{"id":"d638d652-d1ae-4c33-a61f-4df585bcb60e","arxiv_id":"2505.05833","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Numerical solutions of the LHY-corrected Gross-Pitaevskii equation show a Bose-Bose droplet becomes porous, then moves out of a random repulsive speckle region, forming a ring in 2D and two fragments in 1D.","lead":"This paper uses computer simulations to study a self-bound Bose-Einstein droplet of two atomic species sitting in a random repulsive laser-speckle potential, in one and two dimensions. It finds that strong disorder pushes the droplet to the edge of the disordered region, forming a ring in 2D and splitting the droplet into two pieces in 1D.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reported ring/split states may be metastable artifacts: with no trap and V=0 outside the finite speckle region, a compact droplet outside the disk has lower energy than the ring or two-fragment states, so the central claim needs a ground-state test.","rationale":"The reader's conditional verdict is justified, but I identify a more load-bearing problem than the Gaussian-vs-real-speckle issue. The Gaussian-bump model is mainly an external-realism concern; the ring/split geometry can be questioned on energy grounds inside the paper's own model. Because the speckle potential is zero outside a finite disk/interval and there is no trap, the translationally degenerate outside region should favor a compact droplet. A ring has much larger surface/gradient energy, so it is not the obvious ground state; the observed shape is consistent with incomplete relaxation or a metastable saddle point inherited from the symmetric initial condition. The manuscript provides no convergence tests, no code or data, and does not specify real versus imaginary time, so this alternative cannot be excluded. The proposed numerical test would settle it. If the test shows the ring is the lowest-energy state, the conditional acceptance can stand; if not, the central claim would have to be rejected. My verdict remains CONDITIONAL, unchanged from the reader's, pending that check.","tokens_in":8396,"tokens_out":17849,"duration_ms":199575,"concrete_test":"Run imaginary-time (gradient-flow) relaxation at the reported high densities (e.g., 2D rho=9.549, g=10, N=100; 1D corresponding case) from three initial states: (i) the reported ring/fragment state; (ii) a compact droplet placed at a random point just outside the speckle region; (iii) a broad low-density background plus weak noise. Evolve until the energy change per step is below, say, 1e-8 and the profile is stationary, then compare energies using Eq. (5)/(10). If the ring/fragment relaxes to a compact blob, or if initial state (ii) converges to a compact blob with lower per-particle energy than the ring, the ring is a metastable artifact and the main claim fails. Repeat for several disorder realizations and azimuthal placements to exclude pinning by the random potential.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central qualitative claim is that at high impurity density the droplet is expelled to the disorder-free boundary and forms a sharp ring in 2D (Fig. 2(g)) and two symmetric droplets in 1D (Fig. 4(c),(f)). In the model of Section II there is no external trap; outside the circular (r=20) or linear (L=140) speckle region the potential is exactly zero. A compact self-bound droplet with N=100 and equilibrium density n0 about 1 has radius about 5.6 in 2D; placed just outside the disk it has V=0 and only its intrinsic surface energy. A ring of the same area around radius 20 would have width about 0.8 and a much larger gradient/surface energy, and two 1D fragments have higher surface energy than one blob on either side of the speckle interval. Thus the ring/split configurations are unlikely to be the ground state of Eq. (2)/(7). They are more plausibly stationary points reached from the centered initial condition under the adiabatic V0 ramp: the exterior V=0 region is translationally flat, so there is no symmetry-breaking mechanism, and the text never states whether the split-step Crank-Nicolson propagation is real or imaginary time or gives convergence criteria (Section II). Until ground-state status is demonstrated, the central claim is not established, even within the paper's own Gaussian-bump disorder model.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a self-bound Bose-Bose quantum droplet in one and two dimensions in the presence of a repulsive random speckle potential. The speckle is modeled as a sum of independent Gaussian repellers of fixed amplitude V0=1 and width chi=1. Solving the extended Gross-Pitaevskii equations with Lee-Huang-Yang corrections numerically, the authors report that as the impurity density is increased the droplet first becomes porous and then is expelled to the disorder-free boundary, forming a sharp ring in 2D and two symmetric fragments in 1D. They compute disorder-averaged energy and chemical potential versus impurity density for several interaction strengths and atom numbers, finding a peak at the porous transition and saturation at high density. The central claim is that this expulsion and ring/fragment formation is the generic response of a self-bound droplet to high-density repulsive disorder.","tokens_in":8655,"tokens_out":7645,"duration_ms":85246,"significance":"If the central claim is correct, the paper extends quantum-droplet physics to disordered environments and argues for a common qualitative response in 1D and 2D: a self-bound liquid-like state remains coherent, becomes porous, and is eventually pushed to the boundary of the disordered region. The manuscript provides a clear derivation of the reduced GP equations and the corresponding energy and chemical potential functionals, and the density-sequence figures are visually consistent with the described progression. A notable strength is that no parameter is fitted to the target outcome; all parameters are taken from the model or the literature. However, the numerical protocol is described too imprecisely to establish that the reported final states are stationary, and the idealized Gaussian-bump disorder model raises questions about the external validity of claims about optical speckle. The main conclusions are therefore conditional on additional numerical and modeling work.","major_comments":[{"comment":"The manuscript does not state whether the split-step Crank-Nicolson propagation is real-time or imaginary-time, and it gives no grid spacing, time-step size, or convergence tolerance. The described procedure of slowly increasing V0 and using the previous wave function as the initial guess produces density profiles that are endpoints of an adiabatic evolution, but it does not by itself demonstrate that the profiles in Figs. 2(g) and 4(c),(f) are stationary solutions of the time-independent GP equation. The authors should run imaginary-time propagation (or an equivalent ground-state solver) from the reported final densities and from alternative initial conditions, and they should report the numerical parameters, to verify that the ring and two-fragment states are local minima of the energy functional.","section":"Section II, Eqs. (2) and (7)"},{"comment":"The ring and two-fragment configurations are unlikely to be the ground state of the model because the external potential is exactly zero outside the circular or linear disordered region. A compact droplet placed outside the disorder region has the same energy as the impurity-free droplet, E0, whereas the authors themselves state that the final energy at high impurity density is higher than the initial value because of the larger surface area of the ring or fragments. Thus the reported states have higher energy than a translated compact droplet, so they cannot be global (or even local, without further analysis) minima. The central claim that the droplet 'goes to the circumference and looks ring-shaped' must be accompanied by an energy comparison with a compact droplet outside the disorder region and a stability test under asymmetric perturbations; otherwise the figures describe transient or metastable states rather than the system's preferred configurations.","section":"Section III, Figs. 2(g) and 4(c),(f)"},{"comment":"The Gaussian-bump disorder model with V0=1 and chi=1 is not a faithful representation of optical speckle, which has exponential intensity statistics and a correlation function dictated by the imaging aperture, and whose amplitude can significantly exceed the droplet energy scales. Since the paper's title and abstract state that the subject is a random speckle potential, the authors should either implement a realistic speckle field (e.g., generated by Fourier filtering of random phases) or explicitly restrict their conclusions to a Gaussian random potential and justify this model as a controlled proxy. This issue is load-bearing because the ring/split response might be an artifact of the idealized disorder statistics rather than a general property of droplets in speckle.","section":"Section II, Eqs. (4) and (9)"}],"minor_comments":[{"comment":"The sentence 'the boundary (60×60) is large enough to affect the system' should read 'large enough not to affect the system'; as written it contradicts the intended meaning.","section":"Section II, page 2"},{"comment":"The statement that NP=20 configurations are sufficient is not quantified; the authors should provide error bars or standard deviations for the disorder-averaged energy and chemical potential in Figs. 3 and 5.","section":"Section II, Eq. (6)"},{"comment":"The density figures lack color bars and explicit axis scales, which makes it difficult to assess the density values and the sharpness of the ring or fragments; adding a common color scale to panels within each figure would improve reproducibility.","section":"Figures 1, 2, and 4"},{"comment":"The sentence 'The crucial aspect is that the BEC remains in the liquid phase in the presence of a high impurity potential' is not tied to a quantitative criterion; the authors should specify what observable or diagnostic (e.g., a flat-top density plateau, superfluid fraction, or phase coherence) identifies the 'liquid phase' in their simulations.","section":"Section III"},{"comment":"The comparison with the square-impurity results of Ref. [45] is purely qualitative; quoting the corresponding energy and chemical potential values or density profiles would make the claimed difference concrete and testable.","section":"Section III, comparison to Ref. [45]"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely question and the model equations are internally consistent, but the central claim about expulsion and ring/fragment formation is not yet supported because the numerical method could be producing metastable transients. The missing ground-state test, the lack of numerical convergence details, and the unrealistic Gaussian-bump model of speckle are the key obstacles. With additional imaginary-time simulations, energy comparisons, and an honest reframing of the results as applying to a Gaussian random potential (or with a realistic speckle implementation), the paper could become publishable. The current scope of claims is too broad for the evidence presented."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a straightforward numerical extension of the Petrov/Li et al. droplet model to a finite disk of random Gaussian repellers, and the qualitative result (droplet becomes porous, then moves to the boundary and forms a ring in 2D / two fragments in 1D) is visually clear and new relative to the square-impurity study [45]. The GP equation and the energy functional are mutually consistent, and the density plots support the stated progression.\n\nThe main problem is that the ring/split states are probably not the ground state of their own model. Outside the speckle disk (or the linear interval) the potential is exactly zero and there is no trap. A compact droplet of the same N and equilibrium density placed just outside the disk has essentially the same bulk energy but much smaller surface energy than a ring at radius 20 (width ~0.8) or two separated fragments. So the ring/split are metastable; they are reached because the simulation starts from the center and ramps up V0. The paper never says whether the split-step CN propagation is real or imaginary time, and it never compares against a droplet placed initially outside or tests for the global minimum. This is the load-bearing gap.\n\nOther soft spots are smaller. The disorder model is three Gaussian bumps, V0=chi=1, which is not real speckle (exponential intensity statistics, correlations). The quantitative energy and chemical potential curves may depend on this choice, though the qualitative porous-to-expelled picture may survive. There are no grid spacing/time step values, no convergence data, no error bars on the disordered average (NP=20 is stated but the spread is not shown), and the data availability statement says no data associated with the manuscript.\n\nCredit where due: the equations are right, the figures are decent, and the comparison to [45] is honest. The physics idea—that a self-bound liquid can be expelled as a whole, retaining its density—is worth knowing. But the paper as written does not establish that the ring/split is the true response; it establishes that those states are attractors in their numerical continuation.\n\nWho is this for? People working on droplets in disorder would want to read it, but more as a pointer than a firm result. I would not cite the ring/split claim until the ground-state question is settled. Recommendation: send to peer review—the issue is addressable and the topic is active. The referee should ask for an imaginary-time global relaxation with the droplet initialized outside, a trap-free energy comparison, and numerical details.","headline":"Plausible-looking droplet-in-speckle numerics, but the headline ring/split states are likely metastable; needs a ground-state comparison before I'd trust the central claim.","tokens_in":9243,"tokens_out":4524,"would_cite":false,"duration_ms":46655,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Repulsive speckle disorder does not destroy a Bose-Bose droplet; it turns the droplet porous and then drives it to the boundary, forming a ring in 2D and two fragments in 1D.","keywords":["quantum droplet","Bose-Bose mixture","random speckle potential","Lee-Huang-Yang correction","Gross-Pitaevskii equation","low-dimensional Bose-Einstein condensate","ring-shaped condensate","disordered ultracold gases"],"falsifier":"Solve the same GP equations with a speckle potential that has exponential intensity statistics and a realistic correlation length, scanning $V_0$ from values far below to far above the droplet's chemical potential; if the droplet does not migrate to the boundary into a ring in 2D and two fragments in 1D before evaporating, the central claim is false.","tokens_in":8154,"feed_emoji":"💧","tokens_out":8124,"duration_ms":79323,"temperature":0.7,"pith_summary":"The paper sets out to show how a self-bound Bose-Bose quantum droplet responds to a repulsive random speckle potential in one and two dimensions. Solving the extended Gross-Pitaevskii equation with the Lee-Huang-Yang correction, the authors find that as impurity density rises the droplet first becomes porous, meaning riddled with density voids, and then moves out of the disordered region, ending as a sharp ring in 2D and as two symmetric fragments in 1D. The disorder-averaged energy per particle and chemical potential both increase with impurity density, peak at the porous transition, and then decrease to a saturated value. The droplet stays in the liquid phase throughout, and stronger interactions allow it to withstand higher impurity density before splitting.","feed_headline":"Speckle disorder pushes quantum droplets into a sharp ring","feed_subtitle":"Simulations show a self-bound mixture stays liquid and migrates to the disorder-free rim.","key_machinery":"The load-bearing object is the extended Gross-Pitaevskii equation for a symmetric Bose-Bose mixture, with the Lee-Huang-Yang (LHY) quantum-fluctuation term included and the disorder entering as a sum of Gaussian repulsive bumps $V(\\mathbf r)=V_0\\sum_i e^{-((x-x_i)^2+(y-y_i)^2)/\\chi^2}$ in 2D, with a one-dimensional analogue in 1D, always at $V_0=\\chi=1$. The Gaussian-bump model with uniformly random impurity positions, combined with split-step Crank-Nicolson propagation and disorder averaging over configurations, is what carries the porous-then-expelled picture. Because the LHY term is dimension-dependent, separate forms are used in 1D and 2D.","core_discovery":"On the paper's own terms, the central discovery is that a self-bound droplet of a Bose-Bose mixture in a repulsive random speckle potential is not destroyed: it remains liquid-like while the disordered region becomes porous, and at high impurity density the condensate is expelled to the boundary of the disorder region. In two dimensions the boundary is circular, so the final density profile is a sharp ring; in one dimension the linear disorder region splits the droplet into two symmetric parts. The disorder-averaged energy per particle and chemical potential grow with impurity density up to a maximum that corresponds to the porous configuration, then fall and saturate at values still above the clean-droplet values. Higher interaction strength $g$ requires larger impurity density to reach the maximum and to cause splitting.","pith_inferences":["Whether real optical speckle, with its exponential intensity statistics and finite correlation length, produces the same ring-and-split response is an open question; replacing the Gaussian-bump disorder with a measured speckle intensity profile in the same solver would settle it.","The strong dependence on boundary shape suggests a geometric control: changing the disorder region from circular to, say, elliptical should deform the final ring accordingly, a prediction that goes beyond the paper's simulations.","The energy maximum at the porous transition could be read as a surface-energy barrier, and extracting its height as a function of $g$ and atom number $N$ would give an effective disorder-induced surface tension for the droplet."],"forward_implications":["A self-bound Bose-Bose droplet remains in a liquid-like state even for high speckle density, so repulsive disorder by itself does not evaporate the condensate.","The energy per particle and chemical potential peak exactly at the porous configuration and then decline, giving a measurable signature of the porous-to-expelled transition.","Droplets with larger interaction strength $g$ need a higher impurity density before they split, so stronger interactions confer disorder robustness.","The final morphology is set by the shape of the disorder boundary: circular disorder yields a ring, linear disorder yields two symmetric fragments.","At saturation, the fragmented or ring-shaped droplet has higher energy than the clean droplet because its surface area is larger."],"supporting_citations":[{"why":"Supplies the 1D reduced GP equation with the LHY term, the backbone of the 1D calculation.","marker":"[17]"},{"why":"Gives the 2D reduced single GP equation and the energy and chemical-potential integrals used in the figures.","marker":"[21]"},{"why":"Provides the 2D droplet model with the logarithmic LHY correction on which equation (2) is built.","marker":"[22]"},{"why":"Establishes optical speckle as a controllable random potential for BECs, the physical referent of the disorder.","marker":"[26]"},{"why":"Supplies the Gaussian form of the disorder potential used in equations (4) and (9).","marker":"[32]"},{"why":"The square-impurity droplet study used for comparison; the new ring result is defined against its non-ring outcome.","marker":"[45]"},{"why":"The split-step Crank-Nicolson method used to solve all GP equations.","marker":"[58–60]"}],"fun_headline_variants":["Speckle disorder forces droplets into a sharp ring","Self-bound droplet migrates to disorder-free rim","Droplet survives speckle, splits or rings","Speckle makes Bose mixture form a ring or split","Disorder drives quantum droplet to clean boundary"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calculations assume the speckle potential is a sum of Gaussian bumps of fixed strength and width ($V_0=1$, $\\chi=1$), so if real speckle's exponential intensity statistics or much larger amplitudes change the droplet's response, the ring-and-split conclusion could be an artifact of that idealized disorder model.","fun_headline_variants_meta":{"raw":{"variants":["Speckle disorder forces droplets into a sharp ring","Self-bound droplet migrates to disorder-free rim","Droplet survives speckle, splits or rings","Speckle makes Bose mixture form a ring or split","Disorder drives quantum droplet to clean boundary"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00099,"raw_usage":{"total_tokens":4135,"prompt_tokens":822,"completion_tokens":3313,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":438,"completion_tokens_details":{"reasoning_tokens":3239}},"tokens_in":438,"tokens_out":3313,"duration_ms":23649,"temperature":1.0,"reasoning_tokens":3239,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:55:43.606711+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the same GP equations with a speckle potential that has exponential intensity statistics and a realistic correlation length, scanning $V_0$ from values far below to far above the droplet's chemical potential; if the droplet does not migrate to the boundary into a ring in 2D and two fragments in 1D before evaporating, the central claim is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the 2D reduced single GP equation and the energy and chemical-potential integrals used in the figures."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the 2D droplet model with the logarithmic LHY correction on which equation (2) is built."},{"cited_title":"Rakshit, T","cited_arxiv_id":null,"evidence_quote":"Establishes optical speckle as a controllable random potential for BECs, the physical referent of the disorder."},{"cited_title":"Pilati and P","cited_arxiv_id":null,"evidence_quote":"Supplies the Gaussian form of the disorder potential used in equations (4) and (9)."},{"cited_title":"D’Errico, E","cited_arxiv_id":null,"evidence_quote":"The square-impurity droplet study used for comparison; the new ring result is defined against its non-ring outcome."}],"review_version":1}