Vorticity-Crystalline Order Coupling in Supersolids: Excitations and Re-entrant Phases

This paper theoretically demonstrates that tuning rotation frequency in Bose-Einstein condensates can induce a superfluid-to-supersolid transition and trigger re-entrant phases through a vortex-driven mechanism where quantized vorticity elevates the Goldstone mode to a finite-energy roton, thereby revealing a fundamental coupling between topological defects and crystalline order.

The Gist

Imagine a group of dancers in a room. In a normal state, they move together in perfect, fluid unison, like a single wave of water. This is what physicists call a superfluid. But under very specific conditions, these same dancers can suddenly arrange themselves into a rigid, repeating pattern—like a crystal lattice—while still maintaining their ability to flow without friction. This strange, dual-natured state is called a supersolid.

Usually, to make this transition happen, scientists have to tweak how strongly the dancers "feel" each other (their interactions). However, this paper proposes a new, surprising way to trigger this change: spinning the room.

Here is the story of what happens when you spin these quantum dancers, explained through simple analogies:

1. The Magic Spin

Think of the room as a giant, invisible turntable. When you start spinning it slowly, the dancers (the atoms) don't just spin with it; they start to feel a "synthetic magnetic field." This breaks the symmetry of time, meaning the physics of the room changes just because it's rotating.

The researchers found that by simply adjusting the speed of the spin, they could force the fluid to turn into a solid crystal pattern, even if the dancers' interactions remained exactly the same. It's like if a spinning carousel could suddenly make a crowd of people stand in a perfect circle formation just by changing its RPM.

2. The Vortex "Traffic Jam"

As the room spins faster, the dancers eventually get so agitated that they form little whirlpools, called vortices. Imagine a single whirlpool forming in the middle of the dance floor.

Here is the twist: The paper discovers that these whirlpools act like a "reset button."

• Phase 1 (Fluid): The room spins, the dancers form a crystal pattern (Supersolid).
• Phase 2 (The Whirlpool): As the spin gets faster, a vortex (whirlpool) suddenly appears.
• Phase 3 (Back to Fluid): The appearance of this vortex actually destroys the crystal pattern, turning the dancers back into a smooth, flowing fluid.

This is the "de-softening" mechanism mentioned in the paper. The vortex lifts the energy of the "crystal-making" mode, effectively telling the dancers, "Stop forming a pattern; just flow again."

3. The "Re-entrant" Dance (Going Back and Forth)

The most exciting part of the discovery is what happens if you keep spinning faster. The process doesn't just stop; it repeats in a cycle:

1. Spin up $\rightarrow$ Crystal forms (Supersolid).
2. Spin faster $\rightarrow$ Vortex appears $\rightarrow$ Crystal melts (Superfluid).
3. Spin even faster $\rightarrow$ A second vortex appears $\rightarrow$ Crystal reforms (Supersolid again!).

The paper calls this "re-entrant" behavior. It's like a light switch that you flip on, then off, then on again, just by turning a single dial (the rotation speed) higher and higher. The crystal order is periodically suppressed and restored by the discrete entry of these topological whirlpools.

4. Two Different Dance Floors

The researchers tested this idea on two different "dance floors" (traps):

• The Donut (Toroidal): A ring-shaped trap. Here, the transition happens at lower spin speeds.
• The Pancake (Oblate): A flat, round trap. Here, it takes a faster spin to create the first vortex, which means the "crystal phase" can exist over a wider range of speeds before the first vortex ruins it.

The Bottom Line

This paper reveals a fundamental, previously unknown link between whirlpools (vortices) and crystal patterns (density modulation). It shows that rotation isn't just a way to stir things up; it's a precise control knob that can toggle a quantum material between a flowing liquid and a rigid solid, and then back again, purely by changing the spin speed.

The authors suggest this could be tested in real experiments with ultracold atoms (like Dysprosium) in the near future, offering a new way to study these exotic states of matter without needing to constantly tweak the atoms' internal interactions.

Technical Summary

Technical Summary: Vorticity-Crystalline Order Coupling in Supersolids

Problem Statement
While rotation is a standard tool for breaking time-reversal symmetry (TRS) in ultracold gases to induce quantized vortices and persistent currents, its specific impact on the collective excitations of supersolids (SS) remains largely unexplored. Current experimental approaches to creating supersolids in dipolar Bose-Einstein condensates (dBECs) rely primarily on tuning anisotropic interparticle interactions to soften a finite-momentum roton mode. This work addresses the gap in understanding how external rotation, acting as a synthetic magnetic field, influences the phase transitions and excitation spectra of dBECs, particularly regarding the interplay between topological defects (vortices/persistent currents) and crystalline order.

Methodology
The authors employ a theoretical framework combining analytical insights with numerical simulations:

• Theoretical Model: They model a dilute gas of ultracold dipolar atoms (specifically $^{162}$Dy) confined in rotationally symmetric potentials (toroidal and oblate harmonic traps) subjected to external rotation with angular velocity $\Omega$. The system is described by an extended Gross–Pitaevskii equation (eGPE) including Lee–Huang–Yang (LHY) beyond-mean-field corrections.
• Excitation Spectra: To analyze stability and phase transitions, the authors linearize the eGPE around the rotational ground state using the Bogoliubov ansatz. This yields the Bogoliubov–de Gennes (BdG) equations, which are solved to obtain excitation frequencies ($\omega_j$) and eigenfunctions.
• Analytical Approach: For toroidal geometries, an analytical model is derived to describe the coupling between angular momentum modes and persistent currents. This involves defining an effective hydrodynamic flow velocity ($\Omega_{\text{eff}} = \Omega - \Omega_{\text{GS}}$) and introducing a shifted quantum number $M = m - q$ to account for the winding number $q$ of the ground state.
• Numerical Simulations: The authors solve the eGPE and BdG equations numerically for specific trap parameters (toroidal and oblate) to map phase diagrams in the $(\Omega, a_s)$ plane. They also perform real-time simulations with phenomenological damping to study the dynamical emergence of supersolidity following a rotation quench.

Key Contributions and Results

1. Rotation-Induced Supersolidity:
   The paper demonstrates that tuning the rotation frequency $\Omega$ can trigger the superfluid-to-supersolid (SF-SS) transition, even in regimes where interparticle interactions alone would not induce it. In toroidal traps, the transition occurs when the roton mode energy vanishes at a critical rotation frequency $\Omega_R^{(0)}$, leading to spontaneous density modulation.

2. Vortex-Driven De-softening Mechanism:
   A central finding is the discovery of a mechanism where the nucleation of topological defects reverses the phase transition.

   • In the absence of vortices, rotation softens a specific roton branch (roton$^+$), driving the system into an SS state characterized by a gapless Goldstone mode.
   • Upon further increasing $\Omega$, the system nucleates a quantized vortex (or persistent current). This nucleation reverses the sign of the effective flow velocity ($\Omega_{\text{eff}}$).
   • This reversal causes a "mode swapping" where the previously soft Goldstone mode is lifted back to a finite-energy roton excitation (de-softening). Consequently, the crystalline order is suppressed, and the system re-enters the superfluid (SF) phase.

3. Re-entrant Supersolid Phases:
   The interplay between rotation and vorticity creates a cyclic sequence of phase transitions: SF $\to$ SS $\to$ SF $\to$ SS. As $\Omega$ increases, the system alternates between supersolid pockets (where density modulation exists) and superfluid phases (where it is suppressed) as discrete vortices enter the system. This re-entrant behavior is observed in both toroidal and oblate harmonic traps, though the critical frequencies differ due to the energetic cost of vortex nucleation in different geometries.

4. Nature of the Phase Transition:
   The authors identify the rotation-induced SF-SS transition as first-order. Unlike interaction-driven transitions in rings which are typically second-order, the breaking of TRS by rotation introduces a linear term in the angular momentum within the energy functional. This leads to a discontinuous jump in the superfluid fraction at the transition point, evidenced by a finite Higgs-mode gap at the critical point.

5. Dynamical Stability:
   Real-time simulations confirm that the rotationally induced supersolid state is dynamically accessible. Starting from a non-rotating superfluid ground state, a sudden quench to a specific rotation frequency induces spontaneous density modulation, stabilizing into a supersolid phase without requiring adiabatic tuning.

Significance and Claims
The paper claims to uncover a fundamental coupling between quantized vorticity and crystalline order in supersolids. The primary significance lies in:

• New Control Parameter: Establishing rotation frequency as a viable control parameter for inducing and manipulating supersolidity, offering an alternative to fine-tuning interaction strengths.
• Topological-Crystalline Coupling: Revealing that topological defects (vortices) do not merely coexist with supersolidity but actively drive phase transitions by de-softening Goldstone modes, leading to re-entrant phases.
• Experimental Feasibility: The results suggest that these phenomena are experimentally accessible in current dipolar gas platforms (e.g., lanthanide atoms like Dysprosium) and potentially in other systems like dipolar molecules or exciton-polariton condensates, provided specific spectral conditions (low-energy angular excitations, concave spectrum, and roton softening as the lowest critical velocity) are met.

The authors conclude that this work opens a route to probe TRS-breaking effects in dipolar supersolids, highlighting a previously unknown mechanism where the discrete entry of vortices periodically suppresses and restores crystalline order.