Quantum Droplets in Curved Space

This Letter investigates the formation of quantum droplets in curved spacetime, highlighting the significant influence of curvature on the formation and properties of these objects. While our computations encompass various dimensions, we primarily focus on two dimensions. Our findings reveal a novel class of curvature-driven quantum effects leading to the formation of quasistable liquid droplets, suggesting a feasible pathway for experimental observation, particularly in microgravity environments.

💡 Research Summary

This paper extends the theory of quantum droplets—self‑bound liquid‑like states that arise in binary Bose‑Einstein condensates due to a delicate balance between mean‑field attraction and Lee‑Huang‑Yang (LHY) quantum fluctuations—to curved spacetime backgrounds. The authors consider a minimal two‑component Bose‑Bose mixture described by a covariant action on a static, generally curved metric (g_{\mu\nu}). The interaction potential contains intra‑ and inter‑species contact couplings (\gamma_{pq}) derived from the s‑wave scattering lengths. In flat space the system is stable when the quartic form is positive‑definite; a suitable choice of couplings (repulsive intra‑species, attractive inter‑species) reduces the leading mean‑field term and makes the LHY correction dominant, producing a non‑zero density minimum—a droplet.

To study curvature effects the fields are split into a static, possibly inhomogeneous background (\bar\phi_p(x)) and quantum fluctuations (\delta\phi_p(t,x)). Expanding the action to second order yields a fourth‑order minimal differential operator (D) whose functional determinant gives the one‑loop effective action (\Gamma = \bar S + \frac12\ln\det D). Assuming identical species (equal masses and couplings) the operator decouples into two scalar operators (D_{\pm}). The determinants are evaluated using ζ‑function regularisation, Mellin‑transform techniques, and modular transformations of the Jacobi theta function, allowing the authors to keep explicit dependence on the Ricci scalar (R) and on the spatial dimension (d).

A systematic heat‑kernel expansion is performed, retaining terms up to mass dimension six (i.e., ignoring (R^3), (\nabla^2R^2), etc.). This truncation limits the analysis to moderately curved, slowly varying backgrounds, which is appropriate for droplet profiles that are smooth in space. The resulting ζ‑functions reveal that for odd spatial dimensions (\zeta(0)=0) (no logarithmic term), while for even dimensions (the case of interest (d=2)) logarithmic contributions appear, reflecting the well‑known dimensional dependence of one‑loop corrections.

The effective potential extracted from the one‑loop action takes the schematic form \