Dynamics of Supersolid state: normal fluid, superfluid, and supersolid velocities

This paper proposes a macroscopic dynamical framework for supersolids using Onsager's irreversible thermodynamics, which introduces a momentum-bearing supersolid (lattice) component to resolve the missing mass problem and predicts distinct normal mode behaviors where the normal fluid velocity transitions from being locked to the lattice at low frequencies to becoming independent at high frequencies.

The Gist

Imagine a material that is a paradox: it is rigid like a solid rock, yet it flows like a frictionless liquid. This is the Supersolid, a strange state of matter that physicists have been hunting for decades.

In this new paper, physicist Wayne Saslow acts like a traffic engineer for this weird material. He wants to understand how the different "cars" inside the supersolid move when you push or shake them.

Here is the breakdown of his findings, translated into everyday language.

1. The Three "Cars" in the Traffic Jam

In a normal liquid (like water), you have two types of flow:

• The Normal Fluid: The "traffic" that gets stuck, slows down, and creates friction (like cars in rush hour).
• The Superfluid: The "ghost cars" that can drive through walls and never stop, flowing without any friction.

In a Supersolid, Saslow argues there is a third type of traffic that was previously missing from the map. He calls this the Supersolid component (or the "Lattice").

• The Analogy: Imagine a dance floor.
  • The Normal Fluid is the crowd of people shuffling around, bumping into each other (friction).
  • The Superfluid is a group of dancers moving in perfect, silent unison, gliding over the floor without touching anyone (frictionless).
  • The Supersolid (Lattice) is the actual floorboards themselves. In a normal solid, the floor is just a static stage. But in a supersolid, the floorboards themselves are "alive" and can move, carrying their own weight and momentum.

2. The Missing Mass Mystery

For a long time, scientists looked at supersolids and noticed something weird: if you spin a container of supersolid helium, it doesn't spin as heavy as it should. It feels "lighter."

• The Old View: Scientists thought, "Okay, some of the mass is just sitting still (the normal part), and some is flowing (the superfluid part)."
• Saslow's New View: He says, "Wait, there's a third part!" He calculates that the "floorboards" (the lattice) are moving too. If you don't count the mass of the moving floorboards, the math doesn't add up. He calls this the Supersolid Density.

3. How They Move: The "Drag" and the "Spring"

Saslow wrote down the rules for how these three components interact. He found that the "floorboards" (the supersolid lattice) are pulled by three distinct forces:

1. The Elastic Spring: Just like a real solid, if you push the floorboards, they want to snap back. This is the force of elasticity.
2. The Chemical Push: Just like the "ghost cars" (superfluid), the floorboards are pushed by a pressure difference called the chemical potential. This means the floorboards are deeply connected to the quantum "ground state" of the material, not just sitting there.
3. The Friction Drag: This is the most interesting part. The moving floorboards have to push through the "crowd" (the normal fluid).
   • The Metaphor: Imagine the floorboards are a giant net, and the normal fluid is water trying to flow through it.
   • Slow Motion (Low Frequency): If you move slowly, the water drags the net along with it. The floorboards and the crowd move together as one big blob.
   • Fast Motion (High Frequency): If you shake the net super fast, the water can't keep up. The floorboards wiggle back and forth on their own, independent of the crowd.

4. The "Crossover" Moment

The paper predicts a specific moment where the behavior changes, like a gear shifting in a car.

• Below the shift: The "floor" and the "crowd" are locked together. They move as one unit.
• Above the shift: They break apart. The floor vibrates independently, and the crowd flows separately.

5. Why Does This Matter?

This isn't just about math; it's about how we test these materials.

• The Ring Test: The author suggests using a ring-shaped container (like a donut) filled with this atomic gas. If you spin the ring, you should be able to see these different "modes" of vibration.
• The Takeaway: By understanding that the "floor" itself has momentum and moves, we can finally explain why supersolids feel lighter than they should and how they vibrate.

Summary in One Sentence

Saslow discovered that in a supersolid, the solid "floor" isn't just a static stage; it's a moving actor that drags the liquid crowd along when things move slowly, but breaks free to dance on its own when things move fast, solving a decades-old mystery about missing mass.

Technical Summary

1. Problem Statement

The paper addresses a theoretical gap in the hydrodynamic description of supersolids.

• The Missing Mass Problem: Landau's theory of superfluids posits that at $T=0$, the normal fluid density ($\rho_n$) vanishes because it arises from thermal excitations. Leggett's theory of supersolids suggests that even at $T=0$, the superfluid fraction ($\rho_s/\rho$) is less than 1 due to non-classical rotational inertia (NCRI).
• The Discrepancy: If $\rho_n = 0$ and $\rho_s < \rho$, there is a "missing" inertial mass ($\rho - \rho_s$) that is not accounted for in the standard two-component (superfluid + normal fluid) or three-component (Andreev-Lifshitz) models.
• The Question: How does this missing mass behave dynamically? Specifically, what is its velocity, and how does it interact with the superfluid and normal fluid components?

2. Methodology

The author employs Onsager's irreversible thermodynamics to derive macroscopic dynamical equations for a supersolid.

• Thermodynamic Framework: The system is defined by mass density ($\rho$), entropy density ($S$), and three momentum-bearing velocities:
  1. Superfluid velocity ($v_{si}$)
  2. Normal fluid velocity ($v_{ni}$)
  3. Supersolid velocity ($v_{Li}$): A new variable introduced to account for the missing mass density $\rho_L \equiv \rho - \rho_s - \rho_n$.
• Energy and Entropy: The author constructs the differential of energy density ($dE$) and the Gibbs-Duhem relation, explicitly including terms for the new current $j_{Li} = \rho_L(v_{Li} - v_{ni})$ and velocity $v_{Li}$.
• Force Analysis: The equations of motion are derived by identifying thermodynamic forces (gradients of chemical potential, temperature, and stress) and fluxes. The author assumes the superfluid velocity $v_{si}$ is Galilean (unlike the lattice-frame definition in some previous theories) to ensure non-dissipative uniform superflow.
• Linearization: The study focuses on small deviations from equilibrium to analyze normal modes (transverse and longitudinal) in an isotropic lattice.

3. Key Contributions

• Introduction of the Supersolid Component ($\rho_L$): The paper formally identifies the "missing mass" as a distinct supersolid component with its own velocity $v_{Li}$. This component is associated with the lattice displacement ($u_i$) but is treated as a momentum-bearing fluid in the hydrodynamic limit.
• Three-Velocity Hydrodynamics: The theory establishes that a supersolid requires three independent velocity fields ($v_{si}, v_{ni}, v_{Li}$), whereas a standard superfluid only requires two.
• Equation of Motion for $v_{Li}$: The author derives a specific equation of motion for the supersolid velocity, showing it is driven by three distinct forces:
  1. Elastic Forces: Arising from the stress tensor (lattice elasticity).
  2. Chemical Potential Gradient: $-\nabla \mu$, identical to the driving force for the superfluid component. This implies the supersolid component is also part of the ground state.
  3. Frictional Drag: A term proportional to $(v_{Li} - v_{ni})$ with a relaxation time $\tau$, representing drag between the lattice and the normal fluid.
• Distinction from Andreev-Lifshitz (AL): While AL theory treats the lattice displacement velocity ($\dot{u}_i$) as equilibrating with the normal fluid via diffusion, this work shows that $\dot{u}_i$ equilibrates with $v_{Li}$ via diffusion, while $v_{Li}$ and $v_{ni}$ equilibrate via frictional drag.

4. Results: Normal Modes and Dynamics

The paper analyzes the dispersion relations for transverse and longitudinal modes, revealing a frequency-dependent crossover behavior governed by the drag time $\tau$.

A. Transverse Modes

• Low Frequency ($\omega \tau \ll 1$): Drag forces lock the supersolid and normal fluid together ($v_{Li} \approx v_{ni}$). They move as a single effective mass ($\rho_n + \rho_L$). The system exhibits one transverse mode driven by elasticity.
• High Frequency ($\omega \tau \gg 1$): Drag is negligible. The supersolid and normal fluid decouple. Two coupled transverse modes exist, one primarily involving the supersolid and the other the normal fluid.
• Superfluid: Does not participate in transverse motion ($v_{si} = 0$).

B. Longitudinal Modes

• Low Frequency: Drag locks $v_{Li}$ and $v_{ni}$ together. With $v_{si}$ as a separate degree of freedom, the system exhibits two coupled longitudinal modes.
• High Frequency: Drag is negligible; all three velocities are independent. The system theoretically supports three propagating modes.
  • Zero Frequency Mode: One solution corresponds to $\omega = 0$, where $v_{ni} = 0$ and current $j=0$, but $v_{si}$ and $v_{Li}$ are non-zero and opposite ($\rho_s v_s + \rho_L v_L = 0$). This represents a static redistribution of the ground state components without net flow.
  • Propagating Modes: The other two roots describe sound-like waves. Unlike standard superfluids (which have first and second sound), these modes involve a mix of elastic restoring forces (from the lattice) and pressure/chemical potential forces.
• Stability: The analysis confirms that for low temperatures, the system is stable (roots are positive).

5. Significance and Implications

• Ground State Association: The finding that the supersolid component is driven by $-\nabla \mu$ (like the superfluid) strongly suggests that the supersolid fraction is a ground-state phenomenon, not merely a thermal excitation.
• Experimental Predictions: The paper predicts a crossover in acoustic response based on frequency. Experiments on atomic gas supersolids (particularly in ring geometries) could test these predictions by observing how the normal fluid and lattice velocities decouple at high frequencies.
• Theoretical Unification: The work bridges the gap between Landau's superfluid theory and Leggett's supersolid theory, providing a consistent macroscopic framework that accounts for the "missing mass" observed in NCRI experiments.
• Comparison with GP Theory: The author notes that while Gross-Pitaevskii (GP) theory describes excitations well, it struggles to incorporate near-equilibrium thermal excitations (normal fluid) that carry momentum. Irreversible thermodynamics fills this gap by explicitly treating entropy production and thermal drag.

In conclusion, Saslow provides a rigorous hydrodynamic framework for supersolids, resolving the "missing mass" paradox by introducing a third velocity component and demonstrating how elastic, chemical, and frictional forces govern the complex interplay between the solid lattice and superfluid components.