Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas

Introduction

Interactions move particles out of the condensate even in a Bose gas at zero temperature. For a dilute three-dimensional gas, Bogoliubov theory predicts both the number of these depleted particles and their distribution in momentum. Establishing these predictions for an exact ground state requires information at arbitrarily long wavelengths: their energy cost tends to zero as the volume grows. This paper determines the leading momentum distribution for hard spheres, with the ordinary thermodynamic limit taken before the dilute limit.

The model and the result

Fix $a > 0$ and write $\Lambda_L = (\mathbb{R}/L\mathbb{Z})^3$ for the flat torus of side $L > 2a$, with torus distance $d_L$. The allowed configuration space is

The hard-sphere Hamiltonian $H^{\mathrm{hc}}_{N,L,a}$ is the operator associated with the quadratic form

Here $H^1_0$ is the $H^1$ closure of smooth functions compactly supported in the allowed configuration set; the ambient torus is already periodic. We use units $\hbar^2/(2m)=1$. The parameter $a$ is both the exclusion distance between centers and the scattering length: the exterior zero-energy scattering solution is $1-a/|x|$. The form domain is nontrivial along every sequence with $N/L^3 \to\rho< 1/(8a^3)$ and $N,L \to\infty$: a cubic grid of spacing at least $2a$ eventually provides $N$ sites, and small neighborhoods of these sites give an allowed open configuration set. Zero extension and compactness on the full torus then give compact resolvent.

Let $P_{N,L,a}$ be the projection onto the lowest bosonic eigenspace, and let $\mathcal{G}_{N,L,a}$ consist of the positive trace-one operators $\Gamma$ satisfying $\Gamma=P_{N,L,a}\Gamma P_{N,L,a}$. After extension by zero to the full periodic configuration space, put

and define

Thus $B_{N,L,a}$ is the occupation fraction of the specified constant orbital, and $1-B_{N,L,a}$ is its depletion. The total occupation is $\sum_k n_{\Gamma,L}(k)=N$. For $t\in\mathbb{R}^3\setminus\{0\}$, set

This is a finite positive measure: $g(t)=O(|t|^{-1})$ near zero and $g(t)=O(|t|^{-4})$ at infinity. Its total mass is $8/(3\sqrt{\pi})$.

Theorem 1.1. For every $a>0$, every bounded continuous real function $f$ on $\mathbb{R}^3$, and every $\varepsilon>0$, there is $\rho_0(a,f,\varepsilon)>0$ with the following property. Fix $0<\rho<\rho_0$ and put $\eta=\rho a^3$. For every sequence $N_j,L_j\to\infty$ with $N_j/L_j^3\to\rho$,

The density is fixed throughout each thermodynamic limit. In particular, the theorem controls every thermodynamic accumulation value of the occupation, without requiring a unique occupation limit at fixed positive $\eta$. Its uniformity includes all normalized, possibly complex ground vectors and the normalized projection onto the entire ground eigenspace.

Corollary 1.2. In the same iterated limit, uniformly over all ground states,

This follows by taking $f=1$. The theorem also determines how the depleted particles are distributed on the momentum scale $\sqrt{8\pi\rho a}$, without assuming translation invariance of an individual ground state.

Historical context and the sources of the method

Bogoliubov’s description of a weakly interacting Bose gas replaces its low-energy excitations by quasiparticles obtained from a linear canonical transformation [7]. The quasiparticle vacuum already has particles in nonzero momentum modes. Lee, Huang, and Yang developed the hard-sphere pseudopotential calculation far enough to determine both the second term of the energy and the leading depletion [19]. Thus the predicted momentum profile and the coefficient $8/(3\sqrt{\pi})$ describe the many-body state, as well as the fluctuations responsible for the energy correction. The distinction matters mathematically: matching the energy does not by itself control the occupation of modes whose energy tends to zero with the inverse size of the system.

The rigorous energy theory began with Dyson’s bounds for hard spheres [11]; Lieb and Yngvason later obtained the matching leading lower bound [23]. At the Lee–Huang–Yang order, Yau and Yin proved the upper bound for smooth repulsive short-range potentials [26]. Basti, Cenatiempo, and Schlein subsequently gave a proof for nonnegative radial compactly supported $L^3$ potentials with an improved error [4]. Fournais and Solovej established the lower bound, first for integrable interactions and then for a class including hard cores [14, 15]. For hard spheres, Basti, Cenatiempo, Giuliani, Olgiati, Pasqualetti, and Schlein bounded the correction by a constant times the Lee–Huang–Yang scale [3]; Basti, Brooks, Cenatiempo, Olgiati, and Schlein obtained the sharp coefficient [1]. Their subsequent work extends this upper-bound method to measurable even nonnegative compactly supported potentials, including hard cores [2]. The hard-sphere upper bound supplies the energy comparison used here.

Rigorous information about the state has developed in several different scaling limits. Lieb and Seiringer proved complete condensation for trapped gases in the Gross–Pitaevskii limit, allowing hard cores; their result includes trace-norm convergence of the normalized one-particle density matrix and the corresponding leading momentum-density limit [20]. For a periodic unit box with scattering length of order $N^{-1}$, Boccato, Brennecke, Cenatiempo, and Schlein determined the next-order depletion for nonnegative radial compactly supported $L^3(\mathbb{R}^3)$ potentials [6], eq:1.14 and Appendix A]. The associated discrete momentum occupations are expressed as a trace-norm expansion of the one-particle density matrix by Brennecke, Lee, and Nam, who also establish its positive-temperature analogue [8], eq:6–eq:8 and Theorem 1.1]. The second-order density-matrix expansion resolves the individual discrete occupations as well as their sum. Its relation between volume and scattering length differs from the thermodynamic limit at fixed density considered here.

Condensation on boxes growing with the dilute parameter was proved by Fournais [12], and the propagation of such estimates to larger scales has been developed further by Junge [18]. These results keep the box size tied to the dilution. In the ordinary thermodynamic limit, the one-particle gap vanishes at each fixed positive density, so an additional estimate is needed for variation across the whole torus. A different comparison is provided by the “simple equation” approximation to the Bose gas: Carlen, Jauslin, and Lieb derived its leading depletion coefficient [9] (Theorem 1.6), and Jauslin derived its Bogoliubov momentum profile on the inverse healing-length scale [17], Theorem 3]. Those are results for an approximate correlation equation. The present occupation measures are those of the exact many-body hard-sphere ground states, and their convergence is weak convergence of finite measures on the rescaled momentum space.

Our local energy analysis builds on the lower-bound methods of Fournais and Solovej [14, 15] and the Neumann-box localization of Haberberger, Hainzl, Nam, Seiringer, and Triay [16]. Fournais, Junge, Girardot, Morin, Olivieri, and Triay combined these approaches in their treatment of strong interactions at low temperature [13]. We use intermediate operator estimates from that work, retaining a positive term that measures Bogoliubov excitations, before taking an observable expectation. This is a use of their operator analysis beyond its scalar free-energy conclusion. The excitation-number localization is related to the large-matrix localization of Lieb and Solovej [22], Appendix A, and the coherent-state treatment of the condensate mode uses the $c$-number substitution of Lieb, Seiringer, and Yngvason [21].

The long-distance comparison uses a stationary ground-state path representation, a form of Doob’s $h$-transform [10]. The spatial count estimates and exact conditional killed-Brownian-path laws used here are supplied by the companion paper [24]. The companion establishes a positive constant-orbital occupation uniformly over the hard-sphere ground space. Here the comparison of particle-deletion measures must be accurate below the depletion scale. Its additional quantitative ingredients are a local fourth-moment comparison and a distant second-moment comparison. These compare the deletion measures obtained by averaging the deleted position over different cells.

The route construction adapts the unpredictable-path method of Benjamini, Pemantle, and Peres, who constructed paths in three dimensions with exponential intersection tails [5]. Our comparison also requires encounters to become rare after the segments near the common endpoint are removed, and it must accommodate finite defective regions. Detours use Kesten’s exterior-boundary connectivity theorem in the formulation of Timár [25], Theorem 4. These geometric inputs are combined with two-copy likelihood estimates that retain the exact conditional normalization; the intersection theorem alone does not give the deletion-measure comparison.

From local excitations to the momentum profile

The proof separates excitations inside moderately large boxes from variation between those boxes. For a partition into cubes of side $\ell$, let $P_\ell$ be the one-particle orthogonal projection onto functions constant on each cube, and let $P_0$ be projection onto the single global constant. Since $P_0 \le P_\ell$, every normalized ground vector satisfies the exact identity

The subscript 1 means that the projection acts on the first particle. Bosonic symmetry gives the same identity with the average over all particles.

The first term can be analyzed through the energy, provided the lower bound retains a positive operator measuring Bogoliubov excitations. The energy boxes contain several of the projection cells of side $\ell$; those cells will in turn be subdivided into small cells for the spatial comparison. An energy expansion alone does not identify this observable. We use the hard-sphere upper bound of Basti, Brooks, Cenatiempo, Olgiati, and Schlein [1], together with intermediate operator estimates of Fournais, Junge, Girardot, Morin, Olivieri, and Triay [13]. Their positive excitation terms give quantitative closeness to the local quadratic vacuum. We extract from them a uniform estimate for one-body operators supported on the local excitations, including operators whose matrices are complex and not diagonal in momentum. This stronger observable estimate will allow us to test translations after the long-wavelength contribution has been controlled.

The second term concerns wavelengths larger than $\ell$. To control it, we first take a normalized nonnegative ground vector $\Phi$. For a cube $v$ in a torus of volume $V$, define a measure on the remaining $N-1$ coordinates $Y'$ by the density

This is a finite smoothed particle-deletion measure: the deleted particle is selected with weight proportional to its presence in $v$, and its position is integrated out. Its mass is the expected particle count in $v$ divided by $(N/V)|v|$; this mass averages to one over equal-volume cells. Comparing the square roots $s_v$ for distant cubes gives control of spatial variation that the energy cannot supply.

The comparison transports the deletion along a chain of particles. Two copies of the configuration are coupled so that deleting the first particle in one copy and the last in the other leaves identical remaining configurations. Short Brownian bridges implement the rearrangement while preserving the exact hard-sphere constraints. The normalization of the conditional bridge law cancels in the likelihood ratio. Near either end of a long chain, the paths in one copy are kept unchanged. These unchanged segments remove endpoint interactions from the errors measured by a second moment. Random routes with sparse intersections then make that error smaller than the depletion scale.

Three estimates connect these deletion measures to the projection. A fourth-moment comparison between neighboring small-cell root mean squares shows that their arithmetic mean is close to their root mean square over a cell of side $\ell$. The occupation estimate on those same small cells compares their root mean squares with arithmetic averages of $\Phi$. A second-moment comparison between distant cells of side $\ell$ makes their root mean squares nearly constant. Together these estimates bound the last term of (1.3) by $o(\sqrt{\rho a^3})$. A decomposition into real and imaginary positive parts extends that bound to every ground vector.