The Simplified approach to the Bose gas without translation invariance

TL;DR

This paper extends the Simplified approach to Bose gases to non-translation invariant systems, enabling the study of observables like momentum distribution and confirming its predictions align with established theories at low densities.

Contribution

It generalizes the Simplified approach to systems without translation invariance, allowing analysis of additional observables such as momentum distribution.

Findings

Prediction matches Bogolyubov's at low densities

Approach works for systems with external potentials

Accurately describes momentum distribution up to inverse healing length

Abstract

The Simplified approach to the Bose gas was introduced by Lieb in 1963 to study the ground state of systems of interacting Bosons. In a series of recent papers, it has been shown that the Simplified approach exceeds earlier expectations, and gives asymptotically accurate predictions at both low and high density. In the intermediate density regime, the qualitative predictions of the Simplified approach have also been found to agree very well with Quantum Monte Carlo computations. Until now, the Simplified approach had only been formulated for translation invariant systems, thus excluding external potentials, and non-periodic boundary conditions. In this paper, we extend the formulation of the Simplified approach to a wide class of systems without translation invariance. This also allows us to study observables in translation invariant systems whose computation requires the symmetry to be broken. Such an observable is the momentum distribution, which counts the number of particles in excited states of the Laplacian. In this paper, we show how to compute the momentum distribution in the Simplified approach, and show that, for the Simple Equation, our prediction matches up with Bogolyubov's prediction at low densities, for momenta extending up to the inverse healing length.

Equations

$$\mathcal{H}:=-\frac{1}{2}\sum _{i=1}^N{\Delta }_i+\sum _{1⩽i<j⩽N}v(x_i,x_j)+\sum _{i=1}^N{P}_i$$

$$P_i:=\text{mathds}{1}^{\otimes i-1}\otimes \varpi \otimes \text{mathds}{1}^{\otimes N-i}.$$

$$\frac{N}{V}=\rho$$

$$\mathcal{H}{\psi }_0=E_0{\psi }_0.$$

$$|v(x,y)|⩽\bar{v}(x,y),{\int }_{{\mathbb{R}}^3}dy\bar{v}(x,y)⩽c$$

$$\int dx\varpi f(x)⩽c.$$

$$g_i(x_1,\cdots ,x_i):=\frac{\int \frac{d{x}_{i+1}}{V}\cdots \frac{d{x}_N}{V}\psi (x_1,\cdots ,x_N)}{\int \frac{d{y}_1}{V}\cdots \frac{d{y}_N}{V}\psi (y_1,\cdots ,y_N)}\equiv V^i\frac{\int d{x}_{i+1}\cdots d{x}_N\psi (x_1,\cdots ,x_N)}{\int d{y}_1\cdots d{y}_N\psi (y_1,\cdots ,y_N)}.$$

$$\int \frac{d{x}_i}{V}{g}_i(x_1,\cdots ,x_i)=g_{i-1}(x_1,\cdots ,x_{i-1}),\int \frac{dx}{V}{g}_1(x)=1.$$

$$g_i(x_1,\cdots ,x_i)=\prod _{1⩽j<l⩽i}{W}_i(x_j,x_l)$$

$$W_i(x,y)=f_i(x)f_i(y)(1-u_i(x,y))$$

$$|u_i(x,y)|⩽{\bar{u}}_i(x,y),\int dy{\bar{u}}_i(x,y)⩽c_i$$

$$\underset{V\to \infty }{\lim }\int d{x}_i{\Delta }_{x_i}{g}_i(x_1,\cdots ,x_i)=0$$

$$\int \frac{dx}{V}{g}_1(x)=1$$

$$\int \frac{dy}{V}{g}_2(x,y)=g_1(x)$$

$$\int \frac{dz}{V}{g}_3(x,y,z)=g_2(x,y)$$

$$\int \frac{dz}{V}\frac{dt}{V}{g}_4(x,y,z,t)=g_2(x,y)$$

$$g_2(x,y)=:g_1(x)g_1(y)(1-u_2(x,y)).$$

$$\left(-\frac{\Delta }{2}+\left(\varpi -⟨\varpi ⟩\right)+2\left(\mathcal{E}(x)-⟨\mathcal{E}(y)⟩\right)+\frac{1}{2}\left(\bar{A}(x)-⟨\bar{A}⟩-\bar{C}(x)\right)\right)g_1(x)+{\Sigma }_1(x)=0$$

$$\begin{array}[]{@{}>{\displaystyle}l@{}}\hphantom{\hskip 423.62pt}\\[-14.22636pt] \left(-\frac{1}{2}(\Delta_{x}+\Delta_{y})+v(x,y)-2\rho\bar{K}(x,y)+\rho^{2}\bar{L}(x,y)+\bar{R}_{2}(x,y)\right)g_{1}(x)g_{1}(y)(1-u_{2}(x,y))+\\ \hfill+\Sigma_{2}(x,y)=0\end{array}$$

$$⟨f⟩:=\int \frac{dy}{V}{g}_1(y)f(y),⟨\varpi ⟩\equiv \int \frac{dy}{V}\varpi g_1(y)$$

$$\bar{S}(x,y):=v(x,y)(1-u_2(x,y)),f_1\bar{\ast }{f}_2(x,y):=\int dz{g}_1(z)f_1(x,z)f_2(z,y)$$

$$\mathcal{E}(x):=\frac{\rho }{2}\int dy{g}_1(y)\bar{S}(x,y),\bar{A}(x):={\rho }^2\bar{S}\bar{\ast }{u}_2\bar{\ast }{u}_2(x,x)$$

$$\bar{C}(x):=2{\rho }^2\int dz{g}_1(z)u_2\bar{\ast }\bar{S}(x,z)+2\rho \int dy{\varpi }_y(g_1(y)u_2(x,y)).$$

$$\bar{K}(x,y):=\bar{S}\bar{\ast }{u}_2(x,y)$$

$$\begin{array}[]{@{}>{\displaystyle}l@{}}\hphantom{\hskip 423.62pt}\\[-14.22636pt] \bar{L}(x,y):=\bar{S}\bar{\ast}u_{2}\bar{\ast}u_{2}(x,y)-2u_{2}\bar{\ast}(u_{2}(u_{2}\bar{\ast}\bar{S}))(x,y)+\\ \hfill+\frac{1}{2}\int dzdt\ g_{1}(z)g_{1}(t)\bar{S}(z,t)u_{2}(x,z)u_{2}(x,t)u_{2}(y,z)u_{2}(y,t)\end{array}$$

$$\begin{array}[]{r@{\ }>\displaystyle l}\bar{R}_{2}(x,y)=&2\left(\mathcal{E}(x)+\mathcal{E}(y)-2\left<\mathcal{E}\right>\right)+\left(\varpi_{x}+\varpi_{y}-2\left<\varpi\right>\right)+\\[8.5359pt] &+\frac{1}{2}\left(\bar{A}(x)+\bar{A}(y)-2\left<\bar{A}\right>-\bar{C}(x)-\bar{C}(y)\right)+2\rho u_{2}\bar{\ast}\left(u_{2}(\mathcal{E}-\left<\mathcal{E}\right>)\right)+\\[8.5359pt] &+\rho\int dz\ \varpi_{z}(g_{1}(z)u_{2}(x,z)u_{2}(y,z))-\rho u_{2}\bar{\ast}u_{2}\left<\varpi\right>\end{array}$$

$${\Sigma }_i{⟶}_{V\to \infty }0$$

$$e:=⟨\mathcal{E}⟩+⟨\varpi ⟩+{\Sigma }_0$$

$$-\Delta u(x)=(1-u(x))\left(v(x)-2\rho K(x)+{\rho }^{2}L(x)\right)$$

$$K:=u\ast S,S(y):=(1-u(y))v(y)$$

Full text

1. 1 Introduction

2. 2 The model and main results

3. 2.1 The Simplified approach without translation invariance

4. 2.2 The momentum distribution

5. 3 The Simplified approach without translation invariance, proof of Theorem 2.1

6. 3.1 Factorization

7. 3.1.1 Factorization of $g_{2}$

8. 3.1.2 Factorization of $g_{3}$

9. 3.1.3 Factorization of $g_{4}$

10. 3.2 Consequences of the factorization

11. 3.3 Sanity check, proof of Corollary 2.1

12. 4 The momentum distribution

13. 4.1 Computation of the momentum distribution, proof of Theorem 2.2

14. 4.2 The simple equation and Bogolyubov theory, proof of Theorem 2.2

1 Introduction

The Bose gas is one of the simplest models in quantum statistical mechanics, and yet it has a rich and complex phenomenology. As such, it has garnered much attention from the mathematical physics community for over half a century. It consists in infinitely many identical Bosons and is used to model a wide range of physical systems, from photons in black body radiation to gasses of helium atoms. Whereas photons do not directly interact with each other, helium atoms do, and such an interaction makes studying such systems very challenging. To account for interactions between Bosons, Bogolyubov [Bo47] introduced a widely used approximation scheme that accurately predicts many observables [LHY57] in the low density regime. Even though Bogolyubov theory is not mathematically rigorous, it has allowed mathematical physicists to develop the necessary intuition to prove a wide variety of results about the Bose gas, such as the low density expansion of the ground state energy of the Bose gas in the thermodynamic limit [Dy57, LY98, YY09, FS20, BCS21, FS22], as well as many other results in scaling limits other than the thermodynamic limit (see [Sc22] for a review, as well as, among many others, [LSY00, LS02, NRS16, BBe18, BBe19, DSY19, BBe20, DS20, NT21, BSS22, BSS22b, HST22, NNe22]). In this note, we will focus on the ground state in the thermodynamic limit.

In 1963, E.H. Lieb [Li63, LS64, LL64] introduced a new approximation scheme to compute properties of the ground state of Bose gasses, called the Simplified approach, which has recently been found to yield surprisingly accurate results [CJL20, CJL21, CHe21, Ja22]. Indeed, while Bogolyubov theory is accurate at low densities, the Simplified approach has been shown to yield asymptotically accurate results at both low and high densities [CJL20, CJL21] for interaction potentials that are of positive type, as well as reproduce the qualitative behavior of the Bose gas at intermediate densities [CHe21]. In addition to providing a promising tool to study the Bose gas, the derivation of the Simplified approach is different enough from Bogolyubov theory that it may give novel insights into longstanding open problems about the Bose gas.

The original derivation of the Simplified approach [Li63] is quite general, and applies to any translation invariant system (it even works for Coulomb [LS64] and hard-core [CHe21] interactions). In the present paper, we extend this derivation to systems that break translation invariance. This allows us to formulate the Simplified approach for systems with external potentials, and with a large class of boundary conditions. In addition, it allows us to compute observables in systems with translation invariance, but whose computation requires breaking the translation invariance. We will discuss an example of such an observable: the momentum distribution.

The momentum distribution $\mathcal{M}(k)$ is the probability of finding a particle in the state $e^{ikx}$. Bose gasses are widely expected to form a Bose-Einstein condensate, although this has still not been proven (at least for continuum interacting gasses in the thermodynamic limit). From a mathematical point of view, Bose-Einstein condensation is defined as follows: if the Bose gas consists of $N$ particles, the average number of particles in the constant state (corresponding to $k=0$ in $e^{ikx}$) is of order $N$. The condensate fraction is defined as the proportion of particles in the constant state. The momentum distribution is an extension of the condensate fraction to a more general family of states. In particular, computing $\mathcal{M}(k)$ for $k\neq 0$ amounts to counting particles that are not in the condensate. This quantity has been used in the recent proof [FS20, FS22] of the energy asymptotics of the Bose gas at low density.

The main results in this paper fall into two categories. First, we will derive the Simplified approach without assuming translation invariance, see Theorem 2.1. To do so, we will make the so-called “factorization assumption”, on the marginals of the ground state wavefunction, see Assumption 2.1. This allows us to derive a Simplified approach for a wide variety of situations in which translation symmetry breaking is violated, such as in the presence of external potentials. Second, we compute a prediction for the momentum distribution using the Simplified approach. The Simplified approach does not allow us to compute the ground state wavefunction directly, so to compute observables, such as the momentum distribution, we use the Hellmann-Feynman technique and add an operator to the Hamiltonian. In the case of the momentum distribution, this extra operator is a projector onto $e^{ikx}$, which breaks the translation invariance of the system. In Theorem 2.2, we show how to compute the momentum distribution in the Simplified approach using the general result of Theorem 2.1. In addition, we check that the prediction is credible, by comparing it to the prediction of Bogolyubov theory, and find that both approaches agree at low densities and small $k$, see Theorem 2.2.

The rest of the paper is structured as follows. In Section 2, we specify the model and state the main results precisely. We then prove Theorem 2.1 in Section 3, Theorem 2.2 in Section 4.1, and Theorem 2.2 in Section 4.2. The proofs are largely independent and can be read in any order.

2 The model and main results

Consider $N$ Bosons in a box of volume $V$ denoted by $\Omega_{V}:=[-V^{\frac{1}{3}}/2,V^{\frac{1}{3}}/2]^{3}$, interacting with each other via a pair potential $v\in L_{1}(\Omega_{V}^{2})$ that is symmetric under exchanges of particles: $v(x,y)\equiv v(y,x)$. The Hamiltonian acts on $L_{2,\mathrm{sym}}(\Omega_{V}^{N})$ as

[TABLE]

where $\Delta_{i}\equiv\partial_{x_{i}}^{2}$ is the Laplacian with respect to the position of the $i$-th particle and $P_{i}$ is an extra single-particle term of the following form: given a self-adjoint operator $\varpi$ on $L_{2}(\Omega_{V})$,

[TABLE]

For instance, if we take $\varpi$ to be a multiplication operator by a function $v_{0}$, then $\sum_{i}P_{i}$ is the contribution of the external potential $v_{0}$. Or $\varpi$ could be a projector onto $e^{ikx}$, which is what we will do below to compute the momentum distribution. Because $P_{i}$ acts on a single particle, it breaks translational symmetry as soon as it is not constant.

We may impose any boundary condition on the box, as long as the Laplacian is self-adjoint. We will consider the thermodynamic limit, in which $N,V\to\infty$, such that

[TABLE]

is fixed. We consider the ground state $\psi_{0}$, which is the eigenfunction of $\mathcal{H}$ with the lowest eigenvalue $E_{0}$:

[TABLE]

(It is a standard argument to prove that $\psi_{0}$ exists, and is both real and non-negative.)

In order to take the thermodynamic limit, we will assume that $v$ is uniformly integrable in $V$:

[TABLE]

where $\bar{v}$ and $c$ are independent of $V$. In addition, we assume that, for any $f$ that is uniformly integrable in $V$,

[TABLE]

2.1 The Simplified approach without translation invariance

The crucial idea of Lieb’s construction [Li63] is to consider the wave function $\psi$ as a probability distribution, instead of the usual $|\psi|^{2}$. Since $\psi\geqslant 0$, this can be done by normalizing $\psi$ by its $L_{1}$ norm. We then define the $i$-th marginal of $\psi$ as

[TABLE]

In particular, for $i\in\{2,\cdots,N\}$,

[TABLE]

Because of the symmetry of $\psi$ under exchanges of particles, $g_{i}$ is symmetric under $x_{i}\leftrightarrow x_{j}$.

Inspired by [Li63], we will make the following approximation.

Assumption 2.1

(Factorization)

For $i=2,3,4$,

[TABLE]

with

[TABLE]

in which $f_{i}$ and $u_{i}$ are bounded independently of $V$ and $u_{i}$ is uniformly integrable in $V$:

[TABLE]

with $c_{i}$ independent of $V$. We further assume that, for $i=1,2,3$,

[TABLE]

in other words, these boundary terms vanish in the thermodynamic limit.

In other words, $g_{i}$ factorizes exactly as a product of pair terms $W_{i}$. The $f_{i}$ in $W_{i}$ allow for $W_{i}$ to be modulated by a slowly varying density, which is the main novelty of this paper compared to [Li63]. The inequality (11) ensures that $u_{i}$ decays sufficiently fast on the microscopic scale. Note that, by the symmetry under exchanges of particles, $u_{i}(x,y)\equiv u_{i}(y,x)$.

Here, we use the term “assumption” because it leads to the Simplified approach. However, it is really an approximation rather than an assumption: this factorization will certainly not hold true exactly. At best, one might expect that the assumption holds approximately in the limit of small and large $\rho$, and for distant points, as numerical evidence suggests in the translation invariant case. In the present paper, we will not attempt a proof that this approximation is accurate, and instead explore its consequences. Suffice it to say that this approximation is one of statistical independence that is reminiscent of phenomena arising in statistical mechanics when the density is low, that is, when the interparticle distances are large. In the current state of the art, we do not have much in the way of an explanation for why this statistical independence should hold; instead, we have extensive evidence, both numerical [CHe21] and analytical [CJL20, CJL21], that this approximation leads to very accurate predictions.

The equations of the Simplified approach are derived from Assumption 2.1, using the eigenvalue equation (4) along with

[TABLE]

[TABLE]

[TABLE]

[TABLE]

(all of which follow from (8)) to compute $u_{i}$ and $f_{i}$.

In the translation invariant case, the factorization assumption leads to an equation for $g_{2}$ alone, as $g_{1}$ is constant. When translation invariance is broken, $g_{1}$ is no longer constant, and the Simplified approach consists in two coupled equations for $g_{1}$ and $g_{2}$. We formulate these in terms of $g_{1}$ and $u_{2}$, with

[TABLE]

Theorem 2.2

If $g_{i}$ satisfies Assumption 2.1, the eigenvalue equation (4) and (13)-(16), then $g_{1}$ and $u_{2}$ satisfy the two coupled equations

[TABLE]

and

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

in which $\varpi_{x}$ is the action of $\varpi$ on the $x$-variable, and similarly for $\varpi_{y}$ and

[TABLE]

pointwise. Furthermore, the prediction for the energy per particle is

[TABLE]

where $\Sigma_{0}\to 0$ as $V\to\infty$.

This theorem is proved in Section 3.

Let us compare this to the equation for $u$ in the Simplified approach in the translation invariant case [CHe21, (5)], [Ja22, (3.15)]:

[TABLE]

[TABLE]

[TABLE]

We will prove that these follow from Theorem 2.1:

Corollary 2.3

(Translation invariant case)

In the translation invariant case $v(x,y)\equiv v(x-y)$ and $\varpi=0$ with periodic boundary conditions, if (18)-(18) has a unique translation invariant solution, then (19) reduces to (29) in the thermodynamic limit.

The idea of the proof is quite straightforward. Equation (19) is very similar to (29), but for the addition of the extra term $\bar{R}_{2}$. An inspection of (26) shows that the terms in $\bar{R}_{2}$ are mostly of the form $f-\left<f\right>$, which vanish in the translation invariant case, and terms involving $\varpi$, which is set to 0 in the translation invariant case. The only remaining extra term is $\bar{C}(x)+\bar{C}(y)$, which we will show vanishes in the translation invariant case due to the identity (14).

Theorem 2.1 is quite general, and can be used to study a trapped Bose gas, in which there is an external potential $v_{0}$. In this case, $\varpi$ is a multiplication operator by $v_{0}$. A natural approach is to scale $v_{0}$ with the volume: $v_{0}(x)=\bar{v}_{0}(V^{-1/3}x)$ in such a way that the size of the trap grows as $V\to\infty$, thus ensuring a finite local density in the thermodynamic limit. Following the ideas of Gross and Pitaevskii [Gr61, Pi61], we would then expect to find that (18) and (19) decouple, and that (19) reduces to the translation invariant equation (29), with a density that is modulated over the trap. However, the presence of $\bar{R}_{2}$ in (19) and $\bar{C}$ in (18) breaks this picture. Further investigation of this question is warranted.

2.2 The momentum distribution

The momentum distribution for the Bose gas is defined as

[TABLE]

where

[TABLE]

and $P_{i}$ is defined as in (2):

[TABLE]

Equivalently,

[TABLE]

where $E_{0}$ is the energy in (4) for the Hamiltonian (1). Using the Simplified approach, we do not have access to the ground state wavefunction, so we cannot compute $\mathcal{M}$ using (32). Instead, we use the Hellmann-Feynman theorem, which consists in adding $\sum_{i}P_{i}$ to the Hamiltonian. However, doing so breaks the translational symmetry. This is why Theorem 2.1 is needed to compute the momentum distribution. (A similar computation was done in [CHe21], but, there, the derivation of the momentum distribution for the Simplified approach was taken for granted.)

By Theorem 2.1, and, in particular, (28), we obtain a natural definition of the prediction of the Simplified approach for the momentum distribution:

[TABLE]

Theorem 2.4

(Momentum distribution)

Under the assumptions of Theorem 2.1, using periodic boundary conditions, if $v$ is translation invariant and $\varpi=0$, then, if $k\neq 0$, in the thermodynamic limit,

[TABLE]

where

[TABLE]

where $K$ and $L$ are those of the translation invariant Simplified approach (30)-(31) and

[TABLE]

We thus compute the momentum distribution. To check that our prediction is plausible, we compare it to the Bogolyubov prediction, which can easily be derived from [LSe05, Appendix A]:

[TABLE]

(this can be obtained by differentiating [LSe05, (A.26)] with respect to $\epsilon(k)$, which returns the number of particles in the state $e^{ikx}$, which we divide by $\rho$ to obtain the momentum distribution). Actually, following the ideas of [LHY57], we replace $\hat{v}$ by a so-called “pseudopotential”, which consists in replacing $v$ by a Dirac delta function, while preserving the scattering length:

[TABLE]

where the scattering length $a$ is defined in [LSe05, Appendix C]. Thus,

[TABLE]

We prove that, for the Simple Equation, as $\rho\to 0$, the prediction for the momentum distribution coincides with Bogolyubov’s, for $|k|\lesssim\sqrt{\rho a}$. The length scale $1/\sqrt{\rho a}$ is called the healing length, and is the distance at which pairs of particles correlate [FS20]. It is reasonable to expect the Bogolyubov approximation to break down beyond this length scale.

The momentum distribution for the Simple equation, following the prescription detailed in [CJL20, CJL21, CHe21, Ja22], is defined as

[TABLE]

where [CJL20, (1.1)-(1.2)]

[TABLE]

where $F$ was defined in (39).

Theorem 2.5

Assume that $v$ is translation and rotation invariant ($v(x,y)\equiv v(|x-y|)$), and consider periodic boundary conditions. We rescale $k$:

[TABLE]

we have, for all $\kappa\in\mathbb{R}^{3}$,

[TABLE]

The rotation invariance of $v$ is presumably not necessary. However, the proof of this theorem is based on [CJL21], where rotational symmetry was assumed for convenience.

3 The Simplified approach without translation invariance, proof of Theorem 2.1

3.1 Factorization

We will first compute $f_{i}$ and $u_{i}$ in Assumption 2.1.

3.1.1 Factorization of $g_{2}$

We start by considering $g_{2}$.

Lemma 3.6

Assumption 2.1 with $i=2$ and (13)-(14) imply that

[TABLE]

Proof: Assumption 2.1 implies

[TABLE]

and by (14),

[TABLE]

**1 - ** Let us first take an expansion to order $V^{-1}$. By (11)

[TABLE]

and so

[TABLE]

Applying $\int\frac{dx}{V}\cdot$ to both sides of (51), we find that

[TABLE]

so (51) yields

[TABLE]

**2 - ** We now push the expansion to order $V^{-2}$. Inserting (53) into (49),

[TABLE]

However, by (14),

[TABLE]

so, by (13),

[TABLE]

and

[TABLE]

Taking $\int\frac{dx}{V}\cdot$ on both sides, we find that

[TABLE]

$\square$

Remark: Note that this proof can easily be generalized to show that $f_{2}=g_{1}(1+O(V^{-n}))$ for any $n$.

3.1.2 Factorization of $g_{3}$

We now turn to $g_{3}$.

Lemma 3.7

Assumption 2.1 with $i=2,3$ and (13)-(15) imply that

[TABLE]

with

[TABLE]

[TABLE]

Proof: Using (15) in (9),

[TABLE]

**3 - ** We first expand to order $V^{-1}$. By (11),

[TABLE]

so, by (10),

[TABLE]

By Lemma 3.1.1,

[TABLE]

We take $\int\frac{dy}{V}\cdot$ on both sides of this equation. By (56) and (63),

[TABLE]

and, integrating once more implies that $\int\frac{dy}{V}f_{3}^{2}(y)=1+O(V^{-1})$. Therefore,

[TABLE]

and

[TABLE]

**4 - ** We push the expansion to order $V^{-2}$: (62) is

[TABLE]

By (67)-(68) and Lemma 3.1.1,

[TABLE]

Therefore, by (56),

[TABLE]

Now, let us apply $\int\frac{dy}{V}\cdot$ to both sides of the equation. Note that, by (11),

[TABLE]

Furthermore, by (56),

[TABLE]

and by (67) and (68),

[TABLE]

We are thus left with

[TABLE]

Taking $\int\frac{dx}{V}\cdot$, we thus find that

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

$\square$

3.1.3 Factorization of $g_{4}$

Lemma 3.8

Assumption 2.1 and (13)-(16) imply that

[TABLE]

with

[TABLE]

where $w_{3}$ is the same as in Lemma 3.1.2.

Proof: Using (16) in (9),

[TABLE]

**5 - ** We expand to order $V^{-1}$. By (11),

[TABLE]

so by (10),

[TABLE]

By Lemma 3.1.1,

[TABLE]

Applying $\int\frac{dy}{V}\cdot$ to both sides of the equation, using (56) and (82),

[TABLE]

Integrating once more, we have $\int\frac{dy}{V}f_{4}^{3}(z)=1+O(V^{-1})$ and

[TABLE]

Therefore,

[TABLE]

**6 - ** We push the expansion to order $V^{-2}$: by (11),

[TABLE]

[TABLE]

so

[TABLE]

By (86), (87), and Lemma 3.1.1,

[TABLE]

By (56),

[TABLE]

We apply $\int\frac{dy}{V}\cdot$ to both sides of the equation. By (72)-(74), we find

[TABLE]

Taking $\int\frac{dx}{V}\cdot$, we find that

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

$\square$

3.2 Consequences of the factorization

**7 - ** We first rewrite (4) as a family of equations for $g_{i}$.

**7-1 - ** Integrating (4) with respect to $x_{1},\cdots,x_{N}$, we find that

[TABLE]

with

[TABLE]

[TABLE]

and $B_{0}$ is a boundary term:

[TABLE]

**7-2 - ** If, now, we integrate (4) with respect to $x_{2},\cdots,x_{N}$, we find

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

in which we use the notation $\varpi_{y}$ to indicate that $\varpi$ applies to $y\mapsto g_{2}(x,y)$, and $B_{1}$ is a boundary term

[TABLE]

**7-3 - ** If we integrate with respect to $x_{3},\cdots,x_{N}$, we find

[TABLE]

where, here again, $\varpi_{y}$ indicates that $\varpi$ applies to the $y$-degree of freedom, whereas $\varpi_{x}$ applies to $x$, with

[TABLE]

[TABLE]

[TABLE]

and $B_{2}$ is a boundary term

[TABLE]

**8 - ** We rewrite (97), (101) and (106) using Lemmas 3.1.1, 3.1.2 and 3.1.3.

**8-1 - ** We start with (97): by (5) and Lemma 3.1.1,

[TABLE]

so

[TABLE]

**8-2 - ** We now turn to (101): by (5) and Lemma 3.1.1,

[TABLE]

and by Lemma 3.1.2,

[TABLE]

(we used (60) to write $u_{3}=u_{2}+O(V^{-1})$; this works fine for $u_{3}(x,y)$ and $u_{3}(x,z)$ because the integrals over $y$ and $z$ are controlled by $v(y,z)w_{3}(x,y)$ and $v(y,z)w_{3}(x,z)$ using (5) and (11); in the first term, it does not work for $u_{3}(y,z)$, as $v(y,z)w_{3}(y,z)$ can only control one of the integrals, and not both; the second term has an extra $V^{-1}$ that lets us replace $u_{3}$ by $u_{2}$) and by (11) and (6),

[TABLE]

The first term in $G_{1}^{(3)}$ is of order $V$:

[TABLE]

in which the only term of order $V$ is the first one, and is equal to the first term of order $V$ in $E_{0}$, and thus cancels out. There is a similar cancellation between the second term of order $V$ in $F_{1}^{(2)}$ and $E_{0}$. All in all,

[TABLE]

with, recalling $\rho:=N/V$,

[TABLE]

and using (61),

[TABLE]

[TABLE]

[TABLE]

Rewriting this using (20)-(23), we find (18) with

[TABLE]

**8-3 - ** Finally, we rewrite (106): by (5) and Lemma 3.1.2,

[TABLE]

and by Lemma 3.1.3,

[TABLE]

[TABLE]

and by (11) and (6),

[TABLE]

The first term in $G_{2}^{(4)}$ is of order $V$: by (80),

[TABLE]

in which the only term of order $V$ is the first one, and is equal to the term of order $V$ in $E_{0}$, and thus cancels out. There is a similar cancellation between the term of order $V$ in $F_{2}^{(3)}$ and $E_{0}$. All in all,

[TABLE]

with

[TABLE]

and by (61),

[TABLE]

[TABLE]

[TABLE]

**8-4 - ** Expanding out $\Pi$, see (125), we find (19) with

[TABLE]

and

[TABLE]

Using (22) and (23), (133) becomes (26).

**9 - ** Finally, (28) follows from (97) with

[TABLE]

$\square$

3.3 Sanity check, proof of Corollary 2.1

Assuming the translation invariance of the solution, $g_{1}(x)$ is constant. By (13),

[TABLE]

Furthermore, $\varpi\equiv 0$. We then have

[TABLE]

(see (30)-(31)). Furthermore,

[TABLE]

[TABLE]

[TABLE]

which vanishes by (14). Thus,

[TABLE]

We conclude by taking the thermodynamic limit. $\square$

4 The momentum distribution

4.1 Computation of the momentum distribution, proof of Theorem 2.2

We use Theorem 2.1 with $\varpi$ as in (33). Note that, by (33),

[TABLE]

which trivially satisfies (6).

**10 - ** We change variables in (19) to

[TABLE]

and find

[TABLE]

In addition, by (28),

[TABLE]

We expand in powers of $\epsilon$:

[TABLE]

in which we used the fact that, at $\epsilon=0$, $g_{1}(x)|_{\epsilon=0}=1$, see (136). In particular, the terms of order [math] in $\epsilon$ are independent of $\xi$. Note, in addition, that, by (13),

[TABLE]

**11 - ** The trick of this proof is to take the average with respect to $\xi$ on both sides of (144). Since we take periodic boundary conditions, the $\Delta_{\xi}$ term drops out. We will only focus on the first order contribution in $\epsilon$, and, as was mentioned above, terms of order [math] are independent of $\xi$. Thus, the average over $\xi$ will always apply to a single term, either $g_{1}^{(1)}$ or $u_{2}^{(1)}$. By (13), the terms involving $g_{1}^{(1)}$ have zero average. We can therefore replace $g_{1}^{(1)}$ by 1. (The previous argument does not apply to the terms in which $\Delta_{\zeta}$ acts on $g_{1}$, but these terms have a vanishing average as well because of the periodic boundary conditions.) In particular, by (14) and Lemma 3.1.1,

[TABLE]

so

[TABLE]

and thus, we can replace $u_{2}$ with $u_{2}^{(0)}$. Thus, using the translation invariant computation detailed in Section 3.3, we find that the average of (144) is

[TABLE]

where $K$ and $L$ are defined in (30) and (31) and $F$ comes from the contribution to $\bar{R}_{2}$ of $\varpi$, see (26):

[TABLE]

Similarly, (145) is

[TABLE]

**12 - ** Furthermore, by (33),

[TABLE]

for any integrable $f$, so

[TABLE]

and

[TABLE]

Now,

[TABLE]

so

[TABLE]

Similarly,

[TABLE]

Thus

[TABLE]

Since $k\neq 0$, the $\delta$ function drops out. We conclude the proof by combining (150), (155) and (159) and taking the thermodynamic limit. $\square$

4.2 The simple equation and Bogolyubov theory, proof of Theorem 2.2

**13 - ** We differentiate (44) with respect to $\epsilon$ and take $\epsilon=0$:

[TABLE]

Let

[TABLE]

(this operator was introduced and studied in detail in [CJL21]). We apply $\mathfrak{K}_{e}$ to both sides and take a scalar product with $-\rho v/2$ and find

[TABLE]

and so, using (43),

[TABLE]

and, by (39),

[TABLE]

Note that

[TABLE]

which is the expression for the uncondensed fraction for the simple equation [CHe21, (38)].

**14 - ** By [CJL21, (5.8),(5.27)],

[TABLE]

Furthermore, by the resolvent identity,

[TABLE]

in terms of which, using the self-adjointness of $\mathfrak{K}_{e}$,

[TABLE]

**15 - ** Now, taking the Fourier transform,

[TABLE]

and so

[TABLE]

and thus

[TABLE]

We recall [CJL20, (4.25)]:

[TABLE]

and, by [CJL20, (4.24)],

[TABLE]

Therefore, if we rescale

[TABLE]

we find

[TABLE]

**16 - ** Now,

[TABLE]

so

[TABLE]

Therefore,

[TABLE]

which, using the $q\mapsto-q$ symmetry, is

[TABLE]

that is,

[TABLE]

in which we rescale

[TABLE]

so, by (172)-(173),

[TABLE]

Therefore, by dominated convergence (using the argument above [CJL21, (5.23)] and the fact that $\mathfrak{K}_{e}$ is positivity preserving), and by [CJL21, (5.23)-(5.24)],

[TABLE]

**17 - ** Inserting (175) and (183) into (168), we find

[TABLE]

Finally, we recall [CJL20, (1.23)]:

[TABLE]

so

[TABLE]

**18 - ** Finally, by (42)

[TABLE]

so by (185),

[TABLE]

This, together with (186), implies (46). $\square$