Nature of Lieb's "hole" excitations and two-phonon states of a Bose gas

TL;DR

This paper demonstrates that at weak coupling, Lieb's 'hole' excitations in a 1D Bose gas are equivalent to multiple phonons, challenging their independence, and explores two-phonon states and wave function structures across different regimes.

Contribution

It proves that Lieb's 'holes' are multiple phonons at weak coupling and analyzes two-phonon states and wave functions in various interaction regimes.

Findings

Lieb's 'holes' are j identical phonons at weak coupling.

Maximum number of phonons equals the number of atoms N.

Wave function structure in the Tonks-Girardeau gas shows unusual properties.

Abstract

It is generally accepted that the ``hole'' and ``particle'' excitations are two independent types of excitations of a one-dimensional system of point bosons. We show for a weak coupling that the Lieb's ``hole'' with the momentum $p=j2\pi/L$ is $j$ identical interacting phonons with the momentum $2\pi/L$ (here, $L$ is the size of the system, and $\hbar=1$). We prove this assertion for $j=1, 2$ by comparing solutions for a system of point bosons with solutions for a system of nonpoint bosons obtained in the limit of the point interaction. The additional arguments show that our conclusion should be true for any $j=1, 2, \ldots, N$. Thus, at a weak coupling, the holes are not a physically independent type of quasiparticles. Moreover, we find the solution for two interacting phonons in a Bose system with an interatomic potential of the general form at a weak coupling and any dimension (1, 2, or 3). It is also shown for a weak coupling that the largest number of phonons in a Bose system is equal to the number of atoms $N$. Finally, we have studied the structure of wave functions for the Tonks--Girardeau gas and found that the properties of quasiparticles in this regime are quite strange.

Equations

$$L{k}_i=2\pi n_i+2\sum _{j=1}^N\arctan \frac{c}{k_i-k_j}{|}_{j\ne i},i=1,\dots ,N,$$

$$L{k}_i=2\pi I_i-2\sum _{j=1}^N\arctan \frac{k_i-k_j}{c},i=1,\dots ,N.$$

$$I_i=n_i+i-\frac{N+1}{2}.$$

$${\Psi }_0({\text{r}}_1,\dots ,{\text{r}}_N)=A_0{e}^{S({\text{r}}_1,\dots ,{\text{r}}_N)},$$

$$S$$

$${\Psi }_{\text{p}}({\text{r}}_1,\dots ,{\text{r}}_N)=A_{\text{p}}{\psi }_{\text{p}}{\Psi }_0,$$

$${\psi }_{\text{p}}$$

$${\rho }_{\text{q}}=\frac{1}{\sqrt{N}}\sum _{j=1}^N{e}^{-i\text{q}{\text{r}}_j},$$

$$\text{q}=2\pi \left(\frac{j_x}{L_x},\frac{j_y}{L_y},\frac{j_z}{L_z}\right),$$

$$a_2(\text{p})\equiv a_2(p)=\frac{1-{\alpha }_{\text{p}}}{2},{\alpha }_{\text{p}}=\sqrt{1+\frac{2\rho \nu (p)}{{\hslash }^2{p}^2/(2m)}},$$

$$E(\text{p})=\frac{{\hslash }^2{p}^2}{2m}(1-2a_2(\text{p}))=\sqrt{{\left(\frac{{\hslash }^2{p}^2}{2m}\right)}^2+2\rho \nu (p)\left(\frac{{\hslash }^2{p}^2}{2m}\right)}\equiv E_B(\text{p}),$$

$$\nu (\text{p})=\underset{-L_x}{\overset{L_x}{\int }}dx\underset{-L_y}{\overset{L_y}{\int }}dy\underset{-L_z}{\overset{L_z}{\int }}dzU(r)e^{-i\text{p}\text{r}}.$$

$$v_s=v_s^{(0)}(1+{\delta }_s),{\delta }_s=-\frac{{\hslash }^2}{32m^2(v_s^{(0)}{)}^2}\frac{1}{N}\sum _{\text{q}\ne 0}\frac{q^2}{{\alpha }_{\text{q}}^3}{\left(\frac{2\rho \nu (q)}{{\hslash }^2{q}^2/(2m)}\right)}^2.$$

$$E(p_1)=\sqrt{p_1^4+4{\rho }^2\gamma p_1^2}\cdot (1+{\delta }_s)=\frac{4\pi \rho \sqrt{\gamma }}{L}\sqrt{1+\frac{{\pi }^2}{\gamma N^2}}\cdot (1+{\delta }_s),$$

$${\delta }_s=-\frac{1}{4N}\frac{1}{1+{\pi }^2/(\gamma N^2)}\sum _{j=1,2,\dots ,\infty }\frac{1}{1+{\pi }^2{j}^2/(\gamma N^2)}\frac{1}{\sqrt{1+\gamma N^2/({\pi }^2{j}^2)}}.$$

$$E_L(p_1)=\frac{4\pi \rho \sqrt{\gamma }}{L}\sqrt{1+\frac{{\pi }^2}{\gamma N^2}}\cdot (1+{\delta }_{sL}).$$

$$E_L(p)=\sum _{i=1}^N({\acute{k}}_i^2-k_i^2),$$

$$p=\sum _{i=1}^N({\acute{k}}_i-k_i)=\frac{2\pi }{L}\sum _{i=1}^N({\acute{n}}_i-n_i).$$

$$-\frac{{\hslash }^2}{2m}\sum _j{△}_j\Psi +\frac{1}{2}\sum _{ij}^{i\ne j}U(|{\text{r}}_i-{\text{r}}_j|)\Psi =E\Psi$$

$$-\frac{{\hslash }^2}{2m}\sum _j\left[{△}_j{\psi }_{{\text{p}}_1{\text{p}}_2}+2({\nabla }_{j}S)({\nabla }_j{\psi }_{{\text{p}}_1{\text{p}}_2})\right]=E_{{\text{p}}_1{\text{p}}_2}{\psi }_{{\text{p}}_1{\text{p}}_2},$$

$${\psi }_{{\text{p}}_1{\text{p}}_2}={\psi }_{{\text{p}}_1}{\psi }_{{\text{p}}_2}+\frac{\delta {\psi }_{{\text{p}}_1{\text{p}}_2}}{\sqrt{N}},$$

$$-\frac{{\hslash }^2}{2m}\sum _j\left[2({\nabla }_j{\psi }_{{\text{p}}_1})({\nabla }_j{\psi }_{{\text{p}}_2})+{△}_j\delta {\psi }_{{\text{p}}_1{\text{p}}_2}+2({\nabla }_{j}S)({\nabla }_j\delta {\psi }_{{\text{p}}_1{\text{p}}_2})\right]=$$

$$=[E({\text{p}}_1)+E({\text{p}}_2)+\delta E]\delta {\psi }_{{\text{p}}_1{\text{p}}_2}+\delta E{\psi }_{{\text{p}}_1}{\psi }_{{\text{p}}_2},$$

$$E_{{\text{p}}_1{\text{p}}_2}=E({\text{p}}_1)+E({\text{p}}_2)+\delta E.$$

$$\delta {\psi }_{{\text{p}}_1{\text{p}}_2}$$

$$0$$

$$C_j({\text{q}}_1,\dots ,{\text{q}}_{j-1};{\text{p}}_1,{\text{p}}_2)=0,j=1,\dots ,N.$$

$$B_1({\text{p}}_1,{\text{p}}_2)\frac{2m}{{\hslash }^2}[E({\text{p}}_1)+E({\text{p}}_2)+\delta E-E_1({\text{p}}_1+{\text{p}}_2)]=$$

$$B_2(\text{q};{\text{p}}_1,{\text{p}}_2)\frac{2m}{{\hslash }^2}[E({\text{p}}_1)+E({\text{p}}_2)+\delta E-E_1(\text{q})-E_1(\text{q}+\text{p})]=$$

$$B_1({\text{p}}_1,{\text{p}}_2)=\frac{{\hslash }^2}{m}\frac{{\text{p}}_1{\text{p}}_2-p_1^2{b}_2({\text{p}}_1;{\text{p}}_2)-p_2^2{b}_2({\text{p}}_2;{\text{p}}_1)}{E({\text{p}}_1)+E({\text{p}}_2)+\delta E-E_1({\text{p}}_1+{\text{p}}_2)}.$$

$$0=-b_2(-{\text{p}}_2;{\text{p}}_1)b_2({\text{p}}_1-{\text{p}}_2;{\text{p}}_2)({\text{p}}_2-{\text{p}}_1{)}^2-b_2(-{\text{p}}_1;{\text{p}}_2)b_2({\text{p}}_2-{\text{p}}_1;{\text{p}}_1)({\text{p}}_2-{\text{p}}_1{)}^2$$

Full text

Nature of Lieb’s “hole” excitations

and two-phonon states of a Bose gas

Maksim Tomchenko

Bogolyubov Institute for Theoretical Physics

14b, Metrolohichna Str., Kyiv 03143, Ukraine

E-mail:mtomchenko@bitp.kiev.ua

*It is generally accepted that the “hole” and “particle” excitations are two independent types of excitations of a one-dimensional system of point bosons. We show for a weak coupling that the Lieb’s “hole” with the momentum $p=j2\pi/L$ is $j$ identical interacting phonons with the momentum $2\pi/L$ (here, $L$ is the size of the system, and $\hbar=1$). We prove this assertion for $j=1,2$ by comparing solutions for a system of point bosons with solutions for a system of nonpoint bosons obtained in the limit of the point interaction. The additional arguments show that our conclusion should be true for any $j=1,2,\ldots,N$. Thus, at a weak coupling, the holes are not a physically independent type of quasiparticles. Moreover, we find the solution for two interacting phonons in a Bose system with an interatomic potential of the general form at a weak coupling and any dimension (1, 2, or 3). It is also shown for a weak coupling that the largest number of phonons in a Bose system is equal to the number of atoms $N$. Finally, we have studied the structure of wave functions for the Tonks–Girardeau gas and found that the properties of quasiparticles in this regime are quite strange.

Keywords: point bosons, hole-like excitations, interaction of phonons.

1 Introduction

This work is devoted to two main problems: the determination of the wave function and the energy of two interacting phonons in a Bose gas with a potential of the general form and the study of the nature of Lieb’s “holes”. The first problem was not solved, to our knowledge, and can help one to solve the second problem.

The elementary excitations of a one-dimensional (1D) system of point bosons are usually separated into two types: particle-like (“particles”) and hole-like (“holes”) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. At the weak coupling, the dispersion law of “particles” coincides with the Bogolyubov law [13, 14] and agrees with Feynman’s solutions [15, 16, 17] and the more later models [18, 19, 20, 21, 22, 23, 24, 25, 26] (other references can be found in reviews [27, 28]) allowing one to describe the microscopic properties of a Bose system at the weak and intermediate couplings. Therefore, it is natural to consider that the particles correspond to Bogolyubov–Feynman quasiparticles. The dispersion law of holes was found only in the approach based on the Bethe ansatz, in the well-known work by Lieb [1]. In this case, Lieb attacked the Bogolyubov’s and Feynman’s approaches and proposed some arguments in favor of that the holes are a physically independent type of elementary excitations [1, 2]. This point of view became traditional. Later on, it was found that the dispersion law of holes is close to that for the soliton solution of the 1D Gross–Pitaevskii equation [29, 30]. This became the main argument in favor of that the holes are a particular independent type of quasiparticles. However, such point of view does not agree with the models [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. It is important that the models [14, 15, 16, 21, 22, 23, 24, 25, 26] work in 1D, since they do not use a condensate (we note that the Bogolyubov’s method also works in 1D at small $\gamma$ and $T$, if $N$ is finite [31]). If the holes would be a separate type of quasiparticles, this would mean the significant shortcoming in the models [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] and in close ones. In addition, if the holes are an independent type of excitations, then they must give a separate contribution to thermodynamic quantities (since holes interact with particles and, therefore, participate in the thermal equilibrium). Such analysis indicates that the question about the nature of holes is very important.

The one-dimensional system differs qualitatively from a three-dimensional (3D) one by that the atom in a 1D system cannot get around another atom. The former can only pass through the latter. Despite this circumstance, Lieb believed that 1D and 3D systems are qualitatively similar [1]. Therefore, he made conclusion [1] that holes can exists also in 3D systems, at least in case of a strong coupling.

In what follows, we will study the structure of the wave functions of “particles” and holes and will show that, at a weak coupling, the hole is a collection of interacting “particles”. It was noted in the literature that the holes are not a mathematically independent type of excitations [1, 7, 33]. This conclusion was based on the Lieb–Liniger equations. However, these equations are not enough to clarify the physical nature of holes.

Let us consider what the Lieb–Liniger equations can say about the nature of holes. These equations describe a periodic 1D system of spinless point bosons [34]. Gaudin wrote them in the form [4, 35]

[TABLE]

where $N$ is the number of bosons, $L$ is the size of the system, and $n_{i}=0,\pm 1,\pm 2,\ldots$. In the literature, the point bosons are usually described by the Lieb–Liniger equations in the Yang–Yang’s form [3]:

[TABLE]

The equations (1) and (2) are equivalent [4, 35]: the formula $\arctan{\alpha}=(\pi/2)sgn(\alpha)-\arctan{(1/\alpha)}$ allows one to rewrite Eqs. (2) in the form (1). In this case,

[TABLE]

The ground state of the system corresponds to the quantum numbers $\{I_{i}\}=(1-\frac{N+1}{2},2-\frac{N+1}{2},\ldots,N-\frac{N+1}{2})$, the particle-like excitation with the momentum $p=2\pi j/L$ corresponds to $\{I_{i}\}=(1-\frac{N+1}{2},\ldots,N-1-\frac{N+1}{2},N-\frac{N+1}{2}+j)$, and a hole with the momentum $p=2\pi l/L$ ($l>0$) corresponds to the quantum numbers $I_{i\leq N-l}=i-\frac{N+1}{2}$, $I_{i>N-l}=1+i-\frac{N+1}{2}$. In the language of Eqs. (1), those states correspond to the following collections of quantum numbers $\{n_{i}\}=(n_{1},\ldots,n_{N})$: $(0,\ldots,0)$, $(0,\ldots,0,j)$, and $(0,\ldots,0,1,\ldots,1)$, where $1$ is repeated $l$ times. In this case, the state $(0,\ldots,0,1)$ is particular: it can be considered as a particle and as a hole. In the last case, any state $(n_{1},\ldots,n_{N})$ can be considered as a collection of interacting holes. If the state $(0,\ldots,0,1)$ is a particle, then any state can be considered as a collection of interacting particles. Therefore, the physical nature of the state $(0,\ldots,0,1)$ is the key point. From physical reasonings, we may expect that the state $(0,\ldots,0,1)$ corresponds to a phonon with the wavelength $\lambda=L$ (indeed, if the state $(0,\ldots,0,1)$ would correspond to a hole, then the phonon with $\lambda=L$ would be absent in the system, which is strange). In this case, each state $(n_{1},\ldots,n_{N})$ can be considered as a collection of interacting phonons. In particular, the state $(0,\ldots,0,j)$ should correspond to one phonon with the momentum $p=2\pi j/L$. As for the state with $n_{j\leq N-l}=0,n_{j\geq N-l+1}=1,$ it should correspond to $l$ interacting phonons, each of them has the wavelength $\lambda=L$ and the momentum $2\pi/L$. However, according to the Lieb’s classification [1], the state with the quantum numbers $n_{j\leq N-l}=0,n_{j\geq N-l+1}=1$ corresponds to a hole with the momentum $p=2\pi l/L>0$. Therefore, such hole should coincide with $l$ interacting phonons, each of them has the momentum $2\pi/L$. This possibility is also seen from the analysis by Lieb [1].

To ascertain the nature of a hole, it is necessary to study the structure of $N$-boson wave functions of a hole and a particle. In what follows, we will prove for a weak coupling that the state $(0,\ldots,0,1)$ corresponds to a phonon, and the hole with the momentum $p=4\pi/L$ coincide with two interacting phonons $(0,\ldots,0,1)$. We will also show that, at a strong coupling, the structure of quasiparticles is more unusual.

2 Phonon with the quantum numbers $\{n_{i}\}=(0,\ldots,0,1)$.

One can investigate the structure of wave functions of a “particle” and a hole in two ways: based on the wave functions of point bosons [4, 34] or on the wave functions of nonpoint bosons (i.e., bosons with nonzero interaction radius) [14, 15, 16, 22, 26, 36, 37, 38, 39], by passing to a point potential in the last case. Let us consider the second way.

Consider a periodic system of $N$ bosons with interatomic potential of the general form $U(\textbf{r}_{j}-\textbf{r}_{l})$. The dimensionality can be equal to $1$, $2$, or $3$. The ground state of a gas is described by the wave function [39]

[TABLE]

[TABLE]

and the wave function of a one-phonon state reads [22]

[TABLE]

[TABLE]

Here, $N$ is the total number of atoms, $\textbf{r}_{j}$ are the coordinates of atoms, $A_{0}$ and $A_{\textbf{p}}$ are the normalization constants, $\rho_{\textbf{q}}$ are the collective variables

[TABLE]

and all wave vectors $\textbf{q}_{l}$, $\textbf{p}_{l}$, p are quantized in the 3D case by the rule

[TABLE]

where $j_{x},j_{y},j_{z}$ are integers, and $L_{x},L_{y},L_{z}$ are the sizes of the system. Equations (5), (7) are exact. Note that, in [22, 39] series (5), (7) tend to infinity (i.e., the sums with $a_{j},b_{j}$ for $j=N+1,\ldots,\infty,$ are taken into account). We will see in Appendix 1 that these series must break down according to (5), (7).

The approximate solutions for the functions $\Psi_{0}$ and $\Psi_{\textbf{p}}$ were obtained by Feynman [15, 16], Bogolyubov and Zubarev [14], and Jastrow [36]. Then these methods were developed in a lot of works (see [19, 20, 21, 22, 23, 24, 25, 26, 37, 38, 39] and reviews [27, 28]). We will base on the collective variables method by Vakarchuk and Yukhnovskii [22, 39]. It allows one to get two exact chains of equations for the functions $a_{j}$ and $b_{j}$ at $N=\infty$. The first equations from those chains are given in Appendix 2.

The wave function (6), (7) with small $|p|$ can be considered as the definition of a phonon (here, $b_{j}\sim N^{0}$ for all $j$). Basic is the zero approximation $\psi_{\textbf{p}}=b_{1}\rho_{-\textbf{p}}$; the corrections can be found from the Schrödinger equation. Such solution for a phonon was studied theoretically in many works, starting from [15, 14, 17, 22, 40], and the results agree with experiments. The properties of collective variables [39] imply that function (6), (7) describes also $l$ interacting phonons with the total momentum $\textbf{p}=\textbf{p}_{1}+\ldots+\textbf{p}_{l}$, if we make the following changes in (6), (7): $\Psi_{\textbf{p}}\rightarrow\Psi_{\textbf{p}_{1}\ldots\textbf{p}_{l}}$, $\psi_{\textbf{p}}\rightarrow\psi_{\textbf{p}_{1}\ldots\textbf{p}_{l}}$, $A_{\textbf{p}}\rightarrow A_{\textbf{p}_{1}\ldots\textbf{p}_{l}}$, $b_{j}(\textbf{q}_{1},\ldots,\textbf{q}_{j-1};\textbf{p})\rightarrow b_{j}(\textbf{q}_{1},\ldots,\textbf{q}_{j-1};\textbf{p}_{1},\ldots,\textbf{p}_{l},N)$ for all $j$ (now, $b_{j}$ depend on $N$, generally speaking). We can verify that, in this case, $-i\hbar\sum\limits_{j}\frac{\partial}{\partial\textbf{r}_{j}}\Psi_{\textbf{p}_{1}\ldots\textbf{p}_{l}}=\hbar\textbf{p}\Psi_{\textbf{p}_{1}\ldots\textbf{p}_{l}}$.

For the weak coupling ($\gamma\ll 1$), we can set $a_{j\geq 3}=0$, $b_{j\geq 2}=0$ (it is the zero approximation; here, $\gamma=\rho^{1/3}cm/\hbar^{2}$ (for 3D), $cm/\hbar^{2}$ (2D), $2cm/(\rho\hbar^{2})$ (1D), $\rho=N/V$ is the particle number density, and $c=\nu(0)/2$, see (12)). The coefficient $b_{1}(\textbf{p})$ is considered to be normalizing: we set $b_{1}(\textbf{p})=1$. Then the equations in Appendix 2 yield [22, 39]

[TABLE]

[TABLE]

[TABLE]

We have obtained the Bogolyubov dispersion law. Formula (64) from Appendix 2 gives the known Bogolyubov solution for the ground-state energy $E_{0}$ [13].

In the zero approximation the sound velocity is $v_{s}=\sqrt{\frac{\rho\nu(0)}{m}}\equiv v_{s}^{(0)}$. In the next approximation the solution is as follows [22]:

[TABLE]

For a 1D system the energy of a phonon with the momentum $\hbar p_{1}=\hbar 2\pi/L$ is $E(p_{1})=\hbar p_{1}v_{s}$. In this case, for a finite system we should set $v_{s}^{(0)}=\sqrt{\frac{\rho\nu(p_{1})}{m}+\frac{\hbar^{2}p_{1}^{2}}{4m^{2}}}$.

Consider a finite 1D system of point bosons ($U(r)=2c\delta(r)$, $\nu(p)=2c$) and set $\hbar=2m=1$, $\gamma=c/\rho$. The above-presented formulae give the energy of a phonon with the momentum $p_{1}=2\pi/L$:

[TABLE]

[TABLE]

These formulae are valid for $N^{-2}\ll\gamma\ll 1$.

Our task is to clarify the nature of the particle $(0,\ldots,0,1)$. It is known that the energy $E_{L}(p)$ of Lieb’s particle for small $p$ is close to the Bogolyubov energy $E_{B}(p)$ (11). The small deviation of the particle energy from $E_{B}(p)$ contains the information about the nature of the particle. Let us represent the energy of the particle with the momentum $p_{1}=2\pi/L$ in the form (14):

[TABLE]

The energy and momentum of the particle is given by the known formulae

[TABLE]

[TABLE]

In our case, the collections $\{\acute{k}_{i}\}$ and $\{k_{i}\}$ are solutions of the Gaudin’s equations (1) for a state with one particle ($\{\acute{n}_{i}\}=(0,\ldots,0,1)$) and for the ground state ($\{n_{i}\}=(0,\ldots,0,0)$), respectively. The quasimomenta $\{\acute{k}_{i}\}$ and $\{k_{i}\}$ can be obtained numerically from Eqs. (1) by the Newton method (the Yang–Yang’s equations (2) give the same solution).

It is seen from Fig. 1 that the small quantity $\delta_{sL}$ obtained from Eqs. (16)-(18), (1) coincides with high accuracy with $\delta_{s}$ (15). The difference of $\delta_{sL}$ and $\delta_{s}$ is about $1\%$ for $\gamma=0.0001$–$0.1$. Since the function $\psi_{\textbf{p}}=\rho_{-\textbf{p}}$ for small $p$ describes a phonon, we conclude that Lieb’s particle $\{n_{i}\}=(0,\ldots,0,1)$ is a phonon. In this case, the Gaudin’s equations (1) imply that the hole with the momentum $p=2\pi l/L$ ($l>1$) should coincide with $l$ interacting phonons with the momentum $2\pi/L$. Let us verify this directly for $l=2$.

3 Two interacting phonons vs a hole with the quantum numbers $\{n_{i}\}=(0,\ldots,0,1,1)$.

In the language of the Lieb–Liniger equations (2), the hole with the momentum $p=4\pi/L$ is characterized by the quantum numbers $\{I_{i}\}=(-\frac{N-1}{2},-\frac{N-3}{2},\ldots,\frac{N-5}{2},1+\frac{N-3}{2},1+\frac{N-1}{2})$. In the language of the Lieb–Liniger equations in the Gaudin’s form (1), such hole is described by the quantum numbers $\{n_{i}\}=(0,\ldots,0,1,1)$. In the previous section we proved that the state $\{n_{i}\}=(0,\ldots,0,1)$ describes a phonon with the momentum $p=2\pi/L$. The state $\{n_{i}\}=(0,\ldots,0,1,0)$ is equivalent to $\{n_{i}\}=(0,\ldots,0,1)$. Therefore, it is obvious that the state $\{n_{i}\}=(0,\ldots,0,1,1)$ is two interacting phonons with the momentum $p=2\pi/L$. We now verify this assumption independently, by using the collective variables method.

Consider a Bose gas with weak coupling and dimensionality of $1,2,$ or $3$. Let us find the wave function and the energy of two interacting phonons with wave vectors $\textbf{p}_{1}$ and $\textbf{p}_{2}$. Feynman noticed that the energy of interaction ($\delta E$) of two phonons should be by $\sim N$ times less than the energy of one phonon [15]. However, the solutions for a wave function and $\delta E$ were not found.

The ground state is described by the wave function (4), (5) satisfying the Schrödinger equation

[TABLE]

with energy $E=E_{0}$. The equations for $E_{0}$ and the functions $a_{j}$ from (5) are given in Appendix 2. If the system contains one phonon, then the wave function is $\psi_{\textbf{p}}\Psi_{0}$ with $\psi_{\textbf{p}}$ (7), and the solutions for the functions $b_{j}$ and the energy of a quasiparticle are given in the previous section. If two phonons with wave vectors $\textbf{p}_{1}$ and $\textbf{p}_{2}$ are present, then the system is described by the wave function $\psi_{\textbf{p}_{1}\textbf{p}_{2}}\Psi_{0}$. We substitute this function in the Schrödinger equation and take into account that $\Psi_{0}=Ae^{S}$ satisfies this equation with energy $E_{0}$. As a result, we obtain the equation for the function $\psi_{\textbf{p}_{1}\textbf{p}_{2}}$:

[TABLE]

where $E_{\textbf{p}_{1}\textbf{p}_{2}}=E-E_{0}$ is the energy of two interacting phonons. Since the interaction of two phonons should be weak, we seek $\psi_{\textbf{p}_{1}\textbf{p}_{2}}$ in the form

[TABLE]

where $\psi_{\textbf{p}_{1}}$ and $\psi_{\textbf{p}_{2}}$ are one-phonon solutions. We substitute $\psi_{\textbf{p}_{1}\textbf{p}_{2}}$ (21) in Eq. (20) and take into account that the one-phonon functions $\psi_{\textbf{p}_{1}}$ and $\psi_{\textbf{p}_{2}}$ satisfy Eq. (20) with the energies $E(\textbf{p}_{1})$ and $E(\textbf{p}_{2}),$ respectively. In this way we get the following equation for $\delta\psi_{\textbf{p}_{1}\textbf{p}_{2}}$:

[TABLE]

[TABLE]

Here, the energy $E_{\textbf{p}_{1}\textbf{p}_{2}}$ of two interacting phonons is represented as a sum of the energies $E(\textbf{p}_{1})$ and $E(\textbf{p}_{2})$ of free phonons and the correction $\delta E$.

The solution for $\delta\psi_{\textbf{p}_{1}\textbf{p}_{2}}$ should have the form $\psi_{\textbf{p}}$ (7) with $\textbf{p}=\textbf{p}_{1}+\textbf{p}_{2}$, since formula (7) describes the state with any number of quasiparticles possessing the total momentum $\hbar\textbf{p}$:

[TABLE]

where $\textbf{p}=\textbf{p}_{1}+\textbf{p}_{2}$. We substitute $\delta\psi_{\textbf{p}_{1}\textbf{p}_{2}}$ (24) in (22). The result is reduced to the form

[TABLE]

($\textbf{p}=\textbf{p}_{1}+\textbf{p}_{2}$). Since $\rho_{-\textbf{p}}$, $\rho_{\textbf{q}}\rho_{-\textbf{q}-\textbf{p}}$, $\rho_{\textbf{q}_{1}}\rho_{\textbf{q}_{2}}\rho_{-\textbf{q}_{1}-\textbf{q}_{2}-\textbf{p}},\ldots$ are independent functions of the variables $\textbf{r}_{1},\ldots,\textbf{r}_{N}$ [39], Eq. (25) is equivalent to the system of $N$ equations

[TABLE]

For the weak coupling, it is sufficient to consider the equations $C_{1}=0$ and $C_{2}=0$. They have the form

[TABLE]

[TABLE]

where $\textbf{p}=\textbf{p}_{1}+\textbf{p}_{2}$, $E_{1}(\textbf{q})=\frac{\hbar^{2}q^{2}}{2m}(1-2a_{2}(\textbf{q}))$, and $\delta_{\textbf{q},-\textbf{p}}$ is the Kronecker delta. In this case, $B_{2}(\textbf{q};\textbf{p}_{1},\textbf{p}_{2})=B_{2}(-\textbf{q}-\textbf{p};\textbf{p}_{1},\textbf{p}_{2})$.

Let us present the functions $\psi_{\textbf{p}_{1}}$, $\psi_{\textbf{p}_{2}},$ and $\delta\psi_{\textbf{p}_{1}\textbf{p}_{2}}$ in (21) in the form of expansions (7) and (24). Then the “leading” term in the expansion of $\psi_{\textbf{p}_{1}\textbf{p}_{2}}$ is $A\rho_{-\textbf{p}_{1}}\rho_{-\textbf{p}_{2}}$. Let us write the functions $\psi_{\textbf{p}_{1}}$, $\psi_{\textbf{p}_{2}}$ in the form $b_{1}(\textbf{p}_{1})\tilde{\psi}_{\textbf{p}_{1}}$, $b_{1}(\textbf{p}_{2})\tilde{\psi}_{\textbf{p}_{2}}$. Then we present $\psi_{\textbf{p}_{1}}\psi_{\textbf{p}_{2}}$ as a series, where the first term is $b_{1}(\textbf{p}_{1})b_{1}(\textbf{p}_{2})\rho_{-\textbf{p}_{1}}\rho_{-\textbf{p}_{2}}$. The corresponding terms in the expansion of $\delta\psi_{\textbf{p}_{1}\textbf{p}_{2}}$ (24) have the form $\frac{B_{2}(-\textbf{p}_{1};\textbf{p}_{1},\textbf{p}_{2})+B_{2}(-\textbf{p}_{2};\textbf{p}_{1},\textbf{p}_{2})}{2N^{1/2}}\rho_{-\textbf{p}_{1}}\rho_{-\textbf{p}_{2}}$. Eventually, the coefficient of $\rho_{-\textbf{p}_{1}}\rho_{-\textbf{p}_{2}}$ in the expansion of the function $\psi_{\textbf{p}_{1}\textbf{p}_{2}}$ (21) is $A=b_{1}(\textbf{p}_{1})b_{1}(\textbf{p}_{2})+\frac{B_{2}(-\textbf{p}_{1};\textbf{p}_{1},\textbf{p}_{2})+B_{2}(-\textbf{p}_{2};\textbf{p}_{1},\textbf{p}_{2})}{2N}$. Let us represent the function $\psi_{\textbf{p}_{1}\textbf{p}_{2}}$ (21) in the form $\psi_{\textbf{p}_{1}\textbf{p}_{2}}=A\tilde{\psi}_{\textbf{p}_{1}\textbf{p}_{2}}$, where $\tilde{\psi}_{\textbf{p}_{1}\textbf{p}_{2}}=\frac{b_{1}(\textbf{p}_{1})b_{1}(\textbf{p}_{2})}{A}\tilde{\psi}_{\textbf{p}_{1}}\tilde{\psi}_{\textbf{p}_{2}}+\frac{\delta\psi_{\textbf{p}_{1}\textbf{p}_{2}}}{A\sqrt{N}}$. Since the interaction of phonons is very weak, the term $\frac{B_{2}(-\textbf{p}_{1};\textbf{p}_{1},\textbf{p}_{2})+B_{2}(-\textbf{p}_{2};\textbf{p}_{1},\textbf{p}_{2})}{2N}$ in $A$ should be less than $b_{1}(\textbf{p}_{1})b_{1}(\textbf{p}_{2})$ by $\sqrt{N}$ or even $N$ times. Therefore, $\frac{b_{1}(\textbf{p}_{1})b_{1}(\textbf{p}_{2})}{A}\approx 1$. As a result, $\tilde{\psi}_{\textbf{p}_{1}\textbf{p}_{2}}=\tilde{\psi}_{\textbf{p}_{1}}\tilde{\psi}_{\textbf{p}_{2}}+\frac{\delta\psi_{\textbf{p}_{1}\textbf{p}_{2}}}{A\sqrt{N}}$. Here, $\tilde{\psi}_{\textbf{p}}$ is a one-phonon function (7) with $b_{1}=1$. In this case, $b_{j\geq 2}$ satisfy the equations from Appendix 2, in which $b_{1}=1$. Represent the term $\delta\psi_{\textbf{p}_{1}\textbf{p}_{2}}/A$ in the form (24). Then we consider the factor $A$ to be normalizing and include it in $A_{\textbf{p}}$ (see (6)). Such transformations lead to the necessity to set $b_{1}(\textbf{p}_{1})=b_{1}(\textbf{p}_{2})=1$ and $B_{2}(-\textbf{p}_{1};\textbf{p}_{1},\textbf{p}_{2})=B_{2}(-\textbf{p}_{2};\textbf{p}_{1},\textbf{p}_{2})=0$ in Eqs. (27), (3) and the equations of Appendix 2.

We consider the coupling to be weak: $\gamma\ll 1$, but $\gamma\gg N^{-2}$ (the latter is necessary for the linearity of the dispersion law at small $p$). In this case, we can seek $\delta E$ and $\delta\psi_{\textbf{p}_{1}\textbf{p}_{2}}$ in the zero approximation. This means [22, 39] that all sums in the chain of equations for $B_{j}$ and $\delta E$ should be neglected. As a result, Eq. (27) takes the form

[TABLE]

Let us set in (3) $\textbf{q}=-\textbf{p}_{1}$. Then Eq. (3) reads

[TABLE]

Equation (3) for $\textbf{q}=-\textbf{p}_{2}$ is also reduced to (30) (to sight this, one needs to consider the relations $a_{2}(-\textbf{p})=a_{2}(\textbf{p})$, $a_{3}(\textbf{p},-\textbf{p}_{1})=a_{3}(\textbf{p},\textbf{p}_{1}-\textbf{p})$ and $b_{3}(\textbf{p}_{1},\textbf{p}_{2};\textbf{p}_{3})=b_{3}(-\textbf{p}_{1}-\textbf{p}_{2}-\textbf{p}_{3},\textbf{p}_{2};\textbf{p}_{3})$). Equations (29), (30) allow us to find $B_{1}(\textbf{p}_{1},\textbf{p}_{2})$ and $\delta E$. From Eq. (3) at $\textbf{q}\neq-\textbf{p}_{1},-\textbf{p}_{2}$ we can determine $B_{2}(\textbf{q};\textbf{p}_{1},\textbf{p}_{2})$.

Consider the case $\textbf{p}_{2}=\textbf{p}_{1}$. According to (7), q in $b_{2}(\textbf{q};\textbf{p})$ must be nonzero. Therefore, if (30) includes the term $b_{2}(0;\textbf{p})$, this term should be dropped. Then relations (29), (30) yield

[TABLE]

[TABLE]

Equations (31) and (32) give a square equation for $\delta E$ with the roots

[TABLE]

where

[TABLE]

At $p_{1}\rightarrow 0$ and $\gamma\ll 1,$ the formulae in Appendix 2 yield

[TABLE]

Using the relation $a_{4}(-\textbf{p}_{1},\textbf{p}_{1},\textbf{p}_{1})\approx 7a_{2}(\textbf{p}_{1})/16$ [22], we get $b_{3}(-\textbf{p}_{1},\textbf{p}_{1};\textbf{p}_{1})\approx-5/32$. Therefore, relations (33), (34) are reduced to

[TABLE]

[TABLE]

where $E(p_{1})=\hbar p_{1}v_{s}$, $a_{2}(\textbf{p}_{1})\approx-\alpha_{\textbf{p}_{1}}/2\approx-\frac{\sqrt{m\rho\nu(p_{1})}}{\hbar p_{1}}$, and $E_{1}(2p_{1})$, $v_{s}$ are determined by formulae (11), (13). At N\gg 1,\gamma\ \lower-1.2pt\vbox{\hbox{\hbox to0.0pt{<\hss}\lower 5.0pt\vbox{\hbox{\sim}}}}\ N^{-1}, the corrections $\frac{9}{2N}\left(\frac{\hbar^{2}p_{1}^{2}}{2m}\right)^{2}a_{2}(\textbf{p}_{1})$ and $\frac{5}{32N}\frac{\hbar^{2}p_{1}^{2}}{2m}$ in (36), (37) are negligible, and solutions (36), (37) take the simple form

[TABLE]

[TABLE]

Since $\delta E_{+}>\delta E_{-}$, namely the solution $\delta E_{-}$ should be realized in Nature. Thus, we have found the energy of interaction, $\delta E$, of two phonons with the same momentum $\hbar p_{1}$ at $p_{1}\rightarrow 0$ and weak coupling ($N^{-2}\ll\gamma\ll 1$). This result is new.

At the considered parameters of the system we have $|\tilde{E}|\sim\frac{\hbar^{2}p_{1}^{2}}{2m}$. Therefore, $\delta E_{-}\sim-E(p_{1})/N$. In this case, relations (32), (35) yield $B_{1}(\textbf{p}_{1},\textbf{p}_{1})\sim-1$. It is natural to expect that $|B_{1}(\textbf{p}_{1},\textbf{p}_{2})|\sim 1$ also at $\textbf{p}_{2}\neq\textbf{p}_{1}$. In this case, Eq. (3) yields $|B_{2}(\textbf{q};\textbf{p}_{1},\textbf{p}_{2})|\sim 1$. That is, the term $\delta\psi_{\textbf{p}_{1}\textbf{p}_{2}}/\sqrt{N}$ in formula (21) is less than the main term $\psi_{\textbf{p}_{1}}\psi_{\textbf{p}_{2}}$ by $\sim N$ times. These estimates show that the interaction of two phonons is indeed very weak.

Let us return to the question about the nature of a hole. In the above equations, we pass to a 1D point potential. Compare $\delta E_{-}$ with the quantity

[TABLE]

equal to the difference of the energy of a hole with the quantum numbers $\{I_{i}\}=(-\frac{N-1}{2},-\frac{N-3}{2},\ldots,\frac{N-5}{2},1+\frac{N-3}{2},1+\frac{N-1}{2})$ and two energies of a free “particle” (phonon) with the quantum numbers $\{I_{i}\}=(-\frac{N-1}{2},-\frac{N-3}{2},\ldots,\frac{N-5}{2},\frac{N-3}{2},1+\frac{N-1}{2})$. The quantities $p=4\pi/L$ and $p=2\pi/L$ in (40) are momenta. The values of $E_{h}(p=4\pi/L)$ and $E_{p}(p=2\pi/L)$ can be found numerically from the Yang–Yang’s equations (2) and formulae (17), (18). The value of $\delta E_{-}$ follows from Eqs. (36) and (37), where we set $\nu(p)=2c$, $\hbar=2m=1$, $c/\rho=\gamma$, and $p_{1}=2\pi/L$.

It is seen from Fig. 2 that the energy of interaction of two phonons ($\delta E_{-}$) is close to $\delta E_{h}$, if N^{-2}\ll\gamma\ \lower-1.2pt\vbox{\hbox{\hbox to0.0pt{<\hss}\lower 5.0pt\vbox{\hbox{\sim}}}}\ N^{-1}. The very small value of $\delta E_{h}$ is an indicator of the nature of a hole. The closeness of the values of $\delta E_{-}$ and $\delta E_{h}$ proves that the hole $\{I_{i}\}=(-\frac{N-1}{2},-\frac{N-3}{2},\ldots,\frac{N-5}{2},1+\frac{N-3}{2},1+\frac{N-1}{2})$ coincides with two interacting phonons, each characterized by the collection $\{I_{i}\}=(-\frac{N-1}{2},-\frac{N-3}{2},\ldots,\frac{N-5}{2},\frac{N-3}{2},1+\frac{N-1}{2})$.

In the region $N^{-1}\ll\gamma\ll 1$ the quantities $\delta E_{-}$ and $\delta E_{h}$ are considerably different, since we found a solution for $\delta E_{-}$ only in zero approximation. The error of the numerical calculation of $\delta E_{h}$ should also be significant in this case.

We note that, to obtain namely a two-phonon solution, it is necessary firstly to set the orders of the quantities $B_{j}$ and $\delta E$. Otherwise, we can arrive at another solution, since the function $\psi_{\textbf{p}_{1}\textbf{p}_{2}}$ (21), (24) can describe any excited state with the total momentum $\hbar(\textbf{p}_{1}+\textbf{p}_{2})$ (see Appendix 1). We took the two-phonon nature of the state into account with the help of the condition $|[B_{2}(-\textbf{p}_{1};\textbf{p}_{1},\textbf{p}_{2})+B_{2}(-\textbf{p}_{2};\textbf{p}_{1},\textbf{p}_{2})]/(2N)|\ll|b_{1}(\textbf{p}_{1})b_{1}(\textbf{p}_{2})|$.

The above two-phonon solution should be contained in Eqs. (64)–(10) of Appendix 2, since any (not only one-phonon) excited state of the system with the total momentum p is described by the function $\psi_{\textbf{p}}\Psi_{0}$ (6), (7) (see Sect. 2). In other words, the Vakarchuk–Yukhnovskii’s equations (see Appendix 2) contain solutions for all excited state of a Bose gas.

We have shown that, at $j=1,2$ the hole with the momentum $j2\pi/L$ is $j$ interacting phonons $(0,\ldots,0,1)$. Let, for some $j=J>2,$ this assertion be wrong. Since the solution of the Lieb–Liniger equations is unique [5], the system of point bosons would not contain the state with $J$ interacting phonons $(0,\ldots,0,1)$. This is strange from the physical point of view and would lead to the violation of the continuous transition from solutions for nonpoint bosons to solutions for point ones. However, such transition should exist [41]. Hence, for all $j=1,\ldots,N,$ the hole with the momentum $j2\pi/L$ is $j$ interacting phonons $(0,\ldots,0,1)$.

4 Additional arguments.

Consider a 1D Bose gas with point interaction. Let us find the limit $c\rightarrow 0$ for the Lieb–Liniger solutions [4, 34]

[TABLE]

[TABLE]

For the state $\{n_{i}\}=(0,\ldots,0,1)$, at $c\rightarrow 0$ we get $\{k_{i}\}=(0,\ldots,0,2\pi/L)$. Relations (41) and (42) yield $a(P)=1$ and

[TABLE]

where $k_{N}=2\pi/L$. For the state $\{n_{i}\}=(0,\ldots,0,1,1),$ we get $\{k_{i}\}\approx(0,\ldots,0,2\pi/L,2\pi/L)$. Then relations (41), (42) yield $a(P)=1$ and

[TABLE]

Here, while calculating $a(P),$ we take into account that $(k_{N}-k_{N-1})|_{c\rightarrow 0}\sim c^{1/2}$. Functions (43) and (44) coincide with the wave functions of a system of free bosons, in which one or two (respectively) atoms have the momentum $2\pi/L$. The normalizing coefficients are $c_{1}=L^{-N/2}$, $c_{11}=\sqrt{\frac{N}{N-1}}c_{1}$ [39]. Since $|\rho_{-k_{N}}|\sim 1$ for the overwhelming majority of configurations $(x_{1},\ldots,x_{N})$, the comparison of $\psi_{11}$ (44) and $\psi_{1}$ (43) shows that in the limit $c\rightarrow 0$ the hole $(0,\ldots,0,1,1)$ is two interacting particles $(0,\ldots,0,1)$, which agrees with the result of the previous section. At $c=0$ we have, of course, free atoms instead of quasiparticles.

The one-phonon and two-phonon solutions (6) and (21) pass at $c=0$ to solutions (43) and (44). To demonstrate this with the formulae in Sections 2 and 3, we take the relations $a_{j}=0$, $b_{j\geq 2}=0$, $B_{j\geq 2}=0$, and $\delta E=0$ into account. Relation (31) yields $B_{1}(p_{1},p_{1})=-1$. Thus, Eqs. (6), (7), and (21) describe free bosons at the zero interaction and phonons at a nonzero one (if the interaction is switched-on, the functions $\psi_{\textbf{p}_{1}}$, $\psi_{\textbf{p}_{1}\textbf{p}_{2}}$ vary negligibly, but the dispersion law $E(p)\sim p^{2}$ transits into $E(p)\approx v_{s}p$ due to a change of $\Psi_{0}$).

It is clear that any Lieb–Liniger solution (41) can be presented in the form (6), (7). It would be of interest to get solutions (6), (7), and (21) directly from (41) at $c\neq 0$. This is a task for the future.

Both in the Gaudin’s numbering and in the collective variables method, each excited state of a 1D system is described by the collection of quantum numbers $\{n_{i}\}\equiv(n_{1},n_{2},\ldots,n_{N})$ corresponding to the collection of quasiparticles with the momenta $p_{1},\ldots,p_{N}$, where $p_{j}=2\pi n_{j}/L$. That is, there is one-to-one correspondence between solutions in the collective variables method at $\nu(p)=2c$ and solutions in the Lieb–Liniger approach. In this case, the uniqueness of a solution for each collection $\{n_{i}\}$ was proved only for the Lieb–Liniger approach [5].

The calculation of the statistical sum of a 1D system of point bosons at $N=\infty$ gives [42]

[TABLE]

where $E_{p}(p_{l})$ is the dispersion law of particles. The calculation [42] involves all states of the system (including the ground state, particles, and holes). Formula (45) is exact at $N=\infty$ and $T\rightarrow 0$. Equation (45) is the known formula for the free energy of an ensemble of noninteracting Bose quasiparticles. The verification [33] indicates that formula (45) and the Yang–Yang approach [3] lead to identical thermodynamic solutions $F,S$. If we consider formally the state $\{n_{i}\}=(0,\ldots,0,1)$ as a hole, then any excited state $(n_{1},\ldots,n_{N})$ can be approximately considered as a collection of noninteracting holes. This leads again to formula (45) with the replacement of $E_{p}(p)$ by the dispersion law of holes $E_{h}(p)$. The analysis of the present work shows that such dualism of holes and particles is, apparently, physical at a strong coupling, but is illusory at the weak coupling.

The analysis of Sections $1$–$4$ shows that, at the weak coupling, the hole is merely a collection of identical interacting phonons with the momentum $\pm 2\pi/L$. This corresponds to the Gaudin’s numbering (see Eq. (1)). Therefore, the introduction of quasiparticles with the help of Gaudin’s numbering [33, 42] is more physical, at least at the weak coupling. In this case, the curve of holes $E_{h}(p)$ describes the excited states with minimum energy for given $p$. The Yang–Yang’s numbering (see Eq. (2)) is also useful: using it, it is easy to find the energy of quasiparticles at a strong coupling.

We recall also the arguments by Feynman [15, 16, 17]. According to them, only the single dispersion law, corresponding to phonons, should be in the region of small $E,p$. Such conclusion is in agreement with our analysis.

5 Regime of infinitely strong repulsion and experiments

Above, we studied the regime of the weak coupling. We now consider the Tonks–Girardeau gas: $\gamma=+\infty$ (see reviews [6, 7] and experimental works [43, 44, 45]). This regime is the most unusual. In this case, the point bosons are impenetrable. Therefore, two bosons cannot stay at a single point, which is similar to fermions. As a result, the system of bosons acquires some fermionic properties [1, 46]. In particular, the energy levels coincide with those of a system of free fermions. It is interesting, since any perturbation of a system of interacting bosons is a collection of oscillatory modes. Hence, for $\gamma=+\infty$, the oscillatory modes reproduce exactly the energy levels of the excited system of free fermions.

For a 1D system of impenetrable bosons, Girardeau obtained the dispersion law [46]

[TABLE]

that is a limiting case of the Lieb’s dispersion law of “particles” [1]. The wave function (WF) of any state $(n_{1},\ldots,n_{N})$ can be represented as

[TABLE]

If $n_{j}\geq 0$ for any $j$, then $\psi_{p}$ is the Schur function $S_{\lambda}$ [47]:

[TABLE]

[TABLE]

Formulae (48) and (49) are equivalent. Here, $z_{q}=e^{ik_{1}x_{q}}$, $k_{1}=2\pi/L$, $h_{j}$ and $e_{j}$ are the $j$th complete symmetric function and the $j$th elementary symmetric function, respectively [47]:

[TABLE]

[TABLE]

where $j\geq 0$, $e_{0}=h_{0}=1$, $h_{1}=e_{1}$ (in all similar sums in this section, we sum over $s_{1},\ldots,s_{j}=1,\ldots,N$). The partitions $\lambda=(\lambda_{1},\lambda_{2},\ldots)$ and $\lambda^{\prime}=(\lambda^{\prime}_{1},\lambda^{\prime}_{2},\ldots)$ are defined in [47]. In particular, for the particle $(0,\ldots,0,j)$ we have $\lambda_{1}=j,\lambda_{i\geq 2}=0$, $\lambda^{\prime}_{1}=1,\ldots,\lambda^{\prime}_{j}=1,\lambda^{\prime}_{i>j}=0$. In this case, Eq. (48) leads to the Girardeau’s solution [46]

[TABLE]

For the hole $(0,\ldots,0,1,\ldots,1)$, where $1$ is repeated $j$ times, we have $\lambda_{1}=1,\ldots,\lambda_{j}=1,\lambda_{i>j}=0$, $\lambda^{\prime}_{1}=j,\lambda^{\prime}_{i\geq 2}=0$. Eq. (49) gives the solution

[TABLE]

This formula can be easily verified for $j=N$:

[TABLE]

The same solution follows from the Lieb–Liniger formulae for any $\gamma\geq 0$. This solution is also true for a nonpoint interatomic potential of the general form. Solution (54) describes the translational motion of a system with the velocity $v=\hbar 2\pi/(mL)$. Interestingly, the solution for $N$ interacting phonon-holes $(0,\ldots,0,1)$ coincides with (54).

We will understand the properties of a system better, if we will determine the structure of WFs for the lowest states. The WF of the particle $(0,\ldots,0,1)$ is [46]

[TABLE]

The wave function (55) corresponds to WF (6), (7) of a phonon with $b_{j\geq 2}=0$. It was mentioned in Introduction that the state $(0,\ldots,0,1)$ can be formally considered as a particle and as a hole. The analysis in Sect. 2 indicates that at $\gamma\ll 1$ the state $(0,\ldots,0,1)$ is a phonon. However, at $\gamma=\infty$ the energy levels of the system coincide with those of a system of free fermions. Therefore, the analogy with a hole becomes also physical. That is, at $\gamma=\infty$ the state $(0,\ldots,0,1)$ can be considered as a phonon-hole.

For the particle $(0,\ldots,0,j)$ with $j=2$ and $j=3$ we get, respectively,

[TABLE]

[TABLE]

For the holes with the momenta $p=4\pi/L$ and $p=6\pi/L,$ we obtain, respectively,

[TABLE]

[TABLE]

We did not verify the normalization for formulae (48)–(59). If $N\gg 1,$ we have $|\rho_{-k}|\sim 1$ for the vast majority of configurations $(x_{1},\ldots,x_{N})$. Therefore, at $N\gg 1$ the last term dominates in solutions (56), (57), (58), and (59), and the remaining terms are small corrections. We can conclude that the states $(0,\ldots,0,2)$ and $(0,\ldots,0,1,1)$ [formulae (56), (58)] correspond to two interacting quasiparticles $(0,\ldots,0,1)$ (55). In Sect. 3 we considered the two-phonon state with similar structure. The states $(0,\ldots,0,3)$ and $(0,\ldots,0,1,1,1)$ [formulae (57), (59)] correspond to three interacting quasiparticles $(0,\ldots,0,1)$.

To pass from the solutions $\psi_{p}$ (52), (53) with the momentum $p>0$ to solutions with $p<0$, it is sufficient to change $k_{i}\rightarrow-k_{i}$ for all $i$ in the Lieb–Liniger equations (1) written for a state with $p<0$. Therefore, the solution $\psi_{p}$ for a hole $(-1,\ldots,-1,0,\ldots,0)$ and for a particle $(-j,0,\ldots,0,0)$ can be found, by replacing $z_{s_{i}}\rightarrow 1/z_{s_{i}}$ for all $i=1,\ldots,j$ in Eqs. (50)–(59). We can also use the relation

[TABLE]

which follows from the representation of WF in the form of a determinant [46]; $\psi_{p_{max}}$ is given by (54). From whence, we get that the states $(-2,0,\ldots,0)$ and $(-1,-1,0,\ldots,0)$ correspond to two interacting quasiparticles $(-1,0,\ldots,0)$, and the states $(-3,0,\ldots,0)$, $(-1,-1,-1,0,\ldots,0)$ correspond to three interacting quasiparticles $(-1,0,\ldots,0)$. It is natural to expect that the particles and holes with higher momenta $|p|$ can also be considered as a collection of interacting quasiparticles $(0,\ldots,0,1)$ or $(-1,0,\ldots,0)$.

Such properties of particles are surprising, because, at the weak coupling, the particles with small $|p|$ correspond to phonons and are indivisible structures. However, at $\gamma=\infty$ all particles turn out to be composite structures, except for phonons $(0,\ldots,0,1)$ and $(-1,0,\ldots,0)$. This means that each state $(n_{1},\ldots,n_{N})$ can be, apparently, considered as a collection of interacting phonon-holes $(0,0,\ldots,0,1)$ and (or) $(-1,0,\ldots,0,0)$. In this case, in the Tonks–Girardeau gas there are only two primary indivisible excitations: the phonon-holes $(0,0,\ldots,0,1)$ and $(-1,0,\ldots,0,0)$. Such system is not characterized by any dispersion law. Therefore, the function $S(k=const,\omega)$ should not have a sharp peak. This agrees with the theory [48] according to which $S(k=const,\omega)=const$ for $\gamma=\infty$. The experiment [44] and the theory [49] testify to a widening of the peak of the function $S(k=const,\omega),$ as $\gamma$ increases.

In this case, at $\gamma=\infty,T\rightarrow 0$ particles and holes interact weakly between themselves and, therefore, are “good” quasiparticles (according to Landau’s arguments [50]). Using Eqs. (2) with $c\rightarrow\infty$ and Eqs. (17), (18), we find that the energy of interaction of two particles with the momentum $p=j2\pi/L>0$ is equal to $\frac{-2E_{p}}{N+j-1}$ (where $E_{p}$ is the energy of one particle with the momentum $j2\pi/L$), and the energy of interaction of two holes with the same momentum is $\frac{2E_{h}}{N-j+1}$ (where $E_{h}$ is the energy of a hole with $p=j2\pi/L$).

Interestingly, the thermodynamic velocity of sound coincides with the microscopic one found from Eq. (46): $v_{s}^{therm}=v_{s}^{mic}$ [46]. Therefore, Girardeau concluded that the low-lying excitations with energy (46) correspond to phonons [46]. However, the equality $v_{s}^{therm}=v_{s}^{mic}$ holds also for holes (because $E_{h}(p)=-\frac{\hbar^{2}p^{2}}{2m}+\frac{\hbar^{2}|p|\pi\rho}{m}\frac{N+1}{N}$ [1]). The curves for holes and particles coincide at the points $p=2\pi/L$ and $p=-2\pi/L$ corresponding to the states $(0,0,\ldots,0,1)$ and $(-1,0,\ldots,0,0)$. Therefore, the equality $v_{s}^{therm}=v_{s}^{mic}$ indicates, apparently, only that the excitations $(-1,0,\ldots,0,0)$ and $(0,0,\ldots,0,1)$ are phonons. The remaining excitations may not be single phonons.

Since the levels of a system of impenetrable bosons coincide with those of free fermions, the creation of a hole (particle) is equivalent to a change in the momentum of one atom by the value of the momentum of a hole (particle). In view of this, the scattering of an external atom on the system can be considered as the scattering on a single atom (but not as the creation of a set of phonon-holes $(0,\ldots,0,1)$). It is an individual process. The probability of such process is greater than that of multiple processes. The calculations [49, 51] show that at $\gamma\gg 1$ the peak of the function $S(k=const,\omega)$ should be located between the dispersion curves of particles and holes. This result agrees with the experiment [44].

At the weak coupling, the phonons are described by WF (6), (7) corresponding to a structureless object, and a hole is a collection of phonons $(0,\ldots,0,1)$. Therefore, the creation of a hole cannon be considered as a change in the momentum of a single atom, but it should be considered as a multiple process. The probability of such processes is very small. Due to this, the peak of $S(k,\omega)$ should be close to the dispersion curve of particles, which is in agreement with the theory [49, 51, 52, 53] and experiment [44].

We note that if the system contains two independent types of excitations, then the atom flying through the system can independently create excitations of both types. As a result, the function $S(k=const,\omega)$ should be characterized by two peaks. However, only one peak was observed in the experiment [44] (even at high $p$, for which $E_{h}(p)$ and $E_{p}(p)$ differ significantly from each other). This means that the system contains the elementary excitations of only one type.

In the experiment [54], the profile of $S(k=const,\omega)$ was measured for $N\simeq 30$ atoms in a $1D$ trap with $\gamma\approx 1$. The experiment agrees better with the theory based on the Bethe ansatz, than with the Bogolyubov theory [13, 14]. This is not surprising, because the Bogolyubov formulae work at $\gamma=1$ [1], only if $N$ is large: N\ \lower-1.2pt\vbox{\hbox{\hbox to0.0pt{>\hss}\lower 5.0pt\vbox{\hbox{\sim}}}}\ 100 and N\ \lower-1.2pt\vbox{\hbox{\hbox to0.0pt{>\hss}\lower 5.0pt\vbox{\hbox{\sim}}}}\ 1000 for periodic and zero boundary conditions, respectively [42, 55]. Moreover, the Bogolyubov approach does not account for the interaction of quasiparticles. However, this interaction is important for the many-quasiparticle states, which make a significant contribution to $S(k,\omega)$. In this case, the Bethe equations work at small $N$ and involve the interaction of quasiparticles. Nevertheless, the approaches by Lieb and Bogolyubov are equivalent at the weak coupling and $N\gg 1$, as shown in the present work.

Note also that the atoms were modeled in the experiments [44, 54] as point ones, though the real atoms have nonzero radii of interaction. The account for a finite size of atoms can explain qualitatively, for small $k$ and \gamma\ \lower-1.2pt\vbox{\hbox{\hbox to0.0pt{<\hss}\lower 5.0pt\vbox{\hbox{\sim}}}}\ 1, the experimental shift of a peak of $S(k,\omega),$ as $\gamma$ increases [44]. It can be seen qualitatively, by using the Bogolyubov formula $E_{B}(p)=\sqrt{\left(\frac{\hbar^{2}p^{2}}{2m}\right)^{2}+2\rho\nu(p)\frac{\hbar^{2}p^{2}}{2m}}$ for the point and nonpoint atoms. The consideration of nonpoint atoms will lead also to a shift of the Bogolyubov peak of $S(k=const,\omega)$ to lower $\omega$ for the experiment [54], which can improve the agreement with the experimental peak.

On the whole, the properties of the Tonks–Girardeau gas are strange and not visual.

6 Elementary excitations

The known physical laws are, in fact, the simplest ways to describe the complex connections in Nature. Therefore, it is natural to introduce elementary excitations [1] so that the description of the properties of systems in their language be the simplest. This implies that every state of the system must uniquely correspond to some collection of elementary excitations (and it should match the structure of wave functions). Moreover, the interaction between excitations should become weak at $T\rightarrow 0$ [50].

Any state of a system of point bosons corresponds uniquely to a collection of quasiparticles in two classifications: if each quasiparticle is considered as a collection of “particles” and if each quasiparticle is represented as a collection of “holes”. If the particles and holes are introduced jointly, then the ambiguity arises for the majority of states. For example, the state $(n_{1},\ldots,n_{N})=(0,\ldots,0,2,2)$ can be considered as two interacting phonons $(0,\ldots,0,0,2)$ or as two interacting holes $(0,\ldots,0,1,1)$.

1. Weak coupling. Consider the state $(0,\ldots,0,2)$. Since the states like $(0,\ldots,0,j)$ correspond to the Bogolyubov dispersion law, we identify them with phonons. In the “only particles” language, the state $(0,\ldots,0,2)$ is a particle. In the “only holes” language, the state $(0,\ldots,0,1)$ should be effectively considered as a hole, and the state $(0,\ldots,0,2)$ should be considered as two interacting holes $(0,\ldots,0,1)$. However, WF of the state $(0,\ldots,0,2)$ is known. It is function (6), (7) with $\textbf{p}=\textbf{i}_{x}4\pi/L$. If the state $(0,\ldots,0,2)$ would be two holes $(0,\ldots,0,1)$, then the solution for $(0,\ldots,0,2)$ would coincide with one of the two-phonon solutions found in Sect. 3. But this is not the case: the state $(0,\ldots,0,2)$ has a different energy, and WF (7) with $\textbf{p}=\textbf{i}_{x}4\pi/L$ corresponds to one phonon and is different by structure from the two-phonon WF $\psi_{\textbf{p}_{1}\textbf{p}_{1}}=\psi_{\textbf{p}_{1}}\psi_{\textbf{p}_{1}}+\frac{\delta\psi_{\textbf{p}_{1}\textbf{p}_{1}}}{\sqrt{N}}$ (21) with $\textbf{p}_{1}=\textbf{i}_{x}2\pi/L$.

This shows that, at the weak coupling, the particles $(0,\ldots,0,j)$ are indivisible elementary excitations. In this case, the language of holes can be used only formally, for example, in the calculation of a partition function (see Sect. 4) or the dynamical structural factor (DSF) $S(k,\omega)$.

It is worth to note that, in the hole approach, the statistics of holes turns out to be contradictory. The partition function leads to formula (45) with the replacement $E_{p}(p)\rightarrow E_{h}(p)$. Such formula corresponds to the Bose statistics. But WF has no corresponding symmetry. For example, we saw that WF of the state $(0,\ldots,0,2)$ cannot be presented as WF of two holes $(0,\ldots,0,1)$. Therefore, if the state $(0,\ldots,0,2)$ is formally considered as two holes $(0,\ldots,0,1)$, then these two holes cannot be permuted. Such discrepancy between the thermodynamic formulae and the symmetry of WFs testifies that the partition into quasiparticles is not consistent with the structure of WFs and, therefore, is not quite physical. In the particle-based approach, no such problem arises.

If we consider the particles and holes jointly, then, for each state $(n_{1},\ldots,n_{N}),$ we should indicate the rule, according to which this state is separated into holes and particles. In the phase space of numbers $(n_{1},\ldots,n_{N}),$ such rule sets the boundary between holes and particles. If the rule is formulated, then we can calculate the partition function and DSF and can indicate which contribution is given by holes or particles. If such rule is not set, then the contributions of holes and particles are unknown. In this case, holes and particles are not defined, and nothing can be said about their statistics. At the joint consideration of holes and particles the Bose statistics for quasiparticles should be violated (e.g., if we consider the state $(0,\ldots,0,1,1)$ as a hole, then the state with two phonons $(0,\ldots,0,1)$ is lost, which violates the properties of Bose quasiparticles). In this case, the statistics of quasiparticles can acquire the fermionic character. That is, establishing the boundary between holes and particles in the space of numbers $(n_{1},\ldots,n_{N})$ is a significant point. However, we did not see the articles, where such boundary is introduced at the joint consideration of holes and particles.

Some of the above mentioned properties and difficulties have already been discussed in work [1]. Taking into account the fermionic properties of a Bose system at $\gamma\gg 1$, Lieb conserved the symmetry between particles and holes at any $\gamma$ and considered them jointly [1]. But the above analysis shows that, at the weak coupling, this symmetry is broken to the favor of particles.

2. Strong coupling: $\gamma=+\infty$. By the analysis in the previous section, at $\gamma=+\infty$ each state can be, apparently, considered as a collection of interacting elementary excitations $(0,\ldots,0,1)$ and (or) $(-1,0,\ldots,0)$. In this case, it is worth talking about the bosonic or fermionic properties of these two indivisible excitations only. These are bosons. It follows from the fact that one state of the system can contain several such excitations, as is seen from solutions (56), (57), (58), (59). Moreover, the state $(-1,0,\ldots,0,1)$ corresponds to

[TABLE]

(To obtain this formula, one needs to extract the multiplier $z_{j}$ from $j$th row of the Slater determinant [46] and then to use Eq. (49) with $l_{2}=\lambda_{1}=2$, $\lambda^{\prime}_{1}=N-1$, $\lambda^{\prime}_{2}=1$). Function (61) describes two interacting quasiparticles: $(-1,0,\ldots,0,0)$ and $(0,0,\ldots,0,1)$. Formula (61) shows that $\psi_{p}$ is invariable under the permutation of the quasiparticles $(-1,0,\ldots,0,0)$ and $(0,0,\ldots,0,1)$, which corresponds to bosons.

At $\gamma=+\infty$ there is a symmetry between particles and holes. We can describe the system in the language of particles or in the language of holes.

7 A hole and a soliton.

The Lieb’s hole is a stationary solution of the $N$-body Schrödinger equation for a cyclic system: $\tilde{\Psi}(x_{1},\ldots,x_{N},t)=e^{-iE_{h}(p)t/\hbar}\Psi(x_{1},\ldots,x_{N})$. This solution is characterized by a constant density: $\rho(x,t)=const$ [56]. However, the quasiclassical dark soliton, as a solution of the 1D Gross–Pitaevskii equation, is a solitary running density wave of the form $\Psi(x,t)=\Psi(x-vt)$, $\rho(x,t)=\rho(x-vt)$ [29, 30]. In this case, the wave package of one-hole states shows the properties of an immovable soliton [56, 57, 58] (though the density profile $\rho(x,t)$ of such package spreads, as $t$ increases, in contrast to a quasiclassical soliton [29, 30]). Moreover, the conditional probability density $\rho_{N}(x)$ in the hole state coincides with the stationary dark soliton profile [59]. Note also that the analysis in [30] refers to an infinite noncyclic system. In this case, classical and quantum momentums of the soliton are different. The dispersion curves of solitons and holes are close at the weak coupling only in the classical definition of the soliton momentum [30]. If such properties hold for a cyclic system too, then a single hole is not a soliton (despite results in [59]), since the quantum definition of the momentum is primary. On the whole, the connection between a hole and a soliton is not quite clear yet [56, 57, 58, 59].

We have shown above for the weak coupling that the hole is a collection of identical interacting phonons with the momentum $p=2\pi/L$. Possibly, the collection of identical phonons with $p=4\pi/L$ (or $p=6\pi/L$, etc.) reveals also solitonic properties at the weak coupling. Most probably, a hole has solitonic properties only for high momenta: in this case, the hole consists of a large number of identical phonons, and the collective effect is possible. The solitonic properties of holes are interesting, it is worth studying them in more details. In our opinion, it is better to use zero boundary conditions, because $\rho(x,t)\neq const$ in this case, and the density wave is possible.

8 Conclusion

We have shown that, in the case of the weak coupling, the hole with the momentum $p=jp_{0}$ is a collection of $j$ identical interacting phonons with the momentum $p_{0}=\pm\hbar 2\pi/L$. In this case, the particles are elementary excitations, and the holes are composite ones. If $j\sim N$, the hole corresponds to the condensate of phonons. Thus, Lieb’s excitations quite agree with the Bogolyubov’s and Feynman’s solutions. The traditional point of view, according to which a holes are an independent type of excitations, has survived for so long since the Lieb–Liniger wave functions was not compared with the wave functions of a system of nonpoint bosons.

At a strong coupling, the system of interacting bosons partially acquires the fermionic properties. This is evidenced by the solutions obtained by Girardeau, Lieb, and subsequent authors. That is what is missing from the Bogolyubov’s and Feynman’s approaches. In this case, the holes and the particles become similar to holes and particles in a Fermi system. The structure of quasiparticles is very unusual at $\gamma=+\infty$: It is found above that the low-lying particles and holes are collections of identical phonon-holes with the momentum $p_{0}=\pm\hbar 2\pi/L$. Apparently, only these two quasiparticles are primary indivisible excitations in this case — phonon-holes with the momenta $\hbar 2\pi/L$ and $-\hbar 2\pi/L$.

The author thanks N. Iorgov for the valuable discussion and the anonymous referees for helpful comments. The present work is partially supported by the National Academy of Sciences of Ukraine (project No. 0116U003191).

9 Appendix 1. The largest number of quasiparticles.

Consider $N=10^{6}$ weakly interacting Bose atoms placed in a vessel. How many quasiparticles can exist in such a system? At first sight, the number of quasiparticles $N_{Q}$ should not be bounded from above, since a quasiparticle is similar to a wave in the probability field. However, it turns out that $N_{Q}\leq N$. This can be proved by two methods.

The most simple way is to use the Lieb–Liniger equations (1). In the Gaudin’s numbering, the creation of a quasiparticle is equivalent to a change in some $n_{j}$ from $n_{j}=0$ to $n_{j}=l\neq 0$. In this case, a Bogolyubov–Feynman quasiparticle with the momentum $p=2\pi l/L$ is created. The largest number of quasiparticles is equal to the number of $n$’s with different $j$: it is the number of equations in system (1), which is equal to the number of atoms $N$. In this case, a hole is several Bogolyubov–Feynman quasiparticles. These properties were noted in [33, 42].

For nonpoint bosons it is necessary to note that a wave function (6), (7) describes not only a state with one quasiparticle, but also the states with any number of quasiparticles. Indeed, the WF of any stationary excited state can be written in the form $f(\textbf{r}_{1},\ldots,\textbf{r}_{N})\Psi_{0}$. The periodic system has a definite momentum. The general form of the WF of a state with the total momentum $\hbar\textbf{p}$ is set by formulae (6), (7) (if the number of quasiparticles $\geq 2$, then it is necessary to make changes in (6), (7) as described in Sect. 2). Therefore, the function $f(\textbf{r}_{1},\ldots,\textbf{r}_{N})$ should coincide with $\psi_{\textbf{p}}$ (7). In this case, $b_{j}$ are different for different states. For the state with one phonon, $b_{j}\sim 1$ for all $j$. For a state with two phonons with the momenta $\hbar\textbf{p}_{1}$ and $\hbar\textbf{p}_{2}$ we should set $\textbf{p}=\textbf{p}_{1}+\textbf{p}_{2}$ in (6), (7). In this case, $b_{j\geq 3}\sim 1$, $b_{1}(\textbf{p}_{1},\textbf{p}_{2},N)\sim N^{-1/2}$, $b_{2}(\textbf{q}_{1};\textbf{p}_{1},\textbf{p}_{2},N)\sim N^{-1/2}$ for $\textbf{q}_{1}\neq-\textbf{p}_{1},-\textbf{p}_{2}$, and $b_{2}(\textbf{q}_{1};\textbf{p}_{1},\textbf{p}_{2},N)\sim N^{1/2}$ for $\textbf{q}_{1}=-\textbf{p}_{1},-\textbf{p}_{2}$. For a state with three phonons we have $\textbf{p}=\textbf{p}_{1}+\textbf{p}_{2}+\textbf{p}_{3}$. The lowest not small coefficients $b_{j}$ should be the coefficients $b_{3}(\textbf{q}_{1},\textbf{q}_{2};\textbf{p}_{1},\textbf{p}_{2},\textbf{p}_{3},N)$ with such $\textbf{q}_{1}$ and $\textbf{q}_{2}$, for which $\rho_{\textbf{q}_{1}}\rho_{\textbf{q}_{2}}\rho_{-\textbf{q}_{1}-\textbf{q}_{2}-\textbf{p}}=\rho_{-\textbf{p}_{1}}\rho_{-\textbf{p}_{2}}\rho_{-\textbf{p}_{3}}$. For a state with $N$ quasiparticles the relation $\textbf{p}=\textbf{p}_{1}+\ldots+\textbf{p}_{N}$ holds, and the coefficients $b_{j\leq N-1}$ are negligible: $b_{j\leq N-1}\sim N^{-a_{j}}$ ($a_{j}>0$). The coefficients $b_{N}(\textbf{q}_{1},\ldots,\textbf{q}_{N-1};\textbf{p}_{1},\ldots,\textbf{p}_{N},N)$ are not small at such $\textbf{q}_{1},\ldots,\textbf{q}_{N-1}$, for which $\rho_{\textbf{q}_{1}}\ldots\rho_{\textbf{q}_{N-1}}\rho_{-\textbf{q}_{1}-\ldots-\textbf{q}_{N-1}-\textbf{p}}=\rho_{-\textbf{p}_{1}}\ldots\rho_{-\textbf{p}_{N}}$.

Formulae (6), (7) imply that the largest number of quasiparticles equals $N$, since series (7) contains the terms $\rho_{-\textbf{q}_{1}}\ldots\rho_{-\textbf{q}_{j}}$ with at most $N$ factors $\rho_{-\textbf{q}}$. The last property is caused by that the functions $1,\rho_{-\textbf{q}_{1}}$, $\rho_{-\textbf{q}_{1}}\rho_{-\textbf{q}_{2}},\ldots,\rho_{-\textbf{q}_{1}}\ldots\rho_{-\textbf{q}_{N}}$ form the complete (nonorthogonal) collection of functions, in which any Bose-symmetric function of the variables $\textbf{r}_{1},\ldots,\textbf{r}_{N}$, which can be presented as the Fourier series, can be expanded [39]. Therefore, the product $\rho_{-\textbf{q}_{1}}\ldots\rho_{-\textbf{q}_{N}}\rho_{-\textbf{q}_{N+1}}\ldots\rho_{-\textbf{q}_{N+M}}$ containing more than $N$ factors $\rho_{-\textbf{q}}$ is reduced to an expansion of the form $\psi_{\textbf{p}}$ (7) with $\textbf{p}=\textbf{q}_{1}+\ldots+\textbf{q}_{N+M}$. For example, for $N=2$ we obtain

[TABLE]

Thus, the largest number of quasiparticles in a Bose gas, being in some pure state $\Psi_{p}$, is equal to $N$. According to quantum statistics, the equilibrium number of quasiparticles for the given temperature $T>0$ is

[TABLE]

where $Z=\sum_{p}e^{-E_{p}/k_{B}T}$, $\{\Psi_{p}(x_{1},\ldots,x_{N})\}$ is the complete orthonormalized set of WFs of a system with a fixed number of atoms $N$, and $N_{Qp}$ is the number of quasiparticles in the state $\Psi_{p}$. According to the above analysis, the value of $N_{Qp}$ is determined by the structure of $\Psi_{p}(x_{1},\ldots,x_{N})$, and $N_{Qp}\leq N$ for any state. Therefore, $\bar{N}_{Q}(T)<N$. At low temperatures, the states with small $N_{Qp}$ make the main contribution to (63). Therefore, the average number of quasiparticles is small. In this case, $\bar{N}_{Q}(T)$ increases with $T$. It is clear that, as $T\rightarrow\infty,$ we have $\bar{N}_{Q}(T)\rightarrow N$. Thus, in the gas at a high temperature, the number of quasiparticles is close to the number of atoms. This shows how a quantum Bose system transforms into a classical one.

10 Appendix 2. Vakarchuk–Yukhnovskii’s equations.

The functions $a_{j}$ and $b_{j}$ from Eqs. (5) and (7) satisfy the Vakarchuk–Yukhnovskii’s equations [22, 39]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here, $E_{1}(\textbf{q})=\frac{\hbar^{2}q^{2}}{2m}(1-2a_{2}(\textbf{q}))$. The equation for the function $a_{4}$ is given in [22, 39]. If one of the arguments of the functions $a_{j}$ or $b_{j}$ in (64)–(10) is zero, then the corresponding $a_{j}$ or $b_{j}$ should be set zero. If we describe the state with $l\geq 2$ quasiparticles with the total momentum $\textbf{p}_{1}+\ldots+\textbf{p}_{l}=\textbf{p}$, then it is necessary to make the following changes in (67)–(10): $E({\textbf{p}})\rightarrow E({\textbf{p}_{1},\ldots,\textbf{p}_{l}}$) and $b_{j}(\textbf{q}_{1},\ldots,\textbf{q}_{j-1};\textbf{p})\rightarrow b_{j}(\textbf{q}_{1},\ldots,\textbf{q}_{j-1};\textbf{p}_{1},\ldots,\textbf{p}_{l},N)$ for all $j$.

The functions $a_{j+1}(\textbf{q}_{1},\ldots,\textbf{q}_{j})$ and $b_{j+1}(\textbf{q}_{1},\ldots,\textbf{q}_{j};\textbf{p})$ are invariant relative to the permutations of two any arguments $\textbf{q}_{l}$, $\textbf{q}_{n}$. The functions $a_{j+1}(\textbf{q}_{1},\ldots,\textbf{q}_{j})$ are also invariant relative to the change $\textbf{q}_{l}\rightarrow-\textbf{q}_{1}-\textbf{q}_{2}-\ldots-\textbf{q}_{j}$ for any $j$ and $l=1,\ldots,j$. As for the functions $b_{j+1}(\textbf{q}_{1},\ldots,\textbf{q}_{j};\textbf{p}),$ they are invariant relative to the change $\textbf{q}_{l}\rightarrow-\textbf{q}_{1}-\textbf{q}_{2}-\ldots-\textbf{q}_{j}-\textbf{p}$ for any $j\geq 1$, $l=1,\ldots,j$.

In works [22, 39] a one-phonon state was considered and Eqs. (64)–(10) were deduced for $b_{1}(\textbf{p})=1$. We write these equations for any $b_{1}(\textbf{p})$, so that the equations can be used to describe the states with the number of phonons $\geq 1$.

Equations (64)–(10) are exact for an infinite system: $N,V=\infty$. For a finite system, the product $\rho_{-\textbf{q}_{1}}\ldots\rho_{-\textbf{q}_{N}}\rho_{-\textbf{q}_{N+1}}\ldots\rho_{-\textbf{q}_{N+M}}$ ($M=1,2,\ldots$) is reduced to a sum of terms, each of which contains at most $N$ factors of the form $\rho_{-\textbf{q}}$ (see Appendix 1). One needs to take this property into account while deriving the equations for $a_{j}$ and $b_{j}$, which will cause the appearance of many additional terms in Eqs. (64)–(10). However, for the weak coupling, these terms should be negligible. Apparently, they are negligible also for a nonweak coupling. Otherwise, the transition from the solutions for a large finite system to solutions for the infinite one would occur by jump. However, we do not expect such a jump. One can verify that the solutions of the Lieb–Liniger equations (1) or (2) do not exhibit such a jump. Those additional terms were not considered in the literature, and we omitted them in Sections 2, 3.

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