Finite-Size Effects with Boundary Conditions on Bose-Einstein Condensation

TL;DR

This paper studies how boundary conditions and finite size influence Bose-Einstein condensation in ideal gases, revealing that smaller systems and certain boundary conditions can enhance the transition temperature and condensate fraction.

Contribution

It provides a numerical analysis of finite-size effects and boundary conditions on Bose-Einstein condensation, highlighting the impact of system size and boundary choices.

Findings

Smaller system sizes increase the characteristic temperature and condensate fraction.

Antiperiodic boundary conditions induce a singularity in the system.

Boundary conditions significantly affect Bose-Einstein condensation properties.

Abstract

We investigate the statistical distribution for ideal Bose gases with constant particle density in the 3D box of volume $V=L^{3}$. By changing linear size $L$ and imposing different boundary conditions on the system, we present a numerical analysis on the characteristic temperature and condensate fraction, and find that the smaller linear size is efficient to increase the characteristic temperature and condensate fraction. Moreover, there is a singularity under the antiperiodic boundary condition.

Tables

$L/L_0$	0.2	0.5	1	10	50	100
(a)$k_B{T}_{c,L}\left(\frac{{\hslash }^2}{m}{n}^{\frac{2}{3}}\right)$	74.5	20.1	9.279	3.67	3.38	3.35
(b)$k_B{T}_{c,L}\left(\frac{{\hslash }^2}{m}{n}^{\frac{2}{3}}\right)$	123.3	30	12.13	3.68	3.37	3.34
(c)$k_B{T}_{c,L}\left(\frac{{\hslash }^2}{m}{n}^{\frac{2}{3}}\right)$	62.4	18.2	9.42	4.168	3.57	3.463

Equations

$$N({ϵ}_i)=\frac{1}{e^{\beta ({ϵ}_i-\mu )}-1}=\frac{z{e}^{-\beta {ϵ}_i}}{1-z{e}^{-\beta {ϵ}_i}},$$

$$n=\frac{1}{V}\sum _i\frac{1}{e^{\beta ({ϵ}_i-\mu )}-1}=\frac{1}{V}\sum _i\frac{z{e}^{-\beta {ϵ}_i}}{1-z{e}^{-\beta {ϵ}_i}},$$

$$n_{ϵ}=n-n_0=\frac{1}{V}\sum _{j=1}^{\infty }{z}^j{\int }_{ϵ}^{\infty }D(ϵ)\exp (-j\beta ϵ)dϵ,$$

$$\frac{1}{V}\frac{z}{1-z}+\frac{1}{{\lambda }^3}{g}_{3/2}(z)=n,$$

$$\frac{2\pi }{h^3}(2m{)}^{\frac{3}{2}}{\int }_0^{\infty }\frac{{ϵ}^{\frac{1}{2}}dϵ}{e^{\frac{ϵ}{k_B{T}_c}}-1}=n.$$

$$\frac{1}{V}\sum _i\frac{1}{z_0^{-1}{e}^{{ϵ}_i/k_B{T}_{c,L}}-1}=n,(i=1,2,\cdots )$$

$$\rho (x,y,z=-L/2)=\rho (x,y,z=L/2).$$

$${ϵ}_{n_1{n}_2{n}_3}=\frac{{\hslash }^2[(2n_1\pi +\alpha {)}^2+(2n_2\pi +\alpha {)}^2+(2n_3\pi +\alpha {)}^2]}{2m{L}^2},$$

$$\frac{1}{L^3}\sum _{n_1}\sum _{n_2}\sum _{n_3}\frac{1}{z_0^{-1}{e}^{\frac{{ϵ}_{n_1{n}_2{n}_3}}{k_B{T}_{c,L}}}-1}=n.$$

$${ϵ}_{n_1{n}_2{n}_3}=\frac{2{\hslash }^2{\pi }^2(n_1^2+n_2^2+n_3^2)}{m{L}^2}.$$

$$\frac{1}{L^3}\sum _{n_1}\sum _{n_2}\sum _{n_3}\frac{1}{e^{\frac{2{\hslash }^2{\pi }^2(n_1^2+n_2^2+n_3^2)}{m{L}^2{k}_B{T}_{c,L}}}-1}=n.$$

$${ϵ}_{n_1{n}_2{n}_3}=\frac{{\hslash }^2[(2n_1\pi +\pi {)}^2+(2n_2\pi +\pi {)}^2+(2n_3\pi +\pi {)}^2]}{2m{L}^2}.$$

$$\frac{1}{L^3}\sum _{n_1}\sum _{n_2}\sum _{n_3}\frac{1}{e^{\frac{{\hslash }^2{\pi }^2[(2n_1+1{)}^2+(2n_2+1{)}^2+(2n_3+1{)}^2-3]}{2m{L}^2{k}_B{T}_{c,L}}}-1}=n.$$

$${ϵ}_{n_1{n}_2{n}_3}=\frac{{\hslash }^2{\pi }^2(n_1^2+n_2^2+n_3^2)}{2m{L}^2}.$$

$$\frac{1}{L^3}\sum _{n_1=1}^{\infty }\sum _{n_2=1}^{\infty }\sum _{n_3=1}^{\infty }\frac{1}{e^{\frac{{\hslash }^2{\pi }^2(n_1^2+n_2^2+n_3^2-3)}{2m{L}^2{k}_B{T}_{c,L}}}-1}=n.$$

Full text

I Introduction

The Bose-Einstein condensation (BEC) is a purely quantum-statistical phase transition characterized by the appearance of macroscopic population in ground state below the critical temperature $T_{c}$ and it plays an important role in condensed matter Cataliotti et al. (2001); Inguscio et al. (2005); Roati et al. (2008); Wu et al. (2011); Adhikari (2019); Deuchert et al. (2019); Urvoy et al. (2019); Schneider et al. (2020); Georgescu (2020), optics Klaers et al. (2010); Hansen et al. (2016), atomic and molecular physics Goldstein and Meystre (1997); Jochim et al. (2003); Zwierlein et al. (2003); Zhang et al. (2020) and among others. It is emphasized that the transition actually occurs at the thermodynamic limit, or when the discrete level structure was approximated by a continuous density of states Anderson et al. (1995); Davis et al. (1995); Ketterle and Van Druten (1996); Bradley et al. (1997); Brankov et al. (2000); Zwierlein et al. (2003); Kristensen et al. (2019).

However, the experimental observations of BEC on cold gases Anderson et al. (1995); Davis et al. (1995); Bradley et al. (1997); Andrews et al. (1997); Myatt et al. (1997); Hall et al. (1998); Miesner et al. (1999); Matthews et al. (1999); Papp et al. (2008); McCarron et al. (2011) were performed in the finite volume that neither of the above approximations methods seems to be inherently justified Grossmann and Holthaus (1995); Ketterle and Van Druten (1996). Subsequently, some scholars wonder whether this will lead to deviations from the theoretical predictions and begin to pay attention to BEC in finite systems Ziff et al. (1977); Grossmann and Holthaus (1995); Ketterle and Van Druten (1996); Franzosi et al. (2010). The boundary conditions are of great importance to finite systems, such as periodic, Neumann, and Dirichlet conditions London (1938); Krueger (1968); Greenspoon and Pathria (1974); Ziff et al. (1977); Grossmann and Holthaus (1995); Holthaus et al. (2002); Brankov et al. (2020). The results imply that the shift of the condensation temperature depends only on the total number of particles Grossmann and Holthaus (1995); Ketterle and Van Druten (1996).

From the theoretical point of view, the only requirement is that the Hamiltonian of the finite system should be Hermitian, and the above boundary conditions are just some special cases. These inspire us to explore the physics of BEC in finite volume systems by focusing on the finite-size behaviors and twisted boundary conditions Byers and Yang (1961). For simplicity, in this paper, we model the finite system with 3D box of volume $V=L^{3}$ that consist of ideal Bose gases, keeping particle density fixed. And we characterize BEC phase transition with characteristic temperature and condensate fraction, then numerically calculate them under different linear sizes and boundary conditions.

This paper is organized as follows. In Section 2, the specific formulas for the statistical distribution of ideal Bose gases are directly given. In Section 3, through numerical calculations, finite-size effects on characteristic temperature and condensate fraction are investigated. In Section 4, the changes on characteristic temperature and condensate fraction are also obtained in the presence of twisted boundary conditions. Finally, in Section 5, conclusions and discussions are briefly given.

II The ideal Bose gases in the cubic box

In this section, the BEC statistics of ideal Bose gases with fixed particle density $n$, which is confined in the cubic box of volume $V=L^{3}$ are directly given. According to the Bose-Einstein distribution, the population $N(\epsilon_{i})$ of a state with energy $\epsilon_{i}$ is:

[TABLE]

the particle density $n$ is:

[TABLE]

where $\beta=1/(k_{B}T)$ and $k_{B}$ denotes the Boltzmann’s constant. And the fugacity $z$ related to chemical potential $\mu$ can be expressed by $z$ = exp$(\beta\mu)$. By splitting off the ground-state particle density $n_{0}$, the finite sum over the excited states $n_{\epsilon}$ is replaced by an integral:

[TABLE]

where $n_{0}=\frac{ze^{-\beta\epsilon_{0}}}{1-ze^{-\beta\epsilon_{0}}}$. $D(\epsilon)$ represents the density of states, usually taken to be $\frac{2\pi V}{h^{3}}(2m)^{\frac{3}{2}}\epsilon^{\frac{1}{2}}$, $h$ and $m$ denote the Planck constant and the mass of boson, respectively. Assuming the ground state $\epsilon_{0}$ as zero, the result is:

[TABLE]

with $\lambda=\frac{h}{\left(2\pi mk_{B}T\right)^{1/2}}$, and the Bose function $g_{n}(z)=\sum_{m=1}^{\infty}z^{m}/m^{n}$. The physical meaning of the second term in Eq. (4) implies that as the fugacity $z$ reaches its maximum $\exp(\beta\epsilon_{0})$, the number of the excited particles reaches the maximum, and all particles that exceeding this maximum must drop into the ground state. Namely, the critical temperature $T_{c}$ can now be found by setting $n_{0}=0$ and $z=1$ as:

[TABLE]

We now consider the extension of an ordinary BEC, by confining ideal Bose gases into a finite cubic box with linear size $L$. For finite volume systems, it is always crossover rather than phase transition. We thus define characteristic temperature Brankov et al. (2020) $T_{c,L}$ of the finite system with linear size $L$ by:

[TABLE]

with $z_{0}$=exp$(\epsilon_{0}/k_{B}T_{c,L})$. This temperature becomes critical temperature only in the limit when system size goes to infinity. Additionally, as for small systems, boundary conditions are of crucial importance as they can affect the symmetries of the system and consequently modify the fundamental properties, such as ground state energies and conserved quantities Zawadzki et al. (2017). The components of energy and momentum are quantized when particular physical boundary conditions are imposed on the finite system, and the discussions of BEC statistics should use summation instead of integral.

For simplicity, we consider particles confined in a 3D cubic box of dimensions $L_{x}$, $L_{y}$, $L_{z}$ $\in$[-$\frac{L}{2}$, $\frac{L}{2}$]. Theoretically, in order to ensure the momentum operator $\hat{p_{x}}$, $\hat{p_{y}}$, $\hat{p_{z}}$ in the systems are Hermitian, the quantum probability density $\rho=|\psi|^{2}$ should be constant on the boundary of the cubic box. Namely,

[TABLE]

Therefore, the arbitrary wave function should meet $\psi_{k}(\frac{L}{2})/\psi_{k}(-\frac{L}{2})=e^{i\alpha}$. These twisted boundary conditions are parameterized by a twist angle $\alpha$ at each boundary, with range $0\leq\alpha\leq 2\pi$. The use of twisted boundary conditions for the BEC in the cubic box, which is equivalent to realizing the BEC in the presence of a constant background magnetic potential coupled with bosons. The twist angles characterizing the twisted boundary condition are only obtained at the discrete values of the eigenstates energies, which corresponds to the magnetic flux quanta $N\phi$. The stronger the magnetic potential is, the larger is the twisted angle. Especially, the discrete twist angles are related to the linear size $L$, energy levels are shifted and finite-volume corrections can be generated Sachrajda and Villadoro (2005). Experimentally, according to the superfluid behavior of a BEC Könenberg et al. (2015, 2016), the achievement of BEC in the cubic of side length $L$ characterized with twisted boundary conditions may be realized in term of the non-classical response of the system to an infinitesimal boost or rotation. The boost can be provided by imposing extremely small velocity field such as slow rotation of the cubic box and the rotation can be equivalently produced by introducing disclinations and screw dislocations Kleman and Friedel (2008). Therefore, the allowed components of particle’s momenta inside the box are $p_{x,y,z}=\frac{2\pi}{L}N+\frac{\alpha}{L}$, where $N$ is an integer. And the corresponding energy eigenvalues of bosons in the cubic box are expressed as Karbowski and Turski (2000); Dalfovo et al. (2005):

[TABLE]

where the quantum states are characterized by the quantum numbers $(n_{1},n_{2},n_{3})$, $(n_{1},n_{2},n_{3}=0,\pm 1,\pm 2,\cdots)$. Consequently, we can absorb the effect of $\alpha$ into twist boundary conditions for the wavefunctions, to study different physical boundary conditions Kirsten and Toms (1998) on BEC in finite volume systems. Substituting Eq. (8) into Eq. (6), with the volume $V=L^{3}$, that as:

[TABLE]

Note that the most significant difference among different boundary conditions is the difference in the ranges Eq. (9) in which the quantum numbers $(n_{1},n_{2},n_{3})$ vary. Particularly, the ground state energy which depends sensitively on $\alpha$ needs to be excluded during the calculation. For clarity, here we define $\alpha=k\pi$ $($k$\in$ [0,2]$$).

(i)$k\in[0,1)$, the ground state $(0,0,0)$ should be excluded.

(ii)$k\in(1,2]$, the ground state $(-1,-1,-1)$ should be excluded.

(iii)$k=1$, the ground state $(-1,-1,-1)$, $(-1,-1,0)$, $(-1,0,-1)$, $(0,-1,-1)$, $(0,0,-1)$, $(0,-1,0)$, $(-1,0,0)$, $(0,0,0)$ should be excluded.

Obviously, the BEC statistics in the finite volume system not only depends on the linear size but also on the boundary conditions. And we will do some detailed calculations in following sections.

III Finite-size effects on Bose-Einstein condensation in the cubic box

In this section, performing the BEC statistics for the finite system of ideal Bose gases in the box traps of different linear sizes under periodic, antiperiodic and Dirichlet boundary conditions, we compare their statistics and conclude on the finite-size effects on them.

(a) Periodic boundary condition ($\alpha=0$)

According to our analyses, the discontinuous energy values of single-state can be given by:

[TABLE]

The characteristic temperature can be determined by:

[TABLE]

The numerical results under the periodic boundary condition are shown in Figs.  1,  2 and TABLE I.

(b) Antiperiodic boundary condition ($\alpha=\pi$)

The quantum state energy is given by:

[TABLE]

As a result, the characteristic temperature can be calculated by:

[TABLE]

Similarly, the corresponding results are shown in Figs.  1,  3 and Table I.

(c) Dirichlet condition

This can be described by requiring that the wave function of the particle is identically zero outside the box. Hence an impenetrable barrier can be interpreted as a boundary condition. The Dirichlet condition requires $\psi=0$ on each side of the cubic box, $\psi_{k}(\frac{L}{2})=\psi_{k}(-\frac{L}{2})=0$ $(k=x,y,z)$. In this case, the single-state energy:

[TABLE]

The characteristic temperature can be derived by:

[TABLE]

Here noted that the ranges of the quantum numbers $(n_{1},n_{2},n_{3})$ under Dirichlet boundary condition are from 1 to $\infty$. Therefore, the nonvanishing ground state $(1,1,1)$, $\epsilon_{0}=\frac{3\hbar^{2}\pi^{2}}{2mL^{2}}$ should be excluded. The Figs.  1,  4 and TABLE I depict the corresponding results.

By calculating numerically, the changes of $T_{c,L}$ with $L$, under periodic, antiperiodic and Dirichlet conditions are shown in Fig.  1. It is easy to check that no matter which boundary condition is taken, at a fixed particle number density, $T_{c,L}$ considerably increases as $L$ decreases in the region $L/L_{0}<1$. Since $T_{c,L}$ describes an approach to condensed state from the non-condensed one, the smaller linear size allows BEC to occur at higher temperature. And when $L>10L_{0}$, the values of $T_{c,L}$ agree with the results obtained within the thermodynamic limit results Holthaus et al. (2002), as shown in TABLE I, where we have chosen $\frac{\hbar^{2}}{mk_{B}}n^{\frac{2}{3}}$ as the $T_{c,L}$ unit. In order to illustrate this clearly, we numerically discuss the particle density for the ground state $n_{0}$ under the three boundary conditions sketched in Figs.  2,  3 and  4, respectively. It is clear that as the linear size decreases the slope of the curves gradually becomes slower, condensate fraction $n_{0}/n$ all appears to increase. Namely, the smaller linear size is effective to increase the condensate fraction. Meanwhile, the intersection point of the curve and the horizontal axis $T_{c,L}$, being shifted to high value. The behavior further illustrates that $T_{c,L}$ describe the crossover behavior between ground and excited states, and $T_{c,L}$ becomes critical temperature $T_{c}$ when the volume of the system tend to infinity. These results due to the fact that the coupling between energy level and linear size as Eq. (9) can significantly alters the nature of BEC in the finite system. The smaller linear size can produce a stronger restrictions, and accordingly lower the mean energy, which can greatly facilitate the realization of BEC Bagnato et al. (1987), increasing $T_{c,L}$ and $n_{0}/n$.

Consequently, the finite-size effects play an important role in the small volume systems, allowing for $L-$dependence of the shift of $T_{c,L}$ and $n_{0}/n$. The smaller the linear size (the larger the confinement), the higher the $T_{c,L}$ and the greater the $n_{0}/n$.

IV The Influences of boundary conditions on Bose-Einstein condensation in a cubic box

In order to study the influence of $\alpha$ on condensation under twisted boundary conditions, we numerically calculate $T_{c,L}$ and $n_{0}/n$ versus with $\alpha$ in the cases of $L=L_{0}$, $L=10L_{0}$, $L=100L_{0}$, respectively. And the numerical results are then plotted in Figs.  5,  6,  7,  8,  9,  10 respectively.

As shown in Figs.  5,  6 and  7, Obviously, in the interval $[0,2\pi]$, the change of $T_{c,L}$ with $\alpha$ is symmetric about $\alpha=\pi$, which can be derived from the Section 2. And in the interval $[0,\pi)$, $T_{c,L}$ gradually becomes lower, but when $\alpha$ reaches $\pi$, $T_{c,L}$ takes the maximum. This singular behavior is due to a significant fact that changes in boundary conditions disproportionately adds or removes single-states with the zero quantum numbers Tarasov et al. (2018), and the contribution from that group of single-state to the non-condensed occupation is quite large. Compared with other boundary conditions, the antiperiodic boundary conditions remove more single-state with the zero quantum numbers, which can be derived in Section II. As a result, the fixed particle number density determines that the characteristic temperature $T_{c,L}$ takes its maximum at $\alpha=\pi$. Additionally, comparing Figs.  5,  6 and  7, as $L$ increases until it is large enough, the influence of the boundary conditions on $T_{c,L}$ becomes negligible.

Apart from this, the condensate fraction $n_{0}/n$ versus $T$ with various $\alpha$ in different linear sizes are presented in Figs.  8,  9,  10. The condensate fraction become less sensitive to the boundary conditions as the $L$ increases. Particularly, Fig.  10 demonstrates that the signature of BEC in finite systems become independent of the boundary conditions when the linear size is large enough. In the case of small volume systems, as $\alpha$ changes from 0 to $\pi$, the major difference is a retarded onset of the occupation of the ground state, while $\alpha=\pi$, antiperiodic boundary condition is more conductive for bosons to condensation.

As a consequence, in a system of small volume, the characteristic temperature and condensate fraction are sensitive to boundary conditions, especially in the case of antiperiodic boundary condition.

V Conclusions

In this paper, we have studied ideal Bose gases with fixed particle density confined in the cubic box. By means of the theoretical analyses, we derived the specific formulas of the Bose distribution confined in the cubic box. Through numerical calculation, we analyzed the influence of the finite-size and boundary conditions on characteristic temperature $T_{c,L}$ and condensate fraction $n_{0}/n$. We found that in the case of finite volume system, the smaller linear size can increase $T_{c,L}$ and $n_{0}$, the crossover behavior between the ground and excited states will be advanced as the linear size decreases. More importantly, the smaller the volume, the more sensitive $T_{c,L}$ and $n_{0}/n$ are to antiperiodic boundary conditions.

Additionally, superconductivity and superfluidity that have a lot in common with BEC can be described by similar theories. Therefore, the finite size effect is expected to have an important impact on superconductivity and superfluidity, which requires further research.

Acknowledgments

This work is jointly supported by the National Nature Science Foundation of China (Grants No. 12075117, No. 51721001, No. 11890702, No. 11625418, No. 11535005, No. 11690030), the National Major state Basic Research and Development of China (Grant No. 2016YFE0129300). Y.-L. W. was funded by the Natural Science Foundation of Shandong Province of China (Grant No. ZR2017MA010).