{"id":"74aa7d8f-a79c-4bb1-a9f5-82fa00e9bd48","arxiv_id":"2506.16501","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A distance-dependence parameter Ω controls how active particles on a lattice orient, yielding aligned or anti-aligned states, stripes, and frustration, via a mapping to an anisotropic spin model.","lead":"Active particles that turn toward or away from neighbors can be described as spins on a lattice, with the lattice structure itself determining how the particle orientations order. The paper shows that a single distance-dependence parameter switches between several orientational states, including aligned stripes, anti-aligned columns, and frustrated patterns.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Adiabatic/spin-lattice reduction is never checked against full Eqs. (2)–(3), and the l/a expansion in Eq. (6) is controlled by |Ω−1| l/a, which can be O(1) for the spring-like torques the paper highlights.","rationale":"The paper's internal mathematics is largely sound: given Eq. (4) and Eq. (6), the torque algebra in Appendix B is correct, Eq. (10) follows, the chain Peierls argument is appropriate, and the spin simulations reproduce the predicted states. The weak point is the bridge from the particle-level model to the spin model. It is not merely that 'fast relaxation' is an idealization; the paper's own control parameter Ω can make the first-order expansion break down even when l/a is small, because the expansion parameter is |Ω−1|l/a. This matters for two of the motivating realizations (dipole torques Ω=−4 and elastic-spring torques with Ω>1). A reader cannot currently decide whether the reported phase diagrams describe the active-crystal equations or only the reduced spin Hamiltonian. The reader's CONDITIONAL verdict was based on the untested adiabatic assumption; this stress-test sharpens that by identifying a concrete, checkable condition. No other concern (code availability, nonreciprocity) is as load-bearing. I recommend keeping the CONDITIONAL verdict: the paper is a useful and credible derivation, but its central claim should not be accepted as a description of the full particle dynamics without the proposed finite-τ_e, finite-Ω test.","tokens_in":16642,"tokens_out":13309,"duration_ms":142934,"concrete_test":"Simulate the full Langevin equations (2)–(3), without the adiabatic and small-l expansions, for a chain and a small square patch using the spring torque f(r)=(r−ℓ)/(a−ℓ) with ℓ/a = 0.9 (Ω = 10) and l/a = 0.1, with translational noise D_t = 0 as in the paper; compare steady-state angle distributions and nematic/polar order parameters with those obtained by integrating the spin dynamics from Eq. (10) at the same J = Γ0 l/a. Also run the same test at Ω = 0 as a control. If the full-particle and spin-model statistics agree within the simulation error, the reduction is validated; if they diverge, the spin-lattice mapping is not the correct physical limit in this advertised regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that polarity-bond torques make orientational order lattice-dependent—rests entirely on the reduction of the particle dynamics (Eqs. (2)–(3)) to the spin Hamiltonian (Eq. (10)). That reduction uses two linked assumptions: fast elastic relaxation τ_e ≪ τ_θ, giving the slaving Eq. (4), and the first-order expansion Eq. (6). Both are asserted, not tested against the full equations. The expansion is more delicate than 'l ≪ a'. The correction in Eq. (6) is (Ω−1) r0·l(n_j−n_i)/a^3; relative to the zeroth-order term 1/a its size is |Ω−1| |n_j−n_i| l/a, i.e. up to 2|Ω−1|l/a. Thus the actual small parameter is |Ω−1|l/a, not l/a. For the paper's own spring-like torques f(r)=(r−ℓ)/(a−ℓ), Ω=(1−ℓ/a)^{−1} (Fig. 1c); with ℓ/a near 1 this is large, and even for the dipole case Ω=−4 the factor |Ω−1| is 5. In those regimes the truncated spin model is not a controlled adiabatic limit of Eq. (5). The numerical evidence is generated from the effective spin dynamics at Ω = −0.5, 0, 0.5 (square lattice) and moderate chain Ω, so it cannot detect this breakdown. If the reduction fails, the predicted orientational states are properties of an auxiliary spin model, not of the active crystals the paper claims to describe.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies active particles held on fixed crystalline lattices by harmonic springs, with mutual turn-towards or turn-away torques of the form Γ_ji = Γ0 f(|r_ij|) n_i × r̂_ij. Under the assumptions of fast elastic relaxation (τ_e ≪ τ_θ) and small displacements l ≪ a, the authors eliminate particle positions in favor of orientations and expand f(|r|)/|r| to first order in l/a. This yields the reduced angular dynamics in Eq. (8) and the effective spin Hamiltonian in Eq. (10), whose three terms describe XY-type alignment, alignment with lattice axes, and mirror alignment. The paper then analyzes one-dimensional chains, square lattices, and triangular lattices, using Brownian dynamics simulations of the angular dynamics and two- and three-particle energy arguments. The central claim is that orientational order in active crystals is controlled by both the lattice structure and the distance-dependence parameter Ω = a f'(a), leading to predicted states such as anti-aligned and aligned chain configurations, striped and polar-domain states on the square lattice, and frustrated states on the triangular lattice.","tokens_in":16976,"tokens_out":14242,"duration_ms":152683,"significance":"If the spin-lattice reduction is valid, this is a valuable contribution: it provides a parameter-free mapping from a non-reciprocal active-particle system to an equilibrium-like spin model, yields concrete and falsifiable predictions for several lattice geometries, and identifies lattice structure as a design handle for orientational order. The algebraic derivation is transparent, Ω is a physical property of the interaction rather than a fitted constant, and the paper includes code availability and finite-size checks. The main weakness is that the central reduction is not validated against the full position-orientation dynamics, and the small-expansion control parameter is parameter-dependent in a way that is not acknowledged. These issues do not make the central idea implausible, but they leave the paper's headline claims insufficiently supported as currently written.","major_comments":[{"comment":"The reduction from the microscopic Langevin equations (2)-(3) to the effective spin Hamiltonian (10) is the load-bearing step of the paper, but it is never checked against the full dynamics. The simulations in the sections 'One-dimensional chain,' 'Square lattice,' and 'Triangular lattice' integrate either the reduced torque Eq. (8) or, equivalently, the gradient dynamics of Eq. (10) with white noise; they do not integrate the translational equation (2). Thus the adiabatic slaving assumption Eq. (4) and the small-displacement expansion leading to Eq. (6) remain untested. If those assumptions fail in the parameter regimes of interest, the predicted orientational states would be properties of an auxiliary spin model rather than of the active crystal the paper claims to describe. Please provide a direct comparison with full position-orientation simulations for representative parameters (for example, the Ω values in Fig. 1c), or an analytical estimate of the elimination error.","section":"Active crystals as spin lattices, Eqs. (4)-(10)"},{"comment":"The expansion of f(|r_ij|)/|r_ij| is controlled by |Ω−1|l/a, not by l/a alone. From Eq. (6), the first-order correction relative to the zeroth-order term is (Ω−1)[r0_ij·l(n_j−n_i)]/a^3 divided by 1/a, whose magnitude can be as large as 2|Ω−1|l/a. For the paper's own motivating examples, Ω=−4 for dipole-dipole torques and Ω=(1−ℓ/a)^{-1} for the spring-like torques shown in Fig. 1c, this factor is 5 or diverges as ℓ→a, so the truncated spin model is not a controlled expansion even when l ≪ a. The simulations use Ω=−0.5, 0, 0.5 on the square lattice and a moderate range on the chain, so they do not probe these regimes. Moreover, the existence of the Hamiltonian in Eq. (10) is demonstrated only for the first-order truncated torque; the exact slaved torque in Eq. (5) is not shown to be a gradient in the angle variables. This should be addressed by restricting the claims to |Ω−1|l/a ≪ 1, by simulating the exact slaved dynamics Eq. (5), or by showing explicitly that higher-order terms do not change the state selection.","section":"Eq. (6) and Appendix A"}],"minor_comments":[{"comment":"The sentence 'We perform Brownian dynamics simulations of Eqs. (3) and (8)' is ambiguous because Eq. (3) originally refers to the full angular equation with the original torque. Please state explicitly that the simulations use Eq. (8) as the torque in the angular equation, so it is clear that positions are not integrated.","section":"One-dimensional chain"},{"comment":"The caption writes 'Γ0 = 10a/l (Γ0 = −10a/l),' which is dimensionally inconsistent with the text's 'Γ0l/a = 10.' The intended condition appears to be Γ0l/a = ±10.","section":"Figure 4 caption"},{"comment":"The phrase 'the energy of the XY with model' contains a typo and should read 'the energy of the XY model.'","section":"Appendix D"},{"comment":"There is a stray 'b' in 'Turn-towards Γ0 > 0 b' and the label 'T urn-towards' has an unwanted space; these typographical issues should be fixed.","section":"Figure 1 caption"},{"comment":"The nearest-neighbor truncation is introduced without discussion for the power-law examples f(r)=a^4/r^4 and f(r)=a^2/r^2. For these interactions next-nearest-neighbor torques are not negligible (for the r^-4 case on a square lattice they are one quarter of the nearest-neighbor torque), so a sentence justifying the truncation for the experimental systems, or explicitly framing the model as a nearest-neighbor-only model, would be helpful.","section":"Discussion of nearest-neighbor interactions"},{"comment":"The statement that the first term of Eq. (7) cancels because the lattice vectors add to zero should explicitly note that this holds for the infinite regular lattices considered; finite boundaries or lattice defects would require retaining that term.","section":"Paragraph after Eq. (8)"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this paper derives a genuinely new effective spin model for active particles on a lattice interacting via turn-towards/turn-away torques, and shows that the distance-dependence parameter Ω = a f'(a) controls which kind of orientational order appears. The decomposition of the torque into XY-like alignment, lattice alignment, and mirror alignment is elegant, and the two-particle analysis explains the simulation states well. I believe the central idea is correct and worth citing.\n\nWhat it does well: the derivation from Eq. (1) to Eq. (10) is algebraically clean and the sign structure checks out. The chain phase diagram and the Peierls-style argument for the absence of polar order are nice. The frustration on the triangular lattice for Ω = -1 is a real observation. The mapping between turn-towards and turn-away on a square lattice in Appendix E is a neat symmetry.\n\nThe soft spots: the load-bearing assumptions are never tested. The paper assumes fast elastic relaxation (τ_e ≪ τ_θ) and small displacement l ≪ a, then simulates only the reduced angular dynamics (Eq. 8). The full particle equations (2)-(3) are never integrated. The stress-test note makes a fair point: the expansion in l/a is actually controlled by |Ω−1| l/a, since the first-order correction is (Ω−1) r0·l / a^3, and for the spring-like f(r) the paper itself highlights, |Ω−1| can be large. For dipole interactions Ω = -4, the factor is 5. So the spin mapping is not uniformly controlled for the physical cases discussed. This doesn't kill the qualitative message—I suspect the lattice-dependent order survives for moderate l/a—but it means the paper's predictions are strictly for the auxiliary spin model until the reduction is validated against the full dynamics.\n\nMinor: the code link is a placeholder, which is sloppy given the data availability statement.\n\nWho this is for: soft matter theorists working on active solids, non-reciprocal interactions, and lattice spin models. It deserves a serious referee, but the referee should ask for at least a few full-particle simulations (even in the deterministic limit) to confirm that the spin model captures the actual crystal behavior. If the reduction breaks down, the authors should delimit its range of validity.\n\nI would bring it to a reading group and cite it for the Ω decomposition, but I would not take the quantitative predictions on faith.","headline":"New spin-lattice mapping for polarity-bond active crystals, but the adiabatic reduction is untested and the expansion parameter is actually |Ω−1| l/a rather than l/a.","tokens_in":17473,"tokens_out":4968,"would_cite":true,"duration_ms":47079,"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":"The orientational order of active crystals is set by the lattice structure and by a single distance-dependence parameter, through an effective spin-lattice mapping.","keywords":["active crystals","orientational order","polarity-bond interactions","spin-lattice mapping","XY model","nematic alignment","geometric frustration","distance-dependence parameter"],"falsifier":"Simulate the full Langevin equations without the $\\tau_e\\ll\\tau_\\theta$ approximation, scanning the elastic constant $k$ (or the ratio $\\tau_e/\\tau_\\theta$) at fixed $\\tilde{\\Gamma}_0$ and $\\Omega$: if the predicted chain states ($\\uparrow\\downarrow$, $\\rightarrow\\leftarrow$, $\\uparrow\\uparrow$, $\\leftarrow\\leftarrow$) and square-lattice stripe patterns persist even when $\\tau_e$ is comparable to $\\tau_\\theta$, then the fast-relaxation premise is not load-bearing; if they disappear or change, the spin-lattice mapping is limited by that timescale. Alternatively, measure $f(r)$ directly in an experimental active crystal and check whether the phase boundaries in the $\\Omega$-$\\tilde{\\Gamma}_0$ plane match the measured $\\Omega$.","tokens_in":16448,"feed_emoji":"🧲","tokens_out":12627,"duration_ms":116723,"temperature":0.7,"pith_summary":"This paper asks what controls the orientations of particles in a crystal made of self-propelled particles that turn toward or away from each other. It establishes that, when the particles stay near their lattice sites via fast elastic relaxation, the orientational dynamics reduce to an effective spin model with three competing interactions: XY-style alignment, alignment to the lattice axes, and alignment to mirror images across lattice bonds. A single dimensionless parameter, $\\Omega \\equiv a f'(a)$, where $f(r)$ is the distance dependence of the turning torque, sets the signs and relative strengths of these terms, while the sign of the torque amplitude $\\tilde{\\Gamma}_0$ inverts the entire energy landscape. The predicted states—local ferro- or antiferromagnetic order on a chain, striped and polar-domain states on a square lattice, and frustrated compromise states on a triangular lattice—match direct Brownian dynamics simulations. If correct, the result means the crystalline lattice itself can be used as a design handle to control orientational order in active crystals.","feed_headline":"Lattice geometry dictates orientational order in active crystals","feed_subtitle":"A single distance-dependence parameter decides whether particles align along, across, or against the lattice.","key_machinery":"The load-bearing mechanism is the spin-lattice mapping: under fast elastic relaxation, position degrees of freedom disappear and the orientations obey $\\mathrm{d}\\theta_i/\\mathrm{d}\\tilde{t}=-\\partial H/\\partial\\theta_i+\\sqrt{2}\\,\\eta_i^r$ with the effective energy $H=\\tilde{\\Gamma}_0(l/a)\\sum_{\\langle i,j\\rangle}[(\\Omega+1)/2\\,H^{\\rm XY}_{ij}+(\\Omega-1)/2\\,(H^{\\rm LA}_{ij}+H^{\\rm MA}_{ij})]$. The dimensionless parameter $\\Omega\\equiv a f'(a)$ is the control knob: it decides which of the three terms dominates and with what sign, while $\\tilde{\\Gamma}_0$ sets the overall scale and its sign inverts the energy landscape. This mapping converts the original non-reciprocal torques into a gradient system, and it is what makes the chain equivalent to an anisotropic XY model in a nematic field, the square lattice symmetric under checkerboard spin flips, and the triangular lattice geometrically frustrated.","core_discovery":"The central claim is that polarity-bond interactions, torques of the form $\\Gamma_{ji}=\\Gamma_0 f(|\\mathbf{r}_{ij}|)\\,\\hat{\\mathbf{n}}_i\\times\\hat{\\mathbf{r}}_{ij}$, on a fixed lattice generate orientational order whose character is fixed by the lattice geometry and by $\\Omega\\equiv a f'(a)$. The argument begins in the fast-relaxation regime $\\tau_e\\ll\\tau_\\theta$, where positions are slaved to orientations, $\\mathbf{r}_i=\\mathbf{r}_i^{(0)}+l\\hat{\\mathbf{n}}_i$. Expanding in $l/a$ and summing over nearest neighbors turns the torques into gradients of an effective energy $H$ containing an XY alignment term $H^{\\rm XY}_{ij}=-\\cos(\\theta_j-\\theta_i)$, a lattice-alignment term $H^{\\rm LA}_{ij}=[\\cos 2(\\phi_{ij}-\\theta_i)+\\cos 2(\\phi_{ij}-\\theta_j)]/2$, and a mirror-alignment term $H^{\\rm MA}_{ij}=-\\cos(2\\phi_{ij}-\\theta_i-\\theta_j)$, weighted by $(\\Omega+1)/2$ and $(\\Omega-1)/2$. The paper shows that this energy accounts for the states observed in simulations: on a chain, turn-towards torques give local ferromagnetic order and, away from $\\Omega=1$, the states $\\uparrow\\uparrow$ or $\\leftarrow\\leftarrow$, while turn-away torques give antiferromagnetic states $\\uparrow\\downarrow$ or $\\rightarrow\\leftarrow$; on a square lattice, varying $\\Omega$ from $-0.5$ to $0.5$ crosses from alternating stripes to polar domains with a weak lattice preference; on a triangular lattice, both antiferromagnetic and mirror-alignment interactions are frustrated. The overarching claim is that positional and orientational order are strongly coupled in such crystals, so the lattice structure controls the orientation.","pith_inferences":["The same effective-energy decomposition should apply to any polarity-bond torque, so one could measure $f(r)$ in an experimental system and then read off the expected orientational state from the predicted phase diagrams without running a full simulation.","A natural next experiment is to place identical active particles on square and triangular optical or grooved lattices: the theory predicts stripe formation on one and frustrated states on the other, isolating the role of lattice geometry from particle chemistry.","Because the two-particle and three-particle energy arguments reproduce the many-body states, the dominant physics may be short-ranged correlations rather than collective long-range effects; this can be tested by comparing small-cluster equilibrium probabilities with $\\exp(-H)$.","If the fast-relaxation assumption is relaxed, finite $\\tau_e$ should generate corrections beyond the spin model; the first testable signature would be a dependence of the ordering thresholds on the elastic constant $k$ (equivalently on $l/a$) that is absent from the current phase diagram."],"forward_implications":["On a one-dimensional chain, global polar order is forbidden for any noise strength $D_r>0$, but global nematic order can survive except at $\\Omega=1$, where the model reduces to the rotationally invariant XY model and loses global order.","The two-particle energy landscape reproduces the many-body chain states, so the selected state can be read from a single bond: for turn-towards torques the most probable states are $\\uparrow\\uparrow$ for $0<\\Omega<1$ and $\\leftarrow\\leftarrow$ for $\\Omega>1$, while for turn-away torques the ground state switches from $\\uparrow\\downarrow$ for $\\Omega<1$ to $\\rightarrow\\leftarrow$ for $\\Omega>1$.","On a square lattice, flipping the orientation of every particle on one checkerboard sublattice and changing $\\Gamma_0\\to-\\Gamma_0$ leaves the energy invariant, so every turn-away state has a turn-towards counterpart with the same ordering physics.","On a triangular lattice, both the antiferromagnetic XY term and the mirror-alignment term are geometrically frustrated, so the system selects compromise stripe states, extending the notion of frustration to active orientational order.","Experimental systems fall across the relevant range of $\\Omega$: metal-dielectric Janus colloids have $\\Omega=-4$, chemotactic particles have $\\Omega=-2$, topological robot interactions have $\\Omega=0$, and spring-coupled robots have $\\Omega=1/(1-\\ell/a)$, so switching the lattice should switch the orientational state without changing the particles."],"supporting_citations":[{"why":"Supplies the active-solid model and the fast-relaxation assumption ($\\tau_e\\ll\\tau_\\theta$) under which Eq. (4) holds, eliminating positions to obtain the spin lattice.","marker":"40"},{"why":"Provides the experimental realization of turn-away torques in metal-dielectric Janus colloids, the class of system the model is built for.","marker":"13"},{"why":"Establishes the electrostatic turn-towards torque in Janus particles and the dipole-dipole distance dependence $f(r)=a^4/r^4$, giving $\\Omega=-4$.","marker":"46"},{"why":"Supplies the chemotactic reorientation distance dependence $f(r)=a^2/r^2$, fixing $\\Omega=-2$ as another experimentally relevant case.","marker":"14"},{"why":"Provides the general self-aligning torque structure used in Eq. (1).","marker":"7"},{"why":"Gives the spring-based distance dependence $f(r)=(r-\\ell)/(a-\\ell)$ that yields $\\Omega=1/(1-\\ell/a)$, covering both positive and negative values.","marker":"11"},{"why":"The Mermin-Wagner theorem used to argue that for $\\Omega=1$ the effective energy is the rotationally invariant XY model, where global orientational order is forbidden.","marker":"51"},{"why":"Provides the landscape-inversion phase transition concept used to interpret the $\\Gamma_0\\to-\\Gamma_0$ sign change that inverts all stable states.","marker":"48"}],"fun_headline_variants":["Lattice dictates spin order in active crystals","Active crystals: lattice geometry rules orientation","How the lattice steers active particle alignment","Orientational order in active crystals follows lattice"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The spin-lattice mapping rests on the assumption that elastic relaxation toward lattice sites is much faster than orientation dynamics ($\\tau_e\\ll\\tau_\\theta$), so positions can be slaved to orientations via Eq. (4); if that separation of timescales fails, the effective energy and the predicted orientational states are not guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Lattice dictates spin order in active crystals","Active crystals: lattice geometry rules orientation","How the lattice steers active particle alignment","Orientational order in active crystals follows lattice"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000236,"raw_usage":{"total_tokens":1583,"prompt_tokens":1102,"completion_tokens":481,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":718,"completion_tokens_details":{"reasoning_tokens":426}},"tokens_in":718,"tokens_out":481,"duration_ms":5303,"temperature":1.0,"reasoning_tokens":426,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:24:51.948993+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the full Langevin equations without the $\\tau_e\\ll\\tau_\\theta$ approximation, scanning the elastic constant $k$ (or the ratio $\\tau_e/\\tau_\\theta$) at fixed $\\tilde{\\Gamma}_0$ and $\\Omega$: if the predicted chain states ($\\uparrow\\downarrow$, $\\rightarrow\\leftarrow$, $\\uparrow\\uparrow$, $\\leftarrow\\leftarrow$) and square-lattice stripe patterns persist even when $\\tau_e$ is comparable to $\\tau_\\theta$, then the fast-relaxation premise is not load-bearing; if they disappear or change, the spin-lattice mapping is limited by that timescale. Alternatively, measure $f(r)$ directly in an experimental active crystal and check whether the phase boundaries in the $\\Omega$-$\\tilde{\\Gamma}_0$ plane match the measured $\\Omega$.","supporting_citations":[{"cited_title":"Hern ´andez-L´opez, P","cited_arxiv_id":null,"evidence_quote":"Supplies the active-solid model and the fast-relaxation assumption ($\\tau_e\\ll\\tau_\\theta$) under which Eq. (4) holds, eliminating positions to obtain the spin lattice."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the experimental realization of turn-away torques in metal-dielectric Janus colloids, the class of system the model is built for."},{"cited_title":"Zhang, R","cited_arxiv_id":null,"evidence_quote":"Establishes the electrostatic turn-towards torque in Janus particles and the dipole-dipole distance dependence $f(r)=a^4/r^4$, giving $\\Omega=-4$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the chemotactic reorientation distance dependence $f(r)=a^2/r^2$, fixing $\\Omega=-2$ as another experimentally relevant case."},{"cited_title":"Baconnier, O","cited_arxiv_id":null,"evidence_quote":"Provides the general self-aligning torque structure used in Eq. (1)."},{"cited_title":"Ferrante, A","cited_arxiv_id":null,"evidence_quote":"Gives the spring-based distance dependence $f(r)=(r-\\ell)/(a-\\ell)$ that yields $\\Omega=1/(1-\\ell/a)$, covering both positive and negative values."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Mermin-Wagner theorem used to argue that for $\\Omega=1$ the effective energy is the rotationally invariant XY model, where global orientational order is forbidden."},{"cited_title":"Alert, J","cited_arxiv_id":null,"evidence_quote":"Provides the landscape-inversion phase transition concept used to interpret the $\\Gamma_0\\to-\\Gamma_0$ sign change that inverts all stable states."}],"review_version":2}