Rotating supersolids challenge the conventional notion of superfluid flow: quantized vortices coexist with crystalline order arising from the system's intrinsic density modulation. Drawing on concepts developed in the context of superfluid helium, this seemingly paradoxical state is often described phenomenologically in terms of a dual response, combining superfluid flow with rigid-body-like motion. Here we show that these apparently distinct behaviors emerge from a single microscopic velocity field, globally constrained by irrotationality. By applying a Helmholtz decomposition, we disentangle a divergence-free incompressible component, which carries vortex circulation, from a curl-free compressible component generated by the intrinsic density modulation. In absence of vortices, the response is entirely encoded in the compressible field: the droplets co-rotate with the external drive, while the interstitial fluid develops an oppositely directed counterflow that preserves vanishing circulation. Above the vortex nucleation threshold, the incompressible component carries the usual quantized circulation, whereas the compressible field counteracts the vortex-induced angular momentum, reducing the total jump with respect to an unmodulated condensate. Our results recast supersolid rotation in terms of a flow-counterflow response, providing a microscopic interpretation of the phenomenological rigid/superfluid partitioning.
We consider the dynamics of a supersolid state that is based on Leggett's idea of local phase flow around the atomic sites of the solid. This leads to a local microscopic superfluid velocity that is not Galilean and is defined only relative to the solid and a
T=0 superfluid fraction ${\ensuremath{\rho}}_{S}/\ensuremath{\rho}<1$ (with $\ensuremath{\rho}$ being the macroscopic mass density). Landau's excitation-based argument for superfluids---that at temperature
T=0 the normal fluid mass density ${\ensuremath{\rho}}_{N}$ is zero---should apply to supersolids. Together these two arguments imply that at low
T there is a missing mass density. We attribute this to a supersolid density ${\ensuremath{\rho}}_{L}$, with ${\ensuremath{\rho}}_{L}\ensuremath{\equiv}\ensuremath{\rho}\ensuremath{-}{\ensuremath{\rho}}_{S}\ensuremath{-}{\ensuremath{\rho}}_{N}$ and a momentum-bearing supersolid velocity
vLi. We derive the macroscopic dynamical equations for this system with three momentum-bearing macroscopic velocities
(vLi,vNi,vSi) using Onsager's irreversible thermodynamics. For slow spatial variations,
vSi accelerates along the negative gradient of the chemical potential or mass, like for an ordinary superfluid, and
vNi is subject to drag against
vLi with a characteristic drag time $\ensuremath{\tau}$. There are three contributions to the acceleration of
vLi: drag against
vNi, along the negative gradient of the chemical potential or mass, and due to the forces of lattice elasticity. For high-frequency motion relative to the drag time $\ensuremath{\tau}$, ${\ensuremath{\rho}}_{L}$ moves independently of ${\ensuremath{\rho}}_{N}$. The normal modes for such a system have a crossover frequency below which the normal fluid velocity
vNi is dragged into motion with
vLi (so the theory reduces to the
vSi,vNi theory of Andreev and Lifshitz) and above which the normal fluid velocity
vNi is an independent variable. For an isotropic lattice we study the transverse and longitudinal responses. A frequency-dependent rotational inertia experiment is predicted to show, at frequencies that are low relative to $\ensuremath{\tau}$, that ${\ensuremath{\rho}}_{L}$ and ${\ensuremath{\rho}}_{N}$ move together but that at high frequencies ${\ensuremath{\rho}}_{L}$ moves much more than ${\ensuremath{\rho}}_{N}$.
Abstract A new class of superfluids and superconductors with spatially periodic modulation of the superfluid density is arising 1–12 . It might be related to the supersolid phase of matter, in which the spontaneous breaking of gauge and translational symmetries leads to a spatially modulated macroscopic wavefunction 13–16 . This relation was recognized only in some cases 1,2,5–9 and there is the need for a universal property quantifying the differences between supersolids and ordinary matter, such as the superfluid fraction, which measures the reduction in superfluid stiffness resulting from the spatial modulation 16–18 . The superfluid fraction was introduced long ago 16 , but it has not yet been assessed experimentally. Here we demonstrate an innovative method to measure the superfluid fraction based on the Josephson effect, a ubiquitous phenomenon associated with the presence of a physical barrier between two superfluids or superconductors 19 , which might also be expected for supersolids 20 , owing to the spatial modulation. We demonstrate that individual cells of a supersolid can sustain Josephson oscillations and we show that, from the current–phase dynamics, we can derive directly the superfluid fraction. Our study of a cold-atom dipolar supersolid 7 reveals a relatively large sub-unity superfluid fraction that makes realistic the study of previously unknown phenomena such as partially quantized vortices and supercurrents 16–18 . Our results open a new direction of research that may unify the description of all supersolid-like systems.
Supersolids are states of matter that spontaneously break two continuous symmetries: translational invariance due to the appearance of a crystal structure and phase invariance due to phase locking of single-particle wave functions, responsible for superfluid phenomena. While originally predicted to be present in solid helium, ultracold quantum gases provided a first platform to observe supersolids, with particular success coming from dipolar atoms. Phase locking in dipolar supersolids has been probed through e.g. measurements of the phase coherence and gapless Goldstone modes, but quantized vortices, a hydrodynamic fingerprint of superfluidity, have not yet been observed. Here, with the prerequisite pieces at our disposal, namely a method to generate vortices in dipolar gases and supersolids with two-dimensional crystalline order, we report on the theoretical investigation and experimental observation of vortices in the supersolid phase. Our work reveals a fundamental difference in vortex seeding dynamics between unmodulated and modulated quantum fluids. This opens the door to study the hydrodynamic properties of exotic quantum systems with multiple spontaneously broken symmetries, in disparate domains such as quantum crystals and neutron stars.
Recent studies have suggested a new phase in the extended Bose-Hubbard model in one dimension at integer filling. In this work, we show that this new phase is phase-separated into a supersolid and superfluid part, generated by mechanical instability. Numerical simulations are performed by means of the density matrix renormalization group algorithm in terms of matrix product states. In the phase-separated phase and the adjacent homogeneous superfluid and supersolid phases, we find peculiar spatial patterns in the entanglement spectrum and string-order correlation functions and show that they survive in the thermodynamic limit. In particular, we demonstrate that the elementary excitations of the homogeneous superfluid with enhanced periodic modulations are phonons, find the central charge to be
c=1, and show that the velocity of sound, extracted from the intrinsic level splitting for finite systems, matches with the propagation velocity of local excitations in dynamical simulations. This suggests that the low-energy spectrum of the phase under investigation is effectively captured by a spinless Luttinger liquid, for which we find consistent results between the Luttinger parameter obtained from the linear dependence of the structure factor and the algebraic decay of the one-body density matrix.
Abstract Within the mean field Gross–Pitaevskii framework, ultracold atomic condensates with long-range interaction are predicted to have a supersolid-like ground state beyond a critical interaction strength. Such a mean field supersolid-like ground state has periodically modulated superfluid density which implies the coexistence of superfluid and crystalline order. An ultracold atomic system in such a mean field ground state can be subjected to an artificial gauge field created either through rotation or by introducing space dependent coupling among hyperfine states of the atoms using Raman lasers. Starting from this Gross–Pitaevskii energy functional that describes such systems at zero temperature, we construct a hydrodynamic theory to describe the low-energy long-wavelength excitations of a rotating supersolid of weakly interacting ultracold atoms in two spatial dimensions for a generic type of long-range interaction. We treat the supersolidity in such a system within the framework of the well known two-fluid approximation. Considering such a system in the fast rotation limit where a vortex lattice in superfluid coexists with the supersolid lattice, we analytically obtain the dispersion relations of collective excitations around this equilibrium state. The dispersion relation gives the modes of the rotating supersolid which can be experimentally measured within the current technology. We point out that this can clearly identify an ultracold atomic supersolid phase in an unambiguous way.