Bulk stabilization does not bind every finite quantum droplet

Key claim

A stable zero-pressure bulk liquid does not imply a self-bound droplet at every particle number.

The Lee–Huang–Yang term can stop mean-field collapse and select a finite bulk density. A finite cloud must also pay gradient energy. Near the binding threshold, much of the cloud is interface, so bulk thermodynamics alone cannot determine the critical particle number.

Theme

Bulk LHY stabilization, surface tension, and finite-particle binding.

Guiding question

Why does the competition

produce a stable density in an infinite mixture while a small untrapped cloud still expands?

Setup

Consider a three-dimensional Bose mixture close to the mean-field collapse line,

At the optimal concentration ratio and within the dilute Bogoliubov regime, the soft density mode can be represented schematically by

The first term is the residual mean-field attraction. The second is the repulsive LHY correction. Coefficients and the precise definition of $n$ depend on the reduction from the two-component mixture.

Petrov derived this stabilization mechanism and the resulting liquid-like state for a collapsing Bose mixture Quantum Mechanical Stabilization of a Collapsing Bose-Bose Mixture.

For an inhomogeneous untrapped state, use the effective functional

Effective theory

The local LHY functional assumes a dilute, weakly interacting mixture and slowly varying densities. The one-field notation also suppresses concentration fluctuations and possible mass imbalance.

We first solve the bulk problem, then ask which parts survive at finite $N$.

Analysis

Derivation: zero pressure selects the bulk density

The chemical potential is

The pressure is

A planar liquid surface in equilibrium with vacuum requires

The nonzero solution is

This condition differs from minimizing the energy density $\mathcal{E}(n)$. A free liquid adjusts its volume at fixed particle number; the resulting variation sets its pressure, not ${\partial }_n\mathcal{E}$, to zero.

At $n_0$,

The isothermal zero-temperature compressibility is positive,

and the bulk sound speed satisfies

The equilibrium chemical potential is

The equality ${\mu }_0=\mathcal{E}(n_0)/n_0$ follows from $P=0$. Negative ${\mu }_0$ means that an infinite amount of this liquid has lower energy per particle than atoms at rest in vacuum.

These statements establish a mechanically stable, bound bulk phase within the assumed equation of state. They contain no information about the cost of making a surface.

Scope and assumptions: finite droplets pay for gradients

For a large spherical droplet with radius $R$ much greater than its interfacial width $\xi$, write

Here $\sigma >0$ is the planar surface tension. Eliminating $R$ gives

The negative volume term scales as $N$. The positive surface term scales as $N^{2/3}$ and dominates the energy per particle at sufficiently small $N$.

The capillary estimate $E(N)=0$ gives

$N>N_E^{\mathrm{c}\mathrm{a}\mathrm{p}}$ makes the liquid-drop trial state lower in energy than $N$ free particles within this approximation.

Estimate, not exact threshold

The capillary expansion assumes $R/\xi \gg 1$. At the actual finite-$N$ boundary one often has $R\sim \xi$, so the full stationary extended GPE must replace the sharp bulk-plus-surface split.

Surface tension itself follows from the spatial functional. For a planar interface at coexistence,

with $n(-\mathrm{\infty })=n_0$ and $n(+\mathrm{\infty })=0$. The interface profile and derivative term therefore supply information absent from the bulk EOS.

A curved surface also changes mechanical equilibrium. The Laplace condition is

Thus a finite spherical droplet has positive internal pressure and, for positive compressibility, a central density shifted above $n_0$. Only the large-$R$ limit approaches the planar zero-pressure density.

Several meanings of “critical”

One can ask when a localized stationary solution appears, when its total energy becomes negative, or when its chemical potential blocks one-particle emission. These criteria need not coincide in a crude capillary model.

For example, differentiating the capillary energy gives

The condition ${\mu }_N<0$ concerns infinitesimal particle evaporation, whereas $E(N)<0$ compares the droplet with complete dissociation. Near threshold neither numerical boundary is controlled by the sharp-interface expression.

The experimentally quoted critical atom number is therefore model- and protocol-dependent. The extended GPE predicts it from a finite-profile problem, and corrections can move it appreciably.

Quantum Monte Carlo-informed work found that finite-range effects shift critical numbers in ${}^{39}$K mixtures relative to the zero-range MF+LHY functional Cikojević, Vranješ Markić, and Boronat.

This sensitivity does not invalidate the bulk LHY mechanism. It shows that a finite threshold probes more of the effective theory than the leading bulk balance.

Physical interpretation: surface modes and finite lifetime

The same gradient layer that raises the ground-state energy supports capillary modes. For an incompressible spherical droplet, the leading liquid-drop result for angular momentum $\ell \ge 2$ is

This formula assumes a sharp interface, small deformations, and negligible coupling to compressional modes. Close to the finite-$N$ boundary, those assumptions fail and the extended GPE spectrum is required.

The 2025 experiment by Cavicchioli and collaborators observed an elongated heteronuclear droplet split into multiple droplets in a manner consistent with capillary instability Dynamical formation of multiple quantum droplets in a Bose-Bose mixture.

That observation probes dynamical surface response. It does not provide a universal measurement of the equilibrium $N_c$ in every mixture or geometry.

Self-bound also does not mean indefinitely long-lived. Three-body recombination changes $N(t)$, and an initially bound droplet may cross its finite-particle stability boundary as atoms are lost.

Excitations above the particle-emission threshold can drive self-evaporation. Petrov’s original analysis found a parameter range in which collective modes lie above that threshold, but the claim depends on the model and droplet size.

Static versus dynamical stability

Positive compressibility tests long-wavelength bulk density fluctuations. It does not exclude surface breakup, particle loss, composition modes, or decay caused by inelastic collisions.

A broader review of mixture and dipolar droplets, including the limits of the LHY description, is given by Böttcher et al..

False claim to diagnose

If $P(n)$ has a stable zero at $n_0>0$, every $N>0$ admits a self-bound droplet.

The premise describes a thermodynamic-limit phase. A small cloud has no extensive bulk interior and can be dominated by quantum-pressure and interface costs.

The valid conclusion is:

A stable zero-pressure solution with ${\mu }_0<0$ supports a bound bulk liquid. Finite-droplet binding requires the full inhomogeneous energy to beat the vacuum continuum.

What follows — and what does not

Statement	Status
$n_0=A^2/(9B^2)$ follows from $P(n_0)=0$.	Exact within the assumed bulk EOS
$\mathrm{d}P/\mathrm{d}n>0$ at $n_0$ gives positive bulk compressibility.	Exact local stability statement
${\mu }_0<0$ gives bulk binding relative to vacuum.	Exact within the stated energy convention
LHY repulsion alone guarantees a bound zero-pressure liquid.	False; the attractive term and ${\mu }_0<0$ also matter
Every finite $N$ inherits the bulk binding.	False because gradients are positive
$N_E^{\mathrm{c}\mathrm{a}\mathrm{p}}$ is the exact critical atom number.	False near $R\sim \xi$
A curved droplet has density exactly $n_0$.	False; Laplace pressure shifts it
Positive compressibility proves dynamical stability against all modes.	False; it addresses a bulk density channel
Finite-range corrections can shift $N_c$.	Supported for specific microscopic mixtures

Exercise

Use

1. Compute $\mu (n)$ and $P(n)$.
2. Find the nonzero zero-pressure density $n_0$.
3. Verify ${\mu }_0<0$ and $\mathrm{d}P/\mathrm{d}n{|}_{n_0}>0$.
4. Derive the capillary estimate $N_E^{\mathrm{c}\mathrm{a}\mathrm{p}}$ from $E(N)=0$.
5. Use the Laplace condition to find the leading density shift $\delta n=n_R-n_0$ for large $R$.

Hint 1

For a free liquid, vary the volume at fixed $N$. This produces $P=n\mu -\mathcal{E}$, rather than the condition $\mu =0$.

Hint 2

Linearize $P(n_R)$ around $n_0$ and use $P(n_0)=0$.

Oral check 1. Why is minimizing $\mathcal{E}(n)$ the wrong way to find the equilibrium density of an isolated bulk liquid?

Oral check 2. If $\sigma$ doubles while $n_0$ and $|{\mu }_0|$ remain fixed, how does the capillary $N_E$ change?

Solution

Differentiation gives

$$\mu =-An+\frac{5}{2}B{n}^{3/2},$$Copy LaTeX

and

$$P=n\mu -\mathcal{E}=-\frac{A}{2}{n}^2+\frac{3}{2}B{n}^{5/2}.$$Copy LaTeX

The nonzero root of $P=0$ is

$$n_0=\frac{A^2}{9B^2}.$$Copy LaTeX

At that density,

$${\mu }_0=-\frac{A^3}{54B^2}<0,{\frac{\mathrm{d}P}{\mathrm{d}n}|}_{n_0}=\frac{1}{4}A{n}_0>0.$$Copy LaTeX

For a spherical sharp-interface droplet,

$$E(N)=-|{\mu }_0|N+(36\pi {)}^{1/3}\sigma n_0^{-2/3}{N}^{2/3}.$$Copy LaTeX

Setting $E=0$ yields

$$N_E^{\mathrm{c}\mathrm{a}\mathrm{p}}=36\pi \frac{{\sigma }^3}{n_0^2|{\mu }_0{|}^3}.$$Copy LaTeX

This estimate scales cubically with $\sigma$.

For the curvature correction, write $n_R=n_0+\delta n$. To leading order,

$$P(n_R)simeq{\frac{\mathrm{d}P}{\mathrm{d}n}|}_{n_0}\delta n=\frac{2\sigma }{R}.$$Copy LaTeX

Therefore

$$\delta n=\frac{2\sigma /R}{A{n}_0/4}=\frac{8\sigma }{A{n}_{0}R}>0.$$Copy LaTeX

The $1/R$ shift vanishes in the thermodynamic limit. It becomes large precisely where the planar capillary approximation becomes unreliable.

Check your understanding

Suppose two effective functionals have the same bulk $n_0$, ${\mu }_0$, and compressibility but different gradient coefficients.

Which bulk observables agree? Which finite-droplet properties can differ, and why?

You may also reply with “deeper,” “too easy,” “too hard,” or your derivation.

Further Reading

• Quantum Mechanical Stabilization of a Collapsing Bose-Bose Mixture
• New states of matter with fine-tuned interactions: quantum droplets and dipolar supersolids
• Finite-range effects in ultradilute quantum drops
• Self-bound droplets of a dilute magnetic quantum liquid
• Dynamical formation of multiple quantum droplets in a Bose-Bose mixture

Connections and next step

• Bulk thermodynamics: zero pressure fixes the coexistence density.
• Local stability: positive compressibility controls the long-wavelength density channel.
• Finite size: gradients generate surface tension and a nonextensive energy cost.
• Curvature: Laplace pressure shifts the density away from its planar value.
• Dynamics: capillary modes, evaporation, and inelastic loss require more than the EOS.
• Next step: solve the planar interface equation to compute $\sigma$ from the extended GPE.
• Revisit: compare the droplet threshold with nucleation, while keeping fixed-$N$ quantum binding distinct from thermal activation.