--- schema_version: 1 id: PHYS-2026-08-02-01 date: 2026-08-02 updated_at: 2026-08-02 title: Bulk stabilization does not bind every finite quantum droplet summary: "How zero pressure and positive compressibility define an LHY-stabilized bulk liquid, while gradients, curvature, and particle emission determine whether a finite Bose mixture is self-bound." language: en entry_kind: daily status: published level: graduate-advanced user_difficulty: unrated domains: - condensed-matter - quantum-theory - statistical-mechanics estimated_minutes: 60 --- ## 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 $$ \mathcal E_{\rm MF}\sim-n^2, \qquad \mathcal E_{\rm LHY}\sim+n^{5/2} $$ 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, $$ \delta g=g_{12}+\sqrt{g_{11}g_{22}}<0, \qquad |\delta g|\ll g_{11},g_{22}. $$ At the optimal concentration ratio and within the dilute Bogoliubov regime, the soft density mode can be represented schematically by $$ \mathcal E(n)=-\frac A2n^2+Bn^{5/2}, \qquad A,B>0. $$ 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](https://arxiv.org/abs/1506.08419). For an inhomogeneous untrapped state, use the effective functional $$ E[\psi]=\int\dd^3x\left[ \frac{\hbar^2}{2m}|\nabla\psi|^2+\mathcal E(|\psi|^2) \right], \qquad N=\int\dd^3x\,|\psi|^2. $$ > [!margin: 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 $$ \mu(n)=\frac{\partial\mathcal E}{\partial n} =-An+\frac52Bn^{3/2}. $$ The pressure is $$ P(n)=n\mu(n)-\mathcal E(n) =-\frac A2n^2+\frac32Bn^{5/2}. $$ A planar liquid surface in equilibrium with vacuum requires $$ P(n_0)=P_{\rm vac}=0. $$ The nonzero solution is $$ \sqrt{n_0}=\frac{A}{3B}, \qquad n_0=\frac{A^2}{9B^2}. $$ 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 $n_0$, $$ \left.\frac{\dd P}{\dd n}\right|_{n_0} =\frac14An_0>0. $$ The isothermal zero-temperature compressibility is positive, $$ \kappa_T=\frac{1}{n_0} \left(\left.\frac{\dd P}{\dd n}\right|_{n_0}\right)^{-1}>0, $$ and the bulk sound speed satisfies $$ c^2=\frac1m\left.\frac{\dd P}{\dd n}\right|_{n_0}>0. $$ The equilibrium chemical potential is $$ \mu_0=\frac{\mathcal E(n_0)}{n_0} =-\frac{A^3}{54B^2}<0. $$ 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 $$ E(N)\simeq\mu_0N+4\pi\sigma R^2, \qquad N\simeq\frac{4\pi}{3}n_0R^3. $$ Here $\sigma>0$ is the planar surface tension. Eliminating $R$ gives $$ E(N)\simeq -|\mu_0|N +(36\pi)^{1/3}\frac{\sigma}{n_0^{2/3}}N^{2/3}. $$ 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 $$ \boxed{ N_E^{\rm cap} =36\pi\frac{\sigma^3}{n_0^2|\mu_0|^3} }. $$ $N>N_E^{\rm cap}$ makes the liquid-drop trial state lower in energy than $N$ free particles within this approximation. > [!margin: Estimate, not exact threshold] > The capillary expansion assumes $R/\xi\gg1$. 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, $$ \sigma=\int_{-\infty}^{\infty}\dd z\left[ \frac{\hbar^2}{2m}|\partial_z\psi|^2 +\mathcal E(n)-\mu_0n \right], $$ with $n(-\infty)=n_0$ and $n(+\infty)=0$. 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 $$ P(n_R)-P_{\rm vac}=\frac{2\sigma}{R}. $$ 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. > [!margin: 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 $$ \mu_N=\frac{\partial E}{\partial N} =-\lvert\mu_0\rvert +\frac23(36\pi)^{1/3}\frac{\sigma}{n_0^{2/3}}N^{-1/3}. $$ 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](https://arxiv.org/abs/2001.09086). 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 $\ell\ge2$ is $$ \omega_\ell^2 =\frac{\ell(\ell-1)(\ell+2)\sigma} {m n_0R^3}. $$ 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](https://arxiv.org/abs/2409.16017). 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. > [!margin: 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.](https://arxiv.org/abs/2007.06391). ## 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 | | $\dd P/\dd n>0$ 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 | | $N_E^{\rm cap}$ 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 $$ \mathcal E(n)=-\frac A2n^2+Bn^{5/2}, \qquad A,B>0. $$ 1. Compute $\mu(n)$ and $P(n)$. 2. Find the nonzero zero-pressure density $n_0$. 3. Verify $\mu_0<0$ and $\dd P/\dd n|_{n_0}>0$. 4. Derive the capillary estimate $N_E^{\rm cap}$ 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 $$ \mu=-An+\frac52Bn^{3/2}, $$ and $$ P=n\mu-\mathcal E =-\frac A2n^2+\frac32Bn^{5/2}. $$ The nonzero root of $P=0$ is $$ n_0=\frac{A^2}{9B^2}. $$ At that density, $$ \mu_0=-\frac{A^3}{54B^2}<0, \qquad \left.\frac{\dd P}{\dd n}\right|_{n_0} =\frac14An_0>0. $$ For a spherical sharp-interface droplet, $$ E(N)=-|\mu_0|N +(36\pi)^{1/3}\sigma n_0^{-2/3}N^{2/3}. $$ Setting $E=0$ yields $$ N_E^{\rm cap} =36\pi\frac{\sigma^3}{n_0^2|\mu_0|^3}. $$ This estimate scales cubically with $\sigma$. For the curvature correction, write $n_R=n_0+\delta n$. To leading order, $$ P(n_R)simeq \left.\frac{\dd P}{\dd n}\right|_{n_0}\delta n =\frac{2\sigma}{R}. $$ Therefore $$ \delta n =\frac{2\sigma/R}{A n_0/4} =\frac{8\sigma}{A n_0R}>0. $$ 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](https://arxiv.org/abs/1506.08419) - [New states of matter with fine-tuned interactions: quantum droplets and dipolar supersolids](https://arxiv.org/abs/2007.06391) - [Finite-range effects in ultradilute quantum drops](https://arxiv.org/abs/2001.09086) - [Self-bound droplets of a dilute magnetic quantum liquid](https://arxiv.org/abs/1607.07355) - [Dynamical formation of multiple quantum droplets in a Bose-Bose mixture](https://arxiv.org/abs/2409.16017) ## 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.