Ground state energy of a Bose gas in the Gross–Pitaevskii regime

In 1957, in a short but very influential paper,¹ Dyson proved the first rigorous estimate for the ground state energy of an interacting Bose gas at low density. He considered a gas of hard spheres, but his approach can be easily extended to more general repulsive, short-range interactions. To leading order, he obtained the correct upper bound for the ground state energy and a lower bound, off by a factor around 14.

Forty years later, in 1997, Lieb–Yngvason² managed to show a lower bound for the ground state energy, matching Dyson’s upper bound. They considered a system of N bosons, moving in the box Λ_L = [0;L]³ (with periodic boundary conditions) and interacting through a repulsive (i.e., non-negative), two body radial potential $V:R3→R$⁠, with integrable decay at infinity. They proved that the ground state energy per unit particle at fixed density ρ = N/L³, in the limit N, L → ∞, is such that

in the dilute limit $ρa3→0$⁠. Here, $a$ denotes the scattering length of the potential V and is defined through the solution of the zero-energy scattering equation

with the boundary condition f(x) → 1, as |x| → ∞, so that $f(x)≃1−a/|x|$ asymptotically for large |x|. For the hard-sphere interaction, $a$ coincides with the radius of the spheres.

The next order corrections to (1.1) have been predicted in the physics literature by Lee–Huang–Yang,³ who derived the expression

again for the dilute limit $ρa3≪1$⁠. It is interesting to observe how (1.1) and also (1.2) only depend on the scattering length $a$⁠, and not on further details of the interaction. A rigorous lower bound matching the Lee–Huang–Yang formula has been recently derived by Fournais–Solovej, first for integrable potentials in Ref. 4 and then also for hard-sphere interactions in Ref. 5. A rigorous upper bound for the ground state energy per particle matching (1.2), on the other hand, has been first derived by Yau–Yin in Ref. 6 for smooth interaction potentials (a quasi-free trial state, which only recovered the correct asymptotics in the limit of weak potential, was previously proposed in Ref. 7). More recently, a simpler trial state has been proposed in Ref. 8 for interaction potentials $V∈L3(R3)$⁠. Still missing is an upper bound matching (1.2) for non-integrable interactions, such as hard-sphere potentials (ironically, the upper bound is still missing exactly for the hard-sphere interaction that was first considered by Dyson, who probably regarded it as the simplest case).

In experiments, Bose gases are often very dilute. This leads us to consider also scaling regimes in which the density is coupled with the number of particles N and tends to 0, as N → ∞. An important example is the Gross–Pitaevskii regime, where the density is 1/N². Rescaling lengths, the Gross–Pitaevskii regime corresponds to systems of N particles moving in the fixed box Λ = [0; 1]³ (with periodic boundary conditions) and interacting through a repulsive (i.e., non-negative) rescaled potential N²V(N·), whose scattering length is given by $a/N$⁠, with $a$ denoting the scattering length of V. Translating (1.1) to the Gross–Pitaevskii limit, it implies that the ground state energy in this limit is given by

as N → ∞.

In the recent years, making use of a rigorous version of Bogoliubov theory,⁹ it was possible, at least for integrable potentials, to go beyond the leading order estimate (1.2) and to resolve the ground state energy and the low-energy excitation spectrum up to errors vanishing in the limit of large N. For particles interacting through a repulsive, radial, and compactly supported interaction $V∈L3(R3)$⁠, the ground state energy in the Gross–Pitaevskii regime was shown in Ref. 10 to satisfy

where

and in particular, the limit can be proven to exist. The second line of (1.3) corresponds, in the thermodynamic limit, to the second term in (1.2) [in this regime, the sum in (1.3) can be replaced by an integral, leading to (1.2); see Eqs. (A.26)-(A.29) in Ref. 11]. The correction $eΛa2$⁠, on the other hand, is a finite volume effect, arising because $a$ is defined through the scattering equation (1.2) on the whole space $R3$⁠, rather than on the box Λ, and it has no analog in (1.2).

Additionally, after subtraction of the ground state energy, the excitation spectrum below a threshold ζ > 0 was proven in Ref. 10 to consist of eigenvalues having the form

with $np∈N$ for all momenta $p∈2πZ3\{0}$⁠. In other words, excited eigenvalues are determined, in good approximation, by the sum of the energies of quantized excitations, labeled by their momentum and characterized by the dispersion law $ε(p)=|p|4+16πap2$⁠. Again, we observe that (1.3) and (1.5) only depend on the interaction potential through its scattering length $a$⁠.

The results that we discussed in the last two paragraphs, concerning particles trapped in the unit torus Λ in the Gross–Pitaevskii limit, can also be extended to Bose gases on $R3$⁠, confined by external trapping potentials V_ext. In this case, Lieb–Seiringer–Yngvason proved in Ref. 12 that the ground state energy per particle is given, to leading order, by

with the Gross–Pitaevskii energy functional

More precise bounds, analogous to (1.3) and (1.5), have been recently established, for integrable potentials, in Refs. 13 and 14.

Estimates (1.3) and (1.5) have been proven in Ref. 10 for $V∈L3(R3)$⁠. As indicated in Ref. 14, it would be relatively easy to extend them to $V∈L1(R3)$⁠. On the other hand, extension to hard-spheres (or to more general non-integrable potentials with a hard-core) requires substantial new ideas. The goal of these notes is to present an upper bound for the ground state energy in the Gross–Pitaevskii regime, for particles interacting through a hard-sphere potential, consistent with (1.3); a complete proof of this estimate will appear in Ref. 15; here, we will only sketch some of the main ideas.

Before stating and discussing our new estimate in Sec. IV, we are going to review some tools that play a crucial role in its proof. In Sec. II, we will briefly recall Dyson’s upper bound for the leading order contribution to the ground state energy of a dilute gas of hard spheres. In Sec. III, on the other hand, we will review some of the main steps of the rigorous version of Bogoliubov theory for integrable potentials that led to (1.3) and (1.5).

In this section, we review Dyson’s upper bound for the ground state energy of hard-spheres in the dilute limit. Since the argument plays an important role in the derivation of our new estimate for the Gross–Pitaevskii regime, we are going to sketch a slightly different proof, compared with Dyson’s original work (in fact, our trial state is closer to the one proposed by Jastrow in Ref. 16).

We consider a gas of N hard spheres in the three dimensional torus Λ_L = [0;L]³. We are looking for an upper bound to the ground state energy

with the infimum taken over all $Ψ∈L2(ΛLN)$⁠, symmetric with respect to permutations of the N particles and satisfying the hard-sphere condition Ψ(x₁, …, x_N) = 0 if there exist i, j ∈ {1, …, N}, i ≠ j, with $|xi−xj|<a$⁠. Here (and in the following), we indicate with |x_i − x_j| the distance between x_i and x_j on the torus. We are interested in the thermodynamic limit, where N, L → ∞ with fixed ρ = N/L³, and we focus on the low density regime, where $ρa3≪1$⁠.

To get an upper bound for (2.1), we have to evaluate the energy of an appropriate trial state. To this end, we modify the non-interacting ground state $ΨN,La=0(x1,…,xN)≡1$ by adding correlations among particles. Since correlations are produced mainly by two-body scattering events, it seems natural to consider trial states having the form

where $a≪ℓ≪L$ is a parameter that will be fixed later and f_ℓ is meant to describe two-body correlations up to distance ℓ. Product trial functions like (2.2) have been first considered by Jastrow in Ref. 16 (Dyson, on the other hand, worked in Ref. 1 with a non-symmetric trial state, describing only nearest neighbor correlations). More precisely, in (2.2), we choose f_ℓ as the ground state solution of the Neumann problem

on the ball |x| ≤ ℓ, with the hard-sphere condition f_ℓ(x) = 0 for $|x|<a$ and with the normalization f_ℓ(x) = 1 for |x| = ℓ. We can then extend f_ℓ to a function over the torus Λ_L, setting f_ℓ(x) = 1 for |x| > ℓ. With standard tools in analysis [see, for example, Ref. 18 (Lemma A.1)], we can study the eigenvalue problem (2.3) and prove that

Moreover, denoting by χ_ℓ the characteristic function of the ball |x| ≤ ℓ, we find

To compute the energy of the trial state (2.2), we observe that

where the sum in the last term runs over i, j, m ∈ {1, …, N} all different. From (2.3), we obtain, with the notation $x=(x1,…,xN)∈ΛLN$⁠,

Thus, using the permutation symmetry, we find

Defining $uℓ(x)=1−fℓ2(x)$ and estimating

in the numerator and in the denominator of A, we can decouple the variable x₁; with (2.4) and (2.5), we obtain

under the assumption that $a/ℓ,ρaℓ2≪1$⁠. Similarly, using again (2.5), we can bound [this estimate could be improved using that, by symmetry, the integral of ∇f_ℓ(x₁ − x₂) · ∇f_ℓ(x₂ − x₃) over x₁, x₂, x₃ vanishes]

Thus, we conclude that

Choosing ℓ = ρ^−1/3, we obtain the upper bound

which captures the correct leading order of the ground state energy E_N,L in the dilute limit.

In this section, we consider a gas of N bosons moving in the unit torus Λ = [0; 1]³ and interacting through a repulsive (i.e., non-negative) potential with scattering length of the order 1/N (Gross–Pitaevskii regime). The Hamilton operator takes the form

and acts on the Hilbert space $Ls2(ΛN)$⁠, the subspace of L²(Λ^N) consisting of functions that are symmetric with respect to permutations of the N particles. Here, we assume $V:R3→R$ to be a repulsive, radial, and compactly supported potential.

In fact, it is convenient to embed (3.1) in a family of Hamilton operators for trapped Bose gases, all acting on $Ls2(ΛN)$⁠, given by

and depending on a parameter β ∈ [0; 1]. For β = 1, we recover the Gross–Pitaevskii Hamiltonian (3.1). For β = 0, on the other hand, (3.2) describes a mean-field scaling, where particles interact weakly through a potential varying on the same length scale L = 1 characterizing the box Λ. Mathematically, the mean-field scaling is more approachable, but from the point of view of physics, the Gross–Pitaevskii regime is, of course, more relevant.

Extending the bound (1.2) to all β ∈ [0; 1], we find that the ground state energy of (3.2) is given, to leading order, by

with $aNβ$ denoting N times the scattering length of the rescaled interaction potential N^3β−1V(N^β·). For β = 1, simple scaling of the zero-energy scattering Eq. (1.2) implies that $aNβ=1=a$⁠; for β < 1, on the other hand, $aNβ$ converges toward its first Born approximation $V̂(0)/8π$ [the Born series for $aNβ$ is given in (3.14)]. We conclude that

In the last decade, there has been progress in the mathematical understanding of the properties of trapped Bose gases described by (3.2), based on rigorous versions of Bogoliubov theory, that made it possible to go beyond the leading order estimate (3.3), at least for certain classes of integrable interactions. In the rest of this section, we are going to describe some of the main new ideas and results in this area. We focus here on integrable interaction potentials, excluding hard-spheres (we will come back to this point at the very end of this section and in Sec. IV).

The first important observation in Bogoliubov theory is that, close to the ground state energy, trapped Bose gases exhibit complete Bose–Einstein condensation. All particles, up to a fraction vanishing as N → ∞, are described by the same orbital φ₀ defined by φ₀(x) = 1 for all x ∈ Λ. In the mean-field limit and assuming the interaction potential to be bounded and positive definite, Bose–Einstein condensation follows from

which immediately gives

This operator inequality implies that for every approximate ground state ψ_N, i.e., for every $ψN∈Ls2(ΛN)$ satisfying

the expected number of orthogonal excitations of the condensate is bounded by

where we introduced the notation $q=1−|φ0〉〈φ0|$ for the projection on the orthogonal complement of the condensate wave function φ and where we used the gap in the Laplace operator to estimate $−Δ≥Cq$⁠. Remark that (3.4) implies condensation in a strong sense, proving that the number of excitations remains bounded, as N → ∞.

Verifying the existence of Bose–Einstein condensation in the Gross–Pitaevskii regime (more generally, for β > 1/3, i.e., if the range of the potential is much smaller than the typical distance between particles) is much more challenging. The first proof was given by Lieb–Seiringer. In Refs. 19 and 20, they showed that every approximate ground state wave function of (3.1) is such that

as N → ∞. A similar result was later obtained in Ref. 21, with different tools. More recently, the stronger estimate (3.4), giving optimal bounds on the number of excitations, was proven to hold in the Gross–Pitaevskii regime and, in fact, for all β ∈ (0; 1], in Refs. 10 and 22. Different proofs and extensions to the case of particles trapped by external potentials have been obtained in Refs. 23–25.

After establishing Bose–Einstein condensation, the next step in Bogoliubov theory consists in factoring out the condensate and in focusing instead on its orthogonal excitations. To this end, we proceed as in Ref. 26, and we observe that every $ψN∈Ls2(ΛN)$ can be written as

with uniquely determined $αj∈L⊥2(Λ)⊗sj$ for j = 0, …, N (here, ⊗_s denotes the symmetric tensor product). Defining U_Nψ_N = {α₀, …, α_N}, we map the Hilbert space $Ls2(ΛN)$ into the truncated Fock space

constructed over the orthogonal complement $L⊥2(Λ)$ of the condensate wave function φ₀. Through U_N, we can define the excitation Hamiltonian $LNβ=UNHNβUN*$⁠, acting on the excitation Hilbert space $F+≤N$⁠.

To compute $LNβ$⁠, it is convenient to rewrite (3.2) in momentum space using the formalism of second quantization with the creation and annihilation operators $ap*,ap$⁠, defined for every $p∈Λ*=2πZ3$⁠, satisfying canonical commutation relations

We find

Denoting by $N+$ the number of particles operator on $F+≤N$ (⁠$N+$ measures the number of excitations of the condensate), we have, from Ref. 26, the rules

for all $p,q∈Λ+*=2πZ3\{0}$⁠. Thus, we obtain

where we defined

In (3.5), we introduced operators $bp*,bp$ creating and, respectively, annihilating an excitation with momentum p (removing and, respectively, adding a particle to the condensate). On states with only few excitations, i.e., with $N+≪N$⁠, we have $bp*≃ap*$ and b_p ≃ a_p.

The cubic and the quartic terms in (3.6) look small in the limit N → ∞. In the mean-field regime, one can prove that, on states with few excitations, their contribution is indeed negligible. In other words, for β = 0, we can approximate

To diagonalize the resulting quadratic operator, we conjugate it with a (generalized) Bogoliubov transformation having the form

On states with few excitations, we expect (3.7) to act almost as a standard Bogoliubov transformation (with $b±p♯$ replaced by $a±p♯$⁠), i.e.,

Choosing $τ∈ℓ2(Λ+*)$ so that $tanh(2τp)=−V̂(p)/(p2+V̂(p))$⁠, we obtain, therefore,

This shows that the wave function $UN*TτΩ∈Ls2(ΛN)$ is a good approximation for the ground state of $HNβ=0$⁠, and it allows us to read off the ground state energy

Moreover, it implies that the low-energy excitation spectrum of $HNβ=0−ENβ=0$ consists (up to corrections vanishing as N → ∞) of the eigenvalues

with $np∈N$⁠, for all $p∈Λ+*$⁠. Rigorous justifications of (3.8) and (3.9) for the mean-field regime have been obtained in Refs. 26–29 (recently, higher order expansions of the energy have been given in Refs. 30 and 31).

For β > 0, the analysis is more difficult. In fact, for β ≥ 1/2, cubic and quartic terms produce relevant contributions to the energy and can certainly not be neglected. The main difference with respect to the case β = 0 is that correlations among particles, which are negligible in the mean-field limit, become here more important. To model correlations, we fix ℓ₀ > 0 small enough (but independent of N), and we consider [similarly to (2.3)] the ground state solution of the Neumann problem

on the ball |x| ≤ ℓ₀, with the normalization $fN,ℓ0(x)=1$ if |x| = ℓ₀. We extend $fN,ℓ0$ to Λ, setting $fN,ℓ0(x)=1$ for |x| > ℓ₀, and we define $wN,ℓ0=1−fN,ℓ0$⁠. For $p∈Λ+*$⁠, we consider the coefficients $ηp=−NŵN,ℓ0(p)$⁠, and we introduce the generalized Bogoliubov transformation

To understand (at least heuristically) the choice of the sequence η, we can write

for a constant $C∈R$⁠, making sure that normalization is preserved. This leads to

and [comparing with a wave function of the form (2.2)] explains the choice of η.

With T_η, we define the renormalized excitation Hamiltonian

Similarly to what we did in (3.6) for $LN,β$⁠, we can also decompose the operator $GN,β$ in constant, quadratic, cubic, and quartic terms. Through conjugation with T_η, we extracted important contributions to the energy from the quartic terms in (3.6). As a consequence, the constant term in $GN,β$ is now closer to the true ground state energy, compared with the constant term in (3.6); the vacuum expectations of $LN,β$ and of $GN,β$ are given, to leading order, by $NV̂(0)/2$ and $4πNaNβ$⁠, respectively; the difference is of order N^β, which is exactly the energy carried by the correlations generated by T_η. [With $η̌$ (x) ≃ C χ (|x| ≤ ℓ₀)/(|x| + N^{− β}), we expect that T_η creates order $‖η‖22$ ≃ 1 excitations, with an energy of order $‖η‖H12$ ≃ N^β.]

For β ∈ (0; 1), it is possible to show that cubic and quartic terms in $GN,β$ are negligible in the limit of large N. Thus, in this case, the renormalized excitation Hamiltonian $GN,β$ can be approximated by its quadratic component and can therefore be diagonalized by means of another (generalized) Bogoliubov transformation T_τ, similar to (3.7), correcting the energy at order one. Following this strategy, it was shown in Ref. 33 that, for β ∈ (0; 1), the wave function $UN*TηTτΩ∈Ls2(ΛN)$ is a good approximation for the ground state vector of $HNβ$⁠, that the ground state energy is given by

and that the low-energy excited eigenvalues of $HNβ−ENβ$ have the form

with $np∈N$ for all $p∈Λ+*$⁠, up to errors that vanish as N → ∞. Here, $aNβ$ denotes N times the scattering length of the potential N^3β−1V(N^β·) and can be recovered through the finite Born series inserted in (3.13).

where the order m_β is chosen so large that the error is much smaller than 1/N and therefore only produces negligible contributions when inserted in (3.13). Comparing with (3.8), we conclude that contributions to $aNβ$ associated with k ≥ 2, which are relevant for β ≥ 1/2, emerge from cubic and quartic terms in (3.6).

In the Gross–Pitaevskii regime (i.e. for β = 1), even after renormalization with the Bogoliubov transformation T_η, cubic and quartic terms in $GNGP=GN,β=1$ are still important, and they cannot be neglected. In this case, following Ref. 33, we need to perform a second renormalization of the excitation Hamiltonian, this time conjugating it with a unitary operator having the form

given by the exponential of a cubic expression in (modified) creation and annihilation operators (here, P_H, P_L are appropriately defined sets of high and low momenta). This leads to the (twice) renormalized excitation Hamiltonian $MNGP=S*Tη*UNHNGPUNTηS$⁠. Through conjugation with S, we extracted additional contributions from the cubic and the quartic terms in $GNGP$⁠. At this point, the remaining cubic and quartic terms in $MNGP$ are small, and we can focus on its quadratic part. Diagonalization through a last Bogoliubov transformation T_τ produces the good ansatz $UN*TηSTτΩ∈Ls2(ΛN)$ for the ground state vector and leads to the bounds (1.3) for the ground state energy and (1.5) for the excited eigenvalues.

The approach that we discussed in this section and that led to the estimates (1.3) and (1.5) requires $V∈L3(R3)$ (as well as the standard assumptions that V is non-negative, radial, and of short range). As recently shown in Ref. 14 (for the case of a Bose gas trapped by an external potential), it could be extended to $V∈L1(R3)$ with a slightly different choice of the coefficients η_p in (3.11) [the stronger condition $V∈L3(R3)$ is only important to control properties of the solution of the Neumann problem (3.10)]. On the other hand, this approach cannot be easily extended to particles interacting through a hard-sphere potential. The problem is that it is difficult to impose the hard sphere condition on states that are defined through the action of the unitary transformations T_η, T_τ, S. Keeping, for example, $η̌=−NwN,ℓ0$⁠, as defined after (3.10), we would have $η̌(x)=−N$ for $|x|<a/N$⁠, and according to the heuristic identity (3.12), $(UN*TηΩ)(x1,…,xN)$ would be far from zero, even when particles are closer than allowed by the hard-sphere condition [in this case, of course, the last approximation of the exponential in (3.12) is not valid].

In Sec. IV, we are going to present an alternative approach that we recently developed to obtain an upper bound for the ground state energy matching (1.3) for Bose gases interacting through hard-sphere potential.

In this section, we consider a gas of N hard spheres, with radius $aN=a/N$⁠, moving on the three-dimensional unit torus Λ = [0; 1]³. We are interested in the ground state energy of the system, which is defined as