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

TL;DR

This work investigates how rotation-induced breaking of time-reversal symmetry affects excitations and phase transitions in dipolar Bose-Einstein condensates, demonstrating that rotation can drive a superfluid-to-supersolid transition even at fixed interparticle interactions. The authors solve the rotating extended Gross–Pitaevskii equation with Lee–Huang–Yang beyond-mean-field corrections and compute Bogoliubov–de Gennes spectra in geometries featuring vortices and persistent currents. A central finding is a vortex-driven de-softening mechanism in which quantized vorticity shifts the Goldstone mode into a finite-energy roton, restoring superfluidity and producing re-entrant supersolid pockets as the rotation frequency is varied. This reveals a fundamental coupling between topological defects and crystalline order, offering rotation as a tunable knob to access supersolid phases in dipolar gases, with experimental accessibility in systems such as dysprosium and related platforms. The results are demonstrated in toroidal and oblate-trap geometries and are underpinned by detailed numerical methods and analytical dispersion analyses, supporting broad relevance for observing re-entrant phases in rotating quantum fluids.

Abstract

Rotation is a natural tool in ultracold gases to break time-reversal symmetry, yet its impact on the collective excitations of supersolids remains largely unexplored. We show theoretically that tuning the rotation frequency, (rather than the interparticle interactions), can trigger the superfluid-to-supersolid transition in Bose-Einstein condensates (dBECs). Computing excitation spectra in the presence of vortices and persistent currents, we uncover a vortex-driven de-softening mechanism whereby quantized vorticity elevates the gapless Goldstone mode to a finite-energy roton, restoring superfluidity. This effect results in re-entrant supersolid phases as a function of rotation frequency, revealing a fundamental coupling between topological defects and crystalline order.

Figures (5)

• Figure 1: Rotation-induced supersolidity. (a)-(b) Three-dimensional density isosurfaces (at $40\%$ peak density) of the SF and SS states realized without and with external rotation $\Omega$, respectively, at fixed scattering length $a_s$, in (a) a toroidal and (b) an oblate harmonic trap. The scattering lengths are $a_s = 92.8a_0$ in (a) and $87.4a_0$ in (b), with critical rotation frequencies $\Omega_{\mathrm{PC}}^{(1)} \approx 3.2\,\mathrm{Hz}$ and $\Omega_{\mathrm{V}}^{(1)} \approx 24.0\,\mathrm{Hz}$ marking the nucleation of a single-quantized persistent current and a unit vortex, respectively. (c) The effective hydrodynamic flow velocity $\Omega_{\mathrm{eff}} = \Omega-\Omega_{\mathrm{GS}}$ (solid blue line) versus external rotation $\Omega$ for a toroidal trap of radius $R$, where $\Omega_{\mathrm{GS}}=\hbar q/(MR^2)$ denotes the ground state angular velocity for a winding number $q$. The dashed black curve $\Omega_{R}^{(q)}$ denotes the critical angular velocity for roton softening (or Goldstone de-softening). The vertical black lines signify $\Omega_{\mathrm{PC}}^{(q)}$, at which vortex nucleation reverses the effective flow, leading to a cyclic alternation between SF and SS phases as $\Omega$ varies. (d) Density contrast $\mathcal{C}$ of the ground state showing the re-entrant supersolid pockets.
• Figure 2: Excitation spectrum and phase diagram in toroidal geometry. (a) BdG excitation energies vs. rotational velocity $\Omega$ at $a_s = 92.8a_{0}$. Black, blue, and orange lines denote the lowest roton ($m_{\rm R} = 6$), superfluid phonon ($m_{\rm Ph} = 1$), and Higgs modes, respectively. Vertical dashed lines indicate the sequence of SF $\to$ SS, vortex nucleation, and SS $\to$ SF transitions. Signs $\pm$ indicate angular momentum orientation relative to $\Omega$. The red curve shows the persistent-current energy $E_{\rm PC}$. (Inset) Bogoliubov mode profile $f$ for the $m_{\rm R} = 6$ roton at $\Omega = 0$ in the $xy$-plane (red/blue indicates negative/positive amplitude). (b) Density contrast $\mathcal{C}$ in the $(\Omega, a_s)$ plane, delineating the SF and SS regimes. See text for other parameters.
• Figure 3: Excitation spectrum and phase diagram in an oblate harmonic trap. (a) Excitation frequencies of rotons (black), Higgs (orange), and all other modes (gray) versus rotational angular velocity $\Omega$. The dashed and solid vertical lines indicate $\Omega_R^{(0)}$ and $\Omega_{1\rm V}$, respectively, while the red lines denote the energy differences of the one- and two-vortex states relative to the ground state, $E_{1\rm V}$ and $E_{2\rm V}$. Inset: mode profile $f$ of the $m_{\rm R}=3$ roton driving the SF$\rightarrow$SS transition. (b) Density-modulation contrast $\mathcal{C}$ as a function of $\Omega$ and the $s$-wave scattering length $a_s$. Other parameters are given in the text.
• Figure 4: Dynamical onset of supersolid order. Time evolution of the density contrast $\mathcal{C}$ for toroidal (red) and oblate harmonic (blue) traps. Following a sudden quench of rotation to $\Omega/2\pi = 3$ Hz (toroidal) and $23$ Hz (oblate harmonic), the nonrotating SF ground state undergoes spontaneous density modulation. (Insets) Snapshots of the integrated density $n(x,y)$ in the oblate harmonic trap at specific time intervals (see markers), refer to Ref. Suplementary for video of the dynamics. Scattering lengths are $a_s = 92.8\,a_0$ (toroidal) and $87.4\,a_0$ (oblate harmonic).
• Figure 5: Mode-Swapping in a rotating toroidal Superfluid. The energies of the six lowest excitations ($\mathcal{M}=\pm6,\pm 5,\pm 1$) for the system from Fig.2 in the main text with $a_s=93a_0$. The modes propagating in the direction of $\Omega$ are shown as solid lines, whereas we indicate modes propagating antiparallel to $\Omega$ as dashed lines. The vertical black lines indicate the critical rotation frequencies at which persistent currents are generated, corresponding to mode swapping.