{"id":"d8b5099e-3cd3-4c85-ad2d-5de85716a876","arxiv_id":"2507.07494","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The experiment shows that a chiral superfluid in a shaken optical lattice melts in two steps, with time-reversal symmetry restored before the superfluid order is lost.","lead":"Ultracold rubidium atoms in a shaken optical lattice form a chiral superfluid whose two broken symmetries melt at two different temperatures, leaving an intermediate phase that is superfluid but no longer chiral. This is the first experimental observation of such vestigial order melting in a cold-atom system, a phenomenon relevant to exotic superconductors.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Ising T_m is extracted from a density-normalized asymmetry whose variance carries a factor f_c^2, so a falling condensate fraction can mimic Z2 restoration and bias T_m downward; a valley-resolved reanalysis is needed.","rationale":"The reader's weakest assumption is exactly the one I consider load-bearing, and I agree with that identification. The qualitative scenario is credible and the manuscript provides a useful effective Hamiltonian plus publicly posted data, but the central claim is quantitative: T_m < T_c throughout the explored regime, with the intermediate paramagnetic-superfluid window widening as k0 increases. That quantitative conclusion rests on the unvalidated factorization of the variance O into a condensate-fraction factor and a valley-imbalance factor. Because f_c decreases significantly over the temperature range of the fits, the reported T_m can be biased downward even if the underlying Z2 transition occurs at higher temperature. The proposed reanalysis uses existing data and does not require a new experiment, and it would settle whether the suspected bias actually changes the phase boundaries. I find no additional internally inconsistent step that outweighs this concern; the limited sensitivity of TOF imaging to phase coherence is related but secondary. Therefore the conditional verdict should stand, pending the valley-resolved reanalysis.","tokens_in":13222,"tokens_out":5277,"duration_ms":64263,"concrete_test":"Reanalyze the posted TOF data (Zenodo 19562635) with a valley-resolved imbalance z = (N_+ - N_-)/(N_+ + N_-), where N_+ and N_- are extracted from bimodal fits of the two valley peaks, and fit Eq. (8) to Var(z) instead of O. If the resulting T_m values agree with the reported ones within error, the confound is benign; if T_m shifts upward by more than about 30 nK or approaches T_c, the vestigial-window claim is not established. As a second cross-check, fit O(T) = f_c^2(T) O_Z2(T) using the independently measured condensate fraction and compare the extracted T_m.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing assumption is that O(T) in Eq. (5) isolates the Z2 symmetry-restoring transition when fitted to Eq. (8). For n(kx)=n_th+n_cond with a symmetric thermal background, Eq. (4) yields m = f_c z, where f_c is the condensate fraction and z = (N_+ - N_-)/(N_+ + N_-) is the valley imbalance. Thus O is approximately f_c^2 Var(z), plus subleading thermal terms. Since f_c(T) falls monotonically toward zero at T_c approximately 315 nK and the fitted T_m approximately 165 nK lies where f_c is still changing, the variance inherits a decreasing multiplicative factor that can pull a power-law fit to zero below the true Z2 transition. This would widen the apparent paramagnetic-superfluid window and can even make T_m < T_c when no vestigial phase exists. The supplementary fits (polynomial, exponential, power-law) all share the confound, so their agreement does not test it; the 3-sigma outlier removal can further discard bimodal tails carrying the Z2 signal. Because the central claim is the existence and systematic widening of the intermediate phase, this normalization bias is the most load-bearing concern.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports an experimental study of a Floquet-engineered double-valley band structure realized with 87Rb bosons in a shaken one-dimensional optical lattice. The authors measure two transition temperatures: an Ising-like transition T_m associated with the restoration of time-reversal (Z2) symmetry, extracted from the variance O of a momentum-space asymmetry m defined in Eq. (4), and the superfluid transition T_c extracted from the condensate fraction f_c. They report T_m ≈ 165 nK and T_c ≈ 315 nK at a shaking frequency of 12.55 kHz, and, by varying the frequency, they find that T_m decreases as the valley separation k0 increases while T_c remains roughly constant, which they interpret as a two-step melting of the chiral superfluid into a paramagnetic superfluid and then into a normal phase. The paper includes supplementary fits and a discussion of the effective field theory, and the data are available on Zenodo.","tokens_in":13437,"tokens_out":8295,"duration_ms":94479,"significance":"The platform and measurement are interesting: a shaken-lattice double-valley system with an s-p hybridized band is a clean setting to look for vestigial order in a bosonic superfluid, and the paper provides a full phase diagram with bootstrap error bars, publicly archived data, and several cross-checks of the fits (Supplementary Fig. S5). If the two-step melting scenario is confirmed, it would be a notable demonstration of fluctuation-driven vestigial order in an ultracold atomic gas, relevant to multi-orbital superconductivity. The effective field theory is used only for motivation, and the experimental measurement of T_c is independent of the specific form of g2≈4g1, so there is no circularity in the determination of the transition temperatures.","major_comments":[{"comment":"The order parameter m defined in Eq. (4) is normalized by the total momentum-space density. If the momentum distribution is decomposed as n(kx)=n_th(kx)+n_cond(kx) with a symmetric thermal background and condensate peaks at ±k0, then m ≈ (N_+ - N_-)/N_tot = f_c z, where f_c is the condensate fraction and z=(N_+ - N_-)/(N_+ + N_-) is the valley imbalance of the condensed fraction. Consequently the variance O = Var(m) ≈ f_c^2 Var(z) up to subleading thermal terms. Since f_c(T) decreases monotonically and vanishes at T_c≈315 nK, the measured O carries a temperature-dependent prefactor f_c^2 that is unrelated to Z2 restoration. The fitted T_m=165(40) nK (Eq. (8)) therefore cannot be interpreted as the Z2 transition temperature without first removing this prefactor; the same confound affects the polynomial, exponential, and power-law fits in Supplementary Fig. S5. I request a valley-resolved reanalysis: define z from the populations of the two condensate peaks and extract the transition from Var(z) directly, or fit O(T) with an explicit f_c(T)^2 factor. Without this, the central claim that T_m < T_c and that the intermediate paramagnetic superfluid window widens with k0 is not supported by the presented data.","section":"§2, Eqs. (4)-(5) and Methods Eq. (8)"},{"comment":"The text states that the data analysis applies a 3σ rule to remove outliers before computing O. Near a spontaneous Z2 symmetry-breaking transition, the shot-to-shot distribution of m is bimodal, and the very shots with large fluctuations of m are the ones that carry the symmetry-breaking signal. If the outlier criterion is applied to quantities correlated with m (e.g., total density or peak visibility), it risks removing the bimodal tails that dominate the variance O, thereby suppressing the measured O and biasing T_m downward. Please demonstrate that the reported T_m is stable with respect to the outlier threshold, for example by repeating the fits without the 3σ cut or with 2σ and 4σ thresholds.","section":"§2 (TOF data analysis)"}],"minor_comments":[{"comment":"There is a typo in the main text: 'More detials' should be 'More details'.","section":"§2"},{"comment":"The symmetry group is stated inconsistently: the abstract says 'U(1) and time-reversal Z2', while the Discussion says 'U(1)×U(1) and time-reversal Z2'. Please clarify which symmetry group is actually broken and restored at each transition.","section":"Abstract and Discussion"},{"comment":"The left and right axes for the shaken and static condensate fractions are easy to confuse; consider using two panels or placing the curves on the same axis with clear labels.","section":"Fig. 3(b)"},{"comment":"In the definition of m, please specify that the integral is over the first Brillouin zone and that n(kx) is the momentum distribution integrated over transverse momenta; also state the value of sgn(kx) at kx=0.","section":"Methods, Eq. (4)"},{"comment":"The data are deposited on Zenodo, which is good; adding the analysis code would further support reproducibility and allow the requested reanalysis to be performed by the community.","section":"Data availability"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely question and the experimental platform is well suited for it, but the extraction of the Ising transition temperature is confounded by the condensate-fraction factor in the order parameter. I would ask the authors to perform a valley-resolved reanalysis before publication; if the reanalysis confirms T_m < T_c and the widening of the intermediate phase, the paper could be published, possibly as a revision with the new analysis."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is the first experiment to claim sequential thermal melting of a chiral superfluid into a paramagnetic superfluid, and the phase diagram versus valley separation is new. The experiment is well executed and the data are public. But the Ising transition temperature is extracted from an order parameter that is normalized by total density, so its variance includes the condensate fraction squared. That confound can bias T_m downward and widen the apparent intermediate phase. I think the stress-test note is right.\n\nThe new element is the thermal melting sequence and the systematic widening of the intermediate window with k0. The authors use two independent observables (condensate fraction and Ising variance) and map out a phase diagram across five shaking frequencies. The supplementary fits (polynomial, exponential, power-law) agree, but they all share the normalization issue, so that agreement does not test the confound. The effective field theory and g2≈4g1 come from prior work, but the experimental measurement is independent, so no circularity.\n\nThe main soft spot: Eq. (4) defines m as the ratio of the first moment to the zeroth moment. When the thermal background is symmetric, m = f_c z, where f_c is the condensate fraction and z is the valley imbalance. Thus O = Var(m) ≈ f_c^2 Var(z). Since f_c falls with temperature and T_m ≈ 165 nK is well below T_c ≈ 315 nK but still in a region where f_c is changing, a power-law fit to O can pull the fitted zero below the true Z2 transition. That could manufacture a vestigial phase even if none exists. The 3σ outlier removal may also discard the bimodal tails that carry the Z2 signal. The paper does not discuss this.\n\nA secondary but real issue: the TOF images cannot distinguish a coherent superposition of valleys from a fragmented condensate. The intermediate phase is characterized by |<φ+>|=|<φ−>|≠0, but the momentum distribution alone does not prove phase coherence between valleys. The authors do not claim to measure relative phase, but the label \"paramagnetic superfluid\" is asserted rather than demonstrated.\n\nWho is this for? Cold-atom and condensed-matter theorists and experimentalists interested in vestigial order and multi-orbital systems. It will get cited as the first cold-atom demonstration, but the quantitative T_m values should not be taken at face value until the order parameter is reanalyzed. I would send it to peer review, but the referee should demand a valley-resolved order parameter—for example, measuring the imbalance without total-density normalization—and a clear equilibration check. With that reanalysis, the qualitative scenario may well hold; without it, the central claim is not established.","headline":"Credible first experiment on vestigial order melting in a chiral atomic superfluid, but the Ising transition temperature is confounded by a density-normalized order parameter and needs a reanalysis.","tokens_in":14023,"tokens_out":2891,"would_cite":false,"duration_ms":27325,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["67.85.Hj","03.75.Lm"],"model":"deepseek-v4-flash","headline":"In a shaken optical lattice, a chiral superfluid melts in two stages—time-reversal symmetry is restored at a lower temperature than the superfluid order—and the vestigial paramagnetic-superfluid window widens as the valley separation…","keywords":["chiral superfluid","vestigial order","shaken optical lattice","orbital order","Floquet band","time-reversal symmetry breaking","Ising transition","quantum gas"],"falsifier":"Recompute the asymmetry order parameter from the raw valley populations without normalizing by total density, or restrict the analysis to atoms inside the condensate peaks; if its fluctuations vanish at the same temperature where the condensate fraction extrapolates to zero, then the reported $T_m < T_c$ ordering and the intermediate paramagnetic superfluid are artifacts of the normalization rather than a real vestigial phase.","tokens_in":12947,"feed_emoji":"🌀","tokens_out":8227,"duration_ms":90294,"temperature":0.7,"pith_summary":"This paper reports an experimental observation of vestigial order melting in a quantum gas. In a shaken one-dimensional optical lattice, rubidium atoms form a chiral superfluid that condenses into one of two degenerate valleys, breaking both the $U(1)$ phase symmetry and a time-reversal $\\mathbb{Z}_2$ symmetry. The authors show that heating does not destroy both orders at once: the time-reversal symmetry is restored first, at an Ising transition temperature $T_m$, leaving a vestigial paramagnetic superfluid that still has phase coherence; only at a higher temperature $T_c$ does the superfluid itself disappear. Across every shaking frequency they explored, $T_m$ stays below $T_c$, and the intermediate paramagnetic-superfluid window widens as the valley separation is increased by driving closer to resonance. This is evidence that thermal fluctuations can produce a sequential, rather than simultaneous, melting of intertwined orders, with a vestigial phase surviving between the two transitions.","feed_headline":"Chiral superfluid melts in two steps, not one","feed_subtitle":"Time-reversal symmetry dies first, then phase coherence; the gap widens as valley separation grows.","key_machinery":"The central object is the Floquet-engineered double-valley band: lattice shaking superposes $s$- and $p$-orbital states so that the effective ground band has two degenerate minima at $\\pm k_0$, whose separation grows as the driving frequency approaches the $s$–$p$ gap. Low-energy physics is described by a two-component field theory for fluctuations at the two valleys, with intravalley and intervalley couplings whose ground state minimizes interaction energy by condensing into one valley. The Ising transition is diagnosed by the momentum-space valley asymmetry $m$ (Eq. 4) and its run-to-run variance $O$ (Eq. 5), fit to a power-law form (Eq. 8) to locate $T_m$; the superfluid transition is located by extrapolating the condensate fraction $f_c$ to zero (Eq. 9).","core_discovery":"Using $^{87}$Rb atoms in a shaken one-dimensional optical lattice, the experiment creates a Floquet band whose lowest dispersion has two degenerate minima at $\\pm k_0$, formed by hybridizing $s$- and $p$-orbital states. At low temperature, condensation into a single valley produces a chiral superfluid with a real-space phase winding, breaking the $U(1)$ phase symmetry and the time-reversal $\\mathbb{Z}_2$ symmetry. The central claim is that heating melts this state in two clearly separated steps, not one: at $T_m = 165(40)$ nK for the main driving frequency, the $\\mathbb{Z}_2$ symmetry is restored—atoms populate both valleys equally—while condensate coherence persists; the condensate fraction only vanishes at the higher $T_c = 315(11)$ nK. The authors find $T_m < T_c$ throughout the investigated range of shaking frequencies, with $T_m$ suppressed near resonance while $T_c$ remains almost unchanged, so the paramagnetic-superfluid window grows with $k_0$. Far from resonance the two transitions merge within error, consistent with a vestigial-order melting scenario rather than a single first-order simultaneous transition.","pith_inferences":["The paper lists a chiral thermal state as an alternative melting route but does not observe it; the same Floquet platform might reach that route by tuning parameters in the opposite direction, which would constitute a separate test of the vestigial scenario.","A direct measurement of first-order coherence inside the intermediate window—not just the condensate fraction—would confirm that the paramagnetic superfluid truly carries phase coherence after the valley choice has been erased.","The systematic widening of the window with $k_0$ suggests that larger valley separation generically stabilizes partial symmetry breaking; a quantitative prediction of $T_m(k_0)$ from the two-valley field theory would be a natural theoretical follow-up."],"forward_implications":["In the explored regime, heating always restores time-reversal symmetry before superfluidity, so a paramagnetic superfluid with equal valley populations exists between $T_m$ and $T_c$.","Tuning the shaking frequency toward resonance (larger $k_0$) suppresses $T_m$ while leaving $T_c$ nearly constant, systematically widening the vestigial-order window; far from resonance the two transitions merge within error.","The shaken-lattice superfluid transition temperature is lower than in the static lattice, consistent with the effective halving of phase-space density when the condensate is distributed over two minima.","The dependence of $T_m$ on valley separation is attributed to a reduction of the effective local interaction as the $p$-orbital fraction of the Floquet band increases.","The observed two-step melting provides a finite-temperature realization of vestigial order in a multi-orbital superfluid, offering a controlled setting for studying intertwined symmetry breaking."],"supporting_citations":[{"why":"Supplies the two-valley effective field theory and the proposed chiral Bose liquid/vestigial melting scenario that this experiment tests.","marker":"[13]"},{"why":"Provides prior evidence for an atomic chiral superfluid and the two-step evaporative cooling technique used to control temperature.","marker":"[20]"},{"why":"Establishes the shaken-lattice platform, the relation between shaking frequency and valley separation, and the interaction estimate $g_2 \\approx 4g_1$.","marker":"[25]"},{"why":"Argues that simultaneous restoration of both symmetries would be first order, motivating the sequential vestigial melting pathway.","marker":"[35]"},{"why":"Supplies the power-law form used to fit the Ising order-parameter variance and extract $T_m$.","marker":"[38]"},{"why":"Provides the condensate-fraction fitting form used to extract the superfluid transition temperature $T_c$.","marker":"[40]"}],"fun_headline_variants":["Two-step melt: time-reversal order dies first, then coherence","Chiral superfluid melts in stages, not all at once","Superfluid melting: two transitions, not one","Atomic superfluid dissolves in two separate steps","Vestigial order seen in chiral superfluid's two-step melt"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the run-to-run variance of the valley-asymmetry parameter is governed by the time-reversal symmetry-restoring transition, even though that parameter is normalized by total density and therefore also shrinks as the condensate fraction drops with temperature.","fun_headline_variants_meta":{"raw":{"variants":["Two-step melt: time-reversal order dies first, then coherence","Chiral superfluid melts in stages, not all at once","Superfluid melting: two transitions, not one","Atomic superfluid dissolves in two separate steps","Vestigial order seen in chiral superfluid's two-step melt"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000483,"raw_usage":{"total_tokens":2423,"prompt_tokens":1019,"completion_tokens":1404,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":635,"completion_tokens_details":{"reasoning_tokens":1322}},"tokens_in":635,"tokens_out":1404,"duration_ms":14459,"temperature":1.0,"reasoning_tokens":1322,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:40:45.893872+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the asymmetry order parameter from the raw valley populations without normalizing by total density, or restrict the analysis to atoms inside the condensate peaks; if its fluctuations vanish at the same temperature where the condensate fraction extrapolates to zero, then the reported $T_m < T_c$ ordering and the intermediate paramagnetic superfluid are artifacts of the normalization rather than a real vestigial phase.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the two-valley effective field theory and the proposed chiral Bose liquid/vestigial melting scenario that this experiment tests."},{"cited_title":"Nature 596(7871), 227–231 (2021) https://doi.org/10.1038/s41586-021-03702-0","cited_arxiv_id":null,"evidence_quote":"Provides prior evidence for an atomic chiral superfluid and the two-step evaporative cooling technique used to control temperature."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the shaken-lattice platform, the relation between shaking frequency and valley separation, and the interaction estimate $g_2 \\approx 4g_1$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Argues that simultaneous restoration of both symmetries would be first order, motivating the sequential vestigial melting pathway."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the condensate-fraction fitting form used to extract the superfluid transition temperature $T_c$."}],"review_version":1}