for full networks; worth refereeing if that check is supplied.","tokens_in":23817,"tokens_out":3514,"would_cite":true,"duration_ms":30841,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"All nonvanishing holographic multipartite entanglement quantities for strips share one large-distance power-law exponent fixed by the IR geometry.","keywords":["multipartite entanglement","long-range entanglement","holography","hyperscaling violation","infrared geometry","Ryu-Takayanagi surface","entanglement wedge cross section","gapless quantum systems"],"falsifier":"Compute, in a concrete asymptotically-AdS-to-HV interpolating geometry, the areas of distinct components of a minimal-surface network (arch, straight legs, EWCS, Steiner junction) at several large strip widths, and check whether each component scales exactly as $\\lambda^{1+\\alpha_V/(1-\\alpha_i+\\alpha_r)}$; if the EWCS or Steiner segment has a different leading $\\lambda$-scaling, Eq. (2.24) fails for multipartite quantities built from those components even though it holds for entanglement entropy.","tokens_in":22738,"feed_emoji":"⚛️","tokens_out":7668,"duration_ms":74015,"temperature":0.7,"pith_summary":"The paper tries to establish a scaling rigidity: in any static, translationally invariant holographic gapless system, every nonvanishing multipartite entanglement quantity built from finite networks of minimal surfaces for strip configurations has the same leading large-distance power-law exponent, namely $1+\\alpha_V/(1-\\alpha_i+\\alpha_r)$, fixed by the deep IR geometry. If true, this means long-distance multipartite entanglement in strongly coupled gapless states is not a collection of independent behaviors but one unified phenomenon whose strength is controlled by the IR scaling exponents. The authors then use hyperscaling-violating IR geometries to show this shared exponent can be tuned: entanglement decays for effective dimension $d_{\\rm eff}>1$, becomes logarithmic at $d_{\\rm eff}=1$, grows subextensively for $0<d_{\\rm eff}<1$, and reaches a volume law at $d_{\\rm eff}=0$. They also analyze an anisotropic $\\mathrm{AdS}_3\\times\\mathbb{R}^2$ IR geometry, where long-range multipartite entanglement survives only along the gapless $z$-direction and becomes area-law in the transverse gapped directions. A sympathetic reader would care because it gives a nonlocal, order-parameter-like diagnostic of the IR universality class from entanglement scaling alone.","feed_headline":"One IR exponent rules all long-range multipartite entanglement","feed_subtitle":"Hyperscaling-violating infrared geometries make the shared exponent tunable, from decay to volume law.","key_machinery":"The load-bearing object is the minimal-surface network together with the IR power-law metric data $(\\alpha_i,\\alpha_r,\\alpha_V)$. For strips, each RT surface is an arch with turning point $r_*$, and $l_i\\propto r_*^{1-\\alpha_i+\\alpha_r}$; the scaling transformation $x_i\\to\\lambda x_i$, $r\\to\\lambda^{-1/(1-\\alpha_i+\\alpha_r)}r$ rescales boundary widths and radial depth together, preserving the network shape. The claim that every segment of the network, including entanglement wedge cross sections and Steiner junctions, scales as $\\lambda^{1+\\alpha_V/(1-\\alpha_i+\\alpha_r)}$ is what carries the unification; for a single RT surface it reduces to the entropy exponent $l^{1-d_{\\rm eff}}$ in hyperscaling-violating geometries.","core_discovery":"The central discovery is Eq. (2.24): for any entanglement quantity $Q$ constructed directly from combinations of minimal surfaces (RT surfaces, entanglement wedge cross sections, Steiner-type junctions) for strip configurations, without leading cancellation, $Q \\propto \\lambda^{1+\\alpha_V/(1-\\alpha_i+\\alpha_r)}$ at large common scale $\\lambda$. The exponents come from the deep-IR power-law metric $f_i\\propto r^{2\\alpha_i}$, $f_V\\propto r^{2\\alpha_V}$, $f_r\\propto r^{2\\alpha_r}$, and the combination is gauge invariant. The proof argument: deep turning points $r_*\\to 0$ make the tail outside the IR region negligible (by dominated convergence), then a rescaling $x_i\\to\\lambda x_i$, $r\\to\\lambda^{-1/(1-\\alpha_i+\\alpha_r)}r$ leaves the entire minimal-surface network shape-invariant, so every segment area scales by the same factor. Consequently all nonvanishing multipartite quantities---entanglement entropy, mutual information, $n$-partite information, Markov gap, $\\kappa$, EWCS, multi-EWCS---share one leading IR exponent; differences appear only in coefficients, subleading terms, transition points, and possible leading cancellations.","pith_inferences":["The omitted shape-invariance verification is the point to stress-test: if it fails for networks containing entanglement wedge cross sections or Steiner junctions, the universality would survive only for quantities built from single RT surfaces, and the claimed unification of $I_3$, Markov gap, and $\\kappa$ would not follow.","The same exponent rigidity may extend beyond strip shapes: for ball-shaped regions, curvature corrections would introduce shape-dependent subleading terms, but the leading IR exponent would likely still be controlled by the same $\\alpha$-data, suggesting a shape-independent IR entanglement fingerprint.","Because the exponent is shared while coefficients are not, a tensor-network or quantum-simulation test for volume-law multipartite entanglement near $d_{\\rm eff}=0$ could distinguish hyperscaling-violating IR phases from semi-local criticality, which has no spatially extensive entanglement channel."],"forward_implications":["In any holographic gapless state with a given IR geometry, measuring one nonvanishing strip-based multipartite quantity (say tripartite information) fixes the leading large-distance scaling of all other multipartite quantities; no independent tunability of IR exponents remains.","For HV IR geometries with effective dimension $d_{\\rm eff}$, long-range multipartite entanglement scales as $l^{1-d_{\\rm eff}}$; choosing $d_{\\rm eff}$ between 0 and 1 produces scale-growing subextensive entanglement, and $d_{\\rm eff}=0$ gives a volume-law contribution to ground-state entanglement.","At $d_{\\rm eff}=1$, balanced quantities such as $I_3$, the Markov gap, and $\\kappa$ become constant while unbalanced ones like entanglement entropy and multi-entropy grow logarithmically; both are manifestations of the same universal exponent after accounting for UV divergences.","In the anisotropic $\\mathrm{AdS}_3\\times\\mathbb{R}^2$ IR geometry, long-range multipartite entanglement along the $z$-direction is CFT$_2$-like (logarithmic or constant), while along the $x,y$ directions it undergoes an RT phase transition to area law, so directional entanglement scaling diagnoses which spatial directions remain gapless.","The large-distance scaling of entanglement quantities can be used to read off the IR scaling exponents $\\alpha_i,\\alpha_r,\\alpha_V$, providing an entanglement-based reconstruction of the leading IR geometry."],"supporting_citations":[{"why":"Ryu-Takayanagi formula: boundary entanglement entropy equals area of a bulk minimal surface, the basic ingredient of every quantity considered.","marker":"[8]"},{"why":"Defines the sign-adjusted $n$-partite information $(-1)^n I_n$ in HEGMEC configurations, used as the concrete $I_3$ example.","marker":"[9]"},{"why":"Defines the holographic tripartite multi-entropy as a minimal Steiner network, needed for $\\kappa$.","marker":"[10]"},{"why":"Defines the Markov gap in terms of reflected entropy and mutual information, evaluated as $2E_W - I$.","marker":"[15]"},{"why":"Defines the multipartite entanglement wedge cross section, the object behind $\\Delta W$ and $g_W$.","marker":"[18]"},{"why":"Supplies the hyperscaling-violating geometry family used as the tunable IR background.","marker":"[22]"},{"why":"Gives the null energy condition constraints on HV parameters that bound the physical parameter regime.","marker":"[37]"},{"why":"Provides numerical confirmation in holographic nodal line semimetals that $l\\propto r_*^{1-\\alpha_i+\\alpha_r}$, supporting the asymptotic power-law relation.","marker":"[31]"},{"why":"Provides the magnetic-brane solution whose IR is $\\mathrm{AdS}_3\\times\\mathbb{R}^2$, used for the anisotropic example.","marker":"[44]"}],"fun_headline_variants":["Single exponent governs all long-range multipartite entanglement","One IR exponent sets the scale for every multipartite entanglement measure","Universal scaling exponent for multipartite entanglement in gapless holography","From decay to volume law: one exponent controls multipartite entanglement"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that when the whole configuration is blown up by $\\lambda$, every piece of the network of minimal surfaces---not just the arch for a single strip---keeps exactly the same shape after rescaling the depth coordinate, a step the paper states but does not display the derivation of.","fun_headline_variants_meta":{"raw":{"variants":["Single exponent governs all long-range multipartite entanglement","One IR exponent sets the scale for every multipartite entanglement measure","Universal scaling exponent for multipartite entanglement in gapless holography","From decay to volume law: one exponent controls multipartite entanglement"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000492,"raw_usage":{"total_tokens":2434,"prompt_tokens":979,"completion_tokens":1455,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":595,"completion_tokens_details":{"reasoning_tokens":1382}},"tokens_in":595,"tokens_out":1455,"duration_ms":12956,"temperature":1.0,"reasoning_tokens":1382,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:17:23.882628+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, in a concrete asymptotically-AdS-to-HV interpolating geometry, the areas of distinct components of a minimal-surface network (arch, straight legs, EWCS, Steiner junction) at several large strip widths, and check whether each component scales exactly as $\\lambda^{1+\\alpha_V/(1-\\alpha_i+\\alpha_r)}$; if the EWCS or Steiner segment has a different leading $\\lambda$-scaling, Eq. (2.24) fails for multipartite quantities built from those components even though it holds for entanglement entropy.","supporting_citations":[{"cited_title":"Charmousis, B","cited_arxiv_id":null,"evidence_quote":"Supplies the hyperscaling-violating geometry family used as the tunable IR background."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides numerical confirmation in holographic nodal line semimetals that $l\\propto r_*^{1-\\alpha_i+\\alpha_r}$, supporting the asymptotic power-law relation."},{"cited_title":"D’Hoker, P","cited_arxiv_id":null,"evidence_quote":"Provides the magnetic-brane solution whose IR is $\\mathrm{AdS}_3\\times\\mathbb{R}^2$, used for the anisotropic example."}],"review_version":1}