Bose-Einstein condensation temperature of finite systems

TL;DR

This paper introduces a novel analytical method using heat kernel expansion and zeta-function regularization to accurately determine Bose-Einstein condensation temperatures in finite systems, addressing divergence issues.

Contribution

The paper develops a general technique for calculating BEC critical temperatures in finite systems, applicable to various geometries and potential confinements, with explicit correction terms.

Findings

Derived analytical expressions for BEC temperature using heat kernel coefficients

Exact sums computed for systems with known spectra like slabs and spheres

Higher-order corrections to critical temperatures for harmonic traps

Abstract

In the studies of the Bose-Einstein condensation of ideal gases in finite systems, the divergence problem usually arises in the equation of state. In this paper, we present a technique based on the heat kernel expansion and the zeta-function regularization to solve the divergence problem, and obtain the analytical expression of the Bose-Einstein condensation temperature for general finite systems. The result is represented by the heat kernel coefficients, in which the asymptotic energy spectrum of the system is used. Besides the general case, for the systems with exact spectra, e.g., ideal gases in an infinite slab or in a three-sphere, the sums of the spectra can be performed exactly and the calculation of the corrections to the critical temperatures is more direct. For the system confined in a bounded potential, the form of the heat kernel is different from the usual heat kernel expansion. We show that, as long as the asymptotic form of the global heat kernel can be found out, our method also works. For Bose gases confined in three- and two-dimensional isotropic harmonic potentials, we obtain the higher-order corrections to the usual results of the critical temperatures. Our method also can be applied to the problem of the generalized condensation, and we give the correction of the boundary on the second critical temperature in a highly anisotropic slab.

Equations

$$\ln \Xi =-\sum _i\ln \left(1-z{e}^{-\beta E_i}\right),$$

$$\ln \Xi =\sum _{\ell =1}^{\infty }\frac{1}{\ell }{z}^{\ell }\sum _i{e}^{-\ell \beta E_i}.$$

$$\left[-{\nabla }^2+\frac{2m}{{\hslash }^2}V\left(\mathbf{x}\right)\right]{\psi }_i=\frac{2m}{{\hslash }^2}{E}_i{\psi }_i,$$

$$K\left(t\right)=\sum _i{e}^{-{\lambda }_{i}t},$$

$$K\left(t\right)\approx \frac{1}{{\left(4\pi t\right)}^{3/2}}\sum _{k=0,\frac{1}{2},1,\cdots }^{\infty }{B}_k{t}^k,\left(t\to 0\right)$$

$$\ln \Xi =\sum _{\ell =1}^{\infty }\frac{1}{\ell }K\left(\ell \frac{{\hslash }^2\beta }{2m}\right)z^{\ell }=\frac{1}{{\lambda }^3}\sum _{k=0,\frac{1}{2},1,\cdots }^{\infty }\frac{B_k}{{\left(4\pi \right)}^k}{\lambda }^{2k}{g}_{5/2-k}\left(z\right),$$

$$g_{\sigma }\left(z\right)=\frac{1}{\Gamma (\sigma )}{\int }_0^{\infty }\frac{x^{\sigma -1}}{z^{-1}{e}^x-1}dx=\sum _{k=1}^{\infty }\frac{z^k}{k^{\sigma }}$$

$$N={\left(z\frac{\partial \ln \Xi }{\partial z}\right)}_{V,T}=\frac{1}{{\lambda }^3}\sum _{k=0,\frac{1}{2},1,\cdots }^{\infty }\frac{B_k}{{\left(4\pi \right)}^k}{\lambda }^{2k}{g}_{3/2-k}\left(z\right),$$

$$n{\lambda }^3=g_{3/2}\left(z\right)+\frac{B_{1/2}}{\sqrt{4\pi }V}\lambda g_1\left(z\right)+\frac{B_1}{4\pi V}{\lambda }^2{g}_{1/2}\left(z\right)+\cdots ,$$

$$g_{\sigma}\left(e^{\beta\mu}\right)\approx\left\{\begin{array}[c]{lll}\zeta\left(\sigma\right),&\left(\sigma\geq\frac{3}{2}\right)&\\ -\ln\left(-\beta\mu\right),&\left(\sigma=1\right)&\\ \Gamma\left(-\sigma+1\right)\frac{1}{\left(-\beta\mu\right)^{-\sigma+1}},&\left(\sigma\leq\frac{1}{2}\right)&~{}~{}~{}\left(\mu\rightarrow 0\right)\end{array}\right.$$

$$n{\lambda }^3\approx \zeta \left(\frac{3}{2}\right)-\frac{B_{1/2}}{\sqrt{4\pi }V}\lambda \ln \left(-\beta \mu \right)+I,$$

$$I=\sum _{k=1,\frac{3}{2},2,\cdots }^{\infty }\frac{B_k}{{\left(4\pi \right)}^{k}V}{\lambda }^{2k}\Gamma \left(k-\frac{1}{2}\right)\frac{1}{{\left(-\beta \mu \right)}^{k-1/2}}$$

$$\Gamma \left(\xi \right)={\int }_0^{\infty }{x}^{\xi -1+s}{e}^{-x}dx.\left(s\to 0\right)$$

$$I=\frac{\sqrt{-\beta \mu }}{V}{\int }_0^{\infty }dx{x}^{s-3/2}{e}^{-x}\sum _{k=1,\frac{3}{2},2,\cdots }^{\infty }{B}_k{\left(\frac{{\hslash }^2}{2m}\frac{x}{-\mu }\right)}^k.$$

$$I$$

$$=\frac{{\lambda }^3}{V}\frac{1}{-\beta \mu }{\int }_0^{\infty }dx{x}^s{e}^{-x}K\left(\frac{{\hslash }^2}{2m}\frac{x}{-\mu }\right)-\Gamma \left(s-\frac{1}{2}\right)\sqrt{-\beta \mu }-\Gamma \left(s\right)\frac{B_{1/2}\lambda }{\sqrt{4\pi }V}.$$

$${\int }_0^{\infty }dx{x}^s{e}^{-x}K\left(\frac{{\hslash }^2}{2m}\frac{x}{-\mu }\right)=\sum _{i=1}^{\infty }{\int }_0^{\infty }dx{x}^s{e}^{-x}{e}^{-E_i\frac{x}{-\mu }}=\Gamma \left(1+s\right)\sum _{i=1}^{\infty }\frac{{\left(-\mu \right)}^{1+s}}{{\left(E_i-\mu \right)}^{1+s}}.$$

$$I=\Gamma \left(1+s\right)\frac{{\lambda }^3}{V}\frac{{\left(-\mu \right)}^s}{\beta }\sum _{i=1}^{\infty }\frac{1}{E_i^{1+s}}-\Gamma \left(s\right)\frac{B_{1/2}\lambda }{\sqrt{4\pi }V}.$$

$$\sum _{i=1}^{\infty }\frac{1}{E_i^{1+s}}={\left(\frac{2m}{{\hslash }^2}\right)}^{1+s}\sum _{i=1}^{\infty }\frac{1}{{\lambda }_i^{1+s}}={\left(\frac{2m}{{\hslash }^2}\right)}^{1+s}{\zeta }_s\left(1+s\right).$$

$$n{\lambda }^3=\zeta \left(\frac{3}{2}\right)-\frac{B_{1/2}\lambda }{\sqrt{4\pi }V}\ln \left(-\beta \mu \right)+\Gamma \left(1+s\right)\frac{{\lambda }^3}{V}\frac{{\left(-\mu \right)}^s}{\beta }\sum _{i=1}^{\infty }\frac{1}{E_i^{1+s}}-\Gamma \left(s\right)\frac{B_{1/2}\lambda }{\sqrt{4\pi }V}.$$

$$N\left(\chi \right)=\frac{1}{2\pi i}{\int }_{c-i\infty }^{c+i\infty }K\left(t\right)\frac{e^{\chi t}}{t}dt.$$

$$N\left(\chi \right)=\frac{1}{{\left(4\pi \right)}^{3/2}}\sum _{k=0,\frac{1}{2},1,\cdots }^{\infty }\frac{B_k}{\Gamma \left(5/2-k\right)}{\chi }^{3/2-k}.$$

$$N\left({\lambda }_i\right)=i$$

$${\lambda }_i\approx {\left(6{\pi }^2\right)}^{2/3}\frac{1}{B_0^{2/3}}{i}^{2/3}-{\left(\frac{3}{4}\right)}^{1/3}{\pi }^{7/6}\frac{B_{1/2}}{B_0^{4/3}}{i}^{1/3}.$$

$$\sum _{i=1}^{\infty }\frac{1}{E_i^{1+s}}\approx \sum _{i=1}^{\infty }{\left[\frac{2m}{{\hslash }^2}{\left(\frac{B_0}{6{\pi }^2}\right)}^{2/3}\right]}^{1+s}\left[\frac{1}{i^{2/3+2s/3}}+\frac{\left(1+s\right)B_{1/2}}{2\cdot 6^{1/3}{\pi }^{1/6}{B}_0^{2/3}}\frac{1}{i^{1+2s/3}}\right]$$

$$={\left[\frac{2m}{{\hslash }^2}{\left(\frac{V}{6{\pi }^2}\right)}^{2/3}\right]}^{1+s}\left[\zeta \left(\frac{2}{3}+\frac{2s}{3}\right)+\frac{\left(1+s\right)B_{1/2}}{2\cdot 6^{1/3}{\pi }^{1/6}{V}^{2/3}}\zeta \left(1+\frac{2s}{3}\right)\right].$$

$$I\approx \frac{4\zeta \left(2/3\right)}{6^{2/3}{\pi }^{1/3}}\frac{\lambda }{V^{1/3}}-\frac{B_{1/2}\lambda }{\sqrt{4\pi }V}\left(\ln \frac{3^{2/3}{\pi }^{4/3}{\hslash }^2}{2^{1/3}m{V}^{2/3}\left(-\mu \right)}-1-\frac{2}{3}{\gamma }_E\right),$$

$$n{\lambda }^3=\zeta \left(\frac{3}{2}\right)+\frac{4\zeta \left(2/3\right)}{6^{2/3}{\pi }^{1/3}}\frac{\lambda }{V^{1/3}}-\frac{B_{1/2}\lambda }{\sqrt{4\pi }V}\left(\ln \frac{3^{2/3}{\pi }^{1/3}{\lambda }^2}{2^{4/3}{V}^{2/3}}-1-\frac{2}{3}{\gamma }_E\right).$$

$$T_0=\frac{2\pi {\hslash }^2{n}^{2/3}}{{\zeta }^{2/3}\left(3/2\right)m{k}_B}.$$

$$\frac{T_c-T_0}{T_0}$$

$$=\left[0.7115-\left(0.06607\ln N+0.05506\right)\frac{B_{1/2}}{V^{2/3}}\right]\frac{1}{N^{1/3}}.$$

$$\frac{T_c-T_0}{T_0}=-\frac{8\zeta \left(2/3\right)}{3\cdot 6^{2/3}{\pi }^{1/3}{\zeta }^{2/3}\left(3/2\right)}\frac{1}{N^{1/3}}=0.7115\frac{1}{N^{1/3}}.$$

Full text

1 Introduction

After the experimental realization of Bose-Einstein condensation (BEC) in ultracold atoms [1, 2, 3], numerous theoretical studies have been devoted to the BEC phase transition in finite systems. Strictly speaking, a phase transition can only occur in the infinite system. In a finite system, all thermodynamic quantities are analytical functions of the temperature, so there does not exist a genuine phase transition. On the other hand, the behavior of a large system is practically the same as the infinite one just as observed in the experiments, so a (quasi-)critical temperature is needed to discuss the property of the finite system. To define the critical temperature of BEC in a finite system, many schemes with different criteria have been discussed, such as a small value of the condensate fraction [4, 5, 6, 7], the maximal fluctuation or the inflexion point of the ground-state occupation number [8], the maximum of the specific heat [9, 10], etc.

To discuss the BEC phase transition, the heat kernel approach [11, 12, 13] is a powerful tool. In the past years, the heat kernel approach has been widely applied in many fields of physics, including quantum field theory [14, 15, 16], quantum gravity [17, 18], string theory [19], and quantum statistical mechanics [11, 20]. However, when applied to the BEC of an ideal gas in a finite system, the heat kernel expansion encounters the problem of divergence. In fact, the problem of divergence is also inevitable in other approaches besides the heat kernel expansion. If one regards that the phase transition occurs at the chemical potential $\mu=0$ just as that in the thermodynamic limit, the equation of state of the gas will be divergent [5, 7, 21]. To avoid this difficulty, the discussion of the phase transition in finite systems usually ignores the divergent terms [4, 9, 10, 11, 22, 23], or, for the systems with exact energy spectra, directly performs the sum of the spectra [5, 24, 25].

In this paper, we will present a technique to solve the divergence problem in the BEC phase transition of nonrelativistic ideal gases confined in finite systems, with the help of the heat kernel expansion and the zeta-function regularization. By using the asymptotic spectrum calculated from the heat kernel expansion, we obtain the analytical expression of the critical temperature for ideal Bose gases in general finite systems, which is expressed only by the heat kernel coefficients. In this result, the effect of the finite number of particles is separated from other factors. We compare it with the numerical result of the specific heat of an ideal Bose gas in a cube with period boundary conditions, and they fit very well. Our result also give the influence of the boundary on the critical temperature for an arbitrary cavity. Furthermore, when the sum of the energy spectrum can be performed exactly, we can replace the asymptotic result by the exact sum and obtain a more precise critical temperature. The critical temperatures of BEC in an infinite slab and in a three-sphere $S^{3}$ are given as examples, and the results are consistent with those calculated by other methods. In a bounded potential, the heat kernel has a different form from the usual heat kernel expansion. Our method also can be applied to such cases. As examples, we give the higher-order corrections to the critical temperatures of BEC in three- and two-dimensional isotropic harmonic potentials. In addition, in highly anisotropic systems the condensation may not occur in the ground state but distributes in a set of single-particle states, i.e., the generalized BEC [26, 27, 28]. The Bose gases in such systems may undergo two kinds of phase transitions. We will also consider the influence of the boundary on the second critical temperature in a highly anisotropic slab.

The paper is organized as follows. In section 2, we discuss the BEC phase transition of an ideal gas in a general finite system and express the critical temperature analytically in terms of the heat kernel coefficients. As examples, we discuss the influences of the finite number of particles and the boundary, respectively. In section 3, we calculate the critical temperatures of BEC in an infinite slab and in a three-sphere. In these cases the sums of the energy spectra are performed exactly. In section 4, we consider the phase transition in three- and two-dimensional isotropic harmonic traps, and obtain the higher-order corrections to the critical temperatures. In section 5, we discuss the correction of the boundary on the second critical temperature in a highly anisotropic slab. The conclusion and some discussions are presented in section 6.

2 Critical temperature of BEC in finite systems

In the grand canonical ensemble, the grand potential of an ideal Bose gas is

[TABLE]

where $\left\{E_{i}\right\}$ is the single-particle energy spectrum, the fugacity $z=e^{\beta\mu}$ with $\mu$ the chemical potential, and $\beta=1/\left(k_{B}T\right)$ with $k_{B}$ the Boltzmann constant. Expanding the logarithmic term gives

[TABLE]

Since the energy spectrum satisfy the eigenvalue equation

[TABLE]

the sum over the spectrum in eq. (2) can be expressed as the global heat kernel of the operator $D=-\nabla^{2}+\left(2m/\hbar^{2}\right)V\left(\mathbf{x}\right)$. Mathematically, the global heat kernel of an operator $D$ is defined as

[TABLE]

where $\left\{\lambda_{i}\right\}$ is the spectrum of the operator $D$. When $t\rightarrow 0$, it can be asymptotically expanded as a series of $t$, which is the heat kernel expansion,

[TABLE]

where $B_{k}$ $\left(k=0,1/2,1,\cdots\right)$ are the heat kernel coefficients. Therefore, with the help of the heat kernel expansion, the grand potential in eq. (2) can be rewritten as

[TABLE]

where

[TABLE]

is the Bose-Einstein integral, and $\lambda=\sqrt{2\pi\beta}\hbar/\sqrt{m}$ is the mean thermal wavelength. For a system with the fixed number of particles, the relation between the fugacity $z$ and the temperature $T$ is given by the equation of the particle number

[TABLE]

or,

[TABLE]

where $n=N/V$ is the particle number density, and we have taken $B_{0}=V$.

Clearly, the first terms in eqs. (6) and (9) give the ordinary equation of state for an ideal Bose gas in the thermodynamic limit. The other terms in these equations describe the difference between the finite system and that in the thermodynamic limit. To discuss the BEC phase transition in the thermodynamic limit, one only needs to set $z=1$ or the chemical potential $\mu=0$ in the equation of the particle number, then the critical temperature can be obtained immediately. However, since the Bose-Einstein integral $g_{\sigma}\left(1\right)$ is divergent at $\sigma\leq 1$, the correction terms in eq. (9) become singular at $\mu=0$. In other words, the traditional method for determining the critical temperature is invalid for the system described by eq. (9).

In the following, we will show that, based on the heat kernel expansion and the zeta-function regularization, we can solve the divergence problem and obtain an analytical expression of the critical temperature of BEC.

At the transition point, the chemical potential $\mu\rightarrow 0$, and the leading term of the asymptotic expression of the Bose-Einstein integral is

[TABLE]

where $\zeta\left(\sigma\right)=\sum_{n=1}^{\infty}n^{-\sigma}$ is the Riemann zeta function. Expanding eq. (9) and keeping the leading term of the Bose-Einstein integrals, we reach its asymptotic expression for $\mu\rightarrow 0$ as

[TABLE]

where

[TABLE]

is introduced for simplification. When $\mu\rightarrow 0$, this is a series in which every term is divergent. However, it can be summed up by use of the heat kernel expansion and eq. (11) will give an analytical result. We will show the procedure in the following.

First, we represent the gamma function in eq. (12) as an integral and introduce a regularization parameter $s$ which will be set to [math] to deal with the divergence problem, i.e.,

[TABLE]

Then eq. (12) becomes

[TABLE]

The sum in the integral differs from the heat kernel expansion eq. (5) just by two extra terms and a common coefficient, so we can express eq. (14) by the global heat kernel as

[TABLE]

As a result, the divergent sum in eq. (12) is converted to the global heat kernel.

Next, the integral in the first term in eq. (15) can be performed by substituting the definition of the global heat kernel eq. (4),

[TABLE]

Our aim is to determine the critical temperature of the BEC, which occurs at $\mu\rightarrow 0$. At this limit, eq. (15) becomes

[TABLE]

Note that the sum of the spectrum is indeed the spectrum zeta function defined as $\zeta_{s}\left(\sigma\right)=\sum_{i=1}^{\infty}\lambda_{i}^{-\sigma}$, which gives

[TABLE]

Eq. (11) now is

[TABLE]

From this result we can find that although it is based on the whole heat kernel expand, only the first two heat kernel coefficients $B_{0}=V$ and $B_{1/2}$ appear in this expression. However, it does not mean that the higher-order heat kernel coefficients are irrelevant to the critical temperature. In eq. (19), there is a term of the sum of the spectrum, and the information of the spectrum is embodied in the heat kernel coefficients.

Finally, we will deal with the sum in eq. (19). For a general system, the exact spectrum is not known, but we can obtain its asymptotic expression on the basis of the heat kernel expansion. As given in Ref. [29], we can first obtain the counting function from the heat kernel expansion, then achieve the asymptotic expansion of the spectrum from the counting function. Specifically, the counting function $N\left(\chi\right)$ is defined as the number of the eigenstates of an operator with the eigenvalue smaller than $\chi$. The relation between the counting function and the global heat kernel is [29]

[TABLE]

From the heat kernel expansion eq. (5), we can calculate the asymptotic expansion of the counting function

[TABLE]

Setting

[TABLE]

will give the asymptotic expansion of the spectrum. The first two terms read [29]

[TABLE]

Thus the spectrum of a general system is represented by the heat kernel coefficients asymptotically.

Since the energy spectrum $E_{i}=\left(\hbar^{2}/2m\right)\lambda_{i}$, from eq. (23) we have

[TABLE]

In this calculation, we have represented the sum by the Riemann zeta function. Note that by the zeta-function regularization, the two divergent sums are replace by zeta functions. However, the second term in eq. (24) still contains a function $\zeta\left(1+\frac{2s}{3}\right)$ which is divergent at $s\rightarrow 0$. Substituting this result into eq. (17) and expanding it around $s=0$ gives

[TABLE]

where $\gamma_{E}=0.5772$ is the Euler constant. We find that the linearly divergent terms of $s$ from the zeta function and the gamma function have been canceled. On the other hand, there is a logarithmically divergent term of $\mu$ in this equation, but this term also will be canceled when substituting eq. (25) into eq. (11):

[TABLE]

By using the above technique, our final result eq. (26) is fully analytical now. This equation holds at the transition point, so the critical temperature can be solved straightforward. Clearly, the first term in the right-hand-side of eq. (26) gives the critical temperature in the thermodynamic limit

[TABLE]

Regarding the last two terms in eq. (26) as small corrections, we can obtain the critical temperature $T_{c}$, satisfying

[TABLE]

This result shows that the correction to the critical temperature of an ideal Bose gas in a finite system is represented by the heat kernel coefficients. Since the typical case of nonvanishing $B_{1/2}$ is the system with a boundary, one direct application of this result is to describe the influence of boundary on the critical temperature. Besides, the first term in eq. (28) is proportional to $1/N^{1/3}$, or, it only related to the particle number. It indicates that this term describes the pure contribution of the finite number of particles.

Eq. (28) only contains two terms since we only consider the first two terms in the asymptotic expression of the spectrum eq. (23). More contributions can be easily obtained by including more terms in eq. (23), and each of them will contribute a term proportional to some $\zeta\left(\alpha\right)$ with $\alpha>1$, so there is no divergence in these terms and the calculation is straightforward. It is easy to check that all these contributions are also proportional to $1/N^{1/3}$.

In the following, we will consider two examples to show the correction in eq. (28) to the critical temperature.

2.1 Effect of the finite number of particles

According to eq. (28), if the only influence on the ideal gas is the finite number of particles, the critical temperature of the BEC will be

[TABLE]

Therefore, the critical temperature increases in a finite system compared with that in the thermodynamic limit.

To check this result, we consider an ideal Bose gas confined in a three-dimensional cube of side length $L$ with period boundary conditions. The single-particle energy spectrum is

[TABLE]

The corresponding global heat kernel can be obtained by repeatedly applying the Euler-MacLaurin formula [30]

[TABLE]

The heat kernel expansion only contains one term just like that in infinite space, but it is not an exact result since some exponentially small terms have been neglected. This form implies that the only factor affecting the critical temperature in such a system is the volume, or, the particle number. In other words, the critical temperature of BEC in this system should satisfy eq. (29).

In figure 1 we plot the exact numerical result of the specific heat versus the temperature for an ideal Bose gas in the cube with period boundary conditions. The critical temperature given by eq. (29) agrees very well with the maximum of the specific heat.

2.2 Effect of the boundary

The influence of the boundary on BEC with different boundary conditions has motivated many studies [31, 32, 33]. In the following, we will discuss the influence of the boundary on the critical temperature according to eq. (28). We know that at a manifold with a boundary, the coefficient $B_{1/2}$ reflects the leading effect of the boundary [12]

[TABLE]

where $S$ is the surface area of the system, the signs $-$ and $+$ correspond Dirichlet and Neumann boundary conditions, respectively. Substituting eq. (32) into eq. (28) gives the critical temperature for the corresponding system.

To check the result, we consider a Bose gas confined in a cube of side length $L$ with Dirichlet boundary conditions. The energy spectrum is

[TABLE]

and the heat kernel coefficients are [29]

[TABLE]

Then from eq. (28), the correction to the critical temperature is

[TABLE]

Clearly, this result contains the contributions from the finite number of particles and the boundary. As mentioned above, in eq. (35) only the first-order correction of the spectrum is included. If included more terms in the asymptotic expansion of the spectrum eq. (23), we can obtain a more precise critical temperature. The final result converges to

[TABLE]

The $\ln N$ term in the parentheses is consistent with Ref. [31].

3 Systems with exact spectra

In section 2, we have given a general discussion on the critical temperature of BEC, in which the asymptotic expansion of the spectrum eq. (23) is taken into account. However, if the spectrum of the system is known and can be summed exactly, we can perform the sum and obtain a more precise critical temperature. In the following we will consider two examples: the infinite slab and the three-sphere $S^{3}$.

3.1 BEC in the infinite slab

Consider a Bose gas between two infinite parallel planes with distance $L$. It can be regarded as a rectangular box of side lengths $L_{x}=L_{y}=a$ $\left(a\rightarrow\infty\right)$ and $L_{z}=L$ with Dirichlet boundary conditions. The single-particle energy spectrum is then

[TABLE]

For such a form of the spectrum, the sum in eq. (18) can be performed exactly. Since $a\rightarrow\infty$, the sums of $n_{x}$ and $n_{y}$ are converted to integrals, we have

[TABLE]

The heat kernel coefficients for the Laplace operator can also be calculated from the spectrum eq. (37),

[TABLE]

At the transition point, $\mu\rightarrow 0$, eq. (19) becomes

[TABLE]

When $s\rightarrow 0$, all of the divergent terms in this expression are canceled, we have

[TABLE]

This equation holds at the transition point. The second term represents the correction from the boundary. Then we can obtain the correction to the critical temperature

[TABLE]

This result is consistent with that in Ref. [24], in which the critical temperature is obtained by use of the Mellin-Barnes transform.

3.2 BEC in the three-sphere

The spectrum of the Laplace operator in a three-sphere $S^{3}$ of radius $R$ is [34]

[TABLE]

with the degeneracy $\left(n+1\right)^{2}$. The global heat kernel is [35]

[TABLE]

where $V=2\pi^{2}R^{3}$ is the volume of $S^{3}$. Therefore, the heat kernel expansion in $S^{3}$ does not contain half-integer powers of $t$. In fact, this result can be obtained directly without the specific form of the heat kernel: Since $S^{3}$ is a smooth manifold without boundary, the heat kernel expansion will not contain half-integer power terms.

According to the analysis in section 2, the critical temperature satisfies eq. (19). For the $S^{3}$ case, the coefficient $B_{1/2}=0$, which makes the divergent terms in the general case vanish, so we can take $s=0$ in the expression, i.e.,

[TABLE]

Then we perform the sum by using the exact spectrum eq. (43). The sum of all the excited states is

[TABLE]

where we have used eq. (215) in Ref. [36]

[TABLE]

Then eq. (45) becomes

[TABLE]

and the correction to the critical temperature is

[TABLE]

In Ref. [24], the authors have also discussed the BEC in $S^{3}$. Their result has a different coefficient from eq. (49), but the reason is that the spectrum in Ref. [24] is approximately taken as $\lambda_{n}=\left(n+1\right)^{2}/R^{2}$. If we use this spectrum too, our result will be the same as that in Ref. [24].

4 BEC in harmonic potentials

In section 2, our discussion is based on the heat kernel expansion eq. (5). On the other hand, if an ideal Bose gas is confined in a bounded potential, the form of the heat kernel will change. In this section, we will show that even if the form of the heat kernel is different from eq. (5), the above method can still be used to determine the critical temperature of BEC. We will take three- and two-dimensional harmonic potentials as examples.

4.1 Three-dimensional harmonic potentials

In the literature, many studies are devoted to the BEC phase transition in three-dimensional ($3D$) harmonic potentials since the realization of the BEC of ultracold atoms is in such potentials. In a harmonic potential, the heat kernel of the Laplace operator has a different form from eq. (5), but it still can be expanded as a series. At the transition point, one also will encounter the problem of divergence. In many theoretical studies on the critical temperature of BEC in harmonic potentials based on the heat kernel approach, the divergent terms are ignored [4, 9, 10, 23]. In this section, we will apply the technique provided in section 2 to consider all the terms in the equation of state. By solving the divergence problem, we will give the analytical result of the critical temperature with the higher-order correction.

The energy spectrum of a particle in a $3D$ isotropic harmonic potential is

[TABLE]

with the degenerate degree $\left(n+1\right)\left(n+2\right)/2$, where the zero-point energy has been suppressed. Then the heat kernel is

[TABLE]

where the coefficients are

[TABLE]

Although the form of the heat kernel expansion is different from eq. (5), we can apply the same procedure to discuss the BEC phase transition in the harmonic potential.

Replacing eq. (5) by eq. (51) and substituting it into the grand potential eq. (2) gives

[TABLE]

Then the number of particles is

[TABLE]

The critical temperature is given by eq. (54) at the limit $\mu\rightarrow 0$. However, at this limit, only the first two terms in the sum are convergent. In the usual treatment, all of the divergent terms are ignored [4, 9, 10, 23], and the result of the critical temperature only contains the first-order correction. In the following, we will give the second-order correction to the critical temperature.

Taking advantage of the asymptotic expression of the Bose-Einstein integral eq. (10), we have

[TABLE]

where we have introduced

[TABLE]

After introducing the integral form of the gamma function with the regularization parameter $s\rightarrow 0$, eq. (13), we have

[TABLE]

where we have used the heat kernel eq. (51) to perform the divergent sum. The second and third terms in eq. (57) contain positive powers of $\mu$, so they will vanish when $\mu\rightarrow 0$. The first term can be changed to a spectrum zeta function, i.e., a sum of the power of the spectrum, by eq. (16). Therefore, when $\mu\rightarrow 0$, we have

[TABLE]

The sum of the excited-state spectrum can be performed exactly as

[TABLE]

When $s\rightarrow 0$, eq. (58) becomes

[TABLE]

where the coefficients eq. (52) have been used. In this result, the linearly divergent terms of $s$ have been canceled. Substituting eqs. (60) and (52) into eq. (55) gives

[TABLE]

Just like the general case discussed in section 2, the logarithmically divergent terms of $\mu$ are also canceled. This is a fully analytical result of the critical temperature. Since $\beta\hbar\omega\ll 1$ at the transition point, the last two terms correspond the first- and second-order corrections. Then the critical temperature is approximately

[TABLE]

Compared with Ref. [4, 9, 10, 23], the first-order correction in eq. (62) is consistent with their results, and we also give the second-order correction.

4.2 Two-dimensional harmonic potentials

After suppressing the zero-point energy, the spectrum of a particle in a two-dimensional ($2D$) isotropic harmonic potential is still given by eq. (50), but the degeneracy is $n+1$, so the heat kernel becomes

[TABLE]

where the expansion coefficients are

[TABLE]

Then the grand potential eq. (2) and the number of particles become

[TABLE]

At the limit $\mu\rightarrow 0$, by using the asymptotic form eq. (10), we can rewrite the expression of the particle number as

[TABLE]

where we have introduced

[TABLE]

Clearly, different from the $3D$ case, the second term in eq. (66) is divergent. By using the integral form of the gamma function eq. (13), we obtain

[TABLE]

When $\mu\rightarrow 0$, the second term vanishes, and the first term becomes a sum of the spectrum,

[TABLE]

where the coefficients eq. (64) have been substituted. The exact result of the sum of the excited-state spectrum is

[TABLE]

Then taking $s\rightarrow 0$ gives

[TABLE]

The particle number eq. (66) becomes

[TABLE]

Just as before, all of the divergent terms have been canceled, so the critical temperature of BEC in the $2D$ harmonic potential is

[TABLE]

5 The second critical temperature of BEC

As is well known, when the BEC phase transition occurs, a large number of particles will fall into the ground state and the ground state will be macroscopically occupied. However, in some anisotropic systems, the behavior of the Bose gas may be more complicated. If there exists a set of quantum states which are very closed to the ground state, a macroscopic number of particles may distribute over the set of states. This phenomenon is called the generalized BEC [26, 27, 28], which can be classified into three types [26, 27, 37]: Type I (II) refers to the case that a finite (infinite) number of single-particle states are macroscopically occupied; type III refers to the case that the occupation of the set of the states is a macroscopic fraction of the total particle number although none of these states is macroscopically occupied. The generalized BEC has been discussed in various geometries and external potentials, with and without interaction [38, 39, 40].

In ref. [40], the authors discuss the generalized BEC in some anisotropic systems in the thermodynamic limit, including slabs, squared beams, and ”cigars”. They show that in these systems, ideal Bose gases will undergo two kinds of phase transitions. Therefore, besides the conventional critical temperature $T_{c}$, there is a second critical temperature $T_{m}$ connected to the second phase transition. In this section, we will take the anisotropic slab as an example to discuss the same problem without the assumption of the thermodynamic limit. and we will give the correction of the boundary on the second critical temperature of BEC.

We consider a Bose gas in a highly anisotropic slab described in section 3.1. As in ref. [40], the side length of the slab takes the form $a=Le^{\alpha L}\gg L$. The total particle number can be expressed as

[TABLE]

where $f\left(n_{x},n_{y},n_{z}\right)=\left(z^{-1}e^{-\beta E\left(n_{x},n_{y},n_{z}\right)}-1\right)^{-1}$ denotes the average particle number in the state $\left(n_{x},n_{y},n_{z}\right)$. In eq. (74) we have divided the total particle number into three parts: the ground-state particle number $N_{0}\left(T\right)$, the number of particles in the states with $n_{z}=1$ (not including the ground state) $N_{1}\left(T\right)$, and the number of particles in the state with $n_{z}\geq 2$, $N_{2}\left(T\right)$. The condition for the conventional BEC is then [40]

[TABLE]

Since at the transition point $T_{c}$, the number of the particles in the states with $n_{z}=1$ is negligible, $N_{2}\left(T_{c}\right)$ can be replaced by the total number of the excited-state particles. It means that the condition eq. (75) is actually equivalent to the condition for the conventional BEC. The critical temperature $T_{c}$ has been given in section 3.1, which satisfies eq. (41), or,

[TABLE]

Different from the usual three-dimensional system, in the highly anisotropic slab, when $T<T_{c}$, no single state is macroscopically occupied, but the entire band of the states with $n_{z}=1$ is macroscopically occupied, i.e., this is a type III generalized condensation. Only when the temperature is lower than the second critical temperature, $T<T_{m}$, the ground state becomes macroscopically occupied. Then there is a coexistence of the type III generalized condensation and the standard type I condensation in the ground state [40]. As a result, the second critical temperature $T_{m}$ satisfies

[TABLE]

In the following we will discuss the influence of the boundary to the second critical temperature.

Since

[TABLE]

when $a\rightarrow\infty$, the sums of $n_{x},n_{y}$ can be converted to integrals, we have

[TABLE]

where we have introduced

[TABLE]

Nearby the second critical temperature, $\beta\Delta\rightarrow 0$, then

[TABLE]

Since its maximum appears at $\beta\Delta=\lambda_{m}^{2}/a^{2}$, the condition for the phase transition eq. (77) becomes

[TABLE]

Substituting $a=Le^{\alpha L}$ gives

[TABLE]

This is the equation that the second critical temperature $T_{m}$ obeyed. Taking advantage into eq. (76), we can obtain the relation between the two critical temperatures

[TABLE]

where

[TABLE]

Easy to check that in the thermodynamic limit $L\rightarrow\infty$, the relation (84) becomes

[TABLE]

which is consistence with the result given in ref. [40].

On the other hand, if we consider an isothermal process, the conditions for the phase transitions will described by the critical particle densities. According to eqs. (76) and (83), in an isothermal process, the two critical densities are

[TABLE]

Then we have

[TABLE]

In the thermodynamic limit $L\rightarrow\infty$, this relation will go back to the result in ref. [40].

In the above we have obtained the correction of the boundary to the second critical temperature of BEC in a slab geometry. The similar method can also be applied to the systems with other anisotropic boundaries or external potentials.

6 Conclusion and discussions

In the above, we discuss the critical temperature of BEC of ideal gases in finite systems in a general framework based on the heat kernel expansion and the zeta-function regularization. Our method gives the analytical expression of the critical temperature only related to the heat kernel coefficients. We consider some specific examples and give the corresponding critical temperatures. Some of them have been obtained by other methods, but we provide a consistent treatment for different systems in this paper. Besides, taking advantage of the asymptotic spectrum, we divide the effect of the finite number of particles on the critical temperature from other factors, and the result agrees with the numerical calculation very well. For the Bose gas in isotropic harmonic traps, we obtain the second-order correction to the critical temperature for $3D$ case and the first-order correction for $2D$ case. In some highly anisotropic systems, besides the conventional BEC, the generalized condensation may occur in a Bose system. We also give the correction of the boundary on the second critical temperature in an anisotropic slab. We hope that our work can help to reveal the nature of the phase transition in finite systems.

In this paper, we only discuss the critical temperature of BEC. Besides these results, our method can also be applied to other aspects of the phase transition. For example, it can be used to analyze the properties of the thermodynamic functions, especially near the transition point. We will leave these for future work.

The author is very indebted to Prof. Wu-Sheng Dai for his help. This work is supported in part by NSF of China, under Project No. 11575125.