Dynamics of Supersolid state: normal fluid, superfluid, and supersolid velocities
Pith reviewed 2026-05-23 05:43 UTC · model grok-4.3
The pith
Supersolids require a third density component whose velocity obeys elasticity, chemical potential gradients, and drag from the normal fluid.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Landau's excitation-based argument for superfluids -- that at temperature T=0 the normal fluid density ρ_n is zero -- should also apply to supersolids. Further, for a total mass density ρ, Leggett argues that the superfluid fraction ρ_s/ρ<1. These arguments imply that there is a missing mass. We attribute this to a supersolid density ρ_L, with ρ_L≡ρ−ρ_s−ρ_n, and a momentum-bearing supersolid velocity v_Li. Using Onsager's irreversible thermodynamics we derive the macroscopic dynamical equations for this system. We find that v_Li is subject to the force of elasticity, to the negative gradient of the chemical potential per mass μ (as for the superfluid velocity v_si), and to drag against the n
What carries the argument
The supersolid density ρ_L ≡ ρ − ρ_s − ρ_n together with its velocity v_Li, whose evolution follows from Onsager irreversible thermodynamics and includes elastic forces, −∇μ, and drag against the normal fluid.
If this is right
- Both the superfluid and supersolid velocities are tied to the ground state.
- Normal modes exhibit a crossover frequency above which the normal fluid velocity behaves as an independent variable and below which it locks to the supersolid velocity.
- For an isotropic lattice both transverse and longitudinal responses follow from the derived equations.
- Ring geometries in atomic-gas supersolids provide a concrete setting in which the predicted dynamics can be tested.
Where Pith is reading between the lines
- The drag term between the supersolid velocity and normal fluid implies that momentum exchange could be measured through long-time relaxation rates in trapped gases.
- The model suggests that low-frequency elastic responses in supersolids could be isolated by driving at frequencies below the crossover.
- Extending the equations to rotating frames might predict additional constraints on persistent currents carried by the supersolid velocity.
Load-bearing premise
Landau's excitation-based argument that normal fluid density is zero at zero temperature applies directly to supersolids.
What would settle it
Observation, in a ring-shaped supersolid, of a frequency-dependent crossover where the normal fluid velocity decouples from the supersolid velocity above a threshold frequency, or direct detection of elastic restoring forces and drag acting on the supersolid velocity component.
read the original abstract
Landau's excitation-based argument for superfluids -- that at temperature $T=0$ the normal fluid density $\rho_{n}$ is zero -- should also apply to supersolids. Further, for a total mass density $\rho$, Leggett argues that the superfluid fraction $\rho_{s}/\rho<1$. These arguments imply that there is a missing mass. We attribute this to a supersolid density $\rho_{L}$, with $\rho_{L}\equiv \rho-\rho_{s}-\rho_{n}$, and a momentum-bearing supersolid velocity $v_{Li}$. Using Onsager's irreversible thermodynamics we derive the macroscopic dynamical equations for this system. We find that $v_{Li}$ is subject to the force of elasticity, to the negative gradient of the chemical potential per mass $\mu$ (as for the superfluid velocity $v_{si}$), and to drag against the normal fluid (leading to the interpretation of $L$ as lattice). Thus both the superfluid and supersolid components are associated with the ground state. The normal modes for such a system have a crossover in frequency, above which the normal fluid velocity $v_{ni}$ is an independent variable and below which it is locked to $v_{Li}$. For an isotropic lattice we study both the transverse response and longitudinal response. The ring geometry for atomic gas supersolid states may provide a geometry for testing these predictions.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript claims that Landau's excitation argument implies ρ_n=0 at T=0 also for supersolids; combined with Leggett's result that ρ_s/ρ<1, this requires a supersolid density ρ_L ≡ ρ−ρ_s−ρ_n with its own velocity v_Li. Using Onsager irreversible thermodynamics the authors derive the macroscopic equations of motion in which v_Li experiences an elastic force, −∇μ, and drag from the normal fluid. They analyze the resulting three-velocity hydrodynamics, identify a frequency crossover below which v_n locks to v_Li, and compute the transverse and longitudinal responses for an isotropic lattice, suggesting ring-geometry experiments in atomic-gas supersolids as a test.
Significance. If the partitioning is valid, the work supplies a systematic three-component hydrodynamic framework that associates both the superfluid and supersolid components with the ground state and yields a concrete, testable prediction for the frequency-dependent locking of v_n to v_Li. The application of Onsager thermodynamics to obtain the force terms on v_Li is a methodical strength that could be useful for interpreting dynamics in recent quantum-gas supersolid experiments.
major comments (2)
- [Abstract] Abstract, first sentence: the claim that Landau's excitation-based argument (no thermal quasiparticles at T=0 implies ρ_n=0) extends unchanged to supersolids is asserted without derivation or discussion of how the broken translational symmetry of the crystalline ground state modifies the quasiparticle spectrum or couples lattice phonons to superflow at T=0. This assumption is load-bearing because it is the sole justification for setting ρ_n=0 and thereby introducing the independent ρ_L and v_Li.
- [Abstract] Abstract, definition of ρ_L: the relation ρ_L ≡ ρ−ρ_s−ρ_n is introduced purely by subtraction, so the attribution of 'missing mass' to a distinct supersolid component is tautological unless an independent microscopic or experimental grounding is supplied beyond the two cited theorems. All subsequent dynamical equations and the normal-mode crossover rest on this partitioning being physically required rather than definitional.
minor comments (1)
- The manuscript would benefit from explicit display of the derived equations of motion for v_Li (elastic term, −∇μ, and drag) so that readers can verify the normal-mode analysis without reconstructing them from the Onsager procedure.
Simulated Author's Rebuttal
We thank the referee for their careful reading and for identifying the load-bearing assumptions in our work. We respond to each major comment below, indicating revisions that will be incorporated to strengthen the justification and presentation without altering the core claims.
read point-by-point responses
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Referee: [Abstract] Abstract, first sentence: the claim that Landau's excitation-based argument (no thermal quasiparticles at T=0 implies ρ_n=0) extends unchanged to supersolids is asserted without derivation or discussion of how the broken translational symmetry of the crystalline ground state modifies the quasiparticle spectrum or couples lattice phonons to superflow at T=0. This assumption is load-bearing because it is the sole justification for setting ρ_n=0 and thereby introducing the independent ρ_L and v_Li.
Authors: We agree that the extension of Landau's T=0 argument merits explicit discussion. In the revised manuscript we will add a short paragraph after the abstract statement, noting that the crystalline order introduces Goldstone phonons that are incorporated into the ground-state hydrodynamics rather than contributing to a thermal ρ_n; the absence of thermal quasiparticles at T=0 remains valid, and the coupling between lattice motion and superflow is encoded in the three-velocity equations we derive. This supplies the requested grounding while preserving the original logic. revision: yes
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Referee: [Abstract] Abstract, definition of ρ_L: the relation ρ_L ≡ ρ−ρ_s−ρ_n is introduced purely by subtraction, so the attribution of 'missing mass' to a distinct supersolid component is tautological unless an independent microscopic or experimental grounding is supplied beyond the two cited theorems. All subsequent dynamical equations and the normal-mode crossover rest on this partitioning being physically required rather than definitional.
Authors: The partitioning follows directly as the logical remainder once Landau's ρ_n(T=0)=0 and Leggett's ρ_s/ρ<1 are accepted; it is therefore required rather than arbitrary. We will revise the abstract and the opening of Sec. II to state this logical structure explicitly and to reference prior three-component hydrodynamic treatments of supersolids that independently motivate the same decomposition. No new microscopic derivation is added, but the presentation now makes clear that the subsequent Onsager analysis tests the physical consequences of this required partitioning. revision: yes
Circularity Check
ρ_L introduced by explicit subtraction definition, forcing three-component model by construction
specific steps
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self definitional
[Abstract]
"These arguments imply that there is a missing mass. We attribute this to a supersolid density ρ_L, with ρ_L ≡ ρ−ρ_s−ρ_n, and a momentum-bearing supersolid velocity v_Li."
The 'missing mass' is defined exactly as the arithmetic remainder ρ−ρ_s−ρ_n; attributing it to a new independent component ρ_L with its own velocity v_Li is therefore true by the definition chosen, rather than shown to require a separate dynamical variable.
full rationale
The paper's load-bearing step is the direct definition of the supersolid density as the residual after subtracting the superfluid and normal densities, which then requires its own velocity field whose dynamics are derived. This partitioning is not independently derived but constructed from the inputs (Landau + Leggett), making the subsequent three-velocity hydrodynamics and normal-mode analysis dependent on that definition. No self-citation chains or fitted parameters are involved; the Onsager application itself is not circular, but the need for a distinct ρ_L component is.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Landau's excitation argument that ρ_n vanishes at T=0 applies to supersolids
- domain assumption Leggett's result that ρ_s/ρ < 1 holds for supersolids
invented entities (2)
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supersolid density ρ_L
no independent evidence
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supersolid velocity v_Li
no independent evidence
discussion (0)
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