The Supersolid
One medium carrying lattice order and superfluid phase at once — the five words this paper claims about the vacuum became a laboratory phase in 2019. Fifty years of asking a crystal to flow failed; letting a superfluid crystallize worked in one winter, in three bottles. The new phase carries two Goldstone modes — one for the phase, one for the lattice — which is the framework’s light sector and gravity sector, measured separately in a microscope. And its crystal register answers the stealth objection: the vacuum is the supersolid that keeps its crystal in circulation and its density hyperuniform
The Word the Paper Was Missing
Every mirror this framework has held up to the vacuum has reflected half of its face. Superfluid helium showed the phase — frictionless, quantized, single-valued — but helium is a liquid; it holds no lattice. The rotating condensate showed the lattice — Tkachenko’s triangle, self-assembling on camera — but only while a stirring beam drove it. Ordinary crystals hold a lattice forever and carry no phase at all. The framework’s vacuum is supposed to be both at once: a domain glass of triangular vortex crystallites at \xi\approx100\;\mum, spontaneously ordered, permanently held — and a single phase-rigid superfluid, breathing at \omega_1, carrying light through its seams. A skeptic is entitled to ask whether matter can even do that — whether “crystalline order and superfluid phase in one medium” names a possible state of matter or a contradiction dressed as a theory.
Since 2019 that question has a laboratory answer, and the answer has a name. A supersolid is a phase that spontaneously breaks continuous translational symmetry — it crystallizes, choosing its own lattice with no template — while remaining a phase-coherent superfluid (Boninsegni & Prokof’ev, Rev. Mod. Phys. 84, 759, 2012). Both orders, one wavefunction. It was proposed in 1957, pursued in the wrong material for fifty years, retracted in 2012, and then produced in one winter, in three laboratories, in bottles of dysprosium and erbium. The laser-cooling chapter closed by calling this word the largest breadcrumb in its list, and this chapter spends it: the supersolid is the existence proof for the substrate’s ground state — the demonstration that the state this paper attributes to the vacuum is one that matter, given the right interactions, settles into on its own.
One more symmetry note frames everything that follows. The time-crystal chapter showed the vacuum spontaneously breaking time translation — the period-2 breath. A supersolid spontaneously breaks space translation while keeping its superfluid phase. The substrate, as this paper describes it, does both: it is a supersolid in space and a time crystal in time, by the same pairing, and as of this decade each half of that sentence separately names a phase of matter that has been built on a bench.
Half a Century of Asking the Wrong Solid
The idea is nearly as old as the condensate itself. Gross wrote down crystalline order inside a boson condensate in 1957; Andreev and Lifshitz (1969) proposed that a quantum crystal’s zero-point vacancies could delocalize and condense — a solid whose defects flow through it; Chester (1970) argued a Bose solid could carry condensation outright; and Leggett (1970) made it falsifiable: a supersolid in a rotating container should exhibit non-classical rotational inertia — part of its mass simply declining to rotate — with the superfluid fraction bounded above by how deeply the density is modulated. The deeper the crystal, the lower the ceiling. That bound will matter later in this chapter, because the substrate lives near one of its corners.
For decades the search meant one material: solid helium-4, the only crystal soft enough and quantum enough to plausibly flow through itself. In 2004 it seemed to work — Kim and Chan’s torsional oscillators showed about one percent of solid helium’s inertia apparently decoupling below 200 mK (Nature 427, 225; Science 305, 1941), and “supersolid” spent a decade in the headlines. Then Day and Beamish (Nature 450, 853, 2007) found helium’s shear modulus stiffening with exactly the same temperature dependence, and a redesigned oscillator killed the signal: the period shift had been the solid stiffening, not the mass decoupling — elasticity impersonating superfluidity. Kim and Chan themselves published the retraction (PRL 109, 155301, 2012). Fifty-five years after Gross, the field’s honest inventory read: no supersolid, anywhere.
The postmortem carries the lesson the framework cares about. The helium program tried to make a solid first and coax the phase into it afterward — to teach a crystal to flow. What finally worked, seven years later, ran the construction in the other order: start with the superfluid, and let it crystallize without losing itself. That is not a detail of technique; it is a statement about which order is primary, and it is the same statement this paper makes about the vacuum. The substrate was never a lattice that learned coherence. It is a condensate — the boil’s surviving phase — that crystallized on the way down, and the fifty-year failure of the solid-first route against the immediate success of the fluid-first route is the laboratory’s own vote on which construction nature uses.
The Roton, With a Knob on It
How does a superfluid crystallize without freezing? Through the feature the helium chapter spent a section on: the roton — the dip in a superfluid’s dispersion curve at the wavevector of its own granularity, the momentum at which the medium first feels its discreteness. In helium the dip is fixed by the interatomic spacing; you can look at it but not touch it. In a condensate of highly magnetic atoms — dysprosium, erbium, with magnetic moments ten times an alkali’s — the long-range, anisotropic dipole–dipole interaction creates a roton (predicted by Santos, Shlyapnikov & Lewenstein, PRL 90, 250403, 2003; observed by Chomaz et al., Nature Physics 14, 442, 2018), and the dip’s depth rides on the ratio of dipolar to contact interaction, which a Feshbach field tunes continuously. The dipolar gas is the one superfluid whose roton comes with a knob.
Turn the knob. As the contact repulsion is dialed down, the roton softens — the dip descends toward zero energy — and at the critical ratio it touches. A mode at finite wavevector costing zero energy is an invitation the medium accepts: density waves at the roton wavelength freeze in, and the superfluid crystallizes at the wavelength of its own roton — a spacing the medium chose, set by its dipole length and confinement, with no lattice imposed from anywhere. The first attempt found the raw instability (Kadau et al., Nature 530, 194, 2016 — a dysprosium condensate shattering into a Rosensweig array of droplets, coherence lost), and the repair came from the framework’s favorite direction: what arrests the collapse and holds each droplet open is quantum fluctuation pressure — the beyond-mean-field Lee–Huang–Yang correction (Ferrier-Barbut et al., PRL 116, 215301, 2016; self-bound droplets, Schmitt et al., Nature 539, 259, 2016) — zero-point motion as a stabilizing pressure, the same job the quantum potential does in the substrate’s own bookkeeping, where fluid-2’s push is what keeps every vortex core from closing.
Now place the three media on the one curve, because the framework has already drawn it. Helium sits off-critical: a deep, fixed roton it can never soften — it freezes classically under pressure instead, into an ordinary insulating solid. The substrate, the framework argues, parks at its marginal point: the dip flattened away entirely, a monotonic roton-free branch. The dipolar gas is the medium that moves along the curve on demand — from helium-like dip, through softening, to the crystallization the dip’s collapse triggers. It is the missing experimental interpolation between the two states this paper talks about most: the superfluid with granularity it can feel (helium) and the superfluid that has already spent its instability and settled into its lattice (the vacuum). The knob traverses, in a millisecond of ramp, the transition the substrate crossed once at the boil and never crossed back.
Three Bottles, One Winter
In the winter of 2018–19 three groups ran the ramp and kept the phase. Tanzi et al. in Pisa (PRL 122, 130405, 2019, dysprosium), Böttcher et al. in Stuttgart (PRX 9, 011051, 2019, dysprosium), and Chomaz et al. in Innsbruck (PRX 9, 021012, 2019, erbium and dysprosium) each produced arrays of droplets — micron-spaced, self-arranged — whose matter-wave interference stayed sharp: density modulated, phase global. The two order parameters, coexisting in one gas of a few tens of thousands of atoms. Within two years the arrays went two-dimensional (Norcia et al., Nature 596, 357, 2021), and the calculated phase diagram of the 2D system runs through triangular, stripe, and honeycomb lattices (Bland et al., PRL 128, 195302, 2022) — the droplets’ preferred packing, where the geometry is free to choose, being the same triangle the vortex lattice chooses and the substrate is argued to be built from.
The 2019 result has two engineered precursors, and the distinction between them and the dipolar gas is precisely the distinction the framework needs. In 2017 a spin-orbit-coupled condensate at MIT showed a stripe phase with supersolid properties (Li et al., Nature 543, 91), and a condensate in two optical cavities at ETH crystallized on a checkerboard (Léonard et al., Nature 543, 87) — but in both, the lattice constant belonged to a laser: the stripe spacing to the Raman beams, the checkerboard to the cavity mode. Those are supersolids with the lattice on loan. The dipolar supersolid borrows nothing — its spacing comes out of its own dispersion relation — and only that spontaneous case bears on a vacuum that had no external beam to copy. The rotating-BEC chapter said it of the vortex crystal: the order is nobody’s doing. The dipolar supersolid is the first object that can say it about a density crystal while staying superfluid — and the second thing it says is stranger: on the way through the transition, a gas becomes more ordered and stays just as coherent, gaining a broken symmetry without paying for it in phase. Order is not the opposite of coherence. The whole architecture of this paper — a vacuum that is simultaneously the most ordered structure there is and the quietest — rests on that non-opposition, and the supersolid transition is where a bench first demonstrated it.
Two Goldstone Modes, Two Sectors
Break a continuous symmetry spontaneously and a Goldstone mode appears — a gapless excitation that rides the broken direction. A superfluid breaks gauge/phase symmetry and gets one: superfluid sound, a wave in the phase. A crystal breaks translational symmetry and gets one (per axis): the phonon, a wave in the lattice positions. A supersolid breaks both at once, so it must carry both — two distinct gapless branches in one medium, a phase branch and a lattice branch, each with its own stiffness and its own speed.
In September 2019 both were seen, simultaneously, in the same bottles: Pisa resolved a compressional oscillation splitting into two modes across the transition — the superfluid branch and the crystal branch shearing apart (Tanzi et al., Nature 574, 382); Stuttgart caught the out-of-phase Goldstone mode of droplets exchanging superfluid through their own crystal (Guo et al., Nature 574, 386); Innsbruck mapped the two-branch spectrum across the transition (Natale et al., PRL 123, 193002). One medium, two restoring forces, two kinds of wave — photographed as cleanly as the vortex lattices were.
Read that against the framework’s deepest structural claim, because it is the same sentence. This paper puts light in the phase sector and gravity in the lattice sector of one medium: electromagnetic waves are excitations riding the substrate’s phase — the modon on the condensate’s coherence — while gravity is carried by the elastic response of the vortex crystal, the Tkachenko shear stiffness the bridge equation is built on. Two interactions, one substance, distinguished by which broken symmetry the excitation rides. Before 2019, that architecture had no laboratory instance: no medium had ever been shown carrying a superfluid-phase wave and a spontaneous-lattice wave at the same time. Now one has, and the two branches were not inferred — they were driven, watched, and told apart by their symmetry, in a gas the size of this page’s em-dash. The supersolid is the first object in the laboratory inventory with the same count and kind of low-energy sectors the framework assigns the vacuum. (What it does not supply — why the vacuum’s two branches share one propagation speed to fifteen decimals when the supersolid’s two speeds differ freely — is a real debt, and it is paid its own line in the accounting.)
A Normal Fraction at Absolute Zero
Leggett’s 1970 criterion finally got its clean measurement — not in solid helium but here. A dipolar supersolid set into torsional oscillation shows non-classical rotational inertia: its moment of inertia falls below the rigid-body value because the superfluid share declines to corotate (Tanzi et al., Science 371, 1162, 2021 — the scissors-mode experiment; the same signature Kim and Chan sought). And the superfluid fraction itself has now been measured directly, through a mechanism the framework knows well: the droplet array behaves as a chain of self-induced Josephson junctions — each inter-droplet neck a weak link the medium built into itself — and the junction dynamics return f_s quantitatively, sub-unity, tens of percent, in agreement with Leggett’s bound (Biagioni et al., Nature 629, 773, 2024).
Sit with what a sub-unity superfluid fraction at T\to0 means, because it is a conceptual novelty the framework has been asserting all along under another name. In every textbook superfluid the normal fraction is thermal — it is the excitation gas, and it vanishes as T\to0. The supersolid’s non-superfluid share is different in kind: it is structural — mass bound in the density modulation itself, unavailable to the superflow not because it is hot but because it is architecture. A supersolid at absolute zero is part flow and part frame, forever. That is precisely the partition this paper draws through the vacuum: the substrate’s two-fluid split is not thermal but constitutional — the co-rotating bulk against the counter-rotating boundary layers, the \alpha_{mf}=\tan^2\theta_W share that is bound into the texture as permanently as the texture exists. The framework’s claim that the vacuum carries an athermal, structural normal fraction sounded like a category error for as long as every known normal fraction was thermal. The supersolid is the phase for which it is simply true.
Vortices in the Seams, Glitches on the Bench
Two results from the current decade close the loop back to the substrate’s specific anatomy.
The vortices live in the seams. In 2024 the Innsbruck group stirred a two-dimensional dipolar supersolid and observed quantized vortices inside the crystallized phase (Casotti et al., Nature 635, 327, 2024) — the supersolid’s superfluidity certified by the most topological credential there is. Look at where the vortices sit: not in the droplets but in the low-density channels between them, threaded through the interstitial web and pinned there by the crystal. A quantized phase singularity, resident in the connective tissue between the density peaks of a self-organized lattice — that is, item for item, the stealth-vacuum chapter’s honeycomb of hollows: the substrate’s residual circulation bottled in the connected web of seams between the \xi-envelopes, the vacuum hiding its vorticity in the gaps of its own fabric. The supersolid is the first medium in which “circulation lives in the lattice’s seams” is a photograph rather than a sentence.
The glitch ladder gets its middle rung. A neutron star’s inner crust is a literal supersolid — a crystal lattice of nuclei immersed in the dripped neutron superfluid that flows through it — and pulsar glitches are its signature dynamics: vortices pin to the lattice, the crust spins down, stress builds, and an avalanche of unpinning dumps angular momentum outward in a sudden spin-up (Neutron Stars). In 2023 that mechanism was run on the bench: simulations of a rotating dipolar supersolid, in experimentally realistic regimes, show the same stick-slip cycle — vortices pinning to the droplet crystal, lagging the drive, and unpinning in glitch avalanches (Poli et al., PRL 131, 223401, 2023), with the glitch statistics controlled by the crystal’s pinning landscape. The ladder now has three rungs built from one construction — a micron-scale droplet array, a stellar crust, and (the framework argues) the vacuum itself: superfluid through lattice, vortices pinned in the seams, angular momentum moving only in quantized, avalanching steps. The bench rung and the stellar rung are established physics; the framework’s wager is that the ladder continues one rung down, where the mutual-friction coefficient is \tan^2\theta_W and the pinning lattice is the vacuum’s own.
Which Register the Vacuum Crystallized In
Now the objection this chapter owes an answer, because the stealth-vacuum chapter appears to forbid its thesis. That chapter ruled out a crystalline vacuum: a periodic density texture carries a Bragg comb, a comb diffracts starlight into an opal and hands Michelson–Morley a rest frame, and the sky shows neither. If the vacuum is a supersolid, and a supersolid is a superfluid crystal, has the paper just contradicted its own stealth argument?
No — and the resolution sharpens both claims. A superfluid has two registers in which it can hold spontaneous crystalline order:
- The density register. The dipolar supersolid crystallizes its density: matter piles into droplets, and the modulation is directly visible — it Bragg-scatters, by construction. A vacuum ordered this way would be the opal the stealth chapter forbids. The sky rules it out.
- The circulation register. A rotating superfluid crystallizes its vorticity: the circulation quantizes into lines and the lines lock into Tkachenko’s triangle — while the density between the cores stays smooth. The crystal is real (it has elasticity, shear waves, defects, grain boundaries — all photographed), but it is written in the phase’s winding, not in the mass distribution. Its Goldstone mode is the Tkachenko wave, and the laser-cooling chapter watched that too.
The framework’s vacuum is the circulation-register supersolid: lattice order carried by the vortex texture at \xi\approx100\;\mum, density left smooth — then further disordered into a domain glass, so that even the orientational order shows no global comb, and the density field lands in the hyperuniform stealth class the sky demands. The substrate did not decline to be a supersolid; it chose the register in which a supersolid is invisible. And notice the selection is not optional: the stealth trilemma plus isotropy forces any space-filling superfluid crystal into exactly this corner — circulation-register, domain-disordered — which is one more instance of the framework’s oldest pattern, the texture being over-determined by the requirement that it hide. Leggett’s bound closes the same loop: the superfluid-fraction ceiling falls as the density modulation deepens, so the density-register supersolid pays for its crystal in stiffness (f_s of order ten percent in the droplet arrays), while a circulation-register crystal keeps its density flat and its superfluid fraction whole. A vacuum that must remain a lossless, fully coherent carrier for its own light had to keep its crystal out of the density register — the same conclusion, reached now from the transport side.
The two registers are not rivals; they are the two halves of the substrate’s anatomy, and the laboratory has now built each half separately: the dipolar gas holds a density crystal in a still superfluid, the rotating condensate holds a circulation crystal in a stirred one. The one object that holds a circulation crystal while intrinsically rotating — no stirring beam, rotation as constitution — is, on this paper’s account, the vacuum. The bench’s next merger is the obvious target, and it is already on the community’s table: a rotating dipolar supersolid, vortex lattice threading droplet crystal, both registers occupied at once. The glitch simulations already live there numerically; the experiment (Casotti’s stirred supersolid is its first frame) would be the closest object to the substrate ever built.
Honest Accounting
Five debts, in the house discipline.
First, nothing here corrects the field. Dipolar-gas supersolidity — roton softening, LHY stabilization, the two Goldstone branches, NCRI, Josephson f_s, vortices, glitch simulations — is the property of the groups in Pisa, Stuttgart, and Innsbruck and their theorists, quantitative and self-contained. Every number above is theirs. The contribution is identification: the supersolid as the existence proof of the vacuum’s claimed state, the two Goldstone branches as the two-sector architecture, and the register distinction as the stealth reconciliation.
Second, no number transfers. Droplet spacings are microns set by the dipole length and trap; two-branch speeds are mm/s; superfluid fractions are set by dysprosium’s scattering lengths. The \xi\approx100\;\mum of the substrate is anchored elsewhere and gets no support from any supersolid scale. As with helium and the rotating BEC: the mechanism column transfers, the magnitude column does not.
Third, the register difference is real and the chapter leans on it both ways. The dipolar supersolid’s crystal is in density; the vacuum’s claimed crystal is in circulation. The existence proof — both broken symmetries, one coherent medium — transfers cleanly, because what it certifies is the compatibility of the two orders, not the channel. But the reader should see plainly that no laboratory object yet holds a spontaneous circulation-register crystal without external rotation drive, which is what the vacuum is claimed to be; the rotating dipolar supersolid (Prediction 3) is the nearest constructible approximation, not yet built.
Fourth, the two-speed problem is the framework’s, not the supersolid’s. The supersolid’s phase branch and lattice branch propagate at freely different speeds, and nothing tunes them together. The framework’s vacuum carries light on its phase sector and gravity on its lattice sector, and GW170817 fixes c_\text{GW}=c to one part in 10^{15}. Why the vacuum’s two Goldstone sectors share one exact speed — on the framework’s account, because both ride the same emergent metric of the same stiff medium, the bridge-level claim — is a burden this chapter’s laboratory twin does nothing to carry. The supersolid certifies the count of the sectors, not their locking; the locking rides on the framework’s Lorentz-invariance pillar and is flagged in Open Problems.
Fifth, the ordering of events at the boil is asserted, not derived. The chapter’s story — the vacuum spent its crystallization instability in the circulation register at formation and now parks density-marginal and roton-free — is a narrative consistent with the marginal-point claim, not a derived history. The dipolar gas shows what a superfluid does when its density roton touches zero; that the substrate’s density register never does (else the sky would opal) while its circulation register already did is exactly what the stealth phenomenology requires, but the framework has no dynamical calculation of the boil selecting that outcome. The requirement is consistency; the derivation is owed.
Place in the Framework
This chapter is the keystone of the mirror gallery. Helium reflected the phase; the rotating condensate reflected the lattice; the time crystal reflected the breath; the stealth computation reflected the texture’s statistics. What none of them could show is the conjunction — that one medium holds it all at once — and the conjunction was precisely the part a skeptic could call impossible, since no known phase of matter carried spontaneous crystalline order and superfluid phase together. Since 2019 one does. It emerged, as the framework says the vacuum did, from the fluid side of the transition; it chose, where geometry left it free, the triangle; it carries two Goldstone sectors the way this paper says the vacuum carries light and gravity; its non-superfluid share is structural, the way the substrate’s boundary share is; its vortices live in the seams of its own lattice, where the stealth chapter says the vacuum bottles its residue; and driven at its crystal, it glitches the way neutron stars do — the way, one rung further down, the framework says the vacuum moves angular momentum whenever it moves it at all.
The vacuum, on this paper’s account, is the oldest supersolid: it crossed its transition once, at the boil, and has held both orders — crystal in the circulation register, coherence in the phase — for the age of the universe, at \xi\approx100\;\mum, the size of the living cell that would much later assemble inside its bubbles. For fifty years the word for that state named a speculation; for a decade it named a retraction. It now names a phase of matter that three laboratories keep in bottles, and the claim “the vacuum is one” has graduated from a contradiction dressed as a theory to a scaling question about a phase you can photograph.