Reentrant melting of scarred odd crystals by self-shear

Domain: physics
Published: 2026-01-16
Editorial outcome: minor_revision

Abstract
Spatial confinement can induce geometrical frustration in condensed phases, giving rise to topological defects that confer materials with new and exotic properties. Here, we experimentally uncover the effect of confinement-induced defect strings termed ‘grain boundary scars’ on the behavior of dense two-dimensional assemblies of granular spinners, a canonical odd elastic solid. The spatial arrangement of these scars fundamentally reshapes the flows triggered by chiral activity. Specifically, they cause the topologically protected edge flows-a ubiquitous feature of confined spinner assemblies-to decouple from the bulk. Strikingly, increasing the net chiral activity of the system by tuning the ratio of counterclockwise to clockwise spinners caused spontaneous self-shearing. The resulting odd radial stresses led to a chiral activity-mediated reentrant melting transition at a fixed areal spinner density. Our findings open new avenues for exploiting geometrical frustration to elicit novel responses from odd elastic solids.

Introduction
Topological defects are often the orchestrators of material behavior1. While their established role lies in tuning the physical properties of passive materials2,3, there is growing interest in extending this paradigm to active matter systems4,5. A simple route to create these defects is through geometrical frustration6, achieved by confining the system within a carefully chosen boundary shape that prevents the preferred local order of the system from tiling space. These defects persist in the system even in its ground state-a consequence of Euler’s theorem7. A classic example is the appearance of twelve pentagonal disclination defects in a triangular lattice on a spherical surface, as seen in viral capsids8, colloidosomes9–11, and the familiar soccer ball. Six such defects emerge on confining the same lattice within a two-dimensional (2D) circular boundary12,13. To relieve the elastic strain energy on exceeding a threshold system size, these disclinations bind to dislocations, forming 1D defect strings called ‘grain boundary scars’9–11,14,15, which in turn govern the yielding behavior of these frustrated crystals16. In the realm of active matter, geometrical frustration effects are more dramatic. In these out-of-equilibrium systems, particularly in those made of elongated building blocks, defects themselves can be motile entities17–21. This has spurred a surge of research exploring how boundary geometry can be harnessed to create and manipulate these defects, ultimately enabling global control over active flows and turbulence22–35.

Inspired by these studies, here we examined how confinement-induced topological defects alter the collective behavior of 2D dense assemblies of active spinners. By breaking both parity and time-reversal symmetries, this system allows for the existence of novel odd material moduli that are forbidden in equilibrium36,37. These odd moduli generate transverse couplings between mechanical perturbations and responses; for example, compressing the assembly can induce a net torque density, driving spontaneous rotation. This behavior originates from transverse forces generated by friction between the spinning elements38,39. While spinners offer a specific realization of oddness, odd mechanics is a more general consequence of chirality and broken time-reversal40,41. The presence of odd moduli results in exotic new phenomena: topologically protected unidirectional edge currents capable of transporting cargo without scattering42, self-sustained chiral waves43, spontaneous rotation and self-kneading of spinner crystals into whorls44, chiral excitation and tilting of force chains45, and emergence of avoid phase46. Although many previous studies have investigated confined spinner packings47–49, the fundamental role of defects arising from geometrical frustration on their emergent dynamics remains unaddressed, highlighting a key gap in our understanding.

Results
Tuning the oddness of dense granular spinner assemblies
To isolate how confinement-induced topological defects affect spinner ensembles, our experimental system must meet two key criteria: it must be able to host a defect-free ground state without confinement, and its odd mechanical response must be tunable. Granular spinners driven by vertical vibrations suit our purpose50,51 (Fig. 1A, see “Methods” section). Our spinners are spherical domes of diameter a = 4.2 mm standing on tilted legs that transduce vertical vibrations into torque and make them spin. The handedness of rotation, i.e., clockwise (⊗) or counterclockwise (⊙), is set by the handedness of the leg tilt. Pairwise spinner interactions comprise a longitudinal hard-core repulsion and a transverse force due to contact friction. The horizontal cross-section of the spinners is circular, and they can readily pack into defect-free triangular lattices. We induced geometrical frustration in the system by confining the spinners within a circular arena of diameter D = 30 cm. Our experiments spanned area fractions 0.68 ≤ ϕ ≤ 0.79, with the lower limit corresponding to the density beyond which the spinners began to form crystalline domains with triangular symmetry. The strength of the odd response scales with the internal torque density36,38,44, which we tuned by adjusting the net chiral activity, χ, of the spinner assembly. Transitioning from a racemic mixture (a 50:50 combination of ⊗ and ⊙ spinners) to a homochiral state (all ⊗ or all ⊙) tunes the internal torque density from zero to the maximum achievable while keeping all other parameters fixed. Here χ = (N⊗ − N⊙)/N with N⊗ and N⊙ being the number of ⊗ and ⊙ spinners, respectively, and N = N⊗ + N⊙. Fig. 1A shows a snapshot of the system for ϕ = 0.72 and χ = 0. For 0 ≤ χ < 1, the spinners remained well-mixed and did not show any tendency to phase separate (Supplementary Fig. 1). At the largest area fraction studied (ϕ = 0.79), the system contains N = 4000 spinners, making it an order of magnitude larger than previous granular experiments48–51.

We complement the experiments with discrete element method (DEM) simulations that capture key features of our granular assemblies. In our minimal model, the spinners are monodisperse disks within a circular confinement (“Methods” section). Each spinner is subject to a torque that enforces a constant angular velocity. The equations of motion for the position and the angular coordinate of the spinners follow underdamped dynamics. Spinners interact via a repulsive normal force and a tangential friction force, computed using the spring-dashpot model.

A chiral activity-mediated reentrant melting transition
Figure 1B captures our main observation. The panels show spinner packings with increasing χ for ϕ = 0.72, with the colors representing the magnitude of the local hexagonal bond-order parameter, |ψ6|.
Here, ψ6 = (1/N) ∑j=1N exp(6iθij) where θij is the angle of the nearest-neighbor j with respect to an arbitrary reference axis. What is striking is the emergence of a near-perfect single crystal in the bulk (region enclosed by the dashed circle) for intermediate values of χ, while for the extreme values, a large pocket of liquid is evident. Tuning χ causes reentrant melting of the bulk, the imprint of which is also present in the pair-correlation function, g(r) (Supplementary Fig. 2). We recall that for equilibrium hard disks in 2D, the liquid phase is found at densities ϕLiq ≤ 0.716, and the crystalline phase is stable for ϕXtal ≥ 0.72; the intervening phase is a hexatic52. While drawing direct comparisons with equilibrium systems is difficult because our system is driven, it is notable that the bulk areal density, ϕBulk < ϕLiq for χ ∈ [0, 1] and ϕBulk ≥ ϕXtal for χ ∈ [0.3, 0.6] (Fig. 1B). The system at ϕ = 0.68 also showed a reentrant behavior. At higher densities, increasing χ enhanced crystallinity, but the system did not remelt (Fig. 1C, see Supplementary Figs. 3–5 for |ψ6| at other ϕ and χ values). In fact, the strength of this reentrance, Δ = ⟨|ψ6|⟩χ=0.3 − ⟨|ψ6|⟩χ=1, appears to be non-monotonic with ϕ, with an apparent maximum near ϕ ≈ 0.72 (Fig. 1D).

Chiral activity causes spontaneous self-shearing
Since reentrant melting of the bulk is driven by chiral activity, the underlying physics may be due entirely to odd effects, which are known to couple azimuthal flows to radial density changes36,38,39,44. Indeed, the radial density, ϕA(r), showed marked changes with χ (see Fig. 1E) for ϕ = 0.72. For χ = 0 (net zero torque density), ϕA(r) is nearly identical to that of a passive disk packing at the same ϕ. The enhancement in density near the boundary is simply due to layering53,54. As χ increases, layering became more pronounced, and ϕA(r) in the bulk (r/R < 0.7) also showed clear changes. Both of which are clear signatures of additional radial stresses at play.

These observations present a natural segue into quantifying the flows that may have induced these density changes. In Fig. 2A, we show the annular angular velocity, ω(r), for different values of χ for ϕ = 0.72. For all χ > 0, we observed edge flows-a ubiquitous feature of confined spinner assemblies. These flows are a consequence of the unbalanced torques on spinners at the edge47,48. Additionally, the edge flows have the same handedness as the spin of the majority spinner species (⊗ spinners), indicating that inter-spinner frictional interactions dominate over spinner-boundary ones48 (see Supplementary Fig. 6). The transverse forces exerted by spinners in the penultimate layer on those in the outermost one drive the edge flow. For χ = 0, these forces cancel since N⊗ ≈ N⊙ in each annulus, and ω(r) fluctuated around zero for all r/R.

Two features in the measured ω(r) stood out. First, the magnitude of the edge flow for χ > 0 dropped precipitously at r/R ≈ 0.9 for 0.68 ≤ ϕ ≤ 0.75. This behavior should be contrasted with previous observations on spinner liquids and solids. In spinner liquids, the edge flow decays exponentially into the bulk since the system lacks a shear modulus, while the finite modulus of solids causes these edge flows to couple with the bulk, resulting in rigid body rotation47. Second, for ϕ = 0.68 and 0.72, the system spontaneously self-shears (also see Supplementary Figs. 7 and 8). Specifically, for ϕ = 0.72, while ω(r) hovered around zero for r/R < 0.9 at χ = 0.3, it became negative and nearly constant in bulk for χ = 0.6 and 1, with the magnitude of counterclockwise rotating bulk flow larger at χ = 1 (see Supplementary Videos 1–4).

At higher densities (ϕ = 0.75 and ϕ = 0.79), the system showed neither reentrant melting nor self-shearing (Supplementary Figs. 7 and 8). For ϕ = 0.75 and all χ > 0, the magnitude of the edge current was smaller than that at ϕ = 0.72 due to stronger inter-spinner interactions, and ω(r) decayed to zero in the bulk, while for ϕ = 0.79, we observed rigid body rotation for χ > 0. These results demonstrate that reentrant melting and self-shearing are concomitant phenomena; below, we explain their common origin.

Grain boundary scars aid in decoupling edge and bulk flows
We first looked for cues in the structure of the spinner packings. In Fig. 2B, we show the Voronoi tessellation of the spinner packing for ϕ = 0.72 and χ = 1. The yellow hexagons represent spinners with six-fold coordination (crystalline particles), while polygons of other colors represent under- and over-coordinated spinners (disclination defects). The large defect clusters in the bulk correspond to the liquid pocket shown in Fig. 1B. Notably, at r/R ≈ 0.9, where ω(r) fell sharply (gray annulus in Fig. 2B), we observed azimuthally aligned grain boundary (GB) scars, which also appear as strings of low |ψ6| particles in Fig. 1B (see Supplementary Fig. 9 for other system sizes). Each scar comprised an excess 5-coordinated disclination decorated by a string of dislocations. The dislocations-neutral disclination dipoles-are not a topological requirement but are present to help screen the strain field of the disclination7,14. We could easily identify six GB scars at intermediate chiral activity values, where crystalline order was prominent, while for extreme values (χ = 0 & 1), some of the scars were part of larger defect clusters, making their identification challenging (Supplementary Fig. 10).

The GB scar configuration is determined by geometry and not activity. Crucially, we observed that vibrated passive disk packings at the same ϕ showed similar scar arrangements, confirming that it is the circular confinement that decides these configurations. Further, as observed in passive particle packings on a sphere, the number of excess dislocations per scar, ND, increased with the system size (D/a) for both spinner and passive disk packings (Supplementary Figs. 9 and 11)9,14. Notably, our simulations of confined spinners accurately reproduced our experimental results (Fig. 2C, Supplementary Fig. 12). Importantly, even in systems where the GB scars were not well-defined due to excessive disorder (ϕ = 0.68), the drop in ω(r) occurred in the annuli containing a higher-than-average defect density (Supplementary Figs. 13 and 14).

GB scars/defects clearly impact the coupling between the edge flow and the bulk. To see why, we note that within a continuum description of a chiral active material with strains ∂u_l and strain-rates ∂u_l, the stress tensor σij = −Pδij + Kijkl ∂uk + ηijkl ∂u_k + σspin_ij (Supplementary Fig. 15)36,44,55. Here, P is the pressure, δij is the Kronecker delta, and Kijkl and ηijkl are the elasticity and the viscosity tensors, respectively, and contain all moduli consistent with broken parity and time-reversal. The last term is the antisymmetric stress due to the imposed drive torque and is given by σspin_ij = 2ηR ϵij Ω, where ηR is the rotational viscosity, Ω is the spin angular velocity field of the particles, and ϵij is the Levi-Civita symbol37. A mismatch between Ω and the 2D vorticity, (∇ × v)z = 2ω(r), where v is the velocity field, results in a frictional stress σfric_ij = ηR ϵij (2Ω − ω)37. Since ηR results from frictional collisions between the spinners, it is proportional to the local spinner density and is expressed as ηR ∝ ϕ2g(r)48,56, where g(r) is the pair-correlation function at contact.

Here, in addition to layering, GB scars also modify the spinner density in the annuli, but only near r/R ≈ 0.9. Figure 2D shows that ϕA(r/R ≈ 0.9) in the sector harboring the scar is nearly 25% smaller than the one without it for ϕ = 0.72 (see Supplementary Fig. 16 for ϕ = 0.68 and ϕ = 0.75). Consequently, the average spinner density, and hence ηR and σfric_ij, are smaller in the scarred annuli compared to a system without the scars. A proxy for σfric_ij that can be easily measured in our simulations is τres(r) = ∑i≠j τij - the resistive torque experienced by an annulus from annuli straddling it. Here, i and j denote particles in the annulus under consideration and in those adjacent, respectively. Fig. 2E shows τres(r) for the same χ values investigated in the experiments for ϕ = 0.72. A pronounced minimum appears in τres(r) at r/R ≈ 0.9 for χ = 1 (see Supplementary Fig. 17 for other ϕ values), indicating that GB scars weaken the coupling between the edge flow and the bulk. On lowering χ, both the magnitude of τres(r) and the depth of the minimum becomes smaller and vanish at χ = 0. This trend arises because increasing the fraction of ⊙ spinners at the expense of ⊗ spinners systematically reduces annulus-averaged spin angular velocity, ⟨Ω(r)⟩, and consequently, τres (see Fig. 2F and Supplementary Figs. 18–20).

We can now explain why the system self-shears. We observe that while the spinners in the edge layers (labeled L1–L3 in Fig. 2G) are largely in the registry and move as a cohesive plug, GB scars in layer L4 disrupt layering. This disruption prevents the registry from extending into the bulk, effectively creating a slip-plane between the edge and the bulk. The clockwise rotation of spinners in layer L3 exerts a counterclockwise tangential force on layer L4, and because ϕA_L3 > ϕA_L4 (Fig. 1D), layer L4 rotates counterclockwise. Spinners in layers further interior fall in the registry again, as the scars have eased the frustration caused by circular confinement. The counterclockwise rotation of layer L4 now couples to this ordered bulk, and the system spontaneously self-shears. On decreasing χ, however, the magnitude of the counterclockwise tangential force exerted by L3 on L4 also decreases, and self-shearing weakens.

Odd stresses arising from self-shear drive reentrant melting
With a better understanding of how tuning χ produces these flows, we now argue that parity-violating inter-spinner collisions generate a radial odd stress (σrr), which drives reentrant melting. Let us first consider the case illustrated in Fig. 3A(i), where all particles spin clockwise (⊗, χ = 1) and bulk and edge flows are absent. Here, the edge spinners (green region) are equally likely to collide with bulk spinners (orange region) from above or below. The transverse collision forces from the edge spinners on the bulk spinners (black arrows) average out to zero along br; only the θ component survives. As a result, σrr = 0, and the spinner density is unchanged. In Fig. 3A(ii), the particle spin is still clockwise, but both the edge and bulk flows have a handedness opposite to the spin. Since ωEdge > ωBulk, edge spinners are more likely to collide with the bulk spinners from below. The transverse forces acquire a net radially outward component. σrr > 0, and consequently, the bulk dilates. Conversely, if the handedness of the azimuthal flows is reversed, i.e., it has the same handedness as the particle spin with ωEdge > ωBulk; collisions from above become more frequent (Fig. 3A(iii)). Here, σrr < 0, and the bulk is under compression. Importantly, σrr breaks parity and is a consequence of odd material moduli. This behavior should be contrasted with passive granular packings, where dilation-induced normal forces are insensitive to the direction of shear. Finally, in mixtures of ⊗ and ⊙ spinners, but with the former in excess (0 < χ < 1, Fig. 3A(iv)), ⊗ azimuthal flows still give rise to a compressive radial stress (−ve σrr). However, collisions with ⊙ spinners offset this effect and reduce its magnitude, resulting in milder changes in density.

The coarse-grained local vorticity, presented in Fig. 3B for ϕ = 0.72 at all values of χ, provides a precise representation of the flow field in our system. We calculated the vorticity solely for the longest-lived flows, since short-lived vortices primarily reflect transient fluctuations and do not appreciably affect the steady-state radial density profile ϕA(r). To directly connect these flow patterns with density changes, we quantified the heights of the radial density peaks, PH, in ϕA(r) (Fig. 1D) and tracked how PH(r) varies with r/R (Fig. 3C). A negative slope of PH(r) in the bulk (r/R < 0.7) indicates that this region is compressed, while a positive slope indicates dilation.

At χ = 0, we observed small vortices of both handedness (⊗ and ⊙), but they are too weak to alter the bulk spinner density; as a result, PH(r) remains essentially flat in the interior (r/R ≤ 0.7). Further, wall-induced layering depletes spinners from the bulk, causing its density (ϕBulk ≈ 0.706) to fall below the equilibrium liquid threshold (ϕLiq = 0.716). The absence of radial stresses for χ = 0 (Fig. 3A(i)) means that the peak widths of PH near the edge, PW, are broader than at other chiral activities (Fig. 3D). For χ = 0.3, there is now an excess of ⊗ spinners. Here, the bulk flow is absent, and the edge flow is clockwise (⊗) (Fig. 2A) and gives rise to a −σrr br. This stress compacts the bulk, increasing its density beyond the crystallization threshold (ϕBulk ≈ 0.725 > ϕXtal) and drives the bulk into a crystalline state; |ψ6| is therefore large (Fig. 1C). In addition, due to finite radial stresses, the edge peaks are narrower than those at χ = 0 (Fig. 3D). At larger chiral activity, χ = 0.6 and 1, the situation is more complex since the system self-shears. The inward compressive stress generated by the ⊗ edge flow competes with an outward dilative stress from the ⊙ bulk flow; PW predictably narrows (Fig. 3D). The contribution from the edge barely wins over and compresses the bulk at χ = 0.6, but this suffices to melt the crystal. At χ = 1, the outward dilative stress from the ⊙ bulk flow dominates, and the bulk density falls below ϕLiq, causing the system to melt again. Taken together, these results reveal a direct and robust link between odd radial stress and bulk density changes. Notably, this link was present at all ϕ and χ values studied here (Supplementary Figs. 21–23).

Methods
Particle fabrication
The spinners were 3D-printed using the PROJET3600 MultiJet 3D printer. The printer has a high resolution of 16 μm, allowing us to print 3D models with small features. These spinners were designed in 3D Builder software. Each spinner has 16 legs attached to the oblate spheroidal dome. We print around 1000 spinners in a batch, which takes about 1−2 h. These spinners are embedded in a wax mold as part of the print process, and the post-processing requires sonicating them in oil at 60 °C to remove wax. After sonication, the spinners are washed with a soap solution to remove excess oil from the surface.

Imaging the particles
A high-speed camera was used to image the spinners from the top. The image acquisition rate was between 40 Hz and 30 Hz. These values were chosen appropriately to track the orientation of each spinner. The spatial resolution of 2464 × 2056 pixels was set during data acquisition. The collected data was processed using custom-written code in MATLAB.

Simulation details
To mimic the experimental system45,57–63, we simulate a minimal model of a dense collection of spinners in 2D. This granular assembly consists of monodisperse spinners confined in a circular arena. The equation of motion for each spinner, for the position ri and angular coordinate θi follow the underdamped equations,

m€ri = −ζT r_i + Fi + ηTi (1)

Ii θ€i = τi a + τi int + ηRi (2)

where m is the mass of the i-th spinner, ζT is the translational damping coefficient, Fi is the net interaction force acting on the i-th spinner. This force, Fi = ∑j≠i (Fnij + Ftij) (summed over all neighbors of i, indexed by j) has two different contributions; one is from a repulsive harmonic normal force Fnij and the other from a tangential frictional force Ftij. Also, we compute the interaction torque as

τi int = ∑j≠i Γij = − ∑j≠i (Ri + Rj) r_ij × Ftij ∧ bz (3)

where Γij is the pairwise frictional torque, Ri, Rj are the radius of spinner i, j respectively, rij = ri − rj and bz is the unit vector along z direction. Here, the spinners are distributed in the x–y plane.

To model these forces, we adopt the spring-dashpot model of contact forces (between particle i and j) given by

Fnij = −kn δrij br_ij − ζn vnij (4)

Ftij = −kt sij − ζt vtij (5)

Where kn and kt are spring constants associated with the normal and transverse harmonic forces. The quantities δrij = δrij br_ij, sij are corresponding displacements for the normal and the transverse direction. Here, δrij = Ri + Rj − rij, the relative distance rij = |rij| = |ri − rj|, and br_ij is the unit vector along the relative distance. Normal and tangential damping coefficients ζn, ζt control dissipation during collisions, and vij is the relative velocity of the point of contact (where subscripts n and t denote normal and tangential components). We compute these components following,

vnij = (vij · br_ij) br_ij (6)

vtij = vij − vnij − (Ri θ_i + Rj θ_j) (bz × br_ij) / (Ri + Rj) (7)

The transverse displacement sij is computed during the period of contact (between spinner i and j), i.e., it is initialized at 0 when the contact is just established and integrated numerically following,

s_ij = vtij − (vtij · sij) rij / r^2_ij (8)

and reset to 0 when the contact is broken. Therefore, the contacts (and correspondingly sij) are history-dependent and cannot be computed from the instantaneous configuration. Using Eq. (5), we calculate Ftij and Coulomb’s law of contact friction is implemented by limiting |Ftij| ≤ μ|Fnij| (after which a contact point starts sliding with constant tangential force).

We also have included a translational noise ηTi whose mean is zero ⟨ηTi(t)⟩ = 0 and its variance is given by ⟨ηTi,α(t) ηTj,β(t0)⟩ = 2DT δij δαβ δ(t − t0) where DT represents the translational diffusion constant associated with this noise.

For the equation describing the angular degrees of freedom, Ii is the moment of inertia and τai is the active torque, modeled as τai = α(ω0 − ωi) where 1/α is the relaxation time scale and ωi = dθi/dt. Similarly, as before, for the angular degrees of freedom we have a rotational noise ηRi whose mean is zero ⟨ηRi(t)⟩ = 0 and its variance is given by ⟨ηRi(t) ηRj(t0)⟩ = 2DR δij δ(t − t0) where DR represents the rotational diffusion constant associated with this noise.

We use Nb = 800 frozen (whose position, momentum, and angular degrees of freedom do not evolve) particles to create circular confinement. Unless stated otherwise, we use the size of the circular confinement r = 20.60, mass m = 7.0, the radius of each spinner Ri = Rj = 0.288, ζT = 0.15, DT = 4 × 10−5, DR = 10−3, ω0 = 20, α = 100, ζn = 4.0, ζt = 2.0, kn = 100, kt = 100. Between all the mobile spinners, we used a friction coefficient μp = 0.60, and between the mobile spinners and the frozen boundary particles, we used a friction coefficient μp = 0.65. For different area fractions, we either use (a) different number of spinners, i.e., N = 3689 for ϕ = 0.72, N = 4050 for ϕ = 0.79, etc., or (b) fixed number of spinners N = 3689 with different spinner radius R0 = 0.279, 0.288, 0.293, and 0.301 for ϕ = 0.68, 0.72, 0.75, 0.79. To explore the system size effects for the passive disks, we performed simulations with N = 500, 1000, and 2000 as well. To analyze the radial profile of different observables, we divided the circular arena into nr = 40 annuli and averaged the observables over the spinners inside each annulus.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "I apologise for the delay in my report. In this article, Tiwari et al construct an experimental system of confined spinners with a mixture of orientations, showing that they exhibit odd self-rheology characterised by a coexistence between crystalline regions and regions with defect scars near the edge. The chiral, self-shearing motion is concentrated in these regions and is sensitive to local and global packing fraction, and to the relative concentration of spinners of different orientations.\nI find the research carefully carried out, and the results interesting. However, my main comment is that the current format with an exceptionally short main text makes the arguments rather vague and difficult to follow. I would support publication if the authors satisfactorily answer my questions below and also if they rewrite their manuscript to include substantial parts of the supplementary in the main.\nSpecific comments:\nEdge flows, self shear direction and reentrant melting: Clearly, there is something complex going on here, and I don't think there is a full explanation in the manuscript.\nThe authors interpret their results within the framework of continuum odd elasticity, which has in recent years been developed for systems with internal chiral torques. That is of course fine, but it does not capture jamming / unjamming or solid / liquid transition in the system.\nThe author observe the scars, and link them to lowered chiral active stresses in this region because of fewer contacts that transfer torques due to lower density, as well as directly through simulations. The authors develop an intricate microscopic interpretation of the torques near the grain boundaries. I could not make sense of Figure 3A and the inferred torque and stress directions, and why some systems rotate counterclockwise. The link between torque directions and the odd moduli coupling (equation Fig. S16) is also pretty hand-waivy.\nIn the manuscript, there is an extremely brief reference to 'simulations', with data shown only in Figure 2C and 2E. In the supplementary, it then emerges that this is a detailed molecular dynamics simulation with Coulomb friction, essentially identical to granular materials models except for the inclusion of the spinning torque.\nThis then also allows for a better computation of the chiral torques compared to the 'frictional collisions' of [37], with a dip at the scar (Fig 2E). To me a proper introduction of the simulation belongs in the manuscript directly.\nIt should also allow for calculating the stresses and their profiles directly? In the manuscript, the authors relate compression / dilation to the peak height of packing fraction peaks, which seems rather indirect and which I've never seen before. Is there not an even approximate pressure / density constitutive law that one could infer here?\nSeen from the granular angle, what the authors have here is to me a shear band, very similar to the ones seen in (driven) granular materials in a couette shear cell geometry (see e.g. Schall P, Van Hecke M. Shear bands in matter with granularity. Annual Review of Fluid Mechanics. 2010;42(1):67-88.).\nThe core feature is an instability triggered by an inversion in the (shear) stress / strain rate curve, which then makes the system separate into jammed and flowing regions. I suspect that this concept could be usefully translated to the active case, possibly with the net internal oddness \\Chi as a tuning parameter of the instability?\nThere is a well developed literature on models of shear bands that take into account density, friction, and constitutive laws which include nonlocal effects on stress (see e.g. S. Mandal et al, Phys. Rev. X 11, 021017 (2021)). It would relate stresses, density and strain rates / vorticity, like the manuscript tries to do.\nCan the authors relate their observations on strain rate inhomogeneities (~vorticity), density fluctuations and stresses to this framework, even tentatively?\nAll in all, after reading this manuscript I feel I still don't understand why this phenomenon happens.", "reviewer_2": "In this work, the authors investigate the melting behavior of confined active spinners under varying activity. Combining experiments and simulations, they report a reentrant melting phenomenon driven by the interplay between odd response, edge flow and confinement-induced defects, where the crystalline order of the bulk is higher at intermediate activity (compared to vanishing mean activity) but is disrupted at larger activity at a fixed spinner density. The finding is novel, and the work is well-organized. Given the current growing interest in chiral active matter and odd response, the manuscript should be seriously considered for publication in NC. I will be happy to review the manuscript once the authors will have addressed my points in the following. However, as for now, I am still not in the position to recommend publication of the manuscript.\n1. In this study, the system is binary mixture of clockwise and counter-clockwise spinners, and the chiral activity is changed by tuning the molar fraction of clockwise spinners. A more direct way to change activity is to use monodisperse spinner and then to tune the intensity of vertical vibration. Why do the authors not employ such monodisperse system? In addition, in experiments, what is the initial configuration; and in the steady state, are the clockwise and counter-clockwise spinners uniformly mixed?\n2. On page 5, the authors state that “the magnitude of the edge flow drops precipitously at r/R ≈ 0.9. This behavior is distinct from observations made hitherto in spinner liquids and solids.” Its structural origin looks to be the existence of the grain boundary scar near this location. Further, the authors describe that “Unlike conventional grain boundaries, which terminate at the system edge, these scars terminate within the system.” An interesting question is why the grain boundary scars emerge at this specific location (around the 4th layer)? Did the authors perform equilibrium simulation of passive disks (with frictional or smooth boundary) in a circular confinement to examine the position distribution of the GBS? Moreover, does the \"self-shear\" and \"scarred defects\" depend on the curvature of the confinement (finite-size effect)?\n3. In Supplementary Figure 18, the angular velocity profile at φ = 0.75 is non-monotonic, differing significantly from those at other packing fractions. Additionally, the profile at φ = 0.75 appears to display a weak self-shear instead of vanishing self-shear. The authors should provide further discussion on these observations.\n4. The data for annular angular velocities (Fig. 2a) and spin velocities (Fig. 2f) exhibit considerable fluctuations. To enhance the data quality, more extensive statistical averaging should be conducted.\n5. The strengths of reentrance Δ plotted in Fig.2c and Fig.2d appear to have different meanings. And, they are not equivalent with the definition in the second paragraph of page 4. Please carefully check this.\n6. A typo: Page 7, second paragraph: “spinners in layers further interior fall in…” should read “Spinners in layers further interior fall in…”."}, "author_response": "Response to reviewer’s comments\nReviewer #1 (Remarks to the Author):\nComment: I apologise for the delay in my report. In this article, Tiwari et al\nconstructanexperimentalsystemofconfinedspinnerswithamixtureoforienta-\ntions, showing that they exhibit odd self-rheology characterised by a coexistence\nbetween crystalline regions and regions with defect scars near the edge. The\nchiral, self-shearing motion is concentrated in these regions and is sensitive to\nlocal and global packing fraction, and to the relative concentration of spinners\nof different orientations.\nI find the research carefully carried out, and the results interesting. However,\nmy main comment is that the current format with an exceptionally short main\ntext makes the arguments rather vague and difficult to follow. I would support\npublication if the authors satisfactorily answer my questions below and also if\nthey rewrite their manuscript to include substantial parts of the supplementary\nin the main.\nResponse: We thank the referee for recognizing that our study is carefully\ncarried out and for finding the results interesting. We are also grateful for the\nconstructive feedback. We apologize for the delay in submitting this revision.\nOur instrument was committed to another set of measurements that had to\nbe completed before we could perform the experiments needed to address the\ncomments raised by Reviewer 2.\nBelow, we provide a detailed point-by-point response. All changes made in the\nmain text and Supplementary Information are highlighted in magenta.\nComment 1 : Edge flows, self shear direction and reentrant melting: Clearly,\nthere is something complex going on here, and I don’t think there is a full ex-\nplanation in the manuscript.\nResponse 1: As the referee points out, there is an intricate interplay between\nself-shear-generatedoddstresses,whichdependonthechiralactivity,andreen-\ntrant melting. In addition, grain boundary scars are responsible for the edge-\nbulk decoupling. We agree with the referee that the manuscript will benefit\nfrom a more detailed explanation. To this end, we have rewritten substantial\nportions of the results section in our revision, as well as improved the overall\nreadability of the manuscript. Please see all the highlighted sections. If the\nreferee feels additional information is required, we will include it.\nComment 2: The authors interpret their results within the framework of con-\ntinuum odd elasticity, which has in recent years been developed for systems with\ninternal chiral torques. That is of course fine, but it does not capture jam-\n1\nming/unjamming or solid/liquid transition in the system.\nResponse 2: Therefereeiscorrect. Oddelasticityisarelativelyrecentframe-\nwork (Ref. [54]) and is yet to be applied to the specific system we present.\nImportantly, we were able to observe chiral-activity–driven reentrant melting\n(Fig. 1B) only because our system operated near an equilibrium liquid–solid\nboundary (ϕ = 0.72, Ref. [51]). Thus, small changes in the control parame-\nter—the net chiral activity χ—suffice to drive the system across this boundary.\nOur explanation of the reentrant melting phenomenon relies on a few key con-\ncepts that were put forth in Refs. [38,54]: in materials with internal torque\ndensity, parity violation gives rise to odd elastic moduli, such that azimuthal\nflowscangenerateradialstresses. Thesignofthesestressesdetermineswhether\nthe material dilates or compresses. Our experiments demonstrate a causal link\nbetween vorticity and density changes—a unique hallmark of odd materials.\nComment 3: The author observe the scars, and link them to lowered chiral\nactive stresses in this region because of fewer contacts that transfer torques due\ntolowerdensity, aswellasdirectlythroughsimulations. Theauthorsdevelopan\nintricate microscopic interpretation of the torques near the grain boundaries. I\ncould not make sense of Figure 3A and the inferred torque and stress directions,\nand why some systems rotate counterclockwise. The link between torque direc-\ntionsandtheoddmodulicoupling(equationFig. S16)isalsoprettyhand-waivy.\nResponse 3: We apologize for the earlier lack of clarity. To illustrate the\ndependenceofstressdirectionsonthehandednessofazimuthalflows,wecanfirst\nconsideradensepassivegranularassemblyunderboundaryshear: suchasystem\ndilates and produces outward normal stresses, regardless of shear direction. In\ncontrast, for spinning particles, azimuthal flows are self-generated, and inter-\nspinner collisions break parity. Thus, the radial stress direction depends on\nwhether the flow handedness aligns or opposes the intrinsic particle spin.\nIn\nFig. 3A,wenowpresentthesecasesmoreclearly,withrevisedexplanationsand\ncaptions (see lines 209–228 and lines 231-254).\nSupplementaryFig. S15schematicallyillustrateshowoddelasticmodulicouple\nstrain and strain rates to stresses (adapted from Ref. [6] of the Supplement,\nor [44] of the main paper). For example, in a system of clockwise spinners,\na finite K1 means clockwise edge flows result in compressive stresses, while\n⊥\ncounterclockwise ones result in dilational stresses. It is these odd coefficients\nthat are also at play in our experiments that result in the observed reentrant\nmelting behavior.\nComment 4: In the manuscript, there is an extremely brief reference to ’sim-\nulations’, with data shown only in Figure 2C and 2E. In the supplementary, it\nthenemergesthatthisisadetailedmoleculardynamicssimulationwithCoulomb\nfriction, essentially identical to granular materials models except for the inclu-\nsion of the spinning torque. This then also allows for a better computation of\n2\nthe chiral torques compared to the ’frictional collisions’ of [37], with a dip at\nthe scar (Fig 2E). To me a proper introduction of the simulation belongs in the\nmanuscript directly.\nResponse 4: We appreciate this suggestion. In our revised manuscript, we\nintroduce the simulations immediately after introducing the experimental sys-\ntem (line nos: 87-93). The complete simulation details are now included in the\nMaterials and Methods section, which is now part of the main manuscript. If\nthe referee would like us to include further details in the main text, we will do\nso.\nComment 5: It should also allow for calculating the stresses and their profiles\ndirectly? In the manuscript, the authors relate compression / dilation to the\npeak height of packing fraction peaks, which seems rather indirect and which\nI’ve never seen before. Is there not an even approximate pressure / density\nconstitutive law that one could infer here?\nResponse 5: We thank the referee for this comment. In our experiments, we\nhave access only to particle positions and not the forces. Hence, we use the\npeaks of the radial density profile ϕ (r) as an indirect measure of the pressure.\nA\nIn regions where the pressure is large, the rattle room available for the particle\nis small, and hence the peaks grow in height and shrink in width. We would\nalso like to note here that for active matter, in general, pressure is not a state\nfunction. Even for the case of achiral active matter, the pressure, for instance,\ndependsonthewallcurvatureandroughness. Thisarisesfromthefactthatac-\ntive particles are persistent, and hence the wall features that promote/suppress\nparticlealignmentmodifythepressure,anditisnotjustdependentonthebulk\nparticle density (see Solon et al., Nature Phys. 11, 673 (2015)). For chiral ac-\ntivematteragain,particlesthatrotatewithawell-definedhandednesshugwalls\ndue to intrinsic torques [Barois et al., Phys. Rev. Lett. 125, 238003 (2020)\nand Kant et al., https://arxiv.org/abs/2509.00729], and we expect pressure to\ndependonthewallproperties. Whenweconsiderthecaseofspinners,theubiq-\nuitous edge flows near a boundary will generate odd radial stresses, which will\ncontribute to the pressure at the wall. The direction of these flows, and hence\nthe pressure, depends on inter-spinner and spinner wall friction. This makes\ninferring a pressure-density constitutive law difficult.\nNonetheless, since in our simulations we have access to both the positions and\nforces, we have attempted to calculate whether the pressure and density are\ncorrelated. We calculate the local virial pressure, p = −1 (cid:80)N (cid:0) x Fx +\ni A j=1 ij ij\nj̸=i\ny\nFy(cid:1)\nto relate pressure to the density changes in our system. Here, A is the\nij ij\nVoronoi cell area obtained from the Voronoi tessellation of the spinners. The\nquantities x = x − x and y = y − y represent the Cartesian distance\nij i j ij i j\nbetween the ith and jth spinner along x and y directions. The terms Fx and\nij\nFy represent, respectively, the x and y components of forces acting on the ith\nij\nspinner due to the jth spinner. Here, N is the number of nearest neighbors.\nIn\nFig. 1A,weshowtheannularvirialpressure,P,andannularareafractionϕ at\nA\nχ=1 and ϕ=0.72. Consistent with intuition and our observations (Line nos:\n229-235), the pressure and density are clearly correlated. In addition, the drop\nin density in the annulus harboring the grain boundary scars is accompanied\nby a drop in the pressure as well. In Fig. 1B, we also show the dependence of\nthe virial pressure, P, on the spinners versus Voronoi cell area of the spinners,\nA, to demonstrate the relationship between local pressure and local area in our\nsystem. If the referee wants us to include this figure in the supplement, we will\ndo so.\nFig. 1: (A) Pressure (black squares) and density (red squares) versus r/R in\nsimulations for χ = 1 at ϕ = 0.72. (B) Pressure versus Voronoi cell area for\nχ=1 at ϕ=0.72.\nComment 6: Seen from the granular angle, what the authors have here is to\nme a shear band, very similar to the ones seen in (driven) granular materials\nin a couette shear cell geometry (see e.g. Schall P, Van Hecke M. Shear bands\nin matter with granularity. Annual Review of Fluid Mechanics. 2010;42(1):67-\n88.).\nResponse 6: We thank the referee for this interesting perspective. We ac-\nknowledge that certain findings of ours invite comparison with shear banding\nindrivengranularmaterials. Forinstance,inboundary-drivengranularmatter,\nshear bands nucleate in regions where a local yield threshold is crossed. In our\nexperiments, the slip between the edge and bulk flows indeed occurs in these\npoorly packed regions where the system is more susceptible to yield. Having\nsaid that, we note that in banded flows in driven passive granular materials,\nthe width of the band is set by non-local terms in the constitutive law. In our\nexperiments, in contrast, it is purely determined by the location of the scars,\nwhich is controlled by boundary geometry.\nWe refer to the phenomenon as self-shear rather than a shear band because it\narises spontaneously without boundary forcing. The boundary, here, is static.\nFurthermore, in boundary-driven passive granular systems, the material re-\nsponse, which includes dilation-induced stresses, does not depend on the shear\n4\ndirection-allstressesareparity-even (seeresponsetocomment3). Inmaterials\nwheretheconstitutivebuildingblocksspinwithawell-definedhandedness,since\nthemicroscopicinteractionsareparity-violating,thisisnotthecase. Thesedis-\ntinctions give rise to odd material moduli, and make our findings distinct from\nboundary-driven granular shear bands.\nComment 7: The core feature is an instability triggered by an inversion in\nthe (shear) stress/strain rate curve, which then makes the system separate into\njammed and flowing regions. I suspect that this concept could be usefully trans-\nlated to the active case, possibly with the net internal oddness as a tuning pa-\nrameter of the instability?\nResponse 7: Unlike in generic shear banding systems where one observes\njammed and flowing regions, in our experiments the edge region rotates as a\ncohesive plug and the bulk region in the immediate vicinity of the scar (0.8 <\nr/R<0.9) is also quite well ordered (Fig. 1B) and rotates as a plug, but with\nopposite handedness, for χ = 1 (Fig. 2A). The scar region acts as a slip plane\nbetween two solid-like regions. It is the outward radial stress due to counter-\nrotatingbulkthatcausesthebulktomeltforr/R<0.7. Infact,forthesmaller\nchiral activity values (χ=0.3 and 0.6), the solid-like bulk and edge regions are\nmore apparent.\nThe referee is indeed correct that net internal oddness, or chiral activity, tunes\ntheedgeandbulkflows. Butwhetherthiscanbemappedtotheinversionseenin\nstressversusstraincurvesinsystemsundergoingshearbandingisnotclear,and\nis definitely worth investigating in future studies. The standard shear banding\ninstability often incorporates a flow-structure feedback mechanism. One key\ndifference that does arise in our system is that on systematically increasing χ,\nwe observe a reentrant change in the structure. To the best of our knowledge,\nwearenotawareofsuchafeaturearisinginsystemsundergoingshearbanding.\nComment 8: Thereisawell-developedliteratureonmodelsofshearbandsthat\ntake into account density, friction, and constitutive laws which include nonlocal\neffects on stress (see e.g. S. Mandal et al, Phys. Rev. X 11, 021017 (2021)).\nIt would relate stresses, density and strain rates / vorticity, like the manuscript\ntries to do. Can the authors relate their observations on strain rate inhomo-\ngeneities ( vorticity), density fluctuations and stresses to this framework, even\ntentatively?\nResponse 8: We thank the referee for pointing us to the above paper and the\nnon-localgranularrheologyliterature. Themechanismthatweinvoketoexplain\nour observations hinges on odd elasticity theory [Ref. 54]. This framework pro-\nvides a local constitutive law between stress and strain (Supp. Fig. 15). When\nthis is combined with the force-balance condition ∇·σ+substrate drag=0 it\nbecomes a boundary-value problem, which produces non-local effects: vorticity\nin one region affects the density and stress elsewhere. This suffices to qualita-\n5\ntively explain all our results. We emphasize that our primary goal here was to\ndemonstrate that odd stresses alone can drive phase changes in materials, and\nour experiments and simulations show this can be achieved.\nIn non-local granular rheology models, the non-locality is built into the con-\nstitutive law itself by introducing a flow screening length. This presents many\nadvantages; for instance, it allows for tuning the width of shear bands or cap-\nturing far-field creep. Odd mechanics is itself a very recent development (Ref.\n[54]). Building non-locality into the constitutive laws for systems with parity-\nodd stresses is an exciting theoretical challenge, but it is beyond the scope of\nthe present work.\nComment 9: All in all, after reading this manuscript I feel I still don’t under-\nstand why this phenomenon happens.\nResponse 9: We thank the referee for this constructive remark. To improve\nclarity, we have substantially revised sections of the manuscript. We believe\nthese revisions have enhanced the overall readability. We hope the referee is\nsatisfied with the changes made.\nReviewer #2 (Remarks to the Author):\nComment : In this work, the authors investigate the melting behavior of con-\nfined active spinners under varying activity. Combining experiments and sim-\nulations, they report a reentrant melting phenomenon driven by the interplay\nbetweenoddresponse,edgeflowandconfinement-induceddefects,wherethecrys-\ntallineorderofthebulkishigheratintermediateactivity(comparedtovanishing\nmean activity) but is disrupted at larger activity at a fixed spinner density. The\nfindingisnovel,andtheworkiswell-organized. Giventhecurrentgrowinginter-\nest in chiral active matter and odd response, the manuscript should be seriously\nconsidered for publication in NC. I will be happy to review the manuscript once\ntheauthorswillhaveaddressedmypointsinthefollowing. However, asfornow,\nI am still not in the position to recommend publication of the manuscript.\nResponse: We thank the referee for finding our study novel and the work\nwell-organized. We also thank the referee for their constructive comments. We\napologizeforthedelayinsubmittingourrevision. Ourinstrumentwascommit-\nted to another set of measurements that had to be completed before we could\nperform the experiments needed for addressing this referee’s comments.\nBelow, we provide a point-by-point response. All the changes made to the text\nin the main manuscript and supplement are colored magenta.\nComment 1: In this study, the system is binary mixture of clockwise and\ncounter-clockwise spinners, and the chiral activity is changed by tuning the mo-\n6\nlar fraction of clockwise spinners. A more direct way to change activity is to\nuse monodisperse spinner and then to tune the intensity of vertical vibration.\nWhy do the authors not employ such monodisperse system? In addition, in\nexperiments, what is the initial configuration; and in the steady state, are the\nclockwise and counter-clockwise spinners uniformly mixed?\nResponse 1: We thank the referee for this question. While tuning the vertical\nvibration amplitude is a potential approach to control activity in homochiral\nsystems, it has limitations. Specifically, reaching χ=0 requires turning off the\nvibration, which halts dynamics entirely due to the absence of thermal motion,\npreventing comparison with passive systems. Moreover, it was not evident a\npriori whether a racemic mixture (χ = 0) would behave like vibrated passive\ndisks. Wealsoobservedthatourparticlesspinconsistentlyonlywithinalimited\nrange of vibration amplitudes and frequencies.\nOur experimental system is assembled by hand and we tried to ensure uniform\nmixing of clockwise and counterclockwise spinners prior to each experiment.\nTo test whether the mixture exhibited a tendency to phase-separate, we ana-\nlyzed the local distribution of spinner types for the χ = 0, ϕ = 0.72 system.\nSpecifically, we measured the number of clockwise-spinning nearest neighbors\nsurrounding a clockwise spinner, and likewise for counterclockwise spinners, at\nboth the beginning and end of the experiment (Supplementary Fig. 1). The re-\nsultingdistributionissharplypeakedatthreeandremainsunchangedovertime.\nGiven that the local coordination number in our experiments is six, this out-\ncome indicates that each particle is, on average, surrounded by equal numbers\nof clockwise and counterclockwise spinners.\nComment 2: Onpage5, theauthorsstatethat“themagnitudeoftheedgeflow\ndrops precipitously at r/R ≈ 0.9. This behavior is distinct from observations\nmade hitherto in spinner liquids and solids.” Its structural origin looks to be\nthe existence of the grain boundary scar near this location. Further, the authors\ndescribethat“Unlikeconventionalgrainboundaries, whichterminateatthesys-\ntem edge, these scars terminate within the system.” An interesting question is\nwhy the grain boundary scars emerge at this specific location (around the 4th\nlayer)? Did the authors perform equilibrium simulation of passive disks (with\nfrictionalorsmoothboundary)inacircularconfinementtoexaminetheposition\ndistribution of the GBS? Moreover, does the ”self-shear” and ”scarred defects”\ndepend on the curvature of the confinement (finite-size effect)?\nResponse 2: Wethanktherefereeforthiscomment. Thespatialconfiguration\nof GB scars in disk packings in 2D remains an open problem and is closely\nrelatedtoThomson’spackingproblemonasphere. Wealsodonotknowasyetif\nfrictionbetweendisksaltersthescarconfiguration. Havingsaidthat,aplausible\nexplanationforourobservationsisthefollowing. ByexpellingtheGBscarsnear\nto the boundary, frustration in the bulk is relieved and the system can adopt\ndefect-freecrystallineconfigurations,wheretherattleroomavailableperparticle\n7\nis large. This is akin to the crystallization of hard spheres, which is driven by\nentropy, especially that arising from vibrational contributions. However, if the\nscarsarepushedallthewaytotheperiphery,frustrationfromcurvaturecanstill\npenetrate into the bulk. The trade-off between these effects, in all likelihood,\nresults in the observed scar configuration.\nFollowingthereferee’ssuggestion,weperformedequilibriumsimulationsofpas-\nsivedisksundercircularconfinementatϕ=0.72fordifferentextentsofconfine-\nment (system sizes). Voronoi tessellations reveal azimuthally aligned scars near\nthe boundary for N = 2000 and N = 1000, but less so for N = 500 (Supple-\nmentary Fig. 11). We also see defects in the bulk for N =2000 across different\nruns. Importantly, like in the case of particle packings on a sphere (Ref. 9 and\n14ofthemainmanuscript),thescarlengthdecreaseswithincreasingcurvature.\nTo probe the impact of curvature of confinement on self-shear and scar config-\nuration, we have performed additional experiments for different system sizes:\nN = 2000, 1000, and 500 particles at ϕ = 0.72 and χ = 1. In all these experi-\nments we observed azimuthally-aligned GB scars near the boundary at r/R ≈\n0.9 (Supplementary Fig. 9). Also, we observed self-shearing for N = 2000\nparticles, but not for the other system sizes. For N = 1000 and N = 500, we\nnonetheless see a sudden drop in ω(r) around r/R = 0.9; there is still a de-\ncoupling between the edge and the bulk mediated by GB scars. But for these\nsmallersystemsizes,thebulkrotatesinthesamedirectionastheedgebutwith\na smaller angular velocity.\nComment 3: In Supplementary Figure 18, the angular velocity profile at\nϕ = 0.75 is non-monotonic, differing significantly from those at other pack-\ning fractions. Additionally, the profile at ϕ = 0.75 appears to display a weak\nself-shear instead of vanishing self-shear. The authors should provide further\ndiscussion on these observations.\nResponse 3: We note that our simulations are a minimal model and do not\ncapture all the subtleties of the experimental observations. The primary ob-\njective of the simulation was to determine if the experimentally observed scar\nconfiguration could be reproduced in our model and then probe its impact on\nthe torque transfer between layers. The dip in torque transfer in the annulus\nharboring GB scars supports our arguments (Fig. 2E and Supplementary Fig.\n17). At the same time, features such as the lift-off of ω(r) for all values of ϕ\nfor r/R < 0.6 seen in simulations arise from features that we do not fully un-\nderstand yet. Possible reasons could be that in our experiments, the particles\nare dome-shaped particles, where some overlap is possible, not disks like in the\nsimulations. Theparticlesarealsoconfinedbetweenatopglassplateandabot-\ntom aluminum plate with different friction coefficients. In addition, our system\nis quasi-2D and not perfectly 2D like in the simulations. The simulations also\nassume that the spin speeds of all particles are identical. This can never be the\ncase in experiments.\nnewexperimentswereportfordifferentsystemsizes(SupplementaryFig. 9). In\nfact, for N =2000 particles, we have reproduced the self-shearing phenomenon\nat χ=1. These new experiments are much longer runs (30 min duration) and\nwestilldonotobservealift-offinω(r)forr/R<0.6asseeninthesimulations.\nComment 4: The data for annular angular velocities (Fig. 2a) and spin ve-\nlocities (Fig. 2f) exhibit considerable fluctuations. To enhance the data quality,\nmore extensive statistical averaging should be conducted.\nResponse 4: Following the referee’s comment, we revisited these data, par-\nticularly looking for the origin behind these large fluctuations in the annular\nangular and spin velocities. Our new analysis, presented in Supplementary Fig.\nS20A and B, revealed that the fluctuations in the annular spin velocity have\na one-to-one correspondence with the variations in net chirality within each\nannulus. Hence, these fluctuations have an underlying structural origin - the\ndistribution of clockwise and counterclockwise spinners within each annulus -\nand cannot be averaged away. This also explains why Ω(r) for χ=1 (Fig. 2f)\nis relatively smooth in comparison.\nFor the annular angular velocity, we once again found a strong correspondence\nwith the annular density fluctuations (Supplementary Fig. S20C, for χ = 1\nand ϕ = 0.72). Here, however, this correspondence is more pronounced for\nr/R > 0.6. For r/R ≤ 0.6, where there are many defects present, ω(r) shows\nlarge fluctuations (circles in Fig. 2A).\nComment 5: Thestrengthsofreentrance∆plottedinFig.2candFig.2dappear\nto have different meanings. And, they are not equivalent with the definition in\nthe second paragraph of page 4. Please carefully check this.\nResponse 5: We apologize for this oversight. The erroneous depiction of ∆\nin Fig.1C has been removed. The definition for strength of reentrance, ∆ =\n⟨|ψ6|⟩χ=0.3, has been incorporated in the revision, which aligns with Fig. 1D.\n⟨|ψ6|⟩χ=1\nComment 6: A typo: Page 7, second paragraph: “spinners in layers further\ninterior fall in...” should read “Spinners in layers further interior fall in...”.\nResponse 6: We have now corrected this in our revision.\nWehopethattherefereefindsourresponsessatisfactoryandthatthemanuscript\nis suitable for publication in Nature Communications.\n9"}, {"round": 2, "reviewer_comments": {"reviewer_2": "I have reviewed the revised manuscript. The authors have basically addressed my questions and comments, while parts of which are not quite satisfactorily addressed. Given that this work is sufficiently interesting, I would like to recommend its publication in Nature communications.", "reviewer_3": "In this manuscript, the authors study a 2D mixture of granular spinners with two different chiralities. By varying the ratio of the two chiralities and their overall density, the authors observe the emergence of a large-scale single crystal at intermediate values of the net chiral activity, in contrast to liquid-like regions which appear at large and small values. Combining experimental observations with simulations, the authors demonstrate that confinement-induced defects are responsible for decoupling the bulk and edge rotations, providing detailed insights into the mechanisms behind the observed structural and dynamical behaviours.\nI believe the authors have addressed the previous reviewers' comments in detail, and the work presents a novel and timely contribution. One of the strength of the work is the combination of experiments and simulations that demonstrate the range of the phenomena in the manuscript. However, I suggest the authors further edit the manuscript for clarity.\nComment 1:\nThe data in Fig.~1D exhibits only modest variation, with significant error bars. The authors claim that ``Δ is non-monotonic with ϕ, with a maximum at ϕ = 0.72,'' but I am not convinced the figure supports this statement. I suggest a rephrasing to ``Δ appears to be non-monotonic with ϕ, with an apparent maximum near ϕ ≈ 0.72.'' I also recommend increasing the thickness and contrast of the error bars for clarity.\nComment 2:\nThe sentence ``We calculated vorticity solely for the longest-lived flows since only these can significantly impact the radial density profile'' is not clear to me and should be rephrased. Why do short-lived vortices not impact ϕ_A(r)?\nComment 3:\nThe Methods introduce translational and rotational diffusion coefficients (D^T, D^R), but their physical origin is not clear. Could the authors clarify the experimental source of these terms and how their values were chosen?\nComment 4:\nSupplementary Fig.~S1 shows the distribution of same-type nearest neighbours at χ=0, ϕ=0.72.\nThe authors should clarify whether these distributions were computed over the entire system or only the bulk region.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Response to reviewer’s comments\nReviewer #2 (Remarks to the Author):\nComment: I have reviewed the revised manuscript. The authors have basically\naddressed my questions and comments, while parts of which are not quite satis-\nfactorily addressed. Given that this work is sufficiently interesting, I would like\nto recommend its publication in Nature communications.\nResponse: We thank the referee for finding our manuscript suitable for publi-\ncation in Nature Communications.\nReviewer #3 (Remarks to the Author):\nComment : In this manuscript, the authors study a 2D mixture of granular\nspinners with two different chiralities. By varying the ratio of the two chiralities\nand their overall density, the authors observe the emergence of a large-scale sin-\ngle crystal at intermediate values of the net chiral activity, in contrast to liquid-\nlikeregionswhichappearatlargeandsmallvalues. Combiningexperimentalob-\nservations with simulations, the authors demonstrate that confinement-induced\ndefects are responsible for decoupling the bulk and edge rotations, providing de-\ntailedinsightsintothemechanismsbehindtheobservedstructuralanddynamical\nbehaviours.\nI believe the authors have addressed the previous reviewers’ comments in detail,\nand the work presents a novel and timely contribution. One of the strength of\nthe work is the combination of experiments and simulations that demonstrate\nthe range of the phenomena in the manuscript. However, I suggest the authors\nfurther edit the manuscript for clarity.\nResponse: We thank the referee for finding that we have addressed the earlier\nreviewers’commentsinathoroughmannerandthattheworkoffersanoveland\ntimely contribution.\nBelow, we provide a detailed point-by-point response.\nComment 1 : The data in Fig. 1D exhibits only modest variation, with sig-\nnificant error bars. The authors claim that “∆ is non-monotonic with ϕ, with\na maximum at ϕ=0.72,” but I am not convinced the figure supports this state-\nment. Isuggestarephrasingto“∆appearstobenon-monotonicwithϕ, withan\napparent maximum near ϕ ≈ 0.72.” I also recommend increasing the thickness\nand contrast of the error bars for clarity.\nResponse1: Wethanktherefereeforthiscomment. Inourrevisedmanuscript,\n1\nwe have rephrased the statement to:“∆ appears to be non-monotonic with ϕ,\nwith an apparent maximum near ϕ≈0.72.” Please see lines 110–111 of the\nrevised main manuscript.\nAdditionally, we have increased the thickness and contrast of the error bars in\nFig. 1D to enhance visual clarity.\nComment 2 : Thesentence“Wecalculatedvorticitysolelyforthelongest-lived\nflows since only these can significantly impact the radial density profile” is not\nclear to me and should be rephrased. Why do short-lived vortices not impact\nϕ (r)?.\nA\nResponse 2: We thank the referee for raising this point. Using the coarse-\ngrained vorticity, we aimed to depict the flow field that is primarily responsible\nforaffectingthesteady-state radialdensityprofilesshowninFig.1E.Short-lived\nvortices may correspond to localized flows that exist only in limited regions of\nthe system. These short-lived vortices primarily reflect short-term fluctuations\nintheemergentbulkandedgeflows;therefore,theydonotaccuratelyrepresent\nthe global emergent flows and, consequently, the steady-state density profile\n(see Ref. [39] of Main Manuscript). However, long-lived vortices correspond to\nlarge, system-spanningvorticesthataccuratelycapture the flowfieldcomputed\nover the experimental duration. We have now rephrased the sentence to: “We\ncalculated the vorticity solely for the longest-lived flows, since short-lived vor-\ntices primarily reflect transient fluctuations and do not appreciably affect the\nsteady-state radial density profile ϕ (r)”.\nA\nComment 3 : The Methods introduce translational and rotational diffusion\ncoefficients (DT, DR), but their physical origin is not clear. Could the authors\nclarifytheexperimentalsourceofthesetermsandhowtheirvalueswerechosen?\nResponse 3: We thank the referee for this comment. Our methods section\nintroduces DT and DR as the translational and rotational diffusion constant,\nrespectively, associatedwithtranslationalandrotationalnoise. Inexperiments,\ntranslational and rotational noise can arise from multiple sources in a quasi-2D\nvertically vibrated systems. Any asymmetric or off-axis particle–plate contacts\nduring vertical vibration, where a spinner collides with the substrate at finite\nangles rather than perfectly normal incidence, can generate random forces and\ntorques on the spinner. In addition, sub-micron level inhomogeneities in the\nsubstrate and spinner surfaces, which locally modify frictional contacts, may\nfurther contribute to translational and rotational noise in the system.\nIn our simulations, the diffusion coefficients DT and DR were selected within a\nparameterrangethatreproducesexperimentallyobservedbehavior,suchasself-\nshearingandstablescarconfigurations,withtheresultsremainingqualitatively\nunchanged over this range.\nComment 4 : SupplementaryFig.S1showsthedistributionofsame-typenear-\n2\nest neighbours at χ = 0, ϕ = 0.72. The authors should clarify whether these\ndistributions were computed over the entire system or only the bulk region.\nResponse 4: We thank the referee for this comment. The distribution shown\nin Supplementary Fig. S1 was computed over the entire system. This has now\nbeen explicitly stated in the revised Supplementary Information.\n3"}]

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