Nature of Lieb’s “Hole” Excitations and Two-Phonon States of a Bose Gas

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.

Appendices

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 $N_{Q}\le N$. 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 $n_{j}=l\ne 0$. 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 $f(\mathbf{r }_1,\ldots ,\mathbf{r }_N)\varPsi _0$. The periodic system has a definite momentum. The general form of the WF of a state with the total momentum $\hbar \mathbf{p }$ is set by formulae (6), (7). (If the number of quasiparticles $\ge 2$, then it is necessary to make changes in (6), (7) as described in Sect. 2.) Therefore, the function $f(\mathbf{r }_1,\ldots ,\mathbf{r }_N)$ should coincide with $\psi _{\mathbf{p }}$ (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 $\hbar \mathbf{p }_{1}$ and $\hbar \mathbf{p }_{2}$, we should set $\mathbf{p }=\mathbf{p }_{1}+\mathbf{p }_{2}$ in (6), (7). In this case, $b_{j\ge 3}\sim 1$, $b_{1}(\mathbf{p }_{1},\mathbf{p }_{2},N)\sim N^{-1/2}$, $b_{2}(\mathbf{q }_{1};\mathbf{p }_{1},\mathbf{p }_{2},N)\sim N^{-1/2}$ for $\mathbf{q }_{1}\ne -\mathbf{p }_{1}, -\mathbf{p }_{2}$, and $b_{2}(\mathbf{q }_{1};\mathbf{p }_{1},\mathbf{p }_{2},N)\sim N^{1/2}$ for $\mathbf{q }_{1}= -\mathbf{p }_{1}, -\mathbf{p }_{2}$. For a state with three phonons, we have $\mathbf{p }=\mathbf{p }_{1}+\mathbf{p }_{2}+\mathbf{p }_{3}$. The lowest not small coefficients $b_{j}$ should be the coefficients $b_{3}(\mathbf{q }_{1},\mathbf{q }_{2};\mathbf{p }_{1},\mathbf{p }_{2},\mathbf{p }_{3},N)$ with such $\mathbf{q }_{1}$ and $\mathbf{q }_{2}$, for which $\rho _{\mathbf{q }_{1}}\rho _{\mathbf{q }_{2}}\rho _{-\mathbf{q }_{1}-\mathbf{q }_{2}-\mathbf{p }} = \rho _{-\mathbf{p }_{1}}\rho _{-\mathbf{p }_{2}}\rho _{-\mathbf{p }_{3}}$. For a state with N quasiparticles, the relation $\mathbf{p }=\mathbf{p }_{1}+\ldots +\mathbf{p }_{N}$ holds, and the coefficients $b_{j\le N-1}$ are negligible: $b_{j\le N-1}\sim N^{-a_{j}}$ ($a_{j}>0$). The coefficients $b_{N}(\mathbf{q }_{1},\ldots ,\mathbf{q }_{N-1};\mathbf{p }_{1},\ldots ,\mathbf{p }_{N},N)$ are not small at such $\mathbf{q }_{1},\ldots ,\mathbf{q }_{N-1}$, for which $\rho _{\mathbf{q }_{1}}\ldots \rho _{\mathbf{q }_{N-1}}\rho _{-\mathbf{q }_{1}-\ldots -\mathbf{q }_{N-1}-\mathbf{p }} = \rho _{-\mathbf{p }_{1}}\ldots \rho _{-\mathbf{p }_{N}}$.

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

$$\begin{aligned} \rho _{\mathbf{q }_{1}}\rho _{\mathbf{q }_{2}} \rho _{\mathbf{q }_{3}}=\frac{1}{\sqrt{N}}(\rho _{\mathbf{q }_{1}+\mathbf{q }_{2}} \rho _{\mathbf{q }_{3}}+\rho _{\mathbf{q }_{1}+\mathbf{q }_{3}} \rho _{\mathbf{q }_{2}}+\rho _{\mathbf{q }_{2}+\mathbf{q }_{3}} \rho _{\mathbf{q }_{1}})-\frac{2}{N}\rho _{\mathbf{q }_{1}+\mathbf{q }_{2}+\mathbf{q }_{3}}. \end{aligned}$$

(62)

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

$$\begin{aligned} {\bar{N}}_{Q}(T)=\frac{1}{Z}\int d \mathbf{r }_{1}\ldots d\mathbf{r }_{N}\sum \limits _{p}e^{-E_{p}/k_{B}T}\varPsi ^{*}_{p}{\hat{N}}_{Qp}\varPsi _{p}= \frac{1}{Z}\sum \limits _{p}e^{-E_{p}/k_{B}T}N_{Qp}, \end{aligned}$$

(63)

where $Z=\sum _{p}e^{-E_{p}/k_{B}T}$, $\{\varPsi _{p}(x_{1},\ldots ,x_{N})\}$ 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 $\varPsi _{p}$. According to the above analysis, the value of $N_{Qp}$ is determined by the structure of $\varPsi _{p}(x_{1},\ldots ,x_{N})$, and $N_{Qp}\le N$ 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.

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]

$$\begin{aligned}&E_{0}=\frac{N-1}{2}n\nu (0)- \sum \limits _{\mathbf{q }\ne 0}\frac{n\nu (q)}{2}-\sum \limits _{\mathbf{q }\ne 0}\frac{\hbar ^{2}q^{2}}{2m} a_{2}(\mathbf{q }), \end{aligned}$$

(64)

$$\begin{aligned}&\frac{mn\nu (q)}{\hbar ^{2}}+q^{2}a_{2}(\mathbf{q })-q^{2}a^{2}_{2}(\mathbf{q })-\frac{1}{N} \sum \limits _{\mathbf{q }_{1}\ne 0}a_{3}(\mathbf{q },\mathbf{q }_{1})\mathbf{q }_{1}(\mathbf{q }+\mathbf{q }_{1}) \nonumber \\&\quad - \frac{1}{2N}\sum \limits _{\mathbf{q }_{1}\ne 0}a_{4}(\mathbf{q },-\mathbf{q }_{1},\mathbf{q }_{1})q_{1}^{2}=0, \end{aligned}$$

(65)

$$\begin{aligned}&0=a_{3}(\mathbf{q }_{1},\mathbf{q }_{2})[E_{1}(\mathbf{q }_{1})+E_{1}(\mathbf{q }_{2})+E_{1}(\mathbf{q }_{1}+\mathbf{q }_{2})] +2\mathbf{q }_{1}\mathbf{q }_{2}a_{2}(\mathbf{q }_{1})a_{2}(\mathbf{q }_{2})\nonumber \\&\quad -2\mathbf{q }_{1}(\mathbf{q }_{1}+\mathbf{q }_{2})a_{2}(\mathbf{q }_{1})a_{2}(\mathbf{q }_{1}+\mathbf{q }_{2}) -2\mathbf{q }_{2}(\mathbf{q }_{1}+\mathbf{q }_{2})a_{2}(\mathbf{q }_{2})a_{2}(\mathbf{q }_{1}+\mathbf{q }_{2}) \nonumber \\&\quad - \frac{1}{N}\sum \limits _{\mathbf{q }\ne 0}a_{5}(\mathbf{q }_{1},\mathbf{q }_{2},\mathbf{q },-\mathbf{q })q^{2}+ \frac{1}{N}\sum \limits _{\mathbf{q }\ne 0}\left[ a_{4}(\mathbf{q }_{1}-\mathbf{q },\mathbf{q }_{2},\mathbf{q }) (\mathbf{q }_{1}-\mathbf{q })\mathbf{q }\right. \nonumber \\&\quad + \left. a_{4}(\mathbf{q }_{1},\mathbf{q }_{2}-\mathbf{q },\mathbf{q }) (\mathbf{q }_{2}-\mathbf{q })\mathbf{q }+a_{4}(\mathbf{q }_{1},\mathbf{q }_{2},-\mathbf{q }_{1}-\mathbf{q }_{2}-\mathbf{q }) (-\mathbf{q }_{1}-\mathbf{q }_{2}-\mathbf{q })\mathbf{q } \right] , \end{aligned}$$

(66)

$$\begin{aligned}&b_{1}(\mathbf{p })E(\mathbf{p })=b_{1}(\mathbf{p })E_{1}(\mathbf{p })- \frac{1}{N}\sum \limits _{\mathbf{q }\ne 0}b_{2}(\mathbf{q };\mathbf{p })\frac{\hbar ^{2}}{2m}(\mathbf{p }+\mathbf{q })\mathbf{q } \nonumber \\&\quad -\frac{1}{N}\sum \limits _{\mathbf{q }\ne 0}b_{3}(\mathbf{q },-\mathbf{q };\mathbf{p })\frac{\hbar ^{2}q^{2}}{2m}, \end{aligned}$$

(67)

$$\begin{aligned}&b_{2}(\mathbf{q };\mathbf{p })\frac{2m}{\hbar ^{2}}[E_{1}(\mathbf{q })+E_{1}(\mathbf{p }+\mathbf{q })-E(\mathbf{p })] +2b_{1}(\mathbf{p })\mathbf{p }\mathbf{q }a_{2}(\mathbf{q })-2b_{1}(\mathbf{p })p^{2}a_{3}(\mathbf{p },\mathbf{q })\nonumber \\&\quad -2b_{1}(\mathbf{p })\mathbf{p }(\mathbf{p }+\mathbf{q })a_{2}(\mathbf{p }+\mathbf{q }) - \frac{1}{N}\sum \limits _{\mathbf{q }_{1}\ne 0}q_{1}^{2}b_{4}(\mathbf{q }_{1},-\mathbf{q }_{1},\mathbf{q };\mathbf{p }) \nonumber \\&\quad + \frac{1}{N}\sum \limits _{\mathbf{q }_{1}\ne 0}b_{3}(\mathbf{q }_{1},\mathbf{q }-\mathbf{q }_{1};\mathbf{p })\mathbf{q }_{1}(\mathbf{q }-\mathbf{q }_{1}) \nonumber \\&\quad + \frac{1}{N}\sum \limits _{\mathbf{q }_{1}\ne 0} b_{3}(\mathbf{q }_{1},-\mathbf{q }-\mathbf{q }_{1}-\mathbf{p };\mathbf{p })\mathbf{q }_{1}(-\mathbf{q }_{1}-\mathbf{q }-\mathbf{p }) =0, \end{aligned}$$

(68)

$$\begin{aligned}&b_{3}(\mathbf{q }_{1},\mathbf{q }_{2};\mathbf{p })\frac{2m}{\hbar ^{2}}[E_{1}(\mathbf{q }_{1})+E_{1}(\mathbf{q }_{2})+E_{1}(\mathbf{p }+\mathbf{q }_{1}+\mathbf{q }_{2}) -E(\mathbf{p })] \nonumber \\&\quad -2b_{1}(\mathbf{p })[p^{2}a_{4}(\mathbf{q }_{1},\mathbf{q }_{2},\mathbf{p })-a_{3}(\mathbf{q }_{1},\mathbf{q }_{2}) \mathbf{p }(\mathbf{q }_{1}+\mathbf{q }_{2})]\nonumber \\&\quad - 2b_{1}(\mathbf{p })[a_{3}(\mathbf{q }_{1}+\mathbf{p },\mathbf{q }_{2}) \mathbf{p }(\mathbf{q }_{1}+\mathbf{p })+a_{3}(\mathbf{q }_{2}+\mathbf{p },\mathbf{q }_{1}) \mathbf{p }(\mathbf{q }_{2}+\mathbf{p })]\nonumber \\&\quad - 2b_{2}(\mathbf{q }_{1};\mathbf{p })a_{3}(\mathbf{q }_{1}+\mathbf{p },\mathbf{q }_{2})(\mathbf{p }+\mathbf{q }_{1})^{2} -2b_{2}(\mathbf{q }_{2};\mathbf{p })a_{3}(\mathbf{q }_{2}+\mathbf{p },\mathbf{q }_{1})(\mathbf{p }+\mathbf{q }_{2})^{2} \nonumber \\&\quad - 2b_{2}(-\mathbf{q }_{1}-\mathbf{q }_{2}-\mathbf{p };\mathbf{p })a_{3}(\mathbf{q }_{1},\mathbf{q }_{2})(\mathbf{q }_{1}+\mathbf{q }_{2})^{2} -\frac{1}{N}\sum \limits _{\mathbf{q }_{4}\ne 0}q_{4}^{2}b_{5}(\mathbf{q }_{4},-\mathbf{q }_{4},\mathbf{q }_{1},\mathbf{q }_{2};\mathbf{p })\nonumber \\&\quad -2a_{2}(\mathbf{q }_{1})\mathbf{q }_{1}[b_{2}(\mathbf{q }_{2};\mathbf{p })(-\mathbf{q }_{2}-\mathbf{p })+ b_{2}(-\mathbf{q }_{1}-\mathbf{q }_{2}-\mathbf{p };\mathbf{p })(\mathbf{q }_{1}+\mathbf{q }_{2})] \nonumber \\&\quad -2a_{2}(\mathbf{q }_{2})\mathbf{q }_{2}[b_{2}(\mathbf{q }_{1};\mathbf{p })(-\mathbf{q }_{1}-\mathbf{p })+ b_{2}(-\mathbf{q }_{1}-\mathbf{q }_{2}-\mathbf{p };\mathbf{p })(\mathbf{q }_{1}+\mathbf{q }_{2})] \nonumber \\&\quad -2a_{2}(\mathbf{q }_{1}+\mathbf{q }_{2}+\mathbf{p })(\mathbf{q }_{1}+\mathbf{q }_{2}+\mathbf{p })[b_{2}(\mathbf{q }_{1};\mathbf{p })(\mathbf{q }_{1}+\mathbf{p })+ b_{2}(\mathbf{q }_{2};\mathbf{p })(\mathbf{q }_{2}+\mathbf{p })]\nonumber \\&\quad + \frac{1}{N}\sum \limits _{\mathbf{q }\ne 0}\mathbf{q }(-\mathbf{q }_{1}-\mathbf{q }_{2}-\mathbf{q }-\mathbf{p }) b_{4}(\mathbf{q }_{1},\mathbf{q }_{2},-\mathbf{q }_{1}-\mathbf{q }_{2}-\mathbf{q }-\mathbf{p };\mathbf{p }) \nonumber \\&\quad + \frac{1}{N}\sum \limits _{\mathbf{q }\ne 0}\mathbf{q }(\mathbf{q }_{1}-\mathbf{q }) b_{4}(\mathbf{q }_{1}-\mathbf{q },\mathbf{q }_{2},-\mathbf{q }_{1}-\mathbf{q }_{2}-\mathbf{p };\mathbf{p }) \nonumber \\&\quad + \frac{1}{N}\sum \limits _{\mathbf{q }\ne 0}\mathbf{q }(\mathbf{q }_{2}-\mathbf{q }) b_{4}(\mathbf{q }_{1},\mathbf{q }_{2}-\mathbf{q },-\mathbf{q }_{1}-\mathbf{q }_{2}-\mathbf{p };\mathbf{p }) =0. \end{aligned}$$

(69)

Here, $E_{1}(\mathbf{q })=\frac{\hbar ^{2}q^{2}}{2m}(1-2a_{2}(\mathbf{q }))$. 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)–(69) is zero, then the corresponding $a_{j}$ or $b_{j}$ should be set zero. If we describe the state with $l\ge 2$ quasiparticles with the total momentum $\mathbf{p }_{1}+\ldots + \mathbf{p }_{l}=\mathbf{p }$, then it is necessary to make the following changes in (67)–(69): $E({\mathbf{p }})\rightarrow E({\mathbf{p }_{1},\ldots , \mathbf{p }_{l}}$) and $b_{j}(\mathbf{q }_{1},\ldots ,\mathbf{q }_{j-1};\mathbf{p })\rightarrow b_{j}(\mathbf{q }_{1},\ldots ,\mathbf{q }_{j-1};\mathbf{p }_{1},\ldots , \mathbf{p }_{l},N)$ for all j.

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

In works [22, 39], a one-phonon state was considered and Eqs. (64)–(69) were deduced for $b_{1}(\mathbf{p })=1$. We write these equations for any $b_{1}(\mathbf{p })$, so that the equations can be used to describe the states with the number of phonons $\ge 1$.

Equations (64)–(69) are exact for an infinite system: $N, V=\infty$. For a finite system, the product $\rho _{-\mathbf{q }_{1}}\ldots \rho _{-\mathbf{q }_{N}}\rho _{-\mathbf{q }_{N+1}}\ldots \rho _{-\mathbf{q }_{N+M}}$ ($M=1,2,\ldots$) is reduced to a sum of terms, each of which contains at most N factors of the form $\rho _{-\mathbf{q }}$ (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)–(69). 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 Sects. 2, 3.